Chapter 6
Game Theory
© 2006 Thomson Learning/South-Western
Overview
1.
2.
3.
4.
5.
2
Games – concepts
Prisoners’ Dilemma
Mixed Strategies
Multiple Equilibria
Tragedy of Commons
1.Games - concepts
3
Basic Concepts

All games have three basic elements:



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4
Players
Strategies
Payoffs
Players can make binding agreements in
cooperative games, but can not in
noncooperative games, which are studied
in this chapter.
Players

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

5
A player is a decision maker and can be
anything from individuals to entire nations.
Players have the ability to choose among a set
of possible actions.
Games are often characterized by the fixed
number of players.
Generally, the specific identity of a player is not
important to the game.
Strategies


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6
A strategy is a course of action available
to a player.
Strategies may be simple or complex.
In noncooperative games each player is
uncertain about what the other will do
since players can not reach agreements
among themselves.
Payoffs



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7
Payoffs are the final returns to the players
at the conclusion of the game.
Payoffs are usually measured in utility
although sometimes also monetarily.
In general, players are able to rank the
payoffs from most preferred to least
preferred.
Players seek the highest payoff available.
Equilibrium Concepts


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8
In the theory of markets an equilibrium
occurred when all parties to the market had
no incentive to change his or her behavior.
When strategies are chosen, an equilibrium
would also provide no incentives for the
players to alter their behavior further.
The most frequently used equilibrium
concept is a Nash equilibrium.
Nash Equilibrium


9
The most widely used approach to
defining equilibrium in games is that
proposed by Cournot and generalized in
the 1950s by John Nash.
A Nash equilibrium is a set of strategies,
one for each player, that are each best
responses against one another.
2. Prisoners’ Dilemma
10
Nash Equilibrium

In a two-player games, a Nash
equilibrium is a pair of strategies (a*,b*)
such that a* is an optimal strategy for A
against b* and b* is an optimal strategy for
B against a*.

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11
Players can not benefit from knowing the
equilibrium strategy of their opponents.
Every game has a Nash equilibrium, and
some games may have several.
The Prisoner’s Dilemma


The Prisoner’s Dilemma is a game in
which the optimal outcome for the players
is unstable – there is a temptation to
deviate.
The name comes from the following
situation.


12
Two people are arrested for a crime.
The district attorney has little evidence but is
anxious to extract a confession.
The Prisoner’s Dilemma


13
The DA separates the suspects and tells
each, “If you confess and your companion
doesn’t, I can promise you reduced (one year)
sentence, whereas your companion will get
ten years. If you both confess, you will each
get a three year sentence.”
Each suspect knows that if neither confess,
they will be tried for a lesser crime and will
receive two-year sentences.
TABLE 6-1: The Prisoner’s Dilemma
B
Confess
A
Silent
14
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
The Prisoner’s Dilemma

The normal form (i.e. matrix) of the game
is shown in Table 6-1.
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15
The confess strategy dominates for both
players so it is a Nash equilibrium.
However, an agreement to remain silent (not
to confess) would reduce their prison terms
by one year each.
This agreement would appear to be the
rational solution.
The Prisoner’s Dilemma:
Extensive Form


16
The representation of the game as a tree
is referred to as the extensive form.
Action proceeds from top to bottom.
Figure 6-1: The Prisoner’s Dilemma:
Extensive Form
.
A
Confess
.
Confess
-3, -3
17
Silent
B
Silent
-10, -1
.
B
Confess
Silent
-1, -10
-2, -2
Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 1
B
Confess
A
Silent
18
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 2
B
Confess
A
Silent
19
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 3
B
Confess
A
Silent
20
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 4
B
Confess
A
Silent
21
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Table 6-2: Solving for Nash Equilibrium in
Prisoner’s Dilemma Using the Underlining
Method
Step 5
B
Confess
A
Silent
22
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Dominant Strategies


23
A dominant strategy refers to the best
response to any strategy chosen by the
other player.
When a player has a dominant strategy
in a game, there is good reason to
predict that this is how the player will
play the game.
3. Mixed Strategies
24
Mixed Strategies


25
A mixed strategy refers to when the
player randomly selects from several
possible actions.
By contrast, the strategies in which a
player chooses one action or another
with certainty are called pure strategies.
Table 6-3: Matching Pennies
Game in Normal Form
B
Heads
A
Tails
26
Heads
Tails
1, -1
-1, 1
-1, 1
1, -1
Figure 6-2: Matching Pennies Game
in Extensive Form
.
A
Heads
.
27
Tails
B
Heads
Tails
Heads
1, -1
-1, 1
-1, 1
.
B
Tails
1, -1
Table 6-4: Solving for Pure-Strategy Nash
Equilibrium in Matching Pennies Game
1/2
1/2
Heads
1/2
Tails
A
28
B
1/2
Heads
Tails
1 , -1
-1, 1
-1, 1
1 , -1
Solving for Mixed-Strategy Equilibrium
Expected payoffs for A:
(1/4)(1)+(1/4)(-1)+(1/4)(-1)+(1/4)(1)=0
and similarly for B.
So, both playing Heads and Tails with equal chance is the
mixed-strategy Nash equilibrium.
If there is a strategy that produces higher payoff, then this is
NOT Nash equilibrium.
29
Assume B plays H with ½ and T with ½ probability. If A plays
only H or only T (or anything in between) A’s payoffs will still
be 0.
Solving for Mixed-Strategy Equilibrium
Show that this is the only mixed-strategy equilibrium (no
other probabilities would work).
Assume B plays H with 1/3 and T with 2/3.
Expected payoff for A from playing H
(1/3)(1)+(2/3)(-1)=-1/3
And from playing T
Examples of mixed strategies:
30
4. Multiple Equilibria
31
Multiple Equilibria
TABLE 6-5: Battle of the Sexes in Normal Form
Both prefer to be together rather than apart. She prefers
ballet, he boxing.
B (Husband)
A (Wife)
32
Ballet
Boxing
Ballet
2, 1
0, 0
Boxing
0, 0
1, 2
TABLE 6-6: Solving for Pure-Strategy
Nash Equilibria in Battle of the Sexes
B (Husband)
A (Wife)
33
Ballet
Boxing
Ballet
2, 1
0, 0
Boxing
0, 0
1, 2
Solving for Mixed Strategy Nash
Equilibrium

Equilibrium probabilities do not end up to be
equal for each action:

w – probability that wife chooses ballet
1-w- that she does not (chooses boxing)

h – probability that husband chooses ballet
 1-h – he chooses boxing
Goal is to compute equilibrium w and h.

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Solving for Mixed Strategy Nash
Equilibrium
w and h are any of the values between (0,1) and we
can not use matrix
 Need to find best response function

The
function which gives the payoff-maximizing
choice for one player in each of a continuum of
actions of the other player is referred to as the bestresponse function.
Compute wife’s BRF:
find w that max her payoff for each of husband’s
strategies
Best Response Function
For a given h she can:
prefer to play Ballet and then her best response is w=1
prefer to play Boxing and her best response is w=0,or
be indifferent and her best response is tie between all the
values (0,1)

Husband plays Ballet with probability h and Boxing 1-h.
TABLE 6-7: Computing the Wife’s Best
Response to the Husband’s Mixed Strategy
B (Husband)
Ballet
A (Wife)
Boxing
Ballet h
Box 1
2, 1
Box 3
0, 0
Boxing 1-h
Box2
0, 0
Box 4
1, 2
(h)(2) + (1 – h)(0)
= 2h
(h)(0) + (1 – h)(1)
=1-h
She prefers Ballet if 2h>1-h or h>1/3, w=1
She prefers Boxing if h<1/3, w=0
She is indifferent when h=1/3 and her best response is w(0,1)
37
Figure 6-4: Best-Response Functions
Allowing Mixed Strategies in the Battle of
the Sexes
h
1
Wife’s bestresponse
function
1/3
.
38
.
Husband’s
best-response
function
Pure-strategy
Nash equilibrium
(both play Boxing)
Pure-strategy
Nash equilibrium
(both play Ballet)
.
Mixed-strategy
Nash equilibrium
2/3
1
w
The Problem of Multiple Equilibria
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39
Which equilibrium to choose (will happen)?
A rule that selects the highest total payoff would
not distinguish between two pure-strategy
equilibria.
To select between these, one might follow T.
Schelling’s suggestion and look for a focal
point…a logical outcome on which to
coordinate, based on information outside the
game.
5. Tragedy of Commons
40
Tragedy of Commons

Benefit A gets from each sheep
120  s A  sB
6.4
Total benefit A gets
s A (120  s A  sB )
6.5
 Marginal benefit of additional sheep
120  2s A  sB
41
6.6
Tragedy of Commons(Continuous Actions)
Marginal benefit=Marginal cost=0 and solve for sA
sA
sB
 60 
2
6.7
sB
sA
 60 
2
6.8
FIGURE 6-8: Best-Response Functions
in the Tragedy of the Commons
SB
120
A’s best-response
function
60
Nash equilibrium
40
B’s best-response
function
40
43
60
120
SA
Tragedy of Commons
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44
Equations for the Tragedy of Commons
After Equilibria are Shifted:
132  s A  sB
6.9
sB
s A  66 
2
6.10
FIGURE 6-9: Shift in Equilibrium
When A’s Benefit Increases
SB
A’s best-response
function shifts out
40
36
Nash equilibrium
shifts
B’s best-response
function
40 48
45
SA