Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction

17 Pages·2010·0.15 MB·English

Pure and Bayes-Nash Price of Anarchy for
Generalized Second Price Auction
Renato Paes Leme E´vaTardos
∗ †
Abstract
GeneralizedSecond Price Auction, also knows as Ad Word auctions, and its variants has been the
mainmechanismusedbysearchcompaniestoauctionpositionsforsponsoredsearchlinks. Inthispaper
westudythesocialwelfareoftheNashequilibriaofthisgame. ItisknownthatsociallyoptimalNash
equilibriaexists(i.e.,thatthePriceofStabilityforthisgameis1).Thispaperisthefirsttoprovebounds
onthepriceofanarchy.
Ourmainresultistoshowthatundersomemildassumptionsthepriceofanarchyissmall. Forpure
Nashequilibriaweboundthepriceofanarchyby1.618,assumingallbiddersareplayingun-dominated
strategies. FormixedNashequilibriaweproveaboundof4underthesameassumption.Wealsoextend
the resulttothe Bayesiansettingwhenbiddersvaluationsarealso random,andprovea boundof8for
thiscase.
Our proofexhibits a combinatorialstructure of Nash equilibria and use this structure to boundthe
priceofanarchy.WhileestablishingthestructureissimpleinthecaseofpureandmixedNashequilibria,
the extensionto the Bayesian setting requiresthe use of novelcombinatorialtechniquesthatcan be of
independentinterest.
∗[email protected]. ofComputerScience,CornellUniversity,Ithaca,NY14853. Supportedinpartby
NSFgrantsCCF-0729006.
†[email protected],CornellUniversity,Ithaca,NY14853SupportedinpartbyNSFgrants
CCF-0910940andCCF-0729006,ONRgrantN00014-98-1-0589,andaYahoo!ResearchAllianceGrant.
1 Introduction
Search engines and other online information sources use sponsored search auction, or AdWord auctions,
to monetize their services. These actions allocate advertisement slots to companies, and companies are
charged pay per click, that is, they are charged afee for any user that clicks on the link associated with the
advertisement. Mehta,Saberi,Vazirani,andVazirani[9,10]consideredAdWordauctionsinthealgorithmic
context, studying the problem ofassigning AdWordstoadvertisers online asthewordshowsupinasearch
query. Since the introduction of the model, there has been much work in the area, see the survey of Lahaie
etal[7].
Here we consider AdWords in a game theoretic context: consider the game played by advertisers in
bidding for an AdWord. The bids are used to determine both the assignment of bidders to slots, and also
the fees charged. The bidders are assigned to slots in order of bids, and the fee for each click is decided
by variant of the so-called Generalized Second Price Auction (GSP), a simple generalization of the well-
known Vickrey auction [14] for a single item (or a single advertising slot). The Vickrey auction [14] for
a single item, and its generalization, the Vickrey-Clarke-Groves Mechanism (VCG) [3, 5], make truthful
behavior(whentheadvertisersrevealtheirtruevaluation)dominantstrategy,andmaketheresultingoutcome
maximizethesocialwelfare.
Generalized Second Price Auction, the mechanism adopted by all search companies, is a simple and
natural generalization oftheVickreyauction forasingle slot, butitisneither truthful normaximizes social
welfare. In this paper we will consider the social welfare of the GSP auction outcomes. Our goal in this
paper is to show that the intuition based on the similarity of GSP to the truthful Vickrey auction is not so
far from truth: weprove that thesocial welfare is within asmallconstant factor ofthe optimal in anyNash
equilibrium undermildassumption thattheplayersuseun-dominated strategies.
We consider both full information games when player valuations are known, and also consider the
Bayesian setting when the values are independent random variables. Our results differ significantly from
the existing work on the price of anarchy in a number of ways. Many of the known results can be summa-
rized via a smoothness argument, as observed by Roughgarden [11]. In contrast, we show in Appendix B
that the GSPgame is not smooth in the sense of [11]. Second, most known price of anarchy results are for
thecaseoffullinformationgames. Thefullinformationsettingassumesthatalladvertisers areawareofthe
valuationsofallotherplayers. Thisisaverystrongassumptionandisnotrealistic. Incontrast,theBayesian
settingrequiresonlymuchweakerassumptionthatvaluationsaredrawnfromindependentdistributions, and
these distributions are known to the other players. Proving the price of anarchy bound for the Bayesian
settingrequires theuseofnovelcombinatorial techniques.
We use a standard model of separable click-though rates: where the probability of clicking on an ad-
vertisement j displayed in slot i is α γ , i.e., the probability is a product of two separable components:
i j
depending on the slot, and on the advertiser respectively. Tosimplify the presentation, for the main part of
thepaper, wewillfocus onthesimple casewhenγ = 1forallj, thatis,theprobability ofaclickdepends
j
only on the slot. In Appendix A we show that it is easy to extend our results to the model with separable
click-through rates.
For both our simple model, and the case of separable click-through rates, it is known that there exists
Nashequilibriathataresociallyoptimal[4,13],i.e.,thatthepriceofstabilityis1. Itisnothardtogivesimple
examplesofNashequilibriawherethesocialwelfareisarbitrarilysmallerthantheoptimum. However,these
equilibria areunnatural, assomebidexceedstheplayersvaluations, andhencetheplayertakesunnecessary
risk. We show that bidding above the valuation is a dominated strategy, and define conservative bidders as
bidderswhowon’tbidabovetheirvaluations. Ourresultsassumethatplayers areconservative.
Ourresults ThemainresultsofthispaperarePriceofanarchyboundsforpure,mixedandBayesianNash
equilibria for the GSP game assuming conservative bidders. To motivate the conservative assumption, we
1
observethatbidding abovetheplayersvaluation isdominated strategyinallsettings.
For each setting we exhibit a combinatorial structure of Nash equilibria that can be of independent
interest. To state this structure we need the following notation. For an advertiser k let v be the value of
k
advertiser k for a click (a random variable in the Bayesian case). For a slot i let π(i) be the advertiser
assigned to slot i in a Nash equilibrium (a random variable, in the case of mixed Nash, or in the Bayesian
setting).
ForthecaseofpureNashequilibriathesocialwelfareinaNashequilibriumwithconservativebidders
•
isatmostafactorof1.618abovetheoptimum. Weachievethisboundviaastructuralcharacterization
of such equilibria: for any two slots i and j, we show that in a Nash equilibrium with conservative
bidders, wemusthavethat
α v
j π(i)
+ 1.
α v ≥
i π(j)
Itisnothard toseethatthis structure implies thatthe assignment cannot betoo farfrom theoptimal:
iftwoadvertisersareassignedtopositionsnotintheirorderofbids,theneither(i)thetwoadvertisers
have similar values for a click; or (ii) the click-through rates of the two slots are not very different,
andhenceineithercasetheirrelativeorderdoesn’t affectthesocialwelfareverymuch.
Wealso bound the quality of mixed Nashequilibria. Fora mixedNash equilibrium π(i) isarandom
•
variable, indicating the bidder assigned to slot i, and similarly let the random variable σ(i) denote
the slot assigned to bidder i. For notational convenience we number players in order of decreasing
valuation, and number slots in order of decreasing click rates. By this notation, bidder i should be
assigned toslot iintheoptimal solution. Wederive the structure ofpure Nash equilibria bythinking
aboutapairofbiddersthatareassignedtoslotsinreverseorder. Suchpairsarehardertodefineinthe
mixed case. Instead, we will consider bider i and his optimal slot i, and get the following condition
formixedNashequilibria.
Eα Ev 1
σ(i) π(i)
+ ,
α v ≥ 2
i i
andusethisinequalitytoshowthatthesocialwelfareofamixedNashequilibriumisatleastanfourth
fraction of the social welfare of the socially optimal assignment. Note that an analogous inequality
with bound 1holds also for pure Nash, which implies abound of 2on the pure price ofanarchy. We
usedadifferentcharacterization abovetobeabletoprovethestronger bound/
We prove a bound of 8 on the price of anarchy for the Bayesian setting, and the valuations v are
k
•
random. We do this via a slightly more complicated, structural property, showing that an expression
similar to the one used in the case of mixed Nash must be at least 1/4th in expectation. However,
establishing this inequality in the Bayesian setting in much harder. In the context of pure and mixed
Nash, the inequality follows from the Nash property by considering a simple deviation by a player.
E.g.,aplayer whowouldbeassigned toslot iinthe optimum, maywanttotrytobid highenough to
takeoversloti. Incontrast, wearenotawareofasingle deviating bidthatcanhelpestablish auseful
structure. Instead, we obtain our structural result by considering many different bids, and show that
theinequalities established bythedifferent bidscanbecombined toshowthestructure.
Intheprocessweuseanumberofnewtechniques ofindependent interest. Thebidsweuseforplayer
i are closely related to 2E[b v ;ν(i) = k], twice the expected value of the bid in slot k where the
πk i
|
expectation is conditioned both on the value v of player i, and the fact that its optimal position is k.
i
These expectations as defined here depend on player i both through the conditioning and though the
positionbidderigetsinthepermutationπ. Thisdualdependencemakesithardtoproveanyproperties
ofthem. Forexample,itwouldbenaturaltoassumethatforanyvaluev thevaluesmonotoneincrease
i
withk,butthatisnotalwaysthecase. Togetaroundthisproblem,wewilluseinsteadslightlysmaller
2
values for the bids. We show via an interesting combinatorial argument using the max-flow min-cut
theorem,thatthemodifiedvaluesdoincreasewithk. Thenweuseanovelaveragingtechnique(using
linearprogramming) tocombinetheresulting inequalities toestablish thesimplestructural property.
Related work Sponsored search has been a very active area of research in the last several years. Mehta
et al. [10, 9] considered AdWord auctions in the algorithmic context. Since the original models, there has
been much work in the area, see the survey of Lahaie et al [7] for a general introduction. Here we use the
game theoretic model of the AdWord auctions of Edelman et al [4] and Varian [13], for a truthful auction
seeAggarwaletall[1].
Weusethemodelofseparable click-though rates, wheretheclickthrough rateforbidder j inslotican
be expressed in a simple product form γ α . Forthese models Edelman et al[4] and Varian [13] show that
j i
thepriceofStabilityforthisgameis1,thatis,thereexistsNashequilibria thataresociallyoptimal.
Lahaie [6] also considers the problem of quantifying the social efficiency of an equilibrium. He makes
thestrongassumptionthatclick-through-rate α decaysexponentially alongtheslotswithafactorof 1,and
i δ
provesapriceofanarchyofmin 1,1 1 . Inthispaper,wemakenoassumptionsontheclick-through-rates.
{δ −δ}
ThompsonandLeyton-Brown[12]studytheefficiencylossofequilibria empirically invariousmodels.
Weassumethatbiddersareconservative,inthesensethatnobidderisbiddingabovetheirownvaluation.
Wecanjustifythisassumptionbynotingthatbiddingabovehisvaluationisadominatedstrategy. Lucierand
Borodin [8] and Christodoulou at al [2] also use the conservative assumption to establish price-of-anarchy
results in the context of combinatorial auctions. Without any additional requirement Nash equilibria, even
in the case of the single item Vickrey auction, can have social welfare that is arbitrarily bad compared to
the optimal social welfare. However, we show that Nash equilibria of conservative bidders is within small
constant factoroftheoptimum.
The paper by Lucier and Borodin [8] on greedy auctions is also closely related to our work. They
analyze the Price of Anarchy of the auction game induced by Greedy Mechanisms. The consider in a
generalcombinatorialauctionsetting,greedyalgorithmswithpaymentsarecomputedusingthecriticalprice.
They show via a type of smoothness argument (see [11]) that of the greedy algorithm is a c-approximation
algorithm, thenthePriceofAnarchyoftheresulting mechanism isc+1-forpureandmixedNashandfor
Bayes-Nashequilibria. TheGeneralized SecondPricemechanism isatypeofgreedymechanism, butisnot
acombinatorial auction, andhenceitdoesnotfittheframeworkofLucierandBorodin,andfurthertheGSP
gamedoesnotsatisfythesmoothness condition. Thekeytoprovingthec+1boundofLucierandBorodin
[8]istoconsiderpossiblebids,suchasasinglemindedbidfortheslotintheoptimalsolution,ormodifyinga
bitbychangingitonlyonasingleslot(theoneallocatedintheoptimalsolution). Thecombinatorialauction
frameworkallowssuchcomplexbids;incontrast,thebidsinGSPhavelimitedexpressivity,asbidisasingle
number, and hence bidders cannot make single-minded declarations for a certain slot, or modify their bid
only on one of the slots. Like the GSP game, many natural bidding languages have limited expressivity,
as typically allowing arbitrary complex bids makes the optimization problem hard. However, the limited
expressivityofthebiddinglanguagecanincreasethesetofNashequilibria(astherearefewerdeviatingbids
toconsider). Itisimportant tounderstand ifsuchnaturalbidding languages resultingreatly increased price
ofanarchy.
2 Preliminaries
We consider an auction with n advertisers and n slots (if there are less slots than advertisers, consider
additionalvirtualslotswithclick-through-rate zero). Wemodelthisauctionasagamewithnplayers,where
each advertiser is one player. The types of the advertisers are given by their valuation v , which expresses
i
theirvalueforoneclick. Thestrategyforeachadvertiser isabidb [0, ).
i
∈ ∞
3
Therearenslotsandbasedonthebids,wedecide wheretoallocate eachadvertiser. Inthemostsimple
model,thek-thslotcontainsα clicksandα isamonotonenon-increasing sequece,i.e.,α α ...
k k 1 2
≥ ≥ ≥
α . We prove our results for this simple model, but they extend naturally to the more realistic model of
n
separable click-through-rates, asweshowinAppendixA.Thegameproceeds asfollows:
1. eachadvertisersubmitsabidb 0,whichishisdeclared valueforaclick
i
≥
2. the advertiser are sorted by their bids (ties are broken arbitrarily). Call π(k) the advertiser with the
k-thhighest bid
3. advertiser π(k)isplacedonslotk andtherefore receivedα clicks
k
4. foreachclick,advertiser k paysb ,whichisthenexthighestbid
π(k+1)
The vector π is a permutation that indicates to which slot each player is assigned - it is completely
determined by the set of bids. We define the utility of a user i when occupying slot j as given by u (b) =
i
α (v b ). Wedefine the social welfare ofthis game asthe total value the bidders get from playing
j i π(j+1)
−
it, which is: α v . Here in this paper we are concerned about bounding the social welfare in an
j j π(j)
equilibrium of this game relative to the optimal. This measure is called Price of Anarchy. We analyze the
P
PriceofAnarchyinthreedifferent settingsofincreasing complexity:
PureNashequilibrium: Thevaluationofeachplayerv isafixedvalue. Weconsiderwithoutlossof
i
•
generality that v v ... v . Each player chooses a pure strategy, i.e., a deterministic bid b .
1 2 n i
≥ ≥ ≥
We say that a set of bids b = (b ,...,b ) is a PureNash Equilibrium if any bidder can change his
1 n
bidanincreasehisutility, i.e.:
u (b ,b ) u (b ,b ), b [0, )
i i −i ≥ i ′i −i ∀ ′i ∈ ∞
To gain some intuition, suppose advertiser i is currently bidding b and occupying slot j. Changing
i
hisbidtosomethingbetweenb andb won’tchangethepermutationπandthereforewon’t
π(j 1) π(j+1)
−
change the allocation nor his payment. So, he could try to increase his utility by doing one of two
things:
– increasing hisbidtogetaslotwithabetterclick-through-rate. Ifhewantstogetaslotk < j he
needstooverbid advertiser π(k),saybybidding b +ǫ. Thiswayhewouldgetslotk forthe
π(k)
priceb perclick,gettingutilityα (v b ).
π(k) k i π(k)
−
– decreasing his bid to get a worse but cheaper slot. If he wants to get slot k > j he needs to
bid below advertiser π(k). This way he would get slot k for the price b per click, getting
π(k+1)
utilityα (v b ).
k i π(k+1)
−
We are interested in bounding the Pure Price of Anarchy, which is the ratio α v / α v ,
j j j j j π(j)
betweenthesocialwelfareintheoptimalandinaNashequilibrium.
P P
Mixed Nash equilibrium: The valuation v are still fixed and we can assume (without loss of gen-
i
•
erality) that v ... v , but each player is allowed to pick a distribution over strategies. We can
1 n
≥ ≥
think that each player chooses a random variable b and the Nash equilibrium means that the chosen
i
random variablemaximizestheexpected utility. Inotherwords:
E[u (b ,b )] E[u (b ,b )], b
i i −i ≥ i ′i −i ∀ ′i
where expectation is with respect to the distribution of bids. Now, the assignment π is a random
variable determined by b and therefore the social welfare is also a random variable (even though the
optimalisfixed). ThePriceofAnarchyistheratio: α v /E[ α v ].
j j j j j π(j)
P P
4
Bayes-Nash equilibrium: In a more realistic model, the player’s don’t know the valuations of other
•
players, but they have beliefs about it. This is modeled as follows: the valuation v are drawn from
i
independent distributions. A player chooses a bid (possibly in a randomized fashion) based on his
ownvaluation. Therefore, the strategy ofplayer iisabidding function b (v )that associates foreach
i i
valuation v a distribution of bids. A set of bidding functions is said to be a Bayes-Nash equilibrium
i
if:
E[u (b (v ),b (v ))v ] E[u (b (v ),b (v )v ], b (v ) v
i i i −i −i | i ≥ i ′i i −i −i | i ∀ ′i i ∀ i
whereexpectations aretakenovervaluesandrandomness usedbyplayers.
TheNashassignmentπisarandomvariable,sinceitisdependentonthebids,whicharerandom. The
optimal allocation is also a random variable, and we define it by ν: let ν(k) be the slot occupied by
playeriintheoptimalassignment. Therefore,νisarandomvariablesuchthatv > v ν(i) < ν(j).
i j
⇒
The optimal social welfare is therefore α v . In this setting the quantity we want to bound is
j ν(j) j
theBayes-NashpriceofAnarchy,givenbytheratio: E[ α v ]/E[ α v ]
P j ν(j) j j j π(j)
P P
2.1 Equilibria withLow SocialWelfare
Even for two slots the gap between the best and the worse Nash equilibrium can be arbitrarily large. For
example,considertwoslotswithclick-through-rates α = 1andα = randtwoadvertiserswithvaluations
1 2
v = 1 and v = 0. It is easy to check that the bids b = 0 and b = 1 r are a Nash equilibrium where
1 2 1 2
−
advertiser 1gets thesecond slotand advertiser 2gets thefirstslot. Thesocial welfare inthis equilibrium is
r whilethe optimal is1. Theprice of anarchy istherefore 1/r. Since r canbeany number from 0to1, the
gapbetweentheoptimalandtheworseNashcanbearbitrarily large.
Notice however that this Nash equilibrium seems very artificial: advertiser 2 is exposed to the risk of
negativeutility: ifadvertiser1(oranotheradvertiser)addsabidsomewherebetween0and1 rthisimposes
−
anegativeutilityonadvertiser 2. Foradvertiser2,bidding anything greaterthanzeroisclearlyadominated
strategy. Itisnothardtoseethatforanyv ,biddingb > v isdominated bythebidv .
i i i i
3 Pure Nash Equilibrium
We say pure bid b for advertiser i is conservative if b v . Next we show that non-conservative bids are
i i i
≤
dominated. Wesay that astrategy b is dominated ifthere is someb such that u (b ,b ) u (b ,b )for
i ′i i i −i ≤ i ′i −i
allb andforatleastonevalueofb itholdsstrictly.
i i
− −
Lemma3.1 Abidb > v isdominatedbyb = v .
i i ′i i
Proof. Decreasing the bid from b to v changes allocation and or payment only if there is a bidder j = i
i i
6
withv < b < v . However,inthiscase, bidder igetting negativeutility foreachclick, andwithbidv ,he
i j j i
cannotgetnegativeutility.
Giventheparametersα,v,wesaythatbisaconservativebidderequilibriumifitisaNashequilibrium
andb v forallbiddersi.
i i
≤
Theorem3.2 For2slots, ifalladvertisers areconservative, thenthepriceofanarchyisexactly 1.25.
Proof. We can suppose without loss of generality (by scaling α and v) that α = 1, α = r and α v +
1 2 1 1
α v = 1. Inanynon-optimal Nashequilibrium b b andbytheNashcondition r(v 0) 1(v b )
2 2 1 2 1 1 2
≤ − ≥ −
andbytheconservativeconditionb v . Substitutingv = 1 rv inthosetwoexpressionsandcombining
2 2 1 2
≤ −
themtoeliminate theb termweget: v 1 r . Therefore thesocialwelfareinanynon-optimal Nash
2 2 ≥ 1 r−(r 1)
isα v +α v = 1v +r(1 rv ) 1+r−(1 −r) 1.25.
1 2 2 1 2 2
− ≥ − ≤
5
3.1 WeaklyFeasibleAssignments
Nextweshowthatequilibria withconservative bidders satisfiesthesimpleproperty mentioned intheintro-
duction. Wewillcalltheassignmentssatisfyingthispropertyweaklyfeasible. Inthenextsectionweanalyze
thewelfareproperties ofweaklyfeasible equilibria.
The equations (??) are not very easy to work with, since they are not very symmetric and they depend
on b. We propose a cleaner form of representing an equilibrium that just uses α, v and the permutation π.
Althoughitisaweakerproperty itstillcaptures mostofthetrade-offs:
1. ifvaluesv areveryclosethentheorderofthebidders doesn’tinfluencethesocialwelfarethatmuch
i
2. ifvaluesv areverywellseparated, thenpermutationsthatwouldproduceabadsocialwelfarearenot
i
feasible becausetheyviolateNashconstraints
Lemma3.3 Givenv,αandaNashpermutation π,ifi < j andπ(i) > π(j)then:
α v
j π(i)
+ 1 (1)
α v ≥
i π(j)
inparticular, αj 1 or vπ(i) 1.
αi ≥ 2 vπ(j) ≥ 2
Proof. SinceitisaNashequilibrium bidder inslotj ishappywithhiscondition anddon’t wanttoincrease
his bid to take slot i, so: α (v b ) α (v b )since b 0and b v then:
j π(j) π(j+1) i π(j) π(i) π(j+1) π(i) π(i)
− ≥ − ≥ ≤
α v α (v v )
j π(j) i π(j) π(i)
≥ −
Inspiredbythelastlemma,givenparametersα,vwesaythatpermutationπisweaklyfeasibleifequation
1holdsforeachi< j,π(i) > π(j). FromLemma3.3weknowthat:
Corollary 3.4 Given α,v, any permutation corresponding to a Nash equilibrium with conservative bids is
weaklyfeasible.
Ourmainresultsfollowfromanalyzingthepriceofanarchyratio α v / α v overallweakly
j j j j j π(j)
feasiblepermutations π.
P P
3.2 PriceofAnarchy Bound
Here we present the bound on the price of anarchy for weakly feasible permutations, and hence for GSP
for conservative bidders. We prove it is bounded by 1.618. We will prove this bound for weakly feasible
permutations and it will automatically be deduced to a bound for feasible permutations. Notice that the
weakly feasible permutation nicely capture the fact that if advertisers i and j are in the ”wrong relative
position” (i.e. different to the one in the optimal) then either their values are close (within a factor of 2)
or their click-through-rates are close (within a factor of 2). The proof of the 1.618 factor can be found in
AppendixC.
Theorem3.5 Forconservative bidders,thepriceofanarchyforpureNashequilibriaofGSPisboundedby
1+√5 1.618.
2 ≈
Proof. Asawarm-up wewillprove that the price of anarchy isbounded by 2, since the proof is easier and
captures the main ideas. We show this by induction on n. For 2 advertisers and 2 slots we know that the
worst possible social welfare for a weakly feasible permutation is at most a 1.25 times the optimum. So,
nowweneedtoprovetheinductivestep. Considerparametersv,αandaweaklyfeasiblepermutationπ. Let
i = π 1(1)betheslotoccupied bytheadvertiser ofhigher valueandj = π(1)betheadvertiser occupying
−
thefirstslot. Ifi = j = 1then wecan apply theinductive hypothesis right away. Ifnot, equation 1tells us
6
that: αi 1 or vj 1. Suppose αi 1 and consider an input with slot i and advertiser 1 deleted. This
α1 ≥ 2 v1 ≥ 2 α1 ≥ 2
inputhasn 1advertisers andn 1slotsandthepermutation π restricted tothoseisstillweaklyfeasible,
− −
sobytheinductivehypothesis:
1
α v (α v +...+α v +α v +...+α v )
k π(k) 1 2 i 1 i i+1 i+1 n n
≥ 2 −
k=i
X6
1
(α v +...+α v +α v +...+α v )
2 2 i i i+1 i+1 n n
≥ 2
therefore:
1 1
α v =α v + α v α v + α v
k π(k) i 1 k π(k) 1 1 k k
≥ 2 2
k k=i k>1
X X6 X
If vj 1 wejustdothesamebutdeleting slot1andadvertiser j fromtheinput. Thisfinishestheboundof
v1 ≥ 2
2.
Theaboveanalysis isnottight. Forexample, whentheratio α /α isequal to1/2, ourinequality states
i j
thatwealsohavev /v 1/2. WeprovethestrongerboundinAppendixCbyamorecarefulanalysisusing
i j
≥
the full strength ofthe inequality. Asbefore, weprove theconclusion for allweakly feasible permutations.
Wedefine asequence ofvalues r sothat fork slots social welfare isatleast anr times theoptimum. We
k k
knowthatr = 1.25,anduseasimilarbutmorecarefulinductionprooftosetuparecursionofr ,andthen
2 k
show(inAppendixC)thatr converges tothedesired boundof 1+√5.
k 2
4 Mixed Nash equilibrium
All our results so far dealt with Pure Nash equilibria of the GSP. Here we prove a bound on the Price
of Anarchy of 4 for the mixed Nash equilibria. Consider the same setting: players with valuations v
1
≥
... v and slots with click-through-rates α ... α . Now, the strategy of player i is a probability
n 1 n
≥ ≥ ≥
distribution on[0,v ]represented byarandomvariableb .
i i
Lemma4.1 Arandomized bidb whereP(b > v ) > 0isdominated byb = min(v ,b ).
i i i ′i i i
Now, the allocation, represented by the permutation π is also a random variable. For notational conve-
nience,letσ = π 1. Arandomvectorb = (b ,...,b )isamixedNashequilibriumifforeachdeterministic
− 1 n
bid b : Eu (b ,b ) Eu (b ,b ). We begin by proving a bound similar to Lemma 3.3 for mixed Nash
′i i i −i ≥ i ′i −i
andthenusingthattoproveaweakerbound. Notethattheboundisdifferentasitinvolvesabidderiandits
location iintheoptimalallocation, ratherthantwobiddersthatareallocated to“wrongrelativepositions”.
Lemma4.2 IftherandomvectorbisamixedNashequilibrium forGSPthenforeachplayeri:
Eα Ev 1
σ(i) π(i)
+
α v ≥ 2
i i
Proof. Bidder i by our notation has the ith highest valuation, and hence would be in the ith slot in the
optimal assignment. We will consider whether player i benefits by deviating to the deterministic b =
′i
min(v ,2Eb ),whereEb istheexpectedvalueofthebidthatgetssloti.
i π(i) π(i)
We claim that with probability at least 1, the bidder gets one of the slots of 1,...,i . If b = v then
2 { } ′i i
it for sure gets at least the i-th slot as our conservative assumption guarantees that only the previous i 1
players can bid more. If b = 2Eb by Markov’s inequality: P(b b ) Ebπ(i) = 1. Therefore−we
′i π(i) π(i) ≥ ′i ≤ b′i 2
have:
7
1
Eα v Eu (b) Eu (b ,b ) α (v b )
σ(i) i ≥ i ≥ i ′i −i ≥ 2 i i− ′i ≥
1 1
α (v 2Eb ) α (v 2Ev )
i i π(i) i i π(i)
≥ 2 − ≥ 2 −
Nowitisjustamatterofrearranging theexpression.
Theorem4.3 ThePriceofAnarchyforthemixedNashequilibria ofGSPwithconservative biddersis 4.
≤
Proof. Theproofisasimpleapplication ofLemma4.2andsomealgebraic manipulation:
1 1 α v
E[ u (b)] = E α v +E α v = E α v σ(i) + π(i) =
i 2 σ(i) i i π(i) 2 i i α v
i " i i # i (cid:18) i i (cid:19)
X X X X
1 Eα Ev 1
σ(i) π(i)
= α v + α v
i i i i
2 α v ≥ 4
i (cid:18) i i (cid:19) i
X X
5 Bayes-Nash equilibrium
Recall that in the Bayesian setting, the values v are independent random variables and their distributions
i
are public knowledge. A strategy for a player i is a bidding function b (v ) (or a probability distribution
i i
of such functions) where b (v ) is the players bid when his value is v . For simplicity of presentation we
i i i
will consider only pure bidding functions, but our results extend to randomized bids also. As before, we
will assume that the bid b (v ) is in the range [0,v ]. Here we are assuming that players don’t overbid (as
i i i
overbidding isdominatedstrategy).
Asbefore,wewilluseπandσ = π 1todenotethepermutationrepresentingtheallocation,andwewill
−
use ν to denote the random permutation (defined by v) such that player i occupies slot ν(i) in the optimal
solution. Theexpected social welfareisE[ α v ] = E[ α v ]andthesocial optimum isgivenby
i i π(i) i σ(i) i
E[ α v ]. Thegoalofthissectionistoboundthepriceofanarchy,theratioofthesetwoexpectations.
i ν(i) i P P
P
5.1 PriceofAnarchy bound
Theorem5.1 Ifasetofbidsb ,...,b areaBayesNashequilibrium inconservative strategies then:
1 n
1
E[ α v ] E[ α v ]
i π(i) ν(i) i
≥ 8
i i
X X
inotherwords,GSPhasaBayes-NashPriceofAnarchyinconservative strategies bounded by8.
The proof of the theorem is based on a structural characterization analogous to the one used for Mixed
Nash equilibria, which is similar in form to the previous ones, but much harder to prove. We can state the
inequality used to bound the price of anarchy for mixed Nash as v Eα +α Ev 1v α . To extend
i σ(i) i π(i) ≥ 2 i i
thisexpression weneedtotakeexpectations overvaluations.
Playeriknowshisownvaluation v ,soinconsidering alternate bidsfortheplayer, weneedtoconsider
i
expectations conditioned on the value v . To be able to deal with the conditioning of the player i on his
i
ownvaluation v wemake the terms on the expression independent of i. Todo this, let πi(k) be the bidder
i
occupying slot k in the case i didn’t participate in the auction, i.e., πi(k) = π(k) if σ(i) > σ(k) πi(k) =
π(k+1)otherwise. Nowwecanstatetheclaimedinequality. Notethattheboundisstrongerthansuggested
bythecaseofmixedNash,asb b .
πi(ν(i)) π(ν(i))
≥
8
Lemma5.2 If b () isaBayes-Nashequilibrium oftheGSPthen:
i i
{ · }
1
v E[α v ]+E[α b v ] v E[α v ]
i σ(i)| i ν(i) πi(ν(i))| i ≥ 4 i ν(i)| i
Asbeforethepriceofanarchyboundfollowseasilyfromthelemma.
ProofofTheorem5.1: Theprooffollowsthelinesoftheproofoftheorem4.3:
1 1
SW = E (α v +α v ) E (α b +α v ) =
i π(i) σ(i) i i π(i) σ(i) i
2 ≥ 2
i i
X X
1 1
= E (α b +α v ) E (α b +α v ) =
2 ν(i) π(ν(i)) σ(i) i ≥ 2 ν(i) πi(ν(i)) σ(i) i
i i
X X
1 1
= E E[α b v ]+v E[α v ] E v α
2 ν(i) πi(ν(i))| i i σ(i)| i ≥ 8 i ν(i)
" # " #
i i
X X
recalling thatπi(k)iseitherπ(k)orπ(k+1)implyingtheinequality inthesecondline.
Thehardpart oftheproof isproving Lemma5.2. Themaindifficulty inthe Bayesian setting isthat the
inequality isnotestablished byasingle deviating bid. InthebaseofpureNashweconsidered thebidder in
slotj bidding b = b forapairofslotsi < j. InthecaseofmixedNash,weconsidered bidderibidding
′ π(i)
b = min(v ,2Eb ). In both cases the structural inequality followed by considering a single deviation.
′ i π(i)
In contrast, weobtain our structural result by considering many different bids, and using anovel averaging
argument, and a structural property to show that the inequalities established by the different bids can be
combinedtoshowasimplestructure.
Itwouldbenaturaltoconsider asetofbids2E[b v ;ν(i) = k]twicetheexpectedvalueofthebidin
π(k) i
|
slot k where theexpectation isconditioned onthe value v ofplayer i, and the fact that its optimal position
i
isk. Thebidsasdefined depend onplayer iboththrough theconditioning andthough theposition bidder i
getsinthepermutationi,whichmakesithardtoproveanyproperties ofthem. Forexample,thesebidsmay
notbemonotone functions ofk. Togetaroundthisproblem,wewilluseinstead theslightly smallerbids
B = 2E[b v ;ν(i) = k]
k πi(k) i
|
which gets rid of this second dependence. Notice that B is defined as a conditional expectation, so it is a
k
function ofv . Noticetherefore thatitisabidding functionandnotaconstant function.
i
The proof of Lemma 5.2, depends on two combinatorial results. The first, is a structural property: we
claimthatthebidsB asnowdefinedaremonotoneink. ThiswillallowsustoarguethatbidB hasagood
k k
chanceoftakingaslotk > k whenν(i) = k ,asB B .
′ ′ k k′
≥
Lemma5.3 Givenbiddingfunctions b ,E[b v ,ν(i) = k]innon-increasing ink
i πiν(i) i
|
Wewillprovethelemmaaboveusingacombinatorial argumentandmax-flowmin-cuttheorem.
Tobeabletocombinetheinequalities wegetbyaconsidering thedifferentbidsB weuseanovelway
k
to combine inequalities via a ’dual averaging argument’. The combination will simultaneously guarantee
thatoneaverageisnottoolow,andadifferentaverageisnottohight. WeexpectthatthisLemma5.4,which
proveusinglinearprogramming dualitycanhaveotherapplications.
Lemma5.4 Givenanypositivevaluesγ andB Therearex 0, x = 1suchthat:
k k k ≥ k k
n n P
1
x γ γ
k j j
≥ 2
k j=k j=1
X X X
9