EC941 - Game Theory
Lecture 2
Prof. Francesco Squintani
Email: f.squintani@warwick.ac.uk
1
Structure of the Lecture

Mixed Strategies

Nash Equilibrium and Rationalizability

Correlated Equilibrium
2
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.
3
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.
4
Example: Matching Pennies
Head
Head
Tail
Tail
-1, 1
1, -1
1, -1
-1, 1
There is no sure way
to win for either of
the players.
A reasonable solution is that both players randomize between
H and T with equal probability. The game results in a tie.
5


1
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
H
1
0
When player 2 mixes between
T and H with equal probability,
player 1 is indifferent between
q T and H.
T
-1
6

q
1
We can write best response correspondences in terms
of mixed strategy:
B1(q) = 1
if q < ½
B2(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.
The Nash Equilibrium is
found by intersecting the
Best Response
correspondences.
B2(p)
½
B1(q)
0
½
1
p
The Nash Equilibrium is
(p, q) = (½, ½).
7
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}.
8
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.
9




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∗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.
10
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 Xit+1 is strictly dominated in
G = (I, (Xit)I, U).
The set XiT is the set of rationalizable strategies of player i.
11
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.
12
In games with 2 players and 2 strategies for each player
(let A1 = {U, D} and A2 = {L, R}), we find the
completely 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.
13
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.
14
Battle of the Sexes
B
B
S
S
2, 1
0, 0
2
0, 0
1, 2
U1
B
1
S
0
1/3
1 q
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.
15
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
B2(p)
B1(q)
1/3
0
2/3
1 p
16
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).
17
Coordination Game
N
N
E
E
0, 0
-c, 0
0, -c
1-c, 1-c
1-c
0
U1
E
N
c
1 q
-c
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.
18
B1(q) = 0
if q < c
B1(q) = [0, 1] if q = c
B1(q) =1
if q > c
B2(p)
B1(q)
B2(p) = 0
if p < c
B2(p) = [0, 1] if p = c
B2(p) =1
if p > c
0
c
1 p
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.
19
Games with Dominated Strategies
D
E
F
2
A
2, 1
0, 2
4, 1
B
1, 3
3, 1
3, 0
C
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.
20
D
F 2
E
A
2, 1
0, 2
4, 1
B
1, 3
3, 1
3, 0
C
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.
21
D
E
F
2
A
2, 1
0, 2
4, 1
B
1, 3
3, 1
3, 0
C
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.
22
D
E
F
2
A
2, 1
0, 2
4, 1
B
1, 3
3, 1
3, 0
C
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.
23
E
F
G
H
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
First, we iteratively delete strictly dominated strategies.
In the first round, only strategy A is deleted.
24
E
F
G
H 2
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
After deleting A, strategy H is strictly dominated.
H is the only strategy deleted in the second round.
25
E
F
G
H 2
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
3
After deleting H, strategy D is strictly dominated.
D is the only strategy deleted in the third round.
26
E 4
F
G
H 2
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
3
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.
27
E 4
F
G
H 2
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
3
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}.
28
E 4
F
G
H 2
A
0, 3
1, 0
0, 2
0, 4
B
3, 1
2, 2
1, 1
1, 1
C
4, 0
1, 1
2, 2
0, 1
D
2, 5
1, 4
0, 2
3, 1
1
3
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.
29
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.
30

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)
31
Correlated Equilibrium: Example
Dove
Dove
Hawk
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.
32
Dove
Dove
Hawk
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 -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?
34
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.
35
For player 1, the private recommendation d is selfenforcing 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.
36
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
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.
38
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.
39
Correlated Equilibrium
Definition 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.
40
Summary of the Lecture

Mixed Strategies

Nash Equilibrium and Rationalizability

Correlated Equilibrium
41
Preview Next Lecture

Bayesian Games and Nash Equilibrium

Bayesian Games and Information

Cournot Duopoly with Private Information

Public Good Provision with Private Information
42