A Theatre Attendance Model

Introduction

Arthur (1994) states that “the type of rationality assumed in economics - perfect, logical, deductive - … breaks down under complication” (p. 406), therefore “in a typical problem that plays out over time, one might set up a collection of agents, probably heterogeneous, and assume they can form mental models, or hypotheses, or subjective beliefs” (p. 406). To illustrate how the inductive reasoning can be modeled, he proposes the Bar Attendance problem: each week, people decide independently whether to go to a bar, which offers a special entertainment each Thursday evening, and it is no fun to go there if it is too crowded, i.e. if more than people attend the bar (with ): in this case have a better time people who stay at home. Therefore, quoting Arthur (1999): “the rule is that if a person predicts that more than 60 (say) [i.e. ] will attend, he or she will avoid the crowds and stay home; if he predicts fewer than 60 [], he will go”.

This problem, as observed by Franke (2003), “has received much attention … as a nice paradigm to discuss issues of learning and bounded rationality” (p. 367), which are really important topics in behavioral economics: I simply recall that the Nobel prize recipient (2002) Daniel Kahneman stated that his work (jointly with Amos Tversky) “explored the psychology of intuitive belief and choices and examined their bounded rationality”, and he acknowledged the importance of Simon's studies, since he “had proposed much earlier that decision makers should be viewed as boundedly rational” (Kahneman, 2003, p. 1449): in brief, when they have to make any decision, rather than behaving as utility maximizers, follow some reasonable procedure.

A first generalization of Arthur's model is presented in Challet & Zhang (1997): they introduce the Minority Game, i.e. a binary game in which “… players have to choose one of the two sides independently and those on minority side win…” (p. 407). This game has been studied in a game theoretical framework in some contributions (e.g. Kets & Voorneveld, 2007), while in many other contributions the analytical research on this game “…employs techniques borrowed from statistical physics in order to describe the game as a spin system, thus enabling the system's properties to be outlined…” (Whitehead, 2008, p. 7). It is worth underlining that, by using statistical physics of disordered systems, Challet et al. (2004) derive a complete understanding of the complex behavior of the El Farol Bar problem on the basis of its phase diagram, showing that all the known results about the Minority Game directly apply to the Bar Attendance problem. Several generalizations of the Minority Game have been proposed and analyzed, such as Minority Games with arbitrary cutoffs (Johnson et al., 1999), Minority Games with different payoff functions (Li et al., 2000; Lee et al., 2003), Multiple Choice Minority Games (Chow & Chau, 2013), Networked Minority Game (Lo et al., 2004). Furthermore, minority game models have also been used in financial applications, for example to explain the market volatility basing on the trading behavior of the agents (e.g. Johnson et al., 1998; Marsili & Challet, 2001).