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Game Theoryand the Nash Equilibrium Part 2

Eponine Lupo

Agenda

- Questions from last time
- 3 player games
- Games larger than 2x2rock, paper, scissors
- Review/explain Nash Equilibrium
- Nash Equilibrium in R
- Instability of NEmove towards pure strategy
- Prisoners Dilemma, Battle of the Sexes, 3rd Game
- Application to Life

3-Player Game

2 L R

2 L R

14 , 24 , 32 8 , 30, 27

30 , 16 , 24 13 , 12, 50

16 , 24 , 30 30 , 16, 24

30 , 23 ,14 14 , 24, 32

L

L

1

1

R

R

Strategy Profile R,L,L is the Solution to this

Game

L

R

3

Rock, Paper, Scissors

Player 2 R P S

0 , 0 -1 , 1 1 , -1

1 , -1 0 , 0 -1 , 1

-1 , 1 1 , -1 0 , 0

R

Player 1

P

S

- No pure strategy NE
- Only mixed NE is (1/3,1/3,1/3),(1/3,1/3,1/3)

Nash Equilibrium

- A strategy profile is a Nash Equilibrium if and

only if each players prescribed strategy is a

best response to the strategies of others - Equilibrium that is reached even if it is not the

best joint outcome

Player 2 L C R

Strategy Profile D,C is the Nash

Equilibrium There is no incentive for either

player to deviate from this strategy profile

4 , 6 0 , 4 4 , 4

5 , 3 0 , 0 1 , 7

1 , 1 3 , 5 2 , 3

U

Player 1

M

D

Mixed Strategy NE

- 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?

Mixed Strategy NE

Player 2 R P S

0 , 0 -1 , 1 1 , -1

1 , -1 0 , 0 -1 , 1

R

Player 1

P

- Once Player 1s strategy of S is taken away,

Player 2s strategy R is iteratively dominated by

strategy P.

Mixed Strategy NE

Player 2 q 1-q P

S

- 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)
- 1p0(1-p) (-1)p1(1-p)
- 1p -2p1
- 3p 1
- p1/3

-1 , 1 1 , -1

0 , 0 -1 , 1

p R

Player 1

1-p 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)

Mixed Strategy NE

Player 2 q 1-q P

S

- 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))
- -1q1(1-q) 0q(-1)(1-q)
- -2q1 q-1
- 3q 2
- q2/3

-1 , 1 1 , -1

0 , 0 -1 , 1

p R

Player 1

1-p P

S2 (2/3 , 1/3)

Mixed Strategy NE

- 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

2s payoff may be affected - Thus, it is a Mixed Strategy Nash Equilibrium.

Nash Equilibrium in R

- 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 2s mixed strategy (q, 1-q),

player 1 adjusts his mixed strategy of (p,1-p) in

order to maximize his payoffs - With player 1s 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

Nash Equilibrium in R

- 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 dont know

Implications

- In life, we react to other peoples 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

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