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,a ¯ = 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’ algo- rithms...... 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 3.2. The overall attendance and the probabilities for each of the M 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 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...... 33 3.7. standard deviation off attendance for budget algorithms...... 34 3.8. fairness and efficiency plots for budget algorithms...... 35
4.1. The overall attendance and the probabilities for each of the M 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 al- gorithms...... 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 ad- jective used to describe nouns which ‘are consisted of different and connected parts’. An even more precise definition is ‘consisted of interconnected or interwo- ven 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 in- teract 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 be- haviour 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 = log2(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 2N possible states, therefore:
Ω = 2N ⇔ N = I (1.1.2)
There are many applications of complex systems, in statistical physics, me- teorology, geology, biology, engineering, economics, even social sciences and psy- chology. 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 gov- ernment. 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 univer- sally 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 com- plex 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 Mex- ico which plays each Thursday Irish music. People enjoy visiting it and hearing some quality music but eventually it becomes overcrowded, so people stop enjoy- ing 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 perfor- mance 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 bi- nary 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 sug- gests 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 4 CHAPTER 1. CHAOS, COMPLEXITY AND IRISH MUSIC 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 ‘predic- tors’. 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’ pre- dictor, 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 re- sults 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 accommo- date, before becoming overcrowded. Arthur wanted to predict the binary action th i of the i customer denoted by a ∈ {0, 1}, whereP 1 stands for going to the bar and 0 is not going. The total attendance is A = ai. As mentioned above, the only information available to the agents is the num- ber of the customers of the bar, the previous d weeks:
It = {A(t − d),...,A(t − 2),A(t − 1)} (1.3.1)
d A predictor is a function I ∈ [0,N]d → [0,N]. There are (N + 1)(N+1) 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: 6 CHAPTER 1. CHAOS, COMPLEXITY AND IRISH MUSIC
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:
A(It) = N − A(t − 1)
3. A fixed predictor:
A(It) = 67
4. A rounded average of the last four weeks:
X4 A(It) = 1/4 A(t − r) r=1
5. A rounded and bound by 0 and N, last 8 weeks trend, computed using the least squares method.:
+ 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]
t t Where, s ∈ [1,S], Ai,s(It) the s h predictor for i 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 ith customer is given by:
si(t) = argmaxs0 Ui,s0 (t) and
ai(t) = Θ[L − Ai,si(t)(It)] where, argmaxxf(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 equilib- rium 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. 8 CHAPTER 1. CHAOS, COMPLEXITY AND IRISH MUSIC
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 nega- tive 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 coop- eration 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:
µ as,i = Θ[L − As,i(It)] (2.2.1) which will depend only on the information
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 µ with distribution: P (a) ≡ Prob{as,i} =aδ ¯ (a − 1) + (1 − a¯)δ(a). For this model to work, it is considered that on average clients attend the bar with a frequency µ L/N. This leads toa ¯ ≈ L/N, wherea ¯ is the average of as,i. Figure 2.1, illustrates what happens to the model when L,a ¯ and m remain fixed, while the number of agents N increases. The top graph shows that when Na¯ ≈ L, the attendance converges to the comfort level, while as m is increasing the area where hAi ≈ 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 aver- age, 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 cooper- ation of the agents is measured by:
σ2 = h(A − L)2i (2.2.3) where h... i is the average on the stationary state.
Figure 2.1: Behaviour of the average attendance (Top) and of the fluctuations (bottom) in the El Farol problem with L = 60 seats,a ¯ = 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, σ2/N is at is peak at the point whereaN ¯ = 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 ith individual their jth predictor’s output is given by: ¯ ¯ ¯ li ¯ ¯ Xj ¯ i ¯ i i ¯ xˆj(n) = round ¯aj(0) + aj(t)x(n − t)¯ (2.2.4) ¯ t=1 ¯
i where x(n − t) is the attendance on week (n − t), lj is the number of lag terms in th i the j predictor of individual i, aj(t) is the coefficient for the lag t steps in the i past, and aj(0) is the constant term of the AR model. The absolute value and the rounded output makes sure that no negative val- ues 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 kth model. Lag in the offsprings from i i parents j is set to be one or ten. If lj = 1 then lj − 1 is not allowed, while if i it is equal to ten, lj + 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. 12 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL
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 2.2. APPROACHES TO EFBP 13
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 14 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL 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 ⇔ b P(N −i ≤ N − 1) = 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 th for each i agent. If k is the iteration counter (time), pi the probability that th i 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: XM 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 2.2. APPROACHES TO EFBP 15 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 ) > 1 (2.2.7) pi(k) − µ(N(k) − N ), otherwise 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 16 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL convergence time . But the conclusion is that only hardware limitations affect its behaviour.
N /M 60/100 600/1000 6000/10000 iterations 2 · 108 2 · 108 2 · 108 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 par- tial 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 · 108 2 · 108 2 · 108 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. 2.2. APPROACHES TO EFBP 17
100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 time (iterations) time (iterations)
Figure 2.5: The overall attendance and the probabilities for each of the M agents for ‘partial information’, ‘full information’ and ‘signs’ algorithms. 18 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL
N /M 60/100 600/1000 6000/10000 iterations 2 · 108 2 · 108 2 · 108 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 Histogram of attendances in full info frequency frequency 0 500 1000 1500 2000 0 50 100 150 200 50 55 60 65 40 50 60 70 80 attendances attendances
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 fair- ness and efficiency are developed. A fair outcome requires that agents with similar utilities have similar probabilities to attend the bar. A more strict defi- nition 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. Util- ity 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 2.2. APPROACHES TO EFBP 19
• 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 atten- dances 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 fair- ness 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 algo- rithms. 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 20 CHAPTER 2. PREVIOUS APPROACHES TO THE EL FAROL 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 prop- erties 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.
partial information partial information frequency frequency 0 10 20 30 40 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 probability of attendance payoff probability
full information full information frequency frequency 0 5 10 15 20 0 1 2 3 4 5 6 7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 probability of attendance payoff probability
partial information with signs partial information with signs frequency frequency 0 5 10 15 20 25 30 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 probability of attendance payoff probability
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 2.2. APPROACHES TO EFBP 21 information’ has a relatively high std, with no signs of improvement.
original algorithms standard deviation
partial full signed 0 1 2 3 4 5 6
0 500 1000 1500 2000 2500 3000 iterations
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 subsec- tion2.2.3, 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 infor- mation’ 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 · 108 2 · 108 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 p (k + 1) = 1, i 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 ) > 1 (3.2.4) pi(k) − µtax(N(k) − N ), otherwise 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.
Histogram of attendances in modified tax partial info Histogram of attendances in tax full info frequency frequency 0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 10 20 30 40 50 60 70 40 45 50 55 60 65 70 attendances attendances
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. 26 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
attendance 40 0.4 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations)
Figure 3.2: The overall attendance and the probabilities for each of the M agents for ’partial’ information tax, modified tax algorithms and the full information tax algorithm. 3.2. TAXING/PAYOFFS ALGORITHMS 27
In the following a set of equations that describe the stationary state behaviour of the probabilities will be derived pi(k). PM The number of agents attending at time k is N(k) = 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 PM N(k) is the number of agents attending at time k is equal to i=1 xi(k), where xi(k) are independent Bernoulli trials. The expectation value is calculated using the following formula: X E(XY ) = fx,y(x, y) x 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: X E(µtax(N(k)) · (N(k) − N )) = (N(k) − N ) · µtax(N(k)) · fN(k)(N(k)) ⇒ N(k)−N X X cµ (N(k) − N ) · fN(k)(N(k)) + µ (N(k) − N ) · fN(k)(N(k)) ⇒ N(k)>N N(k)≤N X X cµ N(k) · fN(k)(N(k)) − cµN fN(k)(N(k))+ N(k)>N N(k)>N X X µ N(k) · fN(k)(N(k)) − µN fN(k)(N(k)) (3.2.6) N(k)≤N N(k)≤N Kolmogorov-Smirnov test indicated that z = N(k) might follow Poisson distri- λz bution f(z) = e−λ, with λ = E(z). Hence: z! X X X E(f(z)) = zf(z) = zf(z) + zf(z) = λ ⇒ z z≤N z>N 28 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
X X zf(z) = λ − zf(z) (3.2.7) z≤N z>N and: X X X X X f(z) = 1 ⇒ f(z) + f(z) = 1 ⇒ f(z) = 1 − f(z) (3.2.8) z z≤N z
X X X X cµ zf(z) − cµN f(z) + µ(λ − zf(z)) − µN (1 − f(z)) = 0 (3.2.9) z>N z>N z>N z>N Since µ 6= 0, (3.2.9) becomes: X X 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:
XM λN(k) A(λ, M) = e−λ N(k)! z>N and XM λN(k) B(λ, M) = N e−λ N(k)! z>N we get the final equation: 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: XM XM XM λ = E(N(k) = E( xj(k)) = E( xj(k)) = E(xj(k)) j=1 j=1 j=1
Since xi(k) are Bernoulli trials with θ = pi(k) then E(xi(k)) = pi(k). Hence: XM λ = pj(k) (3.2.12) j=1
PM It must be noted that c = 1 in equation (3.2.11) leads to j=1 pj = N . So for this value of c the original result, mentioned in [10] was recovered. 3.2. TAXING/PAYOFFS ALGORITHMS 29
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 algo- rithms. 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 atten- dance 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 al- gorithm, are visible in (Figure: 3.4). The behaviour of the third taxing algorithm is almost identical with the behaviour of the original. 30 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
partial information with taxing partial information with taxing frequency frequency 0 10 20 30 40 50 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 probability of attendance payoff probability
modified partial information with taxing modified partial information with taxing frequency frequency 0 10 20 30 40 50 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 probability of attendance payoff probability
full information with taxing full information with taxing frequency frequency 0 1 2 3 4 5 6 7 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 0.00 0.05 0.10 0.15 0.20 0.25 probability of attendance payoff probability
Figure 3.3: fairness and efficiency plots for taxing algorithms. 3.3. BUDGET ALGORITHMS 31
taxing algorithms
taxing partial modified taxing partial taxing full standard deviation 0 5 10 15
0 500 1000 1500 2000 2500 3000 iterations
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: Xb ∀ i ∈ [1,M], k ∈ [1, n], if 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 · 108 iterations in almost 3 days, this one for 210˙ 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 32 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
N /M 60/100 60/100 60/100 iterations 5000 4 · 104 2 · 106 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).
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 Histogram of attendances in modified budget with b=10,w=6 frequency frequency 0 200 400 600 0 200 400 600 800 1000 35 40 45 50 55 60 65 70 40 50 60 70 attendances attendances
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. 3.3. BUDGET ALGORITHMS 33
1.0 100
0.8 80
0.6 60
0.4 attendance 40 probabities for each agent 0.2 20
0.0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 time (iterations) time (iterations) 1.0 100
0.8 80
0.6 60
0.4 attendance 40 probabities for each agent 0.2 20
0.0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 time (iterations) time (iterations) 1.0 100
0.8 80
0.6 60
0.4 attendance 40 probabities for each agent 0.2 20
0.0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 time (iterations) time (iterations)
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. 34 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
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 algo- rithms.
budget algorithms
0 days at home 3 days at home 6 days at home standard deviation 0 2 4 6 8 10
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 3.3. BUDGET ALGORITHMS 35
budget algorithm budget algorithm frequency frequency 0 10 20 30 40 50 0 5 10 15 0.0 0.2 0.4 0.6 0.8 0.00 0.05 0.10 0.15 0.20 probability of attendance payoff probability
budget algorithm with 3 days at home budget algorithm with 3 days at home frequency frequency 0 10 20 30 40 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 probability of attendance payoff probability
budget algorithm with 6 days at home budget algorithm with 6 days at home frequency frequency 0 10 20 30 40 0 10 20 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 probability of attendance payoff probability
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
100 1.0
80 0.8
60 0.6
40 0.4 attendance for Bar 1 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
40 0.4 attendance for Bar 2 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations) 100 1.0
80 0.8
60 0.6
40 0.4 attendance for Bar 3 probabities for each agent 20 0.2
0 0.0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time (iterations) time (iterations)
Figure 4.1: The overall attendance and the probabilities for each of the M agents for each bar. 39
300
250
200
150
overall attendance 100
50
0 0 500 1000 1500 2000 2500 3000 time (iterations)
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 Bar 2 59.201667 2.207228 Bar 3 59.297667 2.524133
Table 4.1: mean and std for all bars.
The shared characteristics with the original algorithms continue when pro- ceeding 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. 40 CHAPTER 4. CASE STUDY: MULTIPLE BARS IN SANTA FE
Bar 1 attendance probability 0.0 0.2 0.4 0.6 0.8 1.0
agents
Bar 2 attendance probability 0.0 0.2 0.4 0.6 0.8 1.0
agents
Bar 3 attendance probability 0.0 0.2 0.4 0.6 0.8 1.0
agents
Figure 4.3: Agent attendances for each bar 41
Agent payoffs for Bar 1 Agent payoffs for Bar 2 frequency frequency 0 50 100 150 200 0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 payoff probability payoff probability
Agent payoffs for Bar 3 Agent payoffs for all bars frequency frequency 0 50 100 150 200 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 payoff probability payoff probability
Figure 4.4: payoff probabilities for each bar and cumulative payoff probability for the 3-bar version 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 ap- proaches 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 al- gorithms 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 aver- age 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 iter- ation 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 conver- gence 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 45 calculate attendances in a network of many nodes (bars) which accepts dynami- cally evolving population. The C code used to generate all the results, is modular enough so even more variations can be included. Also the mathematical formu- lation 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. 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
/*** *** 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 49 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 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] /* Uncomment for medium shuffle */ 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; k /*** *** Functions section *** ***/ void m_al() //Intro menu to select algorithm { printf("Please select algorithm\n"); printf("\n"); printf("1. Partial information\n"); printf("2. Full information\n"); printf("3. Partial info with signs\n"); printf("4. Partial info with taxing\n"); 56 APPENDIX A. C CODE FOR THE EL FAROL BAR PROBLEM 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 [1] Arthur W. B. Inductive reasoning and bounded rationality: The el farol problem. Amer. Econ. Rev., 84:406–411, May 1994. [2] Marsili M. Challet D. and Ottino G. Shedding light on el farol. Game Theory and Information 0406002, f. EconWPA, June 2004. available at http://ideas.repec.org/p/wpa/wuwpga/0406002.html. [3] Chellapilla K Fogel D. B. and Angeline P. J. Inductive reasoning and bounded rationality reconsidered. IEEE Transactions on Evolutionary Com- putation, 3(2), 1999. [4] Strogatz S. H. Nonlinear Dynamics and Chaos. Westview Press, 2000. [5] Goertzel B. From Complexity to Creativity, Computational Models of Evo- lutionary, Autopoietic and Cognitive Dynamics. Plenum Press, 1997. [6] Bar-Yam Y. Dynamics of complex systems. Westview Press, July 1993. [7] Waldrop M. M. Complexity: The Emerging Science at the Edge of Order and Chaos. Penguin Books Ltd, January 1994. [8] Arthur W. B. Complexity and the economy. Science, 284:107–109, April 1999. [9] Nie N. and Comaniciu C. Adaptive channel allocation spectrum etiquette for cognitive radio networks, 2006. [10] Sethares W. A. Bell A. M. and Bucklew J. A. Coordination failure as a source of congestion in information networks. IEEE Trans. Signal Processing, 51(3):875–885, March 2003. [11] Garogalo M. The ’el farol bar problem’ in netlogo. Unpublished Draft. [12] Osborne M. J. and Rubinstein A. A Course in Game Theory. MIT Press, 1994. [13] Hargreaves-Heap S. P. and Varoufakis Y. Game Theory, A critical text. Routledge, 2004. 59 60 BIBLIOGRAPHY [14] Challet D. and Zhang Y.-C. Emergence of cooperation and organization in an evolutionary game. Physica A, 246:407–418, 1997. [15] Bell A. M. and Sethares W. A. The el farol problem and the internet: Congestion and coordination failure. Computing in Economics and Finance 1999 812, Society for Computational Economics, March 1999. available at http://ideas.repec.org/p/sce/scecf9/812.html. [16] Greenwald A. Farago J. and Hall K. Fair and efficient solutions to the santa fe bar problem. http://www.clsp.jhu.edu/~khall/pubs/santafe_02.pdf. [17] Sivanadyan T. and Sethares W. A. A decentralized optimal solution to the ‘minority game’. preprint submitted to Physica A, February 2006. http: //homepages.cae.wisc.edu/~thiagars/pdfs/MinorityGame.pdf. [18] Bell A. M. and Sethares W. A. Avoiding global congestion using decentral- ized adaptive agents. IEEE Transactions in Signal Processing, 49(11):2873– 2879, November 2001.