Lec2-10.pdf

Lecture 10: Incomplete Information Games

Yosuke YASUDA

Osaka University, Department of Economics

[email protected]

January 6, 2015

Incomplete Information

Many strategic settings are interesting because players have different information (asymmetric information) at various junctures in a game.

Our analysis in preceding lectures covers strategic settings in which there is asymmetric information only regarding players’ actions.

In what follows, we will introduce a framework that can analyze broader settings in which players have private information about other things than players’ actions.

In a game of incomplete information, at least one player is uncertain about what other players know, i.e., some of the players possessprivate information, at the beginning of the game.

Ex For example, a firm may not know the cost of the rival firm, a bidder does not know her competitors’ valuations in an auction.

Bayesian Games

Following Harsanyi (1967), we can translate any game of incomplete information into a Bayesian game in which a Nash equilibrium is naturally extended to a Bayesian Nash equilibrium:

(1) Nature draws atype vector

t(=t1× · · · ×tn)∈T(=T1× · · · ×Tn), according to a prior

probability distribution p(t).

(2) Nature revealsi’s type to playeri, but not to any other player.

(3) The players simultaneously choose actions ai ∈Ai for

i= 1, ..., n.

(4) Payoffsui(a;ti) fori= 1, .., n are received.

By introducing the fictional moves by “nature” in steps (1) and (2), we have described a game of incomplete information as a game of imperfect information: in step (3) some of the players do not know the complete history of the game, i.e., which actions (types) of other players were chosen by nature.

Cournot Game with Unknown Cost (1)

Firm 1’s marginal cost is constant (c1), while firm 2’s marginal

cost is private information:

high (cH₂ ) with probability θ, or low (cL₂) with prob. 1−θ.

Assume each firm tries to maximize an expected profit given this information structure of the game.

Different types (of player 2) as separate players.

Firm 1’s strategy is a quantity choice, but firm 2’s strategy is to specify her quantity choice in each possible marginal cost.

Letq₂H(=q2(cH₂ ))andq₂L(=q2(cL₂))be the quantity selected by player 2 for each realization of the cost. Then, the optimization problem for each player is described as follows:

max

q1

θπ1(q1, q2H) + (1−θ)π1(q1, q2L)

max

qH

2

π2(q1, q₂H), and max

qL

2

π2(q1, qL₂).

Cournot Game with Unknown Cost (2)

Assuming a linear (inverse) demand,p=a−(q1+q2), the profit

function becomes

πi(q1, q2) = [a−(q1+q2)−ci]qi for i= 1,2, i6=j.

Putting this profit function into the above optimization problems,

dπ1 dq1

=θ[a−2q1−q2H −c1] + (1−θ)[a−2q1−q2L−c1] = 0.

dπ2

dqH₂ =a−q1−2q

H

2 −cH2 = 0, dπ2

dq₂L =a−q1−2q

L

2 −cL2 = 0.

Solving the simultaneous equations give us the following (Bayesian) Nash Equilibrium:

q₁∗ = 1

3[a−2c1+θc

H

2 + (1−θ)cL2].

q∗₂(cH₂ ) = 1 3[a−2c

H

2 +c1] +

1−θ

6 (c

H

2 −cL2).

q₂∗(cL₂) = 1 3[a−2c

L

2 +c1]− θ

6(c

H

Bayesian Nash Equilibrium (1)

Note that firm 2 will produce more (/less) than she would in the complete information case with high (/low) cost, since firm 1 does not take the best response to firm 2’s actual quantity but

maximizes his expected profit.

A (pure) strategy for playeriis a complete action plan

si(ti) :Ti →Ai, which specifies her action for each of her

possible type.

A belief about other players’ types is a conditional probability distribution of other players’ types given the player’s

knowledge of her own type pi(t−i|ti).

When nature reveals ti to player i, she can compute the belief

pi(t−i|ti) using Bayes’ rule:

pi(t−i|ti) =

p(t−i, ti)

p(ti)

= P p(t−i, ti) t−i∈T−ip(t−i, ti)

.

Bayesian Nash Equilibrium (2)

In a Bayesian game, the strategiess∗ = (s∗₁, ..., s∗_n) are a (pure-strategy)Bayesian Nash equilibrium (BNE) if for each playeriand for each ofi’s types ti in Ti,s∗i(ti)solves:

max

ai∈Ai

X

t−i∈T−i

ui(s∗1(t1), . . . , s∗i−1(ti−1), ai, s∗i+1(ti+1),

. . . , s∗_n(tn);t)pi(t−i|ti).

In spite of the notational complexity of the definition, the central idea is both simple and familiar:

Each player’s strategy given her type must be a best response to the other players’ strategies (in expectation).

A BNE is simply a Nash equilibrium in a Bayesian game when each type of every player is treated as separate player.

Simple Example

Ex The nature selects Awith prob. 1/2and B with prob. 1/2. Before the players select their actions, player1 observes nature’s choice, but player2 does not know it. Then, what is the BNE?

12 L R

U 1,1 0,0

D 0,0 2,2

A

12 L R

U0 0,1 1,0

D0 2,0 0,2

B

There is a unique Bayesian Nash equilibrium in which player 1 choosesDU0 and player 2 chooses R. Note that the best reply function for each player is derived as follows:

R1(L) =U D0, R1(R) =DU0.

R2(U U0) =L, R2(U D0) =R, R2(DU0) =R, R2(DD0) =R.

Clearly,(DU0, R)is a unique combination of mutual best responses, i.e., a (Bayesian) Nash equilibrium.

Bilateral Trade: Model

Consider the following bilateral trade with incomplete information.

There are a buyer and a seller whose valuation of the good are denoted by vb andvs, respectively.

These valuations are private information and are drawn from independent uniform distributions on [0, 1].

The seller names an asking price, ps∈R+, and the buyer

simultaneously names an offer price,pb∈R+.

Ifpb≥ps, then trade occurs at the average price,p= pb+ps

2 .

The associated payoffs becomevb−pandp−vsin this case. Ifpb< ps, then no trade occurs. Both players receive 0 payoff.

Rm Each of these utility functions measures thechange in the player’s utility. If there is no trade, then there is no change in utility. It would makeno difference to define, say, the seller’s utility to bepif there is trade at price pand vs if there is no trade.

Bilateral Trade: Equilibrium Conditions

A pair of strategies (pb(vb), ps(vs)) is a BNE if the following two

conditions hold. For eachvb∈[0,1],pb(vb) solves

max

pb

(vb−E[p|pb ≥ps(vs)]) Pr{pb≥ps(vs)}

⇒max

pb

vb−

pb+E[ps|pb ≥ps(vs)]

2

Pr{p_b ≥ps(vs)}

whereE[ps|pb ≥ps(vs)]is the expected price the seller will

demand, conditional on the demand being less than the buyer’s offer ofpb. For eachvs∈[0,1],ps(vs) solves

max

ps

(E[p|ps≤pb(vb)]−vs) Pr{ps≤pb(vb)}

⇒max

ps

ps+E[pb |ps≤pb(vb)]

2 −vs

Pr{ps≤pb(vb)}

Bilateral Trade: Linear Equilibrium

Suppose in a BNE, both players take increasing strategies:

pb(vb) =ab+cbvb

ps(vs) =as+csvb.

whereab, as≥0 andcb, cs >0.

Rm We arenot restricting the players’ strategy spaces to include only linear strategies. We allow the players to choosearbitrary

strategies but ask whether there is an equilibrium that is linear.

Solving the maximization problems (seeGibbons, section 3.2.C),

pb(vb) =

1 12 +

2 3vb

ps(vs) =

1 4 +

2 3vb.

are derived as a BNE. That is,ab =

1

12, as = 1

4, cb =cs = 2 3.

Revelation Principle

The revelation principle, due to Myerson (1979) and others is an important tool for designing games (or mechanisms) when the players have private information.

Definition 2

Adirect mechanism is a static Bayesian game in which each player’s only action is to submit a message (mi ∈Mi) about her

type. That is, strategy space satisfiesMi =Ti for every player i.

Theorem 3 (Revelation Principle)

Any BNE (of any Bayesian game) can be attained by a truth-telling BNE of some direct mechanism.

Rm When no direct mechanism can achieve some outcome in a truth-telling BNE, then there exists no mechanism (no matter how it were general or complicated) that can achieve the outcome.

Revelation Principle: Proof

Lets∗:T →Abe the BNE of the original Bayesian game. Consider the direct mechanism which selects the corresponding equilibrium outcome given reported types.

The outcome of the direct mechanism is set equal tos∗(m)

for any combination of revealed types of the players m∈M. Then, it is easy to show that truth-telling,mi=ti for all i,

must be a BNE of this direct mechanism. Suppose not, then for somei, there exists an action

a0_i =s∗_i(t0_i)6=s∗_i(ti) such that

X

t−i∈T−i

ui(a0i, s∗−i(t−i);ti)pi(t−i|ti)

> X

t−i∈T−i

ui(s∗i(ti), s∗−i(t−i);ti)pi(t−i|ti),