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I am just wondering whether the following question has already been answered in game theory: Fix an arbitrary (say, finite) two-player strategic form game. Under what condition, does one player (say, Player 1) want to be the first-mover than the second mover?

For example, we know that moving first is better in Cournot competition and moving second is better in Rock-paper-scissors. Is there a general condition on the payoffs of two-player games under which we can tell whether Player 1 should go first or second, if he could choose?

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    $\begingroup$ It may seem a little circular, but in an impartial combinatorial game, the game is a second player win when its nimber is 0 and a first player win otherwise. More generally, including non-impartial games, those games with a value of 0 are second player wins while those which are fuzzy with 0 are first player wins $\endgroup$ – Henry Aug 19 '18 at 21:20
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One nice reference that seeks to answer your question is the following paper,

https://link.springer.com/content/pdf/10.1007/BF01076104.pdf

The paper's main results are contained in the following theorem (Theorem I):

a. If both players have a dominant strategy in the basic game, the game is the same. (where the basic game is just the 2x2 bimatrix game)

b. If only one player has a dominant strategy in the basic game, then the extended game has a unique equilibrium in un-dominated strategies. It's different from the basic game outcome iff that outcome is Pareto dominated by another pure strategy outcome. The extended game equilibrium achieves these Pareto dominating payoffs by having the player with the dominant strategy play first and the other second.

Think of this as a way to achieve a Pareto superior vector of payoffs.

(C) If the unique simultaneous play equilibrium of the basic game is in mixed strategies, then like in part (B) there won't be a change unless there's a Pareto superior P.S. vector. If Pareto dominance does not exist, both players will wait until the second period and play simultaneously. If Pareto dominance does exist, the unique extended game equilibrium attains the Pareto dominating payoffs by selecting one of the sequential move games.

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