12/29/11 Game Theory Lecture 12 Symmetric independent private value auc:ons model (SIPV) •  Every player knows his/her own value for the object being sold •  The belief about other players’ values is the same for all players, its common knowledge and it is the following: –  Every other player’s value is drawn from the same probability distribu:on –  The distribu:ons (iden:cal) of other player’s values are sta:s:cally independent •  All players are risk neutral 1 12/29/11 This auc:on game as a Bayesian game • 
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Set of players Set of types (values) Set of ac:ons Beliefs: –  Other players’ values are drawn independently from the distribu:on F –  The distribu:on F is strictly increasing and con:nuous. •  Payoff func:ons for each player: where P(a) is a price paid by the winner, if a is a bidding profile. Second price auc:on •  Bidding my value weakly dominates higher bids •  Bidding my value weakly dominates lower bids 2 12/29/11 First price auc:on • 
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The highest bid wins and the winner pays her bid. Is it profitable to bid your value? –  If you win, you get zero profit. What happens if you bid less than your value. –  If you win, you get posi:ve profit. –  But chances of winning are lower –  Op:mal bid must balance out these two effects Bidding lower than your value is known as bid shading An example with uniform distribu:on •  There is n players •  You are Player 1 and your value is v>0 •  You believe that other players’ values are drawn indpenendently from a uniform [0,1] distribu:on. •  You guess that your opponents play: •  Your expected payoff if you bid b: •  We have to maximize it with respect to b 3 12/29/11 First price auc:on with uniform values •  First order condi:on for objec:ve func:on •  We obtain: •  Hence: •  Which auc:on will bring more revenue to the auc:oneer? –  Second price auc:on •  Players bid their values •  Revenue the second highest bid –  First price auc:on •  Players shade their bids •  Revenue – the highest bid Bayes rule •  Bayes rule: •  Condi:onal probability: •  Monty Hall problem: •  Ini:ally Picks door 1 4 12/29/11 Modified bable of sexes ● 
Two types Daisy
● 
Donald is not sure whether:
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● 
● 
Daisy is a good mood and
wants to meet him
Or Daisy is angry at him and
wants to avoid him
Daisy knows Donald’s type
Modified bable of sexes ● 
Donald knows from experience that
● 
● 
Daisy wants to go out with him with probability ½
(playing the game on the left)
Daisy does not want to go out with him with
probability ½ (playing the game on the right)
Soccer Ballet
2,1
0,0
Soccer
0,0
1,2
Ballet
Soccer Ballet
2,0
0,2
Soccer
0,1
1,0
Ballet
5 12/29/11 Modified bable of sexes ● 
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In order to make a good decision, Donald has to form beliefs about
the action of each type of Daisy
After evaluating these actions, Donald will be able to calculate the
expected value from each of his actions and will choose optimally
For example, if Donald believes that irrespective of her mood Daisy
chooses Soccer, then his expected payoffs are as follows:
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Donald’s payoff from choosing Soccer: 0.5*2+0.5*2=2
● 
Donald’s payoff from choosing Ballet: 0.5*0+0.5*0=0
Soccer
Ballet
Soccer Ballet
2,1
0,0
0,0
1,2
Soccer
Ballet
Soccer Ballet
2,0
0,2
0,1
1,0
Modified abble of sexes ● 
Another example: if Donald believes that Daisy in a good
mood chooses Soccer and Daisy in a bad mood chooses
Ballet then:
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Donald’s payoff from choosing Soccer: 0.5*2+0.5*0=1
● 
Donald’s payoff from choosing Ballet: 0.5*0+0.5*1=0.5
Soccer
Ballet
● 
Soccer Ballet
2,1
0,0
0,0
1,2
Soccer
Ballet
Soccer Ballet
2,0
0,2
0,1
1,0
A Bayesian Nash equilibrium for this game::
– 
Donald’s action is optimal given the actions of both types of Daisy
given Donald’s belief about Daisy’s type
– 
The action of each type of Daisy is optimal given the Donald’s
action
6 12/29/11 Modified bable of sexes S
B
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S,S
2, 1, 0
0, 0, 1
S,B
1, 1, 2
0.5, 0, 0
B,S
1, 0, 0
0.5, 2, 1
B,B
0, 0, 2
1, 2, 0
In each cell:
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The first number – Donald’s payoff
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The second number – Daisy in a good mood payoff
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The third number – Daisy in a bad mood payoff
Bayesian Nash Equilibrium (S,(S,B))
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Given Donald’s beliefs and actions of both types of Daisy, Donald is playing the best
response
Given Donald’s action, both types of Daisy are playing best response
Modified bable of sexes ● 
Interpretation of equilibrium if Daisy is in a good
mood:
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● 
Daisy wants to meet Donald and chooses Soccer
Donald chooses Soccer and believes that if Daisy is
in a good mood she chooses Soccer and if she is in
a bad mood she chooses Ballet
Interpretation of equilibrium when Daisy is a bad
mood:
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● 
Daisy does not want to meet Donald and chooses
Ballet
Doanald chooses Soccer and believes that is Daisy
is a good mood she chooses Soccer and if she is a
bad mood she chooses Ballet
7 12/29/11 Modified bable of sexes 2 ● 
Daisy knows from experience that:
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● 
Donald wants to meet her (good mood) with probability
2/3 (playing top game)
Donald avoids her (bad mood) with probability 1/3
(playing bottom game)
Soccer Ballet
2,1
0,0
Soccer
0,0
1,2
Ballet
Soccer Ballet
2,0
0,2
Soccer
0,1
1,0
Ballet
Soccer Ballet
0,1
2,0
Soccer
1,0
0,2
Ballet
Soccer Ballet
0,0
2,2
Soccer
1,1
0,0
Ballet
Modified bable of sexes 2 ● 
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Before we had two types of Daisy and
hence two states
Now we have two types of Daisy and two
types of Donald, hence four states
Donald does not know Daisy’s type but
knows his own type
Daisy does not know Donald’s type but
knows her own type
8 12/29/11 Soccer Ballet
2,1
0,0
Soccer
0,0
1,2
Ballet
Soccer Ballet
2,0
0,2
Soccer
0,1
1,0
Ballet
Soccer Ballet
0,1
2,0
Soccer
1,0
0,2
Ballet
Soccer Ballet
0,0
2,2
Soccer
1,1
0,0
Ballet
S
S
S
S
S
2
0
1
0
S
B
2
1
2/3 1/3
B
S
0
0
1/3 2/3 1/2
B
B
0
1
0
1
B
1
1
1
1/2 2/3 1 1/3 1
1
1/2 1/2
1
B
2
1
1/3 2/3 1/2
0
0
S
1
2
0
1/2 2/3 1/3
0
0
2/3 1 1/3
1 1 1/3 2/3
1
2 1 1/3 2/3
1
0
2
0
B
0
1/2 1/2
0
B
1
2
2
0
Two Bayesian Nash Equilibria: ((S,S),(S,B)) and ((B,S),(B,B)). 9 12/29/11 Modified bable of sexes 2 ● 
Interpretation of equilibrium:
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● 
● 
Both Daisy and Donald make a plan what to
do before they realize what type they are
Each type of Donald chooses optimal action
given action of Daisy and his beliefs about
Marge
Each type of Daisy chooses optimal action
given action of Donald and her beliefs about
Donald
10 12/29/11 11