Complete information

In economics and game theory, complete information is a term used to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions, payoffs, strategies and "types" of players are thus common knowledge.

Inversely, in a game with incomplete information, players may not possess full information about their opponents. Some players may possess private information that the others should take into account when forming expectations about how a player would behave. A typical example is an auction: each player knows his own utility function (= valuation for the item), but does not know the utility function of the other players. See ^[1] for more examples.

Games of incomplete information arise most frequently in social science rather than as games in the narrow sense. For instance, John Harsanyi was motivated by consideration of arms control negotiations, where the players may be uncertain both of the capabilities of their opponents and of their desires and beliefs.

It is often assumed that the players have some statistical information about the other players. E.g., in an auction, each player knows that the valuations of the other players are drawn from some probability distribution. In this case, the game is called a Bayesian game.

Complete vs. perfect information

Complete information is importantly different from perfect information. In a game of complete information, the structure of the game and the payoff functions of the players are commonly known but players may not see all of the moves made by other players (for instance, the initial placement of ships in Battleship); there may also be a chance element (as in most card games). Conversely, in games of perfect information, every player observes other players' moves, but may lack some information on others' payoffs, or on the structure of the game.^[2] A game with complete information may or may not have perfect information, and vice versa.

• Examples of games with imperfect but complete information are card games, where each player's cards are hidden from other players but objectives are known, as in contract bridge and poker.^[3][4] 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 (slighly) 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.

Games of incomplete information can be converted into games of complete but imperfect information under the "common prior assumption." This assumption is commonly made for pragmatic reasons, but its justification remains controversial among economists.

See also

• Bayesian game
• Handicap principle
• Market impact
• Screening game
• Signaling game
• Small talk
• Trash-talk

References

1. Levin, Jonathan (2002). "Games with Incomplete Information" (PDF). Retrieved 25 August 2016.
2. Osborne, M. J.; Rubinstein, A. (1994). "Chapter 6: Extensive Games with Perfect Information". A Course in Game Theory. Cambridge M.A.: The MIT Press. ISBN 0-262-65040-1.
3. Thomas, L. C. (2003). Games, Theory and Applications. Mineola N.Y.: Dover Publications. p. 19. ISBN 0-486-43237-8.
4. Osborne, M. J.; Rubinstein, A. (1994). "Chapter 11: Extensive Games with Imperfect Information". A Course in Game Theory. Cambridge M.A.: The MIT Press. ISBN 0-262-65040-1.

• Fudenberg, D. and Tirole, J. (1993) Game Theory. MIT Press. (see Chapter 6, sect 1)
• Gibbons, R. (1992) A primer in game theory. Harvester-Wheatsheaf. (see Chapter 3)
• Ian Frank, David Basin (1997), Artificial Intelligence 100 (1998) 87-123. "Search in games with incomplete information: a case study using Bridge card play".