Transcript 6.853: Topics in Algorithmic Game Theory Lecture 22 Fall 2011 Constantinos Daskalakis Characterizations of Incentive Compatible Mechanisms.

6.853: Topics in Algorithmic Game Theory
Lecture 22
Fall 2011
Constantinos Daskalakis
Characterizations of Incentive Compatible Mechanisms
Characterizations
What social choice functions can be implemented?
Only look at incentive compatible mechanisms (revelation principle)
When is a mechanism incentive compatible?
Characterizations of incentive compatible mechanisms.
Maximization of social welfare can be implemented (VCG). Any others?
Basic characterization of implementable social choice functions.
Direct Characterization
Direct Characterization
A mechanism is incentive compatible iff it satisfies the following conditions
for every i and every
:
(i) pi depends on
only through the alternative
i.e., for fixed
, there is an advertised price
per alternative
bidder is free to affect the chosen alternative and through that the
corresponding price that she’ll pay;
(ii)
i.e., for every , we have alternative
where the quantification is over all alternatives in the range of
; the
Direct Characterization (cont’d)
Proof (if part):
Denote
,
where respectively
is the bidder’s true value and
is a potential lie.
Since the mechanism optimizes for i, the utility he receives when telling the
truth is not less than the utility he receives when lying.
Direct Characterization (cont’d)
Proof (cont):
(only if part; (i)) Suppose that for some
but
,
. WLOG, assume that
. Then a player with type
will increase his utility by declaring .
(only if part; (ii)): Suppose
instead
Now a player with type
declaring .
and let
s.t.
will increase his utility by

Weak Monotonicity
Weak Monotonicity
●The direct characterization involves both the social choice function and the
payment functions.
● Weak Monotonicity provides a partial characterization that only involves
the social choice function.
Weak Monotonicity (WMON)
Def: A social choice function satisfies Weak Monotonicity (WMON)
if for all i, all
we have that
i.e. WMON means that if the social choice changes when a single
player changes his valuation, then it must be because the player
increased his value of the new choice relative to his value of the old
choice.
Weak Monotonicity
Theorem: If a mechanism
is incentive compatible then
satisfies WMON. If all domains of preferences are convex sets (as
subsets of an Euclidean space) then for every social choice function that
satisfies WMON there exists payment function
such that
is incentive compatible.
Remarks: (i) We will prove the first part of the theorem. The second part is
quite involved, and will not be given here.
(ii) It is known that WMON is not a sufficient condition for
incentive
compatibility in general non-convex domains.
Weak Monotonicity (cont’d)
Proof: (First part) Assume first that
is incentive
compatible, and fix i and
in an arbitrary manner. The direct
characterization implies the existence of fixed prices for all
(that do not depend on ) such that whenever the
outcome is then i pays exactly .
Assume
is incentive compatible, we have
Thus, we have
. Since the mechanism
Minimization of Social Welfare
We know maximization of social welfare function can be implemented.
How about minimization of social welfare function?
No! Because of WMON.
Minimization of Social Welfare
Assume there is a single good. WLOG, let
case, player 1 wins the good.
If we change to , such that
Now we can apply the WMON.
. In this
. Then player 2 wins the good.
The outcome changes when we change player 1’s value. But according
to WMON, it should be the case that
. But
.
Contradiction.
Weak Monotonicity
WMON is a good characterization of implementable social choice
functions, but is a local one (i.e. a collection of local conditions).
Is there a global characterization of what functions can be
implemented, e.g. maximization of social welfare, etc.?
Weighted VCG
Affine Maximizer
Def: A social choice function is called an affine maximizer if for
some subrange
, for some weights
and for
some outcome weights
, for every
, we have that
weighted social welfare
i.e. maximization
only over A’
+ bonus per alternative
Payments for Affine Maximizer
Proposition: Let
on
be an affine maximizer. Define for every i,
where
is an arbitrary function that does not depend
. Then,
is incentive compatible.
the appropriate generalization of the VCG payment rule
Payments for Affine Maximizer
Proof: First, we can assume wlog
. The utility of player i if
alternative is chosen is
. By
multiplying by
this expression is maximized when
is maximized which is what happens when i
reports truthfully.

Roberts Theorem
Theorem [Roberts 79]: If
, is onto ,
for every i, and
is incentive compatible then is an affine maximizer.
Remark: The restriction
is crucial (as in Arrow’s theorem); for the case
there do exist incentive compatible mechanisms beyond VCG.
|A|=2

,
single-parameter
domains
the valuation function of each bidder is described by
1 parameter; i.e, for every bidder i there exists subset
Wi  A (winning set) and a parameter ri such that the
bidder receives value ri for any outcome in Wi and 0
otherwise; e.g. single-item/multi-unit auctions, etc.
Bayesian Mechanisms
Bayesian Mechanism Design
Def: A Bayesian mechanism design environment consists of the following:
setup
+ for every bidder a distribution
is known; bidder i’s type is sampled from it
Def: A Bayesian mechanism consists of the following:
mech
The utility that bidder i receives if the players’ actions are x1,…,xn is :
Strategy and Equilibrium
Def: A strategy of a player i is a function
Def: A profile of strategies
if for all i, all
, and all
.
is a Bayes Nash equilibrium
we have that
expected utility of bidder i for using si, where
the expectation is computed with respect to the
actions of the other bidders assuming they are
using their strategies
expected utility if bidder i uses a different
action xi’; still the expectation is computed
with respect to the actions of the other bidders
assuming they are using their strategies
First Price Auction
Theorem: Suppose that we have a single item to auction to two bidders
whose values are sampled independently from [0,1]. Then the strategies
s1(t1)=t1/2 and s2(t2)=t2/2 are a Bayes Nash equilibrium.
Remark: In the above Bayes Nash equilibrium, social welfare is
optimized, since the highest winner gets the item.
Proof: On the board.
Expected Payoff?
E[ max(X/2, Y/2)], where X, Y are independent U[0,1] random variables.
=1/3
Expected Payoff of second price auction?
E[ min(X, Y)], where X, Y are independent U[0,1] random variables.
=1/3 !
Revenue Equivalence Theorem
All single item auctions that allocate (in Bayes Nash equilibrium) the item
to the bidder with the highest value and in which losers pay 0 have identical
expected revenue.
Generally:
Given a social choice function
we say that a Bayesian
mechanism implements if for some Bayes Nash equilibrium
we
have that for all types
:
Revenue Equivalence Theorem: For two Bayesian-Nash implementations of the
same social choice function f, * we have the following: if for some type t0i of player
i, the expected (over the types of the other players) payment of player i is the same
in the two mechanisms, then it is the same for every value of ti.
In particular, if for each player i there exists a type t0i where the two mechanisms
have the same expected payment for player i, then the two mechanisms have the
same expected payments from each player and their expected revenues are the same.
Revenue Equivalence Theorem (cont.)
All single item auctions that allocate (in Bayes Nash equilibrium) the item to the
bidder with the highest value and in which losers pay 0 have identical expected
revenue.
So unless we modify the social choice function we won’t increase revenue.
E.g. increasing revenue in single-item auctions:
- two uniform [0,1] bidders
- run second price auction with reservation price ½ (i.e. give item to
highest bidder above the reserve price (if any) and charge him the
second highest bid or the reserve, whichever is higher.
Claim: Reporting the truth is a dominant strategy equilibrium. The
expected revenue of the mechanism is 5/12 (i.e. larger than 1/3).