Budgetary Effects on Pricing Equilibrium in Online Markets

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1 Budgetary Effects o Pricig Equilibrium i Olie Markets Alla Borodi Uiversity of Toroto Toroto, Caada bor@cs.toroto.edu Omer Lev Uiversity of Toroto Toroto, Caada omerl@cs.toroto.edu Tyroe Stragway Uiversity of Toroto Toroto, Caada tyroe@cs.toroto.edu ABSTRACT Followig the work of Babaioff et al [4], we cosider the pricig game with strategic vedors ad a sigle buyer, modelig a sceario i which multiple competig vedors have very good kowledge of a buyer, as is commo i olie markets. We add to this model the realistic assumptio that the buyer has a fixed budget ad does ot have ulimited fuds. Whe the buyer s valuatio fuctio is additive, we are able to completely characterize the differet possible pure Nash Equilibria (PNE) ad i particular obtai a ecessary ad sufficiet coditio for uiqueess. Furthermore, we characterize the market clearig (or Walresia) equilibria for all submodular valuatios. Surprisigly, for certai mootoe submodular fuctio valuatios, we show that the pure NE ca exhibit some couterituitive pheomea; amely, there is a valuatio such that the pricig will be market clearig ad withi budget if the buyer does ot reveal the budget but will result i a smaller set of allocated items (ad higher prices for items) if the buyer does reveal the budget. It is also the case that the coditios that guaratee market clearig i Babaioff et al [4] for submodular fuctios are ot ecessarily market clearig whe there is a budget. Furthermore, with respect to social welfare, while without budgets all equilibria are optimal (i.e. POA = POS = 1), we show that with budgets the worst equilibrium may oly achieve equilibrium. Keywords 1 pricig, price of aarchy, budget, Nash equilibrium of the best 1. INTRODUCTION The questio of how to price items i a market has existed for milleia, ever sice people bega tradig with oe aother. Market pricig is clearly a cetral area of study i Ecoomics. Oe of the most importat aspects of these studies is market clearig (Walrasia) equilibria, iitiated by Walras i (See traslatio by Willam Jaffe [13]). The advet of olie markets has chaged pricig i may respects, both to the vedors beefit ad to their detrimet. O the oe had, vedors ca gai vast iformatio about Appears i: Proceedigs of the 15th Iteratioal Coferece o Autoomous Agets ad Multiaget Systems (AAMAS 016), J. Thagarajah, K. Tuyls, C. Joker, S. Marsella (eds.), May 9 13, 016, Sigapore. Copyright c 016, Iteratioal Foudatio for Autoomous Agets ad Multiaget Systems ( All rights reserved. their customers, ad ca tailor their prices to each particular customer. O the other had, the istat availability of a large amout of competitors allows customers to pick ad choose amog the items they wat. The speed ad size of olie markets ecessitates havig simple rules for settig prices ad uderstadig the ature of pricig equilibria. This world, where vedors compete with each other i olie marketplaces (e.g., Amazo or E-Bay), ad are able to adapt their prices to their customer, ispired a recet lie of research [4, 1], utilizig a game theoretical toolbox to address these scearios. Here self iterested agets (the vedors) post prices for their wares ad a buyer cosumes a subset of the items based o their valuatio for the items ad the posted prices. The vedors aim is to maximize their idividual profits (we assume o trasactio or productio costs) ad they will post prices as high as possible. If a vedor does ot wish to participate i the market they may price their items at ifiity. How vedors best positio themselves i the market relative to other vedors ad the valuatio of the buyer creates iterestig ad o trivial scearios. While we use buyer - seller termiology, this problem is applicable to other allocatio scearios that may arise i a multi-aget eviromet. We add to the settig of [4, 1] the vital compoet of budgets. Previous work assumed the buyer had ulimited purchasig power, their decisio limited oly by the valuatios. Now we caot assume the buyer will purchase ay budle at a price lower tha its value. The buyer will cosume the set of items with highest utility (i.e., et value, takig prices ito accout), that is withi their budget, while the strategies of the vedors is determiig the prices of the items they offer. This brigs about a departure from the previous results, as we show how the buyer s valuatio ad budget ifluece the market. Not oly are some previous results ow redered impossible as they exceed the budget, but this more realistic assumptio materially chages the structure of pricig from the ituitios formed i earlier research, as a more complex strategic behaviour eeds to cosidered. 1.1 Our Cotributios We begi by studyig additive buyer valuatio fuctios. I the o budgeted world of Babaioff et al. [4] ad Lev et al. [1] this is a uiterestig case where there is a simple ad uique equilibrium i which every item is sold. We show, aturally, that a budgeted buyer ca o loger ecessarily purchase all of the items for sale. We idetify a coditio o the valuatio fuctio ad the budget which is sufficiet to esure a market clearig equilibrium exists. 95

2 This equilibrium is iterestig sice each item is priced to provide idetical utility to the buyer. We prove that uder this coditio the market clearig equilibrium is uique. We ext examie what happes whe the uiqueess coditio does ot hold i the additive case. We provide a method to idetify a subset of the sellers whose existece i some sese geeralizes the coditio for uiqueess ad thereby allows for a partial characterizatio of all PNE. We fially prove our coditio for market clearace is ecessary. We the exted our equilibrium aalysis to the case of mootoe submodular valuatio fuctios. We exted the otio of our coditios i the additive case to submodular valuatios provig that a similar market clearig equilibrium exists. We also prove that the coditio is agai ecessary to esure market clearace. We ext provide a example which shows a rather surprisig cosequece of the budget. We show that at certai equilibrium a buyer who has o budget ca pay less ad receive more items tha their budgeted couterparts. We also show that our otio of geeralizig the coditio o the valuatio ad budget to reach a equalutility equilibrium does ot work for XOS valuatios. Fially, we explore the budget effects o social welfare. We show that while it is kow [4] that without a budget the price of aarchy is 1, with a budget the price of aarchy ca be arbitrarily high. 1. Previous work As already metioed, the study of market pricig ad equilibria is oe of the most classical areas of Ecoomics. I particular, the study of Walrasia equilibria cosiders the questio of which markets have market clearig equilibria; that is, a assigmet of prices to items such that whe all agets take their preferred allocatio i this pricig (i.e. a allocatio i their demad set), all items are sold. We are iterested especially i settigs where there are distict sellers ad buyers, which are termed Fisher markets [6, 9]. A lot of work has goe ito examiig this whe all items are divisible (e.g., commodities, such as oil, grais, etc.), but we focus o the case of idivisible items, which is how most items are sold i say olie markets. Oe of the foudatioal moder papers i this regard is that of Gul ad Stacchetti [10] who show that for idivisible items the class of Gross Substitutes, a strict subfamily of submodular fuctios, is the largest class of fuctios cotaiig uit demad buyers that always possess a Walrasia equilibrium. There have bee several papers that follow this work [9, 8,, 3, 1, 7]. The basic emphasis i these papers is the existece ad covergece to such equilibria without explicit represetatio of a pricig fuctio. Moreover, while budgets are metioed i some of these papers, the usual assumptio is that the budgets are sufficietly large. Recetly, as the ability of vedors to aalyze idividual buyers ad persoalize their prices i olie markets has icreased [14], research i aalyzig competitive pricig aalysis i this sceario has icreased. The model of Babaioff et al. [4] is most similar to ours, with may strategic vedors each holdig a sigle item, ad a sigle buyer. The mai differece from their work is that while they cosider buyers with ulimited purchasig power, we limit ourselves to budget costraied buyers. They prove that i ay game with mootoe buyer valuatios there always exists a pure Nash equilibrium that clears the market ad maximizes social welfare. Furthermore, they show that for buyers with submodular valuatios there is a uique equilibrium that maximizes social welfare, ad that the price of aarchy (ad hece price of stability) is 1. While we do ot apply the budgetary costrait to their settig, Lev et al. [1] geeralized [4] by allowig sellers to sell multiple items, ad lettig them choose which items to offer ad for what price. They prove that a equilibrium is ot guarateed to exist for submodular valuatios. Whe a equilibrium does exist the price of aarchy ad price of stability are roughly log where is the umber of items offered.. PRELIMINARIES Our settig has a set of vedors N ( N = ) i which vedor i sells a sigle distict item i. A strategy profile of the vedors is a price vector p = (p 1,..., p ) R 0 where item i is priced at p i (p i 0). We use p(s) = i S pi to represet the cost to the buyer for purchasig a set S N. We write p i to represet the set of prices excludig p i, we write (p i, p i) to represet the full set of prices p. Opposite the vedors we have a sigle buyer, represeted by a valuatio fuctio v(), who faced with price vector p cosumes a set of items which maximises his utility..1 The Buyer The buyer will be represeted by a valuatio fuctio ad a budget. The valuatio fuctio v : N R 0 gives o egative value to each subset of N. We assume the fuctio is ormalized ad mootoe; that is, v( ) = 0 ad v(s) v(t ) for S T N. The budget B R >0 is simply the maximum capital the buyer has available to purchase items; the budget caot be exceeded. We shall discuss 3 types of valuatio fuctios: Additive This fuctio is defied by a vector of per item valuatios v = (v 1,..., v ) ad for ay S N the valuatio is v(s) = i S vi. Submodular This fuctio is characterized by the idea of decreasig margial utilities. That is, for a submodular fuctio we have v(s) v(s \ {a}) v(t ) v(t \ {a}) for S T N ad a S. XOS A XOS (Exclusive Or of Sigletos) fuctio is defied by a fiite set of additive fuctios F = {f 1,..., f k } such that v(s) = max fi F f i(s) for every S N. The set of additive fuctios is a strict subset of the set of mootoe submodular fuctios which i tur is a strict subset of the set of XOS fuctios. It should be oted all of these sets of fuctios are strictly cotaied i the set of subadditive fuctios which are defied as: v(s) + v(t ) V (S T ) for S, T N. For a more i depth look at these fuctios see [11]. We assume the buyer has a budget bouded quasi-liear utility fuctio. That is, for some budle S N ad pricig vector p we have: u b (S, p) = { v(s) p(s) if p(s) B otherwise To ease readig ad otatio, we will write u b (i, p) i place of u b ({i}, p) ad omit the p if it is uambiguous. Followig otatio i [4], the demad correspodece of a buyer 96

3 with valuatio fuctio v facig pricig vector p is the family of sets that maximize the buyer s utility: D(v, p) = {S N : u b (S) u b (T ) T N} The buyer will cosume a set X(v, p) D(v, p). We assume the buyer has access to a demad oracle which allows the buyer to fid X(v, p). For ease of otatio we usually write X(v, p) as X p Ofte we will fid D(v, p) = 1 i the cases we study. A commo assumptio [4] i cases where D(v, p) > 1 is that the buyer will opt for a set S N with the largest size. We call such a buyer maximal.. Vedor Utility ad Equilibrium As metioed before, the utility of the vedors is simply the paymet they receive for their items. That is, if the buyer cosumes X(v, p) whe preseted with prices p the utility to vedor i is: u i(x(v, p)) = { p i if i X(v, p) 0 otherwise I a slight abuse of otatio we will ofte omit the X(v, p) from u i(x(v, p)) istead we write u i(p) implicitly assumig the X( ) ad v are i place. A pricig vector p is defied to be a pure Nash equilibrium (PNE) if there does ot exist ay agets that ca improve their utility by uilaterally modifyig their price. That is, i N such that u i(p i, p i) > u i(p i, p i) = u i(p) for some p i p i. We ote that the sellers also have access to the demad oracle ad are able to determie what X(v, p) is. We say a price vector p is market clearig if X(v, p) = N ad mi i p i > 0. That is every item is bought ad each vedor receives positive utility. The game as described is etirely parametrized by the budget B ad valuatio profile v(). The agets will price themselves presetig a p that maximizes their idividual utility. We are iterested i studyig the properties of stable pricig schemes, i.e., prices that are a PNE. I particular, whe are the PNE uique, how must agets price their items, ad what is cosumed by the buyer. Note this is a game of full iformatio: the agets are assumed to kow B ad v ad ca see the prices p posted by each of the agets. 3. ADDITIVE VALUATIONS We begi our study with a very simple class of buyer valuatios, additive fuctios. Recall v( ) is additive if there exists a set of item values (v 1,..., v ) ad v(s) = i S vi for S N. As metioed earlier, previous work [4, 1] does ot explicitly study the additive case, as there is a very simple ad uique solutio i their settig: assumig a maximal o budgeted buyer, the seller holdig item k should simply price the item at v k. Hece, the maximal buyer will purchase the uiverse of items ad social welfare is maximized. Sice each item is beig cosumed, oe have a icetive to lower their price. If p k < v k the the aget holdig k should be able to icrease p k to v k ad still be cosumed by the maximal buyer. Now cosider the followig istace of our budgeted game (B = 1, (v 1, v ) = (, 1 )). It is easy to verify that the oly PNE are of the form: p = (1, x), for x R 0, ad X p = v 1. The budget has eabled the vedor with the more valuable item to demad the etire budget excludig the other vedor from participatig i the game. 3.1 A Uique PNE We ow provide a sufficiet coditio (later, i Theorem, to be show ecessary) o the buyer s valuatio to esure a uique market clearig PNE i which each vedor participates i the game. More specifically we show that uder this coditio there is a uique PNE that clears the market providig each vedor with positive utility. Throughout this sectio we assume V (N) > B, otherwise this settig reduces to the settig of [4]. Defiitio 1. The relative valuatio costrait for set S N, valuatios (v 1,..., v ) ad budget B is that for each v i S, v i > v j S\{i} S 1 Whe cosiderig the costrait o the full set of items (N) this is equivalet to statig it as a costrait o the budget: for every i N, B > j i (vj vi). We ca see that this costrait does ot imply that the valuatio is a gross substitutes valuatio. Namely, the additive fuctio v(s) = i S vi for 3 items (v1, v, v3) with the values (,, ) correspodigly ad a budget of 1. The fuctio satisfies the relative valuatio costrait ad is ot gross substitutes: the pricig (0., 0.4, 0.4) allows the buyer to buy all items, while the pricig (0.3, 0.4, 0.4) forces the buyer to give up v or v 3 despite their prices stayig the same, hece ot gross substitute 1. Theorem 1. Give a additive valuatio profile (v 1,..., v ), where i vi > B ad the valuatios for N follow the relative valuatio costrait for all items, there is a uique PNE p where mi i(p i) > 0, i pi = B, (i, j), vi pi = vj pj ad X(v, p) = N. Before we prove Theorem 1 we will fid the followig lemma useful: Lemma 1. Give a additive valuatio profile (v 1,..., v ) with budget B ad sets A ad U where A U N. If for j U\{i} i U, v i > v j B j A the v U 1 i >. A Proof. Let i = arg mi j U\A v j, let D = U \ (A {i}). From the coditio o v i: = ( U 1)v i j D v i > = j U\{i} vj B U 1 j D vj + j A vj B U 1 v j > j A v j B (( U 1) D )v i ( U 1)v i j D v j > j A v j B 1 A differet example, show i Lehma et al. [11], is a budget-additive fuctio v(s) = mi{b, i S vi} for 3 items (v 1, v, v 3) = (1, 1, ) ad a budget B =, which satisfies the relative valuatio costrait ad is ot gross substitutes: the prices (0, 1, 1), i which v1 ad v are purchased, i compariso to prices (1, 1, 1) i which oly v3 would be bought. 97

4 Where the last lie is because v i was chose to be the smallest value amogst those i U \ A. So ow we have that: (( U 1) D )v i > j A v j B j A = v i > vj B ( U 1) D j A = vj B A Now we are ready to prove Theorem 1: Proof. We will divide the proof ito two parts. First we will defie a PNE p which meets our criteria. We the will show that this p is the oly possible PNE whe N satisfies the relative valuatio costrait. Existece of PNE p: Let p i = B+( 1)v i j i v j. First j i 1 we ote sice v i > we get that p i > 0. It is also easy to see that i pi = B. We ow argue v i p i = v j p j for ay pair of vedors i ad j. Fix some vedor k: v k p k = v k B + ( 1)v k j k vj = v k B ( 1)v k + j k vj i = vi B v(n) p(n) = That is, each seller i is providig exactly 1 of v(n) p(n), so each item is priced so they provide idetical utility to the buyer. Next we show v i > p i for each item i. Suppose for cotradictio there is some i where p i v i. Sice v i p i = v j p j for ay item j it must be that p j v j. Thus we get p(n) v(n) > B, which cotradicts p(n) = B by costructio. Now we ca prove p is a PNE: Because i pi = B ad 0 < pi < vi each seller has their item chose. Thus oe have a icetive to lower their price. Say seller i icreases their price by of ɛ > 0. Sice (p i + ɛ) + j i pj > B the buyer will cosume at most 1 items. Sice v i p i = v j p j for ay pair of items i ad j we get v i (p i + ɛ) < v j p j, so item i is ot cosumed leavig its seller with 0 utility. So i should ot icrease its price. Thus p is a PNE. Uiqueess of PNE p: Give that the relative valuatio costrait holds for N, we ow argue that p is the oly PNE. Assume we have aother PNE p = (p 1,..., p ). First we show i p i = B: Suppose i p i < B, the clearly X p = N sice p is a PNE. Sice i vi > B it must be that p j < v j for some item j. Let 0 < ɛ mi{b ( i p i), v j p j }. It is easy to see (p j + ɛ, p j) is a better solutio for seller j. By the first term i the mi the ew pricig vector is ot over budget by the secod term v j < p j so the buyer still receives positive utility from item j. Thus item j is still cosumed ad we see that seller j should icrease their price by ɛ. Suppose i p i > B, the uder this pricig scheme there is a aget i, i X(v, p ). We ow argue i is able to offer a positive price where this item will be picked. For the remaider of this sectio whe referrig to X p we exclude items which are priced at zero. We ca assume that j X p p j = B, if this were ot the case seller i could offer a positive price ad still be cosumed. Let s = arg mi u j Xp b(j), that is s is the seller providig the least amout of utility amogst those who are picked. If v i > u b (s) seller i ca offer a price of mi{ v i u b (s), B j X p \s p j }. The first term i the mi esures that the buyer gets more utility from i tha s, the secod term esures the buyer ca afford the curret chose set whe s will be replaced by i. Thus vedor i ca replace vedor s i the chose set. We ow prove v i > u b (s). We kow that u b (s) is at most the average utility provided by a seller i X p. That is u b (s) j X p j X p X p. So we must show show v i >. This is simply a direct result of Lemma 1, X p where we take the sets X(v, p ) ad N to be A ad U i the lemma respectively. Thus at ay PNE p each seller offers a positive price is bought ad the budget is cosumed. We fially argue u b (i) = u b (j) for all sellers i ad j at ay PNE p. Let i = arg max k u b (k) ad j = arg mi k u b (k). Assume, for cotradictio, u b (i) > u b (j). Let 0 < ɛ mi{ B k j v k } ad cosider the poit (p i + ɛ, p i). Sice p is a PNE, i p i = B so uder (p i + ɛ, p i) at least oe item will ot be cosumed. By the first term i the mi, p i + ɛ + k i,j p k < B, so the buyer ca afford all of the items except for j. By the secod term i the mi, v i (p i + ɛ) = u b (i) ɛ > u b (j) so the buyer gets more utility from seller i tha from seller j. Sice seller j is still providig the least utility, it will be the oe left out of the chose set. Thus we see that seller i should icrease the, u b(i) u b (j) price by ɛ ad that p is ot a PNE. Thus it must be that u b (i) = u b (j) for all sellers i ad j. Thus whe the relative valuatio costrait holds for N we get that all PNE must take the form of p as defied above. 3. No Market Clearig PNE ad a Complete Characterizatio of Market Clearig PNE I this sectio we will explore what happes whe the relative valuatio costrait does ot hold for N. Without loss of geerality, let the elemets of N be i decreasig order of valuatios; i.e. if i < j the v i v j. Defiitio. A PNE base set is a set L N, which is costructed by orderig the elemets of N i o-icreasig order of valuatios. Startig with the first (most valuable) elemet of N iteratively add the ext elemet to L util j L. reachig a elemet i s.t. v i Note we are ot coutig v i i the above sum or the size of L. I a sese L is a maximal set of the largest items. We ote that a o-empty L must exist sice L = {1} certaily 98

5 satisfies the defiitio. Furthermore, ote that for several equal valued items, if oe of them is i L, the all of them are i L. We ow prove some additioal useful facts about L: Lemma. i L vi > B Proof. If L = N the sice we assumed i vi > B we are doe. Let us suppose L N the there must be a elemet j L s s.t. v s. Rearragig we get that v s j L vj B, ad sice vs > 0 we get that j L vj B > 0. Thus we see that L cotais strictly more value tha the budget. Lemma 3. j L, v j > k L\j v k B 1 Proof. Let i be the last elemet added to L, that is v i is the smallest value amogst items i L. Whe we added i to L it was because v i > j L L where L is L \ {i}. Sice 1 = L we get v i >. Now trade v i with ay elemet i the sum; the value o the left of the iequality ca t decrease sice v i was miimal ad the value o the right ca t icrease sice we are swappig i v i for some larger elemet v j. Thus the iequality still holds. j L\i 1 It is easy to see that L as defied above is the uique j L\i maximal set such that i L, v i >. If this 1 was ot the case ad some set M existed s.t. i M, v i > ad L M. Lemma 1, where L ad M take j M\i M 1 the place of A ad U i the lemma, implies that for each j L i M v i >. Thus whe costructig L ad cosiderig elemet ( + 1) M we should have added it to L. Thus L is the largest set for which the relative valuatio costrait holds. Lemma 4. There is always a PNE p where X p = L ad v i p i = v j p j for all pairs of elemets i L. Proof. Let p = (p 1,..., p, 0..., 0) where p i = B+( 1)v i j L\i v j. By Lemma 3 we kow p i > 0. By the costructio of each p i, i pi = B ad vi pi = vj pj = j L for all pairs i, j L. As i Theorem 1 o oe aget i L has a icetive to lower their price sice they are beig sold. Furthermore, if oe of the agets i L were to icrease their price, they would be removed from X p as they ow provide the least utility i L, ad prices are ow over budget. Fially combiig Lemma ad a similar argumet foud i Theorem 1 we get that p i v i for each i L. Now we argue that oe of the agets i N \ L have a icetive to icrease their price. Let v s be the largest value amogst the agets i N \ L. By the costructio of L, v s j L. If aget s were to offer a price p s > 0 it would ot be chose sice v s p s <. Sice we are ow over budget ad the utility provided by s is lower tha the utility provided by ay of the agets i L, s is ot chose. Sice v s was the largest value amogst the agets ot i X p oe of the agets outside of X p have a sufficiet valuatio to be chose. j L Ufortuately we do ot get a ice geeralisatio of Theorem 1 where at all PNE v i p i = v j p j for each pair i, j L. Example 1. Let the budget B = 1, the valuatios v = (, 1.5, 0.6, 0.6) ad fially the prices p = (0.6, 0.4, 0.3, 0.3). The utility gaied from each aget is simply u = (1.4, 1.1, 0.3, 0.3). It is easy to see that L = {1, } ad this is a o market clearig PNE where X = L. If the first aget icreased their price the ew cosumed set would be {, 3, 4}. If the secod aget icreased their price the ew cosumed set would be {1, 3} or {1, 4}. Ad if the third or fourth aget dropped their price to ay positive value the cosumed set would still be L = {1, }. We ca, however, obtai a partial characterizatio showig that at ay PNE p the base set L must be a part of X p. Lemma 5. Let p be some PNE ad X p be the set cosumed (ot icludig those sellig for free) it must be that L X p. Proof. Assume for cotradictio that L X p at some PNE p We break this ito two cases. Case 1 (X p (N \ L) ): That is, there are elemets of N \ L i X p. Let s be the elemet of X p (N \ L) providig the least utility to the buyer. Let i be some elemet of L \ X p. Sice L cotaied the largest valued elemets of N it must be that v i > v s. The iequality must be strict for if v i = v s, s would be a member of L. Lettig p i = p s, at this price aget i provides strictly more utility tha aget s, who is providig the least, so the buyer will replace s with i i the cosumed set. So p is ot a PNE. Case (X p (N \ L) = ): That is, X p is a strict subset of L. By makig a argumet similar to the oe i Theorem 1 usig Lemma 1, with X p ad L replacig A ad U i the lemma, we get p is ot a PNE. It is, however, possible that at a PNE p, X p L. Example. Let the budget B = 1, the valuatios v = (.5, 1.5, 1.4) ad the prices p = (0.9, 0.1, 0.9). The per aget utility is u = (1.6, 1.4, 0.5). Obviously L = {1} ad X p = {1, } L. If aget 1 raises the price the ew cosumed set will be {, 3}, if aget icreases the price the ew set will be {1} ad fially if aget 3 drops the price the cosumed set will remai {1, }. Thus p is a PNE with L X p. We coclude our discussio of additive valuatios by provig that havig a market clearig PNE requires that the relative valuatio costrait holds for N ad hece the PNE is uique ad must be of the form described i Theorem 1. Theorem. Cosider a additive valuatio fuctio v ad budget B for the buyer. Let p be some market clearig PNE, such that is X p = N ad mi i p i > 0. It must be that relative valuatio costrait holds for v ad B; equivaletly, the PNE base set L = N. Proof. Assume, for cotradictio, L N. Let s be the elemet providig the least utility to the buyer uder the pricig p. 99

6 j L Sice L X p there must be a elemet i L where j L v i p i >. This is because whe the cosumed set is oly L the average utility provided by a aget i L is. Now we have agets outside of L receivig utility, to accommodate them at least oe aget i L must offer a lower price providig more utility. By the costructio of j L L each elemet of N \L has value at most. Sice s is providig the least utility over all agets, ad o aget j L outside of L ca provide at most utility, v s p s < j L. So we get v i p i > v s p s. Thus we see that aget i ca icrease the price by mi{ (v i p i ) (v s p s), B j N\s pj}. The first term esures aget i still provides more utility tha aget s while the secod esures N \ s is budget feasible. Uder the ew pricig scheme at most 1 items will be bought. Sice s is still providig the least utility we get that N \ s is the ew cosumed set. Thus p was ot a PNE, a cotradictio. This theorem, alogside Example 1, meas that the Babaioff et al. [4] result showig that there is always a market clearig equilibrium for all motoe valuatio is ot true whe itroducig budgets to the model ad eve fails for some additive valuatios. From Theorem we coclude that ay market clearig PNE p must be of the form of the oe described i Theorem 1. That is: Corollary 1. If a PNE p is market clearig, the it is uique, ad for all pairs (i, j) v i p i = v j p j ad i pi = B. 4. MONOTONE SUBMODULAR VALUATION FUNCTIONS We ow focus o submodular valuatio fuctios. Recall submodular valuatios are characterized by dimiishig margial values, that is for S T ad x T we have v(s {x}) v(s) v(t {x}) v(t ). Defie v T (S) = v(t ) v(t \ S) for S T N, that is the margial value of set S for completig set T. To simplify readig, for a set S with sigle item s, we will use the otatio v T (s) to deote v T ({s}). We also defie u i b(s) = v S(i) p i, that is the seller i s margial cotributio to the buyers utility for completig set S. We will fid the followig lemmas regardig submodular ad mootoe fuctios useful i this sectio: Lemma 6. For a submodular valuatio v ad pricig scheme p, if for all vedors i, p i v D(i) the v(d) p(d) v(c) p(c) for sets C D. Proof. Cosider some seller j D \ C. Sice v is submodular ad p j v D(j) we get p j v C {j} (j), so v(c {j}) p(c {j}) v(c) p(c). Sice C {j} D we ca take aother seller j C {j} \ D ad agai coclude due to p j v D(j ) ad the submodularity of v v(c {j, j }) p(c {j, j }) v(c) p(c). We ca repeat this process for the rest of the elemets i C \ D cocludig v(d) p(d) v(c) p(c) for sets C D. Lemma 7. For a mootoe valuatio v ad pricig scheme p, if for vedors a ad c v N (a) p a = v N (c) p c the v(n \ c) p(n \ c) = v(n \ a) p(n \ a). Proof. Assumig v N (a) p a = v N (c) p c: v N (a) p a = v N (c) p c = v(n) v(n \ a) p a = v(n) v(n \ c) p c = v(n \ c) p a = v(n \ a) p c = v(n \ c) p a p k = v(n \ a) p c = v(n \ c) i c k a,c p i = v(n \ a) i a p i k a,c Theorem 3. Give a submodular valuatio fuctio v j i where i N, v N (i) > v N (j) B there is a market clearig PNE p where i 1 pi = B ad vn (i) pi = vn (j) pj for all pairs of vedors i ad j. Proof. Let p i, this value is positive because of the coditio o v N (i). As i the proof = B+( 1)v N (i) j i v N (j) p k j N v N (j) B of Theorem 1 it is easy to see v N (i) p i = so v N (i) p i is the same for each vedor i. It is also easy to see i pi = B. Combiig these two facts we see that as i the proof of Theorem 1 p i < v N (i). Usig these facts ad applyig Lemma 6 to N we see each aget must be chose ad oe have a icetive to lower their price. Suppose p is ot a PNE It must be that some aget has a icetive to icrease their price, call this aget a. If a icreases its price by ɛ > 0, sice i pi = B at least oe of the agets must ot be chose uder the ew pricig scheme. Call this other aget c. Let p be the ew pricig scheme where a icreases its price. First we argue that v(n\c) i c pi > v(x p ) i X pi p ɛ. Note v(x p ) i X p pi ɛ < v(x p ) i X pi. This p is ow a simple applicatio of Lemma 6 where C ad D i the lemma are replaced with X p ad N \ c respectively. Thus we get v(n \ c) i c pi v(x p ) i X p pi > v(x p ) i X pi ɛ. p Next we get directly from Lemma 7 v(n \ c) i c pi = v(n \ a) i a pi. Thus we get v(n \ a) i a pi = v(n \ c) i c pi > v(x p ) i X p pi ɛ. Sice i a pi < B we see that the ew cosumed set will be all agets except a; thus it should ot icrease the price. So we see p was a ideed a equilibrium. Similar to Theorem from the additive case we show that i the submodular case the oly market clearig PNE are of the equal margial utility kid. That is the all market clearig PNE must be of the form of the oe described i Theorem 3 Theorem 4. For a submodular valuatio v give ay PNE p where X(v, p) = N ad u i > 0 for all vedors i, it must be that v N (k) p k = v N (j) p j for ay pair of vedors k ad j. Proof. Suppose at PNE p X p = N. Let a = argmax i N v N (i) p i, that is, a is providig the most margial utility, so v N (a) p a v N (i) p i for every other vedor i. Assume for cotradictio, that i at least oe case this iequality is strict. Let vedor c = arg mi i v N (i) p i so v N (a) p a > v N (c) p c. 100

7 We ow argue a has a icetive to icrease its price by ɛ < mi{ ua b (N), ua b (N) uc b (N), mi j a B p(n \ j)} (recall u i b(n) = v N (i) p i). The first term is positive sice sice v N (c) p c 0 thus v N (a) p a > 0 so v N (a) > p a. The secod positive sice v N (a) p a > v N (c) p c. The third is positive because i pi B. Let (pa + ɛ, p a) = p. Sice X p = N we kow that p i v N (i) for each vedor i ad because of the first term i the above mi, p i v N (i). We ca apply lemma 6 takig set D i the lemma to be ay K N where K = N 1 gettig v(k) p (K) v(j) p (J) for J K. So we see uder p the buyer will cosume a set of size N 1, assumig it is uder budget. By the secod term i the mi, v N (a) p a ɛ > v N (c) p c. Usig this we ca modify the proof of Lemma 7, replacig the equalities with iequalities, gettig v(n \a) i a pi < v(n \ c) i c pi ɛ. Thus the buyer will prefer to exclude c istead of a from X p, assumig the chose set with a is uder budget. The third term i the mi esures ay set of size N 1 that a is a part of will remai uder budget. Thus we see a ca raise its price by ɛ ad p ca t be a PNE. We ote that we caot exted the idea of the PNE base set to the submodular case i the same way while a maximal set of items that adhere to the coditio ca be foud, as show i Example 3, the set is ot uique, ad hece ot all equilibria are a extesio of the same set. Example 3. We have a set of 4 items N = {a, b, c, d}. v(a) = v(b) = 1; v(c) = v(d) = 1. v({a, b}) = 3, v({c, d}) = 1, v({a, c}) = 3. v({a, b, c}) = 7, v({a, c, d}) = 1.5 ad 4 v({a, b, c, d}) = The values for the rest of the sets are defied by this as a ad c ca be replaced with b ad d, respectively (if they were ot i the origial set). The budget is 0.3. Both sets {a, b, c} ad {a, b, d} coform to the relative valuatio costrait (their margial values with respect to the set are equal). However, a, b, c, d does ot. 4.1 Whe the Sum of Margial Values is Below the Budget Whe the sum of margial values of items is below the budget, the equilibrium show i [4] is both market clearig ad lets the buyer keep some of their moey. However, that equilibrium is uique oly whe the budget is ot kow to the sellers. Whe it is kow, other equilibria arise, ad some of them provide less utility to the buyer tha the market clearig oe, ad reduces the social welfare. This shows buyers are worse off aoucig their budget, as Example 4 shows. Example 4. We have a set of 4 items N = {a, b, c, d}. v(a) = v(b) = 1; v(c) = v(d) = 1. v({a, b}) = 7, v({c, d}) = 4 4 1, v({a, c}) = 9. v({a, b, c}) = 1.76, v({a, c, d}) = ad v({a, b, c, d}) = The values for the rest of the sets are defied by this as a ad c ca be replaced with b ad d, respectively (if they were ot i the origial set). The budget is 1. The margial values (i N) of a ad b are 0.7, of c ad d 0.1, so pricig the items with these values is both uder budget ad a equilibrium, as [4] claim, ad will result i all items beig purchased by the seller. However, aother equilibrium is pricig a ad b at 1 each, ad c ad d at 1. 4 Uder these prices, the buyer will oly buy items a ad b (as they exhaust the budget). This is a equilibrium, as either a or b ca raise their prices, as that precludes buyig both of them, ad buyig the oe that did t raise its price with c ad d is more beeficial for the buyer. Items c ad d ca t lower their prices to aythig above 0, as there is o budget to purchase them, ad eve with oly oe of a or b, they still do ot provide as much utility as purchasig both a ad b. Fially, as Example 5 shows, ulike submodular valuatio fuctios, we caot always guaratee equal utility equilibria for XOS fuctios eve whe the valuatios adhere to the relative valuatio costrait: Example 5. We have a set of 3 items N = {a, b, c}. The XOS valuatio fuctio is defied usig additive valuatios: The first is (, 1, 1) respectively for (a, b, c), ad the secod is (3, 0, 0). The budget is 1.5. The margial value of item a is, ad b ad c have margial value 1, so they comply with the relative valuatio costrait. A equal margial utility cotributio would mea pricig item a at 7 ad items b ad c at 1 each. However, i this 6 6 case item a ca icrease its price to 4, as without it, by 3 buyig oly b ad c the buyer has a utility of 5, but by buyig 3 a ad b, the buyer gets a utility of.5, beig obviously better off, so this is ot a equilibrium. 5. SOCIAL WELFARE While we have show price equilibria i various settigs, we have yet to ivestigate the value of the sold set to the buyer. As the sellers get the moey from the buyer, this is equivalet to aalyzig the social welfare the amout of utility created i the equilibrium. The commo measures of the impact of equilibrium o the welfare are price of aarchy (PoA), ad, to a lesser extet, price of stability (PoS), which are, respectively, ratio of the social welfare i the worst (respectively, the best) equilibria ad the optimal welfare. However, the optimal welfare is, of course, that all items are give to the buyer for free, ad therefore, the social welfare is the value of the whole set, amely v(n). I [4] the authors show that for mootoe submodular valuatios the PoA is always 1. Furthermore they show for ay mootoe fuctio the PoS is 1, although the PoA ca be ifiite. I the multi item settig of [1] the authors show whe PNE do exist for mootoe submodular valuatios both the PoA ad PoS are approximately log(m) where m is the umber of items. The usage of PoA is iteded to aid us i uderstadig the effects of a equilibrium compared to the optimal state. Assumig sellers get othig is ot oly urealistic, it also does ot let us cosider the effect of a equilibrium uder the budget costraits (as the budget plays o role). Therefore, we would like to have some measure that takes ito accout the existece of a budget, as well as the desire of sellers to be paid. To deal with this problem we use a somewhat differet metric: P oa, i.e., the ratio betwee the social welfare i the worst equilibrium ad the social welfare i the best equilibrium. I order to avoid items beig give away ad artificially addig to the utility, we do ot cout the social welfare garered from items priced at 0. Our measure shows the differeces betwee differet equilibria uder the same 101

8 budgetary costrait, ad is, therefore, a idicator of how bad equilibria ca become. Surprisigly. ad i cotrast to the results i [4], which showed that without a budget the P oa is 1, we discover the P oa ca be arbitrarily bad for submodular fuctios. Moreover, with budgets, eve i the additive case, the PoS ca be greater tha 1, as is show by Theorem, ad examples such as Example 1. Therefore, we ca have P oa > P oa > 1. Theorem 5. The P oa (ad hece the PoA) ca be ubouded ad approachig eve for additive valuatio fuctios. A correspodig upper boud is, eve for submodular fuctios. Proof. We first show that the ratio is ubouded for the additive case ad approachig, ad the show the upper boud of for submodular fuctios. Let v be a additive fuctio. Let the value of item 1 be, the value of item 1 ad be 0.55, ad all the rest of the items (from to ) have a value of 1. The budget is 1. It is easy to see the set L oly cotais the first item, ad therefore there is a equilibrium i which all the budget goes to this item, ad all others items priced at 0. Hece the utility of this equilibrium is (ote that as ay equilibrium must iclude L, this is the worst possible equilibrium). Now cosider the followig pricig: item 1 is priced at 1, 1 ( 3) items valued at 1 have a price of, ad the fial two items are priced at 0.5 each. This is a Nash equilibrium i which all items but the last two are sold. Noe of the items valued at 1 ca icrease their price (as the they will be the item providig the least utility ad be throw out), ad either ca the items valued at 0.55 lower their price to be purchased, as eve the they provide too little utility. If item 1 tries to raise its price (which ecessitates throwig out oe of the items valued at 1), it is better for the buyer ot to purchase item 1, ad istead add both items 1 ad to the purchases set. The value of this equilibrium is + ( 3). Thaks to Theorem, we kow there is o market clearig equilibrium, ad as is show i the ext paragraph, there ca be o equilibrium with 1 items. Hece P oa = +( 3), which is ubouded. Notice that replacig the valuatio of the top item to ay k N ad the ext 3 items to k 1, does ot chage the existece of both equilibria, ad P oa k( ) 0.1 k+1 k Lookig at a upper boud, ote that P oa is actually value of best NE. The best equilibrium is, at most, the value value of worst NE of the whole set, v(n), ad thaks to submodularity, v(n) max i N v({i}). Furthermore, i ay equilibrium, the buyer gets at least a value max i N v({i}). If ot, suppose i is the value maximizig v({i}), ad there is a equilibrium where the buyer purchases a set S (N \ {i }) at price b B, ad v(s) < v({i }). Vedor i ca improve its situatio (a utility of 0), by chagig its price to b, ad beig bought istead of set S, provig that buyig set S for a price of b is ot a equilibrium. Hece the utility of the worst equilibrium is at least max i N v({i}). Therefore, P oa. 6. CONCLUSION AND FUTURE WORK The additio of budgets to the work of [4] itroduces iterestig ad strategic behaviour amogst agets. We idetify the relative valuatio costrait ad prove it is both sufficiet ad ecessary to esure market clearace with submodular valuatios, ad i the additive case, we show the market clearig equilibrium is uique. This market clearig equilibrium is coceptually iterestig, as i it each vedor provides equal margial utility to the buyer for completig the uiverse of items. We also show how the costrait does ot exted to XOS valuatios where a equal margial utility equilibrium is o loger guarateed to exist. Furthermore we geeralize the relative valuatio costrait to subsets of the vedors for the additive case, ad usig this geeralizatio, we idetify a maximal base set of items (values) that ca form a equilibrium similar to the market clearig oe excludig other sellers. Fially we provide examples illustratig how the budget ca ifluece the game i uexpected ways. Perhaps most strikig of which is example 4 which ca be iterpreted as showig that a buyer aoucig their budget may iduce ew ad iferior equilibrium where they pay more ad receive less compared to ot aoucig the budget at all. Geeralizig our model to may items per seller, as i [1], would be a likely logical extesio to our work. Whe sellers hold may items the problem domai is sigificatly more complex. It is ukow eve with simple additive valuatios what Nash equilibrium this settig permits. It would also be iterestig to see if with a budgeted ad submodular buyer we also ecouter the issue of o equilibria. Also, while [4, 1] are able to avoid the issue of item cost, due to the lack of budget, this issue becomes more iterestig i our settig, ad is yet to be explored. Aother directio that requires more attetio is the multi buyer settig. The simplicity of our model where there is oe idealized buyer leds itself to simple aalysis. It is kow [10] that with multiple buyers there is o Walrasia equilibrium uless the valuatio class is essetially the class of Gross Substitutes [10, 5]. However, there may still be iterestig results for a bouded, small umber of buyers. Aother ope problem is whether the budgeted model may preclude ay Nash equilibria (ot just market clearig oes) i certai settigs. We suspect our results for the relative valuatio costrait should geeralize to XOS valuatios. While the PNE would ot be a equal margial utility oe, as our example shows is impossible, we do believe it is a ecessary ad sufficiet coditio for market clearace. Beyod this it would be iterestig to study more geeral valuatio fuctios, eve oes that exhibit complemets. All of these directios are reasoable itermediate steps towards the most geeral pricig model with sellers holdig may items ad may buyers with complex valuatios ad differig budgets. 7. ACKNOWLEDGEMENTS The authors thak Breda Lucier for his isightful commets ad clarificatios. This work was supported by NSERC grat

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