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Lecture 6 Nash Equilibrium
14.12 Game Theory
Muhamet Yildiz
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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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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.
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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.
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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.
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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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