Download Pure-strategy Nash equilibrium

Chapter 6
Game Theory
© 2006 Thomson Learning/South-Western
Basic Concepts

All games have three basic elements:




2
Players
Strategies
Payoffs
Players can make binding agreements in
cooperative games, but can not in
noncooperative games, which are studied
in this chapter.
Players




3
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 play is not
important to the game.
Strategies



4
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




5
Payoffs are the final returns to the players
at the conclusion of the game.
Payoffs are usually measure in utility
although sometimes measure monetarily.
In general, players are able to rank the
payoffs from most preferred to least
preferred.
Players seek the highest payoff available.
Equilibrium Concepts



6
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


7
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.
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*.


8
Players can not benefit from knowing the
equilibrium strategy of their opponents.
Not 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.
The name comes from the following
situation.


9
Two people are arrested for a crime.
The district attorney has little evidence but is
anxious to extract a confession.
The Prisoner’s Dilemma


10
The DA separates the suspects and tells
each, “If you confess and your companion
doesn’t, I can promise you a 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.
The Prisoner’s Dilemma

The normal form (i.e. matrix) of the game
is shown in Table 6-1.



11
The confess strategy dominates for both
players so it is a Nash equilibria.
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.
TABLE 6-1: The Prisoner’s Dilemma
B
Confess
A
Silent
12
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
The Prisoner’s Dilemma:
Extensive Form


13
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
14
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
15
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
16
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
17
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
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 5
B
Confess
A
Silent
19
Confess
Silent
-3, -3
-1, -10
-10, -1
-2, -2
Dominant Strategies


20
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.
Mixed Strategies


21
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
22
Heads
Tails
1, -1
-1, 1
-1, 1
1, -1
Figure 6-2: Matching Pennies Game
in Extensive Form
.
A
Heads
.
23
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
B
Heads
A
Tails
24
Heads
Tails
1 , -1
-1, 1
-1, 1
1 , -1
TABLE 6-5: Battle of the Sexes in
Normal Form
B (Husband)
A (Wife)
25
Ballet
Boxing
Ballet
2, 1
0, 0
Boxing
0, 0
1, 2
Figure 6-3: Battle of the Sexes Game
in Extensive Form
.
A (Wife)
Ballet
.
Ballet
2, 1
26
Boxing
.
B (Husband)
B (Husband)
Boxing
0, 0
Ballet
0, 0
Boxing
1, 2
TABLE 6-6: Solving for Pure-Strategy
Nash Equilibria in Battle of the Sexes
B (Husband)
A (Wife)
27
Ballet
Boxing
Ballet
2, 1
0, 0
Boxing
0, 0
1, 2
Best-Response Function

28
The function which gives the payoffmaximizing choice for one player in each
of a continuum of actions of the other
player is referred to as the best-response
function.
TABLE 6-7: Computing the Wife’s Best
Response to the Husband’s Mixed Strategy
B (Husband)
Ballet
A (Wife)
29
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
Figure 6-4: Best-Response Functions
Allowing Mixed Strategies in the Battle of
the Sexes
h
1
Wife’s bestresponse
function
1/3
.
30
.
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


31
A rule that selects the highest total payoff
would not distinguish between two purestrategy 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.