Game Theory: introduction and applications to computer networks

Game Theory: introduction and applications to computer networks Introduction Giovanni Neglia INRIA – EPI Maestro 3 February 2014 Part of the slides are based on a previous course with D. Figueiredo (UFRJ) and H. Zhang (Suffolk University)

Always an equilibrium with small Loss of Efficiency? • Consider only affine cost functions, i.e. cα(x)=aα+bαx • We will use the potential to derive a bound on the social cost of a NE • P(f) <= CS(f) <= 2 P(f) Cα (x) Pα(fα) Cα(fα) x fα

Always an equilibrium with small Loss of Efficiency? • Consider only affine cost functions i.e. cα(x)=aα+bαx • We will use the potential to derive a bound on the social cost of a NE • P(f) <= CS(f) <= 2 P(f) • P(f) = Σ ε E P=Σ ε E Σt=1,…f c(t) <= <=Σ ε E Σt=1,…f c(f)=Σ ε E fc(f)=CS(f) • P(f) = Σ ε E P=Σ ε E Σt=1,…f(aα+bαt)= =Σ ε E fαaα+bαfα(fα+1)/2 >= Σ εEfα(aα+bαfα)/2 =CS(f)/2

Always an equilibrium with small Loss of Efficiency? • Consider only affine cost functions i.e. cα(x)=aα+bαx • P(f) <= CS(f) <= 2 P(f) • Let’s imagine to start from routing fOpt with the optimal social cost CS(fOpt), • Applying the BR dynamics we arrive to a NE with routing fNE and social cost CS(fNE) • CS(fNE) <= 2 P(fNE) <= 2 P(fOpt) <= 2 CS(fOpt) • The LoE of this equilibrium is at most 2

Same technique, different result • Consider a network with a routing at the equilibrium • Add some links • Let the system converge to a new equilibrium • The social cost of the new equilibrium can be at most 4/3 of the previous equilibrium social cost (as in the Braess Paradox)

Loss of Efficiency, Price of Anarchy, Price of Stability • Loss of Efficiency (LoE) • given a NE with social cost CS(fNE) • LoE= CS(fNE) / CS(fOpt) • Price of Anarchy (PoA) [Koutsoupias99] • Different settings G (a family of graph, of cost functions,...) • Xg=set of NEs for the setting g in G • PoA = supgεGsupNEεXg{CS(fNE) / CS(fOpt)} => “worst” loss of efficiencyin G • Price of Stability (PoS) [Anshelevish04] • PoS = supgε GinfNEεXg{CS(fNE) / CS(fOpt)} => guaranteedloss of efficiency in G

Stronger results for affine cost functions • We have proven that for unit-traffic routing games the PoS is at most 2 • For unit-traffic routing games and single-source pairs the PoS is 4/3 • For non-atomic routing games the PoA is 4/3 • non-atomic = infinite players each with infinitesimal traffic • For other cost functions they can be much larger (even unbounded)

Performance Evaluation Sponsored Search Markets Giovanni Neglia INRIA – EPI Maestro 4 February 2013

Google • A class of games for which there is a function P(s1,s2,…sN) such that • For each i Ui(s1,s2,…xi,…sN)>Ui(s1,s2,…yi,…sN) if and only if P(s1,s2,…xi,…sN)>P(s1,s2,…yi,…sN) • Properties of potential games: Existence of a pure-strategy NE and convergence to it of best-response dynamics • The routing games we considered are particular potential games

How itworks • Companies bid for keywords • On the basis of the bids Google puts their link on a given position (first ads get more clicks) • Companies are charged a given cost for each click (the cost depends on all the bids)

Somenumbers • ≈ 95% of Google revenues (46 billions$) fromads • investor.google.com/financial/tables.html • 87% of Google-Motorola revenues (50 billions$) • Costs • "calligraphypens" $1.70 • "Loan consolidation" $50 • "mesothelioma" $50 per click • Click fraudproblem

Outline • Preliminaries • Auctions • Matchingmarkets • Possible approaches to adspricing • Google mechanism • References • Easley, Kleinberg, "Networks, Crowds and Markets", ch.9,10,15

Types of auctions • 1stprice & descendingbids • 2ndprice & ascendingbids

Game Theoretic Model • N players (the bidders) • Strategies/actions: biisplayeri’sbid • For player i the good has value vi • piisplayeri’spayment if hegets the good • Utility: • vi-pi if player i gets the good • 0 otherwise • Assumptionhere: values vi are independentand private • i.e. veryparticulargoods for whichthereis not a referenceprice

Game Theoretic Model • N players (the bidders) • Strategies: biisplayeri’sbid • Utility: • vi-bi if player i gets the good • 0 otherwise • Difficulties: • Utilities of otherplayers are unknown! • Better to model the strategyspaceas continuous • Most of the approacheswestudied do not work!

2ndpriceauction • Player with the highestbidgets the good and pays a priceequal to the 2ndhighestbid • There is a dominant strategies • I.e. a strategythatis more convenientindependentlyfromwhat the otherplayers do • Be truthful, i.e. bid how muchyouevaluate the good (bi=vi) • Social optimality: the bidderwho value the good the mostgetsit!

bi=viis the highestbid bids bids bi’>bi bi bk bk Ui=vi-bk>vi-bi=0 Ui’=vi-bk bh bh bn bn Bidding more than viis not convenient

bi=viis the highestbid bids bids bi bk bk bi’<bi Ui=vi-bk>vi-bi=0 Ui’=0 bh bh bn bn Biddinglessthan viis not convenient (maybeunconvenient)

bi=viis not the highestbid bids bids bi’>bi Ui’=vi-bk<vi-bi=0 bk bk bi Ui=0 bh bh bn bn Bidding more than viis not convenient (maybeunconvenient)

bi=viis not the highestbid bids bids bk bk bi bi’<bi Ui’=0 Ui=0 bh bh bn bn Biddinglessthan viis not convenient

Seller revenue • N bidders • Values are independentrandom values between 0 and 1 • Expectedithlargest utility is(N+1-i)/(N+1) • Expected seller revenue is(N-1)/(N+1)

1stpriceauction • Player with the highestbidgets the good and pays a priceequal to her/hisbid • Beingtruthfulis not a dominant strategyanymore! • How to studyit?

1stpriceauction • Assumption: for eachplayer the other values are i.i.d. random variables between 0 and 1 • to overcome the factthat utilities are unknown • Player i’sstrategyis a function s() mapping value vi to a bid bi • s() strictlyincreasing, differentiablefunction • 0≤s(v)≤v  s(0)=0 • Weinvestigate if thereis a strategy s() common to all the playersthat leads to a Nash equilibrium

1stpriceauction • Assumption: for eachplayer the other values are i.i.d. random variables between 0 and 1 • Player i’sstrategyis a function s() mapping value vi to a bid bi • Expectedpayoff of player i if all the playersplays s(): • Ui(s,…s,…s) = viN-1 (vi-s(vi)) i’spayoff if he/shewins prob. i wins

1stpriceauction • Expectedpayoff of player i if all the playersplay s(): • Ui(s,…s,…s) = viN-1 (vi-s(vi)) • What if i plays a differentstrategyt()? • If all playersplaying s() is a NE, then : • Ui(s,…s,…s) = viN-1(vi-s(vi)) ≥ viN-1 (vi-t(vi)) = = Ui(s,…t,…s) • Difficult to check for all the possible functionst() differentfrom s() • Help from the revelationprinciple

The RevelationPrinciple • All the strategies are equivalent to bidder i supplying to s() a different value of vi s() vi bi t() s() vi' vi bi' bi'

1stpriceauction • Expectedpayoff of player i if all the playersplays s(): • Ui(v1,…vi,…vN) = Ui(s,…s,…s) = viN-1 (vi-s(vi)) • What if i plays a differentstrategyt()? • By the revelationprinciple: • Ui(s,…t,…s) = Ui(v1,…v,…vN)= vN-1(vi-s(v)) • If viN-1 (vi-s(vi)) ≥ vN-1(vi-s(v)) for each v (and for each vi) • Then all playersplaying s() is a NE

1stpriceauction • If viN-1 (vi-s(vi)) ≥ vN-1(vi-s(v)) for each v (and for each vi) • Then all playersplaying s() is a NE • f(v)=viN-1 (vi-s(vi)) - vN-1 (vi-s(v)) isminimized for v=vi • f’(v)=0 for v=vi, • i.e. (N-1) viN-2 (vi-s(v)) + viN-1 s’(vi) = 0 for each vi • s’(vi) = (N-1)(1 – s(vi)/vi), s(0)=0 • Solution: s(vi)=(N-1)/N vi

1stpriceauction • All playersbiddingaccording to s(v) = (N-1)/N v is a NE • Remarks • They are not truthful • The more they are, the highertheyshouldbid • Expected seller revenue • ((N-1)/N) E[vmax] = ((N-1)/N) (N/(N+1)) = (N-1)/(N+1) • Identical to 2ndpriceauction! • A general revenue equivalenceprinciple

MatchingMarkets goods buyers v11,v21, v31 1 1 v12, v22, v32 2 2 v12, v22, v32 3 3 vij: value thatbuyer j gives to good i How to match a set of differentgoods to a set of buyerswithdifferentevaluations

MatchingMarkets p1=2 12, 4, 2 1 1 8, 7, 6 p2=1 2 2 7, 5, 2 p3=0 3 3 Whichgoodsbuyerslikemost? Preferred seller graph How to match a set of differentgoods to a set of buyerswithdifferentevaluations

MatchingMarkets • Given the prices, look for a perfectmatching on the preferred seller graph • There is no suchmatching for this graph p1=2 12, 4, 2 1 1 8, 7, 6 p2=1 2 2 7, 5, 2 p3=0 3 3 Whichgoodsbuyerslikemost? Preferred seller graph

MatchingMarkets • But withdifferentprices, thereis p1=3 12, 4, 2 1 1 8, 7, 6 p2=1 2 2 7, 5, 2 p3=0 3 3 Whichgoodsbuyerslikemost? Preferred seller graph

MatchingMarkets • But withdifferentprices, thereis • Suchprices are market clearing prices p1=3 12, 4, 2 1 1 8, 7, 6 p2=1 2 2 7, 5, 2 p3=0 3 3 Whichgoodsbuyerslikemost? Preferred seller graph

Market Clearing Prices • Theyalwaysexist • And canbeeasilycalculated if valuations are known • They are socially optimal in the sensethattheymaximize the sum of all the payoffs in the network (bothsellers and buyers)

Adspricing Ads positions companies r1 v1 1 1 v2 r2 2 2 v3 r3 3 3 vi: value thatcompany i gives to a click ri: click rate for an ad in position i (assumed to beindependent from the ad and known a priori) How to rankadsfromdifferentcompanies

Adspricing as a matchingmarket Ads positions companies • Problem: Valuations are not known! • … but wecould look for something as 2ndpriceauctions r1 v1r1, v1r2, v1r3 1 1 v2r1, v2r2, v2r3 r2 2 2 v3r1, v3r2, v3r3 r3 3 3 vi: value thatcompany i gives to a click ri: click rate for an ad in position i (assumed to beindependent from the ad and known a priori)

The VCG mechanism • The correct way to generalize 2ndpriceauctions to multiple goods • Vickrey-Clarke-Groves • Everybuyersshouldpay a priceequal to the social value loss for the othersbuyers • Example: consider a 2ndpriceauctionwith v1>v2>…vN • With 1 present the othersbuyersget 0 • Without 1, 2 would have got the good with a value v2 • then the social value loss for the othersis v2

The VCG mechanism • The correct way to generalize 2ndpriceauctions to multiple goods • Vickrey-Clarke-Groves • Everybuyersshouldpay a priceequal to the social value loss for the othersbuyers • If VBSis the maximum total valuation over all the possible perfectmatchings of the set of sellers S and the set of buyers B, • If buyer j gets good i, he/sheshouldbecharged VB-jS- VB-jS-i

VCG example Ads positions companies r1=10 v1=3 1 1 v2=2 r2=5 2 2 v3=1 r3=2 3 3 vi: value thatcompany i gives to a click ri: click rate for an ad in position i (assumed to beindependent from the ad and known a priori)

VCG example Ads positions companies 30, 15, 6 1 1 20, 10, 4 2 2 10, 5, 2 3 3

VCG example Ads positions companies • This is the maximum weightmatching • 1 gets 30, 2 gets 10 and 3 gets 2 30, 15, 6 1 1 20, 10, 4 2 2 10, 5, 2 3 3

VCG example Ads positions companies • If 1 weren’tthere, 2 and 3 wouldget 25 instead of 12, • Then 1 shouldpay 13 30, 15, 6 1 1 20, 10, 4 2 2 10, 5, 2 3 3

VCG example Ads positions companies • If 2 weren’tthere, 1 and 3 wouldget 35 instead of 32, • Then 2 shouldpay 3 30, 15, 6 1 1 20, 10, 4 2 2 10, 5, 2 3 3

VCG example Ads positions companies • If 3 weren’tthere, nothingwould change for 1 and 2, • Then 3 shouldpay 0 30, 15, 6 1 1 20, 10, 4 2 2 10, 5, 2 3 3

The VCG mechanism • Everybuyersshouldpay a priceequal to the social value loss for the othersbuyers • If VBSis the maximum total valuation over all the possible perfectmatchings of the set of sellers S and the set of buyers B, • If buyer j gets good i, he/sheshouldbecharged VB-jS- VB-jS-i • Under thispricemechanism, truth-tellingis a dominant strategy

Google’s GSP auction • Generalized Second Price • Once all the bids are collected b1>b2>…bN • Company i pays bi+1 • In the case of a single good (position), GSP isequivalent to a 2ndpriceauction, and also to VCG • But why Google wanted to implementsomethingdifferent???

Game Theory: introduction and applications to computer networks