Lecture 10: Incomplete Information Games
Advanced Microeconomics II
Yosuke YASUDA
Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp
January 6, 2015
1 / 13
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.
✞
✝
☎
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.
2 / 13
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.
3 / 13
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. 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
π2(q1, q2H), and max
qL2
π2(q1, qL2).
4 / 13
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 − cL2). q2∗(cL2) = 1
3[a − 2c
L
2 + c1] −
θ 6(c
H 2 − c
L
2). 5 / 13
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)
.
6 / 13
Bayesian Nash Equilibrium (2)
Definition 1
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.
7 / 13
Simple Example
✞
✝
☎
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.
8 / 13
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.
✞
✝
☎
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.
9 / 13
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)}
10 / 13
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.
✞
✝
☎
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.
11 / 13
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
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. 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.
12 / 13
Revelation Principle: Proof
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 that s∗ is a BNE of the original game. 13 / 13