Math Intro: Matrix Form Games and Nash Equilibrium

Let’s start with an informal discussion of what a game is and when it’s useful…

Components of a game:

PlayersE.g., animals, people, firms, countriesStrategiesE.g., attack Syria, feed brother, have sonsPayoffsE.g., offspring, $, happinessDepend onYour strategyOthers’ strategies

Games are used to describe situations in which payoffs don’t just depends on your actions, but on others’ too

Games are useful: whether to read Freakonomics or the Selfish Gene

Not needed: whether to choose chocolate or vanilla soft-serve

Let’s add a bit of formalization

Let’s start by making some simplifying assumptions (some of which we’ll relax later in the course; and all of which can be relaxed)…

We’ll restrict to a finite number of strategies

Yes: Should I press the gas pedal or the break?Not: How much should I push the gas pedal?

In this class, almost always, we’ll have just 2 players

Players move simultaneously

Yes: Moshe and Erez both decide what color shirt to wear each morningNot: Erez decides what color shirt to wear after seeing whether Moshe wore black again

Then, can present using payoff “matrix”

Games that can be presented this way are called “Matrix Form Games”

Let’s do this for three famous examples…

Prisoners’ Dilemma

3 is the cost of cooperation5 is the benefit if partner cooperates

2, 2 -3,5

5, -3 0, 0

CD

C D

Prisoners’ Dilemma

c>0 is the cost of cooperationb>c is the benefit if partner cooperates

b-c, b-c -c, b

b, -c 0, 0

CD

C D

Prisoners’ dilemma useful for studying “cooperation”

E.g.,

VotingLoveCharity“Prosocial”“Altruism”

Coordination Game

Both better off if play the same: a > c, d > bCan have d > a or vice versa

a,a b,c

c,b d,d

LR

L R

Coordinate game useful for studying

E.g.,

Should we attack Syria?Should we enforce a norm against chemical

weapons?Should we drive on the left?Innuendos

Hawk-Dove

Object worth v>0Cost of fighting c>vGet object if only H, o/w split

(v/2)-c v

0 v/2

HD

H D

Hawk-Dove useful for studying

E.g.,

TerritorialityRightsApologies

Next, let’s see how we solve matrix form games…

What makes this hard is that player 1’s optimal choice depends on what player 2 does

And player 2’s optimal choice depends on what player 1 does

So where do we start?

Nash solved this

Nash equilibrium specifies a strategy pair such that no one benefits from deviating…

… provided no one else deviated

I.e., ceteris parabus, or holding everyone’s actions fixed

Let’s go back and solve for the Nash equilibria of our three example games

Prisoners’ Dilemma

c>0 is the cost of cooperationb>c is the benefit if partner cooperates

b-c, b-c -c, b

b, -c 0, 0

CD

C D

Coordination Game

Both better off if play the same: a > c, d > bCan have d > a or vice versa

a,a b,c

c,b d,d

LR

L R

Hawk-Dove

Object worth v>0Cost of fighting c>vGet object if only H, o/w split

(v-c)/2 v

0 v/2

HD

H D

Note that Nash doesn’t pick out socially optimal solution

E.g., In PD, NE is (D,D) even though social optimum is

(C,C)

In coordination game, both (L,L) and (R,R) are equilibria even if one yields higher payoffs for everyone

What this means for us…

Recall our thesis: Preferences/ideologies that are learned will end up being consistent with Nash

This means that if Nash is inefficient, our preferences/ideologies will be as well!

More generally, if Nash has some weird, counterintuitive property, so will our preferences. This will explain many of our puzzles

Finally, let’s discuss some notation that you should be familiar with, and which we will use towards the end of the class

s1 is a strategy available to player 1

E.g., CooperateRightHawk

S1 is the set of all available strategies available to player 1

E.g., {Cooperate, Defect}{Left, Right}{Hawk, Dove}

u1 is the payoff to player 1

Remember that it depends on player 1’s strategy and player 2’s strategy

So we write, u1(s1, s2)

E.g., u1(C,D) = -cu2(H,D) = 0

Finally, we’re ready to define Nash

A Nash equilibrium is a strategy pair (s1, s2) such that:

u1(s1,s2) ≥ u1(s1’,s2) for any s1’ in S1

andu2(s1,s2) ≥ u1(s1,s2’) for any s2’ in S2

That’s enough formalization for now. We’ll add a little more here and there as we need it

When the time comes, we’ll learn how to deal with games where:

Players don’t move simultaneouslyThe game repeatsPlayers don’t have “complete information”

While we won’t need to go here, you should just be aware that game theory generalizes further. E.g.,

Continuous strategies More than two players“Mixing”