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UNIT II: The Basic Theory

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UNIT II: The Basic Theory

- Zero-sum Games
- Nonzero-sum Games
- Nash Equilibrium: Properties and Problems
- Bargaining Games
- Review
- Midterm 3/21

3/14

Review

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

Review

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

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

Player 2

Compare best response and prudent strategies.

Review

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

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

Player 2

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

Both are correct

Find all the NE of the game.

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

Battle of the Sexes

Review

O F

P2

2

1

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

1 2

P1

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

Mixed Nash Equilibrium

Battle of the Sexes

Review

O F

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

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

q=1 q=0

Player 1’s Expected Payoff

EP1

2/3

0

2

EP1 = 2q +0(1-q)

2,1 0,0

0, 0 1, 2

p=1

p=0

q

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

Battle of the Sexes

Review

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

p=0

q

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

Battle of the Sexes

Review

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

EP1 = 0q+1(1-q)

q

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

Battle of the Sexes

Review

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

Opera

2,1 0,0

0, 0 1, 2

p=1

p=0

Fight

q

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

Battle of the Sexes

Review

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

p=1

0<p<1

p=0

0<p<1

q

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

Battle of the Sexes

Review

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

q=1 q=0

EP1

2/3

1/3

2

0

2,1 0,0

0, 0 1, 2

p = 2/3

p=1

p=0

4/3

p=1

p=0

q

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

Battle of the Sexes

Review

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 q=0

EP1

2/3

1/3

2/3

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

4/3

p=1

p=0

q = 1/3

q

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

Battle of the Sexes

Review

O F

Find the prudent strategy for each player.

q* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

Prudent strategies: 1/3; 2/3

Battle of the Sexes

Review

O F

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* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

Prudent strategies: 1/3; 2/3

Battle of the Sexes

Review

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

O F

EP1

2/3

1/3

2/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q

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

Battle of the Sexes

Review

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

O F

EP1

2/3

1/3

2/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/3 2/3

q

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

Battle of the Sexes

Review

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

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

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

Battle of the Sexes

Review

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

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

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

Battle of the Sexes

Review

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

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

Is the pair of prudent strategies an equilibrium?

Battle of the Sexes

Review

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

O F

EP1

2/3

1/3

2/3

2

0

Opera

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/32/3

q

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

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).

Review

SADDLEPOINT 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

Problems of Nash Equilibrium

- Indeterminacy: Nash equilibria are not usually unique.
2. Inefficiency: Even when they are unique, NE are not always efficient.

Review

Problems of Nash Equilibrium

T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 0,1

1,0 3,3

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?

Review

Problems of Nash Equilibrium

T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 -99,1

1,-99 3,3

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?

Review

Dominant 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.

Review

Saddlepoint: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.

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