Lecturer: Moni Naor

Algorithmic Game TheoryUri Feige Robi Krauthgamer Moni NaorLecture 10: Mechanism Design Lecturer:Moni Naor

Announcements • January: course will be 1300:-15:00 • The meetings on Jan 7th, 14th and 21st 2009

Recap social choice • Social choice: collectively choosing among outcomes or aggregate preferences • Arrow’s Impossibility Theorem • Gibbard-Satterthwaite Theorem: There exists no social choice functionf for more than 2 alternatives that is simultaneously: • Onto: for every candidate, there are some votes that make the candidate win • Nondictatorial • Incentive compatible

Proof of Arrow’s Theorem: Find the Dictator Claim: For any a,b 2 A consider sets of profiles ab ba ba … ba ab ab ba … ba ab ab ab … ba … … … ab ab ab ba Voters Hybrid argument 1 • Change must happen at some profile i* • Where voter i* changed his opinion 2 … n Claim: this i* is the dictator! 0 1 2 n aÁb bÁa Profiles

Single-peaked preferences [Black 48] • Suppose alternatives are ordered on a line • Every voter prefers alternatives that are closer to her most preferred alternative • Choose the median voter’s peak as the winner • Strategy-proof! v5 v4 v2 v1 v3 a1 a2 a3 a4 a5

What about Probabilistic Voting Schemes? Electing the Doge in the Republic of Venice 1268-1797 • A sequence of electoral colleges, where at each stage: • A sub-college is selected at random (lottery) • The sub college elects the next electoral college by approval voting. • Final college elects the Doge Lottery Approval

Probabilistic Voting Schemes Can do something ``non trivial” to get truthful voting • Elect a random leader/dictator • Choose at random a pair of alternatives and see which one is preferred by the majority. But this all we can do: Any scheme has to be a combination of such rules

Range Voting • Each voter ranks the candidates in a certain range (say 0-99) • The votes for all candidates are summed up and the one with highest total score wins Can be considered as a generalization of approval voting from the range 0-1 No incentive for voter to rate a candidate lower than a candidate they like less.

Mechanism Design • Mechanisms • Recall: We want to implement a social choice function • Need to know agents’ preferences • They may not reveal them to us truthfully • Example: • One item to allocate: • Want to give it to the participant who values it the most • If we just ask participants to tell us their preferences: may lie • Can use payments result is also a payment vector p=(p1,p2, … pn)

The setting • Set of alternatives A • Who wins the auction • Which path is chosen • Who is matched to whom • Each participant: a value function vi:A  R • Can pay participants: valuation of choice a with payment pi is vi(a)+pi Quasi linear preferences

Example: Vickrey’s Second Price Auction Despite private information and selfish behavior compute “reliably” the max function! • Single item for sale • Each player has scalar value wi – willingness to pay • If he wins item and has to pay p: utility wi-p • If someone else wins item: utility 0 Second price auction: Winner is the one with the highest declared value wi. Pays the second highest bid p*=maxj  i wj Theorem (Vickrey): for any every w1, w2,…,wn and every wi’. Let ui be i’s utility if he bids wi and u’i if he bids wi’. Then ui¸u’i..

Direct Revelation Mechanism A direct revelation mechanism is a social choice function f: V1 V2 …  Vn  A and payment functions pi: V1 V2 …  Vn R • Participant i pays pi(v1, v2, … vn) A mechanism (f,p1, p2,… pn) is incentive compatible if for every v=(v1, v2, …,vn), i and vi’ 2 V1: if a = f(vi,v-i) and a’ = f(v’i,v-i) then vi(a)-pi(vi,v-i)¸vi(a’)-pi(v’i,v-i) v=(v1, v2,… vn) v-i=(v1, v2,… vi-1 ,vi+1 ,… vn) Prefer telling the truth about vi

Vickrey Clarke Grove Mechanism A mechanism (f,p1, p2,… pn ) is called Vickrey-Clarke-Grove (VCG) if • f(v1, v2, … vn)maximizes i vi(a) over A • Maximizes welfare • There are functions h1, h2,… hn where hi: V1 V2 …  Vn R does not depend on vi we have that: pi(v1, v2, … vn) = hi(v-i) - j  i vj(f(v1, v2,… vn)) Depends only on chosen alternative v=(v1, v2,… vn) v-i=(v1, v2,… vi-1 ,vi+1 ,… vn) Does not depend on vi

Example: Second Price Auction Recall: f assigns the item to one participant and vi(j) = 0 if j  i and vi(i)=wi • f(v1, v2, … vn) = i s.t. wi =maxj(w1, w2,… wn) • hi(v-i) = maxj(w1, w2, … wi-1, wi+1 ,…, wn) • pi(v) = hi(v-i) - j  i vj(f(v1, v2,… vn)) If i the winner pi(vi) = hi(v-i) = maxj  i wj and for j  i pj(vi)= wi – wi = 0 A={i wins|I 2 I}

VCG is Incentive Compatible Theorem: Every VCG Mechanism (f,p1, p2,… pn) is incentive compatible Proof: Fix i, v-i, viand v’i. Let a=f(vi,v-i) and a’=f(v’i,v-i). Have to show vi(a)-pi(vi,v-i)¸vi (a’)-pi(v’i,v-i) Utility of i when declaring vi: vi(a) + j  i vj(a) - hi(v-i) Utility of i when declaring v’i: vi(a’)+ j  i vj(a’)- hi(v-i) Since amaximizessocial welfare vi(a) + j  i vj(a) ¸ vi(a’) + j  i vj(a’) maximizes i vi(a) over A

Clarke Pivot Rule What is the “right”: h? Individually rational: participants always get non negative utility vi(f(v1, v2,… vn)) - pi(v1, v2,… vn) ¸ 0 No positive transfers: no participant is ever paid money pi(v1, v2,… vn) ¸ 0 Clark Pivot rule: Choosing hi(v-i) = maxb 2 Aj  i vj(b) Payment of i when a=f(v1, v2,…, vn): pi(v1, v2,… vn) = maxb 2 Aj  i vj(b) -j  i vj(a) i pays an amount corresponding to the total “damage” he causes other players: difference in social welfare caused by his participation Social welfare (of others) when he participates Social welfare when he does not participate

Rationality of Clarke Pivot Rule Theorem: Every VCG Mechanism with Clarke pivot payments makes no positive Payments. If vi(a) ¸0 then it is Individually rational Proof: Let a=f(v1, v2,… vn) maximizessocial welfare Let b 2 A maximizej  i vj(b) Utility of i: vi(a) + j  i vj(a) - j  i vj(b) ¸j vj(a) - j vj(b) ¸ 0 Payment of i: j  i vj(b) -j  i vj(a) ¸ 0 from choice of b maximizes i vi(a) over A

Examples: Second Price Auction Second Price auction: hi(v-i) = maxj(w1, w2,…, wi-1, wi+1,…, wn) = maxb 2 Aj  i vj(b) Multiunit auction: if k identical items are to be sold to k individuals. A={S wins |S ½ I, |S|=k} and vi(S) = 0 if i2S and vi(i)=wi if i 2 S Allocate units to top k bidders. They pay the k+1th price Claim: this is maxS’ ½ I\{i} |S’| =k j  i vj(S’)-j  i vj(S)

Generalized Second Price Auctions Multiunit auction: if k identical items are to be sold to k individuals. A={S wins |S ½ I, |S|=k} and vi(S) = 0 if i2S and vi(i)=wi if i 2 S Allocate units to top k bidders. The jth highest bidder pays bid j+1. Common in web advertising Claim: this isnot incentive compatible

Examples: Public Project Want to build a bridge: • Cost is C (if built) • Value to each individual vi • Want to built iff  i vj¸ C Player with vj¸ 0 pays only if pivotal j  i vj < C but  j vj¸C in which case pays pj = C- j  i vj In general:  i pj < C Payments do not cover project cost’s • Subsidy necessary! A={build, not build} Equality only when  i vj = C

Buying a (Short) Path in a Graph A Directed graph G=(V,E)where each edge e is “owned” by a different player and has cost ce. Want to construct a path from source s to destination t. • How do we solicit the real cost ce? • Set of alternatives: all paths from s to t • Player e has cost: 0 if enot on chosen path and –ce if on • Maximizing social welfare: finding shortest s-t path: minpathse2 path ce A VCG mechanism that pays 0 to those not on path p: pay each e02 p: e2p’ ce- e2p\{e0} ce where p’ is shortest path withouteo Set A of alternatives: all s-t paths

Clarke mechanism is not perfect • Requires payments & quasilinear utility functions • In general money needs to flow away from the system • Strong budget balance = payments sum to 0 • Impossible in general [Green & Laffont 77] • Vulnerable to collusions • Maximizes sum of players’ utilities (social welfare) • not counting payments) But: sometimes the center is not interested in maximizing social welfare: • E.g. the center may want to maximize revenue

Games with Incomplete Information Game defined by having for every playeri2 I • A set of actions Xi • A set of typesTi. The value ti2Ti is the private informationi knows. • A utility function ui: Ti X1 X2 …  Xn  R where ui(ti, x1, x2, … xn) is the utility obtained by i if his private information is ti and the profile of actions taken by all players is (x1, x2, … xn). Player i chooses his action knowing ti but not other values

…Games with Incomplete Information A strategy for player i2 I is si:Ti X1 A strategy si is (weakly) dominant if for all ti2Ti we have that si(ti) is a dominant strategy in the full information game defined by the ti’s: for all ti’s and all x=(x1, x2, xi-1,x’i, xi+1 … xn) we have that ui(ti, si(ti), x-i) ¸ ui(ti, x) Alternative play

Games and Mechanisms A mechanism is given by • Types T1, T2, … Tn • Actions X1, X2, …, Xn • An alternative set A and outcome function a: X1 X2 …  Xn A • Player’s valuation functions vi: T1 A  R • Payment functions pi: X1 X2 …  Xn R • The utility of player i ui(ti, x1, x2, … xn) = ui(ti, a(x1, x2, … xn)) - pi(x1, x2, … xn) A mechanism implements a social choice functionf f: T1 T2 …  Tn  A in dominant strategies if for some dominant strategiess1, s2, … sn (of the induced game) for all t1, t2, … tn f(t1, t2, … tn ) = a(s1(t1), s2(t2), … sn(tn)) Quasi linear preferences

The Revelation Principle Theorem: if there exists an arbitrary mechanism implementing a social choice function f in dominant strategies, then there exists an incentive compatible mechanism that implements f The payments of the players in the incentive compatible mechanism are identical to those obtained at equilibrium in the original mechanism Proof: by simulation

Revelation Principle: Intuition Constructed “direct revelation” mechanism Strategy s1(t1) Original “complex” “indirect” mechanism Player 1: t1 Strategy . . . . . Outcome a,p1,…,pn . Strategy sn(tn) Player n: tn Strategy

Revelation Principle: Proof • Since si is dominant for player i, then for all ti, x: vi(ti, a(si(ti), x-i)) - pi(si(ti), x-i) ¸ vi(ti,a(x))-pi(x) • In particular for all x-i = s-i (t-i) and xi = si (t’i) To understand mechanism: can think of the equivalent direct revelation mechanism

Direct Characterization • A mechanism is incentive compatible iff the following hold for all i and all vi • The payment pi does not depend on vi but only on the alternative chosen f(vi, v-i) • the payment of alternative a is pa • The mechanism optimizes for each player: f(vi, v-i) 2 argmaxa (vi(a)-pa)

Bayesian Nash Implementation • There is a distribution Di on the typesTi of Player i • It is known to everyone • The value ti2DiTi is the private informationi knows • A profile of strategis si is a Bayesian Nash Equilibrium if for i all ti and all x’i Ed-i[ui(ti, si(ti), s-i(t-i) )] ¸ Ed-i[ui(ti, s-i(t-i)) ]

Bayesian Nash: First Price Auction • First price auction for a single item with two players. • Each has a private value t1 and t2 in T1=T2=[0,1] • Does not make sense to bid true value – utility 0. • There are distributions D1 and D2 • Looking for s1(t1) and s2(t2) that are best replies to each other • Suppose both D1 and D2are uniform. Claim: In the strategies s1(t1)= ti/2 are in Bayesian Nash Equilibrium t1 Win half the time Cannot win

Expected Revenues Expected Revenue: • For first price auction: max(T1/2, T2/2) where T1 and T2 uniform in [0,1] • For second price auction min(T1, T2) • Which is better? • Both are 1/3. • Coincidence? Theorem [Revenue Equivalence]: under very general conditions, every twoBayesian Nash implementations of the same social choice function if for some player and some type they have the same expected payment then • All types have the same expected payment to the player • If all player have the same expected payment: the expected revenues are the same

Lecturer: Moni Naor