Mechanism Design and Auctions Jun Shu EECS228a, Fall 2002 UC - - PowerPoint PPT Presentation

Mechanism Design and Auctions

Jun Shu EECS228a, Fall 2002 UC Berkeley

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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

• utcomes (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

• f 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 ui

M(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<0 assuming no outside financing

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VCG Mechanism Defined

• MVCG= (θ1,…, θn, g(•)) is a direct revelation

mechanism under the quasilinear environment, in which the outcome function is a social choice function, g(θ)=f(θ), and the choice function

where – s.t. –

C t x f ∈ = )) ( ), ( * ( ) ( θ θ θ

∑

= ∈ ∀

=

n i i i X x

x v x

1

) , ( max arg ) ( * θ θ       −       =

∑ ∑

≠ − − ≠ i j j i i i i j j i i

x v x v t ) ), ( * ( ) ), ( * ( ) ( θ θ θ θ θ ) ( = Φ x

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Intuition of VCG Mechanism

• A direct revelation mechanism
• Feasible and Efficient Allocation
• Money Transfer
• Internalize the Externality

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Features of VCG

• Dominant Strategy Incentive Compatible

– The best a designer could ask for – The proof uses the revelation principle.

• Not Budget Balanced

– Can generate money

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Participation Constraint

• When participation in a mechanism is

voluntary, the social choice function implemented must not be only IC but also must satisfy participation constraints.

• Types of Constraints

– Ex Post : – Interim : – Ex Ante :

) ( ) ), , ( (

i i i i i i

u f u θ θ θ θ ≥

−

[ ]

) ( | ) ), , ( (

i i i i i i i

u f u E

i

θ θ θ θ θ

θ

≥

−

−

[ ] [ ]

) ( ) ), , ( (

i i i i i i

u E f u E

i

θ θ θ θ

θ

≥

−

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Applications of Mechanism Design

• An application must consider

– A principal and a set of agents – An objective function:

• For the principal (e.g. revenue maximizing), or
• For the system (e.g. Pareto efficiency)

– Decision variables: the solution/allocation – Constraints

• Individual rationality
• Incentive compatibility

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Public Good

• The Problem: to build a project if and only if the total of the

individual’s valuation of the project exceeds the cost.

• The Implementation: VCG M

– Decision: x=1 to build, x=0 not to build – Agents’ strategy: θ’i – Agents’ utility: ui(x,t)=θix(θ’) + ti+mi – Solution: x(θ’)=1 if Σiθ’i >=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

• n 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 < θi + mi
• If θ’I<θi , and the project is not built, utility: mi < θi + mi

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Vickery Auction

• The Problem: assign an indivisible good to
• ne of two agents in a Pareto efficient way

(i.e. both agents are happy with the result).

• The Implementation: ask the agents to bid
• n the good and award the good to the

highest bidder at the second highest price.

• Features of Vickery auction: IC and IR.

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Intuition behind Vickery Auction

• Assuming two agents, whose values are v1

and v2, and whose bids are b1 and b2.

• Agent’s payoff

– P[b1>b2] (v1 – b2)

• Agent’s best response

– v1 > b2, P[b1>b2] =1 b1 = v1 – v1 < b2, P[b1>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

• n 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

• bject 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