Optimal mechanism design when both allocative inefficiency and expenditure inefficiency matter

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    sRandom assignmentWinner-take-all assignmentRank-order rulesExpenditure inefficiency

    1. Introduction

    People spend valuable resources to obtainwhat they desire, andthis expenditure is not always efficiently invested. Education isprobably one of the most famous examples after Spence (1973):over-investments or even completely wasteful investments insignaling occur to obtain diplomas. Rent-seeking is another famousexample after Tullock (1980): rents are dissipated due to wastefulbehaviors. In these situations, two sources of inefficiency mayoccur. The first is inefficiency from the (at least partial) waste ofvaluable resources. The second is allocative inefficiency that mayresult if the objects of interest are not assigned to those who valuethem most or who can utilize them most productively.

    A simple example will help to clarify the main theme of thispaper.1 There is one object to be assigned to either one of twoplayers. Assume first that both players attribute a value of onedollar to the object, and that this fact is common knowledge. Ifthe object is assigned to the player who throws more pennies intoa big pond,2 each player will waste 50 cents in expectation. Thetotal inefficiency is one dollar, and the rent is fully dissipated. On

    Tel.: +82 2 3290 2222; fax: +82 2 928 4948.E-mail address: kiho@korea.ac.kr.URL: http://econ.korea.ac.kr/kiho.

    1 Though this example postulates complete information for illustrative purpose,this paper in fact studies the incomplete information setting.2 This is an all-pay auction under complete information, for which F(x) = x for

    x [0, 1] is the cumulative distribution function of the unique equilibrium strategy.

    the other hand, if the object is assigned randomly, no inefficiencyresults. Hence, random assignment performs better. Assume nextthat one of the players attributes a value of one dollar, while theother attributes zero to the object. Players know their respectivevaluations, but others do not know whose valuation is one. Then,the player with a low valuation throws no penny obviously, whilethe player with a high valuation throws one penny to get theobject. This leads to (almost) no inefficiency. On the other hand,if the object is assigned randomly then the player with zero valuewill get the object with probability 0.5, resulting in an allocativeinefficiency of 50 cents. In this case, the money-throwing methodperforms better.

    This example shows that, faced with two kinds of inefficiency,allocative inefficiency and expenditure inefficiency, uncertaintyregarding true valuations matters for the relative performance ofalternative assignment rules. There exists no value uncertainty inthe first case, thus no inefficiency from mis-allocation occurs. Allweneed to take care is the inefficiency fromwasteful expenditures.Value uncertainty is significant in the second case, and we have toconsider both expenditure inefficiency and allocative inefficiency.

    We approach this problem from a mechanism design perspec-tive under incomplete information. Hence, we assume that playershave private information regarding their valuations for the object,and look for optimal assignment rules that minimize the sum ofallocative inefficiency and expenditure inefficiency. In particular,we focus on the rank-order assignment rules that respect the or-der but not the exact level of players expenditures. The rank-orderrule, in which only relative (as opposed to absolute) performancematters, is widely used in practice: elections, sports competitions,Journal of Mathematical Eco

    Contents lists available a

    Journal of Mathem

    journal homepage: www.e

    Optimal mechanism design when both ainefficiency matterKiho Yoon Department of Economics, Korea University, Anam-dong, Sungbuk-gu, Seoul, 136-701, Re

    a r t i c l e i n f o

    Article history:Received 10 September 2010Received in revised form30 August 2011Accepted 11 September 2011Available online 18 September 2011

    Keywords:Mechanism

    a b s t r a c t

    We characterize the structexpenditure inefficiency (e.gdepends on how the hazardprobabilistic assignment rulvaluation. We also find that tthe value distribution increa0304-4068/$ see front matter 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2011.09.002nomics 47 (2011) 670676

    t SciVerse ScienceDirect

    atical Economics

    sevier.com/locate/jmateco

    locative inefficiency and expenditure

    ublic of Korea

    re of optimal assignment rules when both allocative inefficiency and., rent-seeking) are present. We find that the optimal structure criticallyrate of the value distribution behaves, and that it is often optimal to uses so that the winner of the object is not always the one with the highesthe inefficiency of the optimal assignment rule decreases as the variability ofes.

    2011 Elsevier B.V. All rights reserved.

  • K. Yoon / Journal of Mathematica

    promotion in organizations, and classroom grading are just a fewexamples. It is a simple scheme to implement, especially when in-dividual performance is hard tomeasure on a cardinal scale. More-over, as the works including Lazear and Rosen (1981) and Greenand Stokey (1983) show, it may in fact be an optimal arrangementfor many economic situations.3

    We find that the optimal structure of assignment rules dependscritically on how the hazard rate of the value distribution behaves,and that it is often optimal to use probabilistic assignmentrules in which the winner of the object is not always the onewith the highest valuation. We also find that the inefficiency ofthe optimal assignment rule decreases as the variability of thevalue distribution increases. These results are obtained in a clearand straightforward fashion by applying the theory of stochasticorders. This theory of stochastic orders among differences of orderstatistics, pioneered by Barlow and Proschan (1966), was recentlyintroduced to various models in economics by Moldovanu et al.(2007, 2008) and Hoppe et al. (2009). In particular, the presentpaper is technically similar to the last paper that studies theassortative two-sided matching with costly signals to see how thechange in either side of the matching affects the relevant variablessuch as the welfare and the signaling efforts. In comparison, westudy the optimal choice of assignment rules depending on thedistribution of one population.4

    The problem of this paper may also be analyzed by the optimalmechanism design approach initiated by Myerson (1981). Then,it is possible to consider more general assignment rules, beyondrank-order assignment rules. Since this approach is quite well-established in the literature, however, we relegate the discussionon the general mechanism design approach to the Appendix. Theanalysis in the Appendix shows that our results (in particular,Propositions 2 and 3) continue to hold even when we considermore general assignment rules. Moreover, the theory of stochasticorders enables us to obtain a meaningful comparative staticresult with respect to the changes in the value distribution(Proposition 4).

    Hartline and Roughgarden (2008), Chakravarty and Kaplan(2009) and Condorelli (2011) adopted the optimal mechanismdesign approach to study similar problems. The last paper, inparticular, derives similar results to Proposition 2. Compared toCondorelli (2011), the current paper encompasses the case whenonly parts of the expenditure are counted as inefficiency as well asestablishes a result as to how the variability of the value distribu-tion affects efficiency. On the other hand, Condorelli discusses het-erogeneous objects case and then the implementation via prioritylists and queues. Also related are Suen (1989), Taylor et al. (2003),and Koh et al. (2006) that compare thewaiting-line auction and thelottery for specific distributions.5

    2. Main results

    2.1. The model

    We consider a situation where one object is to be assigned toone of the players in N = {1, . . . , n}.6 The object may be tangible

    3 Frankel (2010) recently shows that the rank-order rule may be an optimalmechanism under certain circumstances if the worst-case optimality criterioninstead of Bayesian optimality criterion is considered.4 In a sense, this paper takes one of the two sides in Hoppe et al. (2009) as a

    decision variable and chooses optimally according to the changes in the remainingside.5 The main analysis of Taylor et al. (2003) deals only with the Beta distribution

    numerically, while Koh et al. (2006) consider 4 specific (power, Weibull, logistic,and Beta) distributions.

    6 It is a straightforward matter to extend the analysis to the case of multiple

    homogeneous objects. See the discussion at the end of this section.l Economics 47 (2011) 670676 671

    such as a product or a specific position; or it may be intangiblesuch as a government contract, political favor, or social status. Itmay also be a mating partner in the case of biological contests.Each player i N has a valuation vi for the object. We postulateincomplete information so that player i knows his valuation vi,while others only know its distribution. We assume that eachplayers valuation is drawn independently from the interval [v, v]with 0 v v according to a common distribution F .7We assume further that F admits a continuous density function fwhich is strictly positive on the interval [v, v].

    Player i exerts an observable expenditure xi R+ to winthe object. This expenditure is assumed to be an unconditionalcommitment of resources.8 That is, each player exerts xi whether ornot he actually gets the object. This expendituremaybe amonetarybid as in all-pay auctions, an effort level in contests, time in awaiting line, or a costly investment as in a signaling context or in abiological context.

    A mechanism is an assignment rule p = (p1, . . . , pn) thatdepends on the expenditure vector x = (x1, . . . , xn), where piis the probability that player i gets the object. To be feasible,the assignment rule must satisfy pi(x) 0 for all i N andn

    j=1 pj(x) = 1, for all x Rn+. Player is payoff is ui pivi xiwhen his valuation is vi and he exerts an expenditure of xi. Wewant to note that the linear disutility of expenditure is not asrestricted as it appears. We may alternatively introduce a generalcost function c(xi, vi) so that ui pivi c(xi, vi), and assumethat c(xi, vi) is strictly increasing in xi. This function can be eitherconvex or concave in xi as well as either increasing or decreasingin vi. It is an easy and interesting exercise to observe that virtuallythe same results hold in the following if we replace xi with c(xi, vi),in particular, in the expenditure inefficiency and the net efficiencydefined below.

    Each player chooses his expenditure to maximize the expectedpayoff, given other players expenditures and the assignment rule.Hence, player js strategy in the mechanism is a function ej :[v, v] R+ that maps his valuation to the expenditure level.Given others strategies, player is problem with valuation vi is

    maxxi

    Evi [pi(xi, ei(vi))vi xi].In the above expression, we follow the convention that the sub-scripti pertains to players other than player i. For example, vi =(v1, v2, . . . , vi1, vi+1, . . . , vn).

    It will be convenient to work with order statistics. Hence, letv1:n v2:n vn:n be the order statistics of v1, . . . , vn.Note that vk:n is the k-th highest among n valuations drawn fromthe common distribution F . The distribution and the density of vk:nare denoted by Fk:n and fk:n, respectively.9 We will also deal withplayers valuations except player is, so we can similarly have theorder statistic vk:n1 and the corresponding functions Fk:n1 andfk:n1 of the k-th highest among (n 1) valuations.

    As discussed in the introduction, we restrict our attention tothe class of assignment rules that respect the order, but not theamounts of expenditure. That is, given a vector (x1, . . . , xn) ofplayers expenditures, a probability 1 of win is given to the playerwith the highest expenditure x1:n, a probability 2 is given to theplayer with the second highest expenditure x2:n, and so on.10 Let

    7 The Appendix contains an analysis of the asymmetric player case.8 As far as the author is aware, Amman and Leininger (1995, 1996) were the

    first to make the distinction between unconditional commitment and conditionalcommitment.9 Fk:n(z) = k1r=0 nr F(z)nr [1 F(z)]r and fk:n(z) = n!(k1)!(nk)! F(z)nk[1

    F(z)]k1f (z).

    10 Ties can be dealt with by combining the relevant probabilities and assigningequal chances.

  • 672 K. Yoon / Journal of Mathematica

    us call these assignment rules the rank-order assignment rules.Wewill henceforth denote amechanism by the rank-order assignmentrule (1, . . . , n), withk being the probability that a playerwhoseexpenditure is the k-th highest wins the object.

    2.2. Analysis

    Since players are symmetric, we begin with a heuristic deriva-tion of symmetric equilibrium strategies. Suppose that playersother than i follow a symmetric, increasing and differentiable equi-librium strategy e(). First, it is straightforward to see that player iwill never optimally exert an expenditure xi > e(v). Second, it isalso easy to see that a playerwith valuation vwill optimally choosean expenditure of zero. Then player is expected payoff when hisvaluation is vi and he exerts an expenditure of e(wi) is

    Ui(vi;wi) = vi1F1:n1(wi)+ 2[F2:n1(wi) F1:n1(wi)]

    + + n1[Fn1:n1(wi) Fn2:n1(wi)]+n[1 Fn1:n1(wi)]

    e(wi)= vi

    n1k=1

    (k k+1)Fk:n1(wi)+ n e(wi).

    In words, if player i has a true valuation of vi but exerts anexpenditure as if his valuation is wi, he will get the object withprobability1whenhis expenditure is the highest, i.e., v1:n1 wi,he will get the object with probability 2 when his expenditure isthe second highest, i.e., v2:n1 wi < v1:n1, . . . , and he willget the object with probability n when his expenditure exceedsno other players expenditures, i.e., w < vn1:n1. Note well thatFk:n1(wi) Fk1:n1(wi) is the probability that wi lies betweenthe (k 1)-th and the k-th highest among (n 1) other playersvaluations, i.e., Prob[vk:n1 wi < vk1:n1].

    The first-order condition for the payoff maximization withrespect towi is

    vi

    n1k=1

    (k k+1)fk:n1(wi) = e(wi).

    Sincewi = vi at a symmetric equilibrium, we have the differentialequation

    e(vi) = vin1k=1

    (k k+1)fk:n1(vi),

    from which we get the equilibrium strategy

    e(vi) =n1k=1

    (k k+1) viv

    w dFk:n1(w).

    While this is only a heuristic derivation, the following propositionshows that this is indeed an equilibrium.

    Proposition 1. Suppose that the rank-order assignment rule (1, . . . ,n) satisfies1 n. Then, the symmetric equilibrium strategyof the mechanism is

    e(vi) =n1k=1

    (k k+1) viv

    w dFk:n1(w)

    for all i N and vi [v, v].Proof. If 1 = 2 = = n, then the equilibrium strategy ise(vi) = 0 for all vi [v, v]. Now, if1 n with at least onestrict inequality, then e(vi) is strictly increasing and continuous.

    We next show that Ui(vi;wi) is maximized by choosing wi = vi.We havel Economics 47 (2011) 670676

    Ui(vi; vi) Ui(vi;wi)

    =n1k=1

    (k k+1)vi[Fk:n1(vi) Fk:n1(wi)]

    viwi

    w dFk:n1(w)

    =n1k=1

    (k k+1) vi

    wi...

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