The Value of Ignorance about the Number of Players

In the game known as the El-Farol bar, each player decides whether to come to the bar or stay at home (Arthur 1994). Normally, every player prefers to enjoy the party, however if the bar becomes congested beyond its capacity, the police closes the place, inflicting severe trauma on all party-goers. A variant of the problem (which is perhaps more realistic) is where the discomfort of the party-goers increases as the bar becomes more congested, even before capacity is exhausted. If the number of players is known, then in equilibrium the bar will always be packed up to the point of full capacity (just enough to avoid police intervention), or until discomfort is sufficiently high so that players are indifferent between partying and staying home. A possible solution is to hide the exact number of players in the game. Facing uncertainty, some players may prefer to stay home so as to avoid the chance of exceeding capacity. Consequently, the bar will only become reasonably congested, so that at least those who arrive can enjoy the party. Minority games applied the El Farol Bar as a metaphor for various economic situations (Challet, Marsili, and Zhang 2001). Our variant above and the Kolkata Paise Restaurant problem (Chakrabarti 2007), where players choose from multiple restaurants, are useful analogies to real problems like congestion of roads and of service providers—typically modeled as congestion games (Rosenthal 1973). Our starting point is the model of Meir et al. (2012), where agents have uncertainty over the actual number of participants in a congestion game, formalized as a prior distribution over subsets of players. In this work we consider the idea of partial information revelation (signaling), for such games where the number of participants is unknown. We expose a counter-intuitive phenomenon, where hiding the number of players may result in a significant improvement in welfare. We construct several examples of congestion games that demonstrate such improvement when hiding all or some information. We complement our results by proving that by hiding information the welfare can increase by a factor of at most n (the number of agents). As our examples demonstrate improvement by a similar factor, n is tight bound (and almost tight under i.i.d. participation).