What's the difference? To illustrate, please give examples of
- A game with perfect but incomplete information; and
- A game with complete but imperfect information.
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You are at a decision point and you don't know what strategy the other player has taken. But you still know that which game you are in, you still know the other players' strategy sets etc. Under incomplete information you might not even know what game you are into.
For example: Two firms are taking up R&D projects sequentially. One is the first mover. It decides whether to take up R&D or not. Now the second firm has to decide but does not know what first firm has decided. This is imperfect information.
Now imagine ypu are playing a card game and need to throw a card but you don't know whether you are playing poker or sweep. This is incomplete information.
Taken directly from Wikipedia:
A game with perfect but incomplete information
Examples of games with incomplete but perfect information are conceptually more difficult to imagine. 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.
A game with complete but imperfect information
Examples of games with imperfect but complete information card games, where each player's cards are hidden from other players but objectives are known, as in contract bridge and poker. The latter claim assumes that all players are risk-neutral and thus only maximizing their expected outcome. However, since each individual might respond differently to risk, one cannot generally know the exact form of the objective function the other players are trying to maximize and thus the way they will respond to different situations. Thus, from a purely theoretical perspective, these games should generally be considered as having imperfect and (slightly) incomplete information.