UNIT II: The Basic Theory
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Transcript of 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 3/23
3/9
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• Review Terms• Counting Strategies• Prudent v. Best-Response Strategies• Graduate Assignment
Review
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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.
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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.
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 2 has 4 strategies:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 2 has 4 strategies:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 2 has 4 strategies:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
GAME 2: Button-Button
Player 2 has 4 strategies:
-2 4 -2 4
2 -1 -1 2
L
R
LL RR LR RL
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
If Player 2 cannot observe Player 1’s choice …
Player 2 will have fewer strategies.
GAME 2: Button-Button
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Counting Strategies
Left Right
L R L R
-2 4 2 -1
Player 1
Player 2
-2 4
2 -1
L R
L
R
GAME 2: Button-Button
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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
Find all the NE of the game.
Prudence v. Best Response
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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
NE = {(1, 1); (0, 0); }
Prudence v. Best Response
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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
Prudence v. Best Response
NE = {(O,O); (F,F); }
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O F
P1
P2
2
1
Battle of the Sexes
Mixed Nash Equilibrium
Prudence v. Best Response
O
F
2, 1 0, 0
0, 0 1, 2 NE (1,1)
NE (0,0)
1 2 NE = {(1, 1); (0, 0); (MNE)}
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O F
2, 1 0, 0
0, 0 1, 2
O
F
NE = {(1, 1); (0, 0); (MNE)}
Battle of the Sexes
Prudence v. Best Response
Let (p,1-p) = prob1(O, F ) (q,1-q) = prob2(O, F )
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O F
2, 1 0, 0
0, 0 1, 2
O
F
NE = {(1, 1); (0, 0); (MNE)}
Battle of the Sexes
Prudence v. Best Response
Let (p,1-p) = prob1(O, F ) (q,1-q) = prob2(O, F )Then
EP1(Olq) = 2q EP1(Flq) = 1-1q
EP2(Olp) = 1p EP2(Flp) = 2-2p
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O F
2, 1 0, 0
0, 0 1, 2
O
F
NE = {(1, 1); (0, 0); (2/3,1/3)} }
Battle of the Sexes
Prudence v. Best Response
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
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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
Prudence v. Best Response
p=1
NE = {(1, 1); (0, 0); (2/3,1/3)}
Player 1’s Expected Payoff
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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
Prudence v. Best Response
NE = {(1, 1); (0, 0); (2/3,1/3)}
EP1 = 2q +0(1-q)
Player 1’s Expected Payoff
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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
Prudence v. Best Response
p=1
p=0
NE = {(1, 1); (0, 0); (2/3,1/3)}
Player 1’s Expected Payoff
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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
Prudence v. Best Response
p=1
NE = {(1, 1); (0, 0); (2/3,1/3)}
EP1 = 0q+1(1-q)
Player 1’s Expected Payoff
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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
Prudence v. Best Response
Opera
Fight
NE = {(1, 1); (0, 0); (2/3,1/3)}
Player 1’s Expected Payoff
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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
Prudence v. Best Response
p=1p=0
0<p<10<p<1
NE = {(1, 1); (0, 0); (2/3,1/3)}
Player 1’s Expected Payoff
q = 1/3
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q=1 q=0
2, 1 0, 0
0, 0 1, 2
q
Battle of the Sexes
2
0
p=1
p=0
Prudence v. Best Response
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 = 1/3
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q=1 q=0
2, 1 0, 0
0, 0 1, 2
q
Battle of the Sexes
2
0
p=1
p=0
Prudence v. Best Response
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 = 1/3NE = {(1, 1); (0, 0); (2/3,1/3)}
4/3
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O F
2, 1 0, 0
0, 0 1, 2
O
F
Battle of the Sexes
Prudence v. Best Response
Find the prudent strategy for each player.
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O F
2, 1 0, 0
0, 0 1, 2
O
F
Battle of the Sexes
Prudence v. Best Response
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
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O F
2, 1 0, 0
0, 0 1, 2
O
F
NE = {(1, 1); (0, 0); (2/3,1/3)}
Battle of the Sexes
Prudence v. Best Response
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
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O F
2, 1 0, 0
0, 0 1, 2
q
Battle of the Sexes
2
0
O
F
Prudence v. Best Response
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
1/3p = 1/3
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O F
2, 1 0, 0
0, 0 1, 2
q
Battle of the Sexes
2
0
O
F
Prudence v. Best Response
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
q = 1/3
If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.
p = 1/3
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O F
P1
P2
2
12/3
Battle of the Sexes
Prudence v. Best Response
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)}
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O F
P1
P2
2
12/3
Battle of the Sexes
Prudence v. Best Response
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)}
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O F
P1
P2
2
12/3
Battle of the Sexes
Prudence v. Best Response
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)}
BATNA: Best Alternative to a Negotiated Agreement
BATNA
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O F
P1
P2
2
12/3
Battle of the Sexes
Prudence v. Best Response
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?
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O F
2, 1 0, 0
0, 0 1, 2
q
Battle of the Sexes
2
0
O
F
Prudence v. Best Response
p=1p=0
p = 2/34/3
NO: 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
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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).
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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
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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
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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?
Problems of Nash Equilibrium
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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
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Bargaining GamesBargaining games are fundamental to understanding the price determination mechanism in “small” markets.
The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.
When information is asymmetric, profitable exchanges may be “left on the table.”
In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).