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.