Lecture 6 Nash Equilibrium

14.12 Game Theory

Muhamet Yildiz

1

Road Map

1. Definition

2. Examples

3. Mixed-strategy Nash Equilibrium

4. Relation to other solution concepts

5. Population Dynamics

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N ash Equilibrium Definition: A strategy-profile s* =(Sj *, .. "sn *) is a

Nash Equilibrium iff, for each player i, and for each strategy Sj, we have

* * * * * ui(Sj , ... ,Si_ pSi ,Si+P'" ,sn)

> * * * * uJsJ , ... ,Si_ pSi ,Si+J'''' ,sn)'

i.e., no player has any incentive to deviate if he knows what the others play.

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4

Hawk-Dove game

(V V /2)(0,V )

(V ,0)⎟⎠⎞

⎜⎝⎛ −−

2,

2 V < c

/2,

c V c V

Image by MIT OpenCourseWare.

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Stag Hunt

(5,5)(0,4)

(4,0)(2,2)

Image by MIT OpenCourseWare.

Equilibrium in Mixed Strategies

What is a strategy? - A complete contingent-plan of a player.

- What the others think the player might do under various contingencies.

- A social convention

What do we mean by a mixed strategy? - The player is randomly choosing his pure strategies.

- The other players are not certain about what he will do.

- The distribution of the behavior in a society.

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Mixed Strategy Nash Equilibrium

• A mixed strategy profile a* =( a 1 *,000 ,an *) is a Nash Equilibrium iff, for each player i, at is a "best response" when all the other players play according to a* 0

• l.eo 0 1 of a j *() SI > 0 'Sj IS 0 a b est response to a_I * 0

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Stag Hunt

(3,3)(0,2)

(2,0)(2,2)

Assume: Player 2 thinks that, with probability p, Player 1 targets for Rabbit.

p

1-p

His payoff from targeting Rabbit: His payoff from targeting Stag:

U2(R;p) = . U2(S;p) =

She is indifferent iff

Image by MIT OpenCourseWare.

Mixed-strategy equilibrium in Stag-Hunt game

u 3

2 r---~~--------~

o o ~--------------~

p ° if p < 113 qBR (p)= q E [0,1] if P = 113

1 if p > 113

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Best responses in Stag-Hunt game q

-------------------,.---------(')

1/3

p

1/3

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Relation to Other Solution Concepts

• Dominant Strategy => Nash Equilibrium

• Nash Equilibrium => Rationalizability

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• h = Pr(2 plays Hawk)

• d = Pr(2 plays Dove) • Indifference of 1 :

(V-c)h/2 +Vd = Vd/2

• h = Vic C

'---_( O_'_V)_....L..-/1,_/2_, /1,_/2---,) • Nash Equilibria = (h = Vic , d= (c-V)/c)

(h=O,d=l)

Hawk-Dove game

(V V /2)(0,V )

(V ,0)⎟⎠⎞

⎜⎝⎛ −−

2,

2

/2,

c V c V

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Image by MIT OpenCourseWare.

Evolution of Hawks and Doves

• There are H hawks and D doves; Hand D large.

• Animals are randomly matched and get "payoffs" as in left.

• The" payoff" of an animal is the number of its offsprings.

• What is the ratio of Hawks 1M years later?

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14.12 Economic Applications of Game TheoryFall 2012

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