EC941 - Game Theory
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EC941 - Game Theory. Lecture 2. Prof. Francesco Squintani Email: [email protected] Structure of the Lecture. Mixed Strategies Nash Equilibrium and Rationalizability Correlated Equilibrium. Mixed Strategies. In a game in strategic form G=(I, S, u), for each player i , S i

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Ec941 game theory

EC941 - Game Theory

Lecture 2

Prof. Francesco Squintani

Email: [email protected]


Ec941 game theory

Structure of the Lecture

  • Mixed Strategies

  • Nash Equilibrium and Rationalizability

  • Correlated Equilibrium


Mixed strategies

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 siis 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.


Ec941 game theory

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.


Example matching pennies

Example: Matching Pennies

HeadTail

Head

Tail

-1, 1

1, -1

There is no sure way

to win for either of

the players.

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.


Ec941 game theory

  • 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).

U1

When player 2 mixes between

T and H with equal probability,

player 1 is indifferent between

T and H.

1

H

1 q

0

T

-1


Ec941 game theory

  • We can write best response correspondences in terms of mixed strategy:

    B1(q) = 1 if q < ½B2(p) = 0 if p < ½

    B1(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.

q

1

½

The Nash Equilibrium is found by intersecting the

Best Response correspondences.

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

B2(p)

B1(q)

0 ½ 1 p


General definitions

General Definitions

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

Definition A mixed strategy siof 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 Sssi (si) = 1}.


Ec941 game theory

Definition When the players play the mixed strategy

profile s, player i’s expected payoff Uiis:

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

Definition The mixed strategy extension of G is a game

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

Note The mixed strategies Sof the game G are the pure

strategies of game G.

Best Response Correspondences, Nash Equilibrium,

Dominance and Rationalizability can be defined for game G.


Ec941 game theory

  • 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 is

    Bi (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∗ibelongs to Bi (s∗−i), for every i.

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


Ec941 game theory

PropositionIf 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 Xit+1 is strictly dominated in

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

The set XiTis the set of rationalizable strategies of player i.


The indifference principle

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 sito which siassigns non-zero probability.

PropositionThe profile s* is a Nash Equilibrium if and only if

Ci(s*i) is a subset of Bi(s*−i), for every player i.


Ec941 game theory

In games with 2 players and 2 strategies for each player

(let A1 = {U, D} and A2 = {L, R}), we find the

completely mixedstrategy 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.


Ec941 game theory

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:

  • s−ibelongs to S−i(Bi) for all players i.

  • Ci(si) = Bi for all players i.


Battle of the sexes

Battle of the Sexes

U1

B S

2

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.

B

B

S

2, 1

0, 0

1

S

0, 0

1, 2

0

1 q

1/3


Ec941 game theory

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 < 1/3

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

B1 (q) =1 if q > 1/3

B2 (p) = 0 if p < 2/3

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

B2 (p) =1 if p > 2/3

q

1

1/3

B2(p)

B1(q)

0 2/3 1 p


Ec941 game theory

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).


Coordination game

Coordination Game

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

U1

1-c

N

E

E

0, 0

0, -c

0

N

c

1 q

-c, 0

1-c,1-c

-c


Ec941 game theory

B2(p)

B1(q)

B1(q) = 0 if q < c

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

B1(q) =1 if q > c

B2(p) = 0 if p < c

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

B2(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) correspond

to 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.

0 c 1 p


Games with dominated strategies

Games with Dominated Strategies

2

DEF

4, 1

A

B

C

2, 1

0, 2

1, 3

3, 1

3, 0

1, 1

1, 1

2, 2

1

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.


Ec941 game theory

2

DEF

4, 1

A

B

C

2, 1

0, 2

1, 3

3, 1

3, 0

1, 1

1, 1

2, 2

1

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.


Ec941 game theory

2

DEF

4, 1

A

B

C

2, 1

0, 2

1, 3

3, 1

3, 0

1, 1

1, 1

2, 2

1

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 when

2 s2(D) = s2(D) + 3[1- s2(D)] i.e., when s2(D) = 3/4.


Ec941 game theory

2

DEF

4, 1

A

B

C

2, 1

0, 2

1, 3

3, 1

3, 0

1, 1

1, 1

2, 2

1

Player 2 is indifferent between D and E when

s1(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.


Ec941 game theory

EFGH

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

2, 5

1, 4

3, 1

First, we iteratively delete strictly dominated strategies.

In the first round, only strategy A is deleted.


Ec941 game theory

EFGH

2

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

2, 5

1, 4

3, 1

After deleting A, strategy H is strictly dominated.

H is the only strategy deleted in the second round.


Ec941 game theory

EFGH

2

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

3

2, 5

1, 4

3, 1

After deleting H, strategy D is strictly dominated.

D is the only strategy deleted in the third round.


Ec941 game theory

EFGH

4

2

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

3

2, 5

1, 4

3, 1

After deleting D, strategy E is strictly dominated.

Eis the only strategy deleted in the third round.

The remaining strategies {B,C}, {F,G} cannot be deleted.


Ec941 game theory

EFGH

4

2

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

3

2, 5

1, 4

3, 1

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}.


Ec941 game theory

EFGH

4

2

A

B

C

D

0, 2

1

0, 3

1, 0

0, 4

3, 1

1, 1

2, 2

1, 1

4, 0

0, 1

1, 1

2, 2

0, 2

3

2, 5

1, 4

3, 1

Player 1 is indifferent between B and C when

2s2(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.


A game with many players

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.


Ec941 game theory

  • 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

Correlated Equilibrium: Example

DoveHawk

Dove

Hawk

0, 0

-2, 1

1, -2

-3,-3

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.


Ec941 game theory

DoveHawk

Dove

Hawk

0, 0

-2, 1

1, -2

-3,-3

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 -1to both players.


Ec941 game theory

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 correlatedequilibrium.

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?


Ec941 game theory

Suppose that they resort to a mediator:

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).

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.


Ec941 game theory

For player 1, the private recommendation d is self-enforcing if

U1(D|d) = u1(D,D)p(DD|d) + u1(D,H)p(DH|d) >

U1(H|d) = u1(H,D)p(DD|d) + u1(H,H)p(DH|d)

and the private recommendation h is self-enforcing if:

U1(H|h) = u1(H,D)p(HD|h) + u1(H,H)p(HH|h) >

U1(D|h) = u1(D,D)p(HD|h) + u1(D,H)p(HH|h)

and the conditions for player 2 are analogous.


Ec941 game theory

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.


Ec941 game theory

  • Substituting in the payoffs, and using symmetry, we obtain:

  • U1(D|d) = -2p(DH|d) > U1(H|d) = p(DD|d) - 3p(DH|d)

  • U1(H|h) = p(HD|h) - 3p(HH|h) > U1(D|h) = -2p(HH|h).

  • Expanding the conditional probabilities, and simplifying

  • 2p(DH) > p(DD) - 3p(DH)

  • p(HD) - 3p(HH) > -2p(HH).

  • Note that this is a set of linear inequalities.


Ec941 game theory

  • 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.


  • Ec941 game theory

    Correlated Equilibrium

    DefinitionA 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.


    Ec941 game theory

    Summary of the Lecture

    • Mixed Strategies

    • Nash Equilibrium and Rationalizability

    • Correlated Equilibrium


    Ec941 game theory

    Preview Next Lecture

    • Bayesian Games and Nash Equilibrium

    • Bayesian Games and Information

    • CournotDuopoly with Private Information

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