GAME THEORY
ANSWERS TO EXERCISES—PART II
COORDINATION GAMES & MULTIPLE EQUILIBRIA
(FROM THE TEXTBOOK OF C. HOLT)
Gabriele Camera - Fall 2014
Economic Science Institute—Chapman University
and
University of Basel
1
PROBLEM 1
Find the pure strategy NE and the mixed strategy NE in the following game
L
L
2, 4
H
1, 0
H
0, 1
4, 5
2
ANSWER TO PROBLEM 1
Notice
• (L, L) and (H, H) are equally profitable for both players
• However, (H, H) is more profitable than (L, L) for both players
The game is a coordination game with aligned interests
3
PURE STRATEGY NASH EQUILIBRIA
Two equilibria: (L, L) and (H, H)
L
L
2, 4
H
1, 0
H
0, 1
4, 5
For both players, L is the best response against opponent’s choice L, and H is
the best response against opponent’s choice H
4
MIXED STRATEGY NASH EQUILIBRIUM
One equilibrium: (q, p) = ( 12 , 35 )
L (prob. q)
H (prob. 1 − q)
L (prob. p)
2, 4
H (prob. 1 − p)
1, 0
0, 1
4, 5
column player chooses L
column player chooses H
z
}|
{
4q + (1 − q)
z
}|
{
0q + 5(1 − q)
Column is indifferent:
row player chooses L
Row is indifferent:
=
row player chooses H
}|
{
z
}|
{
z
2p + (1 − p) = 0p + 4(1 − p)
5
1
⇒ q=
2
3
⇒ p=
5
PROBLEM 2
Find the pure strategy NE and the mixed strategy NE in the following game
L
L
1, 1
H
3, 2
H
2, 3
1, 1
6
ANSWER TO PROBLEM 2
Conversely, to the game in Problem, now
• (L, H) is more profitable for the Row player
• (H, L) is more profitable for the Column player
Notice that (L, L) and (H, H) generate the lowest payoffs for both players
The game is a coordination game with conflicting interests
7
PURE STRATEGY NASH EQUILIBRIUM
Two pure strategy NE: (L, H) and (H, L)— Players prefer different outcomes
L
L
1, 1
H
3, 2
H
2, 3
1, 1
One mixed strategy NE: (q, p) = ( 23 , 23 )
column player chooses L
column player chooses H
}|
{
z
q + 3(1 − q)
z
}|
{
2q + (1 − q)
Column is indifferent:
=
row player chooses L
Row is indifferent:
z
}|
{
p + 3(1 − p) =
8
⇒
row player chooses H
z
}|
{
2p + (1 − p)
⇒
2
q=
3
2
p=
3
PROBLEM 3
Find the mixed strategy NE in the following game
L
L
1, 0
H
5, 4
S
2, 2
H
6, 5
0, 1
0, 2
9
ANSWER TO PROBLEM 3
Asymmetric action sets: Arow = {L, H} and Acol = {L, H, S}
Column player gets 2 if she chooses S, independent of Row player’s choices
But Row player does not have such a safe option
10
NASH EQUILIBRIA
Two pure strategy NE: (L, H) and (H, L)
Mixed strategy NE: how to proceed?
• It may be possible that some strategies are dominated
• If so, reduce set of possible NE strategies by iterated deletion of dominated
strategies
11
MIXED STRATEGY NE (USING ITERATED DELETION)
Step 1. Find the best response of Column player by comparing his payoff πcol for
each of his action against all possible mixed strategies of Row player {q, 1 − q}
L (prob.q)
L
1, 0
H
5, 4
S
2, 2
H (prob.1 − q)
6, 5
0, 1
0, 2
• If column player selects L:
πcol = 0q + 5(1 − q)
• If column player selects H:
πcol = 4q + (1 − q)
• If column player selects S:
πcol = 2
12
BEST RESPONSES
L is a best response if q ≤ 12 . Why?




z choose
}| H {
5(1 − q) ≥ 4q + (1 − q)
choose S


 5(1 − q) ≥ z}|{
2
⇒q≤
1
2
⇒q≤
3
5
⇒q≥
1
2
⇒q≥
1
3
⇒ q ≤ min
1 3
,
2 5
=
1
2
H is a best response if q ≥ 21 . Why?




zchoose
}| L{
4q + (1 − q) ≥ 5(1 − q)
choose S


 4q + (1 − q) ≥ z}|{
2
13
1 1
⇒ q ≥ max ,
2 3
1
=
2
BEST RESPONSES
S is never a best response. Why?




zchoose
}| L{
2 ≥ 5(1 − q)
choose H

z
{

 2 ≥ 4q + }|
(1 − q)
⇒q≥
3
5
⇒q≤
1
3
14
1 3
⇒ empty set since <
3 5
FINDING DOMINATED STRATEGIES


L is the best response if
q<



 H is the best response if
q>

{L, H} are both best responses if q =



 S is never a best response
1
2
1
2
1
2
Since S is never a best response, column player mixes between L and H
So, consider the reduced game in which we delete the last column
L
L
1, 0
H
5, 4
H
6, 5
0, 1
15
THE REDUCED GAME
L
L
1, 0
H
5, 4
H
6, 5
0, 1
One mixed strategy NE: (q, p) = ( 12 , 12 )
column player chooses L
column player chooses H
z
}|
{
0q + 5(1 − q)
z
}|
{
4q + (1 − q)
Column is indifferent:
row player chooses L
Row is indifferent:
=
⇒
row player chooses H
z
}|
{
}|
{
z
p + 5(1 − p) = 6p + 0(1 − p)
⇒
Hence, the mixed strategy NE is the probability profile {( 12 , 12 ), ( 21 , 12 , 0)}
16
1
q=
2
1
p=
2
ADDITIONAL PROBLEMS ON PART 2
(FROM THE TEXTBOOK OF C. HOLT)
17
PROBLEM 4
Consider a minimum-effort coordination game where each player i = 1, 2 chooses
efforts in E = {100, 110, 120, . . . , 170} and has payoff:
πi = min(ei, e−i) − cei,
Use the logit formula
i = 1, 2 and c > 0
exp(πje/µ)
Pj = P
e /µ)
exp(π
j
j
to calculate the average efforts in the following cases:
(i) c = 0.75 and µ = 1, 5, 10, 20
(ii) c = 0.25 and µ = 1, 5, 10, 20
Use initial belief probabilities of 1/8 (or 0.125) for each of the elements in E.
Then discuss the effects of the error parameter µ on these predictions
18
HINT
There are n = 8 choices since E = {100, 110, . . . , 170}
The probability distribution (Pj )j∈E over E reflects i’s beliefs about e−i
That is, Pj = probability that i’s opponent selects e−i = j ∈ E.
For instance, P100 is the probability that player i’s opponent chooses 100
19
EXPECTED PAYOFFS
Expected payoff if i chooses ei = 100:
e
π100
= 100 − c100
Expected payoff if i chooses ei = 110:
e
π110
= (100 − c110)P100 + (110 − c110)P110 + (110 − c110)(1 − P100 − P110)
Expected payoff if i chooses ei = X > 110:
X
e
πX =
(j − cX)Pj + (X − cX)(1 −
j∈{100,...,X}
X
j∈{100,...,X}
20
Pj )
PROBLEM 5
Find the mixed-strategy equilibrium for the Kreps game shown below
Top
Bottom
Left
200, 50
0, −250
Middle
0, 45
10, −100
21
Non-Nash
10, 30
30, 30
Right
20, −250
50, 40
ANSWER TO PROBLEM 5
Use iterated deletion: Best response for Column against Row player’s randomization: play Top with probability q
Column player selects Left: probability distribution over actions is pL = (1, 0, 0, 0)
• Payoff:
πc(q, pL) = 50q + (−250)(1 − q)
Column player selects Middle: pM = (0, 1, 0, 0)
• Payoff:
πc(q, pM ) = 45q + (−100)(1 − q)
Column player selects Non-Nash: pN N = (0, 0, 1, 0)
• Payoff:
πc(q, pN N ) = 30
Column player selects Right: pR = (0, 0, 0, 1)
• Payoff:
πc(q, pR) = (−250)q + 40(1 − q)
22
BEST RESPONSE: STEP 1
• Check if any pure strategy of column player is dominated by another pure strategy
Left is the best response if:
πc(q, pL) ≥ max πc(q, pM , πc(q, pN N ), πc(q, pR)
15
15 14 29
⇒ q ≥ max 16 , 15 , 59 = 16
Middle is the best response if:
πc(q, pM ≥ max πc(q, pL), πc(q, pN N ), πc(q, pR)
26
26 28
⇒ max 29 , 87 = 29 ≤ q ≤ 15
16
23
BEST RESPONSE: STEP 1
Non-Nash is the best response if:
πc(q, pN N ) ≥ max πc(q, pL), πc(q, pM ), πc(q, pR)
26
1
26 14
⇒ 29 ≤ q ≤ min 29 , 15 = 29
Right is the best response if:
πc(q, pR) ≥ max πc(q, pL), πc(q, pM ), πc(q, pN N )
29 28 1
1
⇒ q ≤ min 59 , 87 , 29 = 29
Note: no pure strategy of column player is strictly dominated by another pure
strategy
24
BEST RESPONSE: STEP 2
Check if any pure strategy of column player is strictly dominated by mixed strategies
Suppose that Column randomizes only between Left and Middle, i.e., p = (p, 1 −
p, 0, 0)
- Row player selects Top, q = 1:
πr (1, p) = 200p
- Row player selects Bottom, q = 0:
πr (0, p) = 10(1 − p)
Row player is indifferent between Top and Bottom if
πr (1, p) = πr (0, p)
25
1
⇒p=
21
BEST RESPONSE: STEP 2
Suppose that row player randomizes selecting Top with probability q
- Column player selects Left:
πc(q, pL) = 50q − 250(1 − q)
- Column player selects Middle:
πc(q, pM = 45q − 100(1 − q)
Column player is indifferent between Left and Middle if
πc(q, pL) = πc(q, pM
30
⇒q=
31
1 20
30
Question: Given q ∗ = 31
, is the probability distribution p∗ = ( 21
, 21 , 0, 0) the
best response for Column player?
26
PAYOFF COMPARISON
Suppose that Column player randomizes using p = (p1, p2, p3, p4)
πc(q ∗, p) = 50qp1 + 45qp2 + 30qp3 + (−250)qp4
+(−250)(1 − q)p1 + (−100)(1 − q)p2 + 30(1 − q)p3 + 40(1 − q)p4
125
125
746
= 10 31 p1 + 31 p2 + 3p3 − 31 p4
(⇐ p4 = 0)
125
(⇐ p3 = 0, since 125
≤ 10 31 p1 + 125
31 p2 + 3p3 ]
31 > 3)
125
≤ 10 31 p1 + 125
31 p2 ]
Hence, the best response to q ∗ =
30
31
is p = (p1, p2, 0, 0)
1 20
Notice: We already saw that p∗ = ( 21
, 21 , 0, 0) makes row player indifferent
30 1
1 20
⇒ Mixed strategy NE: ( 31 , 31 ), ( 21 , 21 , 0, 0)
27