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It is hard to imagine such a game. Here is an example from wikipedia:

Suppose you are playing a game of chess against an opponent who will be paid some substantial amount of money if a particular event happens (an arrangement of pieces, for instance), but you do not know what the event is. In this case you have perfect information, since you know what each move of the opponent is. However, since you do not know the payoff function of the other player (which will affect its behavior even if it does not alter your own victory conditions), it is a game of incomplete information.

Do such game situations exist in real life?

What kind of equilibria can we observe?

Is there a dominant strategy that could be adopted in such a game? Is there a way to maximize your gain even though you don't know all the details?

Still in the chess example, you might start forming some beliefs after a few moves, if you notice for instance that the opponent is always sacrificing his pawns for nothing, maybe in his desired position he needs to have a certain amount of pawns... Or if given the opportunity to exchange queens and in a classical game of chess it would have been advantageous, but he doesn't do it, assuming he takes rational decisions, perhaps it is absolutely necessary for him to keep the queen in order to win with the altered rules.

You might assign certain probabilities to the different possible goals of your opponent and update them with every move he makes.

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  • $\begingroup$ Information is incomplete in most real life decision problems, as you do not know all the details. I don't know what you mean by "What kind of equilibria can we observe?" Please elaborate! $\endgroup$ – Giskard Jan 12 at 8:29
  • $\begingroup$ @denesp I added some clarifications $\endgroup$ – John Jan 12 at 10:36
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    $\begingroup$ Possible duplicate of Imperfect vs incomplete information $\endgroup$ – Kenny LJ Jan 13 at 7:48
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A classical example of a game with perfect but incomplete information is an English auction. We know what all the bids have been up to this point but we don't know everyone's valuation of the good being auctioned.

In general, formulating a strategy in this case involves some beliefs about your opponents' payoffs. Typically these are exogenous to the game but I guess you could have some revealed preference throughout an extensive game.

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Perfect information as the lack of uncertainty about what happened in the past (there were no hidden moves by any player, or by nature), and incomplete information as uncertainty about your opponent's payoffs, or the structure of the game. Situations like that occur all the time. Suppose that you go to a date and of course, you know all the interactions you have had before, but you don't know if the other person is will like you physically, if he/she prefers taller people, or people who are introverted, etc. You just need to think of a situation where you know all the actions that were done in the past, but you don't know the goals of other players in the game or you have uncertainty about the structure of the game itself, so forming beliefs about what their strategy will be is more tricky.

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