This paper gives a critical account of the minority game literature. The minority game is a simple congestion game: players need to choose between two options, and those who have selected the option chosen by the minority win. The learning model proposed in this literature seems to differ markedly from the learning models commonly used in economics. We relate the learning model from the minority game literature to standard game-theoretic learning models, and show that in fact it shares many features with these models. However, the predictions of the learning model differ considerably from the predictions of most other learning models. We discuss the main predictions of the learning model proposed in the minority game literature, and compare these to experimental findings on congestion games.
Congestion games are ubiquitous in economics. In a congestion game (Rosenthal, 1973), players use several facilities from a common pool. The costs or benefits that a player derives from a facility depends on the number of users of that facility. A congestion game is therefore a natural game to model scarcity of common resources. Examples of such systems include vehicular traffic (Nagel et al., 1997), packet traffic in networks (Huberman and Lukose, 1997), and ecologies of foraging animals (DeAngelis and Gross, 1992). Similar coordination problems are encountered in market entry games (Selten and Güth, 1982).
Congestion games are also interesting from a theoretical point of view. In congestion games, players need to coordinate to differentiate. This seems to be more difficult than coordinating on the same action, as any commonality of expectations is broken up. For instance, when commuters have to choose between two roads A and B and all believe that the others will choose road A, nobody will choose that road, invalidating beliefs. The sorting of players predicted in the pure-strategy Nash equilibria of such games violates the common belief that in symmetric games, all rational players will evaluate the situation identically, and hence, make the same choices in similar situations (see Harsanyi and Selten, 1988, p. 73). Moreover, in congestion games, players may obtain asymmetric payoffs in equilibrium which may complicate attainment of equilibrium, as coordination cannot be achieved through tacit coordination based on historical precedent (cf. Meyer et al., 1992). Finally, congestion games often have many equilibria, so that players also face the difficulty of coordinating on the same equilibrium.
Nevertheless, the theory of learning in games provides sharp predictions on players’ behavior in congestion games. As congestion games belong to the class of potential games (Monderer and Shapley, 1996b), all results that have been derived for potential games apply to the class of congestion games. 1 Experimental evidence, however, is not always in line with these predictions. Though several experimental studies have shown that players are remarkably successful at learning to coordinate in congestion games, 2 regularities on 1 See e.g. Hofbauer and Hopkins (2005), Hofbauer and Sandholm (2002), Monderer and Shapley (1996b), and Sandholm (2001Sandholm ( , 2007)). Kets and Voorneveld (2007) study the convergence of play under different learning processes in the minority game.
2 For instance, interacting players rapidly achieve a “magical” degree of tacit coordination in market entry games, which is accounted for on the aggregate level by the Nash equilibrium solution (Kahneman, 1988;Rapoport, 1995;Sundali et al., 1995;Erev and Rapoport, 1998;Rapoport et al., 1998Rapoport et al., , 2000)). See e.g. Meyer et al. (1992) and Selten et al. (2007) for similar results on related games.
the aggregate level generally conceal non-equilibrium behavior at the individual level. Even though aggregate play is close to the Nash equilibrium, individual players generally do not play equilibrium strategies. 3 Moreover, providing players with more information does not always lead to better outcomes. 4 These experimental findings are hard to explain with standard learning models. This paper discusses the literature on the minority game, a simple congestion game based on the El Farol bar problem of Arthur (1994). Players have to choose between two alternatives. Only those who have chosen the minority side get a positive payoff. The minority game literature proposes a learning model that is able to account for many of the experimental findings listed above. We relate this learning model to the standard learning models in economics, and compare its predictions to experimental results on congestion games. The contribution of the current paper is that it relates the literature on the minority game, which has been largely developed in physics, to the literature on learning in game theory and to the literature in experimental economics on congestion games. 5 The outline of this paper is as follows. In Section 2, we introduce the minority game and discuss its equilibria. The learning model proposed in the minority game literature is discussed in Section 3. In Section 4, we discuss the main predictions from the learning model. These predictions are compared to experimental results on congestion games in Section 5. Section 6 concludes.
The minority game is a game in which an odd number of players have to choose between two actions; for instance, players either go to a bar or stay home, either buy or sell an asset, etcetera. Players want to distinguish themselves from the crowd: their aim is to take a different action than the majority of players.
Following the notation of Tercieux and Voorneveld (2005), we denote the set of players by N = {1, . . . , 2k + 1}, with k ∈ N. Each player i ∈ N has a set of pure strategies A i = {-1, +1}: agents
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