講義 8：不完備情報とメカニズムデザイン
上級ミクロ経済学—財務省理論研修
安田 洋祐
政策研究大学院大学（GRIPS）
2012年6月18日
Incomplete Information | 不完備情報
In many strategic settings, players have different information (asymmetric information (非対称情報)) 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 (不完備情報), at least one player is uncertain about what other players know, i.e., some of the players possess private information (私的情報), 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.
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Bayesian Games | ベイジアンゲーム
Following Harsanyi (1967), we can translate any game of incomplete information into a Bayesian game (ベイジアンゲーム) in which a Nash equilibrium is naturally extended to a Bayesian Nash equilibrium:
(1) Nature (自然) draws a type (タイプ) 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.
Cournot Game with Unknown Cost | コストが不明のクールノーゲーム (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 qH2 (= 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, q2H) + (1 − θ)π1(q1, qL2) max
qH2
π2(q1, qH2), and max
qL2
π2(q1, q2L).
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Cournot Game with Unknown Cost | コストが不明のクールノーゲーム (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− q H
2 − c1] + (1 − θ)[a − 2q1− qL2 − c1] = 0. dπ2
dq2H = a − q1− 2q
H 2 − c
H 2] = 0,
dπ2
dqL2 = a − q1− 2q
L 2 − c
L 2] = 0.
Solving the simultaneous equations give us the following (Bayesian) Nash Equilibrium:
q1∗= 1
3[a − 2c1+ θc
H
2 + (1 − θ)c L 2].
q2∗(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).
Bayesian Nash Equilibrium | ベイジアンナッシュ均衡 (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 (信念) 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 tito player i, she can compute the belief pi(t−i|ti) using Bayes’ rule (ベイズルール):
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 | ベイジアンナッシュ均衡 (2)
Def In a Bayesian game, the strategies s∗= (s∗1, ..., s∗n) are a (pure-strategy) Bayesian Nash equilibrium (ベイジアンナッシュ均衡, BNE) if for each player i and for each of i’s types tiin 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 must be a best response to the other players’ strategies.
◮ That is, a Bayesian Nash equilibrium is simply a Nash equilibrium in a Bayesian game.
Revelation Principle | 顕示原理
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 (直接メカニズム) 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 Any Bayesian Nash equilibrium (of any Bayesian game) can be attained by a truth-telling Bayesian Nash equilibrium of some direct mechanism.
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Revelation Principle: Proof | 顕示原理：証明
Let s∗: T → A be the BNE 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= tifor 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.
Applications | 応用
✄
✂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?
1 2 L R
U 1, 1 0, 0
D 0, 0 2, 2
A
1 2 L R
U′ 0, 1 1, 0 D′ 2, 0 0, 2
B
There is a unique BNE 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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Auctions | オークション
Imagine that there is a (potential) seller who has a painting that is worth nothing to her personally. She hopes to make some money by selling the art through an auction.
◮ Suppose there are two potential buyers, called bidders 1 and 2.
◮ Let v1and v2denote the valuations of the two bidders.
◮ If bidder i wins the painting and has to pay x for it, then her payoff is vi− x.
◮ where v
1and v2are chosen independently by nature, and
◮ each of which is uniformly distributed between 0 and 1.
The bidders observe their own valuations before engaging in the auction. The seller and the rival do not observe a bidder’s valuation; they only know the distribution.
Let us analyze two prominent sealed-bid auctions: a first-price auction (1位価 格オークション) and a second-price auction (2位価格オークション).
◮ The former is commonly used while the latter has great theoretical
importance and started to be applied in reality. 11 / 15
First Price Auction | 1 位価格オークション (1)
Bidders simultaneously and independently submit bids b1and b2.
◮ The painting is awarded to the highest bidder i∗with max bi ,
◮ who must pay her own bid, b∗i.
To derive a Bayesian Nash equilibrium, we assume the bidding strategy in equilibrium is i) symmetric, and ii) linear function of vi. That is, in equilibrium, player i chooses
bi= c + θvi. (1)
Now suppose that player 2 follows the above equilibrium strategy, and we shall check whether player 1 has an incentive to choose the same linear straetegy (1). Player 1’s optimization problem, given she received a valuation v1, is
max
b1
(v1− b1) Pr{b1> b2}. (2)
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First Price Auction | 1 位価格オークション (2)
Since v2is uniformly distributed on [0, 1] by assumption, we obtain Pr{b1> b2} = Pr{b1> c + θv2}
= Pr{b1− c
θ > v2} = b1− c
θ .
The first equality comes from the linear bidding strategy (1), the third equality is from the uniform distribution. Substituting it into (2), the expected payoff becomes a quadratic function of b1.
maxb1 (v1− b1)
b1− c θ
Taking the first order condition, we obtain du1
db1
= 1
θ[−2b1+ v1+ c] = 0 ⇒ b1=c 2+
v1
2. (3)
Comparing (3) with (2), we can conclude that c = 0 and θ =12 constitute a Bayesian Nash equilibrium.
Second Price Auction | 2 位価格オークション
Bidders simultaneously and independently submit bids b1and b2.
◮ The painting is awarded to the highest bidder i∗with max bi ,
◮ at a price equal to the second-highest bid, maxj6=i∗bj.
Unlike the first-price auction, there is a weakly dominant strategy for each player in this game.
◮ b
i= viweakly dominates all other possible strategies.
Since the combination of weakly dominant strategies always becomes a Nash equilibrium, bi= vifor all i is a BNE.
✄
✂Rm Note that there are other equilibria. For example, v✁ 1= 1 and v2= 0 constitute a Bayesian Nash equilibrium.
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Revenue Equivalence Theorem | 収入同値定理
As we derived, the two sealed-bid auctions, first-price and second-price auctions, have different equilibrium bidding strategies.
Then, the natural question perhaps is “which auction can yield higher (expected) revenue?”. When you calculate the expected revenue in the above examples, you will find both revenues coincide.
Interestingly, this is not by chance; the revenue equivalence result (often called as “revenue equivalence theorem”) is known to hold in much more general situations. See auction textbooks or related survey articles for the formal analysis on revenue equivalence theorem.
