Trembling-Hand Perfection and Correlation in Sequential Games

Abstract

We initiate the study of trembling-hand perfection in sequential (i.e., extensive-form) games with correlation. We introduce the extensive-form perfect correlated equilibrium (EF-PCE) as a refinement of the classical extensive-form correlated equilibrium (EFCE) that amends its weaknesses off the equilibrium path. This is achieved by accounting for the possibility that players may make mistakes while following recommendations independently at each information set of the game. After providing an axiomatic definition of EFPCE, we show that one always exists since any perfect (Nash) equilibrium constitutes an EFPCE, and that it is a refinement of EFCE, as any EFPCE is also an EFCE. Then, we prove that, surprisingly, computing an EFPCE is not harder than finding an EFCE, since the problem can be solved in polynomial time for general n-player extensive-form games (also with chance). This is achieved by formulating the problem as that of finding a limit solution (as ǫ → 0) to a suitably defined trembling LP parametrized by ǫ, featuring exponentially many variables and polynomially many constraints. To this end, we show how a recently developed polynomial-time algorithm for trembling LPs can be adapted to deal with problems having an exponential number of variables. This calls for the solution of a sequence of (non-trembling) LPs with exponentially many variables and polynomially many constraints, which is possible in polynomial time by applying an ellipsoid against hope approach.

Trembling-Hand Perfection and Correlation in Sequential Games

“…There are only a few computational works on NE refinements. For instance, Miltersen and Sørensen (2010) characterize quasi-perfect equilibria of 2-player EFGs using the in (Marchesi and Gatti 2020) for more details.…”

“…Then, it follows that µ ∈ ∆ Π is an EFCE if, for every i ∈ N , player i's expected utility when following recommendations is at least as large as the expected utility that any (I, a, μi )-trigger agent for player i can achieve (assuming the opponents' do not deviate). A formal statement is provided in Appendix B in (Marchesi and Gatti 2020).…”

“…Computing EFCEs in n-player EFGs The algorithm of Huang and von Stengel (2008) relies on the following LP formulation of the problem of finding an EFCE, which has exponentially many variables and polynomially many constraints (see also Appendix C in (Marchesi and Gatti 2020)).…”

“…Thus, any EFPCE of the game (as well as any EFCE) must recommend a at I with probability 1. Then, in the sub-game reached when playing a at I, it is easy to check that recommending the pairs of actions (c, m), (c, n), and (d, m) each with probability 1 3 is an equilibrium, as no player has an incentive to deviate from a recommendation, even with trembles (see Appendix E in (Marchesi and Gatti 2020)). The correlation device described so far is sufficient to define an EFCE, as recommendations at infosets Y, K, and L are not relevant given that they do not influence players' utilities at the equilibrium (b is never recommended).…”

“…These are crucial for the derivation of the LP exploited by our algorithm. Our characterization is inspired by that of EFCEs based on trigger agents (see Lemma 4 in Appendix B in (Marchesi and Gatti 2020)). However, the presence of trembles in extended games requires some key changes, which we highlight in the following.…”