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Center for Mathematical Economics Working Papers No. 536
The so-called El Farol problem describes a prototypical situation of interacting agents making binary choices to participate in a non-cooperative environment or to stay by themselves and choosing an outside option. In a much cited paper Arthur (1994) argues that persistent on-converging sequences of rates of participation with permanent forecasting errors occur due to the non-existence of a prediction model for agents to forecast the attendance appropriately to induce stable rational expectations solutions. From this he concludes the need for agents to use boundedly rational rules. This note shows that in a large class of such models the failure of agents to find rational prediction rules which stabilize is not due to a non-existence of perfect rules, but rather to the failure of agents to identify the correct class of predictors from which the perfect ones can be chosen. What appears as a need to search for boundedly rational predictors originates from the non existence of stable confirming self-referential orbits induced by predictors selected from the wrong class. Specifically, it is shown that, within a specified class of the model and due to a structural non-convexity (or discontinuity), symmetric Nash equilibria of the associated static game may fail to exist generically depending on the utility level of the outside option. If they exist, they may induce the least desired outcome while, generically, asymmetric equilibria are uniquely determined by a positive maximal rate of attendance. The sequential setting turns the static game into a dynamic economic law of the Cobweb type for which there always exist nontrivial ϵ-perfect predictors implementing ϵ-perfect steady states as stable outcomes. If zero participation is a Nash equilibrium of the game there exists a unique perfect predictor implementing the trivial equilibrium as a stable steady state. In general, Nash equilibria of the one-shot game are among the ϵ-perfect foresight steady states of the dynamic model. If agents randomize over indifferent decisions the induced random Cobweb law together with recursive predictors becomes an iterated function system (IFS). There exist unbiased predictors with associated stable stationary solutions for appropriate randomizations supporting nonzero asymmetric equilibria which are not mixed Nash equilibria of the one-shot game. However, the least desired outcome remains as the unique stable stationary outcome for ϵ = 0 if it is a Nash equilibrium of the static game.
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