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Minority Games
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Minority Games A Complex Systems Project
Going to a concert… • But which night to pick? Friday or Saturday? • You want to go on the night with the least people. So you try and guess which night. • You can base your guess on previous attendence. But so does everyone else. • This is a minority game. • What happens if the concert is on every night?
Properties of Minority Games • Participants try to pick the least common choice. • Communication between participants is only through results of previous attempts. • Each participant thus makes decisions based upon their private strategies and the public history. • Example: The Stock Market
Why are Minority Games Complex Systems? • Large numbers of agents, each with their own strategy sets. • The system adapts to new information each round. • The history is important. • The system is frustrated: the more successful a strategy is, the worse it gets.
Choose a Number! • Pick an integer number between 1 and 10, and try to get the least common number. • We used the first year physics class as our system. • The experiment was repeated, but each time the results of the last round were left up. • 6 rounds were run in total.
The jaggedness Parameter (a) 3 2 1 4 a=57.4
Our Models • 100 agents • Each agent chooses a number according to a strategy • Calculate histogram of results • Rank each number from 1-10 based on popularity. Example • Strategies got 1 point if they led to the least popular number and 0 otherwise.
Model 1 Distribution of Variance Variance
Model 1 Distribution of Variance Variance Variance with Time Variance
Previous round 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 What is happening here? a=5.6
Model 2 • During the Trial: • Agents choose a number and stick with it. • Every p rounds, consider changing. • When changing, change to best number of previous round. Before the Trial: Agents given patience parameter, p.
And then this happened… Insert alpha distribution.
Model 3 • During the Trial: • Agents choose a number and stick with it. • Every round, agents have a probability .02 that they will consider changing. • When changing, change to best number of previous round.
Good for wealth distribution too. Average wealth = 4542 Average wealth =2733
Formalism 2 choice minority game • extension of the El Farol Bar problem N players, two choices: 0 or 1 Rules: At each turn, every agent chooses a side (0 or 1), those that end up in the minority side, win.
Formalism • Memory: the bit-string of past winning outcomes. eg. minority side trial one – 1, trial two -1, trial three – 0, trail four - 1, trial five - 0. M = {1,1,0,1,0} with length: m = 5.
Strategy Space Strategies defined in an abstract way - no psychology. Strategy s – a ‘card’ with a prediction for each possible past history. eg. Strategy1 Strategy2 Strategy 3 etc…
Minority Game Structure • Odd number (N) of agents, each given two or more strategies selected at random from the strategy space. • At each turn, the strategies are evaluated; a point is awarded to each strategy that predicts the correct minority result. The strategy with the highest number of points is chosen for the next turn. • Evaluated for large number of time steps, for different memory lengths m.
Complex behaviour Savit et al, 1999. Region 1: Small number of strategies, crowded behaviour – similar to model 1 of the MCMG. Region 3: Very large strategy space. Less probability of cooperation. Region 2: More available strategies. Able to cooperate to achieve low variance.
Conclusions • Interesting behaviour • Possible to do better-than-random but with abstract/slightly contrived strategies • Further study to see the stability of better-than-random systems with plausible strategy sets.