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Assumption: We have a machine that plays games perfectly.

  1. The machine plays a game of Chess. Chess is a game of perfect information and complete information. We (assuming we have enough computational power for that) build the complete game tree for a game of chess and show that every move the machine made was minimizing the possible loss for the worst case scenario (comparison to minimax) and as a result, prove that it was playing perfectly.

  2. The machine plays a game of Backgammon. Backgammon is a game of imperfect information, but still complete information since both players have the same amount of information available to them. We do the same thing as with chess, including the dice rolls and probabilities for every possible outcome and show that every move that the machine made was, once again, minimizing the possible loss for the worst case scenario (comparison to expectiminimax) and as a result, prove that it was playing perfectly.

  3. Now the machine plays a game of Poker. Poker is a game of imperfect information and also incomplete information. Once we're reviewing the game, can we mathematically prove that the moves the machine made were perfect? Or is the fact that bluffing as a concept exists already means that it's sometimes impossible to compare two moves with each other and decide if there was anything better that could've been done?

If not, does that also mean that the assumption is wrong and it's impossible to play perfectly with incomplete information?

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    $\begingroup$ What is the perfect play? Consider a finite two player game of perfect information in which player 1 can force a win. That doesn't preclude the possibility that player 2 can win and though all behavior of player 2 will be, vacuously, minmax behavior, there are arguably better and worse things to do. $\endgroup$ – Michael Greinecker Jan 3 at 11:48
  • $\begingroup$ Well if both players are playing perfectly then the player 1 will win every time. Or am I missunderstanding? Player two can win only and only if the first player doesn't play perfectly. Perfect play is, where you can logically and mathematically show that any other move is either just as good or worse implying that every turn the perfect move (or one of the perfect moves) was played. Simply take tic-tac-toe, you should be easily able to tell if some player played perfectly or missed a forced win/got forced into a loss. $\endgroup$ – Mantas Kandratavicius Jan 3 at 14:17
  • $\begingroup$ "where you can logically and mathematically show that any other move is either just as good or worse" is not actually a mathematical definition. $\endgroup$ – Michael Greinecker Jan 3 at 14:20
  • $\begingroup$ It is absolutely not. Did you want one? $\endgroup$ – Mantas Kandratavicius Jan 3 at 14:22
  • $\begingroup$ Depending on how you define perfect play, the answer can be yes or no. $\endgroup$ – Michael Greinecker Jan 3 at 14:23

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