Mechanism Design and Auctions
Jun Shu
EECS228a, Fall 2002
UC Berkeley
Class Objectives
• To introduce you to the basic concepts of
mechanism design
• To interest you in using mechanism design
as a tool in networking research
• To give you a list of references for further
study
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Outline
• Mechanism Design Basics
• VCG Mechanism
• Sample Applications
• Auctions
• Recommended Papers
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Presentation Style
• Intuition
• Math
• Example
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MD in a Nutshell
• Given
– A set of choices
– A group of people (agents) with individual preference over the
choices
– A group preference based on individual preference according to
some rule
• Ask
– A planner (principal) must make a decision over the choices
without knowing the individual’s preferences
• Approach
– Design a game for the individuals to play so that the stable
outcomes (equilibriums) of the game is the decision the principal
would have made had she known individual’s preferences.
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Questions in MD
• What kinds of “individual preferences” are
possible?
• What kinds of “group preferences” are possible
(according to “some rules”)?
• Why would an individual (the agents and the
principal) want to participate in a game?
• Why would an agent reveal his/her true preference
to the principal?
• What kinds of “stable outcomes”?
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Relevance to Networks
• A live network is the result of combined actions of
its users and components, all of which are
autonomous.
• MD and Network Mapping
– Agents: end-users, applications, devices, etc.
– Principals: network designer, network provider,
government, etc.
– Outcomes: network load, network performance,
network behavior
• Think outside the box.
• A Very New Approach.
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Social Choice Theory
• Preference Relation (individual)
Suppose there are n agents and a set of social choices
C={c1, …, cm}. The preference relation >>i over C is
defined as the ordering of set C according to the
preference of agent i.
• Social Welfare Functional (group)
A function >> that assigns a rational social preference
relation, >>(>>1, …, >>n), to any profile of individual
rational preference in the admissible domain.
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Arrow’s Impossibility Theorem
• Arrow’s Conditions
– Unanimity: >> is consistent with all the unanimous decisions of the
group members
– Pair-wise Independent: >> over any two choices depends only on
the individual preferences over these choices
– Non-dictatorial: there does not exist a dictator
• Arrow’s Impossibility Theorem
– If |C|>2, then there is no social welfare functional that satisfies all
of the above three conditions
• Implication
– Without any constraints, a collectivity does not behavior with the
kind of coherence that we may hope from an individual.
Institutional detail and procedures matter.
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MD Defined
• Environment: E is a triplet (N, C, U)
– W.L.G., replace U with agents’ type space Θ. An
agent’s utility function is ui(•,θ).
• Social Choice Rule: F:U→2C
• Social Choice Function: f: Θ→C
• Mechanism
– A mechanism M=(S1,…,Sn, g(•)) is a collection of n=|N|
strategy sets (S1,…,Sn) and an outcome function g:
S1x…xSn→C.
– M induces a set of games, each of which has a payoff
function uiM(s1,…,sn)≡ui(g(s1,…,sn)).
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Solution Concepts
• Solution Concept
– S denotes a subset of the strategy space which produces
certain kinds of unspecified equilibrium outcomes in a
game induced by M under E.
• Kinds of Solution Concept
– Dominant Strategy Equilibrium
– Bayesian Nash Equilibrium
– Nash Equilibrium
• Not very useful in mechanism design.
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Implementation
• Implementation
– M S-implements F in E if, when M played,
• S is not empty and ∀(s1,…,sn)∊S , g(s1,…,sn)∊F(u1,…,un) .
• Weak Implementation
– ∃(s1,…,sn) ∊ S , g(s1,…,sn) ∊ F(u1,…,un)
• Implementation of Social Choice Function
• Types of Implementation
– DOM-Implementation
– Bayesian-Nash-Implementation
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Truth-telling Solution Concept
• Direct Revelation Mechanism
– A mechanism in which Si= Θi for all i and
g(θ)=f(θ) for all θ ∊ Θ .
• Truthful Implementation
– A weak implementation is truthful if in the
direct revelation mechanism, telling the truth is
an equilibrium (of some sort) strategy.
– Other term: incentive compatible
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General Results:
Implementable Choice Functions
• Good News: we can focus on the truthful implementation
– Revelation Principle (Theorem)
• If F is DOM-implementable in E, then there exists a weak truthful
implementation in dominant strategies.
• Bad News: without any constraints, little is implementable
– Gibbard-Satterthwaite Impossibility Theorem
If finite |C|>2 and U includes all utility functions, only binary and
dictatorial choice rules are DOM-implementable.
• Constraints: a way out
– Type of environment
– Type of choice functions
– Type of implementation
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VCG Mechanism
• More Restrictive Environment
• DOM-Implementation
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Quasilinear Environment
• n agents
• C=X×Rn, each outcome is c=(x,t), where
– x ∊ X is a feasible solution if Φ(x)=0; and
– t ∊ Rn is a profile of transfer to the agents
• U::=2Θ. Agent i’s exact utility is unknown; however
it takes the form
ui(c)=vi(x,θi) + ti+mi where
• vi(•) is known to at least the principal
• θi is private
• mi is a constant
• Σiti=K, otherwise x(θ’)=0
– Agents’ payment: max(0, K-Σj≠iθ’j)
• Intuition
– An agent’s payment depends on her action only through the action’s effect
on the solution; otherwise, it depends on others’ action
– An agent action matters only if it make a difference in solution
– The dominant strategy for each agent is θ’i=θi
• If θ’I>θi , and the project is built, utility: θi – K + Σj≠iθ’j + mi b2] (v1 – b2)
• Agent’s best response
– v1 > b2, P[b1>b2] =1 b1 = v1
– v1 b2] =0 b1 = v1
– v1 = b2, any action is optimal
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Auction
• A Direct Revelation Mechanism
– Thanks to the revelation principle
• Basic Models
• Revenue Equivalence Theorem
• Basic Types
• Walrasian Auction
• Simultaneous Ascending Auction
• Combinatorial Auction
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Basic Models of Auction
• Private-value
– Each bidder knows know much she values the object(s)
for sale, but her value is private information
• Common-value
– A bidder’s value of the object depends to some extent
on other bidders’ signals
• Pure common-value (almost common value)
– A special common-value case in which all bidders’
actual values are identical functions to the signals.
– Information Dynamics: how to extract public
knowledge (as in market research)
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Revenue Equivalence Theorem
• Consider an auction setting with n risk neutral buyers, in
which buyers’ valuations are drawn from an interval and
has a strictly positive density, and in which buyers’ types
are statistically independent. Suppose that a given pair of
Bayesian Nash equilibriums of two different auction
procedures are such that for every buyer i :
– For each possible realization of valuations, buyer i has identical
probability of getting the good in the two auctions; and
– Buyer i has the same expected utility level in the two auctions
when his valuation for the object is at its lowest possible level
Then these equilibriums of the two auctions generate the
same expected revenue for the seller.
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Four Types of Traditional Auction
• Ascending-bid
• Descending-bid
• First-price Sealed-bid
• Second-price Sealed-bid
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Ascending-bid Auction
• Open, oral, English, open-second-price
– The price is successively raised until only one bidder
remains, and that bidder wins the object at the final
price.
– In private-value model, a dominant strategy is to stay in
the bidding until the price reaches your value. The next-
to-last person will drop out when her value is reached,
so the person with the highest value will win at price of
the second-highest bidder.
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Descending-bid Auction
• Dutch, open-first-price
– The auctioneer starts at a very high price, and then
lowers the price continuously. The first bidder who
calls out that she will accept the current price wins the
object at that price. Used in the sale of flowers in
Netherlands, and so then name.
– This game is strategically equivalent to the first-price
sealed-bid auction, and players’ bidding functions are
exactly the same. Thus the name ”open first-bid”
auction.
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Sealed-bid Auction
• First-price Sealed-bid Auction
– Each bidder independently submits a single bid,
without seeing others’ bids, and the object is
sold to the bidder who makes the highest bid.
The winner pays her bid.
• Second-price Sealed-bid Auction
– Vickery Auction
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Combinatorial Auction
• Bids on combinations of items
• Complementary and Substitutive Relation among
items
• Basic Problems
– Bid Expression
– Winner Determination
• Integer Program
• NP-hard
– IC and IR
– Optional: stopping rules
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Recommended Papers
You may want to familiarize yourself with game theory before you start to
read the following.
• Allan Gibbard, “Manipulation of Voting Schemes: A General Result.”
Econometrica, 41(4):587-601, Jul. 1973.
– Gibbard-Satterthwaite Impossibility Theorem
• Roger Myerson, “Incentive Compatibility and the Bargaining
Problem.” Econometrica, 47:61-73, 1979
– One of the original paper on Revelation Principle
• Roger Myerson, “Optimal Auction Design.” Mathematics of
Operations Research, 6:58-73, 1981
• Wiliam Vickery, “Counterspeculation, Auctions, and Competitive
Sealed Tenders,” Journal of Finance, 16(1):8-37, Mar.1961
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