Week 2
Nash equilibrium and Rationalizability
Reading: Osborne: Ch 1,2.1-2.5, 12
Nash equilibrium
A strategy pro le stfst.se sat is a Nash equilibrium if for every player ien
u st.si si sntzvils sat si
sntlfsies ie.steBRils
So, when others are playing s player i has no strategy s which gives her a strictly higher payo
than sit
MIXED STRATEGY
Example
A B
oe it d
C
i 1 o.o it
iii i
q
bio.azo.i.iIo.zi
i
i pg ti 1st o.o
u
an pqpitpta.pe
c z indeed
steBR s Hi s eBRi s
p 42161071.2
Underlinedarebest
replycorrespondences BR s
n
a up sp p'ssp
Bottomrowstrictlydominatedwhen
Mixed strategies sp l
sp a p l
A mixed strategy of player i ismeals mils.ttiiagtngisiunaerm ptssp i s
Take a pure strategy sesimics Is they probability that player i will play s if he follows
A pro le of strategies is m m mnexien si
Players randomise independently of each otherpram mils
Then player i’s expected payo is
Icu
mi.milsEsuilsi.s.l
Inmjcs
However, we will think in terms of the payo from a pure strategy when others play mixed ones:
Icu si.milJeguisis iTmjls
Nash equilibrium
Nash’s Theorem:
Let G be a game with nitely many players, where each player’s strategy set is nite. G has an NE,
possibly in MS
strategiesso csiandsiarebothinsupportoemi
1 Its si mics oandmils o iethen
itproeplayingtnese
t he sands
prayerm ustb eindictmentb etween
ucsism.ilucsi.mil
ieittheyneverchoosesi.tnentheapectedpayo from
2Hsis missilemissile
ucsim.psuicsi.mg sih asaweakly utility
greaterexpected
Nash equilibrium and Rationalizability
Reading: Osborne: Ch 1,2.1-2.5, 12
Nash equilibrium
A strategy pro le stfst.se sat is a Nash equilibrium if for every player ien
u st.si si sntzvils sat si
sntlfsies ie.steBRils
So, when others are playing s player i has no strategy s which gives her a strictly higher payo
than sit
MIXED STRATEGY
Example
A B
oe it d
C
i 1 o.o it
iii i
q
bio.azo.i.iIo.zi
i
i pg ti 1st o.o
u
an pqpitpta.pe
c z indeed
steBR s Hi s eBRi s
p 42161071.2
Underlinedarebest
replycorrespondences BR s
n
a up sp p'ssp
Bottomrowstrictlydominatedwhen
Mixed strategies sp l
sp a p l
A mixed strategy of player i ismeals mils.ttiiagtngisiunaerm ptssp i s
Take a pure strategy sesimics Is they probability that player i will play s if he follows
A pro le of strategies is m m mnexien si
Players randomise independently of each otherpram mils
Then player i’s expected payo is
Icu
mi.milsEsuilsi.s.l
Inmjcs
However, we will think in terms of the payo from a pure strategy when others play mixed ones:
Icu si.milJeguisis iTmjls
Nash equilibrium
Nash’s Theorem:
Let G be a game with nitely many players, where each player’s strategy set is nite. G has an NE,
possibly in MS
strategiesso csiandsiarebothinsupportoemi
1 Its si mics oandmils o iethen
itproeplayingtnese
t he sands
prayerm ustb eindictmentb etween
ucsism.ilucsi.mil
ieittheyneverchoosesi.tnentheapectedpayo from
2Hsis missilemissile
ucsim.psuicsi.mg sih asaweakly utility
greaterexpected
