Lecture 6
Nash Equilibrium
14.12 Game Theory
Muhamet Yildiz
1
Road Map
1.
2.
3.
4.
5.
Definition
Examples
Mixed-strategy Nash Equilibrium
Relation to other solution concepts
Population Dynamics
2
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)
>
*
*
*
*
uJs J , ... ,Si_ pSi ,Si+J'''' ,sn)'
i.e., no player has any incentive to deviate if he
knows what the others play.
3
Hawk-Dove game
⎛V − c V − c ⎞
,
⎟
⎜
2 ⎠
⎝ 2
(0,V)
(V,0)
V<c
(V/2,V/2)
Image by MIT OpenCourseWare.
4
Stag Hunt
(2,2)
(4,0)
(0,4)
(5,5)
Image by MIT OpenCourseWare.
5
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.
6
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*
l.eo 1of a j *()
SI > 0'Sj IS a b est response to a_I *
0
•
0
0
0
7
Stag Hunt
Assume: Player 2 thinks
that, with probability p,
Player 1 targets for Rabbit.
(2,2)
(2,0)
p
(0,2)
(3,3)
1-p
His payoff from targeting Rabbit:
U2(R;p) = .
His payoff from targeting Stag:
U2(S;p) =
She is indifferent iff
Image by MIT OpenCourseWare.
8
Mixed-strategy equilibrium in Stag-Hunt game
u
3
2 r---~~--------~
o
~--------------~
o
p
qBR (p)= q E
°
[0,1]
if p < 113
if P= 113
1
if p > 113
9
Best responses in Stag-Hunt game
q
-------------------, . - - - - - ----(')
1/3
p
1/3
10
Relation to Other Solution
Concepts
• Dominant Strategy => Nash Equilibrium
• Nash Equilibrium => Rationalizability
11
• h = Pr(2 plays Hawk)
• d =game
Pr(2 plays Dove)
Hawk-Dove
• Indifference of 1:
(V-c)h/2 +Vd = Vd/2
• h
=
Vic
C
, /1,_
/2---,) • Nash Equilibria =
'---_(O_'_V)_....L..-/1,_/2_
⎛V − c V − c ⎞
,
⎟
⎜
2 ⎠
⎝ 2
(0,V)
(h (V
= Vic , d= (c-V)/c)
,0)
(h=O,d=l)
(V/2,V/2)
Image by MIT OpenCourseWare.
12
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?
13
MIT OpenCourseWare
http://ocw.mit.edu
14.12 Economic Applications of Game Theory
Fall 2012
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