Eponine Lupo
 Questions


from last time
3 player games
Games larger than 2x2—rock, paper, scissors
 Review/explain
Nash Equilibrium
 Nash Equilibrium in R


Instability of NE—move towards pure strategy
Prisoner’s Dilemma, Battle of the Sexes, 3rd Game
 Application
to Life
2
L
L
14 , 24 , 32
2
R
L
8 , 30, 27
1
R
L
16 , 24 , 30
30 , 16, 24
R
30 , 23 ,14
14 , 24, 32
1
R
30 , 16 , 24
Strategy
Profile: {R,L,L}
is the Solution
to this Game
13 , 12, 50
L
R
3
Player 1
R
Player 2
P
S
R
0,0
-1 , 1
1 , -1
P
1 , -1
0,0
-1 , 1
S
-1 , 1
1 , -1
0,0
•No pure strategy NE
•Only mixed NE is {(1/3,1/3,1/3),(1/3,1/3,1/3)}
 “A
strategy profile is a Nash Equilibrium if
and only if each player’s prescribed strategy
is a best response to the strategies of
others”

Equilibrium that is reached even if it is not the
best joint outcome
L
Player 1
Player 2
C
R
U
4,6
0,4
4,4
M
5,3
0,0
1,7
D
1,1
3,5
2,3
Strategy Profile:
{D,C} is the Nash
Equilibrium
**There is no
incentive for
either player to
deviate from this
strategy profile

Sometimes there is
NO pure Nash
Equilibrium, or there
is more than one pure
Nash Equilibrium

In these cases, use
Mixed Strategy Nash
Equilibriums to solve
the games

Take for example a
modified game of
Rock, Paper, Scissors
where player 1
cannot ever play
“Scissors”
What now is the Nash
Equilibrium?
Put another way, how
are Player 1 and
Player 2 going to
play?
R
Player 2
P
S
R
0,0
-1 , 1
1 , -1
P
1 , -1
0,0
-1 , 1
Player 1

Once Player 1’s strategy of S is taken away,
Player 2’s strategy R is iteratively dominated by
strategy P.
Player 2
p
q
1-q
P
S
R
-1 , 1
1 , -1
P
0,0
-1 , 1

Player 1 wants to have a
mixed strategy (p, 1-p) such
that Player 2 has no
advantage playing either
pure strategy P or S.

u2((p, 1-p),P)=u2((p, 1-p),S)
1p+0(1-p) = (-1)p+1(1-p)
1p = -2p+1
3p = 1
p=1/3
Player 1
1-p

Now the game has been
cut down from a 3x3 to
2x2 game

There are still no pure
strategy NE

From here we can
determine the mixed
strategy NE
S1 = (1/3 , 2/3)
Player 2
p
q
1-q
P
S
R
-1 , 1
1 , -1
P
0,0
-1 , 1

Likewise, Player 2 wants to
have a mixed strategy (q, 1-q)
such that Player 1 has no
advantage playing either pure
strategy R or P.

u1(R,(q, 1-q))=u1(P,(q, 1-q))
-1q+1(1-q) = 0q+(-1)(1-q)
-2q+1 = q-1
3q = 2
q=2/3
Player 1
1-p
S2 = (2/3 , 1/3)

Therefore the mixed strategy:


Player 1: (1/3Rock , 2/3Paper)
Player 2: (2/3Paper , 1/3Scissors)
is the only one that cannot be “exploited” by either
player.

The values of p and q are such that if Player 1
changes p, his payoff will not change but Player 2’s
payoff may be affected

Thus, it is a Mixed Strategy Nash Equilibrium.
The Nash Equilibrium is a very unstable point
 If you do not begin exactly at the NE, you cannot
stochastically find the NE


Theoretically you will “shoot off” to a pure strategy:
(0,0) (0,1) (1,0) or (1,1)


(similar for n players)
Consider the following:




2 players randomly choose values for p and q
Knowing player 2’s mixed strategy (q, 1-q), player 1
adjusts his mixed strategy of (p,1-p) in order to
maximize his payoffs
With player 1’s new mixed strategy in mind, player 2
will adjust his mixed strategy in order to maximize his
payoffs
This see-saw continues until both players can no
longer change their strategies to increase their
payoffs
 Unfortunately,
I was unable to find a way to
discover a mixed strategy NE in R for any
number of players



Is my code wrong?
Is there simply no way to find the NE in R?
I don’t know
 In
life, we react to other people’s choices in
order to increase our utility or happiness


Ignoring a younger sibling who is irritating
Accepting an invitation to go to a baseball game
 Once
we react, the other person reacts to
our reaction and life goes on

One stage games are rare in life
 Very
rarely are we in a “NE” for any aspect
of our lives

There is almost always a choice that can better
our current utility