The El Farol Bar Problem for next generation systems

The El Farol Bar Problem for next generation 

systems 

Athanasios Papakonstantinou 

Dissertation submitted for the MSc in Mathematics 

with Modern Applications 

Department of Mathematics 

August 2006 

Supervisor Dr. Maziar Nekovee, BT Research

Contents 

1. Chaos, Complexity and Irish Music 1 

1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

1.2. The El Farol Bar Problem (EFBP) . . . . . . . . . . . . . . . . . 4 

1.3. Modelling the original problem . . . . . . . . . . . . . . . . . . . 5 

1.4. Game Theory Definitions . . . . . . . . . . . . . . . . . . . . . . . 7 

2. Previous Approaches to the El Farol 9 

2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.2. Approaches to EFBP . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.2.1. Minority Game . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.2.2. Evolutionary Learning . . . . . . . . . . . . . . . . . . . . 11 

2.2.3. Stochastic Adaptive Learning . . . . . . . . . . . . . . . . 13 

3. Analysis and Extension of the Stochastic Algorithm 23 

3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2. Taxing/Payoffs Algorithms . . . . . . . . . . . . . . . . . . . . . . 23 

3.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2.2. Fairness and Efficiency . . . . . . . . . . . . . . . . . . . . 29 

3.3. Budget Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

3.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

3.3.2. Fairness and Efficiency . . . . . . . . . . . . . . . . . . . . 34 

4. Case Study: Multiple Bars in Santa Fe 37 

5. Conclusions 43 

A. C Code for the El Farol Bar Problem 47 

ii

List of Figures 

1.1. Bar attendance in the first 100 weeks [1]. . . . . . . . . . . . . . . 5 

2.1. Behaviour of the average attendance (Top) and of the fluctuations 

(bottom) in the El Farol problem with L = 60 seats, ā = 1/2 and 

m = 2, 3, 6 from left to right [2]. . . . . . . . . . . . . . . . . . . . 10 

2.2. Mean weekly attendance for all 300 trials[3]. . . . . . . . . . . . . 12 

2.3. The attendance in a typical trial[3]. . . . . . . . . . . . . . . . . . 12 

2.4. The normalised one-step transition matrix[3]. . . . . . . . . . . . 13 

2.5. The overall attendance and the probabilities for each of the M 

agents for ‘partial information’, ‘full information’ and ‘signs’ algorithms. 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

2.6. histograms for partial and full info algorithms. . . . . . . . . . . . 18 

2.7. fairness and efficiency plots for original algorithms. . . . . . . . . 20 

2.8. standard deviation off attendance for original algorithms. . . . . . 21 

3.1. histograms for partial and full info taxing algorithms. . . . . . . . 25 


agents for ’partial’ information tax, modified tax algorithms and 

the full information tax algorithm. . . . . . . . . . . . . . . . . . 26 

3.3. fairness and efficiency plots for taxing algorithms. . . . . . . . . . 30 

3.4. standard deviation off attendance for taxing algorithms. . . . . . . 31 

3.5. histograms for budget algorithms (b = 10, w = 0/6). . . . . . . . . 32 


agents for budget (budget= 10) and modified budget (budget= 10, 

wait=3, 6) algorithms. . . . . . . . . . . . . . . . . . . . . . . . . 33 

3.7. standard deviation off attendance for budget algorithms. . . . . . 34 

3.8. fairness and efficiency plots for budget algorithms. . . . . . . . . . 35 


agents for each bar. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

4.2. The cumulative attendance for all three bars. . . . . . . . . . . . . 39 

4.3. Agent attendances for each bar . . . . . . . . . . . . . . . . . . . 40 

4.4. payoff probabilities for each bar and cumulative payoff probability 

for the 3-bar version . . . . . . . . . . . . . . . . . . . . . . . . . 41 

iv

List of Tables 

1.1. The Prisoner’s Dilemma in matrix form with utility pay-offs. . . . 8 

2.1. Behaviour of the ‘partial information’ algorithm. . . . . . . . . . . 16 

2.2. Behaviour of the ‘full information’ algorithm. . . . . . . . . . . . . 16 

2.3. Behaviour of the ‘signs’ algorithm. . . . . . . . . . . . . . . . . . 18 

2.4. standard deviation and mean of payoff probabilities for original 

algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

3.1. Behaviour of the ‘tax’ algorithm with ctax = 6. . . . . . . . . . . . 24 

3.2. Simulation results for various values of ctax, cp. . . . . . . . . . . . 25 

3.3. standard deviation and mean of payoff probabilities for taxing algorithms. 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

3.4. Behaviour of the ‘modified budget’ algorithm for budget=10 and 

staying at home=6. . . . . . . . . . . . . . . . . . . . . . . . . . . 32 

3.5. Simulation results for budget algorithms. . . . . . . . . . . . . . . 32 

3.6. standard deviation and mean of payoff probabilities for budget 

algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 

4.1. mean and std for all bars. . . . . . . . . . . . . . . . . . . . . . . 39 

vi

Acknowledgements 

I would like to dedicate this dissertation to my parents Costantinos and Valentini 

Papakonstantinou . I want to thank them for giving me the opportunity to study 

in the UK and for always supporting me in all my decisions. 

Special thanks to my supervisor in BT Research, Maziar Nekovee, for his help 

and guidance. He was a very valuable source of information and always available 

when I needed him. Also Keith Briggs in BT Research was very helpful and this 

is very much appreciated. 

Futhermore, I would like to thank all the friends in York and in Ipswich for 

this last year, as well as all friends in Greece. 

vi

Chapter 1 

Chaos, Complexity and Irish 

Music 

1.1 Overview 

There is a common misconception that complexity and chaos are synonymous. 

Besides the nonlinearities that occur in both systems, the other properties those 

two areas of mathematics share are disambiguity in definitions and having many 

interesting and different applications. 

Although a definition of chaos that everyone would accept does not exist, 

almost everyone agrees to three properties a chaotic system should have: Chaos 

is aperiodic long-term behaviour in a deterministic system that exhibits sensitive 

dependence on initial conditions[4]. A time evolving property such as the move of 

tectonic plates or planets, the temperature or any weather characteristic or even 

the price movements in stock markets or dreams and emotions [5] may display 

chaotic behaviour. 

Chaos is often related to complexity, but does not follow from it in all cases. 

Chaos might be occurring when studying phenomena as they progress in time, 

but when the same phenomena are examined from a microscopic point of view 

then, the interaction of the various parts of which the system is consisted creates 

patterns and not ‘erratic’ chaotic behaviour. This is where complexity enters. 

It is rather difficult to define complexity in mathematical terms, although 

there is a measure of complexity there is no other way to give mathematical 

definition. Dictionaries might be useful in this quest for defining complexity. 

According to an online Dictionary by Oxford University Press, complex is an adjective 

used to describe nouns which ‘are consisted of different and connected 

parts’. An even more precise definition is ‘consisted of interconnected or interwoven 

parts.’[6] That means, that in order to understand the behaviour of a complex 

system we should understand the behaviour of each part, as well as how they interact 

to form the behaviour of the whole. Our incapacity to describe the whole 

1

2 CHAPTER 1. CHAOS, COMPLEXITY AND IRISH MUSIC 

without describing each part combined with the necessity to relate each part with 

another when describing makes the study of complex systems very difficult. 

Finally, based on [6] an attempt to formalise all the above definitions can be 

made: a complex system is a system formed out of many components whose behaviour 

is emergent,that is, the behaviour of the system cannot be simply inferred 

only from the behaviour of its components. The amount of information needed 

to describe the behaviour of such a system is a measure of its complexity. If the 

number of the possible states the system could have is Ω and it is needed to 

specify in which state it is in, then the number of binary digits needed to specify 

this state is related to the number of the possible states: 

I = log 2(Ω) (1.1.1) 

In order to realise which state the system is in, all the possible states must 

be examined. The fact that the unique representation of each state requires 

a number equal to the number of the states, leads to the conclusion that the 

number of states of the representation is equal to the number possible states of 

the system. For a string of N bits, there are 2 N possible states, therefore: 

Ω = 2 N ⇔ N = I (1.1.2) 

There are many applications of complex systems, in statistical physics, meteorology, 

geology, biology, engineering, economics, even social sciences and psychology. 

In all sciences we could find systems that could be dismantled in their 

core components and study each part and the system as a whole simultaneously. 

It is very interesting trying to examine and forecast the behaviour of systems 

that consist of human beings, systems like a family, or a business, or even a government. 

Humans have the capacity to learn and to constantly evolve, making 

models that deal with them unrealistic and of no use if this property is not taken 

into consideration. Therefore the need to model this human behaviour created 

the complex adaptive systems (CAS). The most common definition and universally 

approved is the one given by one of its founders, John H. Holland of Santa 

Fe Institute: ‘A Complex Adaptive System (CAS) is a dynamic network of many 

agents (which may represent cells, species, individuals, firms, nations) acting in 

parallel, constantly acting and reacting to what the other agents are doing. The 

control of a CAS tends to be highly dispersed and decentralised. If there is to 

be any coherent behaviour in the system, it has to arise from competition and 

cooperation among the agents themselves. The overall behaviour of the system is 

the result of a huge number of decisions made every moment by many individual 

agents’[7]. It is even more intriguing that CAS do not appear only in human 

networks but wherever there is a system with interacting elements. It could be 

cells in a cellular automaton, ions in a spin glass or even cells in an immune 

system[8]. A complex adaptive system besides the property of complexity, has 

the properties of emergence and self-organisation.

1.1. OVERVIEW 3 

Emergence occurs when agents that operate in the same environment start 

to interact which each other. The number of the interactions increases when 

the number of agents increases, this leads to the appearance of new types of 

behaviour. This process can result to an increase of the complexity and since it 

is an internal property of the system and not managed by an outside source, it 

is a ‘self-organised’ process. 

The subject of this dissertation is the study and expansion of a famous complex 

adaptive system known as El Farol Bar Problem which was introduced by 

the economist W. B. Arthur in 1994[1]. El Farol is a bar in Santa Fe in New Mexico 

which plays each Thursday Irish music. People enjoy visiting it and hearing 

some quality music but eventually it becomes overcrowded, so people stop enjoying 

themselves. Each customer decides independently whether to attend or not, 

based on a set of predictors. This scenario provides a simplified mathematical 

model of a class of congestion and coordination problems that arise in modern 

Information and Communications Technology (ICT) systems. 

One application of great interest is networks of cognitive radios, where agents 

compete with each other for the same resource (RF spectrum). Cognitive radios 

are autonomous agents that have the ability to sense the external environment, 

learn from history and make intelligent decisions in order to optimise their performance 

and adjust better to the environment[9]. Another application is internet 

when a large number of people try to visit the same web page or access the same 

ftp server simultaneously and independently. 

In this first chapter, the original Arthur’s EFBP is defined and analysed. In 

the end of the chapter, some basic Game Theory concepts are explained and 

defined. 

In chapter two, various different approaches to the El Farol Bar problem are 

reviewed. First it is viewed as a minority game and various techniques from 

statistical mechanics are implemented. Also strategies are redefined using a binary 

approach as an attempt to reduce complexity. The next approach tries to 

overcome the restrictive strategies using an evolutionary learning algorithm and 

viewing the problem as a Markov stochastic process. The last approach suggests 

a very simple adaptive algorithm which is based on the maximisation of the 

probability of attendance for each agent. There are no specific strategies to guide 

agents during their decision process, only their intention to attend the bar. 

In chapter three, the last algorithm is analysed in depth. In this original work, 

the stochastic adaptive learning algorithm is extended and several derivatives of 

it are examined as an attempt to deal with the unfairness or low efficiency issues 

that occurred in some cases with the original algorithm. Considerable effort was 

put in order to define the stationary state of one variation. Also fairness and 

efficiency are defined and measured both from the bar management’s and agent’s 

point of view. 

In chapter four, it is examined whether three bars in the same town would 

affect the agents ways of entertainment. They attempt to enter the bars in a


random order, but their decision is made using the original algorithm. 

1.2 The El Farol Bar Problem (EFBP) 

Arthur in his paper, tries to predict the bar’s attendance. He assumed that the 

number of the resindents of Santa Fe or prospective clients of the bar is 100 and 

60 as the maximum number of the clients the bar should have so it could not be 

overcrowded. Then he answered the above question mentioning that if a person 

expects fewer than or equal to 60 to show up, he attends the bar, otherwise he 

stays at home. Each person cannot communicate with others, so no one has any 

information about the intentions of everybody else. They only have access to the 

number of the clients of the previous weeks. 

But there is more than one model, based on the numbers of the previous 

weeks, which can be used to predict the current week’s attendance. This makes 

a rational solution impossible and the problem from the client’s point of view 

ill-defined. But even if there was only one model, or due to mysterious reasons 

the clients managed to have common forecasts, the model would fail. If most of 

the people believe that the bar will be overcrowded, then they will not attend 

leaving the bar almost empty. The opposite will happen if most of them think 

that the bar will have less than 60 customers. 

In order to overcome problems such as this, Arthur issued the use of ‘predictors’. 

A predictor is a function that maps the information of d-recent attendances 

into a prediction for the next attendance. Arthur suggested that although there 

are many predictors, each individual has a set of k predictors in his disposal which 

will guide him through the decision process. Each client will decide whether to 

go to the bar or not, according to the most accurate predictor in his set, which 

will be called ‘active predictor’. Inaccurate predictors do not affect the long term 

behaviour of the system, since they will rarely achieve the status of ‘active’ predictor, 

therefore will be rarely used from the clients. The predictors that were 

actually used in the original problem are described in subsection 1.3. 

The results of the computational experiment are shown in fig 1.1. These results 

indicate a tendency of the mean attendance to converge to 60. It seems that 

the predictors self-organise into an equilibrium pattern or ‘ecology’[1]. The active 

predictors forecast above 60 with propability 0.4 and below 60 with propability 

0.6. In terms of game theory, the above mixed strategy is a Nash equilibrium. 

(Game theory terms like Nash equilibrium, strategies, repeated games and other 

are defined in subsection 1.4.) 

The EFBP is a congested resource problem, because the final decision of an 

agent (this is how clients, customers or individuals who decide whether or not 

they should attend El Farol Bar are going to be referred from now on) depends 

on the decision of the other agents. In other words they compete for a resource. 

This congestion appears in many real life systems like the internet where the

1.3. MODELLING THE ORIGINAL PROBLEM 5 

Figure 1.1: Bar attendance in the first 100 weeks [1]. 

users compete for bandwith or roads and highways. The source of congestion, 

in a deterministic framework like EFBP, is the inability of agents to coordinate 

their actions[10], since there is a lack of a centralised mechanism that could guide 

them. In order to understand better and analyse such systems the traditional 

perfect, logical, deductive rationality has given its place to bounded rationality. 

The agents make their decision based on incomplete knowledge, they know that 

they have access to limited information and do their best to fight this uncertainty 

using a combination of rational rules and empirical evidence. 

1.3 Modelling the original problem 

Now that Arthur’s original EFBP is explained (section 1.2) the model can be 

summarised as following: 

Suppose that there are N agents that have to decide whether they will or not 

attend the bar, and L is the maximum number of clients the bar can accommodate, 

before becoming overcrowded. Arthur wanted to predict the binary action 

of the i th customer denoted by a i ∈ {0, 1}, where 1 stands for going to the bar 

and 0 is not going. The total attendance is A = a i . 

As mentioned above, the only information available to the agents is the number 

of the customers of the bar, the previous d weeks: 

It = {A(t − d), . . . , A(t − 2), A(t − 1)} (1.3.1) 

A predictor is a function I ∈ [0, N] d → [0, N]. There are (N + 1) (N+1)d 

predictors [2] The number of the possible predictors that could be used is rather 

large, that is why a selection of S predictors is used. Arthur used the following 

predictors:


1. The same attendace as k weeks ago: 

A(It) = A(t − k), 

where k in the original models 1, 2 and 5. (k-period cycle detectors) 

2. A mirror image around 50% of last week’s attendance: 

3. A fixed predictor: 

A(It) = N − A(t − 1) 

A(It) = 67 

4. A rounded average of the last four weeks: 

A(It) = 1/4 

4 

A(t − r) 

5. A rounded and bound by 0 and N, last 8 weeks trend, computed using the 

least squares method.: 

r=1 

A(It) = min([trend{A8}] + , N)[11] 

Each predictor has a score associated to it, which evolves according to: 

Ui,s(t + 1) = Ui,s(t) + Θ{[Ai,s(It) − L][A(t) − L]}[2] 

Where, s ∈ [1, S], Ai,s(It) the s t h predictor for i t h customer and Θ is the Heaviside 

function (Θ(x) = 0 for x < 0 and Θ(x) = 1 for x ≥ 0). 

Also the predictor s used by i th customer is given by: 

and 

si(t) = argmax s ′Ui,s ′(t) 

ai(t) = Θ[L − Ai,si(t)(It)] 

where, argmax xf(x) is the value of x for which, f(x) gets its maximum value.

1.4. GAME THEORY DEFINITIONS 7 

1.4 Game Theory Definitions 

Games and solutions 

A game is a description of strategic interaction that includes the constraints on the 

actions the players can take and the players interests, without actually specifying 

the actions that the players do take[12]. A solution is how rational people play 

the game. A ‘rational solution’ corresponds to a set of unique strategies (plans 

for player’s actions) for each player that they will have no rational reasons to 

regret choosing[13]. 

Best reply strategy 

A strategy for player, Ri is best reply(or best response) to C’s strategy Cj if it 

gives R the largest pay-off, provided that C has played the game. 

Pure and mixed strategies 

Pure strategy is the simplest kind of strategy, where someone chooses a specific 

course of action. However there might be a case where there is uncertainty 

about which best pure strategy to choose, due to lack of information or any 

other reason. At those cases, the pure strategy is chosen following a random 

probability distribution. This type of strategy is called mixed strategy. A more 

strict definition follows: 

If a player has N available pure strategies (S1, S2, . . . , SN), a mixed strategy M 

is defined by the probabilities (p1, p2, . . . , pN) of each strategy to be selected[13]. 

For M to be well defined the sum of the probabilities should be equal to one. 

Nash equilibrium 

The outcome of strategies Ri for player R and Cj for player C is a Nash equilibrium 

of the game, and thus a potential solution if Ri and Cj are the best solutions 

to each other. In a way player R chooses Ri because he is expecting C to chose 

Cj (and vice verse). When people select Nash equilibrium strategies there is no 

guarantee that they will be happy. There is however a guarantee that they will 

have no reason to change it. 

Nash equilibrium in pure strategies 

Let G be a game, which involves N players. Each player chooses among a finite 

set of strategies Si: That is, player i (i = 1, . . . , N) has access to strategy set 

Si from which he/she chooses strategy σi ∈ Si. A set of pure strategies S = 

(σ1, σ2, . . . , σi, . . . , σN) constitutes a Nash equilibrium if and only if pure strategy 

σi is a best reply to the combination of the strategies of all other players in S for 

all i = 1, . . . , N[13]. 

Nash equilibrium in mixed strategies 

Mixed strategies are in Nash equilibrium, when there is not any strategy available 

the player could choose in order to improve his/her expected utility.


Example of Nash equilibrium:Prisoner’s Dilemma 

This is a very known example in game theory about two arrested suspects for a 

crime. They are put in different cells and are promised that if someone confesses, 

he will be freed and used as a witness against the other, who will be sentenced 

to four years. If they both confess, they receive a three year sentence and if 

nobody confesses they will be convicted for a year, due to lack of evidence. The 

problem is represented in a matrix form in table 1.1, using utility payoffs. For 

Cooperate Defect 

Cooperate 3,3 0,4 

Defect 4,0 1,1 

Table 1.1: The Prisoner’s Dilemma in matrix form with utility pay-offs. 

each player action ‘D’ (Defect) dominates action ‘C’ (Cooperate). Comparing 

the first numbers in each column or the second numbers in each row, shows that 

no matter what player 1 (column) chooses, player 2 (row) will win by choosing 

Defect, since the reward (utility payoff) is higher. This demonstrates that action 

(D,D) is a unique Nash equilibrium. 

Pareto improvement and efficiency 

Pareto improvement is when an agent chooses a strategy that will have no negative 

effects on the others. A system is Pareto efficient or Pareto optimal, when no 

Pareto improvements can be made. In other words, in a Pareto efficient system 

no individual can make an improvement without worsening the others. 

Repeated games 

Features of ‘one-shot’ games like ‘Prisoner’s Dilemma’ are the total lack of cooperation 

and the inability to study how each player’s actions affect the others as 

time progresses. The model of a repeated game is designed to examine long term 

interaction, based on the idea that a player will take into account other players 

estimated future behaviour, when planning his current strategy. This theory tries 

to isolate types of strategies that support mutually desirable outcomes in every 

game and punishes players with undesirable behaviour. ‘Folk’ theorems [12] give 

the conditions under which the set of the payments, that are acquired when in 

equilibrium, will consist of nearly all reasonable payoff profiles. With the use of 

these theorems it is proved acceptable results cannot be sustained if player are 

‘short-sighted’ and only look after their own interests.

Chapter 2 

Previous Approaches to the El 

Farol 

2.1 Overview 

In this section, various different approaches to El Farol Bar Problem are reviewed. 

Furthermore, three ways to extend the original model and analyse it using a 

different perspective, are introduced. 

2.2 Approaches to EFBP 

2.2.1 Minority Game 

In this approach, results known from minority games and tools of statistical 

mechanics are used for EFB model analysis. A minority game is a binary game 

where N (N:odd) players must choose one of the two sides independently and 

those on the minority side win. Players use a set of strategies, based on the past, 

to make their selections [14]. The greatest difference of this model with Arthur’s 

is the introduction of strategies instead of predictors. A strategy is a function 

a(I) from [0, N] m like predictors, but to {0, 1} instead to [0, N]. In other words 

strategies estimate if an agent should visit or not the bar, based on the previous 

history of attendance but in terms of ones (below the level of attendance) and 

zeros (above it). This is of great importance since the number of strategies is 

2N+1d which is significantly less from the number of the predictors for large N. 

This could be denoted as: 

which will depend only on the information 

a µ 

s,i = Θ[L − As,i(It)] (2.2.1) 

µ(t) = {Θ[L − A(t − 1)], . . . , Θ[L − A(t − d)]} (2.2.2) 

9

10 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL 

There are only 22d [2] strategies of this type, which is even less from 2N+1d and 

independent from N. Each agent is assigned S strategies drawn from the pool 

with distribution: P (a) ≡ Prob{a µ 

s,i } = āδ(a − 1) + (1 − ā)δ(a). For this model 

to work, it is considered that on average clients attend the bar with a frequency 

L/N. This leads to ā ≈ L/N, where ā is the average of a µ 

s,i . 

Figure 2.1, illustrates what happens to the model when L, ā and m remain 

fixed, while the number of agents N increases. The top graph shows that when 

Nā ≈ L, the attendance converges to the comfort level, while as m is increasing 

the area where 〈A〉 ≈ L is shrinking. 

Another feature of this approach is that it measures wasted resources which 

appear when the bar is under or over utilised. Although A(t) equals L on average, 

the amount of unexploited resources (A(t) < L) or over-exploited resources 

(A(t) > L), is equal to the distance |A(t) − L|. Thus, the quality of the cooperation 

of the agents is measured by: 

where 〈. . . 〉 is the average on the stationary state. 

σ 2 = 〈(A − L) 2 〉 (2.2.3) 

Figure 2.1: Behaviour of the average attendance (Top) and of the fluctuations 

(bottom) in the El Farol problem with L = 60 seats, ā = 1/2 and m = 2, 3, 6 

from left to right [2]. 

On the bottom part of figure 2.1 it is shown that for small m, σ 2 /N is at is 

peak at the point where āN = L, while as m increases the maximum is getting 

shallower until it is disappeared for m = 6. This leads to the conclusion that, for 

larger values of m, the efficiency increases. More information about this model 

can be found in [2].

2.2. APPROACHES TO EFBP 11 

2.2.2 Evolutionary Learning 

This approach is based on the fact, that since EFBP is a case where inductive 

reasoning and bounded rationality are experienced, models based on a closed set 

of strategies are inadequate. It introduces a stochastic element, since new models 

are created by randomly varying existing ones, as well as a selective process which 

eliminates the ineffectual models. 

Suppose that, like in Arthur’s original experiment, each agent is given k = 10 

predictive models. For simplicity these models are autoregressive and their output 

unsigned and rounded. According to [3] for the i th individual their j th predictor’s 

output is given by: 

ˆx i ⎛ 

 

 

j(n) = round ⎝ 

 

aij(0) + 

l i j 

t=1 

a i ⎞ 

 

 

j(t)x(n − t) ⎠ 

 

(2.2.4) 

 

where x(n − t) is the attendance on week (n − t), l i j is the number of lag terms in 

the j th predictor of individual i, a i j(t) is the coefficient for the lag t steps in the 

past, and a i j(0) is the constant term of the AR model. 

The absolute value and the rounded output makes sure that no negative values 

are assigned and all predictions above 100 were set to 100 according to the 

original model. For each individual the number of lag terms is chosen uniformly 

from the integers {1, . . . , 10} [3]. Before the prediction of the attendance of the 

current week, each individual evolves its set of models for ten generations. This 

procedure, analysed in [3], is synopsised in the following five steps: 

1. As it was mentioned above, each agent chooses from 10 models. In this stage 

an offspring is created for each agents k th model. Lag in the offsprings from 

parents j is set to be one or ten. If l i j = 1 then l i j − 1 is not allowed, while if 

it is equal to ten, l i j + 1 is not allowed. The AR coefficients of the offspring 

are generated with the addition of a zero mean Gaussian variable with 

standard deviation (std) equal to 0.1. Any newly generated AR coefficients, 

are chosen by sampling form N(0, 0.1). In the end of this stage, there are 

ten parent and ten offspring AR models assigned to each individual. 

2. In this stage, the 20 models assigned to each agent are evaluated based 

on the sum of the squared errors made during the prediction of the bar’s 

attendance in the last 12 weeks. 

3. For each agent, the ten models with the least error are selected and set as 

parents for the next generation. 

4. If less than ten generations are conducted, then it starts over again from 

stage 1. Otherwise the best model for each agent is used to predict current 

week’s bar attendance.


5. If the maximum number of weeks is achieved the algorithm ends, if else, 

the predicted attendance is recorded and the simulations starts over again 

from stage 1. 

Figure 2.2: Mean weekly attendance for all 300 trials[3]. 

Figure 2.3: The attendance in a typical trial[3]. 

The results of this procedure being repeated 300 times, are shown in figure 

2.2. The mean weekly attendance for the first 12 weeks was 59.5, but for the 

next 50 weeks, large oscillations appeared until week 100. From 100 to 982 (end) 

weeks, the behaviour of the model could be described as ‘transient’. The mean 

attendance was 56.3155 and std was 1.0456, which is statistically significantly 

different (p < 0.01)[3] from the results of the Arthur’s original paper. None of 

the 300 trials showed convergence to 60 and the results of each trial were similar 

with those illustrated in figure 2.3. 

The dynamics of this system do not provide useful results about the model’s 

overall behaviour. That is why stochastic models based on Markov chains were


Figure 2.4: The normalised one-step transition matrix[3]. 

used. The weekly attendance is the system’s ‘state’ in a simple first-order random 

process. Each of the attendance transitions from week to week for all the 300 

trials were tabulated and the transition matrix in fig 2.4 was formed. It was also 

proved that the system has the Markov property by executing 300 additional 

trials and recording each final weekly attendance at week 982. The cumulative 

distribution of these attendances is similar to the cumulative distribution function 

after the summing of the limiting probability masses calculated from raising the 

transition matrix to a large number. 

2.2.3 Stochastic Adaptive Learning 

In Arthur’s original paper each agent tries to predict how many others will attend 

El Farol Bar. Each individual decides based on a set of strategies (predictors) 

which estimate the attendance of the Bar. Approaches 1 and 2 mentioned in 

subsections 2.2.1 and 2.2.2 also use similar methods, although they try to refine 

the decision process. In this approach it is shown that there is no need for the 

agents to use different strategies and change them trying to find which is the more 

accurate. The problem is considered in stochastic terms instead of deterministic 

and a simple adaptive learning algorithm is implemented. The main advantage 

of this method is that the algorithm is more simple and the decision process is 

less complex[10] since the agents do not decide based on the decision of all the 

others but based only on their recent experiences in the Bar. 

The agents have identical payoffs, b is the payoff for attending a noncrowded 

bar, g for attending a crowded bar and f for staying at home. Without loss 

of generality h is considered to be zero. There are two strategies: either the 

agent attends the bar and receives payment b or g according to the attendance


of the bar, or he stays at home and receives no payment. In a mixed strategy 

equilibrium the expected payment of following one action is equal to the expected 

payment of following the other. This could be denoted as following: 

g P(N −i ≤ N − 1) + b P(N −i > N − 1) = 0 ⇔ 

P(N −i ≤ N − 1) = b 

b − g 

where, b,g are the payoffs, M the total number of players, N the total observed 

attendance, N −i the observed attendance without agent i and N the maximum 

capacity of an uncrowded bar. 

Using a deterministic setting where agents would only use pure strategies 

has the disadvantage that each agent must predict the attendance, based on the 

predictions of the others. This generates results with a high level of noise and high 

deviation. In this bounded rational model, the adaptive learning rule depends 

only on the history of the decisions of each single one agent. 

To overcome this problem a common sense concept is taken into consideration: 

people in general prefer to experience good times, tend to repeat the enjoyable 

and minimise the unpleasant[10]. So according to this if an agent initially attends 

the bar p percent of time, he will increase this if the bar is uncrowded or will 

decrease this the bar is crowded. As time goes by , agents gather information 

about the attendance of the bar, in the form of the parameter p which differs 

for each ith agent. If k is the iteration counter (time), pi the probability that 

ith agent attends and µ the parameter that defines the degree of change of pi 

according to the attendance, then the number of agents attending at time k is 

given by: 

M 

N(k) − xi(k) (2.2.5) 

i=1 

where xi(k) are independent Bernoulli random variables that are equal to one 

with probability pi(k) and equal to zero otherwise[10]. The following simple 

algorithm describes the evolution of pi(k): 

⎧ 

⎪⎨ 0, pi(k) − µ(N(k) − N )xi(k) < 0 

pi(k + 1) = 1, pi(k) − µ(N(k) − N )xi(k) > 1 

⎪⎩ 

pi(k) − µ(N(k) − N )xi(k), otherwise 

(2.2.6) 

At each timestep k the agent attends the Bar with a probability pi(k) after 

tossing a biased coin. If the bar is uncrowded N(k)−N is added to pi(k), while if 

it is crowded N(k) − N is subtracted. Also pi(k) ∈ [0, 1] since it is a probability. 

If the agent does not attend the bar, xi(k) = 0 leads to pi(k + 1) = pi(k).For 

now on the algorithm 2.2.6 will be referred as ‘partial information’ because the 

agents make their decision relying only on their previous experience. But this


algorithm could be modified in an attempt to generate results close to Arthur’s 

original algorithm. In the following ‘full information’ algorithm the decisions are 

made after having a full record of attendance. 

⎧ 

⎪⎨ 0, pi(k) − µ(N(k) − N ) < 0 

pi(k + 1) = 1, 

⎪⎩ 

pi(k) − µ(N(k) − N ), 

pi(k) − µ(N(k) − N ) > 1 

otherwise 

(2.2.7) 

Both these algorithms rely solely on attendance and not on payoffs. A way to 

implement payoffs is setting the payoff for attending an uncrowded bar to µ and 

−µ for attending a crowded bar. If the payoff for staying at home is set to 0, the 

following algorithm depends on payoffs. 

⎧ 

⎪⎨ 0, pi(k) − µsgn(N(k) − N )xi(k) < 0 

pi(k + 1) = 1, pi(k) − µsgn(N(k) − N )xi(k) > 1 

⎪⎩ 

pi(k) − µsgn(N(k) − N )xi(k), otherwise 

(2.2.8) 

where sign is the following function: 

⎧ 

⎪⎨ −1, (N(k) − N ) < 0 

sgn(N(k) − N ) = 0, (N(k) − N ) = 0 

⎪⎩ 

1, (N(k) − N ) > 0 

The general behaviour of the ‘partial information’ algorithm can be seen in 

the first two plots of figure 2.5. The simulation is run for M = 100, N = 60, 

µ = 0.01 and the initial probabilities follow a random uniform distribution. After 

many iterations, the agents are separated into two groups with those of the first 

group attending every day, while those of the other do not attend at all, or attend 

very rarely. The observed attendance after many iterations is slightly below the 

Nash equilibrium point which is always equal to the maximum capacity of the 

uncrowded bar. Partial information algorithm converges to a value near that 

point N − 1 and never reaches N , which is a Pareto efficient point for that 

algorithm since the only way for an agent to be in a better position is to worsen 

somebody else[15]. 

This algorithm is heavily dependant on the value of µ. Its convergence is 

strongly affected by this parameter, for µ = 0.1, M = 100 and N = 60 it 

converges after 800 iterations, while for µ = 0.001, 13000 iterations were not 

enough. In order to explore the behaviour and the limitations of the algorithm 

a lighter version of the original C program was used. It is different from that in 

the Appendix, since it produces no files for plots and the only outputs are the 

average and standard deviation. A conclusion that could be drawn only from the 

simulations (Table: 2.1) is that the algorithm can handle relatively large numbers 

such as M = 10000, N = 6000, provided that µ is very small, in expense of


convergence time . But the conclusion is that only hardware limitations affect its 

behaviour. 

N /M 60/100 600/1000 6000/10000 

iterations 2 · 10 8 2 · 10 8 2 · 10 8 

µ 0.01 0.001 0.001 

average 59.00020 599.000142 5898.745966 

std 0.006524 0.052298 74.159661 

Table 2.1: Behaviour of the ‘partial information’ algorithm. 

The comparison of the first two with the next two plots of figure 2.5 leads to 

the conclusion that agents coordinate successfully when they have access to partial 

information instead of full information. This happens due to the congestion, 

which comes as a result of the similar response of the agents who have available 

all the information. 

The results of the full information algorithm are very similar to Arthur’s 

original simulation (Figure:1.1). The mean attendance is 60 but the variation 

never settles down. Although the probabilities bounce randomly, they increase 

or decrease simultaneously. This proves the assumption made above, that when 

agents have access to full information, they tend to have the same behaviour. 

The behaviour of the algorithm is summarised in table 2.2 

N /M 60/100 600/1000 6000/10000 

iterations 2 · 10 8 2 · 10 8 2 · 10 8 

µ 0.01 0.001 0.001 

average 60.00000 599.99999 5999.999994 

std 6.454943 18.371018 57.254445 

Table 2.2: Behaviour of the ‘full information’ algorithm. 

The first and third plots in figure 2.5 show that ‘partial information’ and 

‘signs’ algorithm have similar behaviour. A close look in the probabilities for each 

agent, reveals that the new algorithm inherits all the properties of the original. 

The only difference is that standard deviation is slightly greater, but that can be 

explained since the new algorithm converges after more iterations. With the same 

constants, more than 3000 iterations are needed for the algorithm to converge. 

The behaviour of the algorithm is summarised in table 2.3. It can be seen that 

this variation does not have any problems with large numbers, probably because 

it is more simple. Instead of using in every iteration the quantity N(k) − N , it 

uses sign(N(k) − N ). 

The histograms of the attendances (figure: 2.6) for partial and full information 

algorithms show that they have a completely different distribution and indicate 

that attendances in full information algorithm may follow normal distribution.


attendance 

attendance 

attendance 

100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 

time (iterations) 

100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 


100 

80 

60 

40 

20 

0 

0 1000 2000 3000 4000 5000 


probabities for each agent 



1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 1000 2000 3000 4000 5000 


Figure 2.5: The overall attendance and the probabilities for each of the M agents 

for ‘partial information’, ‘full information’ and ‘signs’ algorithms.


frequency 

0 500 1000 1500 2000 

N /M 60/100 600/1000 6000/10000 

iterations 2 · 10 8 2 · 10 8 2 · 10 8 

µ 0.01 0.001 0.001 

average 59.000049 599.059126 5998.999914 

std 0.011685 0.286160 1.078615 

Table 2.3: Behaviour of the ‘signs’ algorithm. 

Histogram of attendances in partial info 

50 55 60 

attendances 

65 

frequency 

0 50 100 150 200 

Histogram of attendances in full info 

40 50 60 

attendances 

70 80 

Figure 2.6: histograms for partial and full info algorithms. 

Except from the fast convergence and the ability to handle large numbers, it 

is very important that some of the quantitive characteristics of the algorithms 

are examined too. It is necessary that tools to measure abstract terms like fairness 

and efficiency are developed. A fair outcome requires that agents with 

similar utilities have similar probabilities to attend the bar. A more strict definition 

would demand exact probabilities of attendance for agents with identical 

payoffs[16]. The algorithm, must also be efficient both for the agents and the bar 

management. 

Before proceeding to the tools used, utility payoff, which as a term was first 

introduced in Prisoner’s Dilemma definition in section 1.4, must be defined. Utility 

payoffs are the rewards or penalties that each agent has, after following a pure 

strategy. In this case, agents who attend a not crowded bar get a reward of 1, 

those who attend a crowded bar get a penalty of −1, while those who stay at 

home get nothing (reward=0): 

⎧ 

⎪⎨ −1, xi(k) = 1, (N(k) − N ) > 0 

U(k) = 1, xi(k) = 1, (N(k) − N ) ≤ 0 

⎪⎩ 

0, xi(k) = 0 

So, fairness and efficiency could be measured using the following methods based 

on utility functions: 

• efficiency is determined by the average payoff. The higher the average 

payoff probability is, the higher is the average reward for each player


• fairness is determined by the distribution of payoffs. A histogram which 

shows the creation of groups is a clear indication that the algorithm is 

unfair, similar conclusions could be drawn from the standard deviation. 

High values of std indicate that a significant number of agents has payoffs 

less than the average, while others have greater. 

There are also other ways to measure fairness and efficiency, which have nothing 

to do with utility functions: 

• system’s efficiency is determined by the attendance’s std. A system is 

considered to be efficient, when the attendances are really close to system’s 

capacity. 

• system’s fairness can be also determined by the distribution of attendances 

for each agent. At each iteration of the algorithm, the decision of 

each agent is recorded. In the end of the algorithm it easy to calculate how 

many times each agent has attended the bar. Histogram or std could used 

in this case also. 

It is called system’s efficiency, because most bar managements have interest in 

keeping a stable attendance, with not many fluctuations which would leave the 

bar some days underutilised and some days overutilised. Although system’s fairness 

gives a good picture of the choices of each agent, the original fairness is more 

important since it is calculated from the utility payoffs. After all, it could be said, 

that what matters is the consequences of each agent’s action and the only way 

to measure it, is in terms of reward or penalty. 

Considering system’s fairness, it can be seen in figure 2.7 that both three 

algorithms are not fair. Agents are divided in two categories: 41 of them rarely 

attend the bar, and the rest of them attend it almost always. Although ‘partial 

information with signs’ algorithm was designed as an improvement of the original 

one it inherits all of its properties. One of those properties is minimal attendance 

for 41 agents and maximal for the rest. It seems that from the agents point of 

view, this algorithm is more fair (Table: 2.4, stdsigns < stdpartial ), but in reality 

it is not. This is a result of the lower profits for the always attending group and 

not of a wider distribution of payoff probability. 

std mean 

partial 0.401591 0.4814233 

signs partial 0.3018023 0.36663 

full 0.02300045 −0.00500668 

Table 2.4: standard deviation and mean of payoff probabilities for original algorithms. 

The most fair algorithm regarding the agents is the full information, indeed 

standard deviation of payoff probability is nearly zero but so is average payoff


probability. Nobody makes profit, nobody is happy, but everybody is content 

and they seem to have no intention of changing their strategy, all these are properties 

of Nash equilibrium points. But in terms of attendance ‘full information 

algorithm’ is not absolutely fair, since again there is a classification of agents in 

two groups, although this time casual bargoers have a much wider distribution. 

frequency 

0 10 20 30 40 

frequency 

0 5 10 15 20 

frequency 

0 10 20 30 

partial information 

0.0 0.2 0.4 0.6 0.8 1.0 

probability of attendance 

full information 

0.2 0.3 0.4 0.5 0.6 0.7 0.8 


partial information with signs 

0.0 0.2 0.4 0.6 0.8 1.0 


frequency 

0 10 20 30 40 50 

frequency 

0 1 2 3 4 5 6 7 

frequency 

0 5 10 15 20 25 30 

partial information 

0.0 0.2 0.4 

payoff probability 

0.6 0.8 

full information 

−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 


partial information with signs 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 


Figure 2.7: fairness and efficiency plots for original algorithms. 

It is obvious that these results are open to different interpretations, and vary 

according to the objectives set each time. In general partial information is the 

most appropriate algorithm if the objective is to maximise the average payoff for 

each agent. It might be the most unfair of all, but even full information is not 

fair enough to justify the minimal profits. 

Considering system’s efficiency, it can be seen in the cumulative plot of the 

running std for the original algorithms (Figure: 2.8), that ‘partial information’ 

has the least standard deviation and a negative slope. On the contrary ‘full


information’ has a relatively high std, with no signs of improvement. 

standard deviation 

0 1 2 3 4 5 6 

original algorithms 

partial 

full 

signed 

0 500 1000 1500 

iterations 

2000 2500 3000 

Figure 2.8: standard deviation off attendance for original algorithms.

22 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL

Chapter 3 

Analysis and Extension of the 

Stochastic Algorithm 

3.1 Overview 

None of the previous three variations was proved to be fair and efficient. Fairness 

and efficiency seem to be contradicting terms. In this chapter, new variations 

of the stochastic adaptive learning algorithms that were presented in subsection2.2.3, 

are examined. The variations could be divided into two categories: the 

taxing/payoff algorithms which have to do with an adaptive change of µ and 

the budget algorithms, where the agents are forced not to attend the bar after 

having attended it consecutively for a number of weeks, as a result of inadequate 

resources. 

3.2 Taxing/Payoffs Algorithms 

3.2.1 Overview 

The basic idea behind this algorithm is that in the system there are three types 

of agents. The ‘selfish’ who attend a bar regardless if it is crowded or not, those 

who attend an uncrowded bar and those who never attend the bar. ‘Partial information’ 

maximises the probability of attendance incrementing a small quantity 

to the original probability when the bar is uncrowded or subtracting it from the 

original probability when the bar is crowded. This quantity depends on µ which 

is constant. In this version (equations: 3.2.1), the parameter µtax is inserted so 

that selfish behaviour can be penalised. When a selfish agent attends a crowded 

bar, the quantity that is subtracted from the original probability is ctax times 

greater than in the ‘partial information’ algorithm. 

Simulations indicated that for rather large values of ctax (ctax ≥ 8) the bar 

is underutilised (mean ≤ 58). The behaviour of this algorithm is summarised in 

23

24 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM 

table 3.1 and in the first two plots of figure 3.2. 

N /M 60/100 600/1000 

iterations 2 · 10 8 2 · 10 8 

µ 0.01 0.001 

average 59.072334 598.99924 

std 0.259638 0.052933 

Table 3.1: Behaviour of the ‘tax’ algorithm with ctax = 6. 

This algorithm could be more fair for the agents if those who do not attend 

the bar were encouraged to attend it. By definition of the algorithm this cannot 

be done through the parameter µ, since there is no change for the probabilities 

pi of those who choose or perhaps are forced not to attend. 

A rather aggressive way to change this, is to multiply these probabilities with 

a number close to 1. This way in every iteration the attendance probability for 

all those that do not attend will slightly increase and finally they will ‘decide’ to 

attend the bar. That leads to the following equations for the partial algorithm 

with taxing: 

⎧ 

0, 

⎪⎨ 

where µppi(k) − µtax(N(k) − N )xi(k) < 0 

pi(k + 1) = 1, 

where µppi(k) − µtax(N(k) − N )xi(k) > 1 

⎪⎩ 

pi(k) − µppi(k) − µtax(N(k) − N )xi(k), otherwise 

(3.2.1) 

µp and µtax are defined as following: 

 

1, xi(k) = 1 

µp = 

(3.2.2) 

cp, xi(k) = 0 

where i = 1 . . . M and c1 a constant which affects the degree pi is changing for 

those agents that do not attend (xi(k) = 0) 

 

µ, N(k) ≤ N 

µtax = 

(3.2.3) 

ctaxµ, N(k) > N 

where c2 > 1 is a constant that defines the degree pi is changing for selfish agents 

who insist to attend even if the bar was crowded the previous time. 

The behaviour of this algorithm is summarised in the 3rd and 4th plots of 

figure 3.2. Simulations showed that in a reasonable taxed system (ctax ≥ 3), 

large values of mp result in an underutilised bar (Table:3.2. An interpretation of 

this is that, although people are encouraged to attend, taxation prevents them

3.2. TAXING/PAYOFFS ALGORITHMS 25 

from doing so. Of course as someone would expect minimal taxation results in 

an overcrowded bar. Another interesting feature of this algorithm is, its almost 

rapid convergence, even when it is compared with the original. 

ctax 6 8 6 1 6 

cp 1 1.01 4 4 n/a (full info) 

average 58.929000 59.682000 49.002333 70.489000 58.165750 

std 1.382448 2.200574 0.437992 4.642286 4.851578 

Table 3.2: Simulation results for various values of ctax, cp. 

For the full information tax algorithm there is no need of the parameter mp: 

⎧ 

⎪⎨ 0, pi(k) − µtax(N(k) − N ) < 0 

pi(k + 1) = 1, 

⎪⎩ 

pi(k) − µtax(N(k) − N ), 

pi(k) − µtax(N(k) − N ) > 1 

otherwise 

(3.2.4) 

The histograms of the attendances (figure 3.1) for partial and full information 

taxing algorithms have much in common with those of the original algorithms 

and very few differences. Both full information algorithms seem to follow the 

normal distribution, although in the partial information taxing algorithm, value 

59 is dominating. 

frequency 

0 500 1000 1500 2000 2500 3000 

Histogram of attendances in modified tax partial info 

10 20 30 40 

attendances 

50 60 70 

frequency 

0 50 100 150 200 250 

Histogram of attendances in tax full info 

40 45 50 55 

attendances 

60 65 70 

Figure 3.1: histograms for partial and full info taxing algorithms. 

In [10] the solutions and convergence properties for the original algorithms 

are examined. After defining the equilibrium points, they are used to derive a 

set of deterministic ODE’s. The results concerning the nature of p(k) in steady 

state, are used in the analysis of the convergence behaviour and the stability 

properties of those ODE’s. Although the outputs of the simulations of the original 

algorithms can be reproduced and proved analytically, there are many reasons 

for making this procedure extremely difficult for the other variations. 

From the last plot in figure 3.2 there is an indication that probabilities might 

converge to a stationary state around 0.6. Although full information algorithms 

are simpler cases, the adaptive behaviour of µtax(k) is a source of complexity.


attendance 

attendance 

attendance 

100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 


100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 


100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 





1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 



for ’partial’ information tax, modified tax algorithms and the full information tax 

algorithm.


In the following a set of equations that describe the stationary state behaviour 

of the probabilities will be derived pi(k). 

The number of agents attending at time k is N(k) = M i=1 xi(k) and xi(k) are 

independent Bernoulli trials given by: 

 

1, pi(k) 

xi(k) = 

(3.2.5) 

0, 1 − pi(k) 

In the stationary state we have: 

pi(k + 1) = pi(k) ∀i ∈ N 

Taking the expectation values results to: 

E(pi(k + 1)) = E(pi(k)) ⇒ E(pi(k + 1)) − E(pi(k)) = 0 ⇒ 

E(µtax(N(k) − N )) = 0 

N(k) is the number of agents attending at time k is equal to M i=1 xi(k), where 

xi(k) are independent Bernoulli trials. 

The expectation value is calculated using the following formula: 

E(XY ) = 

fx,y(x, y) 

where X,Y random variables and fx,y(x, y) joined mass function. In this case 

Y = µtax and X = N(k) − N . By definition, Y = µtax is a function of N(k). So, 

Y = g(X) and fx,y(x, y) = fx(x, g(x)) = fx(x). Hence: 

E(µtax(N(k)) · (N(k) − N )) = 

(N(k) − N ) · µtax(N(k)) · fN(k)(N(k)) ⇒ 

cµ 

N(k)>N 

N(k)−N 

(N(k) − N ) · fN(k)(N(k)) + µ 

cµ 

N(k)>N 

µ 

N(k)≤N 

x 

N(k)≤N 

N(k) · fN(k)(N(k)) − cµN 

(N(k) − N ) · fN(k)(N(k)) ⇒ 

N(k)>N 

N(k) · fN(k)(N(k)) − µN 

N(k)≤N 

fN(k)(N(k))+ 

fN(k)(N(k)) (3.2.6) 

Kolmogorov-Smirnov test indicated that z = N(k) might follow Poisson distribution 

f(z) = λz 

z! e−λ , with λ = E(z). Hence: 

E(f(z)) = 

zf(z) = 

zf(z) + 

zf(z) = λ ⇒ 

z 

z≤N 

z>N


 

zf(z) = λ − 

zf(z) (3.2.7) 

z≤N 

and: 

 

f(z) = 1 ⇒ 

f(z) + 

f(z) = 1 ⇒ 

f(z) = 1 − 

f(z) (3.2.8) 

z 

z≤N 

zN 

z>N 

z>N 

Since µ = 0, (3.2.9) becomes: 

 

zf(z)(c − 1) − N 

f(z)(c − 1) = N − λ (3.2.10) 

z>N 

z>N 

Since f(z) follows the Poisson distribution and after replacing z with N(k) and: 

and 

we get the final equation: 

A(λ, M) = 

B(λ, M) = 

M 

z>N 

M 

z>N 

z>N 

z≤N 

λ N(k) 

N(k)! e−λ 

N λN(k) 

N(k)! e−λ 

z>N 

z>N 

B(λ, M)(c − 1) − N A(λ, M)(c − 1) = N − λ (3.2.11) 

After solving equation (3.2.11), a λ should be found, which for M = 100 

and c = 8 should be equal or close to the average of attendances. With a 

numerical approximation of λ available, the values of pi in equilibrium point 

could be determined following a very simple procedure: 

M 

M 

λ = E(N(k) = E( xj(k)) = E( xj(k)) = 

j=1 

j=1 

M 

E(xj(k)) 

Since xi(k) are Bernoulli trials with θ = pi(k) then E(xi(k)) = pi(k). Hence: 

λ = 

j=1 

M 

pj(k) (3.2.12) 

j=1 

It must be noted that c = 1 in equation (3.2.11) leads to M 

j=1 pj = N . So 

for this value of c the original result, mentioned in [10] was recovered.


3.2.2 Fairness and Efficiency 

Like ‘partial information with taxing’ algorithm also here two groups of agents 

are created. Again 41 of them hardly ever attend and 59 almost always and for 

those who do attend the profits are very close to 1 while for those who do not, 

profits are close to 0. In the modified version, although not attending agents are 

encouraged to attend with a slight increase of the probability pi the results are 

not significantly different. This time 40 stay at home almost always and 60 go 

to the bar. A larger pi will lead to more agents attending the bar, but there is 

also the potential danger of overcrowding. The second algorithm has a slightly 

better behaviour since most agents have a marginal raise in the profits and the 

differences in std are of no significance (Table: 3.3). Although these algorithms 

std mean 

partial info with taxing 0.4748791 0.57201 

modified partial info with taxing 0.4863357 0.5930867 

full info with taxing 0.04937645 0.1766167 

Table 3.3: standard deviation and mean of payoff probabilities for taxing algorithms. 

seem to be as unfair as the original, there is an indication that they might be 

slightly more efficient. The average value of payoff probabilities are larger in the 

taxing variation. 

But the suprise comes from the ‘full information with taxing’ algorithm. As 

it can been seen in figure 3.3 one group has 99 agents with probabilities of attendance 

varying from 0.46 to 0.76 and only one agent never goes to the bar. Also 

compared to the original full information algorithm, this one is more efficient 

from the agents point of view, since the payoff probability for those who attend 

varies from 0.08 to 0.27 and has an average of 0.1766167. On the contrary in the 

original full information, the payoff probability varies from −0.05 to 0.04 with an 

average of −0.005 (figure: 2.7, table: 2.4). It can be said that the improvement 

is quite significant. 

As expected, the first two algorithms are better than the third in terms of 

systems efficiency, although are worse when compared with original ones. That 

happens because of the relatively high distribution. The two first have almost 

identical behaviour, although the effects of the slight increase of pi in the 2nd algorithm, 

are visible in (Figure: 3.4). The behaviour of the third taxing algorithm 

is almost identical with the behaviour of the original.


frequency 

0 10 20 30 40 50 

frequency 

0 10 20 30 40 50 

frequency 

0 2 4 6 8 

partial information with taxing 

0.0 0.2 0.4 0.6 0.8 1.0 


modified partial information with taxing 

0.0 0.2 0.4 0.6 0.8 1.0 


full information with taxing 

0.0 0.2 0.4 


0.6 0.8 

frequency 

0 10 20 30 40 50 

frequency 

0 10 20 30 

frequency 

0 1 2 3 4 5 6 7 

partial information with taxing 

0.0 0.2 0.4 0.6 0.8 1.0 


modified partial information with taxing 

0.0 0.2 0.4 0.6 0.8 1.0 


full information with taxing 

0.00 0.05 0.10 0.15 0.20 0.25 


Figure 3.3: fairness and efficiency plots for taxing algorithms.

3.3. BUDGET ALGORITHMS 31 


0 5 10 15 

taxing algorithms 

taxing partial 

modified taxing partial 

taxing full 

0 500 1000 1500 

iterations 

2000 2500 3000 

Figure 3.4: standard deviation off attendance for taxing algorithms. 

3.3 Budget Algorithms 

3.3.1 Overview 

In this algorithm agents have a limited budget which does not allow them to 

attend the bar every night. In another version (‘modified budget algorithm’) of 

this algorithm they are forced to stay at home after attending the bar several 

consecutive nights. This could be denoted as following: 

∀ i ∈ [1, M], k ∈ [1, n], if 

b 

xi(k − j) = b ⇒ 

j=0 

xi(k + 1) = · · · = xi(k + w) = 0 (3.3.1) 

where b ∈ N is the constant that determines how many nights i-agent can attend 

the bar consecutively and w ∈ N the maximum nights an agent must stay indoors 

after having attended the bar for b nights 

The algorithm settles down after almost 2000 iterations, but it is very CPU 

intensive. In table 3.4 there are some results as well as the time needed for them. 

Although the so far algoririthms have given results for 2 · 10 8 iterations in almost 

3 days, this one for 2˙10 6 after 2 weeks has not given any result yet. The reason 

of this increase in CPU power is that instead of using an array of attendances in 

terms of 1 and 0 which is is initialised in every iteration and its size is equal to the 

number of the agents, it uses a matrix which contains the attendances for every 

iteration. That is a matrix with number of agents×iterations elements. Despite 

the fact that this algorithm is the most fair and efficient of all (see subsection: 

3.3.2), it might not be suitable for networks of hundreds of nodes, or in cases of 

large numbers, because in every iteration the columns of this matrix are scanned 

for sequences of 1’s. The next element or sequence of elements in the ‘modified 

budget’ is set to 0 reflecting the inability of the agents to attend the bar. 

As the ratio of budget to ‘nights staying at home’ reaches 1, the bar becomes 

underutilised. It is as if the town is hit by recession: when people experience


N /M 60/100 60/100 60/100 

iterations 5000 4 · 10 4 2 · 10 6 

µ 0.01 0.01 0.001 

average 59.133400 58.161900 n/a yet 

std 0.999281 1.673139 n/a yet 

time 6 sec 5 min 2 weeks... 

Table 3.4: Behaviour of the ‘modified budget’ algorithm for budget=10 and 

staying at home=6. 

financial difficulties they do not attend bars. For values of this ratio much less 

than 1 the algorithm settles down almost instantly. If the budget is large the 

results are similar to the original ‘partial information’ algorithm (Table: 3.5 and 

Figure: 3.6). 

frequency 

0 200 400 600 

budget 10 100 10 10 10 

waiting 0 0 3 10 6 

average 59.385200 59.287200 59.292000 48.981200 58.165750 

std 2.916301 1.132950 1.861538 7.423848 4.851578 

Table 3.5: Simulation results for budget algorithms. 

Histogram of attendances in budget with b=10 

35 40 45 50 

attendances 

55 60 65 70 

frequency 

0 200 400 600 800 1000 

Histogram of attendances in modified budget with b=10,w=6 

40 50 

attendances 

60 70 

Figure 3.5: histograms for budget algorithms (b = 10, w = 0/6). 

Although budget algorithms are a variation of the partial information algorithm, 

their histograms (figure: 3.5) are quite different. Especially the first which has a 

wider distribution than the original one.





1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 1000 2000 3000 4000 5000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 

1.0 

1000 2000 


3000 4000 

0.8 

0.6 

0.4 

0.2 

0.0 

0 1000 2000 3000 4000 5000 


attendance 

attendance 

attendance 

100 

80 

60 

40 

20 

0 

0 1000 2000 3000 4000 5000 


100 

80 

60 

40 

20 

0 

0 

100 

1000 2000 


3000 4000 

80 

60 

40 

20 

0 

0 1000 2000 3000 4000 5000 


Figure 3.6: The overall attendance and the probabilities for each of the M 

agents for budget (budget= 10) and modified budget (budget= 10, wait=3, 6) 

algorithms.


3.3.2 Fairness and Efficiency 

In the first variation of the ‘budget algorithms’ the formation of the two groups 

persists, although it is clear that this is the fairest partial information algorithm 

considered so far, since only 34 agents decided to stay at home. The modified 

version of this algorithm improves it by resolving most of fairness and efficiency 

issues. Indeed, the more days agents have to stay at home in order to save 

resources, more people have thew opportunity to attend. The number of the 

always no attending agents goes from 23 in the 2nd example, to 6 in the 3rd one. 

The improved version is also more efficient for the agents (Figure: 3.8). For 

the first time, an algorithm can become fairer without becoming less efficient and 

vice versa. As days at home increase, standard deviation diminishes and payoff 

probability is raising (Table: 3.6). Like it was expected, forcing agents to stay 

at home after systematic visits, gives a chance for those that would have never 

thought to attend it and increases everybody’s profits. The only setback seems 

to be that, this increase of profits is made in expense of the bar management, 

since the average of attendance is around 58. 

days waiting std mean 

0 0.0953156 0.132056 

3 0.232645 0.42336 

6 0.1286092 0.471414 

Table 3.6: standard deviation and mean of payoff probabilities for budget algorithms. 


0 2 4 6 8 10 

budget algorithms 

0 days at home 



0 1000 2000 3000 4000 5000 

iterations 

Figure 3.7: standard deviation off attendance for budget algorithms. 

The last two algorithms are more efficient when the interests of the bar are 

considered. They have almost identical behaviour, although in figure: 3.7 it can 

be seen that the algorithm behaves slightly better for smaller values of the ‘staying 

at home’ constant. The std of the attendance for the original budget algorithm, 

seems to be stable, with no indication of improvement. The only problem is that


frequency 

0 10 20 30 40 50 

frequency 

0 10 20 30 40 

frequency 

0 10 20 30 40 

budget algorithm 

0.0 0.2 0.4 0.6 0.8 


budget algorithm with 3 days at home 

0.0 0.2 0.4 


0.6 0.8 


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 


frequency 

0 5 10 15 

frequency 

0 5 10 15 20 25 30 35 

frequency 

0 10 20 30 

budget algorithm 

0.00 0.05 0.10 


0.15 0.20 


0.0 0.1 0.2 0.3 


0.4 0.5 0.6 


0.0 0.1 0.2 0.3 0.4 0.5 


Figure 3.8: fairness and efficiency plots for budget algorithms. 

although the fluctuations in standard deviations are lower when agents stay many 

days at home, the bar is underutilised. Considering system’s fairness and based 

on the histograms of figure:3.5 the first case, where w = 0 is the fairest, since the 

distribution of the attendances is wider.

36 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM

Chapter 4 

Case Study: Multiple Bars in 

Santa Fe 

What if the citizens of El Farol had a chance to choose from a pool of more than 

one bar? Would the same algorithms behave differently? In this chapter, the case 

of three bars is studied. The main intention is to see, if the extremely efficient 

‘partial information’ algorithm can become more fair when three bars instead of 

one are available. Also this problem tends to be more realistic, since in most cases 

more than one choices exist. The rule used in the decision process by each agent, 

is based on a rather simplistic assumption that everybody prefers going out to 

staying at home. The decision is similar to the process described in subsection 

2.2.3. Only in this case, instead of using a biased coin to decide whether to attend 

the bar or not, the agent uses three distinct and slightly biased coins. Each coin 

corresponds to a bar, for example, if the first coin shows he should not attend, 

he tosses the second, if it shows not to attend again, he tosses the third and if it 

shows not to attend he stays at home. 

Instead of using only one fixed sequence of the three bars, 3! were used. At 

each iteration of the algorithm, after the probabilities of attendance have been 

calculated, a sequence of bars is chosen randomly. 

Simulations showed similar results to the one bar version. In order to achieve 

consistency the 6/10 ration is preserved and again µ is set to be 0.01. Santa Fe is 

now consisted of M = 300 citizens and has three bars, each one having a capacity 

of N = 60 for each one of them. Overall Nall = 180 which is the 60% of 300. As 

it is seen in table 4 and in figures 4.1 and 4.2 the behaviour of the algorithm is 

almost identical. 

37

38 CHAPTER 4. CASE STUDY: MULTIPLE BARS IN SANTA FE 

attendance for Bar 1 



100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 


100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 


100 

80 

60 

40 

20 

0 

0 500 1000 1500 2000 2500 3000 





1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 


1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0 500 1000 1500 2000 2500 3000 



for each bar.

overall attendance 

300 

250 

200 

150 

100 

50 

0 

0 500 1000 1500 2000 2500 3000 


Figure 4.2: The cumulative attendance for all three bars. 

µ = 0.01 mean std 

All Bars 177.689333 3.052926 

Bar 1 59.190000 2.360598 



Table 4.1: mean and std for all bars. 

The shared characteristics with the original algorithms continue when proceeding 

to the fairness and efficiency analysis. Indeed, figure 4.3 indicates that 

there are no agents that would want to attend more than one bar regularly. Each 

bar has its set of very devoted customers, despite the randomised decision process. 

This plots (barplots) are used in [17] as a token of unfairness. This unfairness is 

even more clear in payoff probability figures (fig:4.4), where it is seen that 120 

agents have payoff probability close to 0, while 179 are very close to 0.8. It seems 

that even in a town with three bars, agents insist on behaving selfishly, when 

using a partial information based algorithm. 

39

40 CHAPTER 4. CASE STUDY: MULTIPLE BARS IN SANTA FE 

attendance probability 



0.0 0.2 0.4 0.6 0.8 1.0 

0.0 0.2 0.4 0.6 0.8 1.0 

0.0 0.2 0.4 0.6 0.8 1.0 

Bar 1 

agents 


agents 


agents 

Figure 4.3: Agent attendances for each bar

frequency 

0 50 100 150 200 

frequency 

0 50 100 150 200 

Agent payoffs for Bar 1 

0.0 0.2 0.4 


0.6 0.8 


0.0 0.2 0.4 


0.6 0.8 

frequency 

0 50 100 150 200 

frequency 

0 20 40 60 80 


0.0 0.2 0.4 


0.6 0.8 

Agent payoffs for all bars 

0.0 0.2 0.4 


0.6 0.8 

Figure 4.4: payoff probabilities for each bar and cumulative payoff probability 

for the 3-bar version 

41

42 CHAPTER 4. CASE STUDY: MULTIPLE BARS IN SANTA FE

Chapter 5 

Conclusions 

As it was mentioned in the first chapter the objective of this dissertation was 

to analyse the El Farol Bar Problem, view the differences between various approaches 

and try to expand some of them. It is a typical complex adaptive system, 

in which agents interact with each other competing for the same resource. 

The first three solutions insist on Arthur’s original concept of predictors with 

minor variation. A closer look reveals, that the only difference they have is 

the definition of predictors. Arthur uses a close set of predictors, while Challet 

et all [2] introduce the use of binary strategies in an attempt to simplify the 

problem and Fogel et all [3] use evolutionary learning algorithms for a more 

precise, although slightly more complicated, definition of the predictors. 

Arthur’s belief that any solution should require agents that pursue different 

strategies[18] is abandoned in the stochastic adaptive solution, proposed by Bell 

et all in [10]. In partial information algorithms and their variations, there is 

no need for the agents to make prediction about the attendance of the Bar. 

They make their decisions based only on their own previous experience. In the 

full information algorithm, agents do know the full record of attendances. The 

behaviour of this algorithm is similar to the first three, although it does not 

require the use of predictors 

Partial information algorithms are more efficient and always converge with 

a very low standard deviation. Unfortunately this happens due to their unfair 

nature, which resembles the ‘greedy’ concept behind this algorithm. Agents who 

take decisions based only on their previous experiences act extremely selfishly 

when competing for the same resource. Fairness and efficiency could be measured 

with the use of utility payoffs. The results showed that in partial information algorithms 

two groups of agents were formed. Those who almost always attended, 

having an average payoff very close to 1.0 and those who almost never attended, 

who had a payoff near 0. It is clear that these behaviour is very unfair, especially 

when compared to full information results, but it is also efficient since the average 

utility payoff for all agents was higher than the one of the full information 

algorithm. 

43

44 CHAPTER 5. CONCLUSIONS 

Another negative aspect of these algorithms, is that they converge to N − 1, 

leaving always El Farol Bar slightly underutilised. But even in that case the losses 

for the bar management are less than the full information algorithm because of 

its almost zero standrad deviation. Partial information algorithms after their 

convergence have almost zero std and mean 59 while full information has a mean 

of 60 but also std of 6.5. 

This algorithm could be divided in three parts. First comes the probability of 

attendance, which is affected by the µ parameter and determines the attendances. 

In order to introduce fairness and efficiency, each single one of those parts was 

altered. 

Directly changing the probabilities of attendance which exceeded a threshold, 

did not give better results. In this variation of the stochastic adaptive algorithms 

after several iterations another value of probability was assigned, if they were 

larger than an upper bound, or lower than a low bound. Despite the fact that this 

algorithm had countless variations, none of the simulations showed significantly 

good results. This aggressive way of changing the probabilities created more 

unfairness and less efficiency and that is why the results are not included in the 

report, although the code is included in the Appendix. 

The results were much better when taxation was implemented. Although 

the partial information algorithms did not become more fair with the use of 

an adaptive µ, they become more effective. Full information algorithms were 

more fair but also significantly more effective. But probably the best results 

came when agents were forced to stay indoors after a succession of attendances. 

Partial information with budget algorithm gave almost perfectly fair results and 

slightly less efficient than the original one. Although the results of the partial 

information with budget were more than acceptable, this algorithm is very CPU 

intensive and demanding. Also the use of a budget meant that the system was 

no more a decentralised one, where agents made their own decisions. Although 

agents still do not need access to full record, a central mechanism which will 

ensure they do not spent their budget and that they will stay indoors as many 

days as needed, is necessary. 

Finally, the problem was reformulated, so it could include three bars. Each 

agent was randomly assigned one sequence of bars, which changed in every iteration 

of the algorithm. Instead of deciding whether to go to El Farol or stay at 

home, he had to decide whether to go to the first bar of the sequence, second, 

third or stay at home. Despite the randomised sequence, the results were almost 

similar to the previous results. 

In all cases, it was shown that the original algorithms could be extended in 

many ways. There is much work left to be done in this field. Papers [10], [17], [18] 

and [15] have mentioned only the first three variations of the stochastic adaptive 

learning algorithms (partial, full, partial with signs) and examined their convergence 

properties. The most important is that they could be used accordingly in 

order to fulfil a very versatile set of tasks. They could be combined or used to

calculate attendances in a network of many nodes (bars) which accepts dynamically 

evolving population. The C code used to generate all the results, is modular 

enough so even more variations can be included. Also the mathematical formulation 

of the algorithms make it possible to study their convergence properties, 

following the steps defined in the original paper[10] and further extended in this 

dissertation for the tax algorithms. 

45

46 CHAPTER 5. CONCLUSIONS

Appendix A 

C Code for the El Farol Bar 

Problem 

/***************************************************************************/ 

/* Program that solves El Farol Bar problem based in paper: */ 

/* Coordination Failure as a Source of Congestion in Information Networks */ 

/* Ann M. Bell, William A. Sethares, James A. Bucklew */ 

/* IEEE Trans. Signal Processing, 2003 */ 

/***************************************************************************/ 

/***************************************************************************/ 

/* This source code is a part of MSc dissertation: */ 

/* The El Farol Bar Problem for next generation systems */ 

/* by Athanasios Papakonstantinou */ 

/***************************************************************************/ 

/* In order to compile and run this program, type the following from 

the directory the source is saved: 

gcc -Wall -lgsl -lgslcblas -lm demo3.c support.c modules.c debug.c */ 

/* Bugs and known issues: 

i. Unfortunately the GSL C library is needed for the random generators. 

You can install it from yours distribution package management system. This 

remains to be fixed in the future, since this code intended to show quick 

results of many algorithms. It is No 1 priority, because I want the program 

to be portable. 

ii. To generate plots you must run the ef.sh script provided, and install the 

plotutils package. 

iii. Some algorithms can be very demanding. Especially budget algorithms. You 

should not use more than 20000 iterations. This is more a feature than a bug 

iv. This is not optimised code, the results for 2*10^8 iterations were calculated 

from a variation of this code, which does not produce any files for plotting 

v. There is a total lack of substantial user’s interface. There was no time for 

47

48 APPENDIX A. C CODE FOR THE EL FAROL BAR PROBLEM 

that, see Bugs and known issues bullet "i". This will be fixed in the future but 

only if the code is uploaded in the internet 

vi. No other bugs are known. This code was compiled with gcc 4.1.1 in a 

Gentoo GNU/Linux box with 2.6.17-gentoo-r4 kernel and with gcc 3.4.6 in 

a MandrakeLinux 10.1 box with 2.6.8.1-12mdksmp kernel. */ 

#include 

#include 

#include 

#include 

#include 

#include "elfarol.h" 

/*** *** prototypes section *** ***/ 

/* function to write integers in files, used for plotting the attendaces, 

mean and running average of them */ 

void writei_mean (char[], int, int[]); 

/* function to write floats in files, used for plotting probabilities */ 

void writed(char[], int,int, int, float[]); 

/* function that is used in fairness plots! */ 

void write_fair(char[],int[], int, int); 

/* module used in partial and full algorithms */ 

float 1eq5 (float, float, int, int); 

/* module used in signed partial algorithm */ 

float eq7 (float, float, float, int, int); 

/* module used in taxing algorithms */ 

float tax (float, float, float, int, int); 

/* utility function - returns {-1 0 1} */ 

int utility (int, int,int); 

/*** ***files section *** ***/ 

#define attendance "att" 

#define propability "prop" 

#define fairness "fair" 

#define payoff "pay" 

/*** *** constant variables *** ***/ 

//*** General Section *** 

const int M=100; //available agents in the system 

const int cp=60; //capacity of the bar (happy number) 

const int nt=3000; //no of iterations 

const float m=0.01; //the \mu parameter 

//** Taxing Algorithms ** 

/* m1 is used for taxing selfish agents and m3 to encourage

those staying at home to attend the bar.*/ 

const float m1=8; 

const float m3=1.01; 

//** Critical Values ** 

/* up and down are the boundaries and d is used in the shuffle 

variation */ 

const float up=0.9; 

const float down=0.1; 

const int d=2; 

//** Budget Algorithms ** 

const int wait=10; //maximum waiting time 

/*** ***Main program *** ***/ 

int main() 

{ 

int i,k; //for 

int *a_agent; //array of attendances 

float *p_agent //array of M propabilities for nt iterations 

int *god; //a_agent for all iterations 

int *a_total; //attendance after each iteration 

int *sgn; //array in partial sign algorithm 

float *tax_agent; //array in tax algorithm 

float *mu; //array full of \mu’s 

int *u_total; //array of total payoffs after each iteration 

int ans1,ans2; //Answers in interface 

int j,sum,l; //used in budget algorithms 

int max,max_out; //maximum attendances in budget 

const gsl_rng_type *T; //random generator 

gsl_rng *r; //random generator 

/* create a generator chosen by the environment variable 

GSL_RNG_TYPE */ 

gsl_rng_env_setup(); 

T = gsl_rng_default; 

r = gsl_rng_alloc (T); 

printf("How many agents should be viewed in probabilities plot?\n"); 

scanf("%d",&ans2); 

/* Initializing variables i=0 and Initial Conditions IC */ 

p_agent = (float *)malloc((M*nt)*sizeof(float)); 

god = (int *)malloc((M*nt)*sizeof(int)); 

a_total = (int *)malloc((nt)*sizeof(int)); 

a_agent = (int *)malloc((M)*sizeof(int)); 

sgn = (int *)malloc((nt)*sizeof(int)); 

tax_agent = (float *)malloc((M)*sizeof(float)); 

mu = (float *)malloc((M)*sizeof(float)); 

u_total = (int *)malloc((M*nt)*sizeof(int)); 

for(i=0; i


god[i]=9; 

} 


eak; 

//Algorithm 2 full information algorithm 

case(2): 



if(p_agent[i*M+k]>gsl_ran_flat(r,0,1)) {a_agent[k]=1;god[i*M+k]=1;} 

else {a_agent[k]=0;god[i*M+k]=0;} 

} 

a_total[i]=sumar(M,a_agent); 

for(k=0; k

* In this bit, budget is implemented, god matrices columns are scanned for 

sequences of 1’s which indicate continuous attendances. All unassigned 

values are equal to 9, so that not any sum of 1’s is equal to budget but 

only those that have the next after element unassigned (=9) This element 

is set equal to 99 */ 

for (k=0;k


/* Uncomment for up/down shuffle */ 

// if (p_agent[i*M+k]>up) {p_agent[i*M+k]=down;} 

// if (p_agent[i*M+k]up) 

{p_agent[i*M+k]=(1.0/d)*p_agent[i*M+k];} 

if (p_agent[i*M+k]

} 

/* This bit calculates the attendances. If the new god element is not 99 

then the attendance is as usual calculated by the probability. If it is to 99, 

it is so because it is after a sequence of attendances and this and the 

following w-1 must be set to 0 */ 

for(k=0; kgsl_ran_flat(r,0,1)) {a_agent[k]=1;god[i*M+k]=1;} 

else {a_agent[k]=0;god[i*M+k]=0;} 

} 

else if(god[i*M+k]==99) {god[i*M+k]=0; a_agent[k]=0;} 

} 

a_total[i]=sumar(M,a_agent); 

for(k=0; k


} 

printf("5. Full info with taxing\n"); 

printf("6. Partial info with budget\n"); 

printf("7. Partial with critical values\n"); 

printf("8. Partial with budget, exotic!\n"); 

void writei_mean (char filename[], int array_len, int x[]) 

{ 

FILE *fptr; 

int i; 

float sum1; //sum1 is the trend, mean changes with considering every attendance 

float sum2; //sum2 is the mean of all iterations attendances 

sum1=0.0; 

sum2=0.0; 

fptr= fopen (filename, "w"); 

if (fptr==NULL) { 

printf("Unable to open file,check directory permissions and try again\n"); 

exit(-1); 

} 

for (i=0; i

void write_fair(char filename[],int a[], int array_len1, int array_len2) 

/* array_len1:M, array_len2:nt */ 

{ 

FILE *fptr; 

int i,j; 

float sum; 

fptr= fopen (filename, "w"); 

if (fptr==NULL) { 

printf("Unable to open file,check directory permissions and try again\n"); 

exit(-1); 

} 

for (i=0; i

58 APPENDIX A. C CODE FOR THE EL FAROL BAR PROBLEM

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