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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:

  1. A set $\Theta=\Theta_1\times\ldots\times\Theta_I$, where $\Theta_i$ is the (finite) set of possible types for player $i$.
  2. A set $S=S_1\times\ldots\times S_I$, where $S_i$ is the set of possible strategies for player $i$.
  3. A joint probability distribution $p(\theta_1,\ldots,\theta_I)$ over types. For finite type space, assume that $p(\theta_i)\gt0$ for all $\theta_i\in\Theta_i$.
  4. Payoff functions $u_i:S\times\Theta\to\Bbb R$.
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  • $\begingroup$ To my understanding, economics is not a rigorous research field, certainly not in the way that mathematics is. But to address your second question, complete information is, I think, when there is no uncertainty about types, that is, thus conceived of, complete information is not the "negation" of incomplete, but a special case. $\endgroup$
    – mathse
    Apr 11 '14 at 18:04
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    $\begingroup$ i.e. complete information would be: $p(\theta_i)=1$ for some $\theta_i\in\Theta_i$, I think. $\endgroup$
    – mathse
    Apr 11 '14 at 18:07
  • $\begingroup$ In incomplete information games, players may or may no know some information about the other players, e.g. their “type”, their strategies, payoffs, or preferences. "Know" here should be understood directly. Usually in incomplete info games, we specify probability measure over something that players do not know e.g. common prior about all players types... $\endgroup$
    – d.k.o.
    Apr 11 '14 at 18:15
  • $\begingroup$ @mathse every informal statement above is made rigorous at some point... one could say a calculus course is not rigorous because it is not analysis. you don't want to hammer an epistemic model (to provide fundations) when students can't even solve for the equilibrium of Bayesian games. $\endgroup$ Apr 11 '14 at 20:04
  • $\begingroup$ @Sergio Parreiras, well, I think that some mathematicians do study game theory - I assume that they will treat their subject matter rigorously. However, having studied a bit of economics, I feel that, no, economics, by itself, is not a rigorous field of research (even if restricted to game theory). And this lack of rigorousness appears to be deeply rooted in the field: in text books, professors' handling of concepts, etc. (not merely in lower level courses - addressing your calculus/analysis distinction). $\endgroup$
    – mathse
    Apr 11 '14 at 20:12
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To be honest "incomple information" is not a good terminology. If players do not know all the relevant information, the game is not really well specified. But for historical reasons, the term has stuck.

When you want to model a situation of "incomplete information" what you do in practice is to use Harsanyi's trick: you replace the incomplete information game by a (Bayesian) game with random moves by Nature (player zero).

Suppose two players are going to bid in an auction for an object but player 1 does not know the valuation of player 2. You can add an initial move by Nature, which is not observed by player 1, where Nature chooses the valuation of player 1 to be high or low with say probability $p$ and $1-p$.

In the above example, everybody knows the probability that player 1 is high valuation. But now suppose you want to model incomplete information also about beliefs and not only payoffs. Suppose you want to model a situation where player 2 is in doubt whether player 1 knows 2's valuations or player 1 is uncertain about 2's valuation. You can accomplish that by using the same trick. Add an additional move by Nature that is not observed by player 2: with prob. $q$ Nature chooses that player 1 knows 2's valuations and with prob. $1-q$ player 1 only thinks that 2's valuations are high and low with prob $p$ and $1-p$.

To answer the second part of your question "what means that a player knows something", one needs to add a knowledge model. You start with a set of states of the world $\Omega$. Each point in $\Omega$ is a complete description of the world (payoffs, players' beliefs about payoffs, players' beliefs about beliefs, and so on...). Then you give each player a partition of $\Omega$. Please see Rubisntein and Osborne's A Course in Game Theory, Chapter 5, Knowledge and Equilibrium. You can dowload freely and legally here http://arielrubinstein.tau.ac.il/books.html

Finally using the definition you gave, a game of complete information is a game of incomplete information where $\Theta$ is a singleton (i.e. $\Theta$ has only one element).

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  • $\begingroup$ Sergio, just as a clarification, how does the set $\Theta$ then look like? And could it also be that $\Theta$ is not a singleton but $p$ is specified appropriately? $\endgroup$
    – mathse
    Apr 12 '14 at 12:26
  • $\begingroup$ @mathse If it is a singleton, it means each $\Theta_i$ is a singleton as well and $p$ puts probability one on the single element. Another way (along the line you suggested ) is to have all the $\theta_i$ perfectly correlated: $\Pr(\theta_i|\theta_j)=0$ or $1$. but that would violate the assumption all $\theta\in\Theta$ have positive probability. $\endgroup$ Apr 12 '14 at 15:40
  • $\begingroup$ Oh yes, thanks, I have overlooked this part of the definition. I think the latter assumption is purely for convenience, though, and can be dropped without loss of generality. I was merely asking because, conceptually, I think you should say that $\Theta_i$ is a singleton (each agent is exactly of one type), from which it follows that also $\Theta$ is a singleton. The set $\Theta$ appears to be defined only after the sets $\Theta_i$ have been defined. $\endgroup$
    – mathse
    Apr 12 '14 at 16:02
  • $\begingroup$ @mathse if $\Theta=\prod \Theta_i$ then it is a singleton if and only if each $\Theta_i$ is a singleton so there is not difference. Conceptually: I prefer to think about $\Theta$ not as the product of the type spaces but rather as the set of the states of the world (which includes the physical description of the environment but also the description of the agents' beliefs and their beliefs about beliefs and so on...). $\endgroup$ Apr 12 '14 at 16:12
  • $\begingroup$ Of course, it is "if and only if"!! But, for me, I think of $\Theta$ merely as a derivative set (and formally, that is how it has been defined) - so one would always put conditions on the $\Theta_i$'s - but, as you indicate, you have a more semantic definition in mind. $\endgroup$
    – mathse
    Apr 12 '14 at 16:20
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The course of a game can be described by a sequence of states connected by moves. The move (which can be either chosen by a player or random) determines the next state. The state determines what possible moves can occur next and, in the case of a random move, what are the probabilities for the various possible moves. For a game of complete information, the current state is available to a player when deciding on a move, in the sense that the player's strategy is allowed to be an arbitrary function of the state. For a game of incomplete information, only part of the state may be available.

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Building a little on what Sergio Parreiras and d.k.o. said in regard to "what means that a player knows something":

In general the states of the world are elements of a probability space $(\Omega, \mathcal{F}, \mu)$. Player $i$'s information structure is a $\sigma$-subalgebra of $\mathcal{F}_i$ of $\mathcal{F}$. Player $i$ knows a random variable (or just a measurable map) $f$ if $f$ is measurable with respect to $\mathcal{F_i}$.

In your particular case, $\mathcal{F}$ is the type space $\Theta$, which is a product

$$ \Theta = \Theta_1 \times \cdots \times \Theta_n. $$

The probability measure $\mu$ specifies the joint distribution of types. Player $i$'s information set is generated by the projection onto the $i$-th coordinate

$$ (\theta_1, \cdots, \theta_n) \mapsto \theta_i. $$

So in general player $i$'s only knows (the realization of) his own type but not the types of players $-i$. Some special cases: If player $j$ has only one type, then everyone knows her type in the interim stage. If the measure is concentrated at a single point, everyone knows everyone's type, i.e. game of complete information.

I'd point out that this makes introducing Nature, Harsanyi's trick, unnecessary.

More generally, players can have different beliefs/priors, given by different measures $\mu_i$. (Common prior is needed for Harsanyi's trick.)

A player's strategy is then a map into his action set that is again measurable with respect to $\mathcal{F}_i$.

Implicit here is that players know each other's information structures. E.g. in your case, upon realization of his type player $1$ knows player $2$ knows player $2$'s type. And player $1$ maximizes his expected utility accordingly, with respect to his prior.

In the more general formulation, player $i$ knows player $j$ knows $F_j \in \mathcal{F}_j$ condition on $F_i \in \mathcal{F}_i$ if

$$ F_j \cap F_i \neq \emptyset. $$

By induction, you can model all orders of knowledge this way.

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