An Introduction to Evolutionary Game Theory

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

**An** **Introduction** **to** **Evolutionary** **Game** **Theory**

Fabio A. C. C. Chalub

The prisoner

dilemma and

the evolution

of cooperation

Universidade Nova de Lisboa

Mathematical Methods and Modeling of Biophysical

Phenomena, Buzios, August 2007

1 /75

Overview

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

I Today: basics;

2 /75

Overview

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

I Today: basics;

II Tomorrow: hot **to**pics.

2 /75

Overview — **to**day

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

1 What evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

3 /75

Overview — **to**day

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

1 What evolution is?

2 **Evolutionary** dynamics

The prisoner

dilemma and

the evolution

of cooperation

3 /75

Overview — **to**day

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

1 What evolution is?

2 **Evolutionary** dynamics

3 **Game**s

3 /75

Overview — **to**day

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

1 What evolution is?

2 **Evolutionary** dynamics

3 **Game**s

4 The prisoner dilemma and the evolution of cooperation

3 /75

One problem

**Evolutionary**

**Game** **Theory**

One problem: how **to** explain the diversity of the world?

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

4 /75

One problem

**Evolutionary**

**Game** **Theory**

FACC Chalub

First step: Isolate characteristics and quantify them.

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

5 /75

One problem

**Evolutionary**

**Game** **Theory**

FACC Chalub

First step: Isolate characteristics and quantify them.

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

5 /75

One problem

**Evolutionary**

**Game** **Theory**

FACC Chalub

First step: Isolate characteristics and quantify them.

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

5 /75

One problem

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Evolution = change of inherited traits of a

population from generation **to** generation.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

6 /75

One problem

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Evolution = change of inherited traits of a

population from generation **to** generation.

Which are the “causes of evolution”?

6 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

2. infinite population size;

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

2. infinite population size;

3. no selection;

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

2. infinite population size;

3. no selection;

4. no mutation;

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

2. infinite population size;

3. no selection;

4. no mutation;

5. no migration;

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

(Early 20Th century) by Godfrey Hardy (English

mathematician), and Wilhelm Weinberg (German physician).

Theorem

If there is

1. random mating within a single population;

2. infinite population size;

3. no selection;

4. no mutation;

5. no migration;

then there is no evolution (gene frequencies are static).

7 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

Proof:

Step 1: Suppose we have at the n-th generation:

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

◮ aa with frequency r n .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

◮ aa with frequency r n .

Then, in the next generation we have

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

◮ aa with frequency r n .

Then, in the next generation we have

◮ AA with frequency

p n+1 = p 2 n + 21 2 p n2q n + 1 4 (2q n) 2 = (p n + q n ) 2 ;

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

◮ aa with frequency r n .

Then, in the next generation we have

◮ AA with frequency

◮ Aa with frequency

p n+1 = p 2 n + 21 2 p n2q n + 1 4 (2q n) 2 = (p n + q n ) 2 ;

2q n+1 = 2 1 2 p n2q n +2p n r n + 1 2 (2q n) 2 +2 1 2 2q nr n = 2(p n +q n )(q n +r n ) ;

8 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Proof:

Step 1: Suppose we have at the n-th generation:

◮ AA with frequency p n ;

◮ Aa with frequency 2q n ;

◮ aa with frequency r n .

Then, in the next generation we have

◮ AA with frequency

◮ Aa with frequency

p n+1 = p 2 n + 21 2 p n2q n + 1 4 (2q n) 2 = (p n + q n ) 2 ;

2q n+1 = 2 1 2 p n2q n +2p n r n + 1 2 (2q n) 2 +2 1 2 2q nr n = 2(p n +q n )(q n +r n ) ;

◮ aa with frequency

8 /75

r n+1 = 1 4 (2q n) 2 + 2 1 2 2q nr n + r 2 n = (q n + r n ) 2 .

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Step 2:

p n+1 + 2q n+1 + r n+1 = (p n + q n ) 2 + 2(p n + q n )(q n + r n ) + (q n + r n ) 2

= (p n + 2q n + r n ) 2 .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

9 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Step 2:

p n+1 + 2q n+1 + r n+1 = (p n + q n ) 2 + 2(p n + q n )(q n + r n ) + (q n + r n ) 2

= (p n + 2q n + r n ) 2 .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Assuming that p 0 + 2q 0 + r 0 = 1, then p n + 2q n + r n = 1 for all

n ∈ N.

9 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Step 3: We prove that

(p n+1 ,q n+1 ,r n+1 ) = (p n ,q n ,r n ) ⇐⇒ q 2 n = p n r n .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

10 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

Step 3: We prove that

(p n+1 ,q n+1 ,r n+1 ) = (p n ,q n ,r n ) ⇐⇒ q 2 n = p n r n .

⇒ p n = (p n + q n ) 2

q n = (p n + q n )(q n + r n )

r n = (q n + r n ) 2 ⎫

⎬

⎭ =⇒ q2 n = p nr n .

The prisoner

dilemma and

the evolution

of cooperation

10 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 3: We prove that

(p n+1 ,q n+1 ,r n+1 ) = (p n ,q n ,r n ) ⇐⇒ q 2 n = p n r n .

⇒ p n = (p n + q n ) 2

q n = (p n + q n )(q n + r n )

r n = (q n + r n ) 2 ⎫

⎬

⎭ =⇒ q2 n = p nr n .

⇐

p n+1 = pn + 2p nq n + n p

}

(1 − p n − 2q n )

{{ }

{ 2 r n

}} {

qn

2

= p 2 n + 2p n q n + p n − p 2 n − p n 2q n = p n .

10 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Step 4: For n ∈ N, we have

q 2 n+1 = (p n + q n ) 2 (q n + r n ) 2 = p n+1 r n+1 .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

11 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 4: For n ∈ N, we have

q 2 n+1 = (p n + q n ) 2 (q n + r n ) 2 = p n+1 r n+1 .

Conclusion: After the first generation, the fraction of each

type remains constant (no evolution).

11 /75

The Hardy-Weinberg law

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 4: For n ∈ N, we have

q 2 n+1 = (p n + q n ) 2 (q n + r n ) 2 = p n+1 r n+1 .

Conclusion: After the first generation, the fraction of each

type remains constant (no evolution).

Observation: The Hardy-Weinberg law can be easily

generalized for the case of polyploidy.

11 /75

Non-random mating

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Violations of random mating are:

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

12 /75

Non-random mating

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Violations of random mating are:

◮ Inbreeding, which causes an increase in homozygosity for

all genes.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

12 /75

Non-random mating

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Violations of random mating are:

◮ Inbreeding, which causes an increase in homozygosity for

all genes.

◮ Assortative mating, when sexually reproducing organisms

tend **to** mate with individuals that are like themselves in

some respect (positive assortative mating, reducing

variation) or dissimilar (negative assortative mating,

increasing variation).

12 /75

Finite population size

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Genetic drift produces random changes in the frequency of

traits in a population. Genetic drift arises from the element of

chance involved in which individuals survive and reproduce

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

13 /75

Finite population size

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Genetic drift produces random changes in the frequency of

traits in a population. Genetic drift arises from the element of

chance involved in which individuals survive and reproduce

Its rate depends strongly on population size (law of large

numbers).

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

13 /75

Finite population size

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Genetic drift produces random changes in the frequency of

traits in a population. Genetic drift arises from the element of

chance involved in which individuals survive and reproduce

Its rate depends strongly on population size (law of large

numbers).

With a small number of individuals, a lucky break for one or

two causes a disproportionately greater deviation from the

expected result (small populations drift more rapidly than large

ones).

13 /75

Finite population size

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

13 /75

Genetic drift produces random changes in the frequency of

traits in a population. Genetic drift arises from the element of

chance involved in which individuals survive and reproduce

Its rate depends strongly on population size (law of large

numbers).

With a small number of individuals, a lucky break for one or

two causes a disproportionately greater deviation from the

expected result (small populations drift more rapidly than large

ones).

The founder effect is the effect of establishing a new

population by a small number of individuals, carrying only a

small fraction of the original population’s genetic variation. As

a result, the new population may be distinctively different, both

genetically and phenotypically, from the parent population from

which it is derived (possible speciation).

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Natural selection is a process that causes heritable traits that

are helpful for survival and reproduction **to** become more

common, and harmful traits **to** become rarer. This occurs

because organisms with advantageous traits pass on more

copies of these traits **to** the next generation.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

14 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

Natural selection is a process that causes heritable traits that

are helpful for survival and reproduction **to** become more

common, and harmful traits **to** become rarer. This occurs

because organisms with advantageous traits pass on more

copies of these traits **to** the next generation.

Artificial selection is the intentional breeding of certain traits,

or combinations of traits, over others.

The prisoner

dilemma and

the evolution

of cooperation

14 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Sexual selection states that the frequency of traits can

increase or decrease depending on the attractiveness of the

bearer.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

15 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Sexual selection states that the frequency of traits can

increase or decrease depending on the attractiveness of the

bearer.

◮ male **to** male combat produces weapons,

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

15 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Sexual selection states that the frequency of traits can

increase or decrease depending on the attractiveness of the

bearer.

◮ male **to** male combat produces weapons,

◮ mate choice produces ornaments

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

15 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Sexual selection states that the frequency of traits can

increase or decrease depending on the attractiveness of the

bearer.

◮ male **to** male combat produces weapons,

◮ mate choice produces ornaments

◮ mate coercion (forced mating).

15 /75

Selection

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Sexual selection states that the frequency of traits can

increase or decrease depending on the attractiveness of the

bearer.

◮ male **to** male combat produces weapons,

◮ mate choice produces ornaments

◮ mate coercion (forced mating).

Cryptic female choice, a phenomenon in internally fertilizing

animals such as mammals and birds, where a female may

simply dispose of a male’s sperm without his knowledge.

15 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Mutation is the process of producing new or altered traits.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

16 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Mutation is the process of producing new or altered traits.

Mutation are generated by

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

16 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Mutation is the process of producing new or altered traits.

Mutation are generated by

◮ copying errors during cell division;

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

16 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Mutation is the process of producing new or altered traits.

Mutation are generated by

◮ copying errors during cell division;

◮ exposure **to** ultraviolet, ionizing radiation, chemical

mutagens, virus;

16 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Mutation is the process of producing new or altered traits.

Mutation are generated by

◮ copying errors during cell division;

◮ exposure **to** ultraviolet, ionizing radiation, chemical

mutagens, virus;

◮ deliberately;

16 /75

Mutation

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Mutation is the process of producing new or altered traits.

Mutation are generated by

◮ copying errors during cell division;

◮ exposure **to** ultraviolet, ionizing radiation, chemical

mutagens, virus;

◮ deliberately;

Mutations can be deleterious, favorable or neutral.

16 /75

Migration

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

Migration is the gene flow from one population **to** another. In

general, allele frequencies will become more homogeneous

among the populations.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

17 /75

Migration

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

Migration is the gene flow from one population **to** another. In

general, allele frequencies will become more homogeneous

among the populations.

Vertical gene transfer (from parents **to** offspring) requires

migration of individuals;

The prisoner

dilemma and

the evolution

of cooperation

17 /75

Migration

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

General

framework

The

Hardy-Weinberg

law

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Migration is the gene flow from one population **to** another. In

general, allele frequencies will become more homogeneous

among the populations.

Vertical gene transfer (from parents **to** offspring) requires

migration of individuals;

Horizontal gene transfer is any process in which an organism

transfers genetic material **to** another cell that is not its

offspring (relevant in long time scales).

17 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

The prisoner

dilemma and

the evolution

of cooperation

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

◮ Introduced in 1961 by P. A. P. Moran.

The prisoner

dilemma and

the evolution

of cooperation

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

◮ Introduced in 1961 by P. A. P. Moran.

◮ Fixed size population.

The prisoner

dilemma and

the evolution

of cooperation

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

◮ Introduced in 1961 by P. A. P. Moran.

◮ Fixed size population.

◮ Asexual evolutionary dynamics.

The prisoner

dilemma and

the evolution

of cooperation

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

◮ Introduced in 1961 by P. A. P. Moran.

◮ Fixed size population.

◮ Asexual evolutionary dynamics.

◮ No mutations.

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Let us now define a simple evolutionary process, that takes in

consideration the natural selection.

◮ Introduced in 1961 by P. A. P. Moran.

◮ Fixed size population.

◮ Asexual evolutionary dynamics.

◮ No mutations.

◮ Two kind of individuals, with fitnesses φ I and φ II .

18 /75

The Moran process

**Evolutionary**

**Game** **Theory**

Step 1:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals,

19 /75

The Moran process

**Evolutionary**

**Game** **Theory**

Step 2:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals, we choose one **to** reproduce,

proportionally **to** the fitness,

19 /75

The Moran process

**Evolutionary**

**Game** **Theory**

Step 3:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals, we choose one **to** reproduce,

proportionally **to** the fitness, and we choose a

(possibly different) second individual **to** be eliminated.

19 /75

The Moran process

**Evolutionary**

**Game** **Theory**

Step 4:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals, we choose one **to** reproduce,

proportionally **to** the fitness, and we choose a

(possibly different) second individual **to** be eliminated.

19 /75

The Moran process

**Evolutionary**

**Game** **Theory**

Step 1 (again):

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals, we choose one **to** reproduce,

proportionally **to** the fitness, and we choose a

(possibly different) second individual **to** be eliminated.

19 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

We associate **to** the type I individual (mutant), the fitness φ I .

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

20 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We associate **to** the type I individual (mutant), the fitness φ I .

We associate **to** the type II individual, the fitness φ II = 1.

The prisoner

dilemma and

the evolution

of cooperation

20 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

We associate **to** the type I individual (mutant), the fitness φ I .

We associate **to** the type II individual, the fitness φ II = 1.

Transitions probabilities for type I individuals c i,j = Θ(j → i),

are given by:

⎧

N−i iφ

N iφ+N−i

if j = i − 1 ,

⎪⎨

1 − c

c i,j = i+1,i − c i−1,i if j = 1 ,

i N−i

N iφ+N−i

if j = i + 1 ,

⎪⎩

0 otherwise .

20 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

The dynamics can be easily computed from the transition

matrix

M = (c i,j ) i,j=0,···,N

.

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

21 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The dynamics can be easily computed from the transition

matrix

M = (c i,j ) i,j=0,···,N

.

P(i,t) = the probability of having i type I individuals at time t .

The prisoner

dilemma and

the evolution

of cooperation

21 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The dynamics can be easily computed from the transition

matrix

M = (c i,j ) i,j=0,···,N

.

P(i,t) = the probability of having i type I individuals at time t .

The evolution equation is given by

P(t + ∆t,i) = c i,i−1 P(t,i − 1) + c i,i P(t,i) + c i+1,i P(t,i + 1)

21 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The dynamics can be easily computed from the transition

matrix

M = (c i,j ) i,j=0,···,N

.

P(i,t) = the probability of having i type I individuals at time t .

The evolution equation is given by

P(t + ∆t,i) = c i,i−1 P(t,i − 1) + c i,i P(t,i) + c i+1,i P(t,i + 1)

We define ⃗ P(t) = (P(0,t),P(1,t), · · · ,P(N,t)).

21 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The dynamics can be easily computed from the transition

matrix

M = (c i,j ) i,j=0,···,N

.

P(i,t) = the probability of having i type I individuals at time t .

The evolution equation is given by

P(t + ∆t,i) = c i,i−1 P(t,i − 1) + c i,i P(t,i) + c i+1,i P(t,i + 1)

21 /75

We define ⃗ P(t) = (P(0,t),P(1,t), · · · ,P(N,t)).

and then ⃗ P(t + ∆t) = M ⃗ P(t).

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Theorem

lim

k→∞ Mk =

⎛

⎞

1 1 − F 1 · · · 1 − F N

0 0 · · · 0

⎜

⎟

⎝ . ⎠ .

0 F 1 · · · F N

22 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 1: We write the transition matrix M

⎛

⎞

1 ∗

0 ∗ ∗ 0

⎛ ⎞

∗ ∗ ∗

1 ∗ 0

. .. . .. . ..

= ⎝0 ˜M 0 ⎠ ,

⎜

⎟

⎝ 0 0⎠

0 ∗ 1

∗ ∗ 1

where ˜M is a real, tri-diagonal matrix, with strictly positive

entries in the super-, sub-, and main diagonal.

23 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 1: We write the transition matrix M

⎛

⎞

1 ∗

0 ∗ ∗ 0

⎛ ⎞

∗ ∗ ∗

1 ∗ 0

. .. . .. . ..

= ⎝0 ˜M 0 ⎠ ,

⎜

⎟

⎝ 0 0⎠

0 ∗ 1

∗ ∗ 1

where ˜M is a real, tri-diagonal matrix, with strictly positive

entries in the super-, sub-, and main diagonal.

Then, ⃗ P 0 := (1,0, · · · ,0) and ⃗ P N := (0, · · · ,0,1) are

eigenvec**to**rs of M associated **to** the eigenvalue λ = 1.

23 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Step 2: The matrix M is tri-diagonal symmetric, then all

eigenvalues are real.

The prisoner

dilemma and

the evolution

of cooperation

24 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Step 2: The matrix M is tri-diagonal symmetric, then all

eigenvalues are real.

Step 3: (Gershgorin theorem:) given a n × n matrix A, we

define R i = ∑ j≠i A ij. Let λ be an eigenvalue of A. Then, there

is an i such that |λ − A ii | ≤ R i .

The prisoner

dilemma and

the evolution

of cooperation

24 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Step 2: The matrix M is tri-diagonal symmetric, then all

eigenvalues are real.

Step 3: (Gershgorin theorem:) given a n × n matrix A, we

define R i = ∑ j≠i A ij. Let λ be an eigenvalue of A. Then, there

is an i such that |λ − A ii | ≤ R i .

This implies that all eigenvalues of ˜M are |λ| < 1.

The prisoner

dilemma and

the evolution

of cooperation

24 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Step 2: The matrix M is tri-diagonal symmetric, then all

eigenvalues are real.

Step 3: (Gershgorin theorem:) given a n × n matrix A, we

define R i = ∑ j≠i A ij. Let λ be an eigenvalue of A. Then, there

is an i such that |λ − A ii | ≤ R i .

This implies that all eigenvalues of ˜M are |λ| < 1.

Step 4: Then the only stationary states ⃗ P ∗ , such that

M ⃗ P ∗ = ⃗ P ∗ are the trivial ones ⃗ P 0 and ⃗ P N .

Furthermore, all other states decay **to** 0.

24 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

⃗P(∞) =

lim ⃗P(k∆t)

k→∞

= lim

k→∞ Mk ⃗ P(0) = (1 − H[p 0 ],0, · · · ,0,H[p 0 ]) .

The prisoner

dilemma and

the evolution

of cooperation

25 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

⃗P(∞) =

lim ⃗P(k∆t)

k→∞

= lim

k→∞ Mk ⃗ P(0) = (1 − H[p 0 ],0, · · · ,0,H[p 0 ]) .

For any initial condition a mutant gene will be

either fixed or lost.

25 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

⃗P(∞) =

lim ⃗P(k∆t)

k→∞

= lim

k→∞ Mk ⃗ P(0) = (1 − H[p 0 ],0, · · · ,0,H[p 0 ]) .

For any initial condition a mutant gene will be

either fixed or lost.

The fixation probability is given by H[p 0 ].

25 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We compute the fixation probability π i of a initial condition

consisting of i mutants.

The prisoner

dilemma and

the evolution

of cooperation

26 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We compute the fixation probability π i of a initial condition

consisting of i mutants.

Clearly, π 0 = 0 and π N = 1.

The prisoner

dilemma and

the evolution

of cooperation

26 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We compute the fixation probability π i of a initial condition

consisting of i mutants.

Clearly, π 0 = 0 and π N = 1. Furthermore

π i = c i−1,i π i−1 + c i,i π i + c i+1,i π i+1 , i = 1, · · · ,N − 1 .

The prisoner

dilemma and

the evolution

of cooperation

26 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We re-write last equation

(1 − c i,i )π i = c i−1,i π i−1 + c i+1,i π i+1 ,

The prisoner

dilemma and

the evolution

of cooperation

27 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We re-write last equation

(1 − c i,i )π i = c i−1,i π i−1 + c i+1,i π i+1 ,

i(N − i) + (N − i)iφ

π i =

N − i + iφ

i(N − i)

N − i + iφ π i−1 +

(N − i)iφ

N − i + iφ π i+1 ,

The prisoner

dilemma and

the evolution

of cooperation

27 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

We re-write last equation

(1 − c i,i )π i = c i−1,i π i−1 + c i+1,i π i+1 ,

i(N − i) + (N − i)iφ

π i =

N − i + iφ

i(N − i)

N − i + iφ π i−1 +

(N − i)iφ

N − i + iφ π i+1 ,

(i(N − i) + (N − i)iφ)π i = i(N − i)π i−1 + (N − i)iφπ i+1 ,

27 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

We re-write last equation

(1 − c i,i )π i = c i−1,i π i−1 + c i+1,i π i+1 ,

i(N − i) + (N − i)iφ

π i =

N − i + iφ

i(N − i)

N − i + iφ π i−1 +

(N − i)iφ

N − i + iφ π i+1 ,

(i(N − i) + (N − i)iφ)π i = i(N − i)π i−1 + (N − i)iφπ i+1 ,

(1 + φ)π i = π i−1 + π i+1 φ .

27 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

We shall solve

− π i−1 + (1 + φ)π i − φπ i+1 = 0 ,

π 0 = 0 , π N = 1 .

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

28 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

We shall solve

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

− π i−1 + (1 + φ)π i − φπ i+1 = 0 ,

π 0 = 0 , π N = 1 .

We look for solutions of the form π i = α i :

−1 + (1 + φ)α − φα 2 = 0 ,

The prisoner

dilemma and

the evolution

of cooperation

28 /75

The Moran Process

**Evolutionary**

**Game** **Theory**

We shall solve

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

− π i−1 + (1 + φ)π i − φπ i+1 = 0 ,

π 0 = 0 , π N = 1 .

We look for solutions of the form π i = α i :

−1 + (1 + φ)α − φα 2 = 0 ,

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

and then

α =

−(1 + φ) ± √ (1 + φ) 2 − 4φ

2φ

=

−(1 + φ) ± (φ − 1)

2φ

=

{ 1

φ

1

28 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

We use the expression

π i = A φ i + B ,

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

and put the boundary conditions

We find

Finally

0 = π 0 = A + B , 1 = π N = A φ N + B .

A = −B =

φN

1 − φ N .

π i = φ N−i 1 − φ i

1 − φ N .

29 /75

The Moran process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

For the neutral case (φ = 1), we find

lim π i = lim φ N−i 1 − φ i

φ→1 φ→1 1 − φ N = i N .

In the neutral case, mutants will fixate with probability

proportional **to** their initial presence.

30 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

Step 1:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals,

31 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

Step 2:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

In a population of N individuals, we create a new population

of N individuals,

31 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

Step 3:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

31 /75

In a population of N individuals, we create a new population

of N individuals, where each newborn is chosen from

the previous generation proportionally **to** the fitness.

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

Step 4:

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

31 /75

In a population of N individuals, we create a new population

of N individuals, where each newborn is chosen from

the previous generation proportionally **to** the fitness.

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

Step 1 (again):

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

31 /75

In a population of N individuals, we create a new population

of N individuals, where each newborn is chosen from

the previous generation proportionally **to** the fitness.

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

We call φ I and φ II the fitnesses of type I and II individuals,

respectively.

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

32 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

We call φ I and φ II the fitnesses of type I and II individuals,

respectively.

The transition probabilities Θ(i → j) are given by

( ) (

) j (

) N−j

N iφ I (N − i)φII

c j,i =

j iφ I + (N − i)φ II iφ I + (N − i)φ II

The prisoner

dilemma and

the evolution

of cooperation

32 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

We call φ I and φ II the fitnesses of type I and II individuals,

respectively.

The transition probabilities Θ(i → j) are given by

( ) (

) j (

) N−j

N iφ I (N − i)φII

c j,i =

j iφ I + (N − i)φ II iφ I + (N − i)φ II

The transition matrix M is such that

⎛

⎞

∗ ∗ ∗ · · · ∗ ∗

0 0 0 · · · 0 0

lim

k→∞ Mk =

.

⎜

. ..

.

⎟

⎝0 0 0 · · · 0 0⎠

∗ ∗ ∗ · · · ∗ ∗

32 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

To obtain the fixation probabilities we shall solve:

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

33 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

To obtain the fixation probabilities we shall solve:

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

π i =

=

N∑

c(j, i)π j

j=0

N∑

( ) (

) j (

) N−j N iφ I (N − i)φII

π j

j iφ I + (N − i)φ II iφ I + (N − i)φ II

j=0

33 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

To obtain the fixation probabilities we shall solve:

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

π i =

=

=

N∑

c(j, i)π j

j=0

N∑

( ) (

) j (

) N−j N iφ I (N − i)φII

π j

j iφ I + (N − i)φ II iφ I + (N − i)φ II

j=0

N∑

( ) ( N

j

j=0

i

i + (N − i)φ

) j ( ) N−j (N − i)φ

π j ,

i + (N − i)φ

33 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

To obtain the fixation probabilities we shall solve:

π i =

=

=

N∑

c(j, i)π j

j=0

N∑

( ) (

) j (

) N−j N iφ I (N − i)φII

π j

j iφ I + (N − i)φ II iφ I + (N − i)φ II

j=0

N∑

( ) ( N

j

j=0

i

i + (N − i)φ

where φ = φ II

φ I

is the relative fitness.

) j ( ) N−j (N − i)φ

π j ,

i + (N − i)φ

33 /75

The Wright-Fisher process

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

To obtain the fixation probabilities we shall solve:

π i =

=

=

N∑

c(j, i)π j

j=0

N∑

( ) (

) j (

) N−j N iφ I (N − i)φII

π j

j iφ I + (N − i)φ II iφ I + (N − i)φ II

j=0

N∑

( ) ( N

j

j=0

i

i + (N − i)φ

where φ = φ II

φ I

is the relative fitness.

We will not solve this equation!

) j ( ) N−j (N − i)φ

π j ,

i + (N − i)φ

33 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Consider two reproductive strategies:

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

34 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Consider two reproductive strategies:

◮ Male: **to** have only male descendants;

The prisoner

dilemma and

the evolution

of cooperation

34 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Consider two reproductive strategies:

◮ Male: **to** have only male descendants;

◮ Female: **to** have only female descendants.

The prisoner

dilemma and

the evolution

of cooperation

34 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Consider two reproductive strategies:

◮ Male: **to** have only male descendants;

◮ Female: **to** have only female descendants.

Consider an strategy I which consists in playing Male with

probability x and Female with probability 1 − x.

The prisoner

dilemma and

the evolution

of cooperation

34 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Consider two reproductive strategies:

◮ Male: **to** have only male descendants;

◮ Female: **to** have only female descendants.

Consider an strategy I which consists in playing Male with

probability x and Female with probability 1 − x.

Consider a mutant strategy J which consists in playing Male

with probability y and Female with probability 1 − y.

34 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The number of descendants in the first-generation is

strategy-independent. Then, we consider the number of

descendants in the second generation.

The prisoner

dilemma and

the evolution

of cooperation

35 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The number of descendants in the first-generation is

strategy-independent. Then, we consider the number of

descendants in the second generation.

Suppose a population of I-strategists at equilibrium, i.e., a

population of xN males and (1 − x)N females.

The prisoner

dilemma and

the evolution

of cooperation

35 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The number of descendants in the first-generation is

strategy-independent. Then, we consider the number of

descendants in the second generation.

Suppose a population of I-strategists at equilibrium, i.e., a

population of xN males and (1 − x)N females.

As anyone has exactly one father and one mother, this means

that each male in the n-th generation has k k

x

sons and

1−x

daughters.

35 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Now, consider a small number of J in the I population.

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

W(J,I) = y × offspring of a son

+(1 − y) × offspring of a daughter

The prisoner

dilemma and

the evolution

of cooperation

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

W(J,I) = y × offspring of a son

+(1 − y) × offspring of a daughter

= y x + 1 − y

1 − x

The prisoner

dilemma and

the evolution

of cooperation

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

W(J,I) = y × offspring of a son

+(1 − y) × offspring of a daughter

= y x + 1 − y y(1 − 2x)

=

1 − x x(1 − x) + 1

1 − x .

The prisoner

dilemma and

the evolution

of cooperation

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

W(J,I) = y × offspring of a son

+(1 − y) × offspring of a daughter

= y x + 1 − y y(1 − 2x)

=

1 − x x(1 − x) + 1

1 − x .

The prisoner

dilemma and

the evolution

of cooperation

For any x ≠ 0, x ≠ 1 and x ≠ 1 2

that W(J,I) > W(I,I).

it is possible **to** find y such

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

Now, consider a small number of J in the I population.

The number of grandchildren (fitness) will be:

W(J,I) = y × offspring of a son

+(1 − y) × offspring of a daughter

= y x + 1 − y y(1 − 2x)

=

1 − x x(1 − x) + 1

1 − x .

The prisoner

dilemma and

the evolution

of cooperation

For any x ≠ 0, x ≠ 1 and x ≠ 1 2

that W(J,I) > W(I,I).

The ESS is given by x = 1 2 .

it is possible **to** find y such

36 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

This question caught Darwin’s attention.

The prisoner

dilemma and

the evolution

of cooperation

37 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

This question caught Darwin’s attention.

Many people noticed that a small number of males and a large

number of females produces a larger growth.

The prisoner

dilemma and

the evolution

of cooperation

37 /75

Why do we have so many men?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

This question caught Darwin’s attention.

Many people noticed that a small number of males and a large

number of females produces a larger growth.

The prevalence of ratio 1:1 shows that the natural selection

acts over genes (in this case, the same as individuals) and not

over groups.

37 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

The fitness is roughly the probability **to** leave descendants.

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

The prisoner

dilemma and

the evolution

of cooperation

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

If we study a sex-dependent characteristic, we need **to** know

the number of descendants in the next two generations.

The prisoner

dilemma and

the evolution

of cooperation

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

If we study a sex-dependent characteristic, we need **to** know

the number of descendants in the next two generations.

Fitness can be

The prisoner

dilemma and

the evolution

of cooperation

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

If we study a sex-dependent characteristic, we need **to** know

the number of descendants in the next two generations.

Fitness can be

◮ frequency independent;

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

If we study a sex-dependent characteristic, we need **to** know

the number of descendants in the next two generations.

Fitness can be

◮ frequency independent;

◮ frequency dependent.

38 /75

Fitness

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

The Moran

process

The

Wright-Fisher

process

Men and women

Fitness

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

The fitness is roughly the probability **to** leave descendants.

If we study a sex-independent characteristic, we need **to** know

only the number of descendants in the next generation.

If we study a sex-dependent characteristic, we need **to** know

the number of descendants in the next two generations.

Fitness can be

◮ frequency independent;

◮ frequency dependent.

In the last case, we use game theory **to** calculate the fitness.

38 /75

What is a game?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Many agents make decisions and the result depends on

(possibly) all decisions taken.

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

39 /75

What is a game?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Many agents make decisions and the result depends on

(possibly) all decisions taken.

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

39 /75

What is a game?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Many agents make decisions and the result depends on

(possibly) all decisions taken.

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

39 /75

What is a game?

**Evolutionary**

**Game** **Theory**

FACC Chalub

Many agents make decisions and the result depends on

(possibly) all decisions taken.

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

39 /75

What is a game?

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Many agents make decisions and the result depends on

(possibly) all decisions taken.

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

39 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

◮ Each game consists of N pure strategies R 1 , R 2 , · · · , R N .

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

40 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Each game consists of N pure strategies R 1 , R 2 , · · · , R N .

◮ Each player can adopt mixed strategies, defined by a

vec**to**r ⃗p = (p 1 , · · · ,p N ). This means that the player

adopts strategy R i with probability p i .

40 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Each game consists of N pure strategies R 1 , R 2 , · · · , R N .

◮ Each player can adopt mixed strategies, defined by a

vec**to**r ⃗p = (p 1 , · · · ,p N ). This means that the player

adopts strategy R i with probability p i .

◮ We define N × N pay-off matrix U, where u ij is the gain

of an R i -strategist against an R j -strategist.

40 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Each game consists of N pure strategies R 1 , R 2 , · · · , R N .

◮ Each player can adopt mixed strategies, defined by a

vec**to**r ⃗p = (p 1 , · · · ,p N ). This means that the player

adopts strategy R i with probability p i .

◮ We define N × N pay-off matrix U, where u ij is the gain

of an R i -strategist against an R j -strategist.

◮ A ⃗p-strategist against a ⃗q-strategist has an expected

pay-off

⃗p · U⃗q .

40 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

◮ Let ⃗ β(⃗q) be the best strategy against ⃗q, i.e., the vec**to**r

that maximizes the function ⃗p ↦→ ⃗p · U⃗q.

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

41 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Let ⃗ β(⃗q) be the best strategy against ⃗q, i.e., the vec**to**r

that maximizes the function ⃗p ↦→ ⃗p · U⃗q.

◮ A ⃗p-strategy is called a Nash equilibrium if it is the best

strategy against itself

⃗p = ⃗ β(⃗p) .

41 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Let ⃗ β(⃗q) be the best strategy against ⃗q, i.e., the vec**to**r

that maximizes the function ⃗p ↦→ ⃗p · U⃗q.

◮ A ⃗p-strategy is called a Nash equilibrium if it is the best

strategy against itself

⃗p = ⃗ β(⃗p) .

◮ All games in the above conditions has at least one Nash

equilibrium.

41 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ A strategy ˆp is an ESS if no rare mutant can invade a

population of ˆp. For ⃗p ≠ ˆp and ε small enough

⃗p · U(ε⃗p + (1 − ε)ˆp) < ˆp · U(ε⃗p + (1 − ε)ˆp) .

42 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ A strategy ˆp is an ESS if no rare mutant can invade a

population of ˆp. For ⃗p ≠ ˆp and ε small enough

⃗p · U(ε⃗p + (1 − ε)ˆp) < ˆp · U(ε⃗p + (1 − ε)ˆp) .

◮ This condition can be re-written in the following form:

42 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ A strategy ˆp is an ESS if no rare mutant can invade a

population of ˆp. For ⃗p ≠ ˆp and ε small enough

⃗p · U(ε⃗p + (1 − ε)ˆp) < ˆp · U(ε⃗p + (1 − ε)ˆp) .

◮ This condition can be re-written in the following form:

◮ (equilibrium) ⃗p · Uˆp ≤ ˆp · U,

42 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ A strategy ˆp is an ESS if no rare mutant can invade a

population of ˆp. For ⃗p ≠ ˆp and ε small enough

⃗p · U(ε⃗p + (1 − ε)ˆp) < ˆp · U(ε⃗p + (1 − ε)ˆp) .

◮ This condition can be re-written in the following form:

◮ (equilibrium) ⃗p · Uˆp ≤ ˆp · U,

◮ (stability) if ⃗p ≠ ˆp e ⃗p · Uˆp = ˆp · U⃗p then ⃗p · U⃗p < ˆp · U⃗p.

42 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

◮ Assume there are n different types in the population,

E 1 , · · · ,E n , with frequencies x 1 , · · · ,x n .

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

43 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Assume there are n different types in the population,

E 1 , · · · ,E n , with frequencies x 1 , · · · ,x n .

◮ The fitness of each type E i depends on the composition of

the population, i.e., φ i = φ i (⃗x).

43 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Assume there are n different types in the population,

E 1 , · · · ,E n , with frequencies x 1 , · · · ,x n .

◮ The fitness of each type E i depends on the composition of

the population, i.e., φ i = φ i (⃗x).

◮ The state of the population is given by a vec**to**r

⃗x(t) = (x 1 , · · · ,x n ) such that ∑ i x i = 1.

43 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Assume there are n different types in the population,

E 1 , · · · ,E n , with frequencies x 1 , · · · ,x n .

◮ The fitness of each type E i depends on the composition of

the population, i.e., φ i = φ i (⃗x).

◮ The state of the population is given by a vec**to**r

⃗x(t) = (x 1 , · · · ,x n ) such that ∑ i x i = 1.

◮ The variation of the relative size of the E i -strategists,

ẋ i /x i , depends on the relative success of the E i strategy.

43 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Let us assume

ẋ i

x i

= fitness ofE i − average fitness of the population .

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

44 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Let us assume

ẋ i

x i

= fitness ofE i − average fitness of the population .

We define the average fitness

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

¯f (⃗x) = ∑ i

x i f i (⃗x) .

44 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Let us assume

ẋ i

x i

= fitness ofE i − average fitness of the population .

We define the average fitness

¯f (⃗x) = ∑ i

x i f i (⃗x) .

and we write the replica**to**r equation:

ẋ i = x i (f i (⃗x) − ¯f (⃗x)) .

44 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Let us assume

ẋ i

x i

= fitness ofE i − average fitness of the population .

We define the average fitness

¯f (⃗x) = ∑ i

x i f i (⃗x) .

and we write the replica**to**r equation:

ẋ i = x i (f i (⃗x) − ¯f (⃗x)) .

44 /75

Property:

( ) ( )

d xi xi

= (f i − f j ) .

dt x j x j

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

We define the n − 1-dimensional simplex:

S n := {⃗x = (x 1 ,x 2 , · · · ,x n ) ∈ R n∣ n∑

∣ x i = 1}

i=1

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

45 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

We define the n − 1-dimensional simplex:

S n := {⃗x = (x 1 ,x 2 , · · · ,x n ) ∈ R n∣ ∣

This equation preservers the simplex:

n∑

x i = 1}

i=1

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

S := ∑ i

x i =⇒ Ṡ = (1 − S)¯f .

45 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

We define the n − 1-dimensional simplex:

S n := {⃗x = (x 1 ,x 2 , · · · ,x n ) ∈ R n∣ ∣

This equation preservers the simplex:

n∑

x i = 1}

i=1

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Assume

S := ∑ i

x i =⇒ Ṡ = (1 − S)¯f .

f i (⃗x) = (V⃗x) i .

45 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

We define the n − 1-dimensional simplex:

S n := {⃗x = (x 1 ,x 2 , · · · ,x n ) ∈ R n∣ ∣

This equation preservers the simplex:

n∑

x i = 1}

i=1

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Assume

S := ∑ i

x i =⇒ Ṡ = (1 − S)¯f .

f i (⃗x) = (V⃗x) i .

Then, the replica**to**r dynamics is given by

ẋ i = x i ((V⃗x) i − ⃗x · V⃗x) .

45 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Consider a game with two strategies and a pay-off matrix given

by

( ) a b

V =

c d

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

46 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Consider a game with two strategies and a pay-off matrix given

by

( ) a b

V =

c d

We have

⃗x = (x,1 − x) ,

A⃗x = (ax + b(1 − x),cx + d(1 − x)) ,

⃗x · A⃗x = ax 2 + bx(1 − x) + cx(1 − x) + d(1 − x) 2 ,

(A⃗x) 1 − ⃗x · A⃗x = (1 − x)((a − c)x + (b − d)(1 − x)) .

46 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Consider a game with two strategies and a pay-off matrix given

by

( ) a b

V =

c d

We have

⃗x = (x,1 − x) ,

A⃗x = (ax + b(1 − x),cx + d(1 − x)) ,

⃗x · A⃗x = ax 2 + bx(1 − x) + cx(1 − x) + d(1 − x) 2 ,

(A⃗x) 1 − ⃗x · A⃗x = (1 − x)((a − c)x + (b − d)(1 − x)) .

This implies in the replica**to**r equation

ẋ = x(1 − x)((a − c)x + (b − d)(1 − x)) .

46 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Definition

Given a dynamics ẋ = Ψ(x) and an initial condition x, the

ω-limit, ω(x) is the set of accumulation points of ω(x).

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

47 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Definition

Given a dynamics ẋ = Ψ(x) and an initial condition x, the

ω-limit, ω(x) is the set of accumulation points of ω(x).

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

ω(x)

47 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Definition

Given a dynamics ẋ = Ψ(x) and an initial condition x, the

ω-limit, ω(x) is the set of accumulation points of ω(x).

Theorem

Let ẋ = Ψ(x) be defined in G ⊂ R. Let V : G → R be

continuously differentiable. If ˙V(x(t)) ≤ 0, for all t ≥ 0, then

ω(x) ∪ G ⊂ {x ∈ G| ˙V(x) = 0}

(the ω-limits are stationary points).

47 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Definition

Given a dynamics ẋ = Ψ(x) and an initial condition x, the

ω-limit, ω(x) is the set of accumulation points of ω(x).

Theorem

Let ẋ = Ψ(x) be defined in G ⊂ R. Let V : G → R be

continuously differentiable. If ˙V(x(t)) ≤ 0, for all t ≥ 0, then

ω(x) ∪ G ⊂ {x ∈ G| ˙V(x) = 0}

(the ω-limits are stationary points).

V is known as a Lyapunov function.

47 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

We study the stability of the stationary points.

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

We study the stability of the stationary points.

We linearize the equation (system) near a stationary point x 0 :

ẋ = Mx, where M is a matrix.

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

We study the stability of the stationary points.

We linearize the equation (system) near a stationary point x 0 :

ẋ = Mx, where M is a matrix.

Suppose that all eigenvalues have non-zero real part.

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

We study the stability of the stationary points.

We linearize the equation (system) near a stationary point x 0 :

ẋ = Mx, where M is a matrix.

Suppose that all eigenvalues have non-zero real part.

1. All eigenvalues have positive real part: x 0 is a source;

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

We study the stability of the stationary points.

We linearize the equation (system) near a stationary point x 0 :

ẋ = Mx, where M is a matrix.

Suppose that all eigenvalues have non-zero real part.

1. All eigenvalues have positive real part: x 0 is a source;

2. All eigenvalues have negative real part: x 0 is a sink;

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

We study the stability of the stationary points.

We linearize the equation (system) near a stationary point x 0 :

ẋ = Mx, where M is a matrix.

Suppose that all eigenvalues have non-zero real part.

1. All eigenvalues have positive real part: x 0 is a source;

2. All eigenvalues have negative real part: x 0 is a sink;

3. Some positive/some negative: x 0 is a saddle.

48 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Theorem

Let ˆx ∈ S n be an ESS for a certain game. Then ˆx is a stable

point of the replica**to**r dynamics.

Proof: We only need **to** show that

P(x) = ∏ i

xˆxi

i

is a Lyapunov function

49 /75

Dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

Example

If A is symmetric (partnership games) the V (x) = 1 2⃗x · A⃗x is

Lyapunov.

˙V (x) = −˙⃗x · A⃗x = − ∑ i

= ∑ i

ẋ i (A⃗x) i

x i ((A⃗x) i − ⃗x · A⃗x)(A⃗x) i

= − ∑ i

x i ((A⃗x) i − ⃗x · A⃗x) 2 ≤ 0 .

50 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

◮ 2 players

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

51 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

◮ 2 players

◮ 2 strategies: Cooperate and Defect

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

51 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

◮ 2 players

◮ 2 strategies: Cooperate and Defect

◮ One pay-off matrix for each player:

( ) R S

T P

T > R > P > S e 2R > T + S.

51 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

◮ 2 players

◮ 2 strategies: Cooperate and Defect

◮ One pay-off matrix for each player:

( ) R S

T P

T > R > P > S e 2R > T + S.

◮ The Nash equilibrium occurs when both players, play

Defect and each one receives P < R, the gain if both had

cooperated.

51 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

This equilibrium is not fully satisfac**to**ry, because altruism

exists in nature.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

52 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

This equilibrium is not fully satisfac**to**ry, because altruism

exists in nature.

Altruism occurs when on individual decreases its fitness **to**

increase someoneelse’s fitness.

52 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

This equilibrium is not fully satisfac**to**ry, because altruism

exists in nature.

Altruism occurs when on individual decreases its fitness **to**

increase someoneelse’s fitness.

In the prisoner dilemma case, altruism corresponds **to** play

Cooperate.

52 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Examples of cooperation/altruism:

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

53 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Examples of cooperation/altruism:

◮ **An**ts in society.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

53 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

◮ **An**ts in society.

◮ A bird singing **to** alert the proximity of a preda**to**r.

53 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

◮ **An**ts in society.

◮ A bird singing **to** alert the proximity of a preda**to**r.

◮ Bird’s flight in V.

53 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

◮ **An**ts in society.

◮ A bird singing **to** alert the proximity of a preda**to**r.

◮ Bird’s flight in V.

◮ Symbionts.

53 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Possible explanation: The Iterated prisoner dilemma (IPD).

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

54 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Possible explanation: The Iterated prisoner dilemma (IPD).

Each **to**urnament consists in a given number of matches

(iterations). The final pay-off is the sum the pay-off of all

rounds.

54 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Possible explanation: The Iterated prisoner dilemma (IPD).

Each **to**urnament consists in a given number of matches

(iterations). The final pay-off is the sum the pay-off of all

rounds.

This allows more complex strategies (including, e.g., the

previous opponents choices).

54 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

R. Axelrod asked many specialists in **Game** **Theory**, from

different fields **to** present strategies for the IPD. Furthermore,

he included a random strategy (playing with probability 1/2

Cooperate or Defect).

55 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

R. Axelrod asked many specialists in **Game** **Theory**, from

different fields **to** present strategies for the IPD. Furthermore,

he included a random strategy (playing with probability 1/2

Cooperate or Defect).

Each strategy played against each other (round-robin

**to**urnament). Each match consisted of 200 iterations of the

prisoner’s Dilemma (but no strategist knew that).

55 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

R. Axelrod asked many specialists in **Game** **Theory**, from

different fields **to** present strategies for the IPD. Furthermore,

he included a random strategy (playing with probability 1/2

Cooperate or Defect).

Each strategy played against each other (round-robin

**to**urnament). Each match consisted of 200 iterations of the

prisoner’s Dilemma (but no strategist knew that).

The final gain is the sum of all gains after 200 rounds.

55 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of strategies:

◮ TIT − FOR − TAT: Cooperate at first and then

reproduces the opponent’s move.

◮ DOWNING: Use Bayesian inference **to** understand

opponents strategy and then tries **to** maximize its own

gain.

◮ JOST: Variation of TFT that plays Defect from time **to**

time.

56 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Nome

Comp.

1 **An**a**to**l Rapoport 4

2 Nicholas Tideman & Paula Chieruzzi 41

3 Rudy Nydegger 23

4 Bernard Grofman 8

5 Martin Schubik 16

6 William Stein & Amnon Rapoport 13

7 James Friedman 13

8 Mor**to**n Davis 6

9 James Graaskamp 63

10 Leslia Downing 33

11 Scott Feld 6

12 Johann Joss 5

13 Gordon Tullock 18

14 Name withheld 77

15 RANDOM 5

57 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Class. Nome Comp. Gain

1 **An**a**to**l Rapoport 4 504.5

2 Nicholas Tideman & Paula Chieruzzi 41 500.4

3 Rudy Nydegger 23 485.5

4 Bernard Grofman 8 481.9

5 Martin Schubik 16 480.7

6 William Stein & Amnon Rapoport 13 477.8

7 James Friedman 13 473.4

8 Mor**to**n Davis 6 471.8

9 James Graaskamp 63 400.7

10 Leslia Downing 33 390.6

11 Scott Feld 6 327.6

12 Johann Joss 5 304.4

13 Gordon Tullock 18 300.5

14 Name withheld 77 282.2

15 RANDOM 5 276.3

57 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The winner was TFT, the simplest strategy in the **to**urnament.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

58 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the **to**urnament.

The eight first strategies were all nice.

58 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the **to**urnament.

The eight first strategies were all nice.

One strategy is nice if it is never the first **to** play Defect.

58 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the **to**urnament.

The eight first strategies were all nice.

One strategy is nice if it is never the first **to** play Defect.

No other strategy were nice.

58 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Axelrod organized a second **to**urnament, with 63 different

strategies.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

59 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod organized a second **to**urnament, with 63 different

strategies.

The number of iterations were not fixed. A new iteration was

played if probability w = .9954.

59 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod organized a second **to**urnament, with 63 different

strategies.

The number of iterations were not fixed. A new iteration was

played if probability w = .9954.

Again, the winner was TFT.

59 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Axelrod did a third **to**urnament, this time simulating an

ecological dynamics.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

60 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third **to**urnament, this time simulating an

ecological dynamics.

After each round, each strategy has an offspring which is

proportional **to** the pay-off obtained in the previous round.

60 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third **to**urnament, this time simulating an

ecological dynamics.

After each round, each strategy has an offspring which is

proportional **to** the pay-off obtained in the previous round.

Initially, all strategies had the same population and the

simulation lasted over many generations.

60 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third **to**urnament, this time simulating an

ecological dynamics.

After each round, each strategy has an offspring which is

proportional **to** the pay-off obtained in the previous round.

Initially, all strategies had the same population and the

simulation lasted over many generations.

Again, TFT won.

60 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third **to**urnament, this time simulating an

ecological dynamics.

After each round, each strategy has an offspring which is

proportional **to** the pay-off obtained in the previous round.

Initially, all strategies had the same population and the

simulation lasted over many generations.

Again, TFT won.

From the fifteen first, 14 were nice. From the fifteen last, 14

were not nice.

60 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Behavioral reasons for the success of TFT:

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Behavioral reasons for the success of TFT:

◮ Niceness: never is the first **to** defect.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Behavioral reasons for the success of TFT:

◮ Niceness: never is the first **to** defect.

◮ Retaliation: pays Defect every time the opponent plays

Defect.

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Behavioral reasons for the success of TFT:

◮ Niceness: never is the first **to** defect.

◮ Retaliation: pays Defect every time the opponent plays

Defect.

◮ Forgiveness: has short memory (1 iteration).

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Behavioral reasons for the success of TFT:

◮ Niceness: never is the first **to** defect.

◮ Retaliation: pays Defect every time the opponent plays

Defect.

◮ Forgiveness: has short memory (1 iteration).

◮ Has no envy: it is not worried with receiving more that the

opponent, but only with the **to**tal value.

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Behavioral reasons for the success of TFT:

◮ Niceness: never is the first **to** defect.

◮ Retaliation: pays Defect every time the opponent plays

Defect.

◮ Forgiveness: has short memory (1 iteration).

◮ Has no envy: it is not worried with receiving more that the

opponent, but only with the **to**tal value.

◮ Clarity: the opponent understands quickly the he/she

should cooperate **to** have a high pay-off.

61 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

Ecological reasons for the success of TFT:

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

62 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Ecological reasons for the success of TFT:

◮ Robustness: TFT succeeds very well in different

environments.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

62 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Ecological reasons for the success of TFT:

◮ Robustness: TFT succeeds very well in different

environments.

◮ Stability: no strategy can invade an environment

dominated by TFT.

62 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Ecological reasons for the success of TFT:

◮ Robustness: TFT succeeds very well in different

environments.

◮ Stability: no strategy can invade an environment

dominated by TFT.

◮ Viability: from a small number TFT-strategists, they,

collaborating **to** each other, can dominate the

environment.

62 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

63 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

When we have perspectives of future iterations:

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

64 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

When we have perspectives of future iterations:

Play TFT.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

64 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

When we have perspectives of future iterations:

Play TFT.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

When there is no (or few) expectations of future iterations:

64 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

When we have perspectives of future iterations:

Play TFT.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

When there is no (or few) expectations of future iterations:

Play AllD.

64 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

Examples of strategy change:

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

65 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of strategy change:

◮ Burkitt lymphoma: oncogenic virus. Moderated in healthy

patients and devastating when conjugated with malaria.

65 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of strategy change:

◮ Burkitt lymphoma: oncogenic virus. Moderated in healthy

patients and devastating when conjugated with malaria.

◮ Chronic diseases × acute infections.

65 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of strategy change:

◮ Burkitt lymphoma: oncogenic virus. Moderated in healthy

patients and devastating when conjugated with malaria.

◮ Chronic diseases × acute infections.

◮ Social evolution: the level of cooperation tends **to** be

higher in smaller societies.

65 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

In animals with complex neural structure, individual recognition

can be done directly. This does not happen with bacteria.

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

66 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

In animals with complex neural structure, individual recognition

can be done directly. This does not happen with bacteria.

Symbiotic bacteria relate with only on individual.

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

66 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

In animals with complex neural structure, individual recognition

can be done directly. This does not happen with bacteria.

Symbiotic bacteria relate with only on individual.

In the human case, individual recognition is made from face

observation. The human brain has an enormous area dedicated

only **to** face recognition. Its misbehavior is called prosopagnosia

and practically do not affect any other brain function

(reasoning, memory, ...). This is probably linked **to** the

evolution of sophisticated social patterns of cooperation.

66 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

We consider an iterated prisoner dilemma, where w is the

probability of a new round.

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

67 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

We consider an iterated prisoner dilemma, where w is the

probability of a new round.

The probability of having n + 1 rounds is w n+1 .

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

67 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

We consider an iterated prisoner dilemma, where w is the

probability of a new round.

The probability of having n + 1 rounds is w n+1 .

The average number of rounds is

∞∑

(n + 1)w n+1 (1 − w) = 1

1 − w .

n=0

67 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

We consider an iterated prisoner dilemma, where w is the

probability of a new round.

The probability of having n + 1 rounds is w n+1 .

The average number of rounds is

∞∑

(n + 1)w n+1 (1 − w) = 1

1 − w .

n=0

Let A n be the pay-off at the n-Th round.

67 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

We consider an iterated prisoner dilemma, where w is the

probability of a new round.

The probability of having n + 1 rounds is w n+1 .

The average number of rounds is

∞∑

(n + 1)w n+1 (1 − w) = 1

1 − w .

n=0

Let A n be the pay-off at the n-Th round.

The **to**tal pay-off is

∞∑

A = A n w n .

n=0

67 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

In a population of AllD, no isolated mutant strategy is able **to**

invade.

68 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

In a population of AllD, no isolated mutant strategy is able **to**

invade.

Proof: **An**y cooperative strategy will be exploited by the strategy of

the majority.

68 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If

w ≥ T − R

T − P

and

w ≥ T − R

R − S ,

then, in a population of TFT-players, no isolated mutant

strategy can invade.

69 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If

w ≥ T − R

T − P

and

w ≥ T − R

R − S ,

then, in a population of TFT-players, no isolated mutant

strategy can invade.

Proof: As TFT has only one round memory, we need only **to**

compare its pay-off with the strategy that plays only Defect and the

one that alternates Cooperate and Defect.

69 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If

w ≥ T − R

T − P

and

w ≥ T − R

R − S ,

then, in a population of TFT-players, no isolated mutant

strategy can invade.

Proof: As TFT has only one round memory, we need only **to**

compare its pay-off with the strategy that plays only Defect and the

one that alternates Cooperate and Defect.

The average pay-off of TFT against itself is ∑ Rw n = R

1−w ,

69 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If

w ≥ T − R

T − P

and

w ≥ T − R

R − S ,

then, in a population of TFT-players, no isolated mutant

strategy can invade.

Proof: As TFT has only one round memory, we need only **to**

compare its pay-off with the strategy that plays only Defect and the

one that alternates Cooperate and Defect.

The average pay-off of TFT against itself is ∑ Rw n = R

1−w ,

...against AllD is T + ∑ Pw n+1 = T + Pw

1−w ,

69 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If

w ≥ T − R

T − P

and

w ≥ T − R

R − S ,

then, in a population of TFT-players, no isolated mutant

strategy can invade.

Proof: As TFT has only one round memory, we need only **to**

compare its pay-off with the strategy that plays only Defect and the

one that alternates Cooperate and Defect.

The average pay-off of TFT against itself is ∑ Rw n = R

1−w ,

...against AllD is T + ∑ Pw n+1 = T + Pw

1−w ,

...against the alternating strategy is ∑ Tw 2n + ∑ Sw 2n+1 = T+wS

1−w

. 2

69 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If a population of nice strategists cannot be invaded by a single

mutant (with a given strategy) then, it cannot be invaded by

any group of mutants (with the same strategy).

70 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

If a population of nice strategists cannot be invaded by a single

mutant (with a given strategy) then, it cannot be invaded by

any group of mutants (with the same strategy).

Proof: The average gain of residents is R. The average gain of

mutants is at most R. The gain of a mutant against a resident is at

most R (otherwise, the mutant could invade alone).

70 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

A population of AllD-strategists can be invaded by a group of

TFT-strategists with relative size larger than

f ∗ :=

(P − S)(1 − w)

R + P − S − T + (S + T − 2P)w .

71 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

A population of AllD-strategists can be invaded by a group of

TFT-strategists with relative size larger than

f ∗ :=

(P − S)(1 − w)

R + P − S − T + (S + T − 2P)w .

Proof: Let f be the relative size of invaders.

71 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

A population of AllD-strategists can be invaded by a group of

TFT-strategists with relative size larger than

f ∗ :=

(P − S)(1 − w)

R + P − S − T + (S + T − 2P)w .

Proof: Let f be the relative size of invaders. The average pay-off of

the invaders is

[

R

f

1 − w + (1 − f ) S + Pw ]

.

1 − w

71 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

A population of AllD-strategists can be invaded by a group of

TFT-strategists with relative size larger than

f ∗ :=

(P − S)(1 − w)

R + P − S − T + (S + T − 2P)w .

Proof: Let f be the relative size of invaders. The average pay-off of

the invaders is

[

R

f

1 − w + (1 − f ) S + Pw ]

.

1 − w

The average pay-off of the residents is

[

f T + Pw ]

P

+ (1 − f )

1 − w 1 − w .

71 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Theorem

A population of AllD-strategists can be invaded by a group of

TFT-strategists with relative size larger than

f ∗ :=

(P − S)(1 − w)

R + P − S − T + (S + T − 2P)w .

If w = 1, the condition reduces **to** f (R − P) > 0, then invasion

is always possible when R > P.

71 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Homogeneous population of AllD and TFT cannot be invaded

by isolated mutants. But a small number of coopera**to**rs,

cooperating also among themselves, is able **to** invade a

population of defec**to**rs. The inverse is not possible.

72 /75

The prisoner dilemma

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Homogeneous population of AllD and TFT cannot be invaded

by isolated mutants. But a small number of coopera**to**rs,

cooperating also among themselves, is able **to** invade a

population of defec**to**rs. The inverse is not possible.

Then, we expect that a population that seldom interacts will

be dominated by defec**to**rs, while frequently interacting

individuals will possibly be dominated by coopera**to**rs.

72 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

If we consider the replica**to**r dynamics associated **to** the

prisoner dilemma, we have (for x, the fraction of coopera**to**rs)

ẋ = x(1 − x)((R − T)x + (1 − x)(S − P)) .

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

73 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

If we consider the replica**to**r dynamics associated **to** the

prisoner dilemma, we have (for x, the fraction of coopera**to**rs)

Define x ∗ =

ẋ = x(1 − x)((R − T)x + (1 − x)(S − P)) .

P−S

(P−S)−(T −R) . If

(P − S) − (T − R) > 0 =⇒ x ∗ > 1 ,

(P − S) − (T − R) < 0 =⇒ x ∗ < 0 .

73 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

If we consider the replica**to**r dynamics associated **to** the

prisoner dilemma, we have (for x, the fraction of coopera**to**rs)

Define x ∗ =

ẋ = x(1 − x)((R − T)x + (1 − x)(S − P)) .

P−S

(P−S)−(T −R) . If

(P − S) − (T − R) > 0 =⇒ x ∗ > 1 ,

(P − S) − (T − R) < 0 =⇒ x ∗ < 0 .

In both cases, we have (0 dominates 1):

0 1

73 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

Now, consider a game between AllD and TFT.

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

74 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

Now, consider a game between AllD and TFT.

The pay-off matrix is given by:

AllD

P

AllD

1−w

TFT S + Pw

1−w

TFT

T + Pw

R

1−w

1−w

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

74 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Now, consider a game between AllD and TFT.

The pay-off matrix is given by:

We have

and

AllD

P

AllD

1−w

TFT S + Pw

1−w

P > S =⇒

T + Pw

1 − w < R

1 − w

TFT

T + Pw

R

1−w

1−w

P

1 − w > s + Pw

1 − w ,

⇐⇒ w > T − R

T − P .

74 /75

The prisoner dilemma: the replica**to**r dynamics

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Now, consider a game between AllD and TFT.

The pay-off matrix is given by:

We have

and

AllD

P

AllD

1−w

TFT S + Pw

1−w

P > S =⇒

T + Pw

1 − w < R

1 − w

TFT

T + Pw

R

1−w

1−w

P

1 − w > s + Pw

1 − w ,

⇐⇒ w > T − R

T − P .

In this case, we have (coordination game):

74 /75

0 x* 1

Tomorrow

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

◮ The Hawk and Dove game;

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

75 /75

Tomorrow

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

◮ The Hawk and Dove game;

◮ Adaptive dynamics;

75 /75

Tomorrow

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

◮ The Hawk and Dove game;

◮ Adaptive dynamics;

◮ Asymmetric conflict;

75 /75

Tomorrow

**Evolutionary**

**Game** **Theory**

FACC Chalub

What

evolution is?

**Evolutionary**

dynamics

**Game**s

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

◮ The Hawk and Dove game;

◮ Adaptive dynamics;

◮ Asymmetric conflict;

◮ Group selection.

75 /75