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Lecturer: Moni Naor Algorithmic Game Theory Uri Feige Robi Krauthgamer Moni Naor Lecture 10: Mechanism Design

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

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

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Change must happen at some profile i* Where voter i* changed his opinion 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 a Á ba Á bb Á ab Á a Claim: this i* is the dictator! Hybrid argumentVoters 1 2 n Profiles 012 … n

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Single-peaked preferences [Black 48] Suppose alternatives are ordered on a line Every voter prefers alternatives that are closer to her most preferred alternative a1a1 a2a2 a3a3 a4a4 a5a5 v1v1 v2v2 v3v3 v4v4 v5v5 Choose the median voter’s peak as the winner Strategy-proof!

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What about Probabilistic Voting Schemes? Electing the Doge in the Republic of Venice 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

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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

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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.

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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=(p 1,p 2, … p n )

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

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Example: Vickrey’s Second Price Auction Single item for sale Each player has scalar value w i – willingness to pay If he wins item and has to pay p : utility w i -p If someone else wins item: utility 0 Second price auction : Winner is the one with the highest declared value w i. P ays the second highest bid p*=max j i w j Theorem (Vickrey): for any every w 1, w 2,…,w n and every w i ’. Let u i be i’s utility if he bids w i and u’ i if he bids w i ’. Then u i ¸ u’ i.. Despite private information and selfish behavior compute “reliably” the max function!

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Direct Revelation Mechanism A direct revelation mechanism is a social choice function f: V 1 V 2 … V n A and payment functions p i : V 1 V 2 … V n R Participant i pays p i (v 1, v 2, … v n ) A mechanism (f,p 1, p 2,… p n ) is incentive compatible if for every v=(v 1, v 2, …,v n ), i and v i ’ 2 V 1 : if a = f(v i,v -i ) and a’ = f(v’ i,v -i ) then v i (a)-p i (v i,v -i ) ¸ v i (a’) -p i (v’ i,v -i ) Prefer telling the truth about v i v=(v 1, v 2,… v n ) v -i =(v 1, v 2,… v i-1,v i+1,… v n )

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Vickrey Clarke Grove Mechanism A mechanism (f,p 1, p 2,… p n ) is called Vickrey- Clarke-Grove (VCG) if f(v 1, v 2, … v n ) maximizes i v i (a) over A –Maximizes welfare There are functions h 1, h 2,… h n where h i : V 1 V 2 … V n R does not depend on v i we have that: p i (v 1, v 2, … v n ) = h i (v -i ) - j i v j (f(v 1, v 2,… v n )) v=(v 1, v 2,… v n ) v -i =(v 1, v 2,… v i-1,v i+1,… v n ) Depends only on chosen alternative Does not depend on v i

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Example: Second Price Auction Recall: f assigns the item to one participant and v i (j) = 0 if j i and v i (i)=w i f(v 1, v 2, … v n ) = i s.t. w i =max j (w 1, w 2,… w n ) h i (v -i ) = max j (w 1, w 2, … w i-1, w i+1,…, w n ) p i (v) = h i (v -i ) - j i v j (f(v 1, v 2,… v n )) If i the winner p i (v i ) = h i (v -i ) = max j i w j and for j i p j (v i )= w i – w i = 0 A={i wins|I 2 I}

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maximizes i v i (a) over A VCG is Incentive Compatible Theorem : Every VCG Mechanism (f,p 1, p 2,… p n ) is incentive compatible Proof : Fix i, v -i, v i and v’ i. Let a=f(v i,v -i ) and a’=f(v’ i,v -i ). Have to show v i (a)-p i (v i,v -i ) ¸ v i (a’) -p i (v’ i,v -i ) Utility of i when declaring v i : v i (a) + j i v j (a) - h i (v -i ) Utility of i when declaring v’ i : v i (a’)+ j i v j (a’)- h i (v -i ) Since a maximizes social welfare v i (a) + j i v j (a) ¸ v i (a’) + j i v j (a’)

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Social welfare (of others) when he participates Clarke Pivot Rule What is the “right”: h ? Individually rational : participants always get non negative utility v i (f(v 1, v 2,… v n )) - p i (v 1, v 2,… v n ) ¸ 0 No positive transfers : no participant is ever paid money p i (v 1, v 2,… v n ) ¸ 0 Clark Pivot rule: Choosing h i (v -i ) = max b 2 A j i v j (b) Payment of i when a=f(v 1, v 2,…, v n ): p i (v 1, v 2,… v n ) = max b 2 A j i v j (b) - j i v j (a) i pays an amount corresponding to the total “ damage ” he causes other players: difference in social welfare caused by his participation Social welfare when he does not participate

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maximizes i v i (a) over A Rationality of Clarke Pivot Rule Theorem : Every VCG Mechanism with Clarke pivot payments makes no positive Payments. If v i (a) ¸ 0 then it is Individually rational Proof : Let a=f(v 1, v 2,… v n ) maximizes social welfare Let b 2 A maximize j i v j (b) Utility of i : v i (a) + j i v j (a) - j i v j (b) ¸ j v j (a) - j v j (b) ¸ 0 Payment of i : j i v j (b) - j i v j (a) ¸ 0 from choice of b

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Examples: Second Price Auction Second Price auction: h i (v -i ) = max j (w 1, w 2,…, w i-1, w i+1,…, w n ) = max b 2 A j i v j (b) Multiunit auction : if k identical items are to be sold to k individuals. A={S wins |S ½ I, |S|=k} and v i (S) = 0 if i 2 S and v i (i)=w i if i 2 S Allocate units to top k bidders. They pay the k+1 th price Claim : this is max S’ ½ I\{i} |S’| =k j i v j (S’)- j i v j (S)

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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 v i (S) = 0 if i 2 S and v i (i)=w i if i 2 S Allocate units to top k bidders. The j th highest bidder pays bid j+1. Common in web advertising Claim : this is not incentive compatible

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

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Set A of alternatives: all s-t paths 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 c e. Want to construct a path from source s to destination t. How do we solicit the real cost c e ? –Set of alternatives: all paths from s to t –Player e has cost: 0 if e not on chosen path and –c e if on – Maximizing social welfare : finding shortest s-t path : min paths e 2 path c e A VCG mechanism that pays 0 to those not on path p : pay each e 0 2 p : e 2 p’ c e - e 2 p\{e 0 } c e where p’ is shortest path without e o

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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

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Games with Incomplete Information Game defined by having for every player i 2 I A set of actions X i A set of types T i. The value t i 2 T i is the private information i knows. A utility function u i : T i X 1 X 2 … X n R where u i (t i, x 1, x 2, … x n ) is the utility obtained by i if his private information is t i and the profile of actions taken by all players is (x 1, x 2, … x n ). Player i chooses his action knowing t i but not other values

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…Games with Incomplete Information A strategy for player i 2 I is s i : T i X 1 A strategy s i is (weakly) dominant if for all t i 2 T i we have that s i (t i ) is a dominant strategy in the full information game defined by the t i ’s: for all t i ’s and all x=(x 1, x 2, x i-1, x’ i, x i+1 … x n ) we have that u i (t i, s i (t i ), x -i ) ¸ u i (t i, x) Alternative play

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Games and Mechanisms A mechanism is given by Types T 1, T 2, … T n Actions X 1, X 2, …, X n An alternative set A and outcome function a: X 1 X 2 … X n A Player’s valuation functions v i : T 1 A R Payment functions p i : X 1 X 2 … X n R The utility of player i u i (t i, x 1, x 2, … x n ) = u i (t i, a(x 1, x 2, … x n )) - p i (x 1, x 2, … x n ) A mechanism implements a social choice function f f: T 1 T 2 … T n A in dominant strategies if for some dominant strategies s 1, s 2, … s n (of the induced game) for all t 1, t 2, … t n f(t 1, t 2, … t n ) = a(s 1 (t 1 ), s 2 (t 2 ), … s n (t n )) Quasi linear preferences

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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

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Revelation Principle: Intuition Player 1 : t 1 Player n : t n... Strategy s 1 (t 1 ) Original “complex” “indirect” mechanism Outcome a,p 1,…,p n Constructed “direct revelation” mechanism Strategy s n (t n )...

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

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

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Bayesian Nash Implementation There is a distribution D i on the types T i of Player i It is known to everyone The value t i 2 D i T i is the private information i knows A profile of strategis s i is a Bayesian Nash Equilibrium if for i all t i and all x’ i E d -i [u i (t i, s i (t i ), s -i (t -i ) )] ¸ E d -i [u i (t i, s -i (t -i )) ]

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Bayesian Nash: First Price Auction First price auction for a single item with two players. Each has a private value t 1 and t 2 in T 1 =T 2 =[0,1] Does not make sense to bid true value – utility 0. There are distributions D 1 and D 2 Looking for s 1 (t 1 ) and s 2 (t 2 ) that are best replies to each other Suppose both D 1 and D 2 are uniform. Claim : In the strategies s 1 (t 1 ) = t i /2 are in Bayesian Nash Equilibrium t1t1 Cannot winWin half the time

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Expected Revenues Expected Revenue: –For first price auction: max(T 1 /2, T 2 /2) where T 1 and T 2 uniform in [0,1] –For second price auction min(T 1, T 2 ) –Which is better? –Both are 1/3. –Coincidence? Theorem [Revenue Equivalence] : under very general conditions, every two Bayesian 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

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