Repeated Budgeted Second Price Ad Auction Asaph Arnon∗

Yishay Mansour†

May 6, 2011

Abstract Our main goal is to abstract existing repeated sponsored search ad auction mechanisms which includes budgets, and study their equilibrium and dynamics. Our abstraction has multiple agents biding repeatedly for multiple identical items (such as impressions in an ad auction). The agents are budget limited and have a value for per item. We abstract the repeated interaction as a one-shot game, which we call budget auction, where agents submit a bid and a budget, and then items are sold by a sequential second price auction. Once an agent exhausts its budget it does not participate in the proceeding auctions. Our main result is that if agents bid conservatively (never bid above their value) then there always exists a pure Nash equilibrium. We also study simple dynamics of repeated budget auctions, showing their convergence to a Nash equilibrium for two agents and for multiple agents with identical budgets.

1

Introduction

Auctions have become the main venue for selling online advertisements. This trend started in the sponsored search advertisements (such as, Google’s AdWords, Yahoo!’s Search Marketing and Microsoft’s AdCenter), and expended to the display advertisement (such as, Double click Ad Exchange [20]). This trend has even propagated to classical advertisement media, such as TV [21]. There are a few features that are shared by many of those auctions mechanisms. First, the price is set using a second price (or a generalized second price (GSP)) with the motivation that users should try to bid their utility rather than search for a optimal bid value. Second, there are daily budgets that cap the advertiser’s total payment in a single period (e.g., day). Our main goal is to abstract a model for such existing auctions, and study its equilibria and dynamics. It is worthwhile to expand on the role of budgets in auctions. The budget allows the advertiser to cap its spending in a given period (day). This is an important feature to the advertiser, for a few reasons. First, many advertisers are brand advertisers, whose goal is to promote their brand and they do not see an immediate return on their advertisement investments. For such an advertiser, a budget is the main tool to control the expense of an ad campaign. Second, even merchants that can test their return on investment (ROI) from ads, many times are budget limited, due to the corporate financial structure and the way budgets are allocated throughout the corporation. There has been an increasing research interest in the role of budgets in auctions, since budgets significantly influences the strategic behavior of agents. A very interesting line of research is constructing incentive compatible mechanism for an auction with budgets [8, 13, 16]. Another line of research has been maximizing the auctioneer’s revenue [1, 19, 4, 22, 3, 6, 11]. In this work we have a very different agenda. We would like to abstract the existing mechanisms and study their equilibria and dynamics. In our abstraction, as with any abstraction, we made a few compromises. The nature of the online advertisement auctions, is that there is a huge number of daily impressions, and agents compete repeatedly for those impressions. In some systems the advertisers input their budget limit explicitly (for example, in the Double Click Ad Exchange [20], Google’s AdWords, TV [21], etc.), and the system bids on their behalf ∗ School

of Computer science, Tel Aviv University and Google Tel Aviv. Email: [email protected] of Computer science, Tel Aviv University. Email:[email protected]. This research was supported in part by the Google Inter-university center for Electronic Markets and Auctions, by a grant from the Israel Science Foundation, by a grant from United States-Israel Binational Science Foundation (BSF), and by a grant from the Israeli Ministry of Science (MoS). † School

until either their budget is exhausted or the day ends. We abstract each day as a single-shot game, where the advertiser sets at the start of the day its bid and budget. (A similar conceptual abstraction to a one-shot game was done for studying the truthfulness of click through rates [5, 12].) More concretely, each agent has a private value (for each item) and a private budget (which caps its total spend in a day). Agents bid for multiple identical divisible items.1 Each day, the agents submit a bid and a budget to the auctioneer, which conceptually runs a sequence of second price auctions with some fixed minimum price. The auction terminates when all items have been sold or all agents have exhausted their budget. This sequential auction is a one-shot budget auction. The one-shot budget auction abstracts a repeated sponsored search auction for a single slot or a single display advertisement [20]. Our main focus is studying the properties and the existence of pure Nash equilibrium in a budget auction. We first observe that in a budget auctions, submitting the true budget is a dominant strategy, while bidding the true value is not a dominant strategy. The existence of pure Nash equilibrium depends on the assumptions regarding the bids of losing agents. For the case of two agents or multiple agents with identical budgets we show that there exists a pure Nash equilibria, even when the losing agents are restricted to bid their true value. For the case of multiple agents with different budgets, if losing agents are restricted to bid their true value then there are cases where no pure Nash equilibrium exists. Our main result is that if we relax this restriction, and assume that the losing agents bid conservatively, i.e., any value between the minimum price and their value, then there always exists a pure Nash equilibrium. We also study simple dynamics of repeated budget auctions with myopic agents. For the dynamics we use the Elementary Stepwise System [23], where in each day one non-best-responding agent modifies his bid to a best response, and bid values are discrete. We prove that these repeated budget auction converge to a Nash equilibrium for the case of two agents and for the case of multiple agents with identical budgets (under some restrictions). To illustrate our results for the repeated budget auction we ran simulations. We observed two distinct bidding patterns: either smooth convergence to an equilibrium or a bidding war cycle. The smoothed convergence is observed for a very wide range of parameters and suggests that the convergence is much wider than we are able to prove. Related Work: The existence of a pure Nash equilibrium in GSP sponsored search auction was shown in the seminal works [15, 24]. (For equilibrium in other related models see [22].) There are many works on dynamics of bidding strategies in sponsored search auctions, including theoretical and empirical works [17, 14, 2, 9, 10]. Paper Outline: Section 2 presents the budget auction model and derives some basic properties of budget auction. Section 3 studies the existence of pure Nash equilibria in budget auctions. Section 4 analyzes the dynamics of budget auctions when played repeatedly, and presents simulations. Appendix A includes the missing proofs.

2

The Model

The budget auction model has a set of k agents, K = {1, ..., k}, bidding to buy N identical divisible items. Each agent i ∈ K has two private values: his daily budget Bˆi , and his value for a single item vi . His utility ui depends on the amount of items he received xi and the price he paid pi , and ui (xi , pi ) = xi (vi − pi ) as long ˆi ), and ui = −∞ if he exceeded his budget2 (i.e., xi pi > B ˆi ). as he did not exceed his budget (i.e., xi pi ≤ B The auction proceeds as follows. The auctioneer sets a minimum price pmin , which is known to the agents (she ignores bids below the minimum price). Each agent i ∈ K submits two values, his bid bi and his budget Bi . Therefore, the auction’s input is a vector of bids ~b = (b1 , b2 , ..., bk ), and a vector of budgets ~ = (B1 , B2 , ..., Bk ). The output of the auction is an allocation ~x = (x1 , x2 , ..., xk ), such that P B i∈K xi ≤ N and prices p~ = (p1 , p2 , ..., pk ), such that pi ∈ [pmin , bi ]. Agent i is charged xi pi for the xi items he receives. The allocation and prices are calculated in the following way. Initially, the auctioneer renames the agents such that b1 ≥ b2 ≥ . . . ≥ bk ≥ pmin , we later refer to this index also as ranking.3 First, agent 1 receives items 1 While technically, advertisement impressions are clearly not divisible, due to the large volume of impressions, this is a very reasonable abstraction. 2 This hard budget approach is used also in [13]. Another way to ensure agents don’t exceed their budget is if agents deposit their budget to the auctioneer (instead of just reporting it), and the auctioneer returns unused budget the end of the auction. 3 Ties between identical bids are broken by lexicographic order, i.e., the auctioneer first sell items to agent with the lower original index.

2

Agent A B C D

Private Budget value 20 2.0 25 1.5 30 1.5 20 0.5

Submitted B b 20 1.0 25 1.0 20 0.5 20 0.3

x 20 50 30 0

Outcome p u type 1.0 20 Winner 0.5 50 Winner 0.3 36 Border 0.0 0 Loser

Table 1: An example of a budget auction with four agents, N = 100 and pmin = 0. B - submitted budget, b submitted bid, x - allocation, p - price per item, u - utility. at price p1 = b2 until he runs out of budget or items, i.e., x1 = min(N, B1 /p1 ). Then, if there are still items left for sale, agent 2 pays a price p2 = b3 , for x2 = max{0, min(N − x1 , B2 /p2 )} items, and generally agent i Pi−1 receives xi = max{0, min(N − j=1 xj , Bi /pi )} items at a price pi = bi+1 , where we define bk+1 = pmin . The auction is completed either when all items are sold, or when all agents exhaust their budgets. Obviously, if all items are sold to agents with higher rank than agent i, then xi = 0 and P ui = 0. Note that by this definition, the allocation of items to agents will never exceed the supply N , i.e., i∈K xi ≤ N . Given the outcome of the budget auction, we can split the agents into three different categories: Winner Agents, Loser Agents and a Border Agent. A Border Agent is the lowest ranked agent that gets a positive allocation, i.e., h is a Border Agent if h = max{i : xi > 0}. Any agent j > h has xj = 0 and is called a Loser Agent. Any agent i < h is called a Winner Agent and has xi = Bi /pi > 0, i.e., winner agents exhaust their budgets. The example in Table 1 illustrates a budget auction with 100 items for sale and a minimum price of 0. The agents are ordered by their bids (note that A is ranked before B although their bids are equal, due to lexicography order). It is worth making a few remarks regrading our model. Our main goal is to abstract a repeated GSP auction with budgets. We make few important simplifying assumptions: First, that the budgets and bids are set once, which simplifies the game to be a one-shot game. This assumption is equivalent to assuming that the agents do not modify their bids. Second, we consider only a single item with a know quantity. Third, we assume that the items are divisible and prices are continuous, which are both a very accurate approximation in a sponsored search setting (where the number of impressions is usually huge, and prices are discritized at a very fine level). Given those assumptions, our model gives the same outcome as a repeated GSP auction. For the most part we assume that agent i submits bids in the range [pmin , vi ]. The assumption that bi ≥ pmin is with out loss of generality, since the auctioneer ignores agents that bid below pmin . The assumption that agents do not bid above their true value, i.e., bi ≤ vi , is a very reasonable and realistic assumption, and was termed conservative bidding in [18, 7].4 In a Pure Nash Equilibrium (PNE) no agent i ∈ K can gain by unilaterally changing his submitted bid bi ~ let ~b−i and B ~ −i be the submitted bids and budget Bi . Formally, for a bid vector ~b and a budget vector B, ~ ~ and budgets, respectively, of all agents except agent i. The pair (b, B) is a PNE, if for each agent i, and any ~ i )) ≤ ui (~b, B). ~ deviation b0i and Bi0 , we have ui ((b0i , ~bi ), (Bi0 , B In the following we define a dominant strategy. Definition 2.1. Submitting budget y is a dominant strategy for agent i if for any bid vector ~b, and any ~ −i we have that ui (xi , pi ) ≥ ui (x0 , p0 ), where xi and pi alternative budget y 0 and budgets of the other agents B i i (x0i and p0i , respectively) are the allocation and price under bids ~b and budgets (B−i , y) ((B−i , y 0 ), respectively). ~ and any alterSimilarly, submitting bid z is a dominant strategy for agent i if for any submitted budgets B, 0 0 0 ~ native bid z and bids of the other agents b−i we have that ui (xi , pi ) ≥ ui (xi , pi ), where xi and pi (x0i and p0i , ~ and bids (~b−i , z) ((~b−i , z 0 ), respectively). respectively) are the allocation and price under budgets B Not surprising, bidding the true value is not a dominant strategy, and agents can bid lower than their true value in order to maximize their utility. We observe that submitting the true budget is a dominant strategy. 4 Theoretically, an agent might profit by over bidding his value, since it increases the price of the agent i − 1 who ranked above him and therefore, decreases his allocation xi−1 . This will leave more items for agent i and might increase his own allocation xi . Nevertheless, such bidding might expose the agent to negative utility, since he might pay more than his value.

3

Figure 1: The Market equilibrium price of the example in Table 1 is peq = 0.75, as at that point the aggregated demand equals the supply. Note that at the vertical drops, the aggregated demand is an interval and not a point. This happens when the price equals the values (vi ) of the different agents (in the example this occurs at 0.5 and at 1.5 and in 2.0).

Claim 2.2. In a budget auction, bidding the true value is not a dominant strategy, while submitting the true budget is a dominant strategy. It is instructive to compare the allocation and prices of the budget auction to the Market Equilibrium Price, which equalizes the supply and demand. Definition 2.3. The demand of agent i at price p is an interval D(p) (or a point), as follows,  if vi > p  Bi /p 0 if vi < p Di (p) =  [0, Bi /p] vi = p The interval (or point) D(p) is the Aggregated Demand of all agents at price p, such that, D(p) = Price peq is the Market Equilibrium Price if N ∈ D(peq ).

P

i∈K

Di (p).

Notice that peq is unique since the function D(p) is strictly decreasing in p, namely for p1 > p2 , for any this monotonicity implies that N ∈ 1 ) and ∀y ∈ D(p2 ), we have x > y. In our setting, P ∀x ∈ D(pP P P [ i∈S Bi /peq , i∈S∪Z Bi /peq ], and in the same way that peq ∈ [ i∈S Bi /N, i∈S∪Z Bi /N ], where S = {i : vi > peq } and Z = {i : vi = peq }. Figure 1 shows the computation of the market equilibrium price, for the example in Table 1. One can impose the market equilibrium price to be the outcome of the budget auction in two simple alternatives. The first is that the auctioneer can set the minimum price to be the market clearing price, i.e., pmin = peq . In this case we will have a PNE where each agent i bids bi = min{vi , peq }. (Clearly, any agent that gets an allocation would pay peq , and the property of market equilibrium price guarantee that all the items are sold.) A similar PNE is to implement this behavior of the auctioneer, by adding a dummy agent, that bids peq with an infinite budget. This agent will not be conservative in both his bid (bidding above his value) and his budget (bidding an infinite budget), however, since this agent will not get any items, his utility would be zero. Both alternatives are not satisfactory. The first one, since if the auctioneer knows the market clearing price, she can simply post this price and sell all the item at that price (intuitively, she is holding an auction exactly because she does not know the market clearing price). The second alternative has the same drawback, assuming that an agent will be non-conservative both in his bid and budget, just to enforce the market clearing price, risking a negative utility. For this reason we would concentrate on conservative biding (both in the bid and budget), which guarantee a non-negative utility. A final remark is regarding Social Welfare maximization. In a budget auction, the allocation that maximizes the social welfare allocates all the N items to the agent with the highest valuation. This allocation is clearly not the goal in our setting. The budget auction does not maximize social welfare and therefore is not efficient, however, a similar issue always exists in the presence of budgets. 4

Agent A B C D

Private Values ˆ i vi B ci 40 2.0 1.143 40 2.0 1.143 40 2.0 1.143 8 1.0 1.0

bi 1.143 1.143 1.143 1.0

bidding at critical bid pi xi ui type 1.143 35 30 winner 1.143 35 30 winner 1.0 30 30 border 0 0 0 loser

bi 1.143 1.143 1.0 −  1.0

under pi 1.143 1.0 0.0 1.0 − 

bidding xi ui 35 30 40 40 17 34 8 0

type winner winner border winner

Table 2: Two possible outcomes of the budget auction with 4 agents, N = 100 and pmin = 0.

3

Pure Nash Equilibrium

In this section we study the existence of a Pure Nash Equilibrium (PNE) in budget auctions. Our main result is that under mild conditions a PNE does exists for every budget auction. We start by showing properties that any PNE in a budget auction must have, we then define the notion of critical bid, which intuitively is the bid which make the agent indifferent between being a winner or a border agent, and we complete by proving that PNE exist in budget auctions.

3.1

Properties of a Pure Nash Equilibrium

We show that in any PNE all winner agents pay the same price, which implies that all winner agents and the border agent bid the same value (maybe with the exception of the top rank winner agent, who can bid higher). In addition, we show that this price is at most the Market Equilibrium price. Claim 3.1. In any PNE, all winner agents pay the same price p, the border agent pays a price p0 ≤ p, and any loser agent j (if it exists) has value vj ≤ p. In addition, p is at most the market equilibrium price, i.e., p ≤ peq . It seems that one of the critical assumptions in our model is regarding the bids of the loser agents. As in the discussion of market equilibrium price, if we do not assume that agents are conservative, we can force a PNE at the market equilibrium price. (See the discussion there regarding the drawbacks of such an equilibrium).

ˆj (at equilibrium, A rather natural assumption is that loser agents bid truthfully, i.e., bj = vj and Bj = B they will get a utility of at most zero with any bid). The example in Table 2 shows a case where there is no PNE when loser agents are restricted to report their true value and true budget, and is summarized in the following claim. The proof of the claim appears in the Appendix, and uses the notion of critical bid which will be define in the next sub-section. Claim 3.2. If the loser agents are restricted to bid their true value and budget, then there exists a budget auction with no pure Nash equilibrium.

3.2

Critical Bid

A critical bid of an agent tries to capture the point in which an agent is indifferent between being a winner agent and a border agent. Intuitively, when other agents bid low, an agent could prefer the top rank (as it is cheap). Similarly, when other agents bid high, then an agent could prefer the bottom rank, and get the ’leftover’ items at the minimum price. The critical bid models the transition point between these two strategies. Specifically, consider the case when all the agents bid the same value, then a critical bid is the that value, for which an agent is indifferent between being a winner agent and a border agent (receiving the remaining items at minimum price). The critical bid plays an important role in our proof of the existence of a PNE. Definition 3.3. Consider an auction with the set of agents K and a minimum price of pmin . The critical bid for agent i is the bid value x = ci (K, pmin ), such that when all agents participating in the auction bid x, i.e., ~b = (x, ..., x), agent i is indifferent between the top rank (being a winner agent) and the bottom rank (being a border agent). 5

Obviously each agent has potentially a different critical bid. When clear from the context we denote the critical bid of agent j by cj . A function that would be of interest is ϕk (pmin ) = min1≤i≤k {ci (K, pmin )} which is the lowest critical bid among agents in the set K with a minimum price pmin . The following lemma shows that the critical bid is between the agent’s value and minimum price, and characterizes its best response when all the agents bid the same values. Lemma 3.4. Let cj (K, pmin ) be the critical bid of agent j, then: (a) cj (K, pmin ) ∈ [pmin , vj ], and (b) if ~b = (x, ..., x) then for x < cj (K, pmin ) agent j prefers the top rank and for x > cj (K, pmin ) agent j prefers the bottom rank. Proof. Assuming that all agents bid the same value x ∈ [pmin , vj ], let’s look at the utility of agent j as a function of x. Let the function fj (x) be agent j’s utility if he is ranked first, and pays x: ( B N (vj − x) if pmin ≤ x < Nj fj (x) = Bj B if Nj ≤ x ≤ vj x (vj − x) Let the function gj be agent’s j utility if he bids x, ranked last, and pays pmin :  if x < x0   0 P B i gj (x) = (N − i6=xj )(vj − pmin ) if x0 ≤ x < x00   Bj (v − p if x00 ≤ x ≤ vj min ) pmin j P

Bi

P

Bi

j where x0 = i6=Nj , and x00 = N −Bi6=j /p min It is easy to verify the following properties: (i) Both functions are continuous in the range [pmin , vj ]. (ii) Function fj is (strictly) decreasing in x, and gj is (weakly) increasing in x. (iii) fj (pmin ) ≥ gj (pmin ), as in both cases agent j pays price pmin , but its allocation at top rank is equal or higher than its allocation at the bottom rank. (iv) gj (vj ) ≥ fj (vj ) = 0. We conclude that functions fj and gj must intersect in a unique point in the given range, this point is the critical bid cj . In addition fj (x) > gj (x) for x < cj , and fj (x) < gj (x) for x > cj .

The next claim relates the critical bid to the market equilibrium price. This property will be useful in the proofs. Claim 3.5. The critical bid value of any agent is at most the Market Equilibrium Price. We now show a few properties of the agents’ incentives. Those properties will help us to eliminate some of the potential deviations. Claim 3.6. Consider a bid vector ~b = (b1 , . . . , bk ). Then: (a) The top ranked agent, or any winner agent j ∈ K, cannot improve his utility by bidding higher, i.e., b0j > bj , (b) The bottom ranked agent, or any loser agent j ∈ K, cannot improve his utility by bidding lower, i.e., b0j < bj , and (c) If every agent i ∈ K bids bi = cj (agent j critical bid), then agent j cannot improve his utility by changing his bid.

3.3

PNE existence: special cases

In this section we prove the existence of a PNE in a budget auction in two special interesting special cases: only two agents and multiple agents with identical budgets. The proofs of these cases would be latter extended in the next subsection to establish the general theorem, that a PNE exists for budget auction with any number of agents. Two Agents: We start by characterizing the PNE in the simple case of only two agents by constructing a bid vector that is indeed a PNE. Theorem 3.7. Assume that we have two agents with c2 ≤ c1 . In the case that B1 /v2 < N then any bids b1 = b2 ∈ [c2 , min{v2 , c1 }] are a PNE. In the case that B1 /v2 ≥ N then any bids b1 ∈ [v2 , v1 ] and b2 ∈ [pmin , v2 ] are a PNE. Those are the only PNEs where agents submit their true budget and bid at most their value.

6

Proof. We start with the case that B1 /v2 < N . Fix bids b1 = b2 ∈ [c2 , min{v2 , c1 }]. By Lemma 3.4 the critical bids c1 ∈ [pmin , v1 ], and c2 ∈ [pmin , v2 ] so both bids are conservative. Since b1 ≥ c2 , agent 2 prefers the bottom rank. This implies that he would not gain by bidding more. Since agent 2 is at the bottom rank, then by Claim 3.6 he would not gain from bidding less. Since b2 ≤ c1 , agent 1 prefers the top rank. This implies that he would not gain by bidding less. Since agent 1 is the top ranked agent, by Claim 3.6 he does not gain from increasing his bid. Therefore, this is a PNE. Assume that there exists another PNE in this case. Since B1 /v2 < N , agent 2 has a positive utility in the PNE. In order to maximize his utility, he would bid as high as he can, namely, b2 = b1 . Similarly, agent 1 will set b1 = b2 to maximize his utility. Therefore, in any PNE we have b1 = b2 . We can not have b2 = b1 < c2 since then both agents prefer the top rank. We can not have b2 = b1 > c1 since then both agents prefer the bottom rank. By assumption, we can also not have b2 > v2 . Therefore, those are the only PNE. For the case that B1 /v2 ≥ N , we have c2 = v2 , otherwise agent 2 would prefer the top rank at c2 . Since c1 ≥ c2 = v2 agent 1 prefers the top rank, and in any PNE it would bid b1 ≥ v2 . Since both agents bid conservatively, it implies that b1 ∈ [v2 , v1 ] and b2 ∈ [pmin , v2 ]. (Note that for any such PNE agent 2 has zero utility.) Those are all the Nash equilibrium in this case. We remark that at any PNE for this setting, the order of the agents would be the same (due to the order of the critical bids). This is true except when c1 = c2 , where b1 = b2 = c1 = c2 is a PNE, and both agents are indifferent between the bottom and top rank, so the order is not important. Agents with identical budgets: We examine the case where all agents have identical budgets, so agents differ only in their private value vi . We start with a simple claim that limits the underbidding of agents. Claim 3.8. Let j be a loser agent, with budget Bj , and let agent i be ranked above him with budget Bi ≤ Bj . If agent i under-bids agent j, i.e., b0i = bj −  for some  > 0, then agent i becomes a loser agent. P Proof. Agent j is a loser agent, so l6=j xl = N . Now, let agent i underbid j, i.e., b0i = bj − , and all other agents keep their bid, i.e., b0−i = b−i . All agents above j pay in ~b0 price lower or equal than in ~b, i.e., for every l 6= i, j, p0l ≤ pl , and p0j = b0i = bj −  < pi . Since Bi ≤ Bj then the aggregated budget of all agents that where ranked above agent j in ~b is lower than the aggregated budget of all agents that are above i in ~b0 . Since P agent P 0 the aggregated budget is higher and the price each agent pays is lower or equal, then l6=i xl ≥ l6=j xl = N , and therefore agent i is a loser agent in ~b0 . We now prove the following theorem. Theorem 3.9. There exists a PNE for any number of agents with identical budgets, where agents submit their ˆi ), bid bi ∈ [pmin , vi ], and loser agents bid bi = vi . true budget (Bi = B Proof. We prove the theorem by induction over the number of agents. By Theorem 3.7 there exists a PNE for two agents, which establishes the base of the induction. For the induction hypothesis, we assume there is a PNE for k − 1 agents with identical budgets and prove there is a PNE for k agents. Let agent j have the lowest critical bid, i.e., cj = ϕk (pmin ). By Lemma 3.4 we have cj ∈ [pmin , vj ]. We first consider whether ~b = (cj , . . . , cj ) is a PNE when agent j is at the bottom rank. From Claim 3.6 we know that in ~b agent j cannot improve his utility. For the other agents, there are two cases, depending on whether agent j is a loser or a border agent. If j is the border agent (xj (~b) > 0), then the other agents are winner agents. Every winner agent i will not increase his bid (Claim 3.6). Moreover, ci ≥ cj so agent i prefers the top rank over the bottom rank so he will not decrease his bid as well, and his current utility as a winner agent is identical to the one if he was at the top rank. Therefore ~b is a PNE. If j is a loser agent (xj (~b) = 0), then uj = 0. Since the agent j utility from top rank and bottom rank in ~b are equal, then cj = vj (otherwise at top rank, agent j have a non-zero utility). There is no PNE at the range [pmin , vj ), since all agents (including j) in this range prefer the top rank. At this point two interesting things happen: (a) Agent j is a loser agent, and cannot increase his bid more since cj = vj . (b) Since agents have identical budget then according to Claim 3.8 we know that no higher ranked agent will underbid agent j, since he is a loser agent in ~b. From Claim 3.5 we know that there cannot be a critical bid above the Market 7

Equilibrium Price peq . Therefore, for every agent i, max{x ∈ D(ci )} ≥ N which implies that for all possible critical bids agent j will indeed stay a loser agent. Therefore, we can set agent j bid to be bj = vj . The remaining agents k − 1 define a new budget auction with a new minimum price pmin = vj (since by Claim 3.8 no agent will underbid vj ) and by the inductive hypothesis, for this new budget auction, there exist a PNE. The fact that agents have identical budgets prevented winner agents from underbidding loser agents (Claim 3.8). This allowed us to assume that loser agents bid truthfully both their budget and value. On the other hand, for the general case of different budgets, if we restrict loser agents to bid their true value and budget, then there are examples where a PNE does not exists (Claim 3.2). The main idea for the general case would be to let the loser agents submit their true budget, and to bid conservatively, i.e., any bid below their value.

3.4

PNE existence: General case

In this subsection we prove that every budget auction with any number of agents has a Pure Nash Equilibrium where the agents submit their true budget and bid at most their value. Assume that, v1 ≥ v2 ≥ ... ≥ vk . For h ≤ k, let Sh = {1, 2, ..., h} and Sk = K. The following claim shows that if there is a critical bid which is lower than the value of all h agents, then there is a PNE. Claim 3.10. If the lowest critical bid is lower than the value of any agent, i.e., cj = ϕh (pmin ) < vh , then ~b = (cj , ..., cj ) is a PNE for Sh , where agent j is the border agent and other agents are winner agents.5 The following claim shows that by modifying the minimum price we can modify the price p that winner agents pay. Claim 3.11. Let ~b1 be a PNE for the set Sh of h agents and minimum price pmin , such that all winner agents pay price p < vh , the border agent pays pmin , and there are no loser agents. Then for every p∗ ∈ [p, vh ] there exists a minimum price p∗min ∈ [pmin , vh ] and an agent j ∈ Sh such that there is a PNE ~b2 in which every agent i 6= j is a winner agent and pays p∗ , agent j is the border agent and pays pj = p∗min , and there are no loser agents. Proof. Let cl = ϕh (pmin ), meaning that agent l has the lowest critical bid when the minimum price is pmin . Since it is a PNE then cl ≤ p, otherwise no agent prefers the bottom rank at price p, and it cannot be PNE. A critical bid of agent i is when he is indifferent between the top rank (being aPwinner agent with a utility of Bi p ·(vi −p))

and the bottom rank (being border agent with a utility of max[0, N −

j6=i

p

Bj

](vi −pmin )). Therefore P

Bj

agent i’s critical bid can be extracted from the following equation: Bcii · (vi − ci ) = (N − j6=cii )(vi − pmin ). Let fi (y) = ci (Sh , y) be the function that maps a minimum price y to a critical bid for agent i. Namely, P ( j6=i Bj ) · (vi − y) + Bi · vi fi (y) = N · (vi − y) + Bi P Let x := (vi − y) and A := j6=i Bj , then, fi (x)

= =

P

Bi vi − BNi A Ax + Bi vi A = + N x + Bi N N x + Bi A Wi + , N N x + Bi P

Bj

Bj

where Wi = Bi vi − BNi A = Bi (vi − j6=Ni ). Since vi ≥ p ≥ j6=Ni , then Wi ≥ 0, which implies that agent i, the function fi (x) is decreasing in x, and hence the function fi (y) is increasing in y, and in addition the function fi is continuous in the range, [pmin , vh ]. Therefore, the function ϕh (y) = min1≤i≤h {ci (Sh , y)} is also continuous and increasing in y for that range which means that the critical bid increases with the minimum price. Since ch (Sh , vh ) = vh then ϕh (vh ) ≤ vh , but the critical bid cannot be lower than the minimum price so ϕh (vh ) = vh . In addition we know that ϕh (pmin ) = cl . Hence, for every p∗ ∈ [cl , vh ] there exists p∗min ∈ [pmin , vh ] such that ϕh (p∗min ) = p∗ . Therefore, by Claim 3.10 there exists another PNE where all winner agents pay price p∗ , the border agent pay price p∗min , and there are no loser agents, as required. 5 We

assume that agent j slightly underbids cj , and we ignore this small perturbation.

8

The following claim is essentially our inductive step in the proof of the PNE. It shows that we can increase the number of agents in a PNE by one, introducing a new agent with a value lower than the value of any of the previous agents. Claim 3.12. Let ~b1 be a PNE with h agents and a minimum price pmin , such that all winner agents pay price p. If there is a new agent h + 1 such that (a) vh ≥ vh+1 , and (b) For every i ∈ Sh the new critical bid ci (Sh+1 , pmin ) ≥ vh+1 , then we can define a ~b2 which is a PNE for Sh+1 with the same minimum price pmin , where agent h + 1 is a loser agent. Proof. We split the proof into two cases, depending on the price p and vh+1 , the value of agent h + 1. (a) Assume that p ≥ vh+1 . First we show that, given that the agents in Sh keep their bid ~b1 then agent h + 1 cannot gain a positive utility for any bid b ∈ [pmin , vh+1P ]. From the fact that ~b1 is a PNE without agent h + 1 at price p and from Claim 3.1 we know that vi ≥p Bi /p ≥ N (since p ≤ peq , then P P 2 vi ≥p Bi /p ≥ vi ≥peq Bi /peq ≥ N ). This means that for any bid bh+1 agent h + 1 utility is zero: If he pays price ph+1 = vh+1 per item then uh+1 = 0, and if he pays price ph+1 < vh+1 then the agents ranked above him (who pay p ≥ vh+1 ) buy all items, and again uh+1 = 0. We now show that if agent h + 1 bids b2h+1 = pmin and the agents in Sh keep their previous bid, i.e., b2−i = b1−i , then ~b2 is a PNE. The utility of the agents in Sh does not change between ~b1 = ~b2 , since agent h + 1 bid equals the minimum price. We already shown that agent h + 1 cannot gain positive utility with any bid in the range [pmin , vh+1 ]. Therefore ~b2 is a PNE. (Note that it is possible that there are loser agents in Sh in the bid vector ~b1 .) (b) Assume that p < vh+1 . Then p < vh , which implies that agent h cannot be a loser agent in ~b1 , and therefore, there are no loser agents at in ~b1 . We show that there exists a PNE ~b2 where agents i ∈ Sh bid b2i = vh+1 and agent h + 1 bids b2h+1 ∈ (pmin , vh+1 ]. In ~b2 all winner agents pay vh+1 , the border agent pays b2h+1 and agent h + 1 is a loser agent. The fact that p < vh+1 implies that originally the lowest critical bid (considering only agents in Sh ) is lower than vh+1 . By Claim 3.11 for some higher minimum price p∗min > pmin the lowest critical bid would be cj = vh+1 = p∗ . So for the h agents in Sh , ~b3 = (p∗ , . . . , p∗ ) is a PNE, where winner agents pay p∗ and minimum price is p∗min . Setting agent h + 1 bid b2h+1 = p∗min , we prove that ~b2 = (vh+1 , ..., vh+1 , p∗min ) is a PNE for Sh+1 with minimum price pmin , by showing that no agent would want to deviate: ~b3 is a PNE for Sh (without agent h + 1) at price • Agent h + 1: Same as in (a) - from the fact P ∗ p = vh+1 and from Claim 3.1 we know that i≤h Bi /p∗ ≥ N , which means that at price p∗ the agents in Sh buy the entire N items. This implies that even if agent h + 1 bids as high as he can, bh+1 = vh+1 , still uh+1 = 0 since if ranked first he pays price ph+1 = vh+1 , and if ranked last he is allocated no items. For bids lower than vh+1 agent h + 1 is ranked last and is a loser agent. • Agents in Sh : Since the additional agent h+1 is a loser agent, there is no change in the utility of the winner agents compared to ~b3 , and since b2h+1 = p∗min the price and utility of the border agent did not change either. The only possible change is that the agents in Sh can now underbid b2h+1 = p∗min as it is no longer the minimum price of the auction (the minimum price is pmin < b2h+1 ). We show that the agents in Sh cannot improve their utility by such underbidding. Agent’s i ∈ Sh critical bid ci ≥ vh+1 , so agent i weakly prefers the top rank over the bottom rank when all other agents pay price vh+1 . If agent i underbids b2h+1 then agent h + 1 and the agent above him will pay a price lower or equal to b2h+1 < vh+1 , and the rest will keep paying price vh+1 . Therefore, underbidding the agent i has a smaller allocation than taking the bottom rank at price vh+1 , since agent i pays pmin in both cases. Therefore, underbidding agent h + 1 reduces agent i utility. Since neither agent h + 1 nor the agents in Sh can improve their utility by deviating, ~b2 is a PNE.

9

The following is our main theorem, regarding the existence of a PNE in a budget auction. It shows that when all the agents bid conservatively, there is always a PNE. ˆi ) Theorem 3.13. There exists a PNE for any number of agents, where agents submit their true budget (Bi = B and bid at most their value (bi ≤ vi ). Proof. The proof is by induction on the number of agents. By Theorem 3.7 there exist a PNE when there are only two agents. For the induction hypothesis, we assume there is a PNE for h agents with minimum price pmin and prove there is a PNE for h + 1 agents with minimum price pmin . Let cj = ϕh+1 (pmin ), so agent j has the lowest critical bid. We consider two cases: 1. If cj < vh+1 ≤ vj , then by Claim 3.10, ~b = (cj , ..., cj ) is a PNE. 2. If cj = vh+1 , then ch+1 = vh+1 . So, we can ’take-out’ agent h + 1 (with the lowest value), and for the auction with the agents in Sh , according to our induction hypothesis, there is a PNE with minimum price pmin . Since we have: (a) vh+1 ≤ vh , and (b) for each i ∈ Sh , we have ci (Sh+1 , pmin ) ≥ vh+1 , then by Claim 3.12 we can add back agent h + 1, to an auction with minimum price pmin , and there will be a PNE for all agents in Sh+1 .

4

Repeated Budget Auction

In this section we analyze the dynamics of the budget auction when it is played multiple times with myopic agents. Our goal is to exhibit simple dynamics that converge to an equilibrium in this repeated setting. We are able to show the convergence of two agents, under rather general assumptions, and the convergence of multiple agents with identical budget, under more restrictive assumptions (the most important is that the agents start with low bids).

4.1

Bidding Strategies and Dynamics

We first outline our assumption regarding the way the agents select their budget and bids. For a one-shot budget auction reporting the true budget is a dominant strategy (Claim 2.2), so we assume agents always report their true budget (although, technically, in the repeated auction setting it is not a dominant strategy anymore). Since even for a one-shot budget auction, bidding true value is not a dominant strategy (Claim 2.2), we should definitely observe agents bidding differently than their value. We assume that bids are from a discrete set, namely bti =  · ` for some integer `. Best Response: We assume that agents are myopic, and when modifying their bid, they are performing a best response to the other agents’ bids. Since there could be many bids which are best response, we specify a unique bid that is selected as BRi , as follows. Let the BRSi (~b−i ) be the set of (discrete) bids that maximizes agent’s i utility given the bids ~b−i of other agents. Let x = l ·  = min{BRSi (~b−i )}. (This implies that for every y = l0 ·  < x, we have ui (~b−i , y) < ui (~b−i , x), and for every y = l0 ·  > x, we have ui (~b−i , y) ≤ ui (~b−i , x).) Let ( x if ui (~b−i , x) > 0 ~ BRi (b−i ) = vi if ui (~b−i , x) = 0 Note that an agent for which any best response yields zero utility, bids his true value. Since the bids are discrete, we need to redefine the critical bid notion. Definition 4.1. Let ~bx−i be the bid vector such that every agent j ∈ {1, . . . , i − 1} bids bj = x = l · , and every agent j ∈ {i + 1, . . . , k} bids bj = x +  = (l + 1). The discrete value x is agent’s i critical bid if: (a) ~ x− ui (~bx−i , x) > ui (~bx−i , x + ), i.e., agent i prefers the bottom rank in ~bx−i , and (b) ui (~bx− −i , x − ) < ui (b−i , x), i.e., agent i prefers the top rank in ~bx− −i . The following claim relates the critical (discrete) bid with the agent preferences.

10

Claim 4.2. Let x be the critical bid of agent i. Then: (a) If all agents bid at least as high as in ~bx−i , then agent i prefers the bottom over the top rank, and (b) If all agents bid lower than what they did in ~bx−i , then agent i prefers the top over the bottom rank. Dynamics: After each daily auction we compute for each agent its best response. If all the agents are performing a best response, the dynamics terminates (in a PNE). Otherwise, a single agent, which is not playing best response, is selected by a centralized Scheduler, and changes his bid using the specific Best Response we described. We use the following notation: bti is the bid of agent i at day t. It is important to note that: (i) Budget ˆi each day, (ii) Agents have full information: restriction is daily - meaning that agent i can spend up to B they know the number of items (N ) the minimum price (pmin ), and after each day they observe the bids (~b) ~ prices (~ budgets (B) p) and allocations (~x) of the previous days. Nevertheless, each agent i true value vi and ˆi are private information. real budget B Scheduler: We model the dynamics as an Elementary Stepwise System (ESS) [23] with a scheduler. The scheduler, after each daily auction, selects a single agent that changes his bid to his best response. We considered the following schedulers: (i) Lowest First - From the set of agents that are not doing best response, the lowest ranked agent is selected. (Intuitively, this mechanism prioritize loser agents over border and winner agents.) (ii) Round Robin - Selects agents by order of index in a cyclic fashion. (iii) Arbitrary scheduler Selects arbitrarily from the set of agents that are not doing best response.

4.2

Convergence

In this section we study the converges of the repeated budget auction to a PNE. We start with two agents, and generalize it to any number of agents with identical budgets. Theorem 4.3. For a repeated budget auction with two agents and discrete bids bti = l ·  ∈ [pmin , vi ], the ESS dynamics with any scheduler and any starting bids converges to a PNE. Proof. For two agents, there is no difference between different schedulers, since no scheduler can select the same agent twice. Therefore, any scheduler alternates between scheduling the two agents until a PNE is reached. We prove the case that c2 ≤ c1 , the other case is similar and the proof is omitted (the other case is not identical since the index of an agent has influence on the tie breaking rule). Let ~b1 = (b11 , b12 ) be the bid vector at the first day, and let agent 1 be the first to move. Notice that the best response of an agent does not depend on his own bid, but rather on the bid of the other agent. We split the proof to three cases, based on agent 2 first bid, b12 : 1. b12 < c2 ≤ c1 . In this case the bids will increase until they reach a PNE, as follows. Since b12 < c1 , by Claim 4.2 agent 1 prefers the top rank which implies that at time t = 2, b21 = BR1 (b12 ) = b12 (equal bids ranks agent 1 at the top). At time t = 3, since b21 < c2 agent 2 also prefers the top rank, so b32 = BR2 (b21 ) = b21 +  (agent 2 needs a strictly higher bid to get the top rank). Both agents will continue to increase their bids till at time t, such that bt1 = c2 . By Definition 4.1, agent 2 will prefer the bottom rank and bt+1 = BR2 (bt1 ) = bt1 = c2 . Since agent 1 still prefers the top rank (even if c1 = c2 ) 2 t+2 t+1 then b1 = BR2 (b2 ) = bt+1 = c2 , and we reached a PNE since both agents best response is to keep 2 their bid. 2. c2 ≤ b12 ≤ c1 . In this case the bids reach a PNE in one move, as follows. Since b12 ≤ c1 then by Claim 4.2, agent 1 prefers the top rank, so b21 = BR1 (b12 ) = b12 . Since b21 ≥ c2 then by Claim 4.2, agent 2 prefers the bottom rank, so b32 = BR2 (b21 ) = b21 = b12 . We reached a PNE since both agents best response is to keep their bid. 3. c2 ≤ c1 < b12 . In this case bids will decrease until they reach PNE, as follows. Since b12 > c1 then by Claim 4.2, agent 1 prefers the bottom rank, so b21 = BR1 (b12 ) = b12 − . Since b21 = b12 −  ≥ c2 then agent 2 also prefers the bottom rank, so b32 = BR2 (b21 ) = b21 = b22 − . Both agents will continue to decrease their bid till at time t, such that bt2 = c1 < c1 + . By Definition 4.1, agent 1 will prefer the top rank, so bt+1 = BR1 (bt2 ) = bt2 = c1 . Since bt+1 ≥ c2 then agent 2 still prefers the 1 1 11

t+1 bottom rank, and bt+2 = BR2 (bt+1 = c1 . We reached a PNE since both agents best response is 2 1 ) = b1 to keep their bid.

Next we prove that a repeated budget auction with any number of agents, with identical budgets and different values, converge to a PNE. However, for our proof we need to make sure that no two critical bids are equal. Definition 4.4. Agents critical bids are -Separated, if for any agents i, j ∈ K, and any minimum price pmin , we have |ci (K, pmin ) − cj (K, pmin )| > . We assume that the aggregated demand at minimum price exceed the supply N . (Otherwise, it is an uninteresting case where all critical bids equal pmin and this is a PNE.) For agents with identical budgets, B, this implies that Bk > N pmin . The following claim shows that if the values are separated then the critical bids would be separated. Claim 4.5. Assume that Bk > N pmin . For every  > 0 there exists δ() > 0 such that for any two agents i, j ∈ K if |vi − vj | > δ() then |ci − cj | > . We can now state the convergence theorem for agents with identical budgets. Theorem 4.6. A repeated budget auction with any number of agents with identical budgets and different values, with starting bids of pmin , the ESS dynamics with Lowest First scheduler, converges to a PNE. Proof. Let agent j have the lowest critical bid cj ∈ [pmin , vj ]. The proof shows that every day the lowest ranked agent increases his bid by  until cj is reached. At this point if agent j is the border agent we claim that a PNE is reached, and if agent j is a loser agent, then the remaining agents continue increasing their bids until the next critical value is reached, and so on. We prove the theorem by induction over the number of agents. According to Theorem 4.3 an auction with only two agents converges to a PNE (regardless of their starting bid). For the induction hypothesis, we assume that an auction with k − 1 agents converges to a PNE after a finite number of days, and prove it converges for k agents as well. Assume that cj > pmin (we handle the case cj = pmin later). At time t = 1, ~b1 = (pmin , . . . , pmin ), and agent k is ranked last, so he is chosen by the scheduler to move. Since pmin < cj ≤ ck , by Claim 4.2, agent k prefers the top rank, and b2k = BRk (~b1−k ) = pmin +  (which ranks him at the top). At time t = 2, ~b2 = (pmin , . . . , pmin , pmin + ), agent k − 1 is ranked at the bottom, and agent k is ranked at the top. Since pmin < cj ≤ ck−1 then b3k−1 = BRk−1 (~b2−k ) = pmin +  (which ranks him at the top). For the same reason, the rest of the agents (in the following order k − 2, k − 3, ..., 1) will increase their bid by  when they are ranked last, and this will rank them at the top. Therefore, after k days ~bk = (pmin + , . . . , pmin + ). This process will continue until at some day t, agents 1, . . . , j bid cj , while agents j + 1, . . . , k bid cj + , so let ~bcj be the matching bid vector (see also Definition 4.1). Note that for the case pmin = cj this happens when t = k − j, before agents 1, . . . , j have increased their bid for the first time. At this point by Definition 4.1, agent j prefers the bottom rank. Since agent j is already ranked at the bottom he is playing best response and the scheduler will not choose him. We now consider the best response of every other agent i 6= j. There are now two cases: • If agent j is the border agent, then the rest are all winner agents. By Claim 3.6, increasing their bid is not best response. On the other hand, since every agent i 6= j, ci > cj then by Claim 4.2 in the bid vector ~bcj every agent i prefers the top rank over the bottom rank, so he will not decrease his bid either. We conclude that every agent i best response is to keep his bid. Since we know that the border agent j keeps his bid as well, it is a PNE. • If agent j is a loser agent then vj < cj +  (otherwise he would have preferred the top rank). Agent j will remain at last rank and keep his bid bj = cj . Since all agents have identical budgets then according to Claim 3.8 no border or winner agents will under-bid cj . Therefore we can consider agent j to be ’out of the game’. The remaining agents define a repeated budget auction with k − 1 agents and a minimum price of pmin = cj . For this setting, according to our inductive hypothesis, the auction converges to a PNE. 12

(a) starting bid is pmin

(b) starting bid is vi

(c) Random starting bid

Figure 2: A repeated budget auction with the 4 agents as described at Table 1. These simulation we used the ’Round Robin’ scheduler and in each simulation agents had a different starting bid. All simulations converge to a PNE, in which agents A and C are winner agents, agent B is the border agent, and agent D is a loser agent. Note that there is a small difference in the starting price. At this point not necessarily all agents starting price is the new minimum price: agents 1, . . . , j − 1 bid pmin = cj , while agents j = 1, . . . , k bid pmin +  = cj + . We can now recalculate the new critical bids for the remaining agents, taking under consideration that agent j is not participating, and the new minimum price (vj ). From Claim 4.5 we know the new lowest critical bid is at least  higher than cj , so agents 1, . . . , j − 1 increase their bid to pmin +  when it is their turn to move, and all ’starting prices’ are equal again.

4.3

Simulations

This section shows simulations of dynamics in budget auctions, which can give some intuition about typical bidding patterns of myopic agents. We simulated an ESS dynamics with a Round Robin scheduler and  = 0.01. Our simulations show two bidding patterns: smoothed convergence to an equilibrium and a bidding war cycle. Similar patterns where observed by Asdemir [2] using an infinite horizon alternating move game, for the case of two symmetrical agents (identical budget and value). Convergence: In our theoretical results, we only managed to prove that the repeated budget auction converges under the following restrictions: (i) All agents have equal budgets, (ii) All agents start by bidding the minimum price, and (iii) The Lowest First scheduler is used. In our simulation, however, we noticed that many repeated budget auctions do converge to a PNE even when we relaxed these restrictions. (See Figure 2 for an example of converge to a PNE with different budgets, different starting bids and a Round Robin scheduler). Bidding War: Auctions that do not converge to an equilibrium follow a ’Bidding War Cycle’ pattern as shown in Figure 3. In this pattern some agents out bid each other, and so bids are rising until at some point (when the price is high enough) one of the agents drops his bid, and the other agents follow by dropping their bid as well (just above the previous agent). Later, the same agents continue to out bid each other till they start a new cycle. This pattern was also spotted in real data collected by Edelman and Ostrovsky [14] from Overture search engine which they referred to as ’Sawtooth’ pattern. It is worth mentioning that Overture used a first price auction mechanism, in which the existence of this pattern is less surprising. This bidding war pattern matches our theoretical results that show cases where there is no PNE if the loser agents is restricted to bid truthfully (Claim 3.2).

References [1] N. Andelman and Y. Mansour. Auctions with budget constraints. In SWAT, pages 26–38, 2004. [2] K. Asdemir. Bidding patterns in search engine auctions. In In Second Workshop on Sponsored Search Auctions, ACM Electronic Commerce. Press, 2006. [3] Y. Azar, B. E. Birnbaum, A. R. Karlin, C. Mathieu, and C. T. Nguyen. Improved approximation algorithms for budgeted allocations. In ICALP (1), pages 186–197, 2008.

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Figure 3: A simulation of the repeated budget auction with agents specified in Table 2. Bids follow a Bidding War Cycle pattern. [4] Y. Azar, B. E. Birnbaum, A. R. Karlin, and C. T. Nguyen. On revenue maximization in second-price ad auctions. In ESA, pages 155–166, 2009. [5] M. Babaioff, Y. Sharma, and A. Slivkins. Characterizing truthful multi-armed bandit mechanisms: extended abstract. In ACM Conference on Electronic Commerce, pages 79–88, 2009. [6] S. Bhattacharya, G. Goel, S. Gollapudi, and K. Munagala. Budget constrained auctions with heterogeneous items. In STOC, pages 379–388, 2010. [7] K. Bhawalkar and T. Roughgarden. Welfare guarantees for combinatorial auctions with item bidding. In SODA, 2011. [8] C. Borgs, J. Chayes, N. Immorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budgetconstrained bidders. In EC ’05: Proceedings of the 6th ACM conference on Electronic commerce, pages 44–51, 2005. [9] C. Borgs, N. Immorlica, J. Chayes, and K. Jain. Dynamics of bid optimization in online advertisement auctions. In In Proceedings of the 16th International World Wide Web Conference, pages 13–723, 2007. [10] M. Cary, A. Das, B. Edelman, I. Giotis, K. Heimerl, A. R. Karlin, C. Mathieu, and M. Schwarz. Greedy bidding strategies for keyword auctions. In ACM Conference on Electronic Commerce, pages 262–271, 2007. [11] D. Chakrabarty and G. Goel. On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and gap. SIAM J. Comput., 39(6):2189–2211, 2010. [12] N. R. Devanur and S. M. Kakade. The price of truthfulness for pay-per-click auctions. In ACM Conference on Electronic Commerce, pages 99–106, 2009. [13] S. Dobzinski, N. Nisan, and R. Lavi. Multi-unit auctions with budget limits. In In Proc. of the 49th Annual Symposium on Foundations of Computer Science (FOCS), 2008. [14] B. Edelman and M. Ostrovsky. Strategic bidder behavior in sponsored search auctions. In In Workshop on Sponsored Search Auctions, ACM Electronic Commerce, pages 192–198, 2005. [15] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. American Economic Review, 97, 2005. [16] A. Fiat, S. Leonardi, J. Saia, and P. Sankowski. abs/1001.1686, 2010.

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[19] A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani. Adwords and generalized online matching. J. ACM, 54(5):22, 2007. [20] S. Muthukrishnan. Ad exchanges: Research issues. In WINE ’09: Proceedings of the 5th International Workshop on Internet and Network Economics, pages 1–12, Berlin, Heidelberg, 2009. Springer-Verlag. [21] N. Nisan. Google’s auction for TV ads. In ESA, page 553, 2009. [22] N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, editors. Algorithmic Game Theory. Cambridge, 2007. [23] A. Orda, R. Rom, and N. Shimkin. Competitive routing in multiuser communication networks. IEEE/ACM Trans. Netw., 1(5):510–521, 1993. [24] H. R. Varian. Position auctions. International Journal of Industrial Organization, 2006.

15

A

Omitted Proofs

Proof of Claim 2.2: To show that an agent can gain by bidding lower than its true value, consider the example in Table 1. If agent B would bid its true value then it would receive only 25 items, and still exhaust its budget, lowering its utility to 12.5 (from 50). Therefore bidding the true value is not a dominant strategy. Now we show that reporting the true budget is a dominant strategy. Agent i ∈ K submits its bid bi and ˆi . Therefore the utility depends on the amount budget Bi , and his utility function is xi (vi −pi ) when xi ·pi ≤ B of items he is allocated, xi , his true value, vi , and the price pi he pays. The budgets in this model do not effect the ranking order, and therefore does not effect the price. The budgets do, however, effect the number of items allocated to agent i, i.e, xi . Let x ˆi and u ˆi be the number of items allocated and the utility of agent i, respectively, in the case he reports ˆi . Let xi and ui be the allocation and utility of agent i when he report a differnt budget his true budget B (Bi ) ˆi , and gets allocated xi items, where xi = min(N − Suppose agent i reported a smaller budget Bi < B Pi−1 Pi−1 ˆ ˆi = min(N − j=1 x ˆj , Bi /pi ). All agents in higher ranks than agent i are not effected j=1 xj , Bi /pi ) and x Pi−1 Pi−1 by the budget reported by agent i, so xj = x ˆj for j < i, which implies that N − j=1 xj = N − j=1 x ˆj . ˆ Since Bi /pi < Bi /pi we have that xi ≤ x ˆi which implies ui ≤ u ˆi . ˆi . If xi = x ˆi (which will happen Now suppose agent i reported higher budget Bi > B ˆi then xi · pi ≤ B when i is a loser or border agent) then the agent does not even exhaust his real budget, regardless of the budget he reports, and ui = u ˆi . Otherwise, if xi > x ˆi (which happens when agent i is a winner agent) it ˆi and xi · pi > B ˆi . Therefore, agent i exceeds his budget and his utility, by definition, implies that x ˆi · pi = B is ui = −∞ < u ˆi . We conclude that an agent cannot improve his utility by reporting a different budget than his true budget. Proof of Claim 3.1: If there is only one winner agent the first claim holds trivially. Otherwise, for contradiction, assume there is a PNE with at least two winner agents paying different prices. The first ranked agent pays p1 , and let j be the highest ranked winner agent that pays pj < p1 . In such a case we will show that agent 1 can improve his utility, in contradiction to the assumption that it is a PNE. Pj Since agents 1 to j are all winner agents, then any agent i ≤ j is allocated xi = Bi /pi and i=1 xi ≤ N . Pj Therefore x1 ≤ N − i=2 xi . If the top agent drops down to rank j (by bidding pj − ) he is allocated Pj Pj x01 = min(B1 /pj , N − i=2 xi ). Now, B1 /pj > B1 /p1 = x1 and N − i=2 xi ≥ x1 so x01 ≥ x1 . Since he now pays strictly less per item and receives at least as many items as before, he strictly increases his utility, which contradicts our assumption that this is a PNE. Therefore, in any PNE all winner agents pay the same price. The Border agent is ranked after all winner agents, so he pays price p0 ≤ p for each item. All loser agents receive no items and have zero utility. If there is a loser agent i with value vi > p then he can become a winner agent (by bidding p + ), and gain a positive utility, which contradict the fact that this is a PNE. Therefore, in an equilibrium any loser agents must have value at most p. Regarding the market equilibrium price peq , for contradiction assume that there exists a PNE with p > peq . The utility of a winner agent j paying price p > peq is at least as much P as if he was the border agent and paid price p0 ≤ p since it is a PNE. Therefore, (Bj /p)(vj − p) ≥ (N − i∈S−{j} Bi /p)(vj − p0 ), when P P S = {i : vi ≥ p}. Since (vj −p) ≤ (vj −p0 ), then Bj /p ≥ N − i∈S−{j} Bi /p, but that leads to i∈S Bi /p ≥ N P and i∈S Bi /p ∈ D(p). Since peq is the Market Equilibrium Price and p > peq , we have that max(D(p)) < N , and we have reached a contradiction. Proof of Claim 3.2: Consider the example in Table 2. We will show that in this example there is no PNE when loser agents are restricted to report their true value and true budget. In the example, agent D has the lowest critical bid (cD = vD = 1.0), so there clearly couldn’t be any PNE below this price, and in any PNE, agent D must be a loser agent. According to Claim 3.1 all winner agent must pay the same price. If agent D bids his true values bD = vD = 1.0, then the only possible PNE is when agents A, B and C bid their critical bid cA = cB = cC = c = 1.143. This is due to the fact that for lower price they all prefer the top rank, and for higher price they all prefer the bottom rank. Therefore, the only bid vector that can yield a PNE is ~b = (c, c, c, 1.0). Table 2 shows at column ’bidding at critical bid’ the outcome when agents bid ~b (in which agents A, B, and C, are indifferent between ranks 1, 2 and 3). Nevertheless, any agent i ∈ {A, B, C} can underbid agent D (by bidding 1.0 − ) which improves his utility (shown in column ’underbidding’ in Table 2), so the bids vector ~b is not a PNE. We conclude that in this example there is no 16

PNE where loser agents report their true value. Proof of Claim 3.5: For contradiction, assume that there exists an agent j with critical bid cj > peq . P Therefore, (Bj /cj )(vj − cj ) = (N − i∈S−{j} Bi /cj )(vj − p0 ), when S = {i : vi ≥ cj }, and p0 is the highest P bid among the loser agents. Since (vj − cj ) ≤ (vj − p0 ), then Bj /cj ≥ N − i∈S−{j} Bj /cj , this leads to P i∈S Bj /cj ≥ N , but since cj > peq it contradicts the definition Market Equilibrium Price. B Proof of Claim 3.6: The utility of winner agent j is uj = xj · (vj − pj ) = pjj · (vj − pj ). By increasing his bid he can only increase his price pj , and decrease his allocation xj , and therefore decrease his utility. As for the agent in the top rank (it can be either a winner or border agent) after increasing his bid, his price and allocation do not change. This proves (a). The bottom ranked agent k ∈ K, if he is a border agent, then the agent ranked above him is a winner agent. To improve his utility, uk , agent k must increase his allocation (his price can not decrease, since he is already paying the minimum price pk = pmin ). Decreasing bk will decrease the price the agent ranked above him, and would increase the allocation of that agent. This could only decrease the allocation of the bottom agent. For any loser agent the claim is trivial, as he is not allocated any items, and by lowering his bid it will remain a loser agent. This proves (b). By definition when ~b = (cj , ..., cj ) agent j is indifferent between being ranked at the top or bottom. According to (a) when ranked at the top he cannot improve his utility by bidding higher, and according to (b) when ranked at the bottom, he cannot improve his utility by bidding lower. Therefore, agent j cannot improve his utility. This proves (c). Proof of Claim 3.10: According to Claim 3.6 agent j cannot improve his utility, and he is indifferent between the bottom and top rank. Since cj < vh and vh ≤ vj , then cj < vj so when agent j is ranked in ~b at the top he pays pj = cj < vj , hence he has positive utility. This means that at bottom rank in ~b he also has positive utility, which implies that all other agents (i.e., Sh − {j}) are winners agents. Each winner agent i 6= j has ci ≥ cj so they all weakly prefer the top rank over the bottom rank, and cannot improve their utility by bidding less than cj . On the other hand, according to Claim 3.6, they cannot improve their utility by bidding more than cj . Therefore ~b is a PNE. Proof of Claim 4.2: (a) If all agents bid exactly as in ~bx−i , then by Definition 4.1 the claim holds. If some agents bid higher, then agent i utility from the top rank strictly decreases (as the price is higher). In addition agent i utility from the bottom rank is at least as high as before since all winner agents pay equal or higher prices, so their allocation is lower or equal, which implies that agent i allocation at the bottom rank is at least as high. Therefore, agent i prefers the bottom rank over the top rank. x− (b) By Definition 4.1, for the bid vecotr ~b−i the claim holds. If some agents bid less than x − , then there are agents who pay less than before, and are allocated more items. This implies that agent i allocation at the bottom rank decreases, and so does his utility (as price stays pmin ). In addition, if the highest bid is now lower than x −  then agent i utility from the top rank increases. In conclusion, agent i prefers the top rank over the bottom rank, as required. Proof of Claim 4.5: Agent i critical bid is when his utility from top rank equals his utility from bottom rank, so for the case of identical budgets we get: B (vi − ci ) ci

=

ci

=

B(k − 1) )(vi − pmin ) ci kBvi − B(k − 1)pmin . N vi − N pmin + B

(N −

The critical bid of agent i depends on his value, vi , and other parameters that are common to all agents. We define f (x) to be the function that maps an agent value x to his critical bid. f (x)

=

f 0 (x)

=

kBx − B(k − 1)pmin N x − N pmin + B B 2 k − BN pmin (N x − N pmin + B)2

17

=

B(Bk − N pmin ) . (N x − N pmin + B)2

Since we assume that at minimum price all items are sold (Bk − N pmin > 0) then f 0 (x) > 0. Moreover, since x > pmin then N x − N pmin + B > 0, which implies that f 0 (x) is decreasing in x. Let vmax be the highest pmin ) 1 value among all agents, so for every x ∈ [pmin , vmax ], f 0 (x) ≥ (N vB(Bk−N 2 = α. We define δ() = α , max −N pmin +B) α so |vi − vj | > δ() implies that |ci − cj | ≥ |vi − vj |α > δ()α = α  = .

18