Question: In game theory, what is the precise definition of a "game of incomplete information"?
What I've found so far:
- In the standard first year graduate economics textbook on microeconomics (MWG), the best I can find is this:
Games in which “players know all relevant information about each other” “are known as games of complete information" (p. 253).
But what does "know" mean? And what does "relevant information" mean?
- And in the standard graduate textbook on game theory (Fudenberg and Tirole), the best I can find is this:
When some players do not know the payoffs of the others, the game is said to have incomplete information (p. 209).
But again, what does "know" mean?
- Briefly Googling, the only precise definition I can find of a game of incomplete information is the below (Levin, 2002, p. 3). However, this definition then prompts the question: "What is a game of complete information?" There does not seem to be any clear way to negate this definition (of a game with incomplete information) to produce a definition of a game with complete information.
Definition $\bf 1$ A game with incomplete information $G=(\Theta,S,P,u)$ consists of:
- A set $\Theta=\Theta_1\times\ldots\times\Theta_I$, where $\Theta_i$ is the (finite) set of possible types for player $i$.
- A set $S=S_1\times\ldots\times S_I$, where $S_i$ is the set of possible strategies for player $i$.
- A joint probability distribution $p(\theta_1,\ldots,\theta_I)$ over types. For finite type space, assume that $p(\theta_i)\gt0$ for all $\theta_i\in\Theta_i$.
- Payoff functions $u_i:S\times\Theta\to\Bbb R$.