An Introduction to Evolutionary Game Theory

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An Introduction to Evolutionary Game Theory

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

Games

The prisoner

dilemma and

the evolution

of cooperation

I Today: basics;

2 /75


Overview

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

I Today: basics;

II Tomorrow: hot topics.

2 /75


Overview — today

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

1 What evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

3 /75


Overview — today

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

1 What evolution is?

2 Evolutionary dynamics

The prisoner

dilemma and

the evolution

of cooperation

3 /75


Overview — today

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

1 What evolution is?

2 Evolutionary dynamics

3 Games

3 /75


Overview — today

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

1 What evolution is?

2 Evolutionary dynamics

3 Games

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

Games

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

Games

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

Games

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

Games

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.

Games

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

Games

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).

Games

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

Games

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;

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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 ;

Games

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 ;

Games

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 .

Games

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

Games

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

Games

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

Games

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

Games

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 .

Games

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 .

Games

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 .

Games

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

Games

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

Games

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 .

Games

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

Games

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

Games

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:

Games

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.

Games

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

Games

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

Games

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).

Games

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

Games

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

Games

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.

Games

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

Games

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.

Games

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,

Games

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

Games

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

Games

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

Games

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.

Games

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

Games

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;

Games

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

Games

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

Games

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

Games

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.

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

eigenvectors 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

Games

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

Games

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

Games

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

Games

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

Games

⃗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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

− 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 ,

Games

The prisoner

dilemma and

the evolution

of cooperation

and then

α =

−(1 + φ) ± √ (1 + φ) 2 − 4φ


=

−(1 + φ) ± (φ − 1)


=

{ 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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

40 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

vector ⃗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

Games

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

vector ⃗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

Games

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

vector ⃗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 vector

that maximizes the function ⃗p ↦→ ⃗p · U⃗q.

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

41 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Let ⃗ β(⃗q) be the best strategy against ⃗q, i.e., the vector

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

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

◮ Let ⃗ β(⃗q) be the best strategy against ⃗q, i.e., the vector

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

Games

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

Games

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

Games

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

Games

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 .

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

43 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

Games

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 vector

⃗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

Games

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 vector

⃗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

Games

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

Games

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

Games

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 replicator equation:

ẋ i = x i (f i (⃗x) − ¯f (⃗x)) .

44 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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 replicator 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

Games

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

Games

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

Games

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

Games

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 replicator 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

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

46 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

Games

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 replicator 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).

Games

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

Games

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

Games

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

Games

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.

Games

Definitions

Dynamics

The prisoner

dilemma and

the evolution

of cooperation

48 /75


Dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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 replicator 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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

This equilibrium is not fully satisfactory, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

This equilibrium is not fully satisfactory, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

This equilibrium is not fully satisfactory, 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:

Games

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

Games

Examples of cooperation/altruism:

Ants 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

Ants in society.

◮ A bird singing to alert the proximity of a predator.

53 /75


The prisoner dilemma

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

Ants in society.

◮ A bird singing to alert the proximity of a predator.

◮ Bird’s flight in V.

53 /75


The prisoner dilemma

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Examples of cooperation/altruism:

Ants in society.

◮ A bird singing to alert the proximity of a predator.

◮ 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).

Games

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Possible explanation: The Iterated prisoner dilemma (IPD).

Each tournament 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Possible explanation: The Iterated prisoner dilemma (IPD).

Each tournament 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

Games

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

Games

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

tournament). 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

Games

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

tournament). 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

Games

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Nome

Comp.

1 Anatol 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 Morton 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Class. Nome Comp. Gain

1 Anatol 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 Morton 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

Games

The winner was TFT, the simplest strategy in the tournament.

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the tournament.

The eight first strategies were all nice.

58 /75


The prisoner dilemma

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the tournament.

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

The winner was TFT, the simplest strategy in the tournament.

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

Games

Axelrod organized a second tournament, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod organized a second tournament, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod organized a second tournament, 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 tournament, this time simulating an

ecological dynamics.

Games

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third tournament, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third tournament, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third tournament, 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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

Axelrod did a third tournament, 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

Games

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.

Games

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

Games

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

Games

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

Games

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 total value.

61 /75


The prisoner dilemma

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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 total 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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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.

Games

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.

Games

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.

Games

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:

Games

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

Games

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

Games

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

Games

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.

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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 total 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

Games

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

Games

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: Any 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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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

Games

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 cooperators,

cooperating also among themselves, is able to invade a

population of defectors. The inverse is not possible.

72 /75


The prisoner dilemma

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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 cooperators,

cooperating also among themselves, is able to invade a

population of defectors. The inverse is not possible.

Then, we expect that a population that seldom interacts will

be dominated by defectors, while frequently interacting

individuals will possibly be dominated by cooperators.

72 /75


The prisoner dilemma: the replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

If we consider the replicator dynamics associated to the

prisoner dilemma, we have (for x, the fraction of cooperators)

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

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

73 /75


The prisoner dilemma: the replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

If we consider the replicator dynamics associated to the

prisoner dilemma, we have (for x, the fraction of cooperators)

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 replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

If we consider the replicator dynamics associated to the

prisoner dilemma, we have (for x, the fraction of cooperators)

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 replicator dynamics

Evolutionary

Game Theory

Now, consider a game between AllD and TFT.

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

The prisoner

dilemma and

the evolution

of cooperation

Basics

The iterated PD

Results

74 /75


The prisoner dilemma: the replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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 replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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 replicator dynamics

Evolutionary

Game Theory

FACC Chalub

What

evolution is?

Evolutionary

dynamics

Games

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

Games

◮ 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

Games

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

Games

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

Games

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

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