Collective behavior of El Farol attendees European Conference on Complex Systems 2007 October 1-6, 2007 – Dresden Photo credit Matthew Bannister, James Gibbs, and Charles d’Autremont

A total number of N players must decide independently whether to attend the bar or not A total number of N players must decide independently whether to attend the bar or not If a player forecasts that the total attendance will exceed the comfort level, L, she will not show up, otherwise she will go If a player forecasts that the total attendance will exceed the comfort level, L, she will not show up, otherwise she will go The attendance data of the last m weeks [a m, a m-1, …, a 1 ] are available to the agents The attendance data of the last m weeks [a m, a m-1, …, a 1 ] are available to the agents Each week we seek to find the number of people who will actually attend the bar, a o Each week we seek to find the number of people who will actually attend the bar, a o WB Arthur Inductive Reasoning and Bounded Rationality American Economic Review 84, 406 (1994) Setting the stage

The algorithm pool The algorithm pool Point-wise hypothesis – the agent uses the attendance data of the kth previous week (1 ≤ k ≤ m) Point-wise hypothesis – the agent uses the attendance data of the kth previous week (1 ≤ k ≤ m) Arithmetic average – the agent uses the average of the last k (1 < k ≤ m) weeks as her prediction Arithmetic average – the agent uses the average of the last k (1 < k ≤ m) weeks as her prediction Weighted average – the agent uses a weighted average of the last k (1 < k ≤ m) weeks, where the more recent a week’s data is, the larger weight it has Weighted average – the agent uses a weighted average of the last k (1 < k ≤ m) weeks, where the more recent a week’s data is, the larger weight it has Trend – the agent makes a least squares fit to the last k weeks’ data (1 < k ≤ m), and uses its extrapolation to the following week as her prediction Trend – the agent makes a least squares fit to the last k weeks’ data (1 < k ≤ m), and uses its extrapolation to the following week as her prediction Total of 4m – 3 algorithms available, where m is the memory of the system

Attendance distribution, P(A) Any memory of attendances? What type of a function is A typical trajectory of attendances  L, does it always?

Stickiness I – change the algorithm randomly if it fails II – the best rule for the last week’s data selected for the next week III – choose from the pool based on cumulative performance of each predictor Johnson NF et al., Physica A 258, 230 (1998)

With increasing memory, distribution shifts from bimodal towards unimodal with a fat tail above the threshold A phase transition from higher to lower congestion levels occurs at a moderate memory Scheme I, L=60, N=100

Cycle detectors are the most frequently used algorithms in all schemes

Strong mean reverting tendency on consecutive weeks At two week intervals, initial increase in correlations, eventually lost at high memory Permanent and slightly significant correlations at three week intervals At transition memory, correlations take the longest to decay Large changes tend to be followed by large changes, of either sign, and small changes by small changes

Convergence slightly affected Most significant shift during transition Agents’ confidence on their algorithms Correlations are accentuated and longer lasting only at transition memory

Convergence remarkably delayed compared to feeding on-time information At low memory – shallow, and approach all-or-none behavior with increased delay At high memory – skewed character, congested tails persist Delayed information allocation Correlations are accentuated and longer lasting at high delays irrespective of memory horizon

Average attendance follows an S-shaped curve depending on the information carrying capacityAverage attendance follows an S-shaped curve depending on the information carrying capacity Whether the average attendance will converge externally provided comfort level or internally determined limiting state depends on the algorithm selection proceduresWhether the average attendance will converge externally provided comfort level or internally determined limiting state depends on the algorithm selection procedures As critical memory is exceeded, algorithm usage probabilities are stabilizedAs critical memory is exceeded, algorithm usage probabilities are stabilized The mean-reverting nature of the problem is also characterized by the correlationsThe mean-reverting nature of the problem is also characterized by the correlations A “phase transition” is observed as a function of information carrying capacity of the agentsA “phase transition” is observed as a function of information carrying capacity of the agents Delaying information dissemination is more consequential than agents’ rigidity on their beliefsDelaying information dissemination is more consequential than agents’ rigidity on their beliefs