Troels Bjerre Sørensen

Computing sequential equilibria for two-player games

Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for two-player extensive-form zero-sum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Koller and Pfeffer pointed out that the strategies obtained by the algorithm are not necessarily sequentially rational and that this deficiency is often problematic for the practical applications. We show how to remove this deficiency by modifying the linear programs constructed by Koller, Megiddo and von Stengel so that pairs of strategies forming a sequential equilibrium are computed. In particular, we show that a sequential equilibrium for a two-player zero-sum game with imperfect information but perfect recall can be found in polynomial time. In addition, the equilibrium we find is normal-form perfect. Our technique generalizes to general-sum games, yielding an algorithm for such games which is likely to be prove practical, even though it is not polynomial-time.

Approximate well-supported nash equilibria below two-thirds

In an e-Nash equilibrium, a player can gain at most e by changing his behaviour. Recent work has addressed the question of how best to compute e-Nash equilibria, and for what values of e a polynomial-time algorithm exists. An e-well-supported Nash equilibrium (e-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most e less than a best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new technique that leads to an improvement to the worst-case approximation guarantee.

The computational complexity of trembling hand perfection and other equilibrium refinements

The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium, (the strategy part of) sequential equilibrium, quasi-perfect equilibrium and CURB. The proofs all use a reduction from the problem of comparing the minmax value of a three-player game in strategic form to a given rational number. This problem was previously shown to be NP-hard by Borgs et al., while a SQRT-SUM hardness result is given in this paper. The latter proof yields bounds on the algebraic degree of the minmax value of a three-player game that may be of independent interest.

Approximability and Parameterized Complexity of Minmax Values

We consider approximating the minmax value of a multi-playergame in strategic form. Tightening recent bounds by Borgs et al.,we observe that approximating the value with a precision ofelogn digits (for any constant e> 0) isNP-hard, where n is the size of the game. On the other hand,approximating the value with a precision of c loglogn digits (forany constant c ≥ 1) can be done inquasi-polynomial time. We consider the parameterized complexity ofthe problem, with the parameter being the number of pure strategiesk of the player for which the minmax value is computed. We showthat if there are three players, k = 2 and there areonly two possible rational payoffs, the minmax value is a rationalnumber and can be computed exactly in linear time. In the generalcase, we show that the value can be approximated with anypolynomial number of digits of accuracy in time n O(k). On theother hand, we show that minmax value approximation is W[1]-hardand hence not likely to be fixed parameter tractable. Concretely,we show that if k-Clique requires time n Ω(k) then so doesminmax value computation.

Approximate Well-supported Nash Equilibria Below Two-thirds

In an

Computing proper equilibria of zero-sum games

\epsilon

Computing a proper equilibrium of a bimatrix game

ϵ-Nash equilibrium, a player can gain at most