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

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 EE228a -- Jun Shu Mechanism Design for Networks 2

Outline • Mechanism Design Basics • VCG Mechanism • Sample Applications • Auctions • Recommended Papers EE228a -- Jun Shu Mechanism Design for Networks 3

Presentation Style • Intuition • Math • Example EE228a -- Jun Shu Mechanism Design for Networks 4

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 Shu Mechanism Design for Networks 5

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 Shu Mechanism Design for Networks 6

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 Shu Mechanism Design for Networks 7

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 Shu Mechanism Design for Networks 8

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 Shu Mechanism Design for Networks 9

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 Shu Mechanism Design for Networks 10

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 Shu Mechanism Design for Networks 11

Implementation • 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 Shu Mechanism Design for Networks 12

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 Shu Mechanism Design for Networks 13

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 Shu Mechanism Design for Networks 14

VCG Mechanism • More Restrictive Environment • DOM-Implementation EE228a -- Jun Shu Mechanism Design for Networks 15

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 Shu Mechanism Design for Networks 16

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 θ = θ θ ∈ f ( ) ( x * ( ), t ( )) C where n ∑ – s.t. Φ x = ( ) 0 θ = θ x * ( ) arg max v ( x , ) i i ∀ ∈ x X = i 1     ∑ ∑ – θ = θ θ − θ θ t ( ) v ( x * ( ), ) v ( x * ( ), )     − − i i j i i i j     ≠ ≠ j i j i EE228a -- Jun Shu Mechanism Design for Networks 17

Intuition of VCG Mechanism • A direct revelation mechanism • Feasible and Efficient Allocation • Money Transfer • Internalize the Externality EE228a -- Jun Shu Mechanism Design for Networks 18

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 Shu Mechanism Design for Networks 19

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 θ θ θ ≥ θ u ( f ( , ), ) u ( ) – Ex Post : − i i i i i i [ ] – Interim : θ θ θ θ ≥ θ E u ( f ( , ), ) | u ( ) θ − i i i i i i i − i [ ] [ ] – Ex Ante : θ θ θ ≥ θ E u ( f ( , ), ) E u ( ) − θ i i i i i i i EE228a -- Jun Shu Mechanism Design for Networks 20

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 Shu Mechanism Design for Networks 21

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 Shu Mechanism Design for Networks 22