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According to Wikipedia, an incomplete-information game can be converted into an imperfect-information game with complete information in extensive form by using the so-called Harsanyi transformation, which means adding chance nodes at the beginning of the game:

It may be the case that a player does not know exactly what the payoffs of the game are or of what type their opponents are. This sort of game has incomplete information. In extensive form, it is represented as a game with complete but imperfect information using the so-called Harsanyi transformation.

My question is: is this transformation feasible only for some kind of incomplete-information games or for all of them? I easily see how it can be done when the distributions of probabilities over the possible types of players are known. In this case, you add one or more chance nodes at the beginning of the game, labeled with said probabilities. But what if the distributions are not known? Can the transformation be made? In this case, I suppose the chance nodes added would not be labeled. Would this still be an imperfect but complete-information game? Would it still be an extensive-form game at all, considering that in the definition of the extensive-form game given by Wikipedia the probabilities of actions by nature (chance nodes) are known?

If the answer to this question is "only certain kinds of incomplete-information games can be converted in imperfect but complete-information games", then another question naturally rises: considering that there is no difference in modeling an imperfect-information game and some particular kinds of incomplete-information games, are imperfect-information games a subset of incomplete-information games?

If the answer is "all the incomplete-information games can be converted in imperfect but complete-information games", then I would ask if imperfect-information games are exactly all of the games with incomplete-information.

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    $\begingroup$ How do you define games of incomplete information? I'm not sure I understand what you mean by " if the distributions are not known." $\endgroup$ – Michael Greinecker Jun 23 at 17:37
  • $\begingroup$ How to define incomplete-information games is part of my question... it is not entirely clear to me. By "distributions" I mean the probability distributions over the possible types of players, which translate to different probability labels on the edges of chance nodes. The type of a player is his own function from outcomes to utilities, which in incomplete-information games (according to all the definitions of incomplete-information game I've seen) is not known with certainty. $\endgroup$ – Camingo Jun 24 at 12:09
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    $\begingroup$ I think in that case, it might be better to look in textbooks; the topic is a bit broad. $\endgroup$ – Michael Greinecker Jun 24 at 12:12
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Two points:

  1. An Incomplete-Information Game is not well-defined without specifying the beliefs of the players. This is specially relevant if you want to use the standard Bayesian equilibrium techniques. There is still significant leeway here - you may define subjective probabilities for each players, players may have inconsistent beliefs (look at Yildiz's excellent papers in a bargaining setup), etc. Recently, there have been some papers that deal with situations where a player is uncertain about the beliefs held by other players (see Gab Caroll's Robustness and Linear Contracts for a brilliant exposition)- but you may need to develop additional tools (like updating rules under ambiguity - see Faruk Gul's paper) to study these situations.

  2. As for Harsanyi's transformation, the critical component of his 1967-68 paper is the common prior assumption (so every player has consistent beliefs regarding the uncertain elements of the game. Thus Harsanyi states that any incomplete information game with consistent beliefs can be remodelled as a game on imperfect information by introducing Nature's move at the beginning. There may have been extensions to the result in the past 30 years, but Im not aware of it.

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