Competition between adaptive agents: from learning to collective efficiency and back

We use the Minority Game and some of its variants to show how efficiency depends on learning in models of agents competing for limited resources. Exact results from statistical physics give a clear understanding of the phenomenology, and opens the way to the study of reverse problems. What agents can optimize and how well is discussed in details. Designed a simplification of Arthur’s El Farol bar problem [1], the Minority Game [2, 3] provides a natural framework for studying how selfish adaptive agents can cope with competition. The major contribution of the Minority Game is not only to symmetrize the problem, which physicists like very much, but also to introduce a well parametrized set of strategies, and more generally to provide a well defined and workable family of models. In this game, N agents have to choose one between two choices at each time step; those who are in the minority win, the other lose. Obviously, it is easier to loose than to win, as the number of winners cannot exceed that of the losers. If the game is played once, only a random choice is reasonable, according to Game Theory [4]. When the game is repeated, it is sensible to suppose that agents will try to learn from the past in order to outperform the other agents, hence, the question of learning arises, as the minority mechanism entails a never-ending competition. Let me first introduce the game and the needed formalism. There are N agents, agent i taking action ai ∈ {−1,+1}. A game master aggregates the individual actions into A = ∑N i=1 ai and gives private payoffs −aig(A) to each agent i = 1, · · · , N . The minority structure of the game implies that g must be an odd function of A. The simplest choice for g may seem to be g(A) = sgn (A), but a linear function is better suited to mathematical analysis. The MG is a negative sum game, as the total payoff given to the agents, is − ∑N i=1 aig(A) = −g(A)A < 0, since g is an odd function. In particular, the linear payoff function gives a total loss of A; when the game is repeated, the average total loss is nothing else than the fluctuations of the attendance σ = 〈A2〉 where the average is over time.