More on Social choice and implementations
Using slides by
Uri Feige
Robi Krauthgamer
Moni Naor
Costas Daskalakis
1
Second Price auction:
hi(t-i) = maxj(w1, w2,…, wi-1, wi+1,…, wn)
= maxb 2 A j  i tj(b)
Multiunit auction: if k identical items are to be sold
to k individuals. A={S wins |S ½ I, |S|=k} and
vi(S) = 0 if i2S and vi(i)=wi if i 2 S
Allocate units to top k bidders. They pay the k+1th
price
Claim: this is
maxS’ ½ I\{i} |S’| =k j  i vj(S’)-j  i vj(S)
Multiunit auction: if k identical items are to be sold
to k individuals. A={S wins |S ½ I, |S|=k} and
vi(S) = 0 if i∉S and vi(i)=wi if i 2 S
VCG with Clarke Pivot Payments:
Allocate units to top k bidders. Each pays bid j+1.
GSP: The k items are not identical (ad slots)
vi(S) = 0 if i∉S and vi(j)=wij if i is given item j
Agents bid one value
i’ th top bidder gets slot i at price of bid i+1
Common in web advertising
Claim: this is not incentive compatible
Want to build a bridge:
◦ Cost is C (if built)
◦ Value to each individual vi
◦ Want to built iff  i vj ¸ C
A={build, not build}
Player with vj ¸ 0 pays only if pivotal
j  i vj < C but  j vj ¸ C
in which case pays pj = C- j  i vj
In general:  i pj < C
Equality only when
 i vj = C
Payments do not cover project cost’s
 Subsidy necessary!
A Directed graph G=(V,E) where each edge e is
“owned” by a different player and has cost ce.
Want to construct a path from source s to
destination t.
Set A of alternatives: all
s-t paths
 How do we solicit the real cost ce?
◦ Set of alternatives: all paths from s to t
◦ Player e has cost: 0 if e not on chosen path and –ce if on
◦ Maximizing social welfare: finding shortest s-t path:
minpaths e 2 path ce
A VCG mechanism pays 0 to those not on path p:
pay each e0 2 p: e 2 p’ ce - e 2 p\{e } ce
where p’ is shortest path without eo
0
If e0 would not have woken up in the morning, what would other
edges earn? If he does wake up, what would other edges earn?
• Requires payments & quasilinear utility functions
• In general money needs to flow away from the system
– Strong budget balance = payments sum to 0
– Impossible in general [Green & Laffont 77]
• Vulnerable to collusions
• Maximizes sum of players’ valuations (social welfare)
– (not counting payments, but does include “COST” of alternative)
But: sometimes [usually, often??] the mechanism is not
interested in maximizing social welfare:
–
–
–
–
E.g. the center may want to maximize revenue
Minimize time
Maximize fairness
Etc., Etc.
Black slides from Costas Daskalakis, MIT
6.853: Topics in Algorithmic Game Theory Fall 2011
Vickrey’s auction and VCG are both single round and direct-revelation
mechanisms.
We now consider a general model of mechanisms. It can model multi-round and
indirect-revelation mechanisms.
Games with Strict Incomplete Information
Def: A game with (independent private values and) strict incomplete
information for a set of n players is given by the following ingredients:
(i)
(ii)
(iii)
Strategy and Equilibrium
Def: A strategy of a player i is a function
Def: Equilibrium (ex-post Nash and dominant strategy)
A profile of strategies
is an ex-post Nash equilibrium
if for all i, all
, and all we have that
A profile of strategies
if for all i, all
, and all
is a dominant strategy equilibrium
we have that
Equilibrium (cont’d)
Proposition: Let
be an ex-post Nash equilibrium of a game
Define
equilibrium in the game
, then
is a dominant strategy
Mechanism
Def: A (general-non direct revelation) mechanism for n players is given by
The game with strict incomplete information induced by the mechanism has
the same type spaces and action spaces, and utilities
Implementing a social choice function
Given a social choice function
A mechanism implements in dominant strategies if for some dominant
strategy equilibrium
of the induced game, we have that for all
,
.
Ex: Vickrey’s auction implements the maximum social
welfare function in dominant strategies, because
a dominant
strategyNash
equilibrium,
and maximum
Similarly weiscan
define ex-post
implementation.
social welfare is achieved at this equilibrium.
outcome of the social choice function
outcome of the mechanism at the equilibrium
Remark: We only requires that for some equilibrium
and
allows other equilibria to exist.
The Revelation Principle
Revelation Principle
We have defined direct revelation mechanisms in previous lectures.
Clearly, the general definition of mechanisms is a superset of the direct
revelation mechanisms.
But is it strictly more powerful? Can it implement some social choice
functions in dominant strategy that the incentive compatible (direct
revelation dominant strategy implementation) mechanism can not?
Revelation Principle
Proposition: (Revelation principle) If there exists an arbitrary mechanism
that implements in dominant strategies, then there exists an incentive
compatible [direct revelation] mechanism that implements . The
payments of the players in the incentive compatible mechanism are
identical to those, obtained at equilibrium, of the original mechanism.
Incentive Compatible
Def: A mechanism
is called incentive compatible, or
truthful , or strategy-proof iff for all i, for all
and
for all
utility of i if he says the truth
utility of i if he lies
i.e. no incentive to lie!
Revelation Principle
Proposition: (Revelation principle) If there exists an arbitrary mechanism
that implements in dominant strategies, then there exists an incentive
compatible mechanism that implements . The payments of the players in
the incentive compatible mechanism are identical to those, obtained at
equilibrium, of the original mechanism.
Proof idea: Simulation
Revelation Principle (cont’d)
new mechanism
original mechanism
Proof of Revelation Principle
Proof: Let
be a dominant strategy equilibrium of the original
mechanism such that
, we define a
new direct revelation mechanism:
Since each
have that
is a dominant strategy for player i, for every
Thus in particular this is true for all
we have that
and any
which gives the definition of the incentive compatibility of the
mechanism.
, we
Revelation Principle (cont’d)
Corollary: If there exists an arbitrary mechanism that ex-post Nash equilibrium
implements , then there exists an incentive compatible mechanism that
implements . Moreover, the payments of the players in the incentive
compatible mechanism are identical to those, obtained in equilibrium, of the
original mechanism.
Proof sketch: Restrict the action spaces of each player. By the previous
proposition, we know in the restricted action spaces, the mechanism
implements the social choice function in dominant strategies. Now we can
invoke the revelation principle to get an incentive compatible mechanism.
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)
i.e., for every
, there exists a price
chosen alternative is , the price is
, when the
(ii)
i.e., for every , we have alternative
where the quantification is over all alternatives in the range of
Direct Characterization (cont’d)
Proof:
(if part) Denote
and
. Since the mechanism optimizes for i,
the utility of i when telling the truth is not less than the utility
when lying.
Direct Characterization (cont’d)
Proof (cont):
(only if part; (i)) If for some
type
,
but
. WLOG, we assume
. Then a player with
will increase his utility by declaring .
(only if part; (ii)) If
Now a player with type
utility by declaring .
, we fix
and
will increase his

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.
Is there a global characterization?
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
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.
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 exists incentive compatible mechanisms beyond VCG.
,
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: ???.
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).




There is a distribution Di on the types Ti of
Player i
It is known to everyone
The value ti 2DiTi is the private information i
knows
A profile of strategis si is a Bayesian Nash
Equilibrium if for i all ti and all x’i
Ed-i[ui(ti, si(ti), s-i(t-i) )] ¸ Ed-i[ui(ti, s-i(t-i)) ]
First price auction for a single item with two
players.
 Each has a private value t1 and t2 in T1=T2=[0,1]
 Does not make sense to bid true value – utility 0.
 There are distributions D1 and D2
 Looking for s1(t1) and s2(t2) that are best replies
to each other
 Suppose both D1 and D2 are uniform.
Claim: The strategies s1(t1) = ti/2 are in Bayesian
Nash Equilibrium

Win half the time
t1
Cannot win
Expected Revenue:
◦ For first price auction: max(T1/2, T2/2) where T1 and
T2 uniform in [0,1]
◦ For second price auction min(T1, T2)
◦ Which is better?
◦ Both are 1/3.
◦ Coincidence?
Theorem [Revenue Equivalence]: under very general
conditions, every two Bayesian Nash implementations of
the same social choice function
if for some player and some type they have the same
expected payment then
◦ All types have the same expected payment to the
player
◦ If all player have the same expected payment: the