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Nash Equilibrium: Theory

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Strategic or Simultaneous-moveGames

Definition: A simultaneous-move gameconsists of:

• A set of players

• For each player, a set of actions

• For each player, a preference relation overthe set of action profiles.

Or• For each player, a payoff function over the

set of action profiles.

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Example I: The Prisoner’s Dilemma

• Two suspects (Bonnie and Clyde) in a crime, held inseparate cells.

• Enough evidence to convict each on minor, but notmajor offense unless one confesses.

• Each can remain silent or can confess.

• Both remain silent: each convicted of minoroffense—1 year in prison.

• One and only one confesses: one who confesses isused as a witness against the other, and goes free;

other gets 4 years.• Both confess: each gets 3 years in prison.

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Example I: The Prisoner’s Dilemma

Strategic game: two players; each player’s

actions {RS, C}; preference orderings

• (C,RS) >1 (RS,RS) >1 (C, C) >1 (RS, C)

• (RS, C) >2 (RS,RS) >2 (C, C) >2 (C,RS).

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Example II: Working on a joint project

• Each of two people can either work hard or goofoff.

• Each person prefers to goof off if the other works

hard—project would be better if worked hard,but not worth the extra effort.

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Example II: Working on a joint project

Person 2

Work hard Goof off

Person 1

Work hard 2, 2 0, 3

Goof off 3, 0 1, 1

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Example III: Time vs. Newsweek

• Two major news stories: – There is an impasse between the House and the

Senate on the budget

– A new drug is claimed to be effective against AIDS• 100 newsstand buyers

– 70 interested on the AIDS story

– 30 interested on the budget story• If both magazines go for the same story the buyers

split equally between them

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Example III: Time vs. Newsweek

Newsweek

AIDS Budget

Time

AIDS 35, 35 70, 30

Budget 30, 70 15, 15

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Example IV: Duopoly

• Two firms producing same good.

• Each firm can charge high price or low price.

• Both firms charge high price: each gets profit of

$1,000

• One firm charges high price, other low: firmcharging high price gets no customers and loses

$200, while one charging low price makes profitof $1,200

• Both firms charge low price: each earns profit of$600

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Example IV: Duopoly

Firm 2

High Low

Firm 1

High 1,000, 1,000 -200, 1,200

Low 1,200, -200 600, 600

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Example V: Nuclear arms race

• Two countries

• What matters is relative strength, and bombs arecostly, so that for each country:

have bombs, other doesn’t > neither country hasbombs > both countries have bombs > don’thave bombs, other does

• The game is Prisoner’s Dilemma, with RS

means don’t build bombs and C means buildbombs.

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Example VI: Battle of the sexes

• In the Prisoner’s dilemma the players agree that(RS,RS) is a desirable outcome, though each has anincentive to deviate from this outcome.

• In BoS the players disagree about the outcome that is

desirable.• Peter and Jane wish to go out together to a concert.

The options are U2 or Coldplay.• Their main concern is to go out together, but one

person prefers U2 and the other person prefersColdplay.• If they go to different concerts then each of them is

equally unhappy listening to the music of either band.

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Example VI: Battle of the sexes

Jane

U2 Coldplay

Peter

U2 2, 1 0, 0

Coldplay 0, 0 1, 2

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Example VII: Matching Pennies

• In the Prisoner’s dilemma and BoS there are aspects

of both conflict and cooperation. Matching Pennies ispurely conflictual.

• Each of two people chooses either the Head or Tail ofa coin.

• If the choices differ, person 1 pays person 2 a dollar.

• If they are the same, person 2 pays person 1 a dollar.• Each person cares only about the amount of money

that she receives.

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Player 2

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

- This is an example of a zero-sum game.

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• In this game the players’ interests are

diametrically opposed (such a game iscalled ―strictly competitive‖): player 1

prefers to take the same action as theother player, while player 2 prefers to takethe opposite action.

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• Dominant Strategy: a strategy that is best for aplayer in a game, regardless of the strategies chosenby the other players.

• Rational players do not play strictly dominatedstrategies and so, once you determine a strategy isdominated by another, simply remove it from

consideration -- it will not be part of ANY equilibrium.

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Strictly Dominated Strategies

A player’s action is ―strictly dominated‖ if it is inferior,

no matter what the other players do, to some otheraction.

Definition: In a strategic game player i’ s action bi strictly dominates her action b’

i if

ui(bi, a-i) > ui(b’i, a-i) for every list a-i of the otherplayers’ actions.

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Iterated Elimination of StrictlyDominated Strategies

For Clyde, ―remain silent‖ is a dominated strategy.

So, ―remain silent‖ should be removed from his

strategy space.

Bonnie

RS C

Clyde

RS 1, 1 4, 0

C 0, 4 3, 3

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Bonnie

RS C

ClydeRS 1, 1 4, 0

C 0, 4 3, 3

Given the symmetry of the game, it is easy to seethat ―remain silent‖ is a dominated strategy for

Bonnie also. So, ―remain silent‖ should be

removed from her strategy space, too.

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Player 2

Player 1

Left Middle Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

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Player 2

Player 1

Left Middle Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

IESDS yields a unique result!

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In many cases, iterated elimination of strictlydominated strategies may not lead to a uniqueresult.

Player 2

Player 1

Left Middle Right

Top 0,4 4,0 5,3

Middle 4,0 0,4 5,3

Bottom 3,5 3,5 6,6

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Definition: an strategy is a complete plan ofaction (what to do in every contingency).

In simultaneous-move games a Pure-Strategy is simply an action.

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Nash Equilibrium

Definition: The strategy profile a* in astrategic game is a Nash equilibrium if, foreach player i and every strategy bi of

player i , a* is at least as good for player i as the strategy profile (bi, a*-i):

ui(ai*, a*-i) ≥ ui(bi, a*-i) for every strategy bi ofplayer i .

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Nash Equilibrium: Prisoner's Dilemma

Bonnie

RS C

Clyde

RS 1, 1 4, 0

C 0, 4 3, 3

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Nash Equilibrium: Prisoner's Dilemma

Bonnie

RS C

Clyde

RS 1, 1 4, 0

C 0, 4 3, 3

(C, C) is the unique Nash Equilibrium in pure-strategies

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Nash Equilibrium: BoS

Jane

U2 Coldplay

Peter

U2 2, 1 0, 0

Coldplay 0, 0 1, 2

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Nash Equilibrium: Matching Pennies

Player 2

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

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Nash Equilibrium: Matching Pennies

Player 2

Head Tail

Player 1

Head 1, -1 -1, 1

Tail -1, 1 1, -1

In Matching pennies there are no Nash equilibria inpure-strategies.

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Strict Nash Equilibrium

Player 2

Left Right

Player 1

Top 1, 1 0, 0

Bottom 0, 0 0, 0

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Strict Nash Equilibrium

Player 2

Left Right

Player 1

Top 1, 1 0, 0

Bottom 0, 0 0, 0

(Top, Left) strict Nash equilibrium

(Bottom, Right) Nash equilibrium, but Not strict

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Nash Equilibrium

Exercise 34.1(Guessing two-thirds of the average)

Each of three people announces an integer from 1 to K. Ifthe three integers are different, the person whose integer isclosest to 2/3 of the average of the three integers wins $1. Iftwo or more integers are the same, $1 is split equally

between the people whose integer is closest to 2/3 of theaverage.

• Is there any integer k such that the strategy profile (k, k, k),in which every person announces the same integer k, is aNash equilibrium? (if k ≥ 2, what happens if a person

announces a smaller number?)• Is any other strategy profile a Nash Equilibrium? (what is the

payoff of a person whose number is the highest of the three?Can she increase this payoff by announcing a differentnumber?)

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Best Response Functions(Correspondences)

For any list of actions a-i of all the players otherthan i , let Bi(a-i) be the set of player i’ s bestactions, given that every other player j chooses

a j :

Bi(a-i) = {ai

Ai : ui(ai, a-i) ≥ ui(bi, a-i) for all bi

Ai}.

Bi is the best response function of player i .

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Best Response Functions(Correspondences)

• Bi is a set-valued function: its values aresets, not points. Every member of the setBi(a-i) is a best response of player i to a-i: if

each of the other players adheres to a-i thenplayer i can do no better than choose amember of Bi(a-i).

a-i

Bi(a-i)

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Alternative definition of Nashequilibrium

Proposition: The strategy profile a* is aNash equilibrium of a strategic game if andonly if every player’s action is a best

response to the other players’ actions:

a*i

Bi(a*-i) for every player i (1)

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Alternative definition of Nashequilibrium

If each player i has a unique best response toeach a-i.

Bi(a-i) = {bi(a-i)}

ai = bi(a-i) for every player i (1’)

a collection of n equations in the n unknownsa*i, where n is the number of players in the

game.

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Example: Synergistic Relationship

• Two individuals

• Each decides how much effort to devote torelationship

• Amount of effort is nonnegative real number• If both individuals devote more effort to the

relationship, then they are both better off; for anygiven effort of individual j, the return to individuali’s effort first increases, then decreases.

• Specifically, payoff of i: ai(c + a j - ai), where c > 0is a constant.

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Payoff to player i: ai(c+aj-ai ).

First Order Condition: c + aj – 2ai = 0.

Second Order Condition: -2 < 0, Max

Solve FOC: ai = (c+aj )/2 = bi(aj )

Symmetry implies: aj = (c+ai )/2 = bj(ai )

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Symmetry of quadratic payoff functions impliesthat the best response of each individual i to a j is

bi(a j ) = 1/2 (c + a j )

Nash equilibria: pairs (a*1, a*2) that solve the twoequations

a1 = 1/2 (c + a2)

a2 = 1/2 (c + a1)Unique solution, (c, c)

Hence game has a unique Nash equilibrium

(a* , a* ) = (c, c)

02 ne theory

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