UNIT II: The Basic Theory

• Zero-sum Games• Nonzero-sum Games• Nash Equilibrium: Properties and Problems• Bargaining Games• Review• Midterm 3/19

3/12

• Prudent v. Best-Response Strategies• Problem Sets 1 & 2• Graduate Assignment

Review

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 1

Player 2

2, 1 0, 0

0, 0 1, 2

O F

O

F

Battle of the Sexes

Review

Compare best response and prudent strategies.

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 1

Player 2

2, 1 0, 0

0, 0 1, 2

O F

O

F

Battle of the Sexes

Review

NE = {(1, 1); (0, 0); }Find all the NE of the game.

NE = {(O,O); (F,F); }

Both are correct

O F

P1

P2

2

1

Battle of the Sexes

Mixed Nash Equilibrium

Review

O

F

2, 1 0, 0

0, 0 1, 2 NE (1,1)

NE (0,0)

1 2 NE = {(1, 1); (0, 0); (MNE)}

O F

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {1/3, 2/3)}

Battle of the Sexes

Review

Let (p,1-p) = prob1(O, F )

(q,1-q) = prob2(O, F )Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q q* = 1/3

EP2(Olp) = 1p

EP2(Flp) = 2-2p

p* = 2/3

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

EP1

2/3

0

2

p=1

p=0

Review

NE = {(1, 1); (0, 0); (2/3,1/3)}

EP1 = 2q +0(1-q)

Player 1’s Expected Payoff

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

EP1

1

2/3

0

2

0

p=1

p=0

Review

p=1

p=0

NE = {(1, 1); (0, 0); (2/3,1/3)}

Player 1’s Expected Payoff

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

EP1

1

2/3

0

2

0

p=1

p=0

Review

p=1

NE = {(1, 1); (0, 0); (2/3,1/3)}

EP1 = 0q+1(1-q)

Player 1’s Expected Payoff

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

EP1

1

2/3

0

2

0

p=1

p=0

Review

Opera

Fight

NE = {(1, 1); (0, 0); (2/3,1/3)}

Player 1’s Expected Payoff

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

EP1

1

2/3

0

2

0

p=1

p=0

Review

p=1p=0

0<p<10<p<1

NE = {(1, 1); (0, 0); (2/3,1/3)}

Player 1’s Expected Payoff

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

2

0

p=1

p=0

Review

p=1p=0

p = 2/34/3

EP1

2/3

1/3

If Player 1 uses her (mixed) b-r strategy (p=2/3), her expected payoff varies from 1/3 to 4/3.

NE = {(1, 1); (0, 0); (2/3,1/3)}

q=1 q=0

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

2

0

p=1

p=0

Review

p=1p=0

EP1

2/3

1/3

2/3

If Player 2 uses his (mixed) b-r strategy (q=1/3), the expected payoff to Player 1 is 2/3, for all p.

q = 1/3

NE = {(1, 1); (0, 0); (2/3,1/3)}

4/3

O F

2, 1 0, 0

0, 0 1, 2

O

F

Battle of the Sexes

Review

Find the prudent strategy for each player.

q* = 2/3

Prudent strategies: 1/3; 2/3

O F

2, 1 0, 0

0, 0 1, 2

O

F

Battle of the Sexes

Review

Let (p,1-p) = prob1(O, F )

(q,1-q) = prob2(O, F )Then

EP1(Olp) = 2p

EP1(Flp) = 1-1p p* = 1/3

EP2(Oiq) = 1q

EP2(Flq) = 2-2q

q* = 2/3Prudent strategies: 1/3; 2/3

O F

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

2

0

O

F

Review

p=1p=0

p = 2/34/3

If Player 1 uses her prudent strategy (p=1/3), expected payoff is 2/3 no matter what player 2 does

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

EP1

2/3

1/3

2/3

p = 1/3

O F

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

2

0

O

F

Review

p=1p=0

p = 2/34/3

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

EP1

2/3

1/3

2/3 q = 1/3 2/3

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

p = 1/3

O F

P1

P2

2

12/3

Battle of the Sexes

Review

O

F

2, 1 0, 0

0, 0 1, 2 NE (1,1)

NE (0,0)

2/3 1 2

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

O F

P1

P2

2

12/3

Battle of the Sexes

Review

O

F

2, 1 0, 0

0, 0 1, 2 NE (1,1)

NE (0,0)

2/3 1 2

If both players use prudent strategies, expected payoff is 2/3 for each.

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

O F

P1

P2

2

12/3

Battle of the Sexes

Review

O

F

2, 1 0, 0

0, 0 1, 2 NE (1,1)

NE (0,0)

2/3 1 2 NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

Is the pair of prudent strategies an equilibrium?

O F

2, 1 0, 0

0, 0 1, 2

q

Battle of the Sexes

2

0

O

F

Review

p=1p=0

p = 2/34/3

Player 1’s best response to Player 2’s prudent strategy (q=2/3) is Opera (p=1).

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

EP1

2/3

1/3

2/3 q = 1/3 2/3

Opera

p = 1/3

Therefore not an equilibrium!

Review[I]f game theory is to provide a […] solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

ReviewSADDLEPOINT v. NASH EQUILIBRIUM

STABILITY: Is it self-enforcing? YES YES

UNIQUENESS: Does it identify an unambiguous course of action?YES NO

EFFICIENCY: Is it at least as good as any other outcome for all players?

--- (YES) NOT ALWAYS

SECURITY: Does it ensure a minimum payoff?YES NO

EXISTENCE: Does a solution always exist for the class of games? YES YES

Review

1. Indeterminacy: Nash equilibria are not usually unique.

2. Inefficiency: Even when they are unique, NE are not always

efficient.

Problems of Nash Equilibrium

Review

T1 T2

S1

S2

5,5 0,1

1,0 3,3

Multiple and Inefficient Nash Equilibria

When is it advisable to play a prudent strategy in a nonzero-sum game?

What do we need to know/believe about the other player?

Problems of Nash Equilibrium

Review

T1 T2

S1

S2

5,5 -99,1

1,-99 3,3

Multiple and Inefficient Nash Equilibria

When is it advisable to play a prudent strategy in a nonzero-sum game?

What do we need to know/believe about the other player?

Problems of Nash Equilibrium

ReviewDominant Strategy: A strategy that is best no matter what the

opponent(s) choose(s).

Prudent Strategy: A prudent strategy maximizes the minimum payoff a player can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Mixed Strategy: A mixed strategy for player i is a probability distribution over all strategies available to player i.

Best Response Str’gy: A strategy, s’, is a best response strategy iff Pi(s’,t) > Pi(s,t) for all s.

Dominated Strategy: A strategy is dominated if it is never a best response strategy.

ReviewSaddlepoint: A set of prudent strategies (one for each

player), s. t. (s*, t*) is a saddlepoint, iff maxmin = minmax.

Nash Equilibrium: a set of best response strategies (one for each player), (s’,t’) such that s’ is a best response to t’ and t’ is a b.r. to s’.

Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs.

Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible.