NonPrice Equilibria in Markets of Discrete Goods


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1 NonPrice Equilibria in Markets of Discrete Goods (working paper) Avinatan Hassidi Hai Kaplan Yishay Mansour Noa Nisan ABSTRACT We study arkets of indivisible ites in which pricebased (Walrasian) equilibria often do not eist due to the discrete nonconve setting. Instead we consider Nash equilibria of the arket viewed as a gae, where players bid for ites, and where the highest bidder on an ite wins it and pays his bid. We first observe that pure Nashequilibria of this gae ecatly correspond to pricebased equilibiria (and thus need not eist), but that iednash equilibria always do eist, and we analyze their structure in several siple cases where no pricebased equilibriu eists. We also undertake an analysis of the welfare properties of these equilibria showing that while pure equilibria are always perfectly efficient ( first welfare theore ), ied equilibria need not be, and we provide upper and lower bounds on their aount of inefficiency.. INTRODUCTION. Motivation The basic question that Econoics deals with is how to best allocate scarce resources. The basic answer is that Google, Tel Aviv. Google, Tel Aviv, and the School of Coputer science, Tel Aviv University. This research was supported in part by a grant fro the Israel Science Foundation (ISF), by a grant fro United StatesIsrael Binational Science Foundation (BSF). School of Coputer science, Tel Aviv University. This research was supported in part by a grant fro the the Science Foundation (ISF), by a grant fro United StatesIsrael Binational Science Foundation (BSF), by a grant fro the Israeli Ministry of Science (MoS), and by the Google Interuniversity center for Electronic Markets and Auctions. School of Coputer Science and Engineering, Hebrew University of Jerusale. Supported by a grant fro the Israeli Science Foundation (ISF), and by the Google Interuniversity center for Electronic Markets and Auctions. Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. Copyright 20XX ACM XXXXXXXXX/XX/XX...$0.00. trade can iprove everyone s welfare, and will lead to a arket equilibriu: a vector of resource prices that clear the arket and lead to an efficient allocation. Indeed, Arrow and Debreu [] and uch further work shows that such arket equilibria eist in general settings. Or do they...? An underlying assuption for the eistence of priceequilibria is always soe notion of conveity. While soe ay feel cofortable with the restriction to conve econoies, arkets of discrete ites arguably the ain object of study in coputerized arkets and auctions are only rarely conve and indeed in ost cases do not have any pricebased equilibria. What can we predict to happen in such arkets? Will these outcoes be efficient in any sense? In this paper we approach this questions by viewing the arket as a gae, and studying its Nashequilibria..2 Our Model To focus on the basic issue of lack of pricebased equilibria, our odel does not address inforational issues, assues a single seller, and does not assue any budget constraints. Our seller is selling heterogeneous indivisible ites to n buyers who are copeting for the. Each buyer i has a valuation function v i specifying his value for each subset of the ites. I.e. for a subset S of the ites v i(s) specifies the value for that buyer if he gets eactly this subset of the ites, epressed in soe currency unit (i.e., the buyers are quasilinear). We will assue free disposal, i.e., that the v i s are onotonically nondecreasing, but nothing beyond that. The usual notion of pricebased equilibriu in this odel is called a Walrasian equilibriu: a set of ite prices p... p and a partition S... S n of the ites aong the n buyers such that each buyer gets his deand under these prices, i.e., S i arga S(v i(s) j S pj). When such equilibria eist they aiize social welfare, i vi(si), but unfortunately it is known that they only rarely eist eactly when the associated integer progra has no integrality gap (see [3] for a survey). We will consider this arket situation as a gae where each player i announces offers b i,... b i, with the interpretation that b ij is player i s bid of ite j. After the offers are ade, independent first price auctions are being ade. That is the utility of each bidder i is given by u i(b) = v i(s i) j S i b ij where S... S n are a partition of the ites with the property that each ite went to a highest bidder on it. Soe care is needed in the case of We use interchangeably the ters: player, bidder and buyer, and all three have the sae eaning.
2 ties two (or ore) bidders i i that place the highest bid b ij = b i j for soe ite j. In this case a tie breaking rule is needed to coplete the specification of the allocation and thus of the gae. Iportantly, we view this as a gae with coplete inforation, so each player knows the (cobinatorial) valuation function of each other player..3 Pure Nash Equilibriu Our first observation is that the pure equilibria of this gae capture eactly the Walrasian equilibria of the arket. This justifies our point of view that when we later allow iednash equilibria as well, we are in fact strictly generalizing the notion of priceequilibria. Theore: Fi a profile of valuations. Walrasian equilibria of the associated arket are in  correspondence with pure Nash equilibria of the associated gae. This holds in the eact sense for soe tiebreaking rule, and holds in the sense of liits of ɛnash equilibria for all tiesbreaking rules. A profile of strategies (bids) in the gae is called a liit of ɛnash equilibria if for every ɛ > 0 there eists a sequence of ɛnash equilibria that approach it. Let us deonstrate this theore with a trivial eaple: a single ite on sale and two bidders who have values of and 2 respectively for it. A Walrasian equilibriu can fi the ite s price p anywhere between and 2, at which point only the second bidder desires it and the arket clears. In the associated gae (with any tie breaking rule), a bid p for the first player and bid p + ɛ for the second player will be an ɛnashequilibriu. In the special case that the tie breaking rule gives priority to the second bidder, an eact purenash equilibriu will have both bidders bidding p on the ite. This theore is soewhat counter intuitive as strategic (nonpricetaking) buyers in arkets ay iprove their utility by strategically reducing deand. Yet, in our setting strategic buyers still reach the basic nonstrategic priceequilibriu. As an iediate corollary of the fact that a Walrasian equilibriu optiizes social welfare ( The first welfare theore ), we get the sae optiality in our gae setting: Corollary A First Welfare Theore For every profile of valuations and every tiebreaking rule, every pure Nash equilibriu of the gae (including a liit of ɛequilibria) optiizes social welfare. In other words, the price of anarchy of pure Nash equilibria is trivial..4 Mied Nash Equilibria As entioned above, since Walrasian equilibria only rarely eists, so do only rarely eist pure Nash equilibria in our gaes. So it is quite natural to consider also the standard generalization, MiedNash equilibria of our arket gaes. The issue of eistence of such ied Nash equilibria is not trivial in our setting as buyers have a continuu of strategies and discontinuous utilities so Nash s theore does not apply. Nevertheless, there has been a significant aount of econoic work on these types of settings and a theore of Sion and Zae [8] provides at least a partial general positive answer: Corollary (to a theore of [8]): For every profile of valuations, there eists soe (ied) tiebreaking rule such that the gae has a iednash equilibriu. It sees that, like in the case of pure equilibria, an ɛ Nash equilibriu should eist for all tie breaking rules, but we have not been able to establish this. Once eistence is established, we turn our attention towards analyzing what these ied equilibria look like. We start with the two basic eaples that are well known not to have a price equilibria: Eaple Copleents and Substitutes Bidders: In this eaple there are two ites and two bidders. The first bidder ( OR bidder ) views the two ites as perfect substitutes and has value of v or for either one of the (but is not interested in getting both). The second bidder ( AND bidder ) views the as copleents and values the bundle of both of the at v and (but is not interested in either of the separately). It is not difficult to see that when v and < 2v or no pure equilibriu eists, however we find specific distributions F and G for the bids of the players that are in iednash equilibriu. Eaple Triangle: In this eaple there are three ites and three players. Each of the players is interested in a specific pair of ites, and has value for that pair, and 0 for any single ite, or any other pair. A pure Nash equilibriu does not eist, but we show that the following is a ied Nash equilibriu: each player picks a bid uniforly at rando in the range [0, /2] and bids this nuber on each of the ites. Interestingly the epected utility of each player is zero. We generalize the analysis to the case of single inded players, each desiring a set of size k, each ite is desired by d players, and no two players sets intersect in at ost a single ite. We generalize our analysis to ore general eaples of these veins. In particular, these provide eaples where the iednash equilibriu is not optial in ters of aiizing social welfare and in fact is far fro being so. Corollary A First NonWelfare Theore : There are profiles of valuations where a iednash equilibriu does not aiize social welfare. There are eaples where pure equilibria (that aiize social welfare) eist and yet a ied Nash equilibriu achieves only O( ) fraction of social welfare (i.e., price of anarchy is Ω( )). There eist eaples where all iednash equilibria achieve at ost O( (log )/) fraction of social welfare (i.e., price of stability is Ω( /(log ))). At this point it is quite natural to ask how uch efficiency can be lost, in general, as well for interesting subclasses of valuations, which we answer as follows. Theore An Approiate First Welfare Theore : For every profile of valuations, every tiebreaking rule, and every iednash equilibriu of the gae we have that the epected social welfare obtained at the equilibriu is at least /α (the price of anarchy ) ties the optial social welfare, where. α 2β if all valuations βfractionally subadditive. (The case β = correponds to fractionally subadditive valuations, also known as XOS valuations. They include the set of subodular valuations.) 2. α = O(log ) if all valuations are subadditive. 3. α = O(), in general.
3 These bounds apply also to correlatednash equilibria and even to coarsecorrelated equilibria. A related PoA result is that of [2] which derive PoA for βfractionally subadditive bidders in a second price siultaneous auction under the assuption of conservative bidding. In this work we use the first price (rather than the second price) and do not ake any assuption regarding the bidding. Finally we etend these results also to a Bayesian setting where players have only partial inforation on the valuations of the other players. We show that for any prior distribution on the valuations and in every Bayesian Nash equilibriu, where each player bids only based on his own valuation (and the knowledge of the prior), the average social welfare is lower by at ost α = O(n) than the optial social welfare achieved with full shared knowledge and cooperation of the players. For a prior which is a product distribution over valuations which are βfractionally subadditive we show that α = 4β, which iplies a bound of 4 for subodular valuations and a bound of O(log )for subadditive valuations. Our proof ethodology for this setting is siilar to that of [2]..5 Open Probles and Future Work We consider our work as a first step in the systeatic study of notions of equilibriu in arkets where price equilibria do not eist. Our own work focused on the iednash equilibriu, its eistence and for, and its welfare properties. It is certainly natural to consider other properties of such equilibria such as their revenue or invariants over the set of equilibria. One ay also naturally study other notions of equilibriu such as those corresponding outcoes of natural dynaics (e.g. coarse correlated equilibria which are the outcoe of regret iniization dynaics). It is also natural to consider richer odels of arkets (e.g. twosided ones, nonquasilinear ones, or ones with partial inforation). Even within the odest scope of this paper, there are several reaining open questions: the eistence of iednash equilibriu under any tiebreaking rule; the characterization of all equilibria for the siple gaes we studied; and closing the various gaps in our price of anarchy and price of stability results. 2. MODEL We have a set M of heterogeneous indivisible ites for sale to a set N of n bidders. Each bidder i has a valuation function v i where for a set of ites S M, v i(s) is his value for receiving the set S of ites. We will not ake any assuptions on the v i s ecept that they are onotone non decreasing (free disposal) and that v i( ) = 0. We assue that the utility of the bidders is quasilinear, naely, if bidder i gets subset S i and pays p i then u i(s i, p i) = v i(s i) p i. We will consider this arket situation as a gae where the ites are sold in siultaneous first price auctions. Each bidder i N places a bid b ij on each each ite j M, and the highest bidder on each ite gets the ite and pays his bid on the ite. We view this as a gae with coplete inforation. The utility of each bidder i is given by u i(b) = v i(s i) j S i b ij where S...S n are a partition of the ites with the property that each ite went to the bidder that gave the highest bid for it. Soe care is required in cases of ties, i.e., if for soe bidders i i and an ite j M we have that b ij = b i j are both highest bids for ite j. In these cases the previous definition does not copletely specify the allocation, and to coplete the definition of the gae we ust specify a tie breaking rule that chooses aong the valid allocations. (I.e. specifies the allocation S,..., S n as a function of the bids.) In general we allow any tie breaking rule, a rule that ay depend arbitrarily on all the bids. Even ore, we allow randoized (ied) tie breaking rules in which soe distribution over deterinistic tie breaking rules is chosen. We will call any gae of this faily (i.e.,with any tie breaking rule) a first price siultaneous auction gae (for a given profile of valuations). 3. PURE NASH EQUILIBRIUM The usual analysis of this scenario considers a arket situation and a pricebased equilibriu: Definition 3.. A partition of the ites S...S n and a nonnegative vector of prices p...p are called a Walrasian equilibriu if for every i we have that S i arga S(v i(s) j S pi). We consider bidders participating in a siultaneous first price auction gae, with soe tie breaking rule. Our first observation is that pure equilibria of a first price siultaneous auction gae correspond to Walrasian equilibiria of the arket. In particular the price of anarchy of pure equilibria is. Proposition A profile of valuation functions v...v n adits a Walrasian equilibriu with given prices and allocation if and only if the first price siultaneous auction gae for these valuations has a pure Nash equilibriu for soe tie breaking rule with these winning prices and allocation. 2. Every pure Nash equilibriu of a first price siultaneous auction gae achieves optial social welfare. Proof. Let S,..., S n and p,..., p be a Walrasian equilibriu. Consider the bids where b ij = p j for all j and let the gae break ties according to S...S n. Why are these bids a pure equilibriu of this gae? Since we are in a Walrasian equilibriu, each player gets a best set for hi under the prices p j. In the gae, given the bids of the other players, he can never win any ite for strictly less than p j, whatever his bid, and he does wins the ites in S i for price p j eactly, so his current bid is a best response to the others 2. Now fi a pure Nash equilibriu of the gae with a given tie breaking rule. Let S,..., S n the allocation specified by the tie breaking rule, and let p j = a i b ij for all j. We clai that this is a Walrasian equilibriu. Suppose by way of contradiction that soe player i strictly prefers another bundle T under these prices. This contradicts the original 2 The reader ay dislike the fact that the bids of loosing players see artificially high and indeed ay be in weakly doinated strategies. This however is unavoidable since, as we will see in the net section, counterintuitively soeties there are no pure equilibria in undoinated strategies. What can be said is that inial Walrasian equilibria correspond to pure equilibria of the gae with strategies that are liits of undoinated strategies.
4 bid of i was a best reply since the deviation bidding b ij = p j + ɛ for j T and b ij = 0 for j T would give player i the utility fro T (inus soe ɛ s) which would be ore than he currently gets fro S i a contradiction. The allocation obtained by the gae, is itself the allocation in a Walrasian equilibriu, and thus by the First Welfare Theore is a socialwelfare aiizing allocation. Two shortcoings of this proposition are obvious: first is the delicate dependence on tiebreaking: we get a Nash equilibriu only for soe, carefully chosen, tie breaking rule. In the net section we will show that this is unavoidable using the usual definitions, but that it is not a real proble: specifically we show that for any tiebreaking rule we get arbitrarily close to an equilibriu. The second shortcoing is ore serious: it is well known that Walrasian equilibria eist only for restricted classes of valuation profiles 3. In the general case, there is no pure equilibriu and thus the result on the price of anarchy is void. In particular, the result does not etend to ied Nash equilibria and in fact it is not even clear whether such ied equilibria eist at all since Nash s theore does not apply due to the noncopactness of the space of ied strategies. This will be the subject of the the following sections. 3. Tie Breaking and Liits of ɛequilibria This subsection shows that the quantification to soe tiebreaking rule in the previous theore is unavoidable. Nevertheless we argue that it is really just a technical issue since we can show that for every tie breaking rule there is a liit of ɛequilibria. A first price auction with the wrong tie breaking rule Consider the full inforation gae describing a first price auction of a single ite between Alice, who has a value of for the ite, and Bob who values it at 2, where the bids, for Alice and y for Bob, are allowed to be, say, in the range [0, 0]. The full inforation gae specifying this auction is defined by u A(, y) = 0 for < y and u A(, y) = for > y, and u B(, y) = 2 y for < y and u B(, y) = 0 for > y. Now coes our ain point: how would we define what happens in case of ties? It turns out that forally this detail deterines whether a pure Nash equilibriu eists. Let us first consider the case where ties are broken in favor of Bob, i.e., u B(, y) = 2 y for = y and u A(, y) = 0 for = y. In this case one ay verify that =, y = is a pure Nash equilibriu 4. Now let us look at the case that ties are broken in favor of Alice, i.e u A(, y) = and u B(, y) = 0 for = y. In this case no pure Nash equilibriu eists: first no y can be an equilibriu since the winner can always reduce his bid by ɛ and still win, then if = y > then Alice would rather bid = 0, while if = y < 2 then Bob wants to deviate to y + ɛ and to win, contradiction. This lack of pure Nash equilibriu doesn t see to capture the essence of this gae, as in soe inforal sense, the 3 When all valuations are substitutes. 4 The bid = is weakly doinated for Alice. Surprisingly, however, there is no pure equilibriu in undoinated strategies: suppose that soe y is at equilibriu with an undoinated strategy <. If yge then reducing y to y = would still ake Bob win, but at a lower price. However, if y < too, then the loser can win by bidding just above the current winner contradiction. correct pure equilibriu is ( =, y = + ɛ) (as well as ( = ɛ, y = )), with Bob winning and paying + ɛ (). Indeed these are ɛequilibria of the gae. Alternatively, if we discretize the auction in any way allowing soe inial ɛ precision then bids close to with inial gap would be a pure Nash equilibriu of the discrete gae. We would like to forally capture this property: that =, y = is arbitrarily close to an equilibriu. Liits of ɛequilibria We will becoe quite abstract at this point and consider general gaes with (finitely any) n players whose strategy sets ay be infinite. In order to discuss closeness we will assue that the pure strategy set X i of each player i has a etric d i on it. In applications we siply consider the Euclidean distance. Definition 3.3. (... n) is called a liit (pure) equilibriu of a gae (u...u n) if it is the liit of ɛequilibria of the gae, for every ɛ > 0. Thus in the eaple of the first price auction, (, ) is a liit equilibriu, since for every ɛ > 0, (, + ɛ) is an ɛequilibriu. Note that if all the u i s are continuous at the point (... n) then it is a liit equilibriu only if it is actually a pure Nash equilibriu. This, in particular, happens everywhere if all strategy spaces are discrete. We are now ready to state a version of the previous proposition that is robust to the tie breaking rule: Proposition For every first price siultaneous auction gae with any tie breaking rule, a profile of valuation functions v...v n adits a Walrasian equilibriu with given prices and allocation if and only if the gae has a liit Nash equilibriu for these valuations with these winning prices and allocation. 2. Every liit Nash equilibriu of a first price siultaneous auction gae achieves optial social welfare. Proof. Let S...S n and p...p be a Walrasian equilibriu. Consider the bids where b ij = p j + ɛ for all j S i and b ij = p j for all j S i. Why are these bids an ɛequilibriu of this gae? Since we are in a Walrasian equilibriu, each player gets a best set for hi under the prices p j. In the gae, given the bids of the other players, he can never win any ite for strictly less than p j, whatever his bid, and player i does win each ite j in S i for price p j + ɛ, so his current bid is a best response to the others up to an additive ɛ for each ite he wins. Now fi a liit Nash equilibriu (b ij) of the gae with soe tie breaking rule and let (b ij) be an ɛequilibriu of the gae with b ij b ij ɛ for all i, j and with no ties; let S...S n the allocation iplied; and let p j = a i b ij for all j. We clai that this is an ɛwalrasian equilibriu. Suppose by way of contradiction that for soe player i and soe bundle T S i, v i(t ) j T pj > vi(si) j S i p j + ɛ. This would contradict the original bid of i being an ɛbest reply since the deviation bidding b ij = p j + ɛ for j T and b ij = 0 for j T would give player i the utility fro T up to ɛ which would be ore than he currently gets fro S i a contradiction. Now let ɛ approach zero and look at the sequence of price vectors p and sequence of allocations obtained as (b ij) approach (b ij). The sequence of price vectors converges to a
5 fied price vector (since they are a continuous function of the bids). Since there are only a finite nuber of different allocations, one of the appears infinitely often in the sequence. It is now easy to verify that this allocation with the liit price vector are a Walrasian equilibriu. 4. GENERAL EXISTENCE OF MIXED NASH EQUILIBRIUM In this section we ask whether such a gae need always even have a iednash equilibriu. This is not a corollary of Nash s theore due to the continuu of strategies and discontinuity of the utilities, and indeed even zerosu twoplayer gaes with [0, ] as the set of pure strategies of each player ay fail to have any iednash equilibriu or even an ɛequilibriu 5. There is soe econoic literature about the eistence of equilibiria in such gaes (starting e.g. with [7, 4]), and a theore of Sion and Zae [8], iplies that for soe (randoized) tie breaking rule, a iednash equilibriu eists. The ain eaple of their (ore general) theore is the following (cf. page 864): Suppose we are given strategy spaces S i, a dense subset S of S = S S n, and a bounded continuous function ϕ : S R n. Let C ϕ : S R n be the correspondence whose graph is the closure of the graph of ϕ, and define Q ϕ(s) to be the conve hull of C ϕ(s) for each s S. We call the correspondence Q ϕ the conve copletion of ϕ. These are Sion and Zae s otivating eaple of gaes with an endogenous sharing rule, and their ain theore is that these have a solution : a pair (q, α), where q is a sharing rule, a Borel easurable selection fro the payoff correspondence Q and α = (α,..., α n) is a profile of ied strategies with the property that each player s action is a best response to the actions of other players, when utilities are according to the sharing rule q. Now to how this applies in our setting: S will be the set of bids with no ties, i.e., where for all j and all i i we have that b ij b i j, which is clearly dense (since bids with ties have easure zero). Here ϕ is siply the vector of utilities of the players fro the chosen allocation which is fully deterined and continuous in S when there are no ties. For b S, we have that C ϕ( b) is the set of utility vectors obtained fro all possible deterinistic tiebreaking rules at b (each of which ay be obtained as a liit of bids with no ties), and Q ϕ is the set of itures (randoizations) over these. The solution thus provides a randoized tiebreaking rule q and ied strategies that are a iednash equilibriu for the gae with this tiebreaking rule. So we get: Corollary 4.. The first price siultaneous auction gae for any profile of valuations has a iednash equilibriu for soe randoized tiebreaking rule. We suspect that the tiebreaking rule is not that significant and that ied ɛnashequilibria (or aybe even eact Nashequilibria) actually eist for every tiebreaking rule, siilarly to the case of pure equilibria in this paper, or as in the soewhat related setting of [6] where an invariance in the tiebreaking rule holds. 5 A well known eaple is having highest bidder win, as long as his bid is strictly less than, in which case he looses (with ties being ties). 5. MIXEDNASH EQUILIBRIA: EXAMPLES In this section we study soe of the siplest eaples of arkets in our setting that do not have a Walresian equilibriu. 5. The ANDOR Gae We have two players an AND player and OR player. The AND player has a value of if he gets all the ites in M, and the OR player has a value of v if he gets any ite in M. Forally, v and (M) = and for S M we have v and (S) = 0, also, v or(t ) = v for T and v or( ) = 0. When v / there is a Walresian equilibriu with a price of v per ite. By Proposition 3.2 this iplies a pure Nash Equilibriu in which both players bid v on each ite, and the AND player wins all the ites. Therefore, the interesting case is when v > /. It is easy to verify that in this case is no Walresian equilibriu. We start with the case that M = 2 and later etend it to the case of arbitrary size. Here is a ied Nash equilibriu for two ites. The AND player bids (y, y) where 0 y /2 according to cuulative distribution F (y) = (v /2)/(v y) (where F (y) = P r[bid y]). In particular, There is an ato at 0: P r[y = 0] = /(2v). The OR player bids (, 0) with probability /2 and (0, ) with probability /2, where 0 /2 is distributed according to cuulative distribution G() = /( ). Note that since the OR player does not have any ass points in his distribution, the equilibriu would apply to any tie breaking rule. We start by defining a restricted ANDOR gae, where the AND player ust bid the sae value on both ites, and show that the above strategies are a ied Nash equilibriu for it. Clai 5.. Having the AND player bid using F and the OR player bid using G is a ied Nash equilibriu of the restricted ANDOR gae for two ites. Proof. Let us copute the epected utility of the AND player fro soe pure bid (y, y). The AND player wins one ite for sure, and wins the second ite too if y >, i.e., with probability G(y). If he wins a single ite he pays y, and he wins both ites he pays 2y. His epected utility is thus G(y)( y) y = 0 for any 0 y /2 (and is certainly negative for y > /2). Thus any 0 y /2 is a bestresponse to the OR player. Let us copute the epected utility of the OR player fro the pure bid (0, ) (or equivalently (, 0)). The OR player wins an ite if > y, i.e., with probability F (), in which case he pays, for a total utility of (v ) F () = v /2, for every 0 /2 (and > /2 certainly gives less utility). Thus any 0 /2 is a bestresponse to the AND player. Net we generalize the proof to the unrestricted setting. Theore 5.2. Having the AND player bid using F and the OR player bid using G is a ied Nash equilibriu of the ANDOR gae for two ites. Proof. We first show that if the AND player plays the ied strategy F then G is a best response for the OR
6 player. This holds since when the AND player is playing F, then all its bids are of the for (y, y) for soe y [0,, /2]. Any bid (, 2) of the OR player, with 2, is doinated by (0, 2), since the AND player is restricted to bidding (y, y). Therefore, G is a best response for the OR player. We now need to show that if the OR player plays the ied strategy G then F is a best response for the AND player. Let Q(, y) be the cuulative probability of the OR player, i.e., Q(, y) = Pr[bid <, bid 2 < y] = 2( ) + y 2( y). for, y [0, ]. The AND utility function, given its distribution P, 2 is: where U AND = E (,y) P [u and (, y)], u and (, y) = Q(, y) (Q(, ) + yq(, y)) We show that for any, y [0, 2 ] we have u and(, y) = 0. This follows since, u and (, y) = Q(, y) (Q(, ) + yq(, y)) ( ) ( = 2( ) + y 2( y) 2( ) + ) 2 ( ) y 2 + y 2( y) = ( ) 2( ) + ( y) y 2( y) 2 y 2 = 0, which copletes the proof. We now etend the result to the ANDOR gae with ites. The AND player selects y using the cuulative probability distribution F (y) = v for y [0, /], and v y bids y on all the ites. The OR player selects using the cuulative probability distribution G() = ( ), where ( ) [0, /], and an i uniforly fro M, and bids on ite i and zero on all the other ites. Theore 5.3. Having the AND player bid using F and the OR player with G is a ied Nash equilibriu. Proof. Let Q(), for [0, /] be the cuulative probability distribution of the bids of the OR player. Given that the OR player bids using G it follows that ( ) i Q() = Pr[ i bid i < i] = i for [0, ]. Let P denote the cuulative probability distribution of the bids of the AND player. Then the utility of the AND player is: where U AND = E P [u and ()], u and () = Q() iq( i, (/) i). We show that for any [0, ] we have u and () = 0. ( k ) u and () = Q() iq( i, (/) i) = = = 0. ( ) i i ( ( ) i i + ( ) ) i ( ) i ( i) i ( ) i This iplies that the ied strategy of the AND player defined by F, is a best response to the ied strategy of the OR player defined by G. We now show that the ied strategy of the OR player defined by G, is a best response to the ied strategy of the AND player defined by F. Recall that P (), for [0, /] is the cuulative probability distribution of the bids of the AND player, and by the definition of the AND player it equals to P () = Pr[ i bid i < i] = v v in i{ i}. (Note that, as it should be, under P the support is the set of all identical bids, i.e., i bid i =. The probability under P of having a vector z is v v.) The utility function of the OR player is: where U OR = E Q[u or()], ( ) u or() = v e() ip ( i, (/) i). where e() = Pr P [ i such that X i < i]. We obtain that for any [0, ] and i [, ] uor(i =, i = 0) = v since u or( i =, i = 0) = v (/) (v ) = v v Furtherore, for any [0, ] we have u or() u or(y), where y keeps only the aial entry in and zeros the rest. This follows since given P, the probability of winning under and y is identical. Clearly the payents under y are at ost those under (since all the bids in are at least the bids in y). We conclude that the OR player s strategy is a best response to the AND player s strategy, and this copletes the proof. 5.2 The Triangle Gae We start with a siple case of three single inded bidders and three ites, where each bidder wants a different set of two ites, and has a value of one for this set. Consider syetric strategies in which each player bids the sae for the pair of ites it wants, naely each player draws their bid fro the sae distribution whose cuulative distribution function is F (). Assuing F () has no
7 atos then the utility of each player is ( 2)F 2 () 2F ()( F ()) = F 2 () 2F () Theore 5.4. If each player draws an fro F () = 2, where 0 /2, and bids on both ites, then it is a ied Nash equilibriu. Proof. Suppose two of the players play according to F () and consider the best response of the third player. For any value 0 /2 if the third player bids (, ), his utility is zero. On the other hand, if it bids y for one ite and z for the other then its utility is F (y)f (z) yf (y) zf (z) = 2(y z) 2 0. Finally, bidding any nuber strictly above /2 is doinated by bidding /2. Consider now a generalization of this gae where each player is single inded and is interested only in a particular set of k ites for which its utility is. We also ake the following assuptions.. Eactly d agents are interested in each ite. 2. For any two bidders i i, we have S i S i. (This iplies that if we fi a player i and consider its set S i of k ites. The other (d )k players who are also interested in these k ites are all different.) Assue each player i draws the sae bid for all ites in its set S i fro the CDF G(). If G() satisfies the equation G (d )k () kg d () = 0 () for all then the utility of a player is zero for every bid. One can easily verify that the function G() = (k) (d ), satisfies Equation () for all. So G() = (k) (d ), 0, fors an equilibriu for the restricted gae, k where in the restricted gae a player has to bid the sae bid on all the ites in his set. The following shows that even if we do not restrict the players to bid the sae then G() is an equilibriu. Theore 5.5. If all players draw a bid for all k ites that they want fro G() = (k) (d ), 0, then k it is a ied Nash equilibriu. Proof. Suppose all the players but one play according to G() and consider the best response of the first player. Suppose its bid is j for the jth ite in S i. Then its utility is Π k k j=(k j) j(k j). j= We clai that this utility is nonpositive for every set of bids,..., k. Indeed this follows since k k k k j k j= k k j= k j by the inequality of arithetic and geoetric eans: k k k k i = j k k j. k j= j= 6. INEFFICIENCY OF MIXED EQUILIBRIA In this section we use our analysis of the eaples given in the previous section to construct eaples where there are large gaps between the efficiency obtained in a iednash equilibriu and the optial efficiency. We first analyze the ANDOR gae with ites, where v /, and hence there is no pure Nash equilibriu. We will analyze the following paraeters: value of the OR player is v = / and the value of the AND player is. Theore 6.. There is a ied Nash equilibriu in the ANDOR gae with the paraeters above whose social welfare is at ost 2/. I.e. for this gae we have P oa /2. Proof. For the PoA consider the equilibriu of Section 5.. Assue that the value of the OR player is v = / and the value of the AND player is. This iplies that the optial social welfare is. The probability that the AND player bids = 0 is v / = /. Therefore with v probability at least / the OR player wins. This iplies that the social welfare is at ost 2/ We now prove the following lea regarding the support of the AND player. Lea 6.2. In any Nash equilibriu the AND player does not have in its support any bid vector b and such that b and,i >. Proof. Assue that there is such a bid vector b and. Since b and,i > the AND player can not get a positive utility, and the only way it can gain a zero utility is by losing all its nonzero bids. This iplies that for any bid vector b or of the OR player, the OR player will win all the ites. Therefore bor,i >. This iplies that the revenue of the auctioneer is larger than (every tie). Since the epected revenue of the auctioneer is larger than, and the optial social welfare is, the su of the epected utilities of the players has to be negative. Hence one of the players has an epected negative utility. This clearly can not occur in equilibriu. It turns out that for this eaple, not only there eist bad equilibria, but actually all equilibria are bad! Theore 6.3. For any Nash equilibriu of the ANDOR gae with the paraeters above the social welfare is at ost 3 (log )/. I.e. the P os / log /3. Proof. Assue we have a Nash equilibriu in which the AND player wins with probability α. This iplies that the epected utility of the OR player u or is at ost ( α)v. Also, the social welfare of the equilibriu is ( α)v + α v + α. By Lea 6.2 the AND player never plays a bid b in which the su of the bids is larger than. This iplies that the AND player can have at ost half of the bids which are larger than 2/. Therefore, if the OR player bid 2/ on log rando ites, it will win soe ite with probability at least /. The OR player utility fro such a strategy is at least ( /)v (log )/. This iplies that in equilibriu, ( α)v u or ( /)v (log )/. For v = (log )/ it iplies that α 2 (log )/. Therefore the social welfare is at ost 3 (log )/.
8 Finally we study eaples in which there are ultiple equilibria, and show that they can be far apart fro one another: Theore 6.4. There is a set of valuations such that in the corresponding siultaneous first price auction there is an efficient (pure) Nash equilibriu, as well as an inefficient one, where the inefficiency is at least by a factor of /2. Equivalently, the corresponding auction has P os = but P oa /2. Proof. Consider = l 2 ites, which are labeled by (i, j) for i, j [, l]. Now we analyze 2l single inded bidder, where for each i [, l] we have a bidder that wants all the ites in (i, ), we call those bidders row bidders. For each j [, l] we have a bidder that want (, j), and we call the colun bidders. All bidders have value l for their set. Note that there is no allocation where both a row and a colun players are satisfied, where a player is satisfied if it is allocated all the ites in his set. The social optiu value is l 2 (satisfying all the row bidders or all the colun bidders). In this gae there is a Walresian equilibriu, where the price of each ite is. Siilarly, there is a pure Nash equilibriu where all bidders bid for each ite and we break the ties in favor to all the row players (or alternatively, to all the colun players). This iplies that the PoS is. Note that this gae has also a ied Nash equilibriu (Section 5.2, Theore 5.5). Since it is a syetric equilibriu, in which every player bids the sae value on all ites, the epected nuber of satisfied players is at ost 2 (since the probability of k satisfied players is at ost 2 k ). This iplies that the PoA is l/2 = /2. 7. APPROXIMATE WELFARE ANALYSIS In this section we analyze the Price of Anarchy of the siultaneous firstprice auction. We start with a siple proof of an O() upper bound on the price of anarchy for general valuations. Then we consider βxos valuations (which are equivalent to βfractionally subadditive valuations) and prove an upper bound of 2β. Since subadditive valuations are O(log ) fractionally subadditive [5, 2] we also get an upper bound of O(log ) on the price of anarchy for subadditive valuations. Assue that in OP T player i gets set O i and receives value o i = v i(o i). Let k i be O i. Let e i be the epected value player i gets in an equilibriu and let u i be the epected utility of player i in an equilibriu. Let r i be the epected su of payents in equilibriu over all ites in O i (these are not necessarily won by player i in equilibriu). Denote the total welfare, revenue, and utility in equilibriu by SW (eq), REV (eq), and U(eq), respectively. By definitions we have: () SW (eq) = i ei, (2) SW (OP T ) = i oi, (3) REV (eq) = i ri SW (eq), (4) U(eq) = i ui = SW (eq) REV (eq). Theore 7.. For any set of buyers the PoA is at ost 4. Proof. We first show that for each buyer i, we have 2u i o i 4k ir i. By Markov, with probability of at least /2 the total su of prices of ites in O i is at ost 2r i. Thus if player i bids 2r i for each ite in O i (and 0 elsewhere) he wins all ites with probability of at least /2, getting epected value of at least o i/2, and paying at ost 2k ir i. Since we were in equilibriu this utility ust be at ost u i. Hence, 2u i o i 4k ir i. Suing over all buyers, and bounding i ki, we get that OP T 2U(eq) + 4Rev(eq) 4SW (eq). A function v is βxos, if there eists an XOS function X such that for any set S we have v(s) X(S) v(s)/β, i.e., if there are nubers λ j,l, j M and l L, such that for any set S we have v(s) a k L λ j,k v(s)/β j S The equivalence of βxos and β fractionally subadditive follows the sae proof as in [5]. Theore 7.2. Assue that the valuations of all the players are βxos. Then the PoA is 2β. Proof. Since v is βxos, there is a k L such that j O i λ j,k v i(o i)/β, and for any set S, we have v(s) j S λ j,k. Let f j be the epected price of ite j. By Markov inequality, with probability of at least /2 the price of ite j is at ost 2f j. Consider the deviation where player i bids bid i,j = in{λ j,k, 2f j} for each ite j O i (and 0 elsewhere). Player i wins each ite j with probability α j and if bid i,j = 2f j then α j /2. Let S i be the set of ite that player i wins with his deviation bids bid i,j. (Note that S i is a rando variable that depends on the rando bids of the other players.) The epected utility of player i fro the deviation is, E[v i(s i) j S i bid i,j] j O i α j(λ j,k bid i,j) 2 (λ j,k bid i,j) j O i 2 (λ j,k 2f j) j O i 2β vi(oi) j O i f j. Since player i was playing an equilibriu strategy, we have that u i E[v i(s i) j S i bid i,j]. Suing over all players i s, and recalling that REV (eq) = j M fj, we get, E[SW (eq)] REV (eq) = which copletes the proof. n u i E[SW (OP T )] REV (eq), 2β 8. BAYESIAN PRICE OF ANARCHY In a Bayesian setting there is a known prior distribution Q over the valuations of the players. We first saple v Q and infor each player i his valuation v i. Following that, each player i draws his bid fro the distribution D i(v i), i.e., given a valuation v i he bids (b i,,..., b i,) D i(v i). The distributions D(v) = (D (v ),..., D n(v n)) are a Bayesian Nash equilibriu if each D i(v i) is a best response of player i, given that its valuation is v i and the valuations are drawn fro Q.
9 We start with the general case, where the distribution over valuations is arbitrary and the valuations are also arbitrary. Later we study product distributions over βxos valuations. Theore 8.. For any prior distribution Q over the players valuations, the Bayesian PoA is at ost 4n + 2. Proof. Fi a Bayesian Nash equilibriu D = (D,..., D n) as described above. Let Q vi be the distribution on v i obtained by conditioning Q on v i as the value of player i. Let u i(v i) be the epected utility of player i when his valuation is v i, i.e., u i(v i) = E bi D i (v i )E v i Q vi E b i D i [v i(s i) j S i b i,j], where S i is the set of ites that player i wins with the set of bids b. Let u i be the epected utility of player i, i.e., E vi Q[u i(v i)]. For any valuation v i for player i, consider the following deviation. Let Rev(v i) be the epected revenue given that the valuation of player i is v i, i.e., Rev(v i) = E v Qvi [ j= a k b k,j ]. Consider the deviation where player i bids 2Rev(v i) on each ite j M. By Markov inequality, he will win all the ites M with probability at least /2. Therefore, his utility fro the deviation is at least v i(m)/2 2Rev(v i) Since this is an equilibriu, we have that u i(v i) v i(m)/2 2Rev(v i) Suing over the players and taking the epectation with respect to v, n E v[u i(v i)] n E v[v i(m)/2 2Rev(v i)] Clearly n Ev[ui(vi)] Ev(SW (D)), where Ev(SW (D)) is the epected social welfare of the Bayesian equilibriu D. Also, n Ev[vi(M)] Ev[SW (OP T (v))]. Finally, for every player i, E v[rev(v i)] = Rev, where Rev is the epected revenue. Therefore, E v[sw (D)] E v[sw (OP T (v))]/2 2nRev Since Rev E v[sw (D)], we have that, (4n + 2)E v[sw (D)] E v[sw (OP T (v))] The following theore show that the Bayesian PoA is at ost 4β when the valuations are liited to βxos and the distribution Q over valuations is a product distribution. The proof uses the ideas presented in [2]. Theore 8.2. For a product distribution Q over βxos valuations of the players, the Bayesian PoA is at ost 4β. Proof. Fi a Bayesian Nash equilibriu D = (D,..., D n) as described above. Let Q vi be the distribution on v i obtained by conditioning Q on v i as the value of player i. Consider the following deviation of player i, given its valuation v i. Player i draws w i Q vi, that is w i are rando valuations of the other players, conditioned on player i having valuation v i. Player i coputes the optial allocation OP T (v i, w i), and in particular his share OP T i(v i, w i) in that allocation. Player i bids 2f j(v i) on each ite j OP T i(v i, w i), where f j(v i) is the epected aiu bid of the other players on ite j in the equilibriu D conditioned on player i having valuation v i, i.e., f j(v i) = E w i Q vi E b i D i (w i )[a k i b k,j]. By Markov inequality player i wins each ite j OP T i(v i, w i) with probability at least half. Since v i is an βxos valuation, its epected value is at least v i(op T i(v i, w i))/(2β) so the utility of player i in this deviation is at least E w i Q vi [v i(op T i(v i, w i))/(2β) 2f j(v i)] j OP T i (v i,w i ) Let u i(v i) be the epected utility of player i when his valuation is v i, i.e., u i(v i) = E bi D i (v i )E v i Q vi E b i D i [v i(s i) j S i b i,j], where S i is the set of ites that player i wins with the set of bids b. Let u i be the epected utility of player i, i.e., E vi Q[u i(v i)]. We get that, u i(v i) E w i Q vi [v i(op T i(v i, w i))/(2β) Takin the epectation with respect to v i, u i = E vi [u i(v i)] j OP T i (v i,w i ) E vi E w i Q vi [v i(op T i(v i, w i))/(2β) 2f j(v i)] j OP T i (v i,w i ) = E v Q[v i(op T i(v))/(2β)] E v Q[ 2f j(v i)] j OP T i (v) = E v Q[v i(op T i(v))/(2β)] 2E v Q[ j M I(j OP T i(v))f j(v i)], 2f j(v i)] where I(X) is the indicator function for the event X. Suing over all the players n n E v Q[v i(op T i(v))/(2β)] u i 2 n E v Q[ j M I(j OP T i(v))f j(v i)] = E v Q[SW (OP T (v))/(2β)] 2 n E v Q[ I(j OP T i(v))f j(v i)] j M Now we use the fact that the distribution Q over the valuations is a product distribution. This iplies that for any valuation v i, we have the sae value f j(v i). Let price(j) be the epected price of ite j M, i.e., price(j) = E v QE b D [ a k b k,j ]. Since price(j) f j(v i) for any buyer i and valuation v i, n u i E v Q[SW (OP T (v))/(2β)] 2 n price(j)e v Q[ I(j OP T i(v))] j M = E v Q[SW (OP T (v))/(2β)] 2 j M price(j),
10 where the last equality follows since ite j is always assigned to soe buyer, therefore, for any v, we have n I(j OP T i(v)) =. Let sw(d) be the epected social welfare of the Bayesian Nash D. Note that n ui = sw(d) j M price(j). Therefore, sw(d) j M price(j) E v Q[SW (OP T (v))/(2β)] 2 j M price(j), which iplies that 2sw(D) sw(d) + j M price(j) E v Q[SW (OP T (v))/(2β)]. This iplies that the PoA of the Bayesian equilibriu D is at ost 4β. 9. REFERENCES [] K. J. Arrow and G. Debreu. Eistence of an equilibriu for a copetitive econoy. Econoetrica, 22: , 954. [2] Kshipra Bhawalkar and Ti Roughgarden. Welfare guarantees for cobinatorial auctions with ite bidding. In SODA, 20. [3] L. Blurosen and N. Nisan. Cobinatorial auctions (a survey). In N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, editors, Algorithic Gae Theory. Cabridge University Press, [4] Fundenberg D. and D. Levine. Liit gaes and liit equilibria. Journal of Econoic Theory, 38:26 279, 986. [5] U. Feige. On aiizing welfare when utility functions are subadditive. SIAM J. Coput., 39():22 42, [6] Matthew O. Jackson and Jeroen M. Swinkels. Eistence of equilibriu in single and double private value auctions. Econoertica, 73():93 39, [7] Dasgupta P. and E. Maskin. The eistence of equilibriu in discontinuous econoic gaes. Review of Econoic Studies, 53: 42, 986. [8] Leo K. Sion and Willia R. Zae. Discontinuous gaes and endogenous sharing rules. Econoetrica, 58(4):86 872, 990.
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