Max Planck Institute for Mathematics in the Sciences

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany Rationality and information in games Jürgen Jost

Collaborators • Nils Bertschinger, Eckehard Olbrich (MPI MIS) • David Wolpert (NASA, Ames) • Mike Harre (U. Sidney) (computer plots)

• Game theory assumes that players know each others’ utility functions and that they can anticipate the actions of their opponents to the extent dictated by their rational intention of utility maximization. • In a mixed Nash equilibrium, however, rationality does not determine a specific action, but only probabilities of actions. Therefore, even if such a game is repeated, opponents’ actions cannot be completely predicted.

Information • Increase: Additional knowledge about actually chosen actions of opponents at mixed equilibria • Decrease: Uncertainty about opponents’ utilities

Rationality Increase of knowledge about actions of opponents at mixed equilibria deviations of their probabilities from NE ones decrease of their rationality Decrease of knowledge by uncertainty about opponents’ utilities less predictable actions of opponents decrease of their rationality

Rationality Increase of knowledge about actions of opponents at mixed equilibria deviations of their probabilities from NE ones decrease of their rationality Decrease of knowledge by uncertainty about opponents’ utilities less predictable actions of opponents decrease of their rationality (Probability distribution of actions might depend on utilities so that very bad mistakes (in terms of pay-offs) are less probable than milder ones ( ! mathematical psychology: ``probabilistic choice'‘)

Rationality Increase of knowledge about actions of opponents at mixed equilibria deviations of their probabilities from NE ones decrease of their rationality ! typically, should be disadvantageous for them, but can be clever in certain games Decrease of knowledge by uncertainty about opponents’ utilities less predictable actions of opponents decrease of their rationality ! typically, should be advantageous for them, but can be harmful in certain games

Alternative interpretation • Opponents randomly selected from a population whose members exhibit some degree of variation, with player only knowing probability distribution of population ( ! econometrics), but not characteristics of individual opponents (each opponent is rational, but their utility functions are not known precisely)

Questions • Can the effects of varying information and rationality be quantified? • How to model the relevant parameters? • Can this be used for purposes of control, that is, can desired effects like Pareto optimality be achieved by tuning those parameters? • If so, should the players themselves decide about those parameter values, or rather some external “well-meaning” controller?

Quantal response equilibria (QREs) (Mc. Kelvey – Palfrey)

• The rationality coefficients are the only parameters, and i may have access only to her own ¯i. Expresses both the direct effect of a variation of ¯i on the utility of i and the indirect effect of the response of –i.

For ¯i 0, player becomes completely irrational and selects all possible actions with equal probability. For ¯i 1, she becomes fully rational. The QREs then converge to Nash equilibria.

For ¯i 0, player becomes completely irrational and selects all possible actions with equal probability. For ¯i 1, she becomes fully rational. The QREs then converge to Nash equilibria. Generically, only saddle-node bifurcations, and there exists a unique path in parameter space from the fully irrational behavior to one specific Nash equilibrium. When the parameter varies in only one direction, there may be hysteresis effects and discontinuous jumps.

For ¯i 0, player becomes completely irrational and selects all possible actions with equal probability. For ¯i 1, she becomes fully rational. The QREs then converge to Nash equilibria. Generically, only saddle-node bifurcations, and there exists a unique path in parameter space from the fully irrational behavior to one specific Nash equilibrium. When the parameter varies in only one direction, there may be hysteresis effects and discontinuous jumps. When both rationality coefficients, ¯i and ¯-i , can be varied, we also see pitchfork bifurcations, and the players can end up at different Nash equilibria, depending on which branch they choose.

An example

p-i+ 1 1/2 0 0 ½ 1 p i+

Varying ¯i equivalent to varying utility ! interpretation as tax rate

How can the ¯ s be varied? 1) Each player sets her value independently, without considering the reactions of her opponents (“Anarchy”) 2) The players play a Nash game for selecting their values within some given range (“Market”) 3) An external controller sets the values, e. g. as tax rates (“Socialism”) In general, the outcomes of the three mechanisms will be different, and which one will achieve the best results in the Pareto sense may depend on the particular game.

Information about opponent

Information about opponent • Analysis analogous to QRE possible when information about opponent is only probabilistic, that is, player receives certain symbol that carries information about opponent move probabilities. This player can then choose the probabilities of the possible reactions to the symbols received.

Information about opponent