Modeling Risk and Ambiguity-on-Nature in Normal-form Games

We propose multi-player frameworks that mitigate decision-theoretical difficulties with the traditional normal-form game, where players are concerned with expected utility functions of their payoffs. We react to Allais’s [1] paradox by concerning players with potentially nonlinear functionals of the payoff distributions they encounter. To counter Ellsberg’s [12] paradox, we let players optimize on vectors of payoff distributions in which every component is a payoff distribution corresponding to one particular nature action. In the preference game we introduce, players merely express preferences over payoff-distribution vectors. Depending on ways in which players’ mixed strategies are verified, there will emerge two equilibrium concepts, namely, the ex post and ex ante types. Conditions for equilibrium existence are identified; also, the unification of the two concepts at the traditional game is explained. When the preference relations lead to real-valued satisfaction functions, we have a satisfaction game. Two notable special cases are one coping with Gilboa and Schmeidler’s [15] ambiguity-averse worst-prior setup and another involving Artzner et al.’s [4] coherent-risk measure with risk-averse tendencies. For both, searching for ex post equilibria boils down to solving sequences of simple nonlinear programs (NLPs).