I’d typically assume that after most card games, the cards become at least semi-ordered in some way, necessitating shuffling. However, after a standard game of war, I can’t quite figure out how the arrangement would become less random, since the winning and losing card stay together. If they’re indeed mathematically “less random,” after the game, why?
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The relevant property you have to look for here is the game being a _bijection_: does every potential ordering of the cards actually result from _some_ game of War, and do two different starting setups result in the exact same order?
If the answer is yes, and the initial shuffling is truly random, with all draws having equal chance, then the same will again hold for the state after a game of War. Otherwise, the chances are uneven for sure; for example, that one (or more) impossible ordering can never come to be.
By things being fine, one can also note that the two halves of the above each imply the other: if every ordering happens at least once, then each comes from exactly one initial arrangement; and if no arrangement happens twice, then none are missing.
**So the question left to check is if absolutely every potential ordering can result as the end state of a game of War.** I would doubt it, and it definitely depends on how you play; in particular, if the winning card ends up on top, or the same player’s card always will be on top, those forbid some arrangements already. If the winner of the previous “fight” has their card out first and thus on bottom, things get a bit more complicated and it is too early here to think this through, but I doubt it works either.
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