The ?-Beauty Contest: Choosing Numbers, Thinking Intervals

The 1-shot ?-beauty contest is a non-equilibrium strategic game under bounded rationality conditions, while equilibrium is approached if the game is played iteratively sufficiently many times. Experimental data of the 1-shot setting of the 0-equilibrium game show a common pattern: The spectrum of announced numbers is a superposition of a skew background distribution and a regime of extra ordinarily often chosen numbers. Our model is capable of quantitatively reproducing this observation in non-equilibrium as well as the convergence towards equilibrium in the iterative setting. The approach is based on two basic assumptions: 1.) Players iteratively update their recent guesses in the sense of eductive reasoning and 2.) Players estimate intervals rather than exact numbers to cope with incomplete knowledge in non-equilibrium. The width of the interval is regarded as a measure for the confidence of the players' respective guess. It is shown analytically that the sequence of guessed numbers approaches a (finite) limit within only very few iterations. Moreover, if all players have infinite confidence in their respective guesses, the asymptotic Winning Number equals the rational Nash equilibrium 0, while if players have only finite confidence in their recent guess, the Winning Number in the 1-shot setting is strictly larger than 0. Our model is also capable of quantitatively describing the �path into equilibrium". Convergence is shown to be polynomial in the number of rounds played. The predictions of our model are in good quantitative agreement with real data for various ?-beauty contest games.