Mechanism Design and Auctions Jun Shu EECS228a, Fall 2002 UC Berkeley

EE228a -- Jun ShuMechanism Design for Networks2 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

EE228a -- Jun ShuMechanism Design for Networks3 Outline Mechanism Design Basics VCG Mechanism Sample Applications Auctions Recommended Papers

EE228a -- Jun ShuMechanism Design for Networks4 Presentation Style Intuition Math Example

EE228a -- Jun ShuMechanism Design for Networks5 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.

EE228a -- Jun ShuMechanism Design for Networks6 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”?

EE228a -- Jun ShuMechanism Design for Networks7 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.

EE228a -- Jun ShuMechanism Design for Networks8 Social Choice Theory Preference Relation (individual) Suppose there are n agents and a set of social choices C={c 1, …, c m }. 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.

EE228a -- Jun ShuMechanism Design for Networks9 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.

EE228a -- Jun ShuMechanism Design for Networks10 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 u i (,θ). Social Choice Rule: F:U→2 C Social Choice Function: f: Θ→C Mechanism –A mechanism M=(S 1,…,S n, g()) is a collection of n=|N| strategy sets (S 1,…,S n ) and an outcome function g: S 1 x…xS n →C. –M induces a set of games, each of which has a payoff function u i M (s 1,…,s n )≡u i (g(s 1,…,s n )).

EE228a -- Jun ShuMechanism Design for Networks11 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.

EE228a -- Jun ShuMechanism Design for Networks12 Implementation – M S -implements F in E if, when M played, S is not empty and ∀ (s 1,…,s n ) ∊ S, g(s 1,…,s n ) ∊ F(u 1,…,u n ). Weak Implementation – ∃ (s 1,…,s n ) ∊ S, g(s 1,…,s n ) ∊ F(u 1,…,u n ) Implementation of Social Choice Function Types of Implementation – DOM -Implementation – Bayesian-Nash -Implementation

EE228a -- Jun ShuMechanism Design for Networks13 Truth-telling Solution Concept Direct Revelation Mechanism –A mechanism in which S i = Θ i 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

EE228a -- Jun ShuMechanism Design for Networks14 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

EE228a -- Jun ShuMechanism Design for Networks15 VCG Mechanism More Restrictive Environment DOM-Implementation

EE228a -- Jun ShuMechanism Design for Networks16 Quasilinear Environment n agents C=X×R n, each outcome is c=(x,t), where – x ∊ X is a feasible solution if Φ(x)=0 ; and – t ∊ R n is a profile of transfer to the agents U::=2 Θ. Agent i ’s exact utility is unknown; however it takes the form u i (c)=v i (x,θ i ) + t i +m i where v i () is known to at least the principal θ i is private m i is a constant Σ i t i <0 assuming no outside financing

EE228a -- Jun ShuMechanism Design for Networks17 VCG Mechanism Defined M VCG = (θ 1,…, θ n, 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. –

EE228a -- Jun ShuMechanism Design for Networks18 Intuition of VCG Mechanism A direct revelation mechanism Feasible and Efficient Allocation Money Transfer Internalize the Externality

EE228a -- Jun ShuMechanism Design for Networks19 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

EE228a -- Jun ShuMechanism Design for Networks20 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 :

EE228a -- Jun ShuMechanism Design for Networks21 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

EE228a -- Jun ShuMechanism Design for Networks22 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: u i (x,t)=θ i x(θ’) + t i +m i –Solution: x(θ’)=1 if Σ i θ’ i >=K, otherwise x(θ’)=0 –Agents’ payment: max(0, K-Σ j≠i θ’ 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 =θ i If θ’ I >θ i, and the project is built, utility: θ i – K + Σ j≠i θ’ j + m i < θ i + m i If θ’ I <θ i, and the project is not built, utility: m i < θ i + m i

EE228a -- Jun ShuMechanism Design for Networks23 Vickery Auction The Problem: assign an indivisible good to one of two agents in a Pareto efficient way (i.e. both agents are happy with the result). The Implementation: ask the agents to bid on the good and award the good to the highest bidder at the second highest price. Features of Vickery auction: IC and IR.

EE228a -- Jun ShuMechanism Design for Networks24 Intuition behind Vickery Auction Assuming two agents, whose values are v 1 and v 2, and whose bids are b 1 and b 2. Agent’s payoff – P[b 1 >b 2 ] (v 1 – b 2 ) Agent’s best response – v 1 > b 2, P[b 1 >b 2 ] =1  b 1 = v 1 – v 1 b 2 ] =0  b 1 = v 1 – v 1 = b 2, any action is optimal

EE228a -- Jun ShuMechanism Design for Networks25 Auction A Direct Revelation Mechanism –Thanks to the revelation principle Basic Models Revenue Equivalence Theorem Basic Types Walrasian Auction Simultaneous Ascending Auction Combinatorial Auction

EE228a -- Jun ShuMechanism Design for Networks26 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)

EE228a -- Jun ShuMechanism Design for Networks27 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.

EE228a -- Jun ShuMechanism Design for Networks28 Four Types of Traditional Auction Ascending-bid Descending-bid First-price Sealed-bid Second-price Sealed-bid

EE228a -- Jun ShuMechanism Design for Networks29 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.

EE228a -- Jun ShuMechanism Design for Networks30 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.

EE228a -- Jun ShuMechanism Design for Networks31 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

EE228a -- Jun ShuMechanism Design for Networks32 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

EE228a -- Jun ShuMechanism Design for Networks33 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): , Jul –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