Refinements of rationalizability for normal-form games
The main idea behind the notion of Nash equilibrium is the rationalizability idea -- that every player is rational, i.e., that every player only uses strategies which are optimal against some of opponent's strategies. Thus, if a strategy is never optimal, it can be safely excluded; now with fewer strategies, we can again exclude those which are not optimal against some rational ones, etc. One of the main drawbacks of this notion is that after all the deletions, e.g., the 1st player may still end up with some strategies \(s_1\) which are weakly dominated by some other strategy \(s_1'\); in precise terms, for all possible strategies \(s_2,\dots\) of all other players, the 1st player's payoff \(u_1(s_1,s_2,\dots)\) resulting from using \(s_1\) is not better than if we use \(s_1'\) (i.e., \(u_1(s_1,s_2,\dots)\leq u_1(s_1',s_2,\dots)\)) and sometimes it is worse, i.e., \(u_1(s_1,s_2,\dots)<u_1(s_1',s_2,\dots)\) for some \(s_2,\dots\) Several refinements of the rationalizability notion have been proposed for which two rational strategies cannot weakly dominate each other. All these refinements are based on the same basic idea: that in reality, strategies are not perfectly implemented, so we should keep strategies which are best responses against small perturbations, and then consider the limit when the perturbations tend to 0. Different reasonable formalizations of this idea lead, however, to slightly different results. The authors provide a complete analysis of the relation between the four known refinements: show when one always leads to the other, and when there is a counterexample.
scientific article; zbMATH DE number 1309802