# Two Player Game Useless Strategy

Let's consider the variant of dominated strategy which is the pure strategy that is not a best response to any mixed strategy of the opponent (two player game). Intuitively it sounds like more stronger notion of dominated strategy, because in this case all mixed strategies of the opponent is taken into account, lets call this kind of dominated strategy as "useless strategy".

Obviously "useless strategy" doesn't participate in mixed strategy of the player, therefore it can be safely excluded from the game.

Is there any algorithm to detect "useless strategies"?

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A strategy that is never a best reply to a mixed stategy of the other player in a two-player game is exactly the same thing as a strictly dominated strategy.

This result goes back to a paper by Pearce on rationalizability. For a simple proof, see Lemma 60.1 in A Course in Game Theory by Osborne and Rubinstein, where it is shown using the minmax-theorem.

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Thank you very much for the answer! Two questions: 1) is never best response is equivalent to useless strategy? Intuitively it seems so, but formally I don't see that it is equivalent (in useless strategy we compare against every mixed strategy of the opponent). 2)How to find the useless strategy? Should we convert to game $G'$ and check that $min_{m_2} max_{a_i} v_1(a_i,m_2) > 0$ like it was suggested in the proof (I am not sure is polynomial time algorithm) or there is another method? –  fog Apr 26 '13 at 19:33
@fo Well, they are defined the same way. Using the transformation might be workable, but algorithmics is not my strong side. The situation is complicated by the problem that a strategy might be strictly dominated by a mixed strategy without being dominated by a pure strategy. –  Michael Greinecker Apr 26 '13 at 23:21