Transcript 6.853: Topics in Algorithmic Game Theory Lecture 21 Fall 2011 Constantinos Daskalakis Review: Direct revelation Mechanisms, VCG.

6.853: Topics in Algorithmic Game Theory
Lecture 21
Fall 2011
Constantinos Daskalakis
Review: Direct revelation Mechanisms, VCG
Direct Revelation Mechanisms
• Setup:
– Set of alternatives A
– n bidders; bidder i has a (private) valuation function
;
– bidder i’s value for alternative a is
;
– if alternative a is chosen and bidder i pays price pi the utility of the
bidder is
(quasi-linear utility).
• Def: A direct revelation mechanism is a collection of functions (f, p1,
p2,…, pn) where:
–
–
chooses an alternative
chooses the payment of bidder i
• “Direct Revelation” because it asks bidders to reveal their whole
valuation function (i.e. doesn’t involve rounds of communication).
• Def: A direct revelation mechanism (f, p1, p2,…, pn) is called Incentive
Compatible iff for all i,
and
:
)
)
)
)
Vickrey-Clarke-Groves Mechanisms
• Def: A mechanism (f, p1, p2,…, pn) is a VCG mechanism if
–
– the payment of bidder i has the form:
(ie chooses SW maximizing alternative)
• Theorem(Vickrey-Clarke-Groves): Any VCG mechanism is IC.
• Def: A payment function pi is called Clarke pivot payment if
)
best social welfare without bidder i
• I.e. bidder pays the harm he causes to the other bidders.
• Theorem: VCG with Clarke pivot payments makes no positive transfers (i.e. sum of
prices charged is always positive). Also if the valuation functions are non-negative, it
is individually rational (bidders never have never negative utility, i.e. value-price is
always non-negative).
VCG Examples
•
•
•
•
Auctioning a single item (Vickrey auction)
Multi-unit Auctions
Reverse auction
Public Project
Power of Non-Direct 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
Formal Definition of Mechanisms
General Mechanisms
Vickrey’s auction and VCG are both single round and direct-revelation
mechanisms.
We will give a general model of mechanisms. It can model multi-round and
indirect-revelation mechanisms.
Mechanism
Def: A (general-non direct revelation) mechanism for n players is defined by
setup
mech
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 require that for some equilibrium
and
allow 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 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 Compatibility (restated)
Def: A direct revelation mechanism 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 about your type! ]
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
So we get that
, we
and any
.
which gives the definition of the incentive compatibility of the direct
revelation mechanism that we obtained above.

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.
Technical Lemma (gedanken experiment: from ex-post to dominant strategy equilibria):
Let
be an ex-post Nash equilibrium of a game
Define new action spaces
. Then
is a dominant strategy
equilibrium of the game
.
Proof sketch: Restrict the action spaces in the original mechanism to the sets
. Our technical lemma implies that now
is a dominant strategy equilibrium. Now invoke the revelation principle on dominant
strategies.
.