# Complete vs Perfect infomation in Combinatorial game theory

In their book "Winning Ways for Your Mathematical Plays", Berlekamp, Conway, and Guy used as the 7th condition for a combinatorial game "Both players know what is going on; There is complete information. There is no occasion for bluffing."

Is the "more recent" notion of "perfect information" generally accepted as part of the correct definition, or is this a point of controversy in CGT? So it would seem that the more recent notion would be that a combinatorial game would have perfect information, but may or may not have complete information. So far as I can tell the trio never explicitly define "complete information", nor do they use it in a way that is inconsistent with "perfect information." I don't remember seeing "perfect information" anywhere in the book.

• According to Wikipedia a distinction made is this: in a game with perfect information, players know all previous events, but may not necessarily have knowledge of all strategies and payoffs; conversely in a game with complete information, all players know all strategies and payoffs, but may not necessarily know all previous events. So make that, what you will. :) – c.w.chambers Jun 24 '14 at 5:24
• After quickly searching one of my game theory books, I suspect this notion of complete information is somehow related to the completeness property of what it calls the preferred outcome relation (applied to decisions with certainty) which says that for any outcome ‘$a$’ and ‘$b$’, either ‘$a\preceq b$’ or ‘$a\succeq b$’, or both are true. If you don't know all outcomes, or can't relate them, then the relation would fail to be complete by this definition. – c.w.chambers Jun 24 '14 at 5:57
• Ah..., I think changing from some sort of mathematical completeness property to "perfect information" in game theory could well be the gist of the change from "Winning Ways" to other "more modern" definitions which use "perfect information" (game theory use). I'd suspect that even within different versions of game theory that "complete information" has slightly different implications. – MaxW Jun 24 '14 at 23:48

The terminology may not be completely standard, but:

In combinatorial game theory, the games considered are those in which the players have complete knowledge of the state of the game, and there are no random events. (There are also some other conditions on the games in CGT that I won't mention here.) A common term in CGT for such games is "game of no chance". This includes chess, go, Othello, etc.

However, "perfect information game" sometimes includes games like Backgammon or Monopoly, which although they have random events (dice rolls), they still do not have any information which is known to one player but not another. (The probabilities of all random events must be known to all players.)

On the other hand, in most card games, you can only see your own cards; thus the players have different information sets, so card games are usually games of imperfect information.

Why do we put Backgammon or Monopoly in with games like chess or go? One important property they have in common is that, at least in theory, bluffing is impossible. In a game like poker, I can bluff because my opponents can't see my cards, thus I can take actions to (mis)convey information about my hand. But in Backgammon, even though there are dice rolls involved, I don't know any more about them than my opponent does, and I cannot influence them, so I cannot bluff. As a result, in a finite, two-player game of perfect information, the optimal strategy is always a pure strategy, even if there are random events involved.

I'm not aware of any difference between "perfect information" and "complete information". The authors just mean (I think) that everyone knows all possible moves of both players, and the resulting position after those moves are made, etc. So all game play sequences are determined in advance, and both know all of them. There are no chance moves, or hidden information (like playing cards one player can see and other cannot) or any unknowns at all at the start of the game (unlike in card games with a closed deck, where both players don't know the deck).

So they model a very limited (in a way) subset of games, but a subset they can do more with. there is no need for "strategies" like in classical, more general, game theory.

• I'm not sure how widely accepted the distinctions are, but some authors do have different meanings for the two terms. See C.W. Chambers comment about the Wikipedia distinction for example. So I'm sort of confused if the terminology has matured, The gang of three was just sloppy, or they had something else in mind. – MaxW Jun 25 '14 at 0:10