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In perfect information games, the agent can see all the moves performed in the past. Besides, it can observe the next action that will be put into practice by the opponent.

In this case, can we say that perfect information games are actually a fully observable environment? If we reach this conclusion, I guess that imperfect information becomes a partially observable environment?

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Not exactly, at least traditionally: in Game Theory, "imperfect information" is most often defined as agents having only partial information about the history of agents' actions, as you correctly noted. But also note that this doesn't refer to the general world facts or state.

But "partial observability" is typically used in terms of systems, e.g. in Markov Decision Processes, where it explicitly refers to world state, which might or might not include the history of other actors' actions.

But of course in the end it depends which exact definitions are used in the context you're looking at - every author is free to define their own concepts, using traditional names or new ones.

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Bastian Blankenburg is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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There is indeed a close parallel here, but the concepts are distinct. Every perfect information game is fully observable, but not every fully observable game is a game of perfect information.

A game of imperfect information is one in which you lack knowledge of any of the following:

  1. The state of the game (e.g. current market prices).
  2. The rewards you will receive from various states (i.e. utility and cost functions).

In contrast, in partially observable process (specifically, a POMDP), the requirement is that you must not know which state you are in.

This is a subtle distinction, so here are some examples:

  • A multi-armed bandit game with stationary distributions. Here, you know which state you are in (in fact, if the distributions are stationary, you know that the state doesn't change, except for the value of your winnings). You are not in a POMDP (the game is fully observable), but you are operating with imperfect information, because you don't know the utility function associated with different actions. You are operating in a regular MDP.

  • The game of chess has perfect information, and is also thus fully observable.

  • The game of poker has imperfect information because you cannot observe the current state of the game (you can't see the cards in your opponent's hand). It is thus a POMDP.
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