On-demand or Spot? Selling the cloud to risk-averse customers

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1 On-demand or Spot? Selling the cloud to rik-avere cutomer Darrell Hoy Nicole Immorlica Brendan Lucier July 29, 2016 Abtract In Amazon EC2, cloud reource are old through a combination of an on-demand market, in which cutomer buy reource at a fixed price, and a pot market, in which cutomer bid for an uncertain upply of exce reource. Standard market environment ugget that an optimal deign ue jut one type of market. We how the prevalence of a dual market ytem can be explained by heterogeneou rik attitude of cutomer. In our tylized model, we conider unit demand rik-avere bidder. We how the model admit a unique equilibrium, with higher revenue and higher welfare than uing only pot market. Furthermore, a rik averion increae, the uage of the on-demand market increae. We conclude that rik attitude are an important factor in cloud reource allocation and hould be incorporated into model of cloud market. 1 Introduction Cloud computing allow client to rent computing reource over the internet to perform a variety of computing tak, from hoting imple web erver to computing complex financial model. By offloading thee tak to the cloud, client avoid the neceity of procuring and maintaining expenive erver and infratructure. The current market leader in thi indutry i Amazon who launched it cloud platform, Amazon Elatic Compute Unit (EC2), in Amazon ue it cloud internally for many of it own computation. Additionally, Amazon contract with large client who reerve intance of cloud reource for long uage period. Due to natural variation in the nature of computing tak from Amazon and it large client, EC2 ha a varying amount of leftover computing reource. Amazon ell thee reource to mall client. Thi lead to a natural quetion: how hould a cloud provider price it reource to thee mall client? The pricing model adopted by Amazon ha two main component: an on-demand market and a pot market. In the on-demand market, client may buy an intance of cloud reource at a fixed reervation price. 1 After reource have been allocated internally, to large client, and to client in the on-demand market, extra upply might till remain. Thi upply i old in the pot market. In a pot market, client place bid for intance, and a price i et o that the available upply equal the total demand at that price. Univerity of Maryland. Part of thi work wa completed while D. Hoy wa an intern at Microoft Reearch. Microoft Reearch Microoft Reearch 1 Thi might more naturally be called a reervation market and we witch to thi terminology in the remainder of the paper; however we tick to the term on-demand for the current dicuion a thi i the term ued by Amazon. 1

2 Viewed through the len of microeconomic theory, the peritence of thi dual market i a curioity at firt glance. Indeed, in tandard economic environment, a rik-neutral, expectedutility-maximizing client who deire a cloud reource hould imply buy it in whichever market i expected to have the lower price typically the pot market. Thi ugget all ale hould happen in the pot market, leaving the on-demand market defunct. That thi i not reflective of reality tem from everal factor. Mot apparent i that client are rarely rik-neutral. For example, it i eay to imagine that a company would have a oft budget et aide for computational cot. They would then pend freely within the confine of thi budget, and extend the budget cautiouly when neceary to meet their computing need. Thi type of behavior ugget a tendency toward rik averion on the part of the client. A the budget i freely available, client might prefer to overpend to guarantee the required reource at the on-demand price. We how in a tylized etting that the preence of heterogenou rik attitude can explain the prevalence of a combined on-demand and pot market. Specifically, thi dual market induce a unique equilibrium in which more rik-avere cutomer (e.g., thoe with higher budget) buy reource in the on-demand market and the other bid in the pot market. We how that thi equilibrium outcome outperform the outcome achieved by running only one of the two type of market on it own in many key objective. 1.1 Reult and Technique In order to highlight the impact of rik averion on the market, we focu our analyi on a imple etting in which there i only one type of computational reource being old (e.g., a erver with one core for one hour), and each buyer demand only a ingle intance of thi cloud reource at any given time 2. Formally, we aume a continuum of buyer, where the type of a buyer conit of a value for an intance and a utility curve that map outcome (i.e., allocation and price paid) to payoff. A alluded to above, the utility curve decribe a buyer attitude toward rik: for example, a buyer with a oft budget (a decribed above) would likely prefer to pend all of their budget all the time than to pend twice their budget half the time, and thi preference i captured by a non-linear utility curve. The buyer type are decribed by a joint common prior. We aume the market i large; i.e., no ingle buyer ha ignificant impact on the market outcome. The market work a follow. Firt, the eller et a price for on-demand intance. Thi price hould be high enough to guarantee that upply exceed demand, motivated by the fact that reource are alway available for purchae in on-demand market in practice. Buyer then realize their type (i.e., value/budget pair) and chooe whether to buy in the on-demand market. After thee deciion have been made, the unold upply receive an exogenou hock, modeling variation in the demand of large client. Any remaining upply i then old to the remaining buyer at a market-clearing price. We prove that thi ytem ha a unique (ubgame-perfect) equilibrium for each choice of the ondemand market price. We do thi by analyzing the relationhip between the pot price ditribution and the ditribution of client type and correponding upply and demand. It turn out that the 2 Of coure, thi model abtract away from many reaonable ource of rik averion in the cloud, uch a client with diminihing marginal return for multiple intance, or the cot of prematurely terminating a long-running tak. Even ignoring thee factor, our model till generate heterogeneou preference toward on-demand veru pot pricing. 2

3 ditribution over pot price up to a certain value v depend only on the ditribution of client type in the range [0, v], and hence one can explicitly olve for the price ditribution recurively. Thi equilibrium atifie a monotonicity property: agent that are more rik-avere are more likely to purchae in the on-demand market, wherea agent that are le rik-avere are more likely to ue the pot market. Furthermore, a the ditribution hift uch that agent become more avere to rik (in the ene of firt-order tochatic dominance), we how more client end up buying intance in the on-demand market and the revenue correpondingly increae. Thi reult i perhap intuitive, but it i not obviou: a client become more avere to rik, they hift toward the on-demand market and hence both decreae upply and demand in the pot market. Thi in turn could caue the pot price to hift either up or down, which would impact purchaing deciion of all client. By further arguing about the equilibrium of the market, we how the hift toward the on-demand market in fact caue the pot price to increae thereby reinforcing the hift toward the on-demand market. Therefore, the equilibrium i monotone with uch hift in the value and budget ditribution. Thi further illutrate the connection between the on-demand market and rik attitude. We leverage our equilibrium characterization to compare the dual market outcome to the outcome of a pot-only or on-demand-only market. We are intereted in the welfare, efficiency, and revenue propertie of thee market. The revenue of a market outcome i imply the um of the payment, and i equal to the cloud provider utility. The welfare of an outcome i the total utility of all market participant including the cloud provider. The efficiency i the total value of the cloud client, ignoring payment. In rik neutral environment, the welfare and efficiency are equal, but with rik-avere client the welfare can be le than the efficiency. It i eay to ee that a pot-only market i more efficient than a dual market, which in turn i more efficient than an on-demand-only market. Thi i becaue the pot market preciely generate the efficient outcome, even with exogenou upply uncertainty: in a pot market, allocation i alway monotone in value and o it i never the cae that a lower-valued client i erved at the expene of a higher-valued one. Surpriingly, thee efficiency comparion do not extend to welfare. A we how, the welfare of the dual market i better than the welfare of the pot market alone, regardle of the price et for the on-demand intance. In particular, thi i true even when the on-demand market price i et to maximize the revenue of the cloud provider. Thi i not trivial: the on-demand market add inefficiency, ince client with high value but low averion to rik may not wih to purchae on-demand, wherea client with lower value but higher rik-averion might. Thi lead to circumtance where lower-valued client win but higher-valued client loe. However, ince the client that are winning in thi cenario are actually more rik-avere, the tranfer of payment to the cloud provider increae welfare. We how that the welfare increae due to additional tranfer from rik-avere client offet any inefficiencie in the allocation. Moreover, ince thi welfare comparion hold at every etting of the on-demand price, it applie in particular to the price that maximize revenue. We how that thi price mut alo generate more revenue than a pot-only market, leading to a imultaneou increae of both welfare and revenue. In ummary, a dual pot/on-demand market imultaneouly improve both the revenue and welfare of a pot-only market. We alo how by example that while an on-demand market i revenue-optimal for rik-neutral buyer, a dual market can generate trictly higher revenue when buyer are rik-avere. Furthermore, while a dual market may ometime generate le revenue than an on-demand-only market, a dual market i alway more efficient. Thi ugget that a cloud 3

4 provider, epecially one that hold a dominant poition in the marketplace, might prefer a dual market ytem. Thi phenomenon i driven by heterogeneou rik attitude that arie naturally in the context of cloud computation, leading u to poit that rik averion i an important element to conider when one model the cloud marketplace. 1.2 Related Work A number of paper explore cloud-computing market deign. Zhang et al. [2013] conider deigning a truthful auction where uncertainty lie in the arrival of demand and value profile of bidder, whether they have a large job with deadline or general demand over time. An et al. [2010] deign a negotiation-baed mechanim for etting price contract in the preence of demand uncertainty. Borg et al. [2014] conider the pricing problem faced by a eller etting on-demand price over multiple time period and uncertain upply, with agent who arrive and have different deadline for their tak. The paradigm of a dual pot+reerve mechanim ha alo received a lot of attention. Wang et al. [2012] ue a Markov deciion proce to model the deigner choice of how to partition upply between the reerve and pot market. Abhihek et al. [2012] model the cloud market a a queuing model, in which a continuum of job arrive and have (private) waiting cot. They find that a fixed cot model provide greater expected revenue than a pot market. Additionally, recent work [Menache et al., 2014; Ma and Huang, 2012] have focued on the problem faced by bidder in uch a market: when to ue the pot market and when to reerve. Ben-Yehuda et al. [2013] analyze the expected pot price in comparion to their reervation price, and find that it i very likely that Amazon i intentionally manipulating the price or upply ditribution o a to provide uer with more uncertainty in the pot market. In all of the model decribed above, agent are rik-neutral and do not have budget. A far a we are aware, our work i the firt to ue budget heterogeneity, or rik-averion more generally, to explain the prevalence of a pot+reerve market. Auction for cloud computation reource hare imilaritie to electricity market, where the plit between a pot market and a o-called future market i common. Indeed, the ue of both market ha been advocated to account for rik-avere buyer and eller (ee e.g., Auubel and Cramton [2010]). One difference i that, unlike the on-demand market for cloud computation, future market for electricity are typically reolved year in advance. While mot work in auction theory aume rik-neutral agent, ome work ha been done for auction with rik-avere bidder. Optimal auction have been characterized for imple etting [Makin and Riley, 1984; Matthew, 1983], but the olution are generally not expreible in cloed form. It i therefore more common to tudy the imple auction ued in practice, with the general finding that econd-price or pot-like auction do poorly for revenue when compared with firt-price auction [Riley and Samuelon, 1981; Hu et al., 2010; Fu et al., 2013]. Matthew [1987] ha looked at the preference of bidder in the auction, and howed that firt-price auction not only can get more revenue than the econd-price auction, but alo can be preferred by bidder due to reduction in uncertainty around the payment. Our model of rik-averion a the preence of a oft budget i cloely related to the capacitated utilitie model of Fu et al. [2013], where the capacity in their model correpond to value minu budget in our model. They how that with capacitated agent, a imple firt-price auction with reerve ha revenue that approximate the revenue of the optimal mechanim. Our reult are of a imilar flavor to the ebay-tyle buy-it-now auction conidered by Mathew and Katzman [2006]. A in our model, they find that adding a buy-it-now option increae revenue, and a agent become more rik-avere, the optimal price increae. Their model differ from our 4

5 in that they conider an explicitly randomized allocation rule, rather than clearing the market at a pot price, in order to incentivize ue of the buy-it-now (i.e., reervation) option. 2 Preliminarie In our model, a ingle cloud provider (the eller) i elling compute reource to a continuum of client (the bidder). Utility tructure Each bidder i ha value v i [0, 1] for a ingle compute intance. A i tandard in the economic literature, we model the rik attitude of bidder i through a utility function u i : R 0 R 0. If the bidder obtain an intance and pay p, then her utility i u i (v i p). We will aume that u i (0) = 0, that u i i continuou and non-decreaing, and that u i i not identically 0. Note that ince u i (0) = 0, we can think of v i a the maximum price at which bidder i i willing to purchae an intance. A bidder that doe not obtain an intance will pay nothing and have utility 0. We will focu our attention on agent that are rik avere. That i, we will aume that utility curve are weakly concave, a i tandard when modeling rik averion. Note that we allow u i to be linear, in which cae bidder i i aid to be rik-neutral. Roughly peaking, an agent with a utility curve that i more concave will be more rik avere, in the ene that they are more likely to prefer guaranteed outcome to uncertain lotterie. More formally, we ay that utility function u i more rik avere than u, and write u u, if for every ditribution L over non-negative real value and every fixed value d 0, if E x L [u(x)] u(d), then E x L [u (x)] u (d). In other word, if an agent with utility curve u prefer a lottery L over a guaranteed payout of d, then an agent with utility curve u would prefer the lottery a well. Thi define a partial order over utility curve. Note that, under thi definition, all (weakly) concave utility curve are (weakly) more rik-avere than a rik-neutral (i.e., linear) curve. Note alo that for twice-differentiable utility curve, u i more rik-avere than u if and only if the tandard Arrow-Pratt meaure of rik-averon i nowhere lower for curve u than for curve u Pratt [1964]. 3 Demand tructure Type are ditributed according to a joint ditribution F on pair (v, u). For eae of expoition, we will aume throughout that F i upported on a finite collection of (v, u) pair. Write V and U for the (finite) et of value and utility curve that upport F, and for (v, u) V U we will write f(v, u) for the probability that an agent ha type (v, u). We will ue F (v) to refer to the induced ditribution over value; that i, F (v) i the probability that an agent value i at mot v. We will aume a large-market condition, which i that the aggregate demand i ditributed exactly according to the type ditribution F. Supply tructure The upply of intance, q, i unknown to the bidder and eller until the intance are to be allocated. The upply i then drawn from a ditribution, Q. We will normalize the upply o that q repreent the fraction of the market that can be imultaneouly erved, hence q [0, 1]. 3 We define rik averion with repect to agent preference directly, rather than via the Arrow-Pratt meaure, to avoid the requirement that utility curve be twice differentiable. 5

6 2.1 Auction Format We will conider three auction format in thi paper: pot auction, reervation auction (previouly referred to a on-demand), and combination (or pot+reervation) auction. Spot (M ) One type of auction to run i a market-clearing auction, or a pot auction. In thi auction, buyer ubmit bid. A market-clearing price p i choen uch that the quantity of bid exceeding p i equal to the upply. Under our unit-demand and large market aumption, it i a dominant trategy for a bidder to ubmit a bid equal to her value; henceforth we aume the bid in the pot market equal the value. We oberve that a market-clearing price alway exit in our market, even in the preence of non-linear utilitie: with available upply q, and ditribution over value F, the market clearing price, written p (q), i preciely 4 the value for which q = 1 F (p (q)). Reervation (M r ) In a reervation-only (or on-demand ) market, the auctioneer et a fixed price p r per intance, in advance of eeing the realization of upply. Price p r need not be a market-clearing price. If there i not enough upply to atify the demand for intance at thi price, then the winning bidder are choen uniformly at random from among thoe who wih to purchae. Spot+Reervation (M +r ) In a pot and reervation market, the auctioneer firt et a fixed price p r and run a reervation auction. The remaining inventory of upply (if any) i then old via a pot auction. The exact timeline of event in the pot and reervation auction M +r i a follow: 1. Auctioneer announce reervation price p r. 2. Bidder realize type (v i, u i ) F (v, u). 3. Each bidder decide whether to purchae an intance in the reervation auction, indicated by a r {0, 1}. Let T = v,u ar (v, u) f(v, u) be the total volume of reerved intance. 4. Auctioneer realize upply q Q, and reerved intance are allocated a in the reervation market decribed above. 5. If q > T, the auctioneer run a market-clearing auction to clear the exce capacity. Let p (q) be the reulting market-clearing pot price. Note that our pecification doe not ak bidder to decide whether or not to participate in the pot market. The fact that bidder are unit demand, and that the pot auction i truthful under our large market aumption, implie that in equilibrium bidder will bid (truthfully) in the pot auction if (and only if) they don t buy an intance in the reervation auction. For a given (implicit) trategy profile for mechanim M +r, we will write S(p ) for the cumulative ditribution function of the reulting pot price. 4 Thi price may not be unique if q = 0 or q = 1. In thee cae we define p (q) to be the upremum of price atifying the written condition, which will be for q = 0. 6

7 Solution concept: ubgame-perfect equilibrium For each of thee auction, the olution concept we apply i ubgame-perfect equilibrium. A trategy profile for a multi-tage game form a ubgame-perfect equilibrium (SPE) if, at every tage t of the game and every poible hitory of action by player in previou tage, no agent can benefit by unilaterally deviating from her precribed trategy from tage t onward. For the pot auction and reervation auction, there i only one tage of the reulting game and hence equilibria are traightforward: each agent chooe to purchae her utility-maximizing quantity of intance given the pecified price. For mechanim M +r, we can characterize the SPE a follow. In the econd (i.e., pot) tage of the mechanim, the equilibrium condition implie that agent alway purchae intance if and only if their value i above the realized pot price. Thu, the only trategic choice to be made by agent i in the firt (i.e., reervation) tage of the mechanim, where each agent mut elect whether to purchae an intance in the reervation market. We will therefore define a trategy profile to be a mapping from a type (v, u) to an action {0, 1}, where (v, u) i interpreted a the number of intance to purchae in the reervation market. Note that the ditribution over market-clearing price in the econd tage i completely determined by the action of agent in the firt tage, and hence i determined by. An equilibrium i then a trategy profile uch that no agent can benefit by unilaterally deviating from trategy (i.e., by reerving more or fewer intance), given the ditribution of pot price implied by. 2.2 Objective We conider three objective when evaluating mechanim: revenue (REV ), welfare (W EL) and efficiency (EF F ). The revenue of a mechanim M, REV (M), i the um of the payment made to the auctioneer. Note that for the pot+reervation mechanim, REV (M +r ) = p r T + E q Q [p (q)(q T )] (1) where we ued the fact that T q with probability 1. The welfare of a mechanim W EL(M) i the um of utilitie of all agent, including the auctioneer (whoe utility i preciely the revenue of the mechanim). The efficiency EF F (M) of a mechanim meaure the value created, without conidering the welfare lot due to the (non-linear) utility function of agent. For the pot+reervation mechanim: EF F (M +r ) = E (vi,u i ) F [v i (v i, u i ) + v i (1 (v i, u i )) S(v i )] If an agent reerve, her value generated i v i. If he doe not reerve, her value generated i v i S(v i ), which i her value time the probability that the pot price i below her value. Note that if agent are rik-neutral (i.e., have the identity function a their utility function), then EF F (M) = W EL(M). 3 Equilibrium Behavior & Analyi In thi ection, we analyze the choice of bidder and ue thi to characterize equilibrium of the pot+reervation market. We begin by noting the relationhip between the pot price ditribution and the ditribution of upply, type, and reervation demand in equilibrium. Recall that Q denote the CDF of the upply ditribution. 7

8 Lemma 1. Fix trategy profile, let S be the ditribution of pot price under, and let T (p) be the volume of reerved intance demanded from agent with value at mot p, under. Then form an equilibrium if and only if, for all p, S(p) = 1 Q (1 F (p) + T (p)), and (2) T (p) = f(v, u) 1 [u(v p r ) E p S [u(max{v p, 0})]]. (3) v V u U v p Proof. The probability that the pot price i at mot p i exactly the probability that the upply i greater than neceary to atify all of the demand for reource from bidder with higher marginal value than p, plu all reervation demand for reource with lower marginal value than p. On the other hand, the volume of reerved reource demanded from agent with value at mot p, at equilibrium, i preciely the probability that uch an agent will prefer the determinitic reervation outcome to the lottery over outcome determined by the ditribution over pot market price. In light of Lemma 1, we will tend to equate equilibria with the reulting ditribution S and T, rather than with an explicit trategy profile. Lemma 2. Purchaing in the reervation tage i monotone in the rik-averion of u i : if a (v i, u i ) bidder (weakly) prefer to reerve an intance, then a (v i, u i ) bidder with u i u i (weakly) prefer reerving. Proof. We begin by conidering the pecial event in which the agent i not allocated an intance even if they reerve, due to the upply being inufficient to honor all reervation and the agent not being elected randomly a a winner. If thi event occur, the bidder utility will necearily be 0, and thi i independent of their utility curve and their choen action (ince, if q < T, no agent that enter the pot market will obtain an intance). It therefore uffice to conider the agent expected utility conditional on the event that the agent will be allocated an intance with certainty if they chooe to reerve. With thi in mind, the utilitie of a unit demand agent from reerving or participating only in the pot market, repectively, are u r (v i, u i ) = u i (v i p r ), u (v i, u i ) = E p S [u i (v i p ) 1 p v i ]. We want to how that if u r (v i, u i ) u (v i, u i ), then we have u r (v i, u i ) u (v i, u i ) a well. Note that, fixing v i, the pot market generate a certain lottery L over value (v i p ), and the reervation market generate a certain value v i p r. Thu, from the definition of rik averion, if an agent with utility curve u i prefer the certain outcome to the lottery L, and u i u i, then an agent with utility curve u i prefer the certain outcome a well. 3.1 Equilibrium Exitence and Uniquene We are now ready to etablih uniquene of equilibrium. One ubtlety about equilibrium uniquene i the manner in which buyer break tie. If a poitive ma of agent i indifferent between the pot and reervation market, there may be multiple market outcome conitent with thoe preference. We will therefore fix ome arbitrary manner in which bidder break tie, which could be randomized and heterogeneou acro bidder. Our claim i that for any uch tie-breaking rule, the reulting equilibrium will be unique. 8

9 Lemma 3. There i a unique equilibrium of M +r. Moreover, thi equilibrium can be computed in time proportional to the ize of the upport of type ditribution f. Proof. A hown in Lemma 1, the challenge of characterizing equilibrium eentially reduce to characterizing the fraction of bidder who reerve at a given price, T (p). Thi i becaue T determine the ditribution S over pot price, and S (together with an arbitrary tie-breaking rule) uniquely determine the trategy profile, ince thi can be inferred from the expected utility when chooing the pot market. Thu, to how uniquene and exitence of equilibrium, it uffice to how uniquene of the function S and T. We will prove that, for all v V, T (v) and S(v) are uniquely determined by the function T and S retricted to value le than v. The reult will then follow by induction on the element of V. Conider firt an agent with value v = min V. Recall that the pot price i alway at leat v. Thu, if p r < v then the agent will alway reerve, if p r > v then the agent will alway chooe the pot market, and if p r = v the agent will be indifferent and apply the fixed tie-breaking rule. In each cae, the value of T (v) i uniquely determined, and thu S(v) i a well. Now chooe v > min V, and uppoe T and S are determined for all element of V [0, v). We claim that the ditribution of the random variable max{v p, 0}, where p i ditributed according to S, i then uniquely determined. Thi i becaue the non-zero value of thi random variable are ditributed according to S retricted to value in V [0, v). But, by Lemma 1, thi random variable determine the value of T (v), which in turn determine the value of S(v). Thu T (v) and S(v) are uniquely determined by S and T on V [0, v), a required. Moreover, thee quantitie can be explicitly computed by evaluating the ummation in Lemma An Example: Soft Budget In thi ection we preent a pecial cae of rik-averion, driven by oft budget, and give an interpretation of our equilibrium characterization for thi cae. Suppoe that each buyer i i characterized by their value v i for a compute intance and a oft budget b i [0, v i ]. We think of b i a a budget of fund that ha been allocated to acquiring a compute intance. If the buyer obtain an intance but pay le than b i, the reidual budget i lot: it i a if they had paid b i. On the other hand, if the buyer pay more than b i, they uffer no additional penalty; they imply incur the cot of their payment. Thi cenario i captured by the following utility curve, which i piecewie linear with two piece: { z if z v i b i u i (z) = v i b i otherwie. To ee that thi utility curve i equivalent to the oft budget cenario decribed above, note that if a buyer with budget b i obtain an intance and pay p i > b i their utility i u i (v i p i ) = v i p i, wherea if they pay p i < b i their utility i u i (v i p i ) = v i b i. Note that for a fixed value v i, an agent with a higher budget i more rik-avere. To ee thi, note that if a buyer trictly prefer a lottery L to a determinitic outcome d, then it mut be that d < v i b i (ince otherwie d mut be utility-maximizing). In thi cae, decreaing the budget can only make the lottery more valuable, while not affecting the utility from the determinitic outcome. Thu, a decreaed budget can only increae the propenity to elect a lottery over a determinitic outcome. 9

10 budget (bi) pot reerve b I (v) value (v i ) Figure 1: With oft-budget, a monotone-decreaing indifference curve partition agent into thoe that reerve an intance and thoe who rely on the pot market. The monotonicity reult in Lemma 2 therefore reult in a partitioning of agent that prefer the pot market to the reervation market and vice vera by an indifference curve over budget. See Figure 1 for an illutration. Any agent with (v i, b i ) below the curve prefer the pot market; any agent above the curve (uch that v i b i ), prefer the reervation market. Lemma 3 how that thi indifference curve i unique, given the ditribution over agent type, and preciely pecifie which agent will chooe to reerve and which will chooe to enter the pot market 4 Comparative Static In thi ection, we firt conider the impact of change to buyer rik attitude. We how that a agent become more rik avere, more agent ue the reervation market and revenue increae, for every etting of the reervation price. Second, we compare the reervation+pot mechanim to the pot market and the reervation market. We firt how that the combination mechanim outcome are more efficient than running a reerve market on it own. We then how that it generate both more revenue and more welfare than running only a pot market. The reult in thi ection hold under two aumption on the reervation price et by the eller. Firt, we will make the aumption that p r i et high enough o that, in the reulting equilibrium, Pr q Q [q < T ] = 0. That i, over the uncertainty in upply, the mechanim can erve the reerved intance with certainty. Thi aumption i motivated by the fact that thee intance are typically viewed a guaranteed by the mechanim. Another natural and practical property i that the reervation price p r be et high enough that it will be greater than the expected pot price. That i, p r i large enough that it i more cotly, in expectation, to reerve a guaranteed intance than to bid for an intance in the pot market. 4.1 Effect of Increaed Rik Averion We conider the impact of an increae in cutomer rik averion. Conider type ditribution F and a type ditribution F + induced by a pointwie tranformation g + (U, V ) (U, V ) applied to each point in F which weakly increae rik averion and doe not affect value. Specifically, for any (u +, v + ) = g + (u, v), v + = v and u + u. In the following lemma, proved in the appendix, we how that uch a change can only increae the fraction of agent who chooe to reerve, and can only increae revenue. Lemma 4. For mechanim M +r, and for any reerve price p r, if rik averion of agent increae then the fraction of agent who purchae in the reervation tage increae, a doe the expected 10

11 revenue of the mechanim. The intuition underlying Lemma 4 i a follow. The firt order effect from a change in rik averion i an increae in T, the fraction of uer who chooe to reerve at a given price. Thi increae in reervation tranlate into higher pot price, ince it reduce the quantity old in the pot market. Higher pot price in turn caue more uer to prefer to reerve, which can only increae pot price further. Thi can be hown by induction over agent value. 4.2 Comparing Mechanim We now compare welfare and revenue of M +r to M and M r. Here we make ue of the two aumption dicued in Section 2.1: 1) the reervation phae i not overubcribed, i.e., the reervation price i et ufficiently high that there will be ufficient upply to fulfill the demand for reerved intance; and 2) the reervation price i ufficiently high to be above the expected pot price. We begin by comparing the revenue of the combination mechanim with the expected revenue of a pot-only market. Note that, trivially, the bet revenue of the combined mechanim i at leat the revenue of a pot market; thi i becaue, in the combined mechanim, the reerve price can be et ufficiently high that all cutomer buy in the pot market. We how omething tronger: for every choice of reervation price, the revenue of the combined mechanim i at leat that of a pot market run in iolation. The formal proof of the following lemma i deferred to the full verion. Lemma 5. For any choice of the reervation price atifying our aumption, the expected revenue of the reervation and pot mechanim i weakly greater than the revenue of the pot-only market. Proof (ketch). A in Lemma 4, a rik averion increae, revenue increae for a fixed reervation price. Fix a reervation price and conider tarting with a ditribution of rik-neutral agent. Thee agent will all bid in the pot market and thu the outcome (and in particular, the revenue) will be identical to the pot-only mechanim. By deforming the utility curve of the agent in a manner that only increae rik averion, until they match the correct ditribution, and applying Lemma 4, we can conclude that the revenue of the dual market only increae while the revenue of the pot-only mechanim, which i unaffected by the utility curve, remain the ame. We next provide an example demontrating that the expected revenue of the reervation+pot mechanim can be trictly greater than the revenue of the reervation market, when agent are not necearily rik-neutral. Example 1. We will conider an example in which agent utility curve are pecified by oft budget, a in Section 3.2. Recall that a budget of 0 correpond to rik-neutrality. Take ɛ > 0 to be ufficiently mall, and conider the following ditribution over buyer type: with probability 0.5 ɛ, (v, b) = (5, 0) with probability 0.5, (v, b) = (10, 10 ɛ) with probability ɛ, (v, b) = (20, 0) The upply i ditributed uch that q = 1 ɛ with probability 0.8, and otherwie q = ɛ/2. The optimal reerve price for the reervation market i 10, which generate a total revenue of (0.8) 10 (0.5 + ɛ) + (0.2) 10 (0.5 + ɛ/2) = 5 + 9ɛ. 11

12 Next conider the reervation and pot mechanim with reervation price 10 ɛ. At equilibrium, the buyer of type (10, 10 ɛ) trictly prefer to reerve (obtaining utility ɛ with probability 1, rather than utility ɛ with probability 0.8), wherea the buyer of type (20, 0) trictly prefer to participate in the pot market (obtaining utility (20 5) with probability 0.8, rather than utility 10 + ɛ with probability 1). The revenue from the reervation market i then (0.5) (10 ɛ), and the revenue from the pot market i (0.8) 5 (0.5 ɛ) + (0.2) 20 (ɛ/2), for a total of 7 5 2ɛ. Thi i greater than 5 + 9ɛ for ufficiently mall ɛ. We next compare the efficiency of the dual mechanim to the efficiency of the reervation-only mechanim. Lemma 6. For any choice of the reerve price atifying the aumption above, the expected efficiency of the reervation+pot mechanim i weakly greater than the efficiency of the reervation market with the ame reervation price. Proof. Recall that the efficiency of a mechanim i the expected value generated by the agent, ignoring the welfare lot due to the nonlinear utility function. For any realized upply q then, weakly more people are erved in M +r, hence efficiency i greater. Finally, we will how that the ocial welfare (um of utilitie) of the combined pot+reervation mechanim i only greater than the welfare of the pot-only mechanim. The formal proof of the following theorem appear in the appendix. Theorem 7. In any equilibrium of the the pot and reervation mechanim where the reervation price i et above the expected pot price, the expected welfare of the reervation and pot mechanim i weakly greater than the expected welfare of the pot-only mechanim. Proof (ketch). Note that, relative to a pot market, introducing a reervation price add inefficiency. Thi i becaue if a bidder i willing to reerve to get a guaranteed intance, any time they would not have one in the pot market, there i a higher valued bidder than them who could be allocated. However, welfare i increaed when a bidder chooe to reerve. Conider an agent with value v i who i willing to reerve at price p r. Reerving increae hi utility becaue he prefer reerving and the auctioneer i receiving more in revenue, becaue the reervation price i greater than the expected pot price (by aumption). The full proof conit of three part. Firt, we define a benchmark B +r that i jut like M +r except agent who reerve pay the pot price intead of the reervation price. Then, we how that the welfare of M +r i greater than the welfare of B +r, which will follow largely becaue the expected reervation price i greater than the expected pot price. Finally, we how that pot price increae when agent chooe to reerve, which lead to the welfare of B +r being greater than the welfare of M, and hence the welfare of M +r i greater than M. Reference Abhihek, V., Kah, I. A., and Key, P. (2012). Fixed and market pricing for cloud ervice. CoRR, ab/

13 An, B., Leer, V., Irwin, D., and Zink, M. (2010). Automated negotiation with decommitment for dynamic reource allocation in cloud computing. In Proceeding of the 9th International Conference on Autonomou Agent and Multiagent Sytem: volume 1-Volume 1, page International Foundation for Autonomou Agent and Multiagent Sytem. Auubel, L. M. and Cramton, P. (2010). deign. Utilitie Policy, 18(4): Uing forward market to improve electricity market Ben-Yehuda, O. A., Ben-Yehuda, M., Schuter, A., and Tafrir, D. (2013). Decontructing Amazon EC2 Spot Intance Pricing. ACM TEAC, 1(3):1 20. Borg, C., Candogan, O., Chaye, J., Lobel, I., and Nazerzadeh, H. (2014). Optimal Multiperiod Pricing with Service Guarantee. Management Science, 60: Fu, H., Hartline, J., and Hoy, D. (2013). Prior-independent Auction for Rik-avere Agent. In EC 13. Hu, A., Matthew, S. A., and Zou, L. (2010). Rik averion and optimal reerve price in firt- and econd-price auction. Journal of Economic Theory, 145(3): Ma, D. and Huang, J. (2012). The Pricing Model of Cloud Computing Service. In EC 12, page Makin, E. and Riley, J. (1984). Optimal auction with rik avere buyer. Econometrica, 52(6): Mathew, T. and Katzman, B. (2006). The role of varying rik attitude in an auction with a buyout option. Economic Theory, 27(3): Matthew, S. A. (1983). Selling to rik avere buyer with unobervable tate. Journal of Economic Theory, 30(2): Matthew, S. A. (1987). Comparing auction for rik avere buyer: A buyer point of view. Econometrica, 55(3): Menache, I., Shamir, O., and Jain, N. (2014). On-demand, Spot, or Both: Dynamic Reource Allocation for Executing Batch Job in the Cloud. In ICAC 14. Pratt, J. W. (1964). Rik Averion in the Small and in the Large. Econometrica, 32(1): Riley, J. and Samuelon, W. (1981). Optimal Auction. The American Economic Review, 71(3): Wang, W., Li, B., and Liang, B. (2012). Toward Optimal Capacity Segmentation with Hybrid Cloud Pricing. In ICDCS 12, page Ieee. Zhang, H., Li, B., Jiang, H., Liu, F., Vailako, A. V., and Liu, J. (2013). A framework for truthful online auction in cloud computing with heterogeneou uer demand. In IEEE INFOCOM, page Ieee. 13

14 A Appendix: full verion proof A.1 Proof of Lemma 4 Proof. Let F be the original ditribution over type, and let F denote a ditribution in which the rik averion of agent increae, for each fixed value v.let T and S refer to the volume of reerved intance and ditribution of pot price under the alternate ditribution F. A before, b I, T, and S correpond to the repective quantitie for the original ditribution F. We will how that for any price p p r, it i the cae that T (p) T (p) and S(p) S(p). To ee how thi implie the lemma, note that thee inequalitie imply that the increae in rik averion of agent reult in (a) more agent reerving, and (b) higher pot price. Becaue the reervation price i greater than the expected pot price, the now-reerving agent pay more than they were paying before and total revenue increae. We will prove the deired inequalitie by induction on p V. Note firt that for any p < p r, an agent with value p will certainly chooe to purchae in the pot market regardle of their rik attitude, and hence T (p) = T (p) and S(p) = S(p). For any p p r, recall that and T (p) = v V v p T (p) = v V v p f(v, u) 1 [u(v p r ) E p S [u(max{v p, 0})]] u U u U [ ] f(v, u) 1 u(v p r ) E p S [u(max{v p, 0})] A in Lemma 3, the dependence on S and S, repectively, i limited to value in V [0, v). Since we aume inductively that S(p) S(p) for all p V [0, v), the ditribution over pot market utilitie under F i tochatically dominated by the ditribution under F, for agent with value v. Since F additionally increae rik averion relative to F, fixing v, we conclude that weakly fewer agent with value v would chooe to reerve under F relative to F. That i, T (p) T (p). Then, ince S(p) = 1 Q (1 F (p) + T (p)) and S(p) = 1 Q ( 1 F (p) + T (p) ) we can immediately conclude S(p) S(p) a well. The deired reult then follow by induction. A.2 Proof of Theorem 7 We now proceed with the proof of Theorem 7: that the welfare of the pot+reervation mechanim i greater than the welfare of the pot-only mechanim. Let W EL q (M) be the welfare of mechanim M at fixed upply q. Then we can write the total welfare of M a W EL(M) = E q Q [W EL q (M)] (4) Write w v i i (p) for the welfare generated by an agent paying price p, hence wv i i (p) = u i(v i p) + p. Note that w i (p) i monotone non-decreaing in p, ince u i i a concave non-decreaing function with u i (0) = 1. 14

15 Let p (q) be the price in the pot-only mechanim M if the realization of upply i q, and p +r (q) the pot price in pot and reervation mechanim M +r. Note that p +r (q) p (q) for all q, no matter how the buyer behave. Thi i becaue p (q) i preciely the minimal price at which a q fraction of the buyer will purchae, and hence if p +r (q) < p (q) then more than a q fraction of buyer mut be purchaing in M +r (ince they would buy in the pot market if they didn t reerve), which i impoible. We now analyze the welfare of M +r, breaking down the welfare generated by thoe bidder who buy in the pot market and who reerve. Let Y (v) be the fraction of bidder with value v who reerve. Then W EL q (M +r ) = wi v (p r )Y (v)f(v) (5) v p +r + (q) v>p +r (q) Note the plit between the two ummation at p +r erved only if they reerve, wherea agent with value above p +r ( w v i (p +r (q))(1 Y (v)) + w v i (p r )Y (v) ) f(v). (6) (q) are (q) are erved whether or not. Call each of (q). Agent with value below p +r they reerve the only quetion for their welfare i whether they pay p r or p +r thee ummation W EL q (M +r ) and W EL + q (M +r ) repectively. We now conider the welfare from a benchmark, B +r. For a given upply q, if agent reerve and have value above p +r (q), we aume that they pay the pot price p +r (q) intead of the reervation price p r. For agent who reerve with value below the pot price p +r (q), we aume they pay the pot price p +r (q) and (magically) get no utility or diutility from doing o, hence the only welfare generated i the welfare the deigner experience from the payment: wi z(p+r (q)) = p +r (q). A we did for the combined mechanim, we can write the welfare of the benchmark a W EL q (B +r ) = W EL q (B +r ) + W EL + q (B +r ), where W EL q (B +r ) denote the contribution to welfare from bidder with value below p +r (q), and W EL q (B +r ) i the contribution to welfare from bidder with value above p +r (q). For bidder with value below the pot price, we have W EL q (B +r ) = The lat line followed becaue T (p +r p +r (q), hence T (p +r the welfare atifie (q)) = v p +r (q) W EL + q (B +r ) = v p +r (q) =p +r p +r (q)y (v)f(v) (q)t (p +r (q)) (q)) i exactly the volume of agent with value at mot Y (v)f(v). For the bidder with value above the pot price, v>p +r (q) wi v (p +r (q))f(v). Whether or not the benchmark welfare for agent with value above the pot price i above or below the actual welfare depend on whether the pot price i above or below the reervation price. But, a the following claim how, the welfare benchmark will be le in expectation than the actual welfare: 15

16 Claim 1. E q Q [ W ELq (M +r ) ] E q Q [ W ELq (B +r ) ] (7) Proof. The expected payment from every agent who reerve in B +r i E q Q [p +r (q)], which by aumption atifie E q Q [p +r (q)] p r. Thu the total revenue in the mechanim M +r i greater than in the benchmark B +r. For all agent who do reerve, we know their utility from reerving i more than their utility if they had not reerved and only participated in the pot market, (u r (v i, u i ) u (v i, u i )). The utilitie of all agent who do not reerve are the ame in both, o they are indifferent. Summing the utilitie of all the agent and the revenue of the deigner give E q Q [ W ELq (M +r ) ] E q Q [ W ELq (B +r ) ]. We now argue that the welfare of the benchmark i an upper bound on the welfare from the pot-only mechanim. Claim 2. W EL q (B +r ) W EL q (M ) (8) Proof. The agent who purchae in M are preciely thoe with value above p (q), the pot price for the pot-only mechanim. Conider eparately the agent with value above and below the pot price p +r (q) for the combined mechanim. Agent with value above p +r (q) are alway allocated in the benchmark and the pot-only mechanim. A the pot price (and hence benchmark payment) i higher in the pot+reervation mechanim than the pot-only mechanim for a given upply, and w i i non-decreaing in p, w i (p +r (q)) w i (p (q)), thu v>p +r (q) w v i (p (q))f(v) v>p +r (q) wi v p +r (q)f(v) = W EL + q (B +r ). (9) Conider now agent with value below p +r (q). If the agent doe not reerve in the pot+reervation mechanim, then we know that the price paid in the benchmark i higher than the welfare generated from receiving the item: v i p +r T (p +r (q)) i the volume of agent with value below p +r v p (q)<v p +r (q) (q), hence we know that wi z(p (q)) p +r (q) who reerve. Thu, w z i (p (q))f(v) Combining equation (12) and (9) give: our deired reult. v p (q)<v p (q) = p +r (q). Recall that p +r (q)f(v) (10) (q)(s(p +r (q)) S(p (q))) p +r (q)t (p +r (q)) (11) = W EL q (B +r ). (12) W EL q (M ) W EL + q (B +r ) + W EL q (B +r ) = W EL q (B +r ), 16

17 We can now combine the bound from Claim 1 and Claim 2 to how that the welfare of the pot and reervation mechanim M +r i greater than the welfare of the pot-only mechanim, M, completing the proof of Theorem 7. Proof of Theorem 7. Taking expectation over the upply and uing the claim above, we have a deired. W EL(M ) W EL(B +r ) W EL(M +r ) 17