Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Incentive Compatible Mechanisms
Arne Hillebrand & Denise Tönissen
May 16, 2011
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
1 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Overview
1
Characterizations of IC Mechanisms
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
2
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
2 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Notation
Because of the Revelation Principle we can once again look at IC
mechanisms M = (f , p1 , . . . , pn ), where:
f :V1 × . . . × Vn → A
Social Choice Function
pi :V1 × . . . × Vn → R
Payment functions
vi :A → R
Arne Hillebrand & Denise Tönissen
Valuation function, vi ∈ Vi
Incentive Compatible Mechanisms
3 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties
A mechanism M is IC if and only if for all i and v−i :
Payment pi does not depend on vi , but only on the chosen
a∈A
M optimizes for each player
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
4 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties
A mechanism M is IC if and only if for all i and v−i :
Payment pi does not depend on vi , but only on the chosen
a∈A
∀a ∈ A : ∃pa ∈ R : ∀vi ∈ Vi : f (vi , v−i ) = a → pi (vi , v−i ) = pa
M optimizes for each player
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
4 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties
A mechanism M is IC if and only if for all i and v−i :
Payment pi does not depend on vi , but only on the chosen
a∈A
∀a ∈ A : ∃pa ∈ R : ∀vi ∈ Vi : f (vi , v−i ) = a → pi (vi , v−i ) = pa
M optimizes for each player
f (vi , v−i ) ∈ arg maxa∈f (v−i ) (vi (a) − pa )
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
4 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties (ctd)
Definition
A mechanism (f , p1 , . . . , pn ) is called IC if for every player i, every
v1 ∈ V1 , . . . , vn ∈ Vn and every vi0 ∈ Vi , if we denote a = f (vi , v−i )
and a0 = f (vi0 , v−i ), then vi (a) − pi (vi , v−i ) ≥ vi (a0 ) − pi (vi0 , v−i ).
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
5 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties (ctd)
Definition
A mechanism (f , p1 , . . . , pn ) is called IC if for every player i, every
v1 ∈ V1 , . . . , vn ∈ Vn and every vi0 ∈ Vi , if we denote a = f (vi , v−i )
and a0 = f (vi0 , v−i ), then vi (a) − pi (vi , v−i ) ≥ vi (a0 ) − pi (vi0 , v−i ).
1) f (vi , v−i ) = a → pi (vi , v−i ) = pa
2) ∀vi : f (vi , v−i ) ∈ arg maxa∈f (v−i ) vi (a) − pa
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
5 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties (ctd)
Definition
A social choice function f satifies Weak Monotonicity if for all i
and all v−i we have :
(f (vi , v−i ) = a 6= f (vi0 , v−i ) = b) → (vi (a) − vi (b) ≥ vi0 (a) − vi0 (b))
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Properties (ctd)
Definition
A social choice function f satifies Weak Monotonicity if for all i
and all v−i we have :
(f (vi , v−i ) = a 6= f (vi0 , v−i ) = b) → (vi (a) − vi (b) ≥ vi0 (a) − vi0 (b))
We have M = (f , p1 , . . . pn ) which is IC
a =f (vi , v−i )
b =f (vi0 , v−i )
a 6=b
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Short recap: Vickery-Clarke-Groves
A mechanism M is called a VCG mechanism if:
f maximizes the social welfare
All players benefit from maximizing f (because that is also the
cheapest option for that player)
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Incentive Compatible Mechanisms
7 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Short recap: Vickery-Clarke-Groves
A mechanism M is called a VCG mechanism if:
f maximizes the social welfare
P
f (v1 , . . . , vn ) ∈ arg maxa∈A i vi (a)
All players benefit from maximizing f (because that is also the
cheapest option for that player)
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
7 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Short recap: Vickery-Clarke-Groves
A mechanism M is called a VCG mechanism if:
f maximizes the social welfare
P
f (v1 , . . . , vn ) ∈ arg maxa∈A i vi (a)
All players benefit from maximizing f (because that is also the
cheapest option for that player)
P
∀v ∈ V n : pi (v ) = hi (v−i ) − j6=i vj (f (v ))
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Incentive Compatible Mechanisms
7 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Affine Maximizer
Definition
A social choice function f is called an affine maximizer if for some
subrange A0 ⊂ A, for some player weights w1 , . . . , wn ∈ R+ and for
some outcome weights ca ∈ R weP
have for every a ∈ A0 that
f (v1 , . . . , vn ) ∈ arg maxa∈A0 (ca + i wi × vi (a))
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Incentive Compatible Mechanisms
8 / 25
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Affine Maximizer
Definition
A social choice function f is called an affine maximizer if for some
subrange A0 ⊂ A, for some player weights w1 , . . . , wn ∈ R+ and for
some outcome weights ca ∈ R weP
have for every a ∈ A0 that
f (v1 , . . . , vn ) ∈ arg maxa∈A0 (ca + i wi × vi (a))
Claim: An affine maximizer
P wj with
pi (v ) = hi (v−i ) − j6=i wi vj (a) −
Arne Hillebrand & Denise Tönissen
ca
wi
is IC.
Incentive Compatible Mechanisms
8 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Single-Parameter Domains
Definition
A single parameter domain Vi is defined by a Wi ⊂ A and a range of
values [x, y ]. Vi is the set of vi such that for some x ≤ t ≤ y ,
∀a ∈ Wi : vi (a) = t and ∀a ∈
/ Wi : vi (a) = 0
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Incentive Compatible Mechanisms
9 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Single-Parameter Domains
Definition
A single parameter domain Vi is defined by a Wi ⊂ A and a range of
values [x, y ]. Vi is the set of vi such that for some x ≤ t ≤ y ,
∀a ∈ Wi : vi (a) = t and ∀a ∈
/ Wi : vi (a) = 0
Definition
A social choice function f on a single parameter domain is called
monotone in vi if for every v−i and every vi ≤ vi0 ∈ R we have that
f (vi , v−i ) ∈ Wi → f (vi0 , v−i ) ∈ Wi .
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
9 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Theorem
A normalized mechanism on a single parameter domain is IC if and
only if:
f is monotone in every vi
Every winning bid pays the critical value
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Theorem
A normalized mechanism on a single parameter domain is IC if and
only if:
f is monotone in every vi
Every winning bid pays the critical value
∀i, vi , v−i : f (vi , v−i ) → pi (vi , v−i ) = ci (v−i )
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
10 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Uniqueness of prices
Theorem
Assume all Vi are connected sets in the usual metric in the Euclidian
space. Let M = (f , p1 , . . . , pn ) be an IC mechanism, then
M 0 = (f , p10 , . . . , pn0 ) is IC if and only if there exists a function
hi (v−i ) such that pi0 (v ) = pi (v ) + hi (v−i )
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Randomized Mechanisms
Definition
A Randomized Mechanism (MR ) is a distribution over a
deterministic mechanism.
A MR is IC in the universal sense if every deterministic
mechanism in the support is incentive compatible
A MR is IC in expectation if truth is a dominant strategy
induced by expectation.
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
12 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Randomized Mechanisms
Definition
A Randomized Mechanism (MR ) is a distribution over a
deterministic mechanism.
A MR is IC in the universal sense if every deterministic
mechanism in the support is incentive compatible
A MR is IC in expectation if truth is a dominant strategy
induced by expectation.
∀i, vi , v−i , vi0 : E [vi (a) − pi ] ≥ E [vi (a0 ) − pi0 ]
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Incentive Compatible Mechanisms
12 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Notation
Properties
The Social Choice Function
Payment functions
Randomized Mechanisms
Randomized Mechanisms (ctd)
Theorem
A normalized MR in a single parameter domain is IC in expectation
if and only if for every i and v−i :
The function wi (vi , v−i ) is monotonically non decreasing in vi
Rv
pi (vi , v−i ) = vi × w (vi , v−i ) − v 0i w (t, v−i )dt
i
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
The Bayesian approach
So far players had no private information of the others and thus
worked under a worst case assumption.
Now there is a commonly known prior distribution.
A player will now optimize in a Bayesian sense according to
that distribution and the information that the player does have.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Definition of the game
The game has independent private values and incomplete
information on a set of n players.
Every player i has its own set of actions Xi , set of types Ti and
a prior distribution Di on Ti .
The value ti ∈ Ti is the private information which player i has
and Di (ti ) is the priori probability that player i gets type ti .
For every player i there is an utility function
ui : Ti × X1 × ... × Xn → R where ui (ti , x1 , ...., xn ) is the utility
achieved by player i.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Strategies
A strategy of a player i is a function si : Ti → Xi .
Definition
A profile of strategies s1 , ..., sn is a Bayesian-Nash equilibrium if for
every player i and every ti we have that si (ti ) is the best response
that i has to s−i () when his type is ti .
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
The Bayesian mechanism
Definition
A mechanism implements a social choice function
f : T1 × ... × Tn → A in the Bayesian sense if for some
Bayesian-Nash equilibrium s1 , ..., sn of the induced game we have
that for all t1 , ..., tn , f (t1 , ..., tn ) = a(s1 (t1 ), ..., sn (tn ))
Here is A like before the alternative set and
a : X1 × ... × Xn → A the outcome function.
The utilities are given by
ui (ti , x1 , ...., xn ) = vi (ti , a(x1 , ...., xn )) − pi (x1 , ...., xn )
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Bayesian Truthfulness
Direct revelation: Ti = Xi .
A mechanism is truthful in the Bayesian sense if it is a direct
revelation and the truthful strategies si (ti ) are a Bayesian-Nash
equilibrium.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Bayesian Truthfulness
Direct revelation: Ti = Xi .
A mechanism is truthful in the Bayesian sense if it is a direct
revelation and the truthful strategies si (ti ) are a Bayesian-Nash
equilibrium.
Revelation principle
If there exist an arbitrary mechanism that implements f in the
Bayesian sense, then there exist a truthful mechanism that
implements f in the Bayesian sense. Moreover the expected
payments of the players in truthful mechanism are identical to those,
obtained in equilibrium in the original mechanism.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
First Price Auction
Private values: Alice a, Bob b.
What would Alice bid?
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
First Price Auction
Private values: Alice a, Bob b.
What would Alice bid?
x < a and if she knew bobs y she would bid x = y + .
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
First Price Auction
Private values: Alice a, Bob b.
What would Alice bid?
x < a and if she knew bobs y she would bid x = y + .
Unfortunately Alice does not know y, but she does know DBob
over b.
Goal: Find a strategy SAlice for Alice given by the function x(a)
and for Bob y (b) that are best replies of each other and thus in
Bayesian-Nash equilibrium.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Symmetric case
Difficult, but easier for the symmetric case where DAlice = DBob .
Lemma
In a first price auction among two players with prior distributions of
the private values a,b uniform over the interval [0,1], the strategies
x(a) = 2a and y (b) = b2 are in Bayesian-Nash equilibrium.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Symmetric case
Difficult, but easier for the symmetric case where DAlice = DBob .
Lemma
In a first price auction among two players with prior distributions of
the private values a,b uniform over the interval [0,1], the strategies
x(a) = 2a and y (b) = b2 are in Bayesian-Nash equilibrium.
Because x < y if and only if a < b the winner is also the player
with the highest private function.
This means that the first prize auction maximizes the social
welfare, just like the second-prize auction.
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Proof
What would be Alice best response to the strategy y (b) = b2 ?
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Incentive Compatible Mechanisms
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Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Proof
What would be Alice best response to the strategy y (b) = b2 ?
uAlice = Pr[Alice wins bid x] .(a − x).
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Incentive Compatible Mechanisms
21 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Proof
What would be Alice best response to the strategy y (b) = b2 ?
uAlice = Pr[Alice wins bid x] .(a − x).
Alice wins if x ≥ b2 , since b is distributed uniformly in [0, 1]
every x above 21 is definitely larger than b.
The chance that Alice wins when x ≤ 12 is 2x, thus we have to
find the maximum of the function 2x.(a − x).
2a − 4x = 0, whose solution is x = 2a .
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
21 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Highest revenue
Which auction has the highest revenue the first price or the
second price auction?
The revenue of the second price auction is given by min(a, b).
E [min(a, b)] = E [b|b < a] + E [a|a < b]
R1Ra
R1R1
1
0 0 b dbda + 0 a a dbda = 3
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Incentive Compatible Mechanisms
22 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Highest revenue
The revenue of the first price auction is given by max( 2a , b2 ).
a
2
> b2 if and only if a > b. Calculates easier but we have to
divide the expected value by 2 later on.
E [max(a, b)] = E [b|b > a] + E [a|a > b]
R1R1
R1Ra
2
0 a a dbda + 0 0 a dbda = 3
Dividing this by two gives an expectancy of
Arne Hillebrand & Denise Tönissen
1
3
as well.
Incentive Compatible Mechanisms
23 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Revenue equivalence principle
The results of the previous slides are no coincidence as two
Bayesian-nash implementations of the same social function generate
the same expected revenue.
Arne Hillebrand & Denise Tönissen
Incentive Compatible Mechanisms
24 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Revenue equivalence principle
The results of the previous slides are no coincidence as two
Bayesian-nash implementations of the same social function generate
the same expected revenue.
The Revenue Equivalence Principle
Under assumption that Vi is convex we have that for every two
Bayesian-Nash implementations of the same social choice function f,
we have that if for some type ti0 of player i, the expected payment of
player i is the same in the two mechanisms, then it is the same for
every value of ti .
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Incentive Compatible Mechanisms
24 / 25
Characterizations of IC Mechanisms
Bayesian-Nash Implementation
Bayesian-Nash Equilibrium
First Price Auction
Revenue Equivalence
Profit Maximization
Only way to increase revenue is changing the social choice
function.
A way to increase the revenue is adding a reservation price of 21 .
This game has an expectancy of
5
12 .
More about this subject will be in the seminar talks of
Wednesday May 25th.
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Incentive Compatible Mechanisms
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