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During the game of war, if you could order the cards in your deck of 26, what strategy should you employ?

  1. Assume Player 2 has a random ordering of 26 cards and is not allowed to change the order in anyway.
  2. Let Player 1 have the advantage of looking at one's own deck of cards and reordering the cards.

Several theories may apply: game theory, combinatorics, simulation, etc. What are some of the ideas that would help one dissect this problem into something useful and begin to solve it? By solve, I mean describe the strategy Player 1 should employ and why he should employ it.

Mark Jones Jr.
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    A plausibly interesting-enough question, but would be more appropriate for MathStackExchange. – paul garrett Jan 15 '18 at 22:02
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    https://mathoverflow.net/questions/11503/does-war-have-infinite-expected-length may be relevant, as in the answers therein include an ordering of an infinitely long game. – Mark S Jan 15 '18 at 22:15
  • I had seen that question, @MarkS. It discusses the outcome under the basic rules, not a variant in which a player orders his deck. – Mark Jones Jr. Jan 15 '18 at 22:36
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    Upon reflection, if this question is construed as addressing the larger game-theoretic issue, then (so far as I know) it is worth keeping on this site. (I did not vote to close, in any case.) – paul garrett Jan 15 '18 at 23:22
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    @MarkJonesJr. any chance you can formalize the question better? – Robert Jan 15 '18 at 23:56
  • It seems to me that a natural formulation is to consider this as a zero sum game and look for a Nash equilibrium in mixed strategies -- if both players choose a probability distribution over orderings what are the equilibrium strategies. Another interesting question is whether the uniform distribution is exploitable -- ie is there a strategy better than the uniform distribution against the uniform distribution? Lastly, how do these issues depend on the actual deals -- ie who got which cards. – ericf Jan 17 '18 at 03:39

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