Lecture 10: Incomplete Information Games
Advanced Microeconomics II
Yosuke YASUDA
National Graduate Institute for Policy Studies
January 16, 2014
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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 possess private information, at the beginning of the game.
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✝
☎
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.
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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 a type vector
t(= t1× · · · × tn) ∈ T (= T1× · · · × Tn), according to a prior probability distribution p(t).
(2) Nature reveals i’s type to player i, but not to any other player. (3) The players simultaneously choose actions ai ∈ Ai for
i = 1, ..., n.
(4) Payoffs ui(a; ti) for i = 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.
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Cournot Game with Unknown Cost (1)
Firm 1’s marginal cost is constant (c1), while firm 2’s marginal cost is private information:
◮ high (cH
2 ) 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. Let q2H(= q2(cH2 )) and q2L(= 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:
maxq1 θπ1(q1, q H
2 ) + (1 − θ)π1(q1, q2L) max
qH2 π
2(q1, q2H), and max
qL2 π
2(q1, qL2).
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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, i 6= 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 − c
L 2).
q2∗(cL2) = 1 3[a − 2c
L
2 + c1] −θ 6(c
H 2 − cL2).
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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 player i is 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(t−i, ti) P
t−i∈T−ip(t−i, ti).
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Bayesian Nash Equilibrium (2)
Def In a Bayesian game, the strategies s∗= (s∗1, ..., s∗n) are a (pure-strategy) Bayesian Nash equilibrium (BNE) if for each player i and for each of i’s types ti in Ti, s∗i(ti) solves:
amaxi∈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.
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Simple Example
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Ex The nature selects A with prob. 1/2 and B with prob. 1/2.✆ Before the players select their actions, player 1 observes nature’s choice, but player 2 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
U′ 0, 1 1, 0 D′ 2, 0 0, 2
B
There is a unique Bayesian Nash equilibrium in which player 1 chooses DU′ and player 2 chooses R. Note that the best reply function for each player is derived as follows:
R1(L) = U D′, R1(R) = DU′.
R2(U U′) = L, R2(U D′) = R, R2(DU′) = R, R2(DD′) = R. Clearly, (DU′, R) is a unique combination of mutual best
responses, i.e., a (Bayesian) Nash equilibrium.
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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 and vs, 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+.
◮ If pb≥ ps, then trade occurs at the average price, p =pb+ ps 2 . The associated payoffs become vb− p and p − vsin this case.
◮ If pb< ps, then no trade occurs. Both players receive 0 payoff.
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Rm Each of these utility functions measures the change in the✆ player’s utility. If there is no trade, then there is no change in utility. It would make no difference to define, say, the seller’s utility to be p if there is trade at price p and vs if there is no trade.
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Bilateral Trade: Equilibrium Conditions
A pair of strategies (pb(vb), ps(vs)) is a BNE if the following two conditions hold. For each vb∈ [0, 1], pb(vb) solves
maxpb
(vb− E[p | pb ≥ ps(vs)]) Pr{pb≥ ps(vs)}
⇒ max
pb
vb−pb+ E[ps| pb ≥ ps(vs)] 2
Pr{pb ≥ ps(vs)} where E[ps| pb ≥ ps(vs)] is the expected price the seller will demand, conditional on the demand being less than the buyer’s offer of pb. For each vs∈ [0, 1], ps(vs) solves
maxps
(E[p | ps≤ pb(vb)] − vs) Pr{ps≤ pb(vb)}
⇒ max
ps
ps+ E[pb | ps≤ pb(vb)]
2 − vs
Pr{ps≤ pb(vb)}
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Bilateral Trade: Linear Equilibrium
Suppose in a BNE, both players take increasing strategies: pb(vb) = ab+ cbvb
ps(vs) = as+ csvb.
where ab, as ≥ 0 and cb, cs > 0.
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Rm We are not restricting the players’ strategy spaces to include✆ only linear strategies. We allow the players to choose arbitrary strategies but ask whether there is an equilibrium that is linear. Solving the maximization problems (see Gibbons, 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.
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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.
Def A direct 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 satisfies Mi = Ti for every player i. Thm (Revelation Principle) Any BNE (of any Bayesian game) can be attained by a truth-telling BNE of some direct mechanism.
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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. In light of the revelation principle, we can restrict our attention to direct mechanisms when searching for some desirable mechanism.
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Revelation Principle: Proof
Let s∗: T → A be 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 to s∗(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 some i, there exists an action a′i = s∗i(t′i) 6= s∗i(ti) such that
X
t−i∈T−i
ui(a′i, 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), which contradicts to the assumption that s∗ is the Bayesian equilibrium of the original game.
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