We present a new dynamic auction for procurement problems where payments are bounded by a hard budget constraint and money does not enter the procurer's objective function

Ensthaler, Ludwig

Giebe, Thomas

English

Open Access LMU

Sonderforschungsbereich/Transregio 15  www.sfbtr15.de Universität Mannheim  Freie Universität Berlin  Humboldt-Universität zu Berlin  Ludwig-Maximilians-Universität München Rheinische Friedrich-Wilhelms-Universität Bonn  Zentrum für Europäische Wirtschaftsforschung Mannheim  Speaker: Prof. Dr. Urs Schweizer.  Department of Economics  University of Bonn  D-53113 Bonn, Phone: +49(0228)739220  Fax: +49(0228)739221                       * Humboldt University at Berlin  ** Humboldt University at Berlin     March 2010      Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. Discussion Paper No. 310   A dynamic auction for multi-object procurement under a hard budget constraint  * Ludwig Ensthaler ** Thomas Giebe   A dynamic auction for multi-object procurementunder a hard budget constraint∗Ludwig Ensthaler†Humboldt University at BerlinDIW BerlinThomas Giebe‡Humboldt University at BerlinMarch 3, 2010AbstractWe present a new dynamic auction for procurement problems wherepayments are bounded by a hard budget constraint and money doesnot enter the procurer’s objective function.JEL D21, D44, D45, D82.Keywords: Auctions, Mechanism Design, Knapsack Problem, Dom-inant Strategy, Budget, Procurement∗Many thanks to Ulrich Kamecke, Roland Strausz, Balázs Szentes, Georg Weizsäcker,and Elmar Wolfstetter for helpful comments. Financial support from the Deutsche For-schungsgemeinschaft through SFB/TR 15 is gratefully acknowledged.†Humboldt University at Berlin, School of Business and Economics, Dept. of Eco-nomics, Spandauer Straße 1, 10178 Berlin, Germany, and Graduate Center of Eco-nomic and Social Research DIW Berlin, Tel: +49-30-2093-1594, Fax: +49-30-2093-5619,lensthaler@diw.de‡(Corresponding author) Humboldt University at Berlin, School of Business and Eco-nomics, Dept. of Economics, Spandauer Straße 1, 10178 Berlin, Germany, Tel: +49-30-2093-5773, Fax: +49-30-2093-5619, thomas.giebe@wiwi.hu-berlin.de.1. introductionA budget-constrained buyer wants to purchase items from a shortlisted set.Items are differentiated by observable quality and sellers have private reserveprices for their items. Sellers quote prices strategically, inducing a knapsackgame. The buyer’s problem is to procure a subset of maximal quality. Westudy a procurement auction that can be viewed as a game-theoretic ex-tension of the greedy-split heuristic for the classic knapsack problem. Theauction exhibits truthtelling in dominant strategies, is ex-post rational, andsatisfies the hard budget constraint. We see many applications for thisprocurement setting, e.g., allocating R&D subidies, scholarships, emissionreductions.Despite its relevance, the problem has received surprisingly little attentionin the literature. We introduced the auction in Ensthaler and Giebe (2009).In independent research, approaching the problem from a different angle,Papadimitriou and Singer (2010) analyze a similar direct mechanism in amore general framework. The allocation problem was first studied by Giebeet al. (2006) in the context of R&D subsidies. Chung and Ely (2002) (ex-tending the work of Maskin, 2002) derive the optimal Bayesian mechanismfor a similar procurement problem under a soft budget constraint.2. the modelLet I := {1, . . . , N}, N ≥ 1, denote the set of sellers where i is a typicalseller. Each seller wants to sell an indivisible item for which he has a pri-vately known reserve price θi. Items differ in quality and quality is commonknowledge. Let wi > 0 denote the quality of seller i’s item. Sellers are facedby a single buyer who has a fixed and finite budget B to spend on theseitems. The buyer’s objective is to purchase a subset of items of maximalquality, where the quality of a set of items is the sum of qualities of itselements. Money does not enter the buyer’s utility function. Sellers’ typessatisfy θi ∈ [0,B]. We assume that the probability of two types being equalis zero and that there is no known statistical relation between sellers’ types,resp. between quality and types.In the following, a mechanism is an open auction or a direct mechanism. Werestrict attention to symmetric deterministic mechanisms, where the sellerssubmit bids bi ≥ 0 (or type reports bi ∈ [0,B]). The vector of sellers’ bids(reports) is b = (bi, b−i) = (b1, . . . , bN ) where a truthful report is bi = θi.For given bids, b, and true type θi, seller i’s ex-post utility isui(b|θi) := ti(b)− qi(b)θi, (1)where ti is a payment to i and qi ∈ {0, 1}. Let A(b) := {i ∈ I|qi(b) = 1}denote the allocation resulting from the play of a mechanism when the vectorof final bids (type reports) is b, and denote the complementary set by Ac.2The allocation is the set of sellers whose objects are procured. For simplicity,denote ui(b) := ui(b|θi) and A := A(b) if confusion is unlikely.The buyer’s utility is the total quality of the procured items,ubuyer(A) =∑j∈Awj =∑i∈Iqiwi, qi ∈ {0, 1}. (2)Under complete information, the buyer faces a simple knapsack problem,argmaxq∑i∈Iwiqi s.t.∑i∈Iθiqi ≤ B, q ∈ {0, 1}N . (3)Call (3) the first-best allocation(s).3. the auctionConsider the following open descending clock auction. A central continuousclock counts down the price-per-quality ratio r, beginning from the highestinitial ratio among all bidders, rmax := maxi∈I B/wi. A bidder can quit theauction at any time, but can never come back. Being active at clock readingr implies that bidder i is willing to accept a payment of rwi in return forhis item.As the clock counts down, each active bidder’s current financial bid, rwi,decreases. As long as the sum of active financial bids exceeds the budget,the countdown continues. As soon as that sum fits in the budget, say, atround rf , the auction ends, and each active bidder i sells his item at theprice rfwi.The sum of active financial bids decreases smoothly as the clock countsdown, until some bidder j exits, at which point that sum is reduced by theamount rwj .Two events can end the auction. First, the end can be triggered by an exit,when before the exit, the sum of active financial bids exceeded the budget,but after the exit the remaining active bids can be accomodated withoutexhausting the budget. Second, it may end because at some point all activebids exactly use up the budget.After the auction, each loser j has a final ratio rj = bj/wj with financialbid bj since j quit the auction at clock reading r = rj . All winners i havethe same final ratio rf = bi/wi with (different) selling prices bi.Lemma 1. The stop rule bi(θi) = θi is a weakly dominant strategy.Proof. Suppose a seller plays the candidate. His profit is either zero, if hehas to quit before the auction ends, or it is positive, if the auction endsbefore he quits. Consider stopping earlier. Either there is no change or hequits but could have made a profit. Consider stopping later. Either there isno change, or he makes a loss if the auction ends after r = θi but before hequits.3In order to give a direct characterization of the auctions’ equilibrium resultwe need the following notation.Let li denote the set of players with final ratio not above that of bidderi. Let rAc denote the smallest ratio among all players who are not in theallocation (unless Ac = ∅).li :={j ∈ I∣∣∣∣∣ bjwj ≤biwi}(4)rAc :=min{bjwj∣∣∣ j ∈ Ac} if Ac 6= ∅Bminj∈I wjif Ac = ∅(5)Lemma 2. In equilibrium (bi = θi), the auction implements allocation Aand payments t whereqi(b) ={1 if biwi∑j∈liwj ≤ B0 otherwise,(6)ti(b) =min{rAcwi,B∑j∈Awjwi}if qi(b) = 10 otherwise.(7)Proof. By Lemma 1, winners’ true ratios are below the final ratio, θi/wi <rf = bi/wi since they were active at the end of the auction. Thus, winnershave lower true ratios than losers. All winners are paid according to thesame ratio, rf , i.e. winner i gets bi = rfwi. Thus, i is a winner iff it isfeasible to pay every player with same or lower final ratio according to i’sfinal ratio, see (6).Losers receive no payment and keep their items. The payments to the win-ners i ∈ A depend on which event ends the auction. If the end is triggeredby an exit, then the exiting player’s final ratio determines the payments,rAcwi, otherwise payments areB∑j∈Awjwi, i.e., they use up the budget.The second line in (5) just makes the definition complete but is never payoff-relevant.Observation 1. Suppose bi ∈ [0,B] are type reports. Then (6) and (7) de-fine an incentive-compatible direct mechanism (q, t) with allocation and pay-ments as in the auction. This mechanism exhibits truthtelling in dominantstrategies, satisfies the hard budget constraint and is ex-post individuallyrational.4. discussionObviously, the auction observes the hard budget constraint. It is ex-postrational since all winners are paid according to a ratio that exceeds theirtrue ratio and losers pay nothing.4Apart from these appealing incentive properties, how well does the auctionperform compared to the first-best solution?For the special case where all items have the same quality, one can easilyshow that the auction is a 2-approximation, which means that in everyinstance, the first-best allocation is at most twice as good as the allocationchosen by the auction.For the general case with different qualities Papadimitriou and Singer (2010)show that a slightly modified direct mechanism is a 6-approximation.Though these worst-case bounds may not sound encouraging, simulationsindicate that the average performance of the auction is likely to be muchbetter.1Is there a mechanism that is better or optimal (on average)? We see plentyof room for further research.ReferencesChung, K.-S. and J. C. Ely (2002): “Mechanism Design with BudgetConstraint,” Working Paper.Ensthaler, L. and T. Giebe (2009): “Subsidies, Knapsack Auctions andDantzig’s Greedy Heuristic,” SFB/TR 15 Discussion Paper No. 254.Giebe, T., T. Grebe, and E. Wolfstetter (2006): “How to allocateR&D (and other) subsidies: An experimentally tested policy recommen-dation,” Research Policy, 35, 1261–1272.Maskin, E. (2002): “How to reduce greenhouse gas emissions - an appli-cation of auction theory,” Nancy L. Schwartz Memorial Lecture, North-western University.Papadimitriou, C. and Y. Singer (2010): “Budget Feasible Mecha-nisms,” Working Paper (University of California at Berkeley).1See Ensthaler and Giebe (2009).5

How to allocate R&D (and other) subsidies: An experimentally tested policy recommendation,”

How to reduce greenhouse gas emissions - an application of auction theory,” Nancy L. Schwartz Memorial Lecture,

Mechanism Design with Budget Constraint,” Working Paper.

A dynamic auction for multi-object procurement under a hard budget constraint

A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by observable quality and sellers have private reserve prices for their items.  Sellers quote prices strategically, inducing a knapsack game. The buyer's problem is to select a subset of maximal quality. We propose a procurement mechanism that can be viewed as a game-theoretic extension of the greedy-split heuristic for the classic knapsack problem. The mechanism exhibits truthtelling in dominant strategies, is ex-post rational, and satisfies the hard budget constraint. Our mechanism obtains an optimal monotone pricing allocation if all items have the same quality. With different qualities, the equilibrium allocation is an optimal proportional pricing allocation

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A dynamic auction for multi-object procurement under a hard budget constraint

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