Introduction to Mechanism Design

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Presentation transcript:

Introduction to Mechanism Design Lirong Xia

Voting game of strategic voters > Alice Strategic vote > Bob Strategic vote > Carol Strategic vote

Game theory is predictive How to design the “rule of the game”? so that when agents are strategic, we can achieve a given outcome w.r.t. their true preferences? “reverse” game theory Example: design a voting rule f so that for every true preference profile D* OutcomeOfGame(f, D*)=Plurality(D*)

Today’s schedule: mechanism design Mechanism design: Nobel prize in economics 2007 VCG Mechanism: Vickrey won Nobel prize in economics 1996 Eric Maskin Roger Myerson Leonid Hurwicz 1917-2008 William Vickrey 1914-1996

Example: second price auction Auction for one indivisible item n bidders Outcomes: { (allocation, payment) } Preferences: represented by a quasi-linear utility function every bidder j has a private value vj for the item. Her utility is vj - paymentj, if she gets the item 0, if she does not get the item suffices to only report a bid (rather than a total preorder) Vickrey auction (second price auction) allocate the item to the agent with the highest bid charge her the second highest bid

Example $ 10 $10 Kyle $ 70 $70 $70 Stan $ 100 $100 Eric

Implementation f * Mechanism f R1* s1 s2 R2* Outcome … … Rn* sn True Profile D* Strategy Profile D Mechanism f R1* s1 R2* s2 Outcome … … Rn* sn A game and a solution concept implement a function f *, if for every true preference profile D* f *(D*) =OutcomeOfGame(f, D*) f * is defined w.r.t. the true preferences f is defined w.r.t. the reported preferences

Mechanism design with money With monetary transfers Set of alternatives: A e.g. allocations of goods Outcomes: { (alternative, payments) } Preferences: represented by a quasi-linear utility function every agent j has a private value vj* (a) for every a∈A. Her utility is uj* (a, p) = vj* (a) - pj It suffices to report a value function vj

Can we adjust the payments to maximize social welfare? Social welfare of a SCW(a)=Σj vj* (a) Can any (argmaxa SCW(a), payments) be implemented w.r.t. DSNE?

The Vickrey-Clarke-Groves mechanism (VCG) The Vickrey-Clarke-Groves mechanism (VCG) is defined by Alterative in outcome: a*=argmaxa SCW(a) Payments in outcome: for agent j pj = maxa Σi≠j vi (a) - Σi≠j vi (a*) negative externality of agent j of its presence on other agents Truthful, efficient

Example: auction of one item Kyle $10 Alternatives = (give to K, give to S, give to E) a* = p1 = 100 – 100 = 0 p2 = 100 – 100 = 0 p3 = 70 – 0 = 70 Stan $70 Eric $100