Apt, K.R. (Krzysztof)

Markakis, V. (Vangelis)

English

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1189Optimal Strategies in Sequential Bidding(Extended Abstract)Krzysztof R. AptCentre for Math andComputer Science (CWI)andUniversity of AmsterdamAmsterdam, the Netherlandsapt@cwi.nlEvangelos Markakis∗Athens University ofEconomics and BusinessPatision 76, 10434Athens, Greecemarkakis@gmail.comCategories and Subject DescriptorsJ.4 [Social and Behavioral Sciences]: Economics1. INTRODUCTIONWe are interested in mechanisms that maximize the finalsocial welfare. In [1] this problem was studied for multi-unit auctions with unit demand bidders and for the publicproject problem, and in each case social welfare undomi-nated mechanisms in the class of feasible and incentive com-patible mechanisms were identified.One way to improve upon these optimality results is byrelaxing the assumption of simultaneity and allowing theplayers to move sequentially. With this in mind, we studyhere sequential versions of two feasible Groves mechanismsused for single item auctions: the Vickrey auction and theBailey-Cavallo mechanism.Because of the absence of dominant strategies in this se-quential setting, we focus on a weaker concept of an optimalstrategy. For each mechanism, we introduce natural optimalstrategies and observe that in each mechanism these strate-gies exhibit different behaviour. However, we then show thatamong all optimal strategies, the one we introduce for eachmechanism maximizes the social welfare when each playerfollows it. The resulting social welfare can be larger thanthe one obtained in the simultaneous setting.2. PRELIMINARIESAssume that there is a finite set of possible outcomes ordecisions D, a set {1, . . ., n} of players where n ≥ 2, andfor each player i a set of types Θi and an (initial) utilityfunction vi. A decision rule is a function f : Θ→D,where Θ := Θ1 × · · · ×Θn.A mechanism is given by a pair of functions (f, t), wheref is the decision rule and t = (t1, ..., tn) is the tax functionthat determines the players’ payments. We assume that the(final) utility function for player i is a function ui definedby ui(d, t1, . . ., tn, θi) := vi(d, θi) + ti.We focus on two special Groves mechanisms (for details onGroves mechanisms see [5]). The first one is the well knownCite as: Optimal Strategies in Sequential Bidding (Short Paper), K.R. Apt, E. Markakis, Proc. of 8th Int. Conf. on AutonomousAgents and Multiagent Systems (AAMAS 2009), Decker, Sich-man, Sierra and Castelfranchi (eds.), May, 10–15, 2009, Budapest, Hun-gary, pp. XXX-XXX.Copyright c© 2009, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.pivotal mechanism1. The Vickrey auction is simply thepivotal mechanism for a single item auction. The second isthe Bailey-Cavallo mechanism, in short BC auction . Itwas originally proposed in [3, 4], where it was also demon-strated that it can be a more appropriate mechanism whenone is interested in maximizing the final social welfare.3. SEQUENTIAL MECHANISMSWe are interested in sequential mechanisms, in which theplayers announce their types sequentially. In this section wereview the relevant concepts, some of which were introducedin [2]. Without loss of generality, we assume the order to be1, . . ., n. Hence each player i knows the types announced byplayers 1, . . ., i − 1, and can use this information to decidewhich type to announce.A strategy of player i in a sequential mechanism is afunctionsi : Θ1 × . . .×Θi →Θi.If the vector of types that the players receive is θ and thevector of strategies that they follow is s(·) := (s1(·), . . ., sn(·)),the vector of the announced types will be denoted by [s(·), θ],where [s(·), θ] is defined inductively by [s(·), θ]1 := s1(θ1)and [s(·), θ]i+1 := si+1([s(·), θ]1, . . ., [s(·), θ]i, θi+1).A strategy si(·) of player i is dominant in the sequentialversion of the mechanism (f, t) if there is no incentive forplayer i to deviate to another strategy no matter what theother players announce.A weaker notion is that of an optimal strategy. We saythat strategy si(·) of player i is optimal if for all θ ∈ Θ andall θ′i ∈ Θiui((f, t)(si(θ1, . . ., θi), θ−i), θi) ≥ ui((f, t)(θ′i, θ−i), θi).Call a strategy of player j myopic if it does not dependon the types of players 1, . . ., j − 1. Then a strategy si(·) isoptimal if for all θ ∈ Θ it yields a best response to all jointstrategies of players j = i in which the strategies of playersi+1, . . ., n are myopic. In particular, an optimal strategy isa best response to the truth-telling by players j = i. Eachdominant strategy is optimal. For player n the concepts ofdominant and optimal strategies coincide.There are two natural ways of maximizing players’ utili-tities. The first one calls for a simultaneous maximizationof other players’ utilities. That is, we say that strategy si(·)1This is sometimes referred to as the VCG mechanism.Cite as: Optimal Strategies in Sequential Bidding, (Extended Abstract), Krzysztof R. Apt, Evangelos Markakis, Proc. of 8th Int. Conf. on Au-tonomous Agents and Multiagent Systems (AAMAS 2009), Decker, Si-chman, Sierra and Castelfranchi (eds.), May, 10–15, 2009, Budapest, Hungary, pp. 1189–1190Copyright © 2009, International F undation for Autonomous Agents and Multiagent Systems (www.ifaamas.org), All rights reserved.AAMAS  2009 • 8th International Conference on Autonomous Agents and Multiagent Systems • 10–15 May, 2009 • Budapest, Hungary 1190of player i is socially maximal if it is optimal and for alloptimal strategies s′i(·) of player i, all θ ∈ Θ and all j = iuj((f, t)(si(θ1, . . ., θi), θ−i), θj) ≥uj((f, t)(s′i(θ1, . . ., θi), θ−i), θj).So a socially maximal strategy of player i simultaneouslyguarantees the maximal utility to every other player, underthe assumption that players i+1, . . ., n use myopic strategies.The second option is to maximize the social welfare. Wesay that strategy si(·) is socially optimal if it is optimaland for all optimal strategies s′i(·) of player i and all θ ∈ ΘPnj=1 uj((f, t)(si(θ1, . . ., θi), θ−i), θj) ≥Pnj=1 uj((f, t)(s′i(θ1, . . ., θi), θ−i), θj).Hence a socially optimal strategy of player i yields themaximal social welfare among all optimal strategies, underthe assumption that players i+1, . . ., n use myopic strategies.Note that each socially maximal strategy is socially optimal.The converse does not hold.For a mechanism where each player i receives a type θi ∈Θi and follows a strategy si(·), we denote the resulting so-cial welfare by SW (θ, s(·)). We are interested in finding asequence of optimal players’ strategies for which the result-ing social welfare is always maximal.3.1 Sequential Vickrey auctionsIn this and the next Section all proofs are omitted. First,we have the following negative result.Theorem 3.1. Consider a sequential Vickrey auction. Fori ∈ {1, . . ., n− 1} no dominant strategy exists for player i.So we shall focus on the weaker notion of optimal strategy.The following natural strategy for player i is an example ofan optimal strategy that deviates from truth-telling:si(θ1, . . ., θi) :=jθi if θi > maxj∈{1,...,i−1} θj ,0 otherwise.(1)When we limit our attention to optimal strategies we getthe following result.Theorem 3.2. Consider a sequential Vickrey auction. Forall θ ∈ Θ, all vectors s(·) of optimal players’ strategies andall optimal strategies s′i(·) of player iui((f, t)([s(·), θ], θi) = ui((f, t)([(s′i(·), s−i(·)), θ], θi).This can be interpreted as a statement that each optimalstrategy is dominant within the universe of optimal strate-gies. However, optimal strategies may differ when the play-ers take into account the utility of other players, in particu-lar, the social welfare.The following result shows that strategy si(·) defined in(1) plays a special role.Theorem 3.3. In the sequential Vickrey auction strategysi(·) defined in (1) is socially maximal for player i.Finally, when each player i follows strategy si(·) of Theo-rem 3.3, maximal social welfare is generated.Theorem 3.4. In the sequential Vickrey auction for allθ ∈ Θ and vectors s′(·) of optimal players’ strategies,SW (θ, s(·)) ≥ SW (θ, s′(·))where s(·) is the vector of strategies si(·) defined in (1).This maximal final social welfare under s(·) is always greaterthan or equal to the social welfare achieved in a Vickreyauction when players bid truthfully. Additionally, for someinputs, it is strictly greater.3.2 Sequential BC auctionsWe can show that in analogy to sequential Vickrey auc-tions no dominant strategies exist except for the last player.In fact we can establish this for a wide class of Groves auc-tions. Details will appear in the full version of this work.We shall thus focus, as in the case of sequential Vickreyauctions, on the weaker notion of optimal strategy. We havevarious natural optimal strategies that deviate from truth-telling, such as the following one:si(θ1, . . . , θi) :=8><>:θi if θi > maxj∈{1,...,i−1} θj(θ1, . . . , θi−1)∗1 if θi ≤ maxj∈{1,...,i−1} θjand i ≤ n− 1(θ1, . . . , θi−1)∗2 otherwise(2)In contrast to the Vickrey auction, the analogue of The-orem 3.2 does not hold for the sequential BC auctions. Inparticular this means that optimal strategies are not domi-nant within the universe of optimal strategies.We now turn to the question of existence of socially opti-mal strategies, which we answer negatively. This is again incontrast to the sequential Vickrey auction.Theorem 3.5. The sequential BC auction does not admitsocially optimal strategies except for the first and last player.The results established so far show that the sequentialVickrey auctions and BC auctions differ in many ways. Weconclude by showing that they do share one property. Namely,within the universe of optimal strategies there exists an op-timal strategy si(·) such that if all players follow it, thenmaximal social welfare is generated for all θ ∈ Θ.Theorem 3.6. In the sequential BC auction for all θ ∈ Θand all vectors s′(·) of optimal players’ strategies,SW (θ, s(·)) ≥ SW (θ, s′(·))where s(·) is the vector of strategies si(·) defined in (2).This maximal final social welfare is always greater thanor equal to the final social welfare achieved in a BC auctionwhen players bid truthfully. Additionally, for some inputs,it is strictly greater.4. REFERENCES[1] K. R. Apt, V. Conitzer, M. Guo, and E. Markakis.Welfare undominated Groves mechanisms. In Proc. 4thInternational Workshop on Internet and NetworkEconomics (WINE 08), pages 426–437, 2008.[2] K. R. Apt and A. Este´vez-Ferna´ndez. Sequentialpivotal mechanisms for public project problems, 2008.[3] M. Bailey. The demand revealing process: To distributethe surplus. Public Choice, 91(2):107–126, 1997.[4] R. Cavallo. Optimal decision-making with minimalwaste: Strategyproof redistribution of VCG payments.In AAMAS ’06: Proc. of the 5th Int. Joint Conf. onAutonomous Agents and Multiagent Systems, pages882–889, 2006.[5] A. Mas-Collel, M. D. Whinston, and J. R. Green.Microeconomic Theory. Oxford University Press, 1995.

Optimal strategies in sequential bidding

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