ec941 - game theory

EC941 - Game Theory

Prof. Francesco SquintaniEmail:

[email protected]

Lecture 2

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Structure of the Lecture

Mixed Strategies

Nash Equilibrium and Rationalizability

Correlated Equilibrium

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Mixed Strategies

In a game in strategic form G=(I, S, u), for each player i, Si

is the set of pure strategies.

A mixed strategy si is a probability distribution over Si.

When playing si , player i operates a randomizing device and

chooses the strategy accordingly.

For example, she flips a coin choosing one strategy if heads

turns up, and another one if tails turns up.

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In general, randomizing devices can be very complex and

induce any probability distribution.

Payoffs and best-response correspondences can be extended

from pure strategies to mixed strategies.

Solution concepts (Nash Equilibrium, Dominance and

Rationalizability) can be extended to mixed strategies.

To play a mixed strategy si, player i must be indifferent

among the possible pure strategies.

This indifference principle is crucial to calculate solutions.

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Example: Matching Pennies

HeadTail

Head

Tail

-1, 1

1, -1

A reasonable solution is that both players randomize between H and T with equal probability. The game results in a tie.

-1, 1

1, -1 There is no sure way to win for either of the players.

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Let p, 0<p<1, be player 1’s probability to play H, and q, 0<q<1, be player 2’s probability to play H.

Player 1’s expected payoff for playing H is: U1 = -1 . q + 1 . (1-q).

Player 1’s payoff for T is U1 = 1 . q - 1 . (1-q).

1 q0

1

-1

H

T

U1 When player 2 mixes between T and H with equal probability, player 1 is indifferent between T and H.

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We can write best response correspondences in terms of mixed strategy:B1(q) = 1 if q < ½ B2(p) = 0 if p < ½B1(q) = [0,1] if q = ½ B2(p) = [0,1] if p = ½ B1(q) = 0 if q > ½ B2(p) = 1 if p > ½.

0 ½ 1 p

q1

½

The Nash Equilibrium is found by intersecting theBest Response correspondences.

The Nash Equilibrium is (p, q) = (½, ½).

B1(q)

B2(p)

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General Definitions

Consider a game in strategic form G = (I, S, u).

Definition A mixed strategy si of a player i is a probability

distribution over player i’s pure strategies Si.

The set of all player i’s possible mixed strategies is

Si ={(si (si)) : si (si) > 0 for all si, and Ss si (si) = 1}.

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Definition When the players play the mixed strategy

profile s, player i’s expected payoff Ui is:

Ui (s) = Ss [ Pi si (si) ] ui (s).

Definition The mixed strategy extension of G is a game

in strategic form G = (I, S, U).

Note The mixed strategies S of the game G are the pure

strategies of game G.

Best Response Correspondences, Nash Equilibrium,

Dominance and Rationalizability can be defined for game G.

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The mixed strategy profile s∗ is a Nash equilibrium if,

for each player i, Ui(s∗) ≥ Ui(si, s∗−i) for

every mixed strategy si of player i.

For every player i, the best response correspondence isBi (s−i) = {si : Ui(si, s−i) ≥ Ui(s’i, s−i) for

every s’i}.

The profile s∗ is a Nash equilibrium if and only if s∗

i belongs to Bi (s∗−i), for every i.

Player i’s mixed strategy si strictly dominates her strategy si if Ui(si, s−i) > ui(si, s−i) for every s−i.

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Proposition If player i’s strategy si is strictly dominated, then any

mixed strategy si which assigns strictly positive probability to si is also

strictly dominated.

The definition of rationalizability follows from the

iterated deletion of strictly dominated strategies.

Given a game G = (I, S, U), let Xi1 = Si.

For each t = 0, . . . , T − 1, Xit+1 is a subset of

Xit such that

every si in Xit that is not in Xi

t+1 is strictly dominated in

G = (I, (Xit)I, U).

The set XiT is the set of rationalizable

strategies of player i.

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The Indifference Principle

Indifference Principle. If the opponents play s−i, player i is

indifferent among all pure strategies in the best response set Bi (s−i).

Definition For any si, let Ci(si) be the support of si, the

set of strategies si to which si assigns non-zero probability.

Proposition The profile s* is a Nash Equilibrium if and only if

Ci(s*i) is a subset of Bi(s*

−i), for every player i.

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13

In games with 2 players and 2 strategies for each player

(let A1 = {U, D} and A2 = {L, R}), we find thecompletely mixed strategy Nash equilibrium

as follows.

From the indifference condition for player 1: sL u1(U,L) + (1- sL) u1(U,R) = sL u1(D,L) + (1- sL) u1(D,R)

we find the mixed strategy sL.

sU u2(U,L) + (1-sU) u2(D,L) = sU u2(U,R) + (1-sU) u2(D,R)

pins down the mixed strategy sU.

If 0<sL<1 and 0<sU<1, then (sL, sU) is Nash Equilibrium.

Definition For any player i, and (non-empty) subset Bi of

the strategy set Si, let S−i (Bi) be the set of mixed strategy

profiles s−i such that Bi = Bi(s−i).

We can find all mixed strategy equilibria as follows.

Consider each profile B of subsets Bi, and calculate the

profile (S−i (Bi))I. The mixed strategy s is a Nash Equilibrium

if:1. s−i belongs to S−i (Bi) for all players i. 2. Ci(si) = Bi for all players i.

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Let p, q be player 1’s and 2’s probability to play B.

U1(B) = 2・ q + 0・ (1 − q) = 2q. U1(S) = 0・ q + 1・ (1 − q) = 1 − q. The condition U1(B) = U1(S), i.e. 2q = 1 − q

yields q = 1/3.

Battle of the SexesB S

B

S

2, 1

0, 0 1, 2

0, 0

1 q

1

2

0

B

S

U1

1/3

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U2(B) = 1・ p + 0・ (1 − p) = p. U2(S) = 0・ p + 2・ (1 − p) = 2 − 2p. The condition U2(B) = U2(S), yields p = 2/3.

B1 (q) = 0 if q < 1/3B1 (q) = [0, 1] if q = 1/3B1 (q) =1 if q > 1/3

B2 (p) = 0 if p < 2/3B2 (p) = [0, 1] if p = 2/3B2 (p) =1 if p > 2/3

q 1

1/3

B1(q)B2(p)

0 2/3 1 p16

The game has three mixed strategy Nash equilibria: (p, q) = (0, 0), (2/3, 1/3), and (1, 1).

The mixed strategy equilibria (0, 0) and (1, 1) correspond to the two pure strategy equilibria.

The expected payoff of the equilibrium (2/3, 1/3) is U1 = 2 ・ 2/9 + 0・ 4/9 + 0・ 1/9 + 1・ 2/9 = 2/3.Likewise, U2 = 2/3.

So, it is Pareto dominated by both pure strategy equilibria, which yield payoffs (2, 1) and (1, 2).

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Let p, q be player 1’s and 2’s probability to play E.

U1(N) = 0, U1(E) = (1 – c)q - c(1 − q) = q - c.

The condition U1(N) = U1(E), yields q = c.

The case of player 2 is symmetric.

N E

N

E

0, 0

-c, 0 1-c, 1-c

0, -c1 q

0

1-c

-c

EN

U1

c

Coordination Game

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B1(q) = 0 if q < cB1(q) = [0, 1] if q = cB1(q) =1 if q > c

B2(p) = 0 if p < cB2(p) = [0, 1] if p = cB2(p) =1 if p > c

The game has three mixed strategy Nash equilibria:

(p, q) = (0, 0), (c, c), and (1, 1).The mixed strategy equilibria (0, 0) and (1,

1) correspondto the pure strategy equilibria (N, N) and

(E, E).The expected payoff of the equilibrium (c, c)

is U1 = c2(1-c) - c(1-c)c + (1-c)0 = 0 = U2.

B1(q)B2(p)

0 c 1 p

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Games with Dominated Strategies

A B C

2, 1

1, 3

1, 1

0, 2

3, 1

1, 1

4, 1

3, 0

2, 2

D E F

Strategy C is dominated by any mixed strategy a1 with 0 < a1(A) < 2/3. Once C is deleted, strategy F is dominated by E.

1

2

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The set of rationalizable strategies is {A, B} and {D, E}.Inspection of the best response correspondence shows that there are no pure-strategy Nash Equilibria.

A B C

2, 1

1, 3

1, 1

0, 2

3, 1

1, 1

4, 1

3, 0

2, 2

D E F

1

2

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With the indifference principle, we find completely mixed strategy N.E. in the game reduced to {A,B}X{D,E}.

Player 1 is indifferent between A and B when2 s2(D) = s2(D) + 3[1- s2(D)] i.e., when s2(D) = 3/4.

A B C

2, 1

1, 3

1, 1

0, 2

3, 1

1, 1

4, 1

3, 0

2, 2

D E F

1

2

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A B C

2, 1

1, 3

1, 1

0, 2

3, 1

1, 1

4, 1

3, 0

2, 2

D E F

1

2

Player 2 is indifferent between D and E whens1(A)+3[1-s1(A)]=2s1(A)+[1-s1(A)] i.e., when s1(A)=2/3.

The unique Nash Equilibrium of the game has C(s)={A,B}X{D,E} with s1(A)=2/3 and s2(D) = 3/4.

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

First, we iteratively delete strictly dominated strategies.

In the first round, only strategy A is deleted.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

After deleting A, strategy H is strictly dominated.

H is the only strategy deleted in the second round.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

2

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

After deleting H, strategy D is strictly dominated.

D is the only strategy deleted in the third round.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

2

3

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

After deleting D, strategy E is strictly dominated.E is the only strategy deleted in the third round.The remaining strategies {B,C}, {F,G} cannot be deleted.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

2

3

4

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

Inspection of best responses shows that (B, F) and (C, G) are pure-strategy Nash Equilibria.We find mixed strategy N.E. in the game on {B,C}X{F,G}.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

2

3

4

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A BC

D

0, 3

3, 1

4, 0

1, 0

2, 2

1, 1

0, 2

1, 1

2, 2

E F G H

Player 1 is indifferent between B and C when2s2(F) + [1-s2(F)] = 2[1-s2(F)] + s2(F), i.e. s2(F)=1/2.By symmetry, player 2 is indifferent when s1(A)=1/2.

2, 5 1, 4 0, 2

0, 4

1, 1

0, 1

3, 1

1

2

3

4

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A Game with Many Players

n people observe a crime.

Each attaches the value v to the police being informed and bears the cost c if calling the police, where v > c > 0.

There are n pure strategy equilibria.

In each one of these equilibria, a player i reports, and the remaining n - 1 players do not.

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Let p be the mixed strategy that a player calls the police.

The symmetric mixed strategy equilibrium p is determined by the indifference condition:

v − c = 0・ Pr{no one else calls}

+ v・ Pr{someone else calls}

v − c = v・ (1- Pr{no one else calls})

v − c = v・ (1 – (1 – p) n-1) or c = – v (1 – p) n-1

p = 1 − (c/v)1/(n−1)

Correlated Equilibrium: Example

DoveHawkDove

Hawk

0, 0

1, -2 -3,-3

-2, 1

In the hawk-dove game, two States confront in a dispute.If a State plays Dove and the other Hawk, the Dove loses the dispute. But if both States are Hawks, a war takes place.

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DoveHawkDove

Hawk

0, 0

1, -2 -3,-3

-2, 1

There are two pure-strategy Nash Equilibria (H, D) and (D, H), with payoffs (1, -2) and (-2, 1).

There is a mixed strategy Nash equilibrium s(D) = 1/2, which yields payoff of -1 to both players. 33

If the players can play (H, D) and (D, H) with equal probability, their expected payoff is -1/2.

Playing the Nash equilibria (H, D) and (D, H) with equal probability, is a correlated equilibrium.

The players coordinate on a different Nash equilibrium on the basis of a fair coin toss. The coin is the correlating device.

Can the players achieve a higher payoff with a more complex correlating device?

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Suppose that they resort to a mediator:

1. The mediator randomly chooses between one of the four outcomes (H, H), (H, D), (D, H), and (D, D) using the probability profile p = (pHH, pHD, pDH, pDD).

2. The mediator does not report her choice. She makes a separate, private recommendation h or d, to each one of the players.

For example, if her choice is (H, D), she recommends h to player 1 and d to player 2.

The probability profile p is a correlated equilibrium if all the private recommendations are self-enforcing.

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To find the symmetric correlated equilibrium which maximizes the sum of payoffs, we solve the program:

max [u1(D,D)+u2(D,D)]p(DD)+[u1(D,H)+u2(D,H)]p(DH) +[u1(H,D)+u2(H,D)]p(HD)+[u1(H,H)+u2(H,H)]p(HH)

s.t. Ui(D|d) > Ui(H|d) Ui(D|h) > Ui(H|h) p(DH) = p(HD)

for both players i = 1, 2. 37

We solve for the optimal correlated equilibrium:

max 0p(DD)+2(1-2)p(DH) - 6p(HH)s.t. p(DH) = p(HD)- 2p(DH) > p(DD) -3p(DH) p(HD) -3p(HH) > -2p(HH)

Simplifying, we obtain: p(DH) = p(HD) > p(DD) and p(HD) = p(DH) > p(HH).

The solution is p(HH)=0, p(HD)=p(DH)=p(DD)=1/3.

The expected payoff of both players i = 1, 2 is:

Ui(p) = 0p(DD) + p(HD) - 2p(DH) = -1/3.

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Correlated EquilibriumDefinition A correlated equilibrium in a game G=(I, S, u) is a probability distribution p over the set S such that, for any player i, and all her strategies si

Ss-i [ui (si, s-i) p(si, s-i)] > Ss-i [ui (s’i, s-i) p(s’i, s-i)], for all s’i.

Proposition Any correlated equilibrium is also a mixed strategy Nash Equilibrium.

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Summary of the Lecture

Mixed Strategies

Nash Equilibrium and Rationalizability

Correlated Equilibrium

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42

ec941 - game theory

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