Research Methods in Political Science Formal Political Theory

Graduate School of Political Science at Waseda University Spring Semester, 2013

Weeks 9-11: “Static Games of Incomplete Information: Beliefs, Conditional Probability, and Bayes’ Rule”

Shuhei Kurizaki Information Structure

In games of imperfect information At some decision node (or info. set), players do not know the complete history of the game.

In games of complete information Players know each other’s payoff functions.

In games of incomplete information Players may not possess full information about the structure of a game. Players may not know other players’ utility functions. Utility functions may be private information. Example: Incomplete Information Game

Two politicians consider whether to run for an election: An incumbent (Player 1) and a potential challenger (Player 2) Player 2 is uncertain about player 1’s costs of electoral campaign.

Player 2 Player 2 Enter DontEnterDont Player 1 Enter 0, −1 2, 0 Enter 1.5, −1 3.5, 0 Dont 2, 1 3, 0 Dont 2, 1 3, 0 High Cost Low Cost Example: Incomplete Information Game

Player 2 Player 2 Enter DontEnterDont Player 1 Enter 0, −1 2, 0 Enter 1.5, −1 3.5, 0 Dont 2, 1 3, 0 Dont 2, 1 3, 0 High Cost Low Cost

Player 1’s strategic problems: If high cost, “Dont” is a dominant strategy for Player 1. If low cost, the optimal strategy depends on his expectation about Player 2’s choice. That is, Player 1’s BR changes depending on its campaign costs, or its type. Example: Incomplete Information Game

Player 2 Player 2 Enter DontEnterDont Player 1 Enter 0, −1 2, 0 Enter 1.5, −1 3.5, 0 Dont 2, 1 3, 0 Dont 2, 1 3, 0 High Cost Low Cost

Player 2’s strategic problems: Entering is profitable only if Player 1 does not enter. But Player 2 does not know whether Player 1 faces a high cost or a low cost. (Player 1’s cost is private information). That is, Player 2 is incompletely informed of Player 1’s type. Player 2’s BR takes into account its belief about Player 1’s type. Analyzing Incomplete Information

Two Key Items in Bayesian Games Knowing one’s own payoff functions is equivalent to knowing its own type. We use the probability distribution, the prior probability,to denote players’ beliefs about other player’s types. Analyzing Incomplete Information

Harsanyi Transformation John Harsanyi’s trick to study incomplete information games. Transform the game of incomplete information into a game of imperfect information. Introduce a prior move by Nature that determines Player 1’s type (i.e., its cost). Player 1 observes Nature’s move but Player 2 can’t But Player 2 knows the probability of Nature’s move. Player 2’s incomplete information about player 1’s type becomes Player 2’s imperfect information about Nature’s move. Analyzing Incomplete Information

Nature

High Cost Low Cost (p) (1 - p) 1 1 Enter Don’t Enter Don’t 2

Enter Don’t Enter Don’t Enter Don’t Enter Don’t

0, -1 2, 0 2, 13, 0 1.5, -1 3.5, 0 2, 1 3, 0 Analyzing Incomplete Information

Common to assume that both players have the same prior beliefs about the probability distribution on Nature’s move. In this case, Nature selects a type for Player 1 according to the prior distribution (i.e., the prior probability that Player 1 is of the high-cost type is p). Posterior beliefs: Beliefs updated in light of new information accumulated in a game. ⇒ This becomes more relevant with sequential moves, where players can send signals and update their prior beliefs. Analyzing Incomplete Information

We can solve this game for Nash equilibrium: Bayesian Nash Equilibrium. One strategy for Player 2. One strategy for Player 1 of the high-cost type. One strategy for Player 1 of the low-cost type. ⇒ Player 1 makes a plan (or has a strategy) for each type: Type-contingent strategies. Bayesian Nash Equilibrium is to add types and beliefs to Nash Equilibrium. Analyzing Incomplete Information

Best response for Player 1 of the high-cost type: Dominant strategy is to not enter the competition. Analyzing Incomplete Information

Best response for Player 1 of the low-cost type: Strictly prefers entering to not entering if

EU1(Enter|L) > EU1(Not|L) 1.5y +3.5(1 − y) > 2y +3(1− y) y < 1/2

where y denotes Player 2’s probability of entry. Analyzing Incomplete Information

Best response for Player 2: Strictly prefers entering to not entering if

EU2(E) > EU2(Don t) | − | > | − | pEU2(E H) + (1 p)EU 2(E L) pEU2(Don t H) + (1 p)EU2(Don t L) High−type Low−type High−type Low−type p(1) + (1 − p)[x(−1) + (1 − x)(1)] > 0 1 − 2x − 2px > 0 1 x < 2(1 − p) where x denotes the probability of entry for the low-cost Player 1 and p denotes the prior probability that Player 1 is of the high-cost type. Analyzing Incomplete Information

Best responses for the low-cost Player 1 and Player 2: ⎧ ∗ ⎨⎪0ify > y ∗ BR1(y|L)=x = [0, 1] if y = y ⎩⎪ 1ify < y ∗

where y ∗ =1/2. ⎧ ∗ ⎨⎪0ifx > x ∗ BR2(x)=y = [0, 1] if x = x ⎩⎪ 1ifx < x∗

∗ 1 where x = 2(1−p) . Analyzing Incomplete Information

y 1 BR2(x)

y* =1/2

BR1(y|L)

x 0 x* = 1/2(1 - p) 1 Analyzing Incomplete Information

Bayesian Nash Equilibrium to Electoral Competition Game Three Bayesian Nash Equilibria, where High-cost Player 1, Low-cost Player 1, and Player 2’s respective equilibrium strategies are: (Dont, Enter; Dont): Separating strategy (Dont, Dont; Enter): Pooling strategy ∗ 1 ∗ 1 (Don t, x = 2(1−p) ; y = 2 ): Semi-separating strategy But wait!

Why a strategy specifies an action for each possible type? See Gibbons’s (1991, 150) discussion. Another Way to Find the Equilibrium

Player 2 Player 2 Enter DontEnterDont Player 1 Enter 0, −1 2, 0 Enter 1.5, −1 3.5, 0 Dont 2, 1 3, 0 Dont 2, 1 3, 0 High Cost p Low Cost (1 − p)

Player 2 Enter Dont Enter, Enter p(0) + (1 − p)1.5, −1 2p +(1 − p)3.5, 0 Enter, Dont p(0) + (1 − p)2, p(−1) + (1 − p)1 2p − (1 − p)3, 0 Dont, Eenter 2p +(1 − p)1.5, p(1) + (1 − p)(−1) 3p +(1 − p)3.5, 0 Dont, Dont 2, 1 3, 0

Too cumbersome → Next slide simplifies the payoffs Another Way to Find the Equilibrium

Player 2 Enter Dont Player 1 Enter, Enter 1.5 − 1.5p, −1 3.5 − 1.5p, 0 Enter, Dont 2 − 2p, 1 − 2p 3 − p, 0 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0

Recall Dont strictly dominates Enter for the high-cost type.

Player 2 Enter Dont Player 1 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0 Another Way to Find the Equilibrium

Player 2 Enter Dont Player 1 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0

If Player 2 chooses Enter, Player 1’s BR is to choose Dont, Dont regardless of p < 1. Hence, Dont, Dont; Enter is a pooling NE. Another Way to Find the Equilibrium

Player 2 Enter Dont Player 1 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0

If Player 2 chooses Enter, Player 1’s BR is to choose Dont, Dont regardless of p < 1. Hence, Dont, Dont; Enter is a pooling NE. For Player 2, Enter dominates Dont if 2p − 1 ≥ 0,or 1 p ≥ . 2 1 So, Don t, Enter; Don t is a separating NE if p ≥ 2 . Another Way to Find the Equilibrium

Player 2 Enter Dont Player 1 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0

In equilibrium, Player 1 mixes so that

u2(Enter)=u2(Don t) σ1(2p − 1) + (1 − σ1)(1) = 0 1 σ1 = . 2(1 − p) Another Way to Find the Equilibrium

Player 2 Enter Dont Player 1 Dont, Enter 1.5+0.5p, 2p − 1 3.5 − 0.5p, 0 Dont, Dont 2, 1 3, 0

In equilibrium, Player 2 mixes so that

u1(Don t, Enter)=u2(Don t, Don t) σ2(1.5+0.5p)+(1− σ2)(3.5 − 0.5p)=σ2(2) + (1 − σ2)(3) 1 σ2 = . 2 Another Way to Find the Equilibrium

Taken together, in addition to two pure-strategy Nash equilibria, there is a mixed-strategy Nash equilibrium, in which , σ 1 Player 1 plays Don t Enter with probability 1 = 2(1−p) . 1 Player 2 plays Enter with probability σ2 = 2 . 1 whenever p < 2 . This is a semi-separating case. Bayesian Nash Equilibrium with Continuous Types

The previous case involves only two types: high-cost type and low-cost type. What about continuous types? Consider the Battle of the Sexes Game with the following modifications: Two-sided incomplete information. Each player has a continuum of types. Player 2 Boxing Nutcracker Player 1 Boxing 2+θ1, 1 0, 0 Nutcracker 0, 0 1, 2+θ2 Bayesian Nash Equilibrium with Continuous Types

Player 2 Boxing Nutcracker Player 1 Boxing 2+θ1, 1 0, 0 Nutcracker 0, 0 1, 2+θ2

Two-Sided Incomplete Information Player 1 is unsure about Player 2’s payoff for coordination on Nutcracker. θ1 ∈ [0, x] is private information for Player 1 Player 2 is unsure about Player 1’s payoff for coordination on Boxing. θ2 ∈ [0, x] is private information for Player 2. Bayesian Nash Equilibrium with Continuous Types

Prior Beliefs For simplicity, we assume that Nature draws (independently) both θ1 and θ2 from a uniform distribution with the support of [0, x] Player 1 believes that θ2 takes a certain value is given by

p1(θ2)=1/x

Player 2 believes that θ2 takes a certain value is given by

p2(θ1)=1/x Bayesian Nash Equilibrium with Continuous Types

We look for a Bayesian equilibrium in which... Player 1 chooses B if θ1 exceeds some critical value x1 and chooses N otherwise. Probability that player 1 chooses B is given by

x1 σ1(θ1)=Pr(θ1 > x1)=1− Pr(θ1 ≤ x1)=1− x

Player 2 chooses N if θ2 exceeds some critical value x2 and chooses B otherwise. Probability that player 2 chooses N is given by

x2 σ2(θ2)=Pr(θ2 > x2)=1− Pr(θ2 ≤ x2)=1− x Bayesian Nash Equilibrium with Continuous Types

Given σ2(θ2), Player 1’s expected payoffs are:

x2 EU1(B)=(1− σ2(θ2))(2 + θ1)+σ2(θ2)(0) = (2 + θ1) x x2 EU1(N)=(1− σ2(θ2))(0) + σ2(θ2)(1) = 1 − x Since B is optimal for Player 1 iff its expected payoff exceeds the expected payoff of N in equilibrium, it must be that

EU1(B) ≥ EU1(N) x2 x2 (2 + θ1) ≥ 1 − x x x θ1 ≥ 1 − − 3 ≡ x1. x2 Bayesian Nash Equilibrium with Continuous Types

Similarly, given σ1(θ1), Player 2’s expected payoffs are:

x1 EU2(B)=(1− σ1(θ1))(0) + σ1(θ1)(1) = 1 − x x1 EU2(N)=(1− σ1(θ1))(2 + θ2)+σ1(θ2)(0) = (2 + θ2) x Since N is optimal for Player 2 iff

EU2(N) ≥ EU2(B) x1 x1 (2 + θ2) ≥ 1 − x x x θ2 ≥ − 3 ≡ x2. x1 Bayesian Nash Equilibrium with Continuous Types

We now have the two critical values to characterize the equilibrium, which should hold simultaneously. x x1 = − 3. x2 x x2 = − 3. x1 Solving this system, the solution is

2 x1 +3x1 − x =0

Using the quadratic equation,weobtain √ −3+ 9+4x x1 = . 2 Bayesian Nash Equilibrium with Continuous Types

The Bayesian Nash Equilibrium to the Battle of the Sexes Player 1’s equilibrium type-contingent (pure) strategy is B if θ1 > x1 s1(θ1)= N if θ1 ≤ x1

Player 2’s equilibrium type-contingent (pure) strategy is B if θ2 ≤ x2 s2(θ2)= N if θ2 ≤ x2

where √ −3+ 9+4x x1 = x2 = . 2