Epistemic Game Theory: Incomplete Information

Abstract

In a game of incomplete information some of the players possess private information which may be relevant to the strategic interaction. Private information is modelled by a type space, in which every type of each player is associated with a belief about the basic issues of uncertainty (like payoffs) and about the other players’ types. At a Bayesian equilibrium each type chooses a strategy which maximizes its expected payoff given the choice of strategies by the other players’ types. Bayesian equilibrium payoffs are often inefficient relative to the equilibrium payoffs that would result had the players been fully informed.

A game of incomplete information is a game in which at least some of the players possess private information which may be relevant to the strategic interaction. The private information of a player may be about the payoff functions in the game, as well as about some exogenous, payoff-irrelevant events. The player may also form beliefs about other players’ beliefs about payoffs and exogenous events, about their beliefs about the beliefs of others, and so forth.

Harsanyi (1967–8) introduced the idea that such a state of affairs can be succinctly described by a type space. With this formulation, T_i denotes the set of player i’s types. Each type t_i ∈ T_i is associated with a belief λ_i(t_i) ∈ Δ(K × T_−i) about some basic space of uncertainty, K, and the combination T_−i of the other players’ types. The basic space of uncertainty K is called the space of states of nature, and Ω = K × Π_{i ∈ I}T_i, where I is the set of players, is called the space of states of the world.

A type space models a game of incomplete information once each state of nature k ∈ K is associated with a payoff matrix of the game, or, more generally, with a payoff function ${u}_i^k$ for each player i ∈ I. This payoff function specifies the player’s payoff ${u}_i^k(s)$ for each combination of strategies s = (s_i)_{i ∈ I} ∈ S = Π_{i ∈ I}S_i of the players. (In the particular case in which k is associated with a payoff matrix, that is, the game is such that each player has finitely many strategies, the payoffs ${u}_i^k(s)$ to the players i ∈ I appear in the entry of the matrix corresponding to the combination of strategies s = (s_i)_{i ∈ I} . ) As usual, the set of strategies S_i of player i ∈ I may be a complex object by itself. For instance, it may be the set of mixed strategies over some set of pure strategies ${S}_i^0$. The payoff function of player i in the state of nature k is ${u}_i^k:S\to \mathrm{\mathbb{R}}$.

Obviously, different types of a player may want to choose different strategies. Thus, a Bayesian strategy of player i in a game of incomplete information specifies the strategy σ_i(t_i) ∈ S_i that the player chooses given each one of her typest_i ∈ T_i.

Given a profile of Bayesian strategies σ = (σ_j : T_j → S_j)_{j ∈ I} of the players, the expected payoff of player i of type t_i is

$$ {U}_i\left(\sigma, {t}_i\right)=\sum \limits_{\left(k,{t}_{-i}\right)\in K\times {T}_{-i}}{u}_i^k\ \left({\sigma}_i\left({t}_i\right),{\sigma}_{-i}(t)\right)\times {\lambda}_i\left({t}_i\right)\left(k,{t}_{-i}\right) $$

where σ_−i(t_−i) = (σ_j(t_j))_{j ≠ i}. If there is a continuum of states of nature and types, the sum becomes an integral:

$$ {\displaystyle \begin{array}{ll}{U}_i\left(\sigma, {t}_i\right)=& {\int}_{K\times {T}_{-i}}{u}_i^k\ \left({\sigma}_i\left({t}_i\right),{\sigma}_{-i}\left({t}_{-i}\right)\right)d{\lambda}_i\left({t}_i\right)\hfill \\ {}& \left(k,{t}_{-i}\right)\hfill \end{array}} $$

(In this case, the expected payoff function U_i(σ, t_i) is well defined if the Bayesian strategies σ_j : T_j → S_j are measurable functions and if the payoff function ${u}_i^{\cdot }:K\times S\to R$ is measurable as well; we omit the details of this technical requirement).

We assume that the players are expected payoff maximizers. Thus, player i prefers the Bayesian strategy σ over σ′ if and only if U_i(σ, t_i) ≥ U_i(σ′, t_i) for each of her types t_i ∈ T_i . It follows that given a Bayesian strategy profile σ_−i of the other players, the Bayesian strategy σ_i is a best reply of player i if for any other strategy σ′ of hers, U_i((σ_i, σ_−i), t_i) ≥ U_i((σ′, σ_−i), t_i) for each of her types t_i ∈ T_i . A Bayes–Nash equilibrium or a Bayesian equilibrium is a profile of Bayesian strategies σ^∗ = (σ_i∗)_{i ∈ I} such that σ_i∗ is a best reply against σ_−i∗ for every playeri ∈ I .

A simple, discrete variant of an example by Gale (1996) may clarify these abstract definitions. There are two investors i = 1, 2 and three possible states of nature k ∈ K = {−1, 0, 1} . Each investor i only knows her own type

$$ {t}_i\in {T}_i=\left\{-10,-6,-2,2,6,10\right\}. $$

Every type t_i of investor i believes that all of the other investor’s types t_j ∈ T_j , j ≠ i , are equally likely, so that each of them has probability $\frac{1}{6}$. Moreover, every type t_i believes that the state of nature is k = 1 when t_i + t_j > 0 ; that the state of nature is k = 0 whent_i + t_j = 0; and that the state of nature is k = − 1 whent_i + t_j < 0 . Formally, the belief λ_i(t_i) of type t_i ∈ T_i is defined by

$$ {\lambda}_i\left({t}_i\right)\left(k,{t}_j\right)=\left\{\begin{array}{cc}\hfill \frac{1}{6}\hfill & \hfill k\kern0.62em \mathrm{has}\kern0.62em \mathrm{the}\kern0.62em \mathrm{same} \operatorname {sign}\kern0.62em \mathrm{as}\kern0.62em {t}_i+{t}_j\hfill \\ {}\hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right. $$

The investors cannot communicate their types to one another. They can invest in at most one of two available investment periods. Each investor has three relevant strategies: invest immediately, in the first period; wait to the second period and invest only if the other investor has invested in the first period; or never invest. The payoff of each of the investors depends on the state of nature k ∈ K = {−1, 0, 1} and on her own investment strategy, but not on the investment strategy of the other investor. The payoffs are as follows:

• Investing immediately when the state of nature is k yields investor i a payoff of k

  $$ {u}_i^k\left({}^{`}{immediately}^{'},\cdotp \right)=k $$

(The · stands for the investment decision of the other investor j ≠ i , which, as we said, does not effect the payoff of investor i.)

• If investor i chooses to wait to the second period and invest only if the other investor has invested in the first period, investor i’s payoff in the state of nature k is

  $$ {u}_i^k\left({}^{`}{wait}^{'},\cdotp \right)=-\frac{3}{4}k. $$

• If the investor never invests, her payoff is 0 irrespective of the state of nature:

  $$ {u}_i^k\left({}^{`}{never}^{'},\cdotp \right)=0. $$

How will the different types behave at a Bayesian equilibrium? The type t_i = 10 assesses that by investing immediately her expected payoff is

$$ Ui\left({}^{`}{immediately}^{'},10\right)=\frac{1}{6}\times 0+\frac{5}{6}\times 1=\frac{5}{6} $$

(immediate investment yields 0 in case t_j = −10, and yields 1 in case t_j = − 6, − 2, 2, 6, 10). This is higher than $\frac{3}{4}$, the maximum payoff she could possibly get by waiting for the second period, and higher than the payoff 0 of never investing. So at a Bayesian equilibrium

$$ {\sigma}_i^{\ast }\ (10)={}^{`}{immediately}^{'},\kern0.48em i=1,2. $$

Next, the expected payoff to the type t_i = 6 from immediate investment is

$$ Ui\left({}^{`}{immediately}^{'},6\right)=\frac{1}{6}\times \left(-1\right)+\frac{1}{6}\times 0+\frac{4}{6}\times 1=\frac{1}{2} $$

(immediate investment yields 1 unless t_j = − 10, in which case the payoff is − 1, or t_j = − 6, in which case the payoff is 0). So investing immediately is preferred for her over never investing. But how about waiting until the second period? That’s an inferior option as well, since the types t_j = − 10, − 6, − 2 will never invest in the first period (this would yield them a negative expected payoff). So only the positive types t_j = 2,6,10 could conceivably invest immediately, with overall probability reaching at most $\frac{3}{6}$. So waiting to see if they invest yields to the type t_i = 6 an expected payoff not higher $\frac{3}{6}\times \frac{3}{4}=\frac{3}{8}$, which is smaller than $\frac{1}{2}$. We conclude that the preferable strategy of t_i = 6 at equilibrium is

$$ {\sigma}_i^{\ast }\ (6)={}^{`}{immediately}^{'},\kern0.48em i=1,2. $$

What about t_i = 2? Immediate investment yields her

$$ Ui\left({}^{`}{immediately}^{'},2\right)=\frac{2}{6}\times \left(-1\right)+\frac{1}{6}\times 0+\frac{3}{6}\times 1=\frac{1}{6} $$

(− 1 is the payoff when t_j = − 10, − 6; 0 is the payoff when t_j = − 2; the payoff is 1 otherwise). However, given that the types t_j = 6,10 invest immediately at equilibrium, and that the negative types t_j = − 10, − 6, − 2 do not invest immediately, the type t_i = 2 figures out that by waiting and investing only if the other investor has invested first would yield her an expected payoff

$$ Ui\left({}^{`}{wait}^{'},2\right)\ge \frac{2}{6}\times \frac{3}{4}=\frac{1}{4}>\frac{1}{6} $$

($\frac{2}{6}$ is the probability assigned by t_i = 2 to the event that t_j ∈ {6, 10} and hence j invests immediately, and $\frac{3}{4}$ is the payoff from the second period investment). Thepreferred strategy of t_i = 2 at equilibrium is therefore

$$ {\sigma}_i^{\ast }(2)={}^{`}{wait}^{'},\kern0.48em i=1,2. $$

We can now compute inductively, in a similar way, that also

$$ {\sigma}_i^{\ast}\left(-2\right)={}^{`}{wait}^{'},\kern0.48em i=1,2{\sigma}_i^{\ast }\ \left(-6\right)={}^{`}{wait}^{'},\kern0.48em i=1,2 $$

and that

$$ {\sigma}_i^{\ast }(10)={}^{`}{never}^{'},\kern0.48em i=1,2. $$

Notice that the equilibrium in the example is inefficient. For instance, when the pair of types is (t₁,t₂) = (2, 2) the investment is profitable, but both investors wait to see if the other one invests, and thus end up not investing at all. In this case, behaviour would become efficient if the investors could communicate their types to each other. Indeed, they would have been happy to do so, because their interests are aligned.

Obviously, there are other strategic situations with incomplete information in which the interests of the players are not completely aligned. For example, a potential seller of an object would like to strike a deal with a potential buyer at a price which is as high as possible, while the potential buyer would like the price to be as low as possible. That’s why the traders might not volunteer to communicate honestly their private valuations of the object, even if they are technically able to do so. Still, in case the buyer values the object more than the seller, they would both prefer to trade at some price in-between their valuations rather than forgoing trade altogether. Therefore, the traders would nevertheless like to avoid a complete lack of communication. Myerson and Satterthwaite (1983) phrase general conditions under which no Bayesian equilibrium of any trade mechanism is ever fully efficient due to this tension between interests alignment and interests mismatch. Under these conditions, even if the traders are able to communicate their private information, at no Bayesian equilibrium does trade take place in all instances in which there exist gains from trade.

In the above variant of Gale’s example we were able to find the unique Bayesian equilibrium using iterative dominance arguments. We have iteratively crossed out strategies that are inferior for some types, which enabled us to eliminate inferior strategies for other types, and so forth. As in games of complete information, this technique is not applicable in general, and there are games with incomplete information in which a Bayesian equilibrium is not the outcome of any process of iterative elimination of dominated strategies (Battigalli and Siniscalchi 2003; Dekel et al. 2007).

Games with incomplete information are discussed in many game theory textbooks (for example, Dutta 1999; Gibbons 1992; Myerson 1991; Osborne 2003; Rasmusen 1989; Watson 2002). Aumann and Heifetz (2002), Battigalli and Bonanno (1999) and Dekel and Gul (1997) are advanced surveys.

See Also

• Epistemic Game Theory: An Overview

• Epistemic Game Theory: Beliefs and Types

• Epistemic Game Theory: Complete Information

• Game Theory