An explicit resolution of the equity-efficiency tradeoff in the random allocation of an indivisible good

Transcription

1 An expicit resoution of the equity-efficiency tradeoff in the random aocation of an indivisibe good Stergios Athanassogou, Gauthier de Maere d Aertrycke January 2015 Abstract Suppose we wish to randomy aocate a singe indivisibe good to a group of agents. Efficiency dictates that the good be aocated with probabiity 1 to the agent vauing it the most. Meanwhie, equity impies that each agent shoud receive the good with equa probabiity. In the current paper, we offer a precise way of formaizing, and utimatey resoving, the tradeoff between these conficting desiderata. We begin by introducing a parsimonious parametric framework, in which an aocation s inequity is defined by its Eucidean distance to the perfecty equitabe equa-probabiity aocation. Subsequenty, using toos from convex and conic programming, given a chosen eve of maximum aowabe inequity, we provide a cosed-form expression for the corresponding maximum achievabe utiity as we as the aocation associated with it. Moreover, we show that this function is differentiabe in the degree of inequity. These resuts aow for the appication of standard methods to sove for the optima eve of inequity baancing the conficting objectives of efficiency and equity. Keywords: random aocation, equity-efficiency tradeoff, second-order cone programming The views expressed herein are purey those of the authors and may not in any circumstances be regarded as stating an officia position of the European Commission. Athanassogou: Corresponding Author. European Commission Joint Research Center, Econometrics and Appied Statistics Unit, athanassogou@gmai.com. De Maere d Aertrycke: EDF Suez and FEEM 1

2 1 Introduction Consider the probem of randomy aocating a singe indivisibe good to a group of agents in an efficient and fair manner. Efficiency dictates that the good be aocated to the agents) having the highest vauation. 1 On the other hand, equity impies that each agent gets an equa a priori probabiity of receiving the good. These objectives are ceary in tension with each other. In this paper, we offer a precise way of formaizing, and utimatey resoving, the tradeoff between these conficting desiderata of efficiency and equity. We begin by introducing a parsimonious parametric framework in which an aocation s inequity is defined via its Eucidean distance to the perfecty equitabe equa-probabiity aocation. This Eucidean metric of inequaity amounts to a specia case of the generaized-entropy cass of inequaity measures Cowe [6]). Using toos from convex and conic programming, we are abe to rigorousy examine its impications in the context of random assignment of a singe good. Specificay, given a chosen eve of maximum inequity, we provide a cosed-form expression for the corresponding maximum achievabe utiity, as we as the aocation associated with it. Moreover, we show that this function is differentiabe in the degree of aowabe inequity. This aows us to use standard cacuus arguments to sove for the optima eve of inequity which reconcies the twin goas of equity and efficiency. In other words, we are abe to rigorousy characterize the corresponding equity-efficiency tradeoff and propose a tractabe way of resoving it. Setting aside its appicabiity to the random assignment context, our expicit soution of the associated second-order cone program coud be of genera technica interest for schoars in OR. 2 In addition, whie the Eucidean nature of the constraints imposes very tight structure to our inquiry, the anaysis of Section 3 may be usefu for the soution of other noninear optimization probems. However, we must stress that this is a specuative caim, and we do not wish to overse the reach of our resuts. Reated iterature. The paper offers a nove and admittedy) idiosyncratic contribution to the wefare-economics iterature on fair division Mouin [9]). It differs from existing work in important ways, conceptua as we as methodoogica. First, concern for fairness is formaized via a parametric constraint on the aowabe deviation from the perfecty-equitabe aocation and not through axiomatic properties such as envy-freeness. Envy-freeness essentiay requires that each agent prefer her aocation to that of her peers. It is a strong property that features prominenty in the axiomatic economics iterature, which in turn seeks to characterize rues 1 To avoid triviaities, we assume throughout that agents have stricty positive vauations for the good. 2 Earier work by Iyengar [8] addressed a simiar optimization probem in passing, in a very different context see Lemma 4.2 in Iyengar [8]). See footnote 5 for more detais. 2

3 on the basis of their efficiency, fairness, and incentive-compatibiity properties [9]. In contrast, our optimization-based approach focuses on finding efficient aocations subject to equity constraints, whose stringency captures society s aversion to inequity. As such, the resoution of the equity-efficiency tradeoff is cast as an optimization probem whose optima soution, we prove, can be expicity computed using famiiar first-order conditions. The operations-research iterature has not deat extensivey with probems of random assignment and fair division. However, a few recent papers by Bertsimas, Farias, and Trichakis [3, 4] are reevant to our work. Bertsimas et a. [4] study a reated variant of the equity-efficiency tradeoff in the aocation of utiities to agents. In their mode, concern for fairness is modeed via a socia panner s or manager s) choice of objective function. Specificay, assuming a constant easticity CES) functiona form for the wefare function, the socia panner is caed to set the inequaity aversion parameter. The higher this parameter is the more she is averse to unequa aocations, and the mode nests the two extremes of utiitarianism and Rawsian max-min. Subsequenty, Berstimas et a. [4] provide a tight worst-case bound on the price of fairness, i.e., the efficiency oss, associated with a particuar choice of inequaity aversion. 2 Mode Description Consider an economy with a set of agents indexed by n = 1, 2,..., and a singe indivisibe good. Each agent n has a stricty positive vauation u n > 0 for the good. An aocation for this economy is given by a vector p = p 1, p 2,..., p ) beonging to the 1-dimensiona simpex { 1 = p R : p 0, p n = 1}, where p n denotes the probabiity that the good is aocated to agent n. An aocation s p s utiity is the expected vaue p nu n. Ceary, maximum utiity is attained if we assign zero probabiity to a agents whose utiities are stricty ess than max n u n. Conversey, an aocation p s inequity, denoted by Ip) is measured via the square of its Eucidean distance to the equa-probabiity vector: Ip) = p n 1 ) 2 1) This way of capturing inequity is nonstandard. Dividing Ip) by 2 equas a specia case of the cass of generaized entropy indices of inequaity Cowe [6]) for α = 2. This famiy of indices has been characterized on the basis of severa desirabe properties, incuding the we-known Pigou-Daton transfer principe [6]. Consider now the set of aocations Pb) Pb) = { p 1 : Ip) b }, 2) 3

4 p 3 0,0,1) b 2 Pb 2 ) b 1 1/3,1/3,1/3) 1,0,0) p 1 Pb 1 ) 0,1,0) p 2 Figure 1: The set Pb) for = 3 and two eves of inequity b 1 < circe of radius b 1, whie Pb 2 ) is the ova-shaped trianguar figure subsuming Pb 1 ). 1 6 < b 2 2 < 3. Pb 1) is the inner where b [ 0, 1 ]. Thus Pb) consists of a aocations whose inequity is no greater than b. Setting b = 0 means that Pb) reduces to the the perfecty-equitabe equa-share singeton, whie b = 1 to the entire simpex.3 Figure 1 provides a graphica visuaization of set Pb) for the case of = 3 agents and different vaues of b. Consider two eves of inequity b 1 and b 2. Assume that b 1 <. In this case, no pair of agents can be assigned a cumuative probabiity of 1, impying that a agents must be assigned positive probabiities. Hence, the set Pb 1 ) is just a circe of radius b 1 that is embedded in the simpex 2 and centered at the equa-probabiity aocation 1/3,1/3,1/3). This circe does not touch the simpex s boundary. Conversey, when b = b 2 such that b 2 > 2 and b 2 < 3, the socia panner a) aows for a share of 1 to be paced on sets of pairs of agents and thus 0 for sets of singe agents), whie b) setting an upper bound of b < 1 on the probabiity than a singe agent can receive. In this case, the set Pb 2 ) has a trianguar ova-ike shape, with parts of it intersecting the boundary of 2. Moreover, the coser b 2 is to 2 3, the coser the curved segments of Pb 2) are to becoming two intersecting straight ines. 2 When b = 3, this transformation is compete and we have Pb) = 2. 3 The atter statement hods in ight of the fact that vaues of b > cannot enarge the feasibe set. This is because the maximizers of ) pn 1 2 over the set of probabiity vectors concentrate a probabiity mass on one agent, eading to a inequity of ) ) ) 1 2 = 1. 4

5 3 Equity-constrained efficient aocations Suppose a socia panner wants to determine a probabiistic aocation of the good to the agents. In doing so, she wishes maximize expected utiity whie aso imposing an upper bound on inequity. To this end, define the function V : [ 0, 1 V b) = max p Pb) ] R+, where p n u n, 3) representing the maximum achievabe utiity, given an aowabe inequity of b. We refer to V b) as the efficiency corresponding to inequity b. We begin by stating a few straightforward properties regarding continuity, monotonicity, and concavity of V. Proposition 1 V is increasing, concave, and continuous. Proof. Foows from standard arguments see Onine Appendix). Before we state our next resut we need to introduce additiona notation. First, et k denote the set of agents sharing the k th order statistic of {u 1, u 2,..., u } and we et k = k. There are a tota of such sets where {1, 2,..., }, With apoogies for the cunky notation, denotes the set of agents sharing the maximum of {u 1, u 2,..., u }. Furthermore, et + k = i=k i, k = k i=1 i and + k = + k, k = k. Our mode structure enabes us to easiy show the foowing Lemma. Lemma 1 Consider the function V of Eq. 3). i) Define b 1 b [ b, 1 ]. 1. V is stricty increasing in b [0, b) and equa to max n u n in ii) Suppose b [0, b]. There exists a unique optima soution for probem 3), denoted by p b), and it must satisfy the quadratic constraint of set 2) with equaity. Proof. Foows from standard arguments see Onine Appendix). Lemma 1 suggests that b is an important threshod. It represents the eve of inequity above which the set Pb) aows for the maximum agent utiity to be attained as an objective function vaue of 3). Moreover, for eves of inequity smaer than this extreme vaue, the optima soutions of probem 3) wi be unique and bind the quadratic inequity constraint associated with set Pb). 4 4 For b b, 1 ] a aocations p b) satisfying p nb) = 0 for n and ) n p n b) 1 2 b 1 2 wi be optima soutions of 3). 5

6 We are now ready to prove the paper s first main resut. Theorem 1 estabishes that function V is differentiabe with respect to b everywhere on 0, 1 ) except at the point b defined in Lemma 1. Moreover, it formaizes a straightforward monotonicity property of the optima soution of 3) that is essentia to the derivation of the vaue function pursued in Theorem 2. In proving Theorem 1 we make extensive use of resuts from conic optimization, in particuar the duaity theory of second-order cone programming see Aizadeh and Godfarb [1]). Theorem 1 i) V is differentiabe everywhere on b 0, 1 ) except b. ii) Let p b) denote an optima soution of 3). The foowing eves of inequity } b k = min {ˆb : b ˆb, we have p n b) = 0 for a n k, 4) where k {1, 2,..., 1}, are we-defined and stricty increasing in k. Part i) of Theorem 1 shows that V is a smooth function of b, except for a singe kink at the minimum eve of inequity at which the maximum efficiency can be obtained. Part ii) impies that b k can be interpreted in the foowing way: it denotes the threshod eve of inequity such that, for a b greater than or equa to it, at optimaity no probabiity mass is ever aocated to agents beonging in k i.e., having a u n that is ess than or equa to the k th order statistic of {u 1, u 2,..., u }). Thus, when b reaches and exceeds this eve, one can safey disregard agents in k. Whie the existence and monotonicity of these inequity threshods may be intuitive, their proofs are reativey invoved, requiring insights from conic duaity [1]. Having estabished differentiabiity, we go on to provide a set of differentia equations that V must satisfy. These differentia equations wi prove vauabe in the subsequent derivation of a cosed-form expression for V. Proposition 2 Suppose agent n k satisfies n k k. V satisfies the foowing differentia equation: 2 d db V b) p n k b) 1 b ) = u nk V b), b 0, b k ). 5) Let u k) denote the k th order statistic of {u 1, u 2,..., u }, where k = 1, 2,...,. Define the foowing quantities: u + k = n + k + k u n, 6) d + k = + k+1 uk) u + k+1) 2, 7) 6

7 where k {1, 2,..., } we set d + 0). The term u + k is simpy an average of the vaues of the set {u 1, u 2,..., u } that are greater than or equa to its k th order statistic. The term d + k measures the squared difference between the k th order statistic of {u 1, u 2,..., u } and the average of those greater than it, adjusted for size of the atter group. ow, we use the above quantities to define b + 0 0, b+ 1 b + k = k + k =k+1 + k+1 d+ k d + + 8), k {1, 2,..., 1}. 9) Thus, we have b + 1 = b. Straightforward agebra yieds the foowing monotonicity property, where k {1, 2,..., 1}: b + k < b+ k+1. We are now ready to state our second main resut and provide a cosed-form expression for V. To prove the foowing Theorem, we expicity sove the systems of differentia equations estabished in Proposition 2. Theorem 2 Consider the optimization probem 3) and the vectors u +, d +, b +) defined in Eqs. 6)-7)-8)-9). The vector b + satisfies b + k = b k k {1, 2,..., 1} where b k is defined in Eq. 4). The function V satisfies V b) = u + k + b k 1 + k ) 1 =k + d +, b [ b + k 1, b+ k ), 10) where k = 1, 2,..., 1. For b b + 1 = b, V b) = max n u n. Theorem 2 shows that V wi be a concatenation of appropriatey-specified square-root-ike functions when k = 1, we set k 1 + k 0).5 These concatenations occur at eves of inequity b + = b which are interpreted by Eq. 4), and can be computed expicity through Eq. 9). The curvature of these functions is driven by the dispersion of agents utiities, as captured by the quantities d + k of Eqs 7). 5 These resuts are consistent with the more genera anaysis of Section 4.2 in Iyengar [8]. However, Iyengar uses different arguments and does not prove differentiabiity in b, nor does he derive and interpret differentia equations and a precise formua for V and its optima soution p we provide the atter in Coroary 1). Instead, his anaysis is concerned with determining the compexity of finding the optima soution. 7

8 The fraction k 1 in the square root represents the minimum eve of b at which it becomes + k possibe to assign zero share to a agents in k 1. Hence, the fact that b k 1 = b+ k 1 > k 1 + k for a k see Eq. 9)), ensures that Eq. 10) is we-defined. We can now combine the various resuts we have estabished to characterize the optima soution of 3). Coroary 1 Suppose b < b + 1 = b. Consider any agent n for some {1, 2,..., }. The unique optima aocation p b) satisfies p 1 + u nb) = + n u + ) k k b k 1 + k i 1 i=k 0 b + i d + i b [ b + k 1, ) b+ k, k = 1, 2,..., [b +, b+ 1 ). ow suppose b b + 1. Then, a vectors p b) satisfying p nb) = 0 for n and p b) Pb) wi be optima. This set is a singeton at b = b + 1 = b. Coroary 1 provides succinct expressions for the optima agent probabiities given an inequity eve b. Exampe 1. Let us now iustrate the anaytic findings of this section with an exampe. Consider the foowing probem instance given in Tabe 1. k u k) k + k k ū + k d + k b + k / /9 Tabe 1: Data for Exampe 1. As we can see from Tabe 1, agent vauations can be sorted into five order statistics so that = 5. In addition, maximum efficiency can be obtained for a inequity eves greater than or equa to b + 4 = b = = 2/9. Appying Eqs. 6)-7) to the probem data, we obtain the u+ k and d + k vaues and use them to cacuate the b+ k threshods via Eq. 9). The resuts are isted in Tabe 1. 8

9 Figure 2: Graphing function V for Exampe 1. Different coors are used to indicate the five different functiona forms of V within the intervas [b + k 1, b+ k ) for k = 1, 2,..., 5. ote that the x-axis is truncated before b + 5 = 8/9.) ow, we are ready to appy Theorem 2, which in turn yieds the foowing expression for V : V b) = b b [0, ), b 1/72) b [0.0422, ), 35/ b 1/18) b [0.0864, ), 32/ b 4/45) b [0.0982, 2/9), 7 b [2/9, 8/9]. Figure 2 graphicay depicts the above agebraic expression of function V. Different coors are used to indicate the five different functiona forms of V within the intervas [b + k 1, b+ k ) for k = 1, 2,..., 5. The differentiabiity of V that was proved theoreticay may be verified visuay as we. Moreover, as predicted by the theory, V s ony kink occurs at b = b + 4 = b = 2/9 where V first reaches its maximum vaue of 7. 4 Baancing equity and efficiency 4.1 Genera cost functions Theorem 2 enabes us to rigorousy investigate the tradeoff between equity and efficiency. For exampe, it aows us to answer the foowing two questions: a) Given an upper bound b on inequity, how cose are we to fu efficiency? This quantity is 9

10 captured by the efficiency ratio corresponding to inequity b, denoted by Eb), where: Eb) = V b) max n u n. b) What is the minimum inequity we shoud toerate if we want to achieve an efficiency ratio of at east x? Denoting this quantity by b x, it satisfies: { b x = min b : V b) x max n The above points a)-b) are interesting questions that can be easiy deat with via Theorem 2. However, they do not expoit the fu range of the previous section s anaysis. In particuar, they do not make use of the differentiabiity of function V. To make use of this resut, we may consider directy modeing the cost of inequity. To wit, suppose an inequity of b entais a cost cb), where c is a twice differentiabe, increasing and convex function. Suppose, further, that a socia panner wishes to baance the competing objectives of efficiency and equity. In doing so, she wishes to find a eve of inequity b OP T where the margina gain in efficiency is equa to the margina cost of inequity. Hence, the objective is to determine an inequity eve b such that u n }. b OP T = arg max b [0, 1 ] {V b) cb)}. 11) Given Lemma 1, when soving for b OP T it is sufficient to restrict ourseves to b [ 0, b ], as vaues of b above this range increase inequity whie having no effect on efficiency. Moreover, in ight of Proposition 1 and Theorem 1, b OP T is unique and can be cacuated by appying first-order conditions on the function V b) cb). At an interior soution, these reduce to: dv db b OP T ) dc b OP T ) = 0. 12) db The derivatives of V can be determined via the expression of Theorem 2. estabishes that for k = 1, 2,... 1: 6 Simpe agebra Since c is increasing and convex, we know that dc db dv db b+ k ) = k+1 u+ k+1 u k)). 13) is positive and weaky increasing. Thus, soving for b OP T reduces to studying the evoution of dc db aong [0, b] and finding the first interva [b + k 1, b+ k ] such that k u+ k u k 1)) > dc db b+ k 1 ) k+1 u+ k+1 u k)) < dc db b+ k ). 14) 6 At k = 1, the expression beow is a eft derivative reca that V is not differentiabe at b + 1 = b). 10

11 When this interva has been found, the optima inequity eve is computed by considering the corresponding expression of V within it, avaiabe via Eq 2), and soving for b OP T by appying the first-order conditions of Eq. 12). 4.2 A specia case: cb) = max n u n b Suppose cb) = max n u n b so that maximizing V b) cb) is equivaent to maximizing b. As mentioned earier, the quantity inequity b. Define the foowing quantities V b) max n u n V b) max n u n represents the efficiency ratio corresponding to q 0 =, q 0 q k = + k+1 d+ k = + k+1 u+ k+1 u k)), k {1, 2,..., 1}. 15) By simpe agebra it is easy to see that q k are stricty decreasing in k. This is unsurprising since, by Eq. 13), we have q k = 2 dv db b+ k ) and V is concave reca Proposition 1. With this choice of cost c, we are abe to competey characterize the optima eves of inequity and efficiency, as we as the corresponding aocations. Theorem 3 Consider the optimization probem of Eq. 11) for cb) = max n u n b. Let k OP T {1, 2,..., } uniquey satisfy: q k OP T 2 max n u n < q k OP T 1. i) The optima inequity eve b OP T, and its efficiency V OP T V b OP T ), are uniquey given by: b OP T = k OP T 1 + k OP T + V OP T = u + k OP T + p OP T = 1 =k OP T d + + 4max n u n ) 2. 1 =k OP T ii) The optima aocation p OP T = p b OP T ) satisfies 1 + k OP T d max n u n. + u + u k OP T 2 max n + k OP T 0 k OP T 1. Thus, we see that the optima inequity eve b OP T impies an aocation p OP T in which an agent is assigned the good with probabiity 0 if and ony he beongs in the set k OP T 1, i.e., u is ess than or equa to the k OP T 1) th ordered statistic of the u n s. This, in turn, impies further structure to the optima probabiities as per the foowing coroary. 11

12 Coroary 2 Consider the optima inequity eve b OP T and its corresponding aocation p OP T. There wi be at most 3 agents such that p OP T = 0. Exampe 1 continued. Appying Theorem 3 to the data in Exampe 1, we first cacuate: q 1 = 32, q 2 = 20, q 3 = 17, q 4 = 4.5. Since 2 max n = 14, we obtain k OP T = 4. Appying Theorem 3 yieds: b OP T = =3 d + + 4max n u n ) 2 = d + 4 V OP T = u =4 + = max n u n = , 14 = and p OP T = = , = , 0 3. Hence, the optima eve of inequity impies that just the five agents having the highest and second highest vauations get assigned the good with positive probabiity. 5 Directions for future research The work presented herein suggests severa fruitfu avenues of future research. First, it woud be interesting to conduct an anaysis simiar to the one appearing in Sections 3 and 4, but repacing the inequity measure of Eq. 1) with aternative, more popuar measures such as the Gini, Thei, or Atkinson indices. As the atter have more compex noninear functiona forms than 1), such an exercise woud be quite chaenging. Cosed-form expressions woud be harder to come by and it is not cear if the differentiabiity properties of the associated V functions woud extend. An aternative, especiay chaenging extension of the current work woud invove the introduction of strategic considerations to the mode. In particuar, we coud assume that agent vauations are private information that is strategicay reveaed to the socia panner. In this environment, we coud attempt to derive an auction-ike mechanism that induces, in ash equiibrium, truthfu reveation of agent vauations to which, in turn, fairness considerations via Eq. 1) are appied. The atter constraint woud ceary infuence the strategic behavior of agents, and so woud need to be taken into account when deriving the auction mechanism. 12

13 Appendix A1 Proofs of main resuts Theorem 1. We begin with part i). Given x = x 0, x) R n+1 we introduce the foowing notation to denote incusion in a second-order cone of dimension n + 1 x 0, x) L 2 n+1 x 0 x 2. We foow Aizadeh and Godfarb [1] to write 3) as a prima conic program Pb) and introduce its dua Db) for carity, next to the prima constraints we indicate the corresponding dua variabes): max p,q,q 0,θ u n p n s.t. p n + q n = 0, n, y n ) p n = 1, y 0 ) Pb) θ n = 0, n, γ n ) q 0 = b + 1, β 0) p n, θ n ) L 2 2, n q 0, q) L 2 n+1, min y 0 + y,y 0,γ,β 0,z p,z q,z q0,z θ s.t. b + 1 β 0 y 0 y n + z pn = u n, n y n + z qn = 0, n γ n + z θn = 0, n Db) β 0 + z q0 = 0 z pn, z θn ) L 2 2, n z q0, z q ) L 2 n+1. Since both the prima and the dua have feasibe stricty interior soutions, strong duaity hods see Theorem 13 of [1]). Without oss of generaity, we can immediatey simpify Db) by setting z θ = γ = 0 and z p 0. Correspondingy, we can eiminate the variabe z q by repacing it with y. Finay, it is evident that at optimaity the quadratic constraint of the dua wi be binding so that zq0 = β 0 = y n) 2 = y2 n. Coecting a of these observations we may re-write the dua in the foowing much simper way: D 1 b) = min y 0 + b + 1 y y,y n 0 2 s.t. u n + y 0 + y n 0, n = 1, 2,...,. 16) Examining 16) we deduce that at optimaity y n = max0, u n y 0 ). Thus we may simpify the dua even further to an unconstrained optimization probem with just one variabe: D 2 b) = min y 0 + b + 1 max0, u n y 0 ) y ) 13

14 By strong duaity the dua optima objective wi be bounded between 1 u n and u ). We immediatey see that soutions satisfying y 0 > u ) resut in stricty greater objective function vaues than y 0 = u ), so that we can safey disregard them. Conversey, soutions satisfying y 0 < 0 yied y 0 + b + 1 max0, u n y 0 ) 2 = y 0 + b + 1 u n y 0 ) 2 > y 0 + b + 1 u 1) y 0 ) = b + 1 u 1) + y 0 1 b + 1). Thus, vaues of y 0 < u 1 b+1 resut in a stricty greater objective function vaue than y 0 = u ) and hence can aso be disregarded. With these observations we may rewrite the dua 17) in the foowing way: D 3 b) = min [ u ) y 0 1 b+1,u ) ] y 0 + b + 1 max0, u n y 0 ) 2 18) The domain of D 3 b) is thus compact, for any b > 0. For vaues of b [0, b) we know that the optima soution of the prima wi be stricty ess than u ). Thus, strong duaity impies that for a b 0, b), any optima soution y b) must satisfy y b) < u 1). However, notice that the objective function of D 3 is stricty convex for y 0 < u 1). Thus, we may deduce that when b 0, b) D 3 b) admits a unique optima soution y 0 b). The above observation impies that we can appy Danskin s theorem see Proposition B.25 in Bertsekas [2]) to concude that the optima dua objective vaue, and therefore by strong duaity V b) as we, is differentiabe at a b 0, b) and that dv db b) = max0, u n y0 b))2, b 0, b ). 19) 2 b + 1 We now show that y 0 b) is stricty increasing in b 0, b). Consider b 1 < b 2 with both beonging in 0, b ) and their optima soutions y0 b 1) and y0 b 2). By uniqueness of y0 b) in this range of b we have y0b 1 ) + b max0, u n y0 b 1)) 2 < y0b 2 ) + b max0, u n y0 b 2)) 2 y0b 2 ) + b max0, u n y0 b 2)) 2 < y0b 1 ) + b max0, u n y0 b 1)) 2. 14

15 Summing the above inequaities and rearranging terms yieds b ) b max0, u n y0 b 1)) 2 max0, u n y0 b 2)) 2 > 0 max0, u n y0 b 1)) 2 max0, u n y0 b 2)) 2 > 0 y0b 2 ) > y0b 1 ). We now discuss now the differentiabiity of V at b = b. ote that the optima soution y 0 b) is not unique; instead it can take any vaue in the interva [u 1), u ) ]. Hence Danskin s theorem impies that the subdifferentia of V b) at b wi consist of a convex combinations of u ) u 1) ) 1 2 b+ and 0. We now prove part ii). Let us go back to the origina prima-dua pair Pb), Db)) and consider a pair of optima soutions of the prima and dua probems. By Lemma 1 the prima optima soution p b), q b), θ b), q0 b)) is unique, whie our reasoning in part i) estabished the uniqueness of the optima dua variabes β 0 b), y b), y 0 b), z pb), z qb)). Appying Theorem 16 and part ii) of the compementarity conditions of Lemma 15 of Aizadeh and Godfarb [1], we arrive at the foowing conditions: q0b)z qnb) + β0b)q nb) = 0 b + 1 y nb) + ynb) 2 p nb) = 0, n = 1, 2,..., b + 1 max0, u n y0b)) + max0, u n y0 b))2 p nb) = 0, n = 1, 2,..., 20). When b < b, strong duaity impies y 0 b) < u 1) which in turn ensures max0, u n y0 b))2 > 0. As mentioned earier, when b = b y0 b) can take any vaue in [u 1), u ) ] so we choose one that again yieds max0, u n y 0 b)) 2 > 0. Hence, the compementarity conditions 20) yied p nb) = 0 u n y 0b) 0, n = 1, 2,...,. 21) Since y 0 b) is stricty increasing in b in 0, b) and im b 0 + y 0 b) = and im b b y b) = u 1), Eq. 21) impies the existence of a set {b 1, b 2,..., b 1 } such that 0 < b 1 < b 2 <... < b 1 = { b } {p nb) = 0 n } k b bk, k = 1, 2,..., 1. 15

16 Proposition 2. Focusing on optimization probem 3), we introduce Lagrangian mutipiers and write the Karush-Kuhn-Tucker KKT) conditions: u n 2λ p n 1 λ ) + µ + ν n = 0, n {1, 2,..., } 22) p n 1 ) ) 2 b = 0, λ 0 23) p n 1 ) 2 b, p n = 1, p 0 24) ν n p n = 0, ν n 0, n {1, 2,..., }. 25) Since our probem is concave with affine equaity constraints and satisfies Sater s condition see section in [5]), strong duaity hods and the KKT conditions 22)-25) wi be necessary and sufficient for both prima and dua optimaity. In other words, the duaity gap is zero and the vector p, ν, λ, µ ) satisfies 22)-25) if and ony p and λ, ν, µ are prima and dua optima respectivey see section in [5]). From Lemma 1 we know that there exists a unique prima optima soution p. By strong duaity, the Lagrangean dua probem admits an optima soution, and we refer to it by λ, ν, µ. 7 Since V b) is differentiabe Theorem 1) and strong duaity hods we foow Section in Boyd and Vandenberghe [5] to deduce the foowing simpe reation: d db V b) = λ b), b 0, b ). 26) Eq. 26) means that we can now focus on cacuating the Lagrange mutipier λ b). Before we do so we note the foowing usefu identity p nb) 1 ) 2 = = p nb) p nb) 1 ) 1 p 1 nb) + 2 p nb) p nb) 1 ). 27) Mutipying both sides of Eq. 22) by p nb) and then summing over a n = 1, 2,.., obtains 27) Lemma?? u n p nb) 2λ b) p nb) p nb) 1 ) + µ b) u n p nb) 2λ b) p nb) 1 ) 2 + µ b) = 0 µ b) = 2λ b) b p nb) = 0 u n p nb). 28) 7 ote that at this point one can manipuate the KKT conditions 22)-25) to show that the Lagrangean dua s optima soution is aso unique. 16

17 ow we consider Eq. 22) for agent n k k. By part b) of Theorem 1 we must have p n k b) > 0 if and ony if b [0, b k ). Substituting the vaue of µ b) obtained in Eq. 28), and appying the compementary sackness condition 25) we obtain u nk 2λ b) p n k b) 1 ) = u n p nb) 2λ b) b 26) 2 d db V b) p n k b) 1 b ) = u nk V b), b 0, b k ). 29) Theorem 2. Reca the definition of b k of Eq. 4). Consider first b 0, b 1 ) so that p nb) > 0 for a b 0, b 1 ) and n. Recaing Proposition 2 and adding Eqs. 29) for a n yieds the foowing differentia equation dv b) 2b = V b) + u n, b 0, b db 1). 30) n Soving differentia equation 30) eads to the foowing expression: V b) = C 1 b + n u n, b [0, b 1), 31) where C 1 is a constant to be determined. Consider now b [b k 1, b k ) for k {2, 3,..., 1}. In this range of b we wi have p nb) > 0 if and ony n + k. Adding Eqs. 29) for a such n + k yieds the foowing differentia equation 2 ) k 1 + k b dv db = n + k Soving differentia equation 32) gives the foowing: V b) = u + k + C k u n + k V b), b [b k 1, b k ) 32) + k b k 1, b [ b k 1, b k ), 33) for k {2, 3,..., 1}, where C k is a constant to determined. Finay since b 1 = b we use Lemma 1 to concude [ V b) = max u n, b b 1, 1 ]. 34) n Putting together Eqs. 31), 33), and 34) we see that V wi equa n un + C 1 b b [0, b 1 ) V b) = u + k + C k + k b k 1 b [ b k 1, k) b, k = 2, 3,..., 1 [ ] max n u n b b 1, 1 35) 17

18 ) for appropriatey chosen constants C 1, C 2,..., C 1 ) and b 1, b 2,..., b 1. By Proposition 1 and Theorem 1, V is continuous everywhere and differentiabe everywhere at 0, 1 ) except b. ) Thus, the vectors C 1, C 2,..., C 1 ) and b 1, b 2,..., b 1 must fufi these criteria of continuity and differentiabiity and are thus uniquey determined by the foowing system of noninear equations 36)-43): Case 1: = 2. n u n + C 1 b 1 = max u n 36) n b 1 = ) Case 2: 3. n u n + C 1 b 1 = u C b ) C 1 b 1 = u + k + C k C ) 2 + b k b k k 1 = u+ k+1 + C k+1 + k+1 b k k, k = 2, 3,..., 2 40) C k + C k k+1 + k+1 =, k = 2, 3,..., 2 41) + k b k k 1 + k+1 b k k u C 1 b 1 = b 1 2 = max n u n 42) 43) It now remains to show that the soution of System 36)-43) wi eventuay ead to the expression of the Theorem. To do this we cacuate expicity the C k and b k s and then show how appying them to formua 35) yieds the desired resut. We begin with Case 1 and = 2. That b 1 = ) is triviay true. Then, Eq. 36) yieds 2 C 1 = u 2) u + ) 1 1 = d ) We now focus on Case 2 and 3. Once again, we have by definition b 1 = 1 1, whence Eq. 42) impies ) C 1 = u ) u + 1 = d + 1.

19 Focusing on Eq. 41) for k {2, 3,..., 2} and soving for C k yieds: C k = + k b k k 1 + k C k+1 + k+1 + k+1 b k k. 46) Pugging 46) into Eq. 40) we obtain: C k+1 + k+1 b k k 1 + k+1 + k k 1 + k b k + k+1 b k k = ū + k ū+ k+1. 47) After some agebra, the eft-hand-side of Eq. 47) can be simpified so that: C k+1 + k+1 k 1 + k k C k+1 + k C k+1 + k+1 b k k k + k+1 + k+1 b k k = + k+1 b k k = n + k un + k n + k+1 un + k+1 u k) + k+1 ) n + u n k k+1 + k + k+1 + = u k) uk+1 48) Combining Eqs. 41) and 48) obtains for k = 2, 3,..., 2: C k = + k+1 + uk) u + k+1) + k b k k 1 49) k ) 2 C k+1 + b u k) u + k k+1 k =, 50) + k+1 which after some simpe agebra eads to the foowing nonhomogeneous inear recursion for the squares of the C k s: C k ) 2 = + k+1 + k C k+1 ) 2 + k + k d + k + k, k = 2, 3,..., 2. 51) Soving recursion 51) backwards with previousy derived) initia vaue C 1, taking square roots, and recaing the positive sign of the C k s, eads to a simpe expression for the C k s: C k = k + d +, k = 2, 3,..., 1. 52) Appying Eq. 52) to Eq. 50) yieds =k b k = k + k =k+1 + k+1 d+ k d + +, k = 2, 3,..., 1. 53) 19

20 Finay pugging C 2 into Eqs. 38)-39) yieds 1 C 1 = =1 + b 1 = d +, 54) 1 =2 + 2 d+ 1 d ) ote that Eqs. 54)-55) are consistent with the resuts for Case 1 as given by Eqs. 44)-45). Thus there is no more need to distinguish between Case 1 and 2. Finay, appying Eqs. 52)-53)-54)-55) to Eq. 35) and performing eementary agebra estabishes the resut. Theorem 3. i) By strict concavity of V there wi be a unique optima soution. Using the expression for V of Theorem 2 and the fact that it is differentiabe in 0, b ) Theorem 1), we may appy first-order conditions on V b) cb). Doing so impies that there wi exist a unique k OP T {1, 2,..., } such that the optima soution b OP T satisfies: bop T = k OP T 1 + k OP T + 1 =k OP T + d + 4max n u n ) 2 and b OP T ) [b +k OP T 1, b+k OP T Looking at the expressions for b + of Eqs. 9) we see that k OP T wi be such that it uniquey satisfies ) 2 b OP T b max + k OP T 1 4 u n + n b OP T < b + k OP T k OP T 1 + k OP T + Agebraic manipuations estabish that 1 =k OP T 56) k OP T d + k OP T 1 57) d + + 4max n u n ) 2 < k OP T + k OP T =k OP T +1 d k OP T +1 d+ k OP T.58) k OP T + k OP T =k OP T +1 d k OP T +1 d+ k OP T = 1 k OP T =k d + OP T + +, k OP T k OP T +1 d+ k OP T so that Eq. 58) is equivaent to ) 2 4 max u n > + n k OP T +1 d+ 59) k OP T Putting Eqs. 58)-59) together yieds: q k OP T < 2 max n u n q k OP T 1. The vaues of V OP T and p OP T foow from the direct appication of Theorem 2 and Coroary 1 to b OP T. 20

21 Coroary 2. Suppose not. Then, by Theorem 3 we must have k OP T { 1, } since otherwise k OP T < 1 woud automaticay impy that at east 3 agents i.e., the ones with the three highest vauations) are granted the good with positive probabiity. Suppose first that k OP T =, so that just agents in = + ) are assigned positive probabiity. By Theorem 3, we must have eading to a contradiction. q 1 2 max u n max u n u 1) ) 2 max n n }{{} u n > 2, n >0 ow, suppose k OP T = 1. Again, by Theorem 3, we must have q 2 2 max u n + 1 u 1) u 2) ) 2 max n }{{}}{{} u n + 1 > 2, n <max n u n eading to a contradiction. >0 21

22 References [1] Aizadeh, F., and Godfarb, D. 2003). Second-order cone programming. Mathematica Programming B, 95, [2] Bertsekas, D. 1999). oninear Programming, Bemont, MA: Athena Scientific. [3] Bertsimas, D., Farias, V. F., and Trichakis,. 2011). The price of fairness. Operations Research, 59, [4] Bertsimas, D., Farias, V. F., and Trichakis,. 2012). On the efficiency-fairness trade-off. Management Science, 58, [5] Boyd, S., and Vandenberghe, L. 2008). Convex Optimization. ew York, Y: Cambridge University Press. [6] Cowe, F. A. 2000). Measurement of inequaity. In Atkinson, A.B. and Bourguignon, F. ed.) Handbook of Income Distribution, orth Hoand, Amsterdam, 1, [7] Foster, J. E. and Sen, A.K. 1997) On Economic Inequaity. After a Quarter Century. Annexe to the Expanded Edition of On Economic Inequaity by A.K. Sen., Carendon Press, Oxford. [8] Iyengar, G. 2005). Robust dynamic programming. Mathematics of Operations Research, 30, [9] Mouin, H. 2004). Fair division and coective wefare. MIT press. 22

23 Onine Appendix - not for pubication Proof of Proposition 1. That V is increasing in b foows by definition. Consider the optimization probems given by the right-hand-side of Eq. 3) for b 1 [0, 1 ] and b 2 b 1 and denote their optima soutions by p b 1 ) and p b 2 ) respectivey. By feasibiity we may note the foowing: p nb 1 ) 1 ) 2 b 1, p nb 2 ) 1 ) 2 b 2. 60) Consider a convex combination of b 1 and b 2 given by bλ) = λb λ)b 2 for some λ [0, 1] and the optimization probem V bλ)) = max p Pbλ)) To prove concavity of V in b it suffices to show that p n u n. 61) V bλ)) λv b 1 ) + 1 λ)v b 2 ). To this end, consider the probabiity vector given by pλ) = λp b 1 ) + 1 λ)p b 2 ). By feasibiity of p b 1 ) and p b 2 ) we immediatey deduce that pλ) 0 and that p nλ) = 1. ow we may write p n λ) 1 ) 2 = triange ineq. 60) λ λ p nb 1 ) 1 ) + 1 λ) p nb 2 ) 1 )) 2 p nb 1 ) 1 ) ) λ) p nb 2 ) 1 ) ) [ λ b λ) b 2 ] 2 [ λb1 + 1 λ)b 2 ] 2 = bλ). 62) 2 By Eq. 62) and the observations immediatey preceding it we can concude that pλ) is feasibe for optimization probem 61). Thus we may write V bλ)) pλ) n u n = λ p nb 1 )u n + 1 λ) p nb 2 )u n = λv b 1 ) + 1 λ)v b 2 ), where the ast equaity foows from the assumed optimaity of p b 1 ) and p b 2 ). We now proceed to show continuity. By concavity V b) wi be continuous on the open interva 0, 1 ) so we need ony consider the endpoints 0 and 1. Since V b) is increasing in b we must 23

24 have im b 1 ). However, if im b 1 contradiction if we appy concavity to 1)/ and other vaues of b. ) V b) V 1 To prove continuity at b = 0 consider an ɛ > 0. ow et δ > 0 and write V δ) V 0) = V δ) V 0) = Höder s ineq. max u n n max u n n Thus, any choice of 0 < δ < ɛ 2 max n u n ) 2 proof. p nδ) 1 [ V b) < V 1 ) ) then we reach a p nδ) 1 ) u n p nδ) 1 ) ] max n u n δ. wi ensure that V δ) V 0) < ɛ, competing the Proof of Lemma 1. i) The function V b) is bounded above by u n for any n. This upper bound is attained by a probabiity vector p if and ony if it satisfies p n = 1, for some n Consider a subset, with cardinaity. The convexity of set Pb) impies that the vaue of b at which it first becomes possibe to assign probabiity 1 to subset is given by b ; ) = 1 1. The minimizer of b ; ) over is the entire set, yieding the desired resut. ow consider b < b and the optima soution p b). As b < b there must exist a j such that p j b) > 0. ow consider increasing b by an amount ɛ. For δ > 0 sma enough the soution p which is identica to p b) except that p j = p j b) δ and p k = p k b) + δ for some k wi be feasibe and resut in a stricty greater objective vaue, so that V b + ɛ) > V b). ii) Suppose first that b = b. It is cear here that the unique optima soution is given by p such that p n = 1/ for a n and p n = 0 otherwise. The quadratic inequity constraint binds by the definition of b. Consider now the case b < b and suppose there exists an optima soution p b) such that the quadratic inequity constraint is sack. As b < b there must exist an j such that p j b) > 0. For ɛ > 0 sma enough the soution p in which p j = p j b) ɛ and p k = p k b) + ɛ for some k wi be feasibe and resut in a stricty greater objective vaue, contradicting p b) s optimaity. Thus, a optima soutions must satisfy the quadratic inequity constraint with equaity. 24

25 We now prove uniqueness. Suppose there exist two optima soutions p,1 and p,2. By the preceding argument they must bind the quadratic equity constraint. Consider the set of probabiity vectors given by their convex combinations pλ) = λp,1 + 1 λ)p,2, λ [0, 1]. For λ 0, 1), pλ) wi satisfy the inequity constraint with strict inequaity, since: p n λ) 1 ) 2 = strict convexity < = λ λ p,1 n 1 ) + 1 λ) p,2 n 1 )) 2 [ λ pn,1 1 ) λ) pn,2 1 ) ] 2 pn,1 1 ) λ) p,2 n 1 ) 2 = λb + 1 λ)b = b. Thus a soutions pλ) are feasibe. That they are optima foows triviay by the assumed optimaity of p,1, p,2 and the inear objective function of 3). But this is a contradiction as a optima soutions must satisfy the quadratic inequity constraint with equaity. The second caim of the Proposition is trivia. 25