Game Applications

1
Game Applications

• Chapter 29

2
Nash Equilibrium

• In any Nash equilibrium (NE) each player chooses
  a best response to the choices made by all of
  the other players.
• A game may have more than one NE.
• How can we locate every one of a games Nash
  equilibria?
• If there is more than one NE, can we argue that
  one is more likely to occur than another?

3
Best Responses

• Think of a 22 game i.e., a game with two
  players, A and B, each with two actions.
• A can choose between actions aA1 and aA2.
• B can choose between actions aB1 and aB2.
• There are 4 possible action pairs(aA1, aB1),
  (aA1, aB2), (aA2, aB1), (aA2, aB2).
• Each action pair will usually cause different
  payoffs for the players.

4
Best Responses

• Suppose that As and Bs payoffs when the chosen
  actions are aA1 and aB1 areUA(aA1, aB1) 6 and
  UB(aA1, aB1) 4.
• Similarly, suppose thatUA(aA1, aB2) 3 and
  UB(aA1, aB2) 5UA(aA2, aB1) 4 and UB(aA2,
  aB1) 3UA(aA2, aB2) 5 and UB(aA2, aB2) 7.

5
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.

6
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If B chooses action aB1 then As best response is
  ??

7
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If B chooses action aB1 then As best response is
  action aA1 (because 6 gt 4).

8
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If B chooses action aB1 then As best response is
  action aA1 (because 6 gt 4).
• If B chooses action aB2 then As best response is
  ??

9
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If B chooses action aB1 then As best response is
  action aA1 (because 6 gt 4).
• If B chooses action aB2 then As best response is
  action aA2 (because 5 gt 3).

10
Best Responses

• If B chooses aB1 then A chooses aA1.
• If B chooses aB2 then A chooses aA2.
• As best-response curve is therefore

aA2
As bestresponse

aA1
Bs action
aB2
aB1
11
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.

12
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If A chooses action aA1 then Bs best response is
  ??

13
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If A chooses action aA1 then Bs best response is
  action aB2 (because 5 gt 4).

14
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If A chooses action aA1 then Bs best response is
  action aB2 (because 5 gt 4).
• If A chooses action aA2 then Bs best response is
  ??.

15
Best Responses

• UA(aA1, aB1) 6 and UB(aA1, aB1) 4UA(aA1,
  aB2) 3 and UB(aA1, aB2) 5UA(aA2, aB1) 4
  and UB(aA2, aB1) 3UA(aA2, aB2) 5 and UB(aA2,
  aB2) 7.
• If A chooses action aA1 then Bs best response is
  action aB2 (because 5 gt 4).
• If A chooses action aA2 then Bs best response is
  action aB2 (because 7 gt 3).

16
Best Responses

• If A chooses aA1 then B chooses aB2.
• If A chooses aA2 then B chooses aB2.
• Bs best-response curve is therefore

aA2
As action
aA1
Bs best response
aB2
aB1
17
Best Responses

• If A chooses aA1 then B chooses aB2.
• If A chooses aA2 then B chooses aB2.
• Bs best-response curve is therefore

Notice that aB2 is astrictly dominantaction for
B.
aA2
As action
aA1
Bs best response
aB2
aB1
18
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria?
As response
As choice
B
A
aA2
aA1
aB2
aB1
Bs choice
Bs response
19
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria? Put
one curve on top
of
the other.
As response
As choice
B
A
aA2
aA1
aB2
aB1
Bs choice
Bs response
20
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria? Put
one curve on top
of
the other.
As response
As choice
B
A
aA2
aA1
aB2
aB1
Bs choice
Bs response
21
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria? Put
one curve on top
of
the other.
As response

aA2
Is there a Nash equilibrium?

aA1
aB2
aB1
Bs response
22
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria? Put
one curve on top
of
the other.
As response

aA2
Is there a Nash equilibrium?Yes, (aA2, aB2). Why?

aA1
aB2
aB1
Bs response
23
Best Responses Nash Equilibria
How can the players best-response curves beused
to locate the games Nash equilibria? Put
one curve on top
of
the other.
As response

aA2
Is there a Nash equilibrium?Yes, (aA2, aB2).
Why? aA2 is a best response to aB2.aB2 is a best
response to aA2.

aA1
aB2
aB1
Bs response
24
Best Responses Nash Equilibria
Player B
Here is the strategicform of the game.
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
aA2 is the only best response to aB2. aB2 is
the only best response to aA2.
25
Best Responses Nash Equilibria
Player B
Here is the strategicform of the game.
aB1
aB2
Is there a 2nd Nasheqm.?
6,4
3,5
aA1
Player A
5,7
4,3
aA2
aA2 is the only best response to aB2. aB2 is
the only best response to aA2.
26
Best Responses Nash Equilibria
Player B
Here is the strategicform of the game.
aB1
aB2
Is there a 2nd Nasheqm.? No, becauseaB2 is a
strictlydominant actionfor Player B.
6,4
3,5
aA1
Player A
5,7
4,3
aA2
aA2 is the only best response to aB2. aB2 is the
only best response to aA2.
27
Best Responses Nash Equilibria
Player B
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
Now allow both players to randomize (i.e.,
mix)over their actions.
28
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1.
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
Now allow both players to randomize (i.e.,
mix)over their actions.
29
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?B1, whatvalue
of ?A1 is bestfor A?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
30
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?B1, whatvalue
of ?A1 is bestfor A?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
EVA(aA1) 6?B1 3(1 - ?B1) 3 3?B1.
31
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?B1, whatvalue
of ?A1 is bestfor A?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
EVA(aA1) 6?B1 3(1 - ?B1) 3 3?B1.EVA(aA2)
4?B1 5(1 - ?B1) 5 - ?B1.
32
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?B1, what value
of ?A1 is best for A?
33
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?B1, what value
of ?A1 is best for A?
34
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?B1, what value
of ?A1 is best for A?
35
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?B1, what value
of ?A1 is best for A?
36
Best Responses Nash Equilibria
As best response is aA1 (i.e. ?A1 1) if ?B1
gt ½ aA2 (i.e. ?A1
0) if ?B1 lt ½
aA1 or aA2 (i.e. 0 ? ?A1 ? 1) if
?B1 ½
As best response
?A1
1
0
?B1
1
0
½
37
Best Responses Nash Equilibria
As best response is aA1 (i.e. ?A1 1) if ?B1
gt ½ aA2 (i.e. ?A1
0) if ?B1 lt ½
aA1 or aA2 (i.e. 0 ? ?A1 ? 1) if
?B1 ½
As best response
?A1
1
0
?B1
1
0
½
38
Best Responses Nash Equilibria
As best response is aA1 (i.e. ?A1 1) if ?B1
gt ½ aA2 (i.e. ?A1
0) if ?B1 lt ½
aA1 or aA2 (i.e. 0 ? ?A1 ? 1) if
?B1 ½
As best response
?A1
1
0
?B1
1
0
½
39
Best Responses Nash Equilibria
As best response is aA1 (i.e. ?A1 1) if ?B1
gt ½ aA2 (i.e. ?A1
0) if ?B1 lt ½
aA1 or aA2 (i.e. 0 ? ?A1 ? 1) if

?B1 ½
As best response
?A1
1
This is As best responsecurve when players
areallowed to mix over theiractions.
0
?B1
1
0
½
40
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?A1, whatvalue
of ?B1 is bestfor B?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
41
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?A1, whatvalue
of ?B1 is bestfor B?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
EVB(aB1) 4?A1 3(1 - ?A1) 3 ?A1.
42
Best Responses Nash Equilibria
Player B
?A1 is the prob. Achooses action aA1. ?B1 is the
prob. Bchooses action aB1. Given ?A1, whatvalue
of ?B1 is bestfor B?
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
EVB(aB1) 4?A1 3(1 - ?A1) 3 ?A1.EVB(aB2)
5?A1 7(1 - ?A1) 7 - 2?A1.
43
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?A1, what value
of ?B1 is best for B?
gtlt
44
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?A1, what value
of ?B1 is best for B?
EVB(aB1) 3 ?A1.EVB(aB2) 7 - 2?A1.3 ?A1
lt 7 - 2?A1 for all 0 ? ?A1 ? 1.
45
Best Responses Nash Equilibria
?A1 is the prob. A chooses action aA1. ?B1 is the
prob. B chooses action aB1. Given ?B1, what value
of ?A1 is best for A?
EVB(aB1) 3 ?A1.EVB(aB2) 7 - 2?A1.3 ?A1
lt 7 - 2?A1 for all 0 ? ?A1 ? 1.Bs best response
is aB2 always (i.e. ?B1 0 always).
46
Best Responses Nash Equilibria
Bs best response is aB2 always (i.e. ?B1 0
always).
?A1
1
This is Bs best responsecurve when players
areallowed to mix over theiractions.
0
?B1
1
0
½
Bs best response
47
Best Responses Nash Equilibria
B
A
As best response
?A1
?A1
1
1
0
0
?B1
?B1
1
0
½
1
0
½
Bs best response
48
Best Responses Nash Equilibria
Is there a Nash equilibrium?
B
A
As best response
?A1
?A1
1
1
0
0
?B1
?B1
1
0
½
1
0
½
Bs best response
49
Best Responses Nash Equilibria
Is there a Nash equilibrium?
B
A
50
Best Responses Nash Equilibria
Is there a Nash equilibrium?
As best response
51
Best Responses Nash Equilibria
Is there a Nash equilibrium? Yes. Just one.
(?A1, ?B1) (0,0) i.e. A chooses aA2 only B
chooses aB2 only.
As best response
52
Best Responses Nash Equilibria
Player B
Lets change the game.
aB1
aB2
6,4
3,5
aA1
Player A
5,7
4,3
aA2
53
Best Responses Nash Equilibria
Player B
Here is a new22 game.
aB1
aB2
3,1
6,4
3,5
aA1
Player A
5,7
4,3
aA2
54
Best Responses Nash Equilibria
Player B
Here is a new22 game. Againlet ?A1 be the
prob.that A chooses aA1and let ?B1 be theprob.
that B choosesaB1. What are the NEof this game?
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
Notice that Player B no longer has a strictly
dominant action.
55
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVA(aA1) ??EVA(aA2) ??
56
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVA(aA1) 6?B1 3(1 - ?B1) 3 3?B1.
EVA(aA2) ??
57
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVA(aA1) 6?B1 3(1 - ?B1) 3 3?B1.
EVA(aA2) 4?B1 5(1 - ?B1) 5 - ?B1.
58
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVA(aA1) 6?B1 3(1 - ?B1) 3 3?B1.
EVA(aA2) 4?B1 5(1 - ?B1) 5 - ?B1. 3
3?B1 5 - ?B1 as ?B1 ½.
gtlt
gtlt
59
Best Responses Nash Equilibria
As best response
?A1
1
0
?B1
1
0
½
60
Best Responses Nash Equilibria
As best response
?A1
1
0
?B1
1
0
½
61
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVB(aB1) ??EVB(aB2) ??
62
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVB(aB1) 4?A1 3(1 - ?A1) 3 ?A1. EVB(aB2)
??
63
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
EVB(aB1) 4?A1 3(1 - ?A1) 4 ?A1. EVB(aB2)
?A1 7(1 - ?A1) 7 - 6?A1.
64
Best Responses Nash Equilibria
Player B
?A1 is the prob. that Achooses aA1.?B1 is the
prob. that Bchooses aB1.
aB1
aB2
3,1
6,4
aA1
Player A
5,7
4,3
aA2
65
Best Responses Nash Equilibria
?A1
1
0
?B1
1
0
Bs best response
66
Best Responses Nash Equilibria
?A1
1
0
?B1
1
0
Bs best response
67
Best Responses Nash Equilibria
B
A
As best response
?A1
?A1
1
1
0
0
?B1
?B1
1
0
1
0
½
Bs best response
68
Best Responses Nash Equilibria
Is there a Nash equilibrium?
B
A
As best response
?A1
?A1
1
1
0
0
?B1
?B1
1
0
1
0
½
Bs best response
69
Best Responses Nash Equilibria
Is there a Nash equilibrium?
B
A
?A1
1
0
?B1
1
0
Bs best response
70
Best Responses Nash Equilibria
Is there a Nash equilibrium?
As best response
?A1
1
0
?B1
1
0
½
Bs best response
71
Best Responses Nash Equilibria
Is there a Nash equilibrium? Yes. 3 of them.
As best response
?A1
1
0
?B1
1
0
½
Bs best response
72
Best Responses Nash Equilibria
Is there a Nash equilibrium? Yes. 3 of them.
(?A1, ?B1) (0,0)
As best response
?A1
1
0
?B1
1
0
½
Bs best response
73
Best Responses Nash Equilibria
Is there a Nash equilibrium? Yes. 3 of them.
Is there a Nash equilibrium?
(?A1, ?B1) (0,0)(?A1, ?B1) (1,1)
As best response
?A1
1
0
?B1
1
0
½
Bs best response
74
Best Responses Nash Equilibria
Is there a Nash equilibrium? Yes. 3 of them.
Is there a Nash equilibrium?
(?A1, ?B1) (0,0)(?A1, ?B1) (1,1) (?A1, ?B1)
( , )
As best response
½
/
?A1
1
0
?B1
1
0
½
Bs best response
75
Some Important Types of Games

• Games of coordination
• Games of competition
• Games of coexistence
• Games of commitment
• Bargaining games

76
Coordination Games

• Simultaneous play games in which the payoffs to
  the players are largest when they coordinate
  their actions. Famous examples are
• The Battle of the Sexes Game
• The Prisoners Dilemma Game
• Assurance Games
• Chicken

77
Coordination Games The Battle of the Sexes

• Sissy prefers watching ballet to watching mud
  wrestling.
• Jock prefers watching mud wrestling to watching
  ballet.
• Both prefer watching something together to being
  apart.

78
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
79
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet. What are
the playersbest-responsefunctions?
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
80
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet. What are
the playersbest-responsefunctions?
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
EVS(B) 8?JB (1 - ?JB) 1 7?JB.
81
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet. What are
the playersbest-responsefunctions?
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
EVS(B) 8?JB (1 - ?JB) 1 7?JB. EVS(MW)
2?JB 4(1 - ?JB) 4 - 2?JB.
82
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet. What are
the playersbest-responsefunctions?
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
EVS(B) 8?JB (1 - ?JB) 1 7?JB. EVS(MW)
2?JB 4(1 - ?JB) 4 - 2?JB. 1 7?JB 4 -
2?JB as ?JB .
gtlt
gtlt
83
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
?SB
Sissy
1
4,8
2,1
MW
EVS(B) 8?JB (1 - ?JB) 1 7?JB. EVS(MW)
2?JB 4(1 - ?JB) 4 - 2?JB. 1 7?JB 4 -
2?JB as ?JB .
gtlt
gtlt
0
?JB
1
0
84
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
?SB
Sissy
Sissy
1
4,8
2,1
MW
EVS(B) 8?JB (1 - ?JB) 1 7?JB. EVS(MW)
2?JB 4(1 - ?JB) 4 - 2?JB. 1 7?JB 4 -
2?JB as ?JB .
gtlt
gtlt
0
?JB
1
0
85
Coordination Games The Battle of the Sexes
?SB
?SB
Sissy
Jock
1
1
0
0
?JB
?JB
1
0
1
0
86
Coordination Games The Battle of the Sexes
The games NE are ??
?SB
?SB
Sissy
Jock
1
1
0
0
?JB
?JB
1
0
1
0
87
Coordination Games The Battle of the Sexes
The games NE are ??
?SB
Sissy
Jock
1
0
?JB
1
0
88
Coordination Games The Battle of the Sexes
The games NE are ??
Sissy
?SB
1
0
?JB
1
0
Jock
89
Coordination Games The Battle of the Sexes
The games NE are (?JB, ?SB) (0, 0) i.e.,
(MW, MW)
Sissy
?SB
1
0
?JB
1
0
Jock
90
Coordination Games The Battle of the Sexes
The games NE are (?JB, ?SB) (0, 0) i.e.,
(MW, MW) (?JB, ?SB)
(1, 1) i.e., (B, B)
Sissy
?SB
1
0
?JB
1
0
Jock
91
Coordination Games The Battle of the Sexes
The games NE are (?JB, ?SB) (0, 0) i.e.,
(MW, MW) (?JB, ?SB)
(1, 1) i.e., (B, B)
(?JB, ?SB) ( , ) i.e., bothwatch
the ballet with prob. 1/9, both watch the
mudwrestling with prob. 4/9, and with prob. 4/9
they watch different
events.
Sissy
?SB
1
0
?JB
1
0
Jock
92
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
93
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
Sissy
4,8
2,1
MW
94
Coordination Games The Battle of the Sexes
Jock
?SB is the prob. thatSissy chooses ballet.?JB
is the prob. thatJock chooses ballet.
B
MW
1,2
8,4
B
So, is the mixedstrategy NE a focalpoint for
the game?
Sissy
4,8
2,1
MW
95
Coordination Games The Prisoners Dilemma

• A simultaneous play game in which each player has
  a strictly dominant action.
• The only NE, therefore, is the choice by each
  player of her strictly dominant action.
• Yet both players can achieve strictly larger
  payoffs than in the NE by coordinating with each
  other on another pair of actions.

96
Coordination Games The Prisoners Dilemma

• Tim and Tom are in police custody. Each can
  confess (C) to a crime or stay silent (S).
• Confession by both results in 5 years each in
  jail.
• Silence by both results in 2 years each in jail.
• If Tim confesses and Tom stays silent then Tim
  gets no penalty and Tom gets 10 years in jail
  (and conversely).

97
Coordination Games The Prisoners Dilemma
Tom
Silent
Confess
-10,0
-2,-2
Silent
Tim
-5,-5
0,-10
Confess
For Tim, Confess strictly dominates Silent.
98
Coordination Games The Prisoners Dilemma
Tom
Silent
Confess
-10,0
-2,-2
Silent
Tim
Confess
-5,-5
0,-10
For Tim, Confess strictly dominates Silent.For
Tom, Confess strictly dominates Silent.
99
Coordination Games The Prisoners Dilemma
Tom
Silent
Confess
-10,0
-2,-2
Silent
Tim
Confess
-5,-5
0,-10
For Tim, Confess strictly dominates Silent.For
Tom, Confess strictly dominates Silent.The only
NE is (Confess, Confess).
100
Coordination Games The Prisoners Dilemma
Tom
Silent
Confess
But (Silence, Silence)is better for both Timand
Tom.
-10,0
-2,-2
Silent
Tim
Confess
-5,-5
0,-10
For Tim, Confess strictly dominates Silent.For
Tom, Confess strictly dominates Silent.The only
NE is (Confess, Confess).
101
Coordination Games The Prisoners Dilemma
Tom
What is needed is ameans of rationallyassuring
commitmentby both players tothe most
beneficialcoordinated actions.
Silent
Confess
-10,0
-2,-2
Silent
Tim
Confess
-5,-5
0,-10
Possible means include future punishments or
enforceablecontracts.
102
Coordination Games Assurance Games

• A simultaneous play game with two coordinated
  NE, one of which gives strictly greater payoffs
  to each player than does the other.
• The question is How can each player give the
  other an assurance that will cause the better
  NE to be the outcome of the game?

103
Coordination Games Assurance Games

• A common example is the arms race problem.
• India and Pakistan can both increase their
  stockpiles of nuclear weapons. This is very
  costly.
• Having nuclear superiority over the other gives a
  higher payoff, but the worst payoff to the other.
• Not increasing the stockpile is best for both.

104
Coordination Games Assurance Games
Pakistan
Dont
Stockpile
1,4
5,5
Dont
India
Stockpile
3,3
4,1
105
Coordination Games Assurance Games
Pakistan
Dont
Stockpile
1,4
5,5
Dont
India
Stockpile
3,3
4,1
The games NE are ??
106
Coordination Games Assurance Games
Pakistan
Dont
Stockpile
1,4
5,5
Dont
India
Stockpile
3,3
4,1
The games NE are (Dont, Dont) and (Stockpile,
Stockpile).Which is the likely NE?
107
Coordination Games Assurance Games
Pakistan
Dont
Stockpile
1,4
5,5
Dont
India
Stockpile
3,3
4,1
The games NE are (Dont, Dont) and (Stockpile,
Stockpile).Which is the likely NE? What if
India moved first? Whataction would it choose?
Wouldnt Dont be best?
108
Coordination Games Chicken

• A simultaneous play game with two coordinated
  NE in which each player chooses the action that
  is not the action chosen by the other player.

109
Coordination Games Assurance Games

• Two drivers race their cars at each other. A
  driver who swerves is a wimp. A driver who
  does not swerve is macho.
• If both do not swerve there is a crash and a very
  low payoff to both.
• If both swerve then there is no crash and a
  moderate payoff to both.
• If one swerves and the other does not then the
  swerver gets a low payoff and the non-swerver
  gets a high payoff.

110
Coordination Games Assurance Games
Dumber
NoSwerve
Swerve
-2,4
1,1
Swerve
Dumb
No Swerve
-5,-5
4,-2
The games NE are ??
111
Coordination Games Assurance Games
Dumber
NoSwerve
Swerve
-2,4
1,1
Swerve
Dumb
No Swerve
-5,-5
4,-2
The games pure strategy NE are (Swerve, No
Swerve) and(No Swerve, Swerve). There is also a
mixed strategy NE inwhich each chooses Swerve
with probability ½.
112
Coordination Games Assurance Games
Dumber
NoSwerve
Can Dumb assurehimself of a payoff of4? Only
by convincingDumber that Dumbreally will choose
NoSwerve. What will beconvincing?
Swerve
-2,4
1,1
Swerve
Dumb
No Swerve
-5,-5
4,-2
The games pure strategy NE are (Swerve, No
Swerve) and(No Swerve, Swerve). There is also a
mixed strategy NE inwhich each chooses Swerve
with probability ½.
113
Some Important Types of Games

• Games of coordination
• Games of competition
• Games of coexistence
• Games of commitment
• Bargaining games

114
Games of Competition

• Simultaneous play games in which any increase in
  the payoff to one player is exactly the decrease
  in the payoff to the other player.
• These games are thus often called constant
  (payoff) sum games.

115
Games of Competition
An example is the game below. What NE can such
agame possess?
116
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up ??
117
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.
118
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left ??
119
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.
120
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis ??
121
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis (Up, Left)
122
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis (Up, Left) and if 0 lt x lt 1the NE is ??
123
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis (Up, Left) and if 0 lt x lt 1the NE is
(Down, Left).
124
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis (Up, Left) and if 0 lt x lt 1the NE is
(Down, Left).If x gt 1 then ??
125
Games of Competition
An example is the game below. What NE can such
agame possess?
If x lt 0 then Up dominatesDown.If x lt 1 then
Left dominatesRight.Therefore, if x lt 0 the
NEis (Up, Left) and if 0 lt x lt 1the NE is
(Down, Left).If x gt 1 then there is no NEin
pure strategies. Is therea mixed-strategy NE?
126
Games of Competition
The probability that 2 choosesLeft is ?L. The
probability that1 chooses Up is ?U. x gt 1.
127
Games of Competition
EV1(U) 2(1 - ?L).EV1(D) x?L 1 - ?L.
The probability that 2 choosesLeft is ?L. The
probability that1 chooses Up is ?U. x gt 1.
128
Games of Competition
EV1(U) 2(1 - ?L).EV1(D) x?L 1 - ?L.
The probability that 2 choosesLeft is ?L. The
probability that1 chooses Up is ?U. x gt 1.
2 - 2?l 1 (x - 1)?Las ?L 1/(1
x).
gtlt
gtlt
129
Games of Competition
EV1(U) 2(1 - ?L).EV1(D) x?L 1 - ?L.
The probability that 2 choosesLeft is ?L. The
probability that1 chooses Up is ?U. x gt 1.
2 - 2?l 1 (x - 1)?Las ?L 1/(1
x).
gtlt
gtlt
EV2(L) - x(1 - ?U).EV2(R) - 2?U - (1 - ?U).
130
Games of Competition
EV1(U) 2(1 - ?L).EV1(D) x?L 1 - ?L.
The probability that 2 choosesLeft is ?L. The
probability that1 chooses Up is ?U. x gt 1.
2 - 2?l 1 (x - 1)?Las ?L 1/(1
x).
gtlt
gtlt
EV2(L) - x(1 - ?U).EV2(R) - 2?U - (1 - ?U).
- x x?U - 1 - ?Uas (x 1)/(1 x)
?U.
gtlt
gtlt
131
Games of Competition
1 chooses Up if ?L gt 1/(1 x) and Down if ?L lt
1/(1 x). 2 chooses Left if ?U lt (x 1)/(1 x)
and Right if ?U gt (x 1)/(1 x).
132
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?L
1
2
1
1
0
0
?U
?L
1
0
0
133
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?L
1
2
1
1
0
0
?U
?L
1
0
0
1/(1x)
(x-1)/(1x)
134
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?L
1
2
1
1
0
0
?U
?L
1
0
0
1/(1x)
(x-1)/(1x)
135
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?L
1
2
1
1
0
0
?U
?L
1
0
0
1/(1x)
(x-1)/(1x)
136
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?L
1
2
1
1
(x-1)/(1x)
0
0
?L
?U
1
0
0
(x-1)/(1x)
1/(1x)
137
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
?U
1
2
1
1
(x-1)/(1x)
0
0
?L
?L
1
0
0
1/(1x)
138
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
1
2
1
(x-1)/(1x)
0
?L
1
0
139
Games of Competition
1 Up if ?L gt 1/(1 x) Down if ?L lt 1/(1
x). 2 Left if ?U lt (x 1)/(1 x) Right if ?U
gt (x 1)/(1 x).
?U
1
When x gt 1 there is onlya mixed-strategy NE
inwhich 1 plays Up withprobability (x 1)/(x
1)and 2 plays Left withprobability 1/(1 x).
(x-1)/(1x)
0
?L
1
0
1/(1x)
140
Some Important Types of Games

• Games of coordination
• Games of competition
• Games of coexistence
• Games of commitment
• Bargaining games

141
Coexistence Games

• Simultaneous play games that can be used to model
  how members of a species act towards each other.
• An important example is the hawk-dove game.

142
Coexistence Games The Hawk-Dove Game

• Hawk means be aggressive.
• Dove means dont be aggressive.
• Two bears come to a fishing spot. Either bear
  can fight the other to try to drive it away to
  get more fish for itself but suffer battle
  injuries, or it can tolerate the presence of the
  other, share the fishing, and avoid injury.

143
Coexistence Games The Hawk-Dove Game
Bear 2
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
Are there NE in pure strategies?
144
Coexistence Games The Hawk-Dove Game
Bear 2
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
Are there NE in pure strategies?Yes (Hawk, Dove)
and (Dove, Hawk).Notice that purely peaceful
coexistence is not a NE.
145
Coexistence Games The Hawk-Dove Game
Bear 2
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
Is there a NE in mixed strategies?
146
Coexistence Games The Hawk-Dove Game
Bear 2
?1H is the prob. that1 chooses Hawk.?2H is the
prob. that2 chooses Hawk. What are the
playersbest-responsefunctions?
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
Is there a NE in mixed strategies?
147
Coexistence Games The Hawk-Dove Game
Bear 2
?1H is the prob. that1 chooses Hawk.?2H is the
prob. that2 chooses Hawk. What are the
playersbest-responsefunctions?
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
EV1(H) -5?2H 8(1 - ?2H) 8 - 13?2H. EV1(D)
4 - 4?2H.
148
Coexistence Games The Hawk-Dove Game
Bear 2
?1H is the prob. that1 chooses Hawk.?2H is the
prob. that2 chooses Hawk. What are the
playersbest-responsefunctions?
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
149
Coexistence Games The Hawk-Dove Game
Bear 2
?1H is the prob. that1 chooses Hawk.?2H is the
prob. that2 chooses Hawk.
Hawk
Dove
8,0
-5,-5
Hawk
?1H
Bear 1
Bear 1
1
4,4
0,8
Dove
0
?2H
1
0
4/9
150
Coexistence Games The Hawk-Dove Game
Bear 1
Bear 2
?2H
?1H
1
1
0
0
?1H
?2H
1
0
1
0
4/9
4/9
151
Coexistence Games The Hawk-Dove Game
Bear 1
Bear 2
?2H
?1H
1
1
4/9
0
0
?1H
?2H
1
0
1
0
4/9
4/9
152
Coexistence Games The Hawk-Dove Game
Bear 1
Bear 2
153
Coexistence Games The Hawk-Dove Game
The game has a NE in mixed-strategies in
whicheach bear plays Hawk with probability 4/9.
?1H
1
0
?2H
1
0
4/9
154
Coexistence Games The Hawk-Dove Game
Bear 2
Hawk
Dove
8,0
-5,-5
Hawk
Bear 1
4,4
0,8
Dove
For each bear, the expected value of the
mixed-strategy NE is(-5) 8
4 , a value between-5 and
4. Is this NE focal?
155
Some Important Types of Games

• Games of coordination
• Games of competition
• Games of coexistence
• Games of commitment
• Bargaining games

156
Commitment Games

• Sequential play games in which
• One player chooses an action before the other
  player chooses an action.
• The first players action is both irreversible
  and observable by the second player.
• The first player knows that his action is seen by
  the second player.

157
Commitment Games
5,9
c
Player 1 has twoactions, a and b. Player 2 has
twoactions, c and d,following a, and two
actions e and ffollowing b.Player 1
chooseshis action beforePlayer 2 choosesher
action.
2
Game Tree
a
d
5,5
Direction of play
1
7,6
e
b
2
f
5,4
158
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
159
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?Yes.
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
160
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?Yes. Is a claim by
Player 2that she will commit tochoosing action
e ifPlayer 1 chooses bcredible to Player 1?
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
161
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?Yes. Is a claim by
Player 2that she will commit tochoosing action
e ifPlayer 1 chooses bcredible to Player
1?Yes.
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
162
Commitment Games
So Player 1 shouldchoose action ??
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
163
Commitment Games
So Player 1 shouldchoose action b.
5,9
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
164
Commitment Games
Change the game.
5,3
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
165
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?
5,3
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
166
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?No. If Player 1
choosesaction a then Player 2does best by
choosingaction d.What should Player 1do?
5,3
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
167
Commitment Games
Is a claim by Player 2that she will commit
tochoosing action c ifPlayer 1 chooses
acredible to Player 1?No. If Player 1
choosesaction a then Player 2does best by
choosingaction d.What should Player 1do?
Still choose b.
5,3
c
2
a
d
5,5
1
7,6
e
b
2
f
5,4
168
Commitment Games
Change the game.
5,3
5,9
c
2
a
d
15,5
5,5
1
7,3
e
b
2
f
5,12
169
Commitment Games
Can Player 1 get 15points?
5,9
c
2
a
d
15,5
1
7,3
e
b
2
f
5,12
170
Commitment Games
Can Player 1 get 15points?If Player 1 chooses
athen Player 2 willchoose c and Player 1will
get only 5 points.
5,9
c
2
a
d
15,5
1
7,3
e
b
2
f
5,12
171
Commitment Games
Can Player 1 get 15points?If Player 1 chooses
athen Player 2 willchoose c and Player 1will
get only 5 points.If Player 1 chooses bthen
Player 2 willchoose f and againPlayer 1 will
get only5 points.
5,9
c
2
a
d
15,5
1
7,3
e
b
2
f
5,12
172
Commitment Games
If Player 1 can changepayoffs so that
acommitment by Player2 to choose d after a is
credible then Player 1s payoff rises from 5to
15, a gain of 10.
5,9
c
2
a
d
15,5
1
7,3
e
b
2
f
5,12
173
Commitment Games
If Player 1 can changepayoffs so that
acommitment by Player2 to choose d after a is
credible then Player 1s payoff rises from 5to
15, a gain of 10.If Player 1 gives 5 ofthese
points to Player2 then Player 2scommitment
iscredible. Player 1 cannot get 15 points.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
174
Commitment Games
Credible NE of thistype are calledsubgame
perfect.What exactly is thisgames SPE?
Itinsists that everyaction chosen isrational
for the playerwho chooses it.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
175
Commitment Games
Credible NE of thistype are calledsubgame
perfect.What exactly is thisgames SPE?
Itinsists that everyaction chosen isrational
for the playerwho chooses it.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
176
Commitment Games
Credible NE of thistype are calledsubgame
perfect.What exactly is thisgames SPE?
Itinsists that everyaction chosen isrational
for the playerwho chooses it.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
177
Commitment Games
Credible NE of thistype are calledsubgame
perfect.What exactly is thisgames SPE?
Itinsists that everyaction chosen isrational
for the playerwho chooses it.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
178
Commitment Games
Credible NE of thistype are calledsubgame
perfect.What exactly is thisgames SPE?
Itinsists that everyaction chosen isrational
for the playerwho chooses it.
5,9
c
2
a
d
10,10
1
7,3
e
b
2
f
5,12
179
Some Important Types of Games

• Games of coordination
• Games of competition
• Games of coexistence
• Games of commitment
• Bargaining games

180
Bargaining Games

• Two players bargain over the division of a pie of
  size 1. What will be the outcome?
• Two approaches
• Nashs axiomatic bargaining.
• Rubinsteins strategic bargaining.

181
Strategic Bargaining

• The players have 3 periods in which to decide how
  to divide the pie else both get nothing.
• Player A discounts next periods payoffs by ?.
• Player B discounts next periods payoffs by ?.
• The players alternate in making offers, with
  Player A starting in period 1.
• If the player who receives an offer accepts it
  then the game ends immediately. Else the game
  continues to the next period.

182
Strategic Bargaining
Period 2B offers x2. A responds.
(x1,1-x1)
(x3,1-x3)
1
1
1
Y
Y
x3
x1
N
(0,0)
B
B
N
x2
A
B
A
A
N
Y
0
0
0
Period 1A offers x1. B responds.
Period 3A offers x3. B responds.
(x2,1-x2)
183
Strategic Bargaining
How should B respond to x3?
(x3,1-x3)
1
Y
x3
N
(0,0)
B
A
0
Period 3A offers x3. B responds.
184
Strategic Bargaining
How should B respond to x3?Accept if 1 x3 0
i.e., accept any x3 1.
(x3,1-x3)
1
Y
x3
N
(0,0)
B
A
0
Period 3A offers x3. B responds.
185
Strategic Bargaining
How should B respond to x3?Accept if 1 x3 0
i.e., accept any x3 1.Knowing this, what
should Aoffer?
(x3,1-x3)
1
Y
x3
N
(0,0)
B
A
0
Period 3A offers x3. B responds.
186
Strategic Bargaining
How should B respond to x3?Accept if 1 x3 0
i.e., accept any x3 1.Knowing this, what
should Aoffer? x3 1.
(1,0)
Y
x31
B
N
(0,0)
A
0
Period 3A offers x3 1. B accepts.
187
Strategic Bargaining
Period 2B offers x2. A responds.
(1,0)
(x1,1-x1)
Y
x31
1
1
Y
B
x1
N
B
(0,0)
N
x2
A
B
A
A
N
Y
0
0
0
Period 1A offers x1. B responds.
Period 3A offers x3 1. B accepts.
(x2,1-x2)
188
Strategic Bargaining
Period 2B offers x2. A responds.
In Period 3 A getsa payoff of 1. Inperiod 2,
whenreplying to Bsoffer of x2,
thepresent-value toA of N is thus ??
(1,0)
Y
x31
1
B
N
(0,0)
x2
B
A
A
N
Y
0
0
Period 3A offers x3 1. B accepts.
(x2,1-x2)
189
Strategic Bargaining
Period 2B offers x2. A responds.
In Period 3 A getsa payoff of 1. Inperiod 2,
whenreplying to Bsoffer of x2,
thepresent-value toA of N is thus ?.
(1,0)
Y
x31
1
B
N
(0,0)
x2
B
A
A
N
Y
0
0
Period 3A offers x3 1. B accepts.
(x2,1-x2)
190
Strategic Bargaining
Period 2B offers x2. A responds.
In Period 3 A getsa payoff of 1. Inperiod 2,
whenreplying to Bsoffer of x2,
thepresent-value toA of N is thus ?.What is
the mostB should offer toA?
1
x2
B
A
N
Y
0
(x2,1-x2)
191
Strategic Bargaining
Period 2B offers x2 ?. A accepts.
In Period 3 A getsa payoff of 1. Inperiod 2,
whenreplying to Bsoffer of x2,
thepresent-value toA of N is thus ?.What is
the mostB should offer toA? x2 ?.
1
x2?
B
A
N
Y
0
(?,1- ?)
192
Strategic Bargaining
Period 2B offers x2 ?. A accepts.
(1,0)
(x1,1-x1)
Y
x31
1
1
Y
B
x1
N
B
x2?
(0,0)
N
A
B
A
A
N
Y
0
0
0
Period 1A offers x1. B responds.
Period 3A offers x3 1. B accepts.
(?,1- ?)
193
Strategic Bargaining
In period 2 A will accept?. Thus B will get
thepayoff 1 - ? in period 2. What is the
present- value to B in period 1 ofN ?
Period 2B offers x2 ?. A accepts.
(x1,1-x1)
1
1
Y
x1
B
x2?
N
A
B
A
N
Y
0
0
Period 1A offers x1. B responds.
(?,1- ?)
194
Strategic Bargaining
In period 2 A will accept?. Thus B will get
thepayoff 1 - ? in period 2. What is the
present- value to B in period 1 ofN ? ?(1 - ?).
Period 2B offers x2 ?. A accepts.
(x1,1-x1)
1
1
Y
x1
B
x2?
N
A
B
A
N
Y
0
0
Period 1A offers x1. B responds.
(?,1- ?)
195
Strategic Bargaining
In period 2 A will accept?. Thus B will get
thepayoff 1 - ? in period 2. What is the
present- value to B in period 1 ofN ? ?(1 -
?).What is the most that Ashould offer to B
inperiod 1?
Period 2B offers x2 ?. A accepts.
(x1,1-x1)
1
1
Y
x1
B
x2?
N
A
B
A
N
Y
0
0
Period 1A offers x1. B responds.
(?,1- ?)
196
Strategic Bargaining
In period 2 A will accept?. Thus B will get
thepayoff 1 - ? in period 2. What is the
present- value to B in period 1 ofN ? ?(1 -
?).What is the most that Ashould offer to B
inperiod 1?1 x1 ?(1 - ?) i.e.x1 1 - ?(1
- ?).B will accept.
Period 2B offers x2 ?. A accepts.
(1-?(1 - ?), ?(1 - ?))
1
1
Y
x1
B
x2?
N
A
B
A
N
Y
0
0
Period 1A offers x1. B responds.
(?,1- ?)
197
Strategic Bargaining
Period 2B offers x2 ?. A accepts.
(1-?(1 - ?), ?(1 - ?))
(1,0)
Y
x31
1
1
x11-?(1-?)
Y
B
N
B
x2?
(0,0)
N
A
B
A
A
N
Y
0
0
0
Period 1 A offersx1 1-?(1 - ?). B accepts.
Period 3A offers x3 1. B accepts.
(?,1- ?)
198
Strategic Bargaining

• Notice that the game ends immediately, in period
  1.
• Player A gets 1 - ?(1 ?) units of the
  pie.Player B gets ?(1 ?) units.Which is the
  larger?
• x1 1 - ?(1 ?) ½ ? ? 1/2(1 - ?)so Player
  A gets more than Player B if Player B is too
  impatient relative to Player A.

199
Strategic Bargaining

• Suppose the game is allowed to continue forever
  (infinitely many periods). Then using the same
  reasoning shows that the subgame perfect
  equilibrium results in Players 1 and 2
  respectively getting and
  pie units.
• Player 1s share rises as ? and ? .Player 2s
  share rises as ? and ? .