Game%20Theory%20and%20the%20Nash%20Equilibrium

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Game Theoryand the Nash Equilibrium Part 2
Eponine Lupo
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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

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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
4
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)

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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
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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?

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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.

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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)
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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)
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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.

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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

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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

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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