An explicit resolution of the equity-efficiency tradeoff in the random allocation of an indivisible good∗ Stergios Athanassoglou, Gauthier de Maere d’Aertrycke

†

January 2015

Abstract Suppose we wish to randomly allocate a single indivisible good to a group of agents. Efficiency dictates that the good be allocated with probability 1 to the agent valuing it the most. Meanwhile, equity implies that each agent should receive the good with equal probability. In the current paper, we offer a precise way of formalizing, and ultimately resolving, the tradeoff between these conflicting desiderata. We begin by introducing a parsimonious parametric framework, in which an allocation’s inequity is defined by its Euclidean distance to the perfectly equitable equal-probability allocation. Subsequently, using tools from convex and conic programming, given a chosen level of maximum allowable inequity, we provide a closed-form expression for the corresponding maximum achievable utility as well as the allocation associated with it. Moreover, we show that this function is differentiable in the degree of inequity. These results allow for the application of standard methods to solve for the “optimal” level of inequity balancing the conflicting objectives of efficiency and equity.

Keywords: random allocation, equity-efficiency tradeoff, second-order cone programming

∗

The views expressed herein are purely those of the authors and may not in any circumstances be regarded

as stating an official position of the European Commission. † Athanassoglou: Corresponding Author. European Commission Joint Research Center, Econometrics and Applied Statistics Unit, [email protected]. De Maere d’Aertrycke: EDF Suez and FEEM

1

1

Introduction

Consider the problem of randomly allocating a single indivisible good to a group of agents in an efficient and fair manner. Efficiency dictates that the good be allocated to the agent(s) having the highest valuation.1 On the other hand, equity implies that each agent gets an equal a priori probability of receiving the good. These objectives are clearly in tension with each other. In this paper, we offer a precise way of formalizing, and ultimately resolving, the tradeoff between these conflicting desiderata of efficiency and equity. We begin by introducing a parsimonious parametric framework in which an allocation’s inequity is defined via its Euclidean distance to the perfectly equitable equal-probability allocation. This Euclidean metric of inequality amounts to a special case of the generalized-entropy class of inequality measures (Cowell [6]). Using tools from convex and conic programming, we are able to rigorously examine its implications in the context of random assignment of a single good. Specifically, given a chosen level of maximum inequity, we provide a closed-form expression for the corresponding maximum achievable utility, as well as the allocation associated with it. Moreover, we show that this function is differentiable in the degree of allowable inequity. This allows us to use standard calculus arguments to solve for the “optimal” level of inequity which reconciles the twin goals of equity and efficiency. In other words, we are able to rigorously characterize the corresponding equity-efficiency tradeoff and propose a tractable way of resolving it. Setting aside its applicability to the random assignment context, our explicit solution of the associated second-order cone program could be of general technical interest for scholars in OR.2 In addition, while the Euclidean nature of the constraints imposes very tight structure to our inquiry, the analysis of Section 3 may be useful for the solution of other nonlinear optimization problems. However, we must stress that this is a speculative claim, and we do not wish to oversell the reach of our results. Related literature. The paper offers a novel and (admittedly) idiosyncratic contribution to the welfare-economics literature on fair division (Moulin [9]). It differs from existing work in important ways, conceptual as well as methodological. First, concern for fairness is formalized via a parametric constraint on the allowable deviation from the perfectly-equitable allocation and not through axiomatic properties such as envy-freeness. Envy-freeness essentially requires that each agent prefer her allocation to that of her peers. It is a strong property that features prominently in the axiomatic economics literature, which in turn seeks to characterize rules 1 2

To avoid trivialities, we assume throughout that agents have strictly positive valuations for the good. Earlier work by Iyengar [8] addressed a similar optimization problem in passing, in a very different context

(see Lemma 4.2 in Iyengar [8]). See footnote 5 for more details.

2

on the basis of their efficiency, fairness, and incentive-compatibility properties [9]. In contrast, our optimization-based approach focuses on finding efficient allocations subject to equity constraints, whose stringency captures society’s aversion to inequity. As such, the resolution of the equity-efficiency tradeoff is cast as an optimization problem whose optimal solution, we prove, can be explicitly computed using familiar first-order conditions. The operations-research literature has not dealt extensively with problems of random assignment and fair division. However, a few recent papers by Bertsimas, Farias, and Trichakis [3, 4] are relevant to our work. Bertsimas et al. [4] study a related variant of the equity-efficiency tradeoff in the allocation of utilities to agents. In their model, concern for fairness is modeled via a social planner’s (or manager’s) choice of objective function. Specifically, assuming a constant elasticity (CES) functional form for the welfare function, the social planner is called to set the inequality aversion parameter. The higher this parameter is the more she is averse to unequal allocations, and the model nests the two extremes of utilitarianism and Rawlsian max-min. Subsequently, Berstimas et al. [4] provide a tight worst-case bound on the “price of fairness”, i.e., the efficiency loss, associated with a particular choice of inequality aversion.

2

Model Description

Consider an economy with a set N of agents indexed by n = 1, 2, ..., N and a single indivisible good. Each agent n has a strictly positive valuation un > 0 for the good. An allocation for this economy is given by a vector p = (p1 , p2 , ..., pN ) belonging to the N − 1-dimensional simplex o n PN p = 1 , where pn denotes the probability that the good is ∆N −1 = p ∈
PN

n=1 pn un .

Clearly, maximum utility is at-

tained if we assign zero probability to all agents whose utilities are strictly less than maxn∈N un . Conversely, an allocation p’s inequity, denoted by I(p) is measured via the square of its Euclidean distance to the equal-probability vector:  N  X 1 2 I(p) = pn − N

(1)

n=1

This way of capturing inequity is nonstandard. Dividing I(p) by 2 equals a special case of the class of generalized entropy indices of inequality (Cowell [6]) for α = 2. This family of indices has been characterized on the basis of several desirable properties, including the well-known Pigou-Dalton transfer principle [6]. Consider now the set of allocations P(b)  P(b) = p ∈ ∆N −1 : I(p) ≤ b , 3

(2)

p3 (0,0,1)

b2

b1

P(b2)

(1,0,0)

p1

(1/3,1/3,1/3)

P(b1) (0,1,0)

p2

Figure 1: The set P(b) for N = 3 and two levels of inequity b1 <

q

1 6

< b2 <

q

2 3.

P(b1 ) is the inner

circle of radius b1 , while P(b2 ) is the oval-shaped triangular figure subsuming P(b1 ).

  where b ∈ 0, NN−1 . Thus P(b) consists of all allocations whose inequity is no greater than b. Setting b = 0 means that P(b) reduces to the the perfectly-equitable equal-share singleton, while b =

N −1 N

to the entire simplex.3

Figure 1 provides a graphical visualization of set P(b) for the case of N = 3 agents and q different values of b. Consider two levels of inequity b1 and b2 . Assume that b1 < 16 . In this case, no pair of agents can be assigned a cumulative probability of 1, implying that all agents must be assigned positive probabilities. Hence, the set P(b1 ) is just a circle of radius b1 that is embedded in the simplex ∆2 and centered at the equal-probability allocation (1/3,1/3,1/3). q This circle does not touch the simplex’s boundary. Conversely, when b = b2 such that b2 > 16 q 2 pairs of and b2 < 3 , the social planner (a) allows for a share of 1 to be placed on sets of q agents (and thus 0 for sets of single agents), while (b) setting an upper bound of 13 + b2 23 < 1 on the probability than a single agent can receive. In this case, the set P(b2 ) has a triangular oval-like shape, with parts of it intersecting the boundary of ∆2 . Moreover, the closer b2 is to q 2 the curved segments of P(b2 ) are to becoming two intersecting straight lines. 3 , the closer q When b = 23 , this transformation is complete and we have P(b) = ∆2 . The latter statement holds in light of the fact that values of b > NN−1 cannot enlarge the feasible set. This is
P 1 2 because the maximizers of N over the set of probability vectors concentrate all probability mass n=1 pn − N

2 1 2 on one agent, leading to a inequity of 1 − N + (N − 1) · N1 = NN−1 . 3

4

3

Equity-constrained efficient allocations

Suppose a social planner wants to determine a probabilistic allocation of the good to the agents. In doing so, she wishes maximize expected utility while also imposing an upper bound on inequity.   To this end, define the function V : 0, NN−1 7→ <+ , where N X

V (b) = max p∈P(b)

pn un ,

(3)

n=1

representing the maximum achievable utility, given an allowable 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. Follows from standard arguments (see Online Appendix). Before we state our next result we need to introduce additional notation. First, let Nk denote the set of agents sharing the k’th order statistic of {u1 , u2 , ..., uN } and we let Nk = ¯ such sets where N ¯ ∈ {1, 2, ..., N }, With apologies for the clunky |Nk |. There are a total of N notation, NN¯ denotes the set of agents sharing the maximum of {u1 , u2 , ..., uN }. Furthermore, + − S Sk − + − let Nk+ = N i=k Ni , Nk = i=1 Ni and Nk = Nk , Nk = Nk . Our model structure enables us to easily show the following Lemma. Lemma 1 Consider the function V of Eq. (3). (i) Define ¯b ≡

1 NN¯

−

1 N.

V is strictly increasing in b ∈ [0, ¯b) and equal to maxn∈N un in

b ∈ [¯b, NN−1 ]. (ii) Suppose b ∈ [0, ¯b]. There exists a unique optimal solution for problem (3), denoted by p∗ (b), and it must satisfy the quadratic constraint of set (2) with equality. Proof. Follows from standard arguments (see Online Appendix). Lemma 1 suggests that ¯b is an important threshold. It represents the level of inequity above which the set P(b) allows for the maximum agent utility to be attained as an objective function value of (3). Moreover, for levels of inequity smaller than this extreme value, the optimal solutions of problem (3) will be unique and bind the quadratic inequity constraint associated with set P(b).4 4

For b ∈ (¯b,

N −1 ] N

all allocations p∗ (b) satisfying p∗n (b) = 0 for n 6∈ NN¯ and

will be optimal solutions of (3).

5

P

n∈NN ¯

p∗n (b) −

1 N


2

≤ b−

N− ¯

N −1

N2

We are now ready to prove the paper’s first main result. Theorem 1 establishes that function
V is differentiable with respect to b everywhere on 0, NN−1 except at the point ¯b defined in Lemma 1. Moreover, it formalizes a straightforward monotonicity property of the optimal solution of (3) that is essential to the derivation of the value function pursued in Theorem 2. In proving Theorem 1 we make extensive use of results from conic optimization, in particular the duality theory of second-order cone programming (see Alizadeh and Goldfarb [1]). Theorem 1
(i) V is differentiable everywhere on b ∈ 0, NN−1 except ¯b. (ii) Let p∗ (b) denote an optimal solution of (3). The following levels of inequity n o b∗k = min ˆb : ∀b ≥ ˆb, we have p∗n (b) = 0 for all n ∈ Nk ,

(4)

¯ − 1}, are well-defined and strictly increasing in k. where k ∈ {1, 2, ..., N

Part (i) of Theorem 1 shows that V is a smooth function of b, except for a single kink at the minimum level of inequity at which the maximum efficiency can be obtained. Part (ii) implies that b∗k can be interpreted in the following way: it denotes the threshold level of inequity such that, for all b greater than or equal to it, at optimality no probability mass is ever allocated to agents belonging in Nk− (i.e., having a un that is less than or equal to the k’th order statistic of {u1 , u2 , ..., uN }). Thus, when b reaches and exceeds this level, one can safely disregard agents in Nk− . While the existence and monotonicity of these inequity thresholds may be intuitive, their proofs are relatively involved, requiring insights from conic duality [1]. Having established differentiability, we go on to provide a set of differential equations that V must satisfy. These differential equations will prove valuable in the subsequent derivation of a closed-form expression for V . Proposition 2 Suppose agent nk satisfies nk ∈ Nk . V satisfies the following differential equation:   1 d ∗ 2 V (b) pnk (b) − − b = unk − V (b), b ∈ (0, b∗k ) . db N

(5)

¯ . Define the Let u(k) denote the k’th order statistic of {u1 , u2 , ..., uN }, where k = 1, 2, ..., N following quantities: P

u+ = k d+ = k

n∈Nk+ un , Nk+ + Nk+1 u(k) −

6

(6) u+ k+1


2

,

(7)

¯ } (we set d+¯ ≡ 0). where k ∈ {1, 2, ..., N N The term u+ k is simply an average of the values of the set {u1 , u2 , ..., uN } that are greater than or equal to its k’th order statistic. The term d+ k measures the squared difference between the k’th order statistic of {u1 , u2 , ..., uN } and the average of those greater than it, adjusted for size of the latter group. Now, we use the above quantities to define + b+ ¯ ≡ 0 ≡ 0, bN

b+ k

N −1 N PN¯ −1

N− = +k + Nk+1 N

(8)

Nl + l=k+1 N + dl l , + Nk+1 d+ k

¯ − 1}. k ∈ {1, 2, ..., N

(9)

¯ Thus, we have b+ ¯ −1 = b. Straightforward algebra yields the following monotonicity property, N ¯ − 1}: where k ∈ {1, 2, ..., N + b+ k < bk+1 .

We are now ready to state our second main result and provide a closed-form expression for V . To prove the following Theorem, we explicitly solve the systems of differential equations established in Proposition 2.
Theorem 2 Consider the optimization problem (3) and the vectors u+ , d+ , b+ defined in Eqs. (6)-(7)-(8)-(9). The vector b+ satisfies ∗ b+ k = bk

¯ − 1} k ∈ {1, 2, ..., N

where b∗k is defined in Eq. (4). The function V satisfies v ! N¯ −1 u − u X Nl  +
Nk−1 + + + t V (b) = uk + b− + + dl , b ∈ bk−1 , bk , Nk N l=k Nl

(10)

¯ − 1. For b ≥ b+¯ = ¯b, V (b) = maxn∈N un . where k = 1, 2, ..., N N −1 Theorem 2 shows that V will be a concatenation of appropriately-specified square-root-like functions (when k = 1, we set

− Nk−1

Nk+ N

≡ 0).5 These concatenations occur at levels of inequity

b+ = b∗ which are interpreted by Eq. (4), and can be computed explicitly through Eq. (9). The curvature of these functions is driven by the dispersion of agents’ utilities, as captured by the quantities d+ k of Eqs (7). 5

These results are consistent with the more general analysis of Section 4.2 in Iyengar [8]. However, Iyengar

uses different arguments and does not prove differentiability in b, nor does he derive and interpret differential equations and a precise formula for V and its optimal solution p∗ (we provide the latter in Corollary 1). Instead, his analysis is concerned with determining the complexity of finding the optimal solution.

7

The fraction

− Nk−1

Nk+ N

in the square root represents the minimum level of b at which it becomes

− possible to assign zero share to all agents in Nk−1 . Hence, the fact that b∗k−1 = b+ k−1 >

− Nk−1

Nk+ N

for

all k (see Eq. (9)), ensures that Eq. (10) is well-defined. We can now combine the various results we have established to characterize the optimal solution of (3). ¯ ¯ Corollary 1 Suppose b < b+ ¯ −1 = b. Consider any agent n ∈ Nl for some l ∈ {1, 2, ..., N }. N The unique optimal allocation p∗ (b) satisfies v  u −  u b− Nk−1   u
 NN+  1 + u − u+ t PN¯ −1 Nki + n + ∗ k Nk pn (b) = i=k N + di  i     0


+ b ∈ b+ k−1 , bk , k = 1, 2, ..., l h  + b ∈ b+ , b ¯ −1 . l N

∗ ∗ ∗ Now suppose b ≥ b+ ¯ and p (b) ∈ P(b) ¯ −1 . Then, all vectors p (b) satisfying pn (b) = 0 for n 6∈ NN N

¯ will be optimal. This set is a singleton at b = b+ ¯ −1 = b. N Corollary 1 provides succinct expressions for the optimal agent probabilities given an inequity level b. Example 1.

Let us now illustrate the analytic findings of this section with an example.

Consider the following problem instance given in Table 1. k

u(k)

Nk

Nk+

Nk−

u ¯+ k

d+ k

b+ k

1

1

1

9

1

41 9

128

0.0422

2

2.5

2

8

3

5

66.67

0.0864

3

3

1

6

4

35 6

57.8

0.0982

4

5.5

2

5

6

6.4

6.75

2/9

5

7

3

3

9

7

0

8/9

Table 1: Data for Example 1. As we can see from Table 1, agent valuations can be sorted into five order statistics so that ¯ = 5. In addition, maximum efficiency can be obtained for all inequity levels greater than or N ¯ equal to b+ 4 = b =

1 3

−

1 9

= 2/9. Applying Eqs. (6)-(7) to the problem data, we obtain the u+ k

+ and d+ k values and use them to calculate the bk thresholds via Eq. (9). The results are listed in

Table 1.

8

Figure 2: Graphing function V for Example 1. Different colors are used to indicate the five + different functional forms of V within the intervals [b+ k−1 , bk ) for k = 1, 2, ..., 5. (Note that the

x-axis is truncated before b+ 5 = 8/9.) Now, we are ready to apply Theorem 2, which in turn yields the following expression for V :  √ 41  + 43.22b b ∈ [0, 0.0422),  9    p    5 + 29(b − 1/72) b ∈ [0.0422, 0.0864),   p V (b) = 35/6 + 12.33(b − 1/18) b ∈ [0.0864, 0.0982),   p    2.7(b − 4/45) b ∈ [0.0982, 2/9), 32/5 +      7 b ∈ [2/9, 8/9]. Figure 2 graphically depicts the above algebraic expression of function V . Different colors + are used to indicate the five different functional forms of V within the intervals [b+ k−1 , bk ) for

k = 1, 2, ..., 5. The differentiability of V that was proved theoretically may be verified visually ¯ as well. Moreover, as predicted by the theory, V ’s only kink occurs at b = b+ 4 = b = 2/9 where V first reaches its maximum value of 7.

4 4.1

Balancing equity and efficiency General cost functions

Theorem 2 enables us to rigorously investigate the tradeoff between equity and efficiency. For example, it allows us to answer the following two questions: (a) Given an upper bound b on inequity, how close are we to full efficiency? This quantity is

9

captured by the efficiency ratio corresponding to inequity b, denoted by E(b), where: E(b) =

V (b) . maxn∈N un

(b) What is the minimum inequity we should tolerate if we want to achieve an efficiency ratio of at least x? Denoting this quantity by bx , it satisfies: n o bx = min b : V (b) ≥ x max un . n

The above points (a)-(b) are interesting questions that can be easily dealt with via Theorem 2. However, they do not exploit the full range of the previous section’s analysis. In particular, they do not make use of the differentiability of function V . To make use of this result, we may consider directly modelling the cost of inequity. To wit, suppose an inequity of b entails a cost c(b), where c is a twice differentiable, increasing and convex function. Suppose, further, that a social planner wishes to balance the competing objectives of efficiency and equity. In doing so, she wishes to find a level of inequity bOP T where the marginal gain in efficiency is equal to the marginal cost of inequity. Hence, the objective is to determine an inequity level b∗ such that bOP T = arg max {V (b) − c(b)} .

(11)

−1 b∈[0, NN ]

  Given Lemma 1, when solving for bOP T it is sufficient to restrict ourselves to b ∈ 0, ¯b , as values of b above this range increase inequity while having no effect on efficiency. Moreover, in light of Proposition 1 and Theorem 1, bOP T is unique and can be calculated by applying first-order conditions on the function V (b) − c(b). At an interior solution, these reduce to: dV OP T
dc OP T
b − b = 0. db db

(12)

The derivatives of V can be determined via the expression of Theorem 2. Simple algebra ¯ − 1:6 establishes that for k = 1, 2, ...N dV + 1 + (bk ) = Nk+1 (u+ k+1 − u(k) ). db 2 Since c is increasing and convex, we know that

dc db

solving for bOP T reduces to studying the evolution of

(13)

is positive and weakly increasing. Thus, dc db

along [0, ¯b] and finding the first interval

+ [b+ k−1 , bk ] such that

dc 1 + + Nk (uk − u(k−1) ) > (b+ ) 2 db k−1 1 + dc Nk+1 (u+ − u(k) ) < (b+ ). k+1 2 db k 6

(14)

¯ − 1, the expression below is a left derivative (recall that V is not differentiable at b+¯ = ¯b). At k = N N −1

10

When this interval has been found, the optimal inequity level is computed by considering the corresponding expression of V within it, available via Eq (2), and solving for bOP T by applying the first-order conditions of Eq. (12).

4.2

A special case: c(b) = maxn∈N un · b

Suppose c(b) = maxn∈N un b so that maximizing V (b)−c(b) is equivalent to maximizing b. As mentioned earlier, the quantity

V (b) maxn un

V (b) maxn un −

represents the efficiency ratio corresponding to

inequity b. Define the following quantities q0 = ∞, qN¯ ≡ 0 q + + + ¯ qk = Nk+1 d+ k = Nk+1 (uk+1 − u(k) ), k ∈ {1, 2, ..., N − 1}.

(15)

By simple algebra it is easy to see that qk are strictly decreasing in k. This is unsurprising since, by Eq. (13), we have qk = 2 ·

dV db

(b+ k ) and V is concave (recall Proposition 1.

With this choice of cost c, we are able to completely characterize the optimal levels of inequity and efficiency, as well as the corresponding allocations. Theorem 3 Consider the optimization problem of Eq. (11) for c(b) = maxn∈N un b. Let k OP T ∈ ¯ } uniquely satisfy: {1, 2, ..., N qkOP T ≤ 2 max un < qkOP T −1 . n∈N

(i) The optimal inequity level bOP T , and its efficiency V OP T ≡ V (bOP T ), are uniquely given by: bOP T

=

Nk−OP T −1 Nk+OP T N

PN¯ −1 +

Nl + l=kOP T N + dl l . 4(maxn∈N un )2

PN¯ −1 V

OP T

=

u+ kOP T

+

Nl + l=kOP T N + dl l

2 maxn∈N un

(ii) The optimal allocation pOP T = p∗ (bOP T ) satisfies  u −u+   +1 + l kOP T 2 maxn∈N N OP T T k pOP = l   0

.

l ∈ Nk+OP T l ∈ Nk−OP T −1 .

Thus, we see that the optimal inequity level bOP T implies an allocation pOP T in which an agent l is assigned the good with probability 0 if and only he belongs in the set Nk−OP T −1 , i.e., ul is less than or equal to the (k OP T − 1)’th ordered statistic of the un ’s. This, in turn, implies further structure to the optimal probabilities as per the following corollary. 11

Corollary 2 Consider the optimal inequity level bOP T and its corresponding allocation pOP T . T = 0. There will be at most N − 3 agents l such that pOP l

Example 1 continued. Applying Theorem 3 to the data in Example 1, we first calculate: q1 = 32, q2 = 20, q3 = 17, q4 = 4.5. Since 2 · maxn∈N = 14, we obtain k OP T = 4. Applying Theorem 3 yields: P4 Nl + − l=3 N + dl 57.8 16 + 6.75 25 4 N l = bOP T = +3 + + = 0.1518. 4(maxn∈N un )2 45 4 · 49 N4 N P4 Nl + 2 l=4 N + dl 6.75 + OP T l = 6.4 + 5 V = u4 + = 6.5929, 2 maxn∈N un 14 and

T pOP l

=

    

1 5

+

7−6.4 14

1

+

5.5−6.4 14

5     0

= 0.2469

l ∈ N5 ,

= 0.1357 l ∈ N4 , l ∈ N3− .

Hence, the optimal level of inequity implies that just the five agents having the highest and second highest valuations get assigned the good with positive probability.

5

Directions for future research

The work presented herein suggests several fruitful avenues of future research. First, it would be interesting to conduct an analysis similar to the one appearing in Sections 3 and 4, but replacing the inequity measure of Eq. (1) with alternative, more popular measures such as the Gini, Theil, or Atkinson indices. As the latter have more complex nonlinear functional forms than (1), such an exercise would be quite challenging. Closed-form expressions would be harder to come by and it is not clear if the differentiability properties of the associated V functions would extend. An alternative, especially challenging extension of the current work would involve the introduction of strategic considerations to the model. In particular, we could assume that agent valuations are private information that is strategically revealed to the social planner. In this environment, we could attempt to derive an auction-like mechanism that induces, in Nash equilibrium, truthful revelation of agent valuations to which, in turn, fairness considerations via Eq. (1) are applied. The latter constraint would clearly influence the strategic behavior of agents, and so would need to be taken into account when deriving the auction mechanism.

12

Appendix A1 Proofs of main results Theorem 1.

We begin with part (i). Given x = (x0 , x ¯) ∈
notation to denote inclusion in a second-order cone of dimension n + 1 (x0 , x ¯) ∈ L2n+1 ⇔ x0 ≥ ||¯ x||2 . We follow Alizadeh and Goldfarb [1] to write (3) as a primal conic program P(b) and introduce its dual D(b) (for clarity, next to the primal constraints we indicate the corresponding dual variables):

max

p,q,q0 ,θ

s.t.

N X

r

un pn

min

y,y0 ,γ,β0 ,zp ,zq ,zq0 ,zθ

n=1

s.t.

−pn + qn = 0, ∀n ∈ N , (yn ) N X

b+

yn + zqn = 0, ∀n ∈ N

−pn = −1, (y0 )

n=1

P(b)

1 β0 N −y0 − yn + zpn = −un , ∀n ∈ N

y0 +

γn + zθn = 0, ∀n ∈ N

θn = 0, ∀n ∈ N , (γn ) r 1 q0 = b + , (β0 ) N (pn , θn ) ∈ L22 , ∀n ∈ N

D(b)

−β0 + zq0 = 0 (zpn , zθn ) ∈ L22 , ∀n ∈ N (zq0 , −zq ) ∈ L2n+1 .

(q0 , q) ∈ L2n+1 ,

Since both the primal and the dual have feasible strictly interior solutions, strong duality holds (see Theorem 13 of [1]). Without loss of generality, we can immediately simplify D(b) by setting zθ = γ = 0 and zp ≥ 0. Correspondingly, we can eliminate the variable zq by replacing it with −y. Finally, it is evident that at optimality the quadratic constraint of the dual will be qP qP N N ∗ ∗ 2 2 binding so that zq0 = β0 = n=1 (−yn ) = n=1 yn . Collecting all of these observations we may re-write the dual in the following much simpler way: v uN r X 1u D1 (b) = min y0 + b + t yn2 y,y0 N n=1

s.t.

−un + y0 + yn ≥ 0, n = 1, 2, ..., N.

(16)

Examining (16) we deduce that at optimality yn∗ = max(0, un − y0 ). Thus we may simplify the dual even further to an unconstrained optimization problem with just one variable: v uN r X 1u D2 (b) = min y0 + b + t max(0, un − y0 )2 . y0 N n=1

13

(17)

By strong duality the dual optimal objective will be bounded between

1 N

PN

n=1 un

and u(N¯ ) . We

immediately see that solutions satisfying y0 > u(N¯ ) result in strictly greater objective function values than y0 = u(N¯ ) , so that we can safely disregard them. Conversely, solutions satisfying y0 < 0 yield r y0 +

v v uN uN r u X X 1t 1u 2 b+ max(0, un − y0 ) = y0 + b + t (un − y0 )2 N N n=1 n=1 √ √ √ > y0 + N b + 1(|u(1) | − y0 ) = N b + 1|u(1) | + y0 (1 − N b + 1).

Thus, values of y0 <

|u(N | √ 1− N b+1

result in a strictly greater objective function value than y0 = u(N¯ )

and hence can also be disregarded. With these observations we may rewrite the dual (17) in the following way: v uN X 1u b+ t max(0, un − y0 )2 N

r D3 (b) =

 y0 ∈

min

|u(N ¯ )| √ ,u ¯ 1− N b+1 (N )

 y0

+

(18)

n=1

The domain of D3 (b) is thus compact, for any b > 0. For values of b ∈ [0, ¯b) we know that the optimal solution of the primal will be strictly less than u(N¯ ) . Thus, strong duality implies that for all b ∈ (0, ¯b), any optimal solution y ∗ (b) must satisfy y ∗ (b) < u(N¯ −1) . However, notice that the objective function of D3 is strictly convex for y0 < u(N¯ −1) . Thus, we may deduce that when b ∈ (0, ¯b) D3 (b) admits a unique optimal solution y0∗ (b). The above observation implies that we can apply Danskin’s theorem (see Proposition B.25 in Bertsekas [2]) to conclude that the optimal dual objective value, and therefore by strong duality V (b) as well, is differentiable at all b ∈ (0, ¯b) and that qP N ∗ 2 dV n=1 max(0, un − y0 (b)) q (b) = , b ∈ (0, b∗ ). db 1 2 b+

(19)

N

We now show that y0∗ (b) is strictly increasing in b ∈ (0, ¯b). Consider b1 < b2 with both belonging in (0, b∗ ) and their optimal solutions y0∗ (b1 ) and y0∗ (b2 ). By uniqueness of y0∗ (b) in this range of b we have v uN r X 1u ∗ ∗ t ∗ 2 max(0, un − y0 (b1 )) < y0 (b2 ) + b1 + y0 (b1 ) + b1 + N n=1 v uN r r uX 1 y0∗ (b2 ) + b2 + t max(0, un − y0∗ (b2 ))2 < y0∗ (b1 ) + b2 + N r

n=1

14

v uN X 1u t max(0, un − y0∗ (b2 ))2 N n=1 v uN X 1u t max(0, un − y0∗ (b1 ))2 . N n=1

Summing the above inequalities and rearranging terms yields v  ! v uN uN r r u u X X 1 1 t b2 + − b1 + max(0, un − y0∗ (b1 ))2 − t max(0, un − y0∗ (b2 ))2  > 0 N N n=1 n=1 v v uN uN uX uX t ∗ 2 max(0, un − y0 (b1 )) − t max(0, un − y0∗ (b2 ))2 > 0 ⇒ y0∗ (b2 ) > y0∗ (b1 ). ⇒ n=1

n=1

We now discuss now the differentiability of V at b = ¯b. Note that the optimal solution y0∗ (¯b) is not unique; instead it can take any value in the interval [u(N¯ −1) , u(N¯ ) ]. Hence Danskin’s theorem implies that the subdifferential of V (b) at ¯b will consist of all convex combinations of √ NN¯ (u(N¯ ) −u(N¯ −1) ) q and 0. 1 2

¯b+ N

We now prove part (ii). Let us go back to the original primal-dual pair (P(b), D(b)) and consider a pair of optimal solutions of the primal and dual problems. By Lemma 1 the primal optimal solution (p∗ (b), q∗ (b), θ ∗ (b), q0∗ (b)) is unique, while our reasoning in part (i) established the uniqueness of the optimal dual variables (β0∗ (b), y∗ (b), y0∗ (b), z∗p (b), z∗q (b)). Applying Theorem 16 and part (ii) of the complementarity conditions of Lemma 15 of Alizadeh and Goldfarb [1], we arrive at the following conditions: v uN uX 1 ∗ ∗ ∗ ∗ ∗ q0 (b)zqn (b) + β0 (b)qn (b) = 0 ⇔ − b + yn (b) + t yn∗ (b)2 p∗n (b) = 0, n = 1, 2, ..., N N n=1 v u r N uX 1 ∗ t max(0, un − y0 (b)) + max(0, un − y0∗ (b))2 p∗n (b) = 0, n = 1, 2, ..., (20) N. ⇔− b+ N r

n=1

P When b < ¯b, strong duality implies y0∗ (b) < u(N¯ −1) which in turn ensures N n=1 max(0, un − y0∗ (b))2 > 0. As mentioned earlier, when b = ¯b y0∗ (b) can take any value in [u(N¯ −1) , u(N¯ ) ] so P ∗ ¯ 2 we choose one that again yields N n=1 max(0, un − y0 (b)) > 0. Hence, the complementarity conditions (20) yield p∗n (b) = 0 ⇔ un − y0∗ (b) ≤ 0, n = 1, 2, ..., N.

(21)

Since y0∗ (b) is strictly increasing in b in (0, ¯b) and limb→0+ y0∗ (b) = −∞ and limb→¯b− y ∗ (b) = u(N¯ −1) , Eq. (21) implies the existence of a set {b1 , b2 , ..., bN¯ −1 } such that 0 < b1 < b2 < .... < bN¯ −1 = ¯b    ∗ − ¯ − 1. pn (b) = 0 ∀n ∈ Nk ⇔ b ≥ bk , ∀k = 1, 2, ..., N

15

Proposition 2. Focusing on optimization problem (3), we introduce Lagrangian multipliers and write the Karush-Kuhn-Tucker (KKT) conditions:   1 + µ + νn = 0, n ∈ {1, 2, ..., N } un − 2λ pn − N !  N  X 1 2 λ pn − − b = 0, λ ≥ 0 N n=1  N  N X X 1 2 pn − ≤ b, pn = 1, p ≥ 0 N n=1

(22) (23)

(24)

n=1

νn pn = 0, νn ≥ 0, n ∈ {1, 2, ..., N }.

(25)

Since our problem is concave with affine equality constraints and satisfies Slater’s condition (see section 5.2.3 in [5]), strong duality holds and the KKT conditions (22)-(25) will be necessary and sufficient for both primal and dual optimality. In other words, the duality gap is zero and the vector (p∗ , ν ∗ , λ∗ , µ∗ ) satisfies (22)-(25) if and only p∗ and λ∗ , ν ∗ , µ∗ are primal and dual optimal respectively (see section 5.5.3 in [5]). From Lemma 1 we know that there exists a unique primal optimal solution p∗ . By strong duality, the Lagrangean dual problem admits an optimal solution, and we refer to it by λ∗ , ν ∗ , µ∗ .7 Since V (b) is differentiable (Theorem 1) and strong duality holds we follow Section 5.6.3 in Boyd and Vandenberghe [5] to deduce the following simple relation: d V (b) = λ∗ (b), b ∈ (0, b∗ ). db

(26)

Eq. (26) means that we can now focus on calculating the Lagrange multiplier λ∗ (b). Before we do so we note the following useful identity    N  N N N X X X 1 2 1 1 X ∗ 1 ∗ ∗ ∗ pn (b) − = pn (b) pn (b) − − pn (b) + N N N N2 n=1 n=1 n=1 n=1   N X 1 = p∗n (b) p∗n (b) − . N

(27)

n=1

Multiplying both sides of Eq. (22) by p∗n (b) and then summing over all n = 1, 2, .., N obtains N X

(27)

⇒

  N X 1 ∗ − 2λ (b) pn (b) − + µ∗ (b) p∗n (b) = 0 N n=1 n=1 n=1   N N 2 X X 1 un p∗n (b) − 2λ∗ (b) p∗n (b) − + µ∗ (b) = 0 N un p∗n (b)

∗

n=1 Lemma ??

⇒

µ∗ (b) = 2λ∗ (b) · b −

N X

p∗n (b)

n=1

N X

un p∗n (b).

(28)

n=1 7

Note that at this point one can manipulate the KKT conditions (22)-(25) to show that the Lagrangean dual’s

optimal solution is also unique.

16

Now we consider Eq. (22) for agent nk ∈ Nk . By part (b) of Theorem 1 we must have p∗nk (b) > 0 if and only if b ∈ [0, b∗k ). Substituting the value of µ∗ (b) obtained in Eq. (28), and applying the complementary slackness condition (25) we obtain   X N 1 ∗ unk − 2λ (b) pnk (b) − = un p∗n (b) − 2λ∗ (b) · b N n=1   d 1 2 V (b) p∗nk (b) − − b = unk − V (b), b ∈ (0, b∗k ) . db N ∗

(26)

⇒

(29)

Theorem 2. Recall the definition of b∗k of Eq. (4). Consider first b ∈ (0, b∗1 ) so that p∗n (b) > 0 for all b ∈ (0, b∗1 ) and n ∈ N . Recalling Proposition 2 and adding Eqs. (29) for all n ∈ N yields the following differential equation −2N b

X dV (b) un , b ∈ (0, b∗1 ). = −N V (b) + db

(30)

n∈N

Solving differential equation (30) leads to the following expression: P √ un V (b) = C1 b + n∈N , b ∈ [0, b∗1 ), N

(31)

¯ − 1}. where C1 is a constant to be determined. Consider now b ∈ [b∗k−1 , b∗k ) for k ∈ {2, 3, ..., N In this range of b we will have p∗n (b) > 0 if and only n ∈ Nk+ . Adding Eqs. (29) for all such n ∈ Nk+ yields the following differential equation ! − X Nk−1 dV 2 − Nk+ b = un − Nk+ V (b), b ∈ [b∗k−1 , b∗k ) N db +

(32)

n∈Nk

Solving differential equation (32) gives the following: s − 
Nk−1 + V (b) = u+ + C N b − , b ∈ b∗k−1 , b∗k , k k k N

(33)

¯ − 1}, where Ck is a constant to determined. Finally since b∗¯ = ¯b we use for k ∈ {2, 3, ..., N N −1 Lemma 1 to conclude  V (b) = max un , b ∈ n∈N

b∗N¯ −1 ,

 N −1 . N

(34)

Putting together Eqs. (31), (33), and (34) we see that V will equal √ + C1 b q + V (b) = uk + Ck Nk+ b −     max n∈N un     

P

n∈N

un

N

− Nk−1 N

b ∈ [0, b∗1 ) 
¯ −1 b ∈ b∗k−1 , b∗k , k = 2, 3, ..., N h i b ∈ b∗N¯ −1 , NN−1

17

(35)

  for appropriately chosen constants (C1 , C2 , ..., CN¯ −1 ) and b∗1 , b∗2 , ..., b∗N¯ −1 . By Proposition 1 and Theorem 1, V is continuous everywhere and differentiable everywhere at (0, NN−1 ) except ¯b.   Thus, the vectors (C1 , C2 , ..., CN¯ −1 ) and b∗1 , b∗2 , ..., b∗N¯ −1 must fulfill these criteria of continuity and differentiability and are thus uniquely determined by the following system of nonlinear equations (36)-(43): Case 1: N = 2. P

n∈N

un

N b∗1 =

+ C1

p

b∗1 = max un

(36)

n∈N

1 1 − . N2 N

(37)

Case 2: N ≥ 3. P

n∈N

un

N

s + C1

p

b∗1 = u+ 2 + C2

C C N+ p 1∗ = q 2 2 b1 N2+ b∗1 − s u+ k

+ Ck

Nk+ b∗k

N2+ b∗1 −

N1− N

(38) (39)

N1− N

N− − k−1 = u+ k+1 + Ck+1 N

s + Nk+1 b∗k −

Nk− , N

¯ −2 k = 2, 3, ..., N

+ Ck+1 Nk+1 Ck Nk+ ¯ −2 q q , k = 2, 3, ..., N = − Nk−1 Nk− + + ∗ ∗ Nk+1 bk − N Nk bk − N s − NN −2 + ∗ = max un u+ ¯ −1 NN ¯ −1 + CN ¯ −1 bN ¯ −1 − N n∈N N 1 1 − b∗N¯ −1 = NN¯ N

(40) (41)

(42) (43)

It now remains to show that the solution of System (36)-(43) will eventually lead to the expression of the Theorem. To do this we calculate explicitly the Ck and b∗k ’s and then show how applying them to formula (35) yields the desired result. We begin with Case 1 and N = 2. That b∗1 =

1 1 − N2 N

(44)

is trivially true. Then, Eq. (36) yields s r
N2 N1 + + C1 = · N · u(2) − u1 = d . N1 N1+ 1 We now focus on Case 2 and N ≥ 3. Once again, we have by definition b∗N¯ −1 = Eq. (42) implies s CN¯ −1 =

 NN¯  · u(N¯ ) − u+ = ¯ N −1 NN¯ −1 18

s

NN¯ −1 + dN¯ −1 . + NN ¯ −1

(45) 1 NN¯

− N1 , whence

¯ − 2} and solving for C ∗ yields: Focusing on Eq. (41) for k ∈ {2, 3, ..., N k q − Nk−1 + + ∗ Ck+1 Nk+1 − N b k k N q . Ck∗ = Nk− Nk+ + ∗ Nk+1 bk − N

(46)

Plugging (46) into Eq. (40) we obtain:   s − Nk−1 + + ∗ − N N b − N + N  Ck+1 Nk+1 · k k =u ¯+ ¯+ b∗k − k 1 − k+1 − k −u k+1 . N Nk+ N + b∗ − Nk k+1 k N

(47)

After some algebra, the left-hand-side of Eq. (47) can be simplified so that: − Nk−1

+ Ck+1 Nk+1

N

Nk+

q

−

Nk− + Nk+1

+ Nk+1 b∗k −

Nk− N

n∈Nk+ un Nk+

= 

⇒

⇒

P

P

+ u(k) Nk+1 −

Nk q = N− + b∗k − Nk Nk+ Nk+1 Ck+1 = u(k) − u+ −q k+1 − N + Nk+1 b∗k − Nk −Ck+1

+ n∈Nk+1 un + Nk+1

− P

+ n∈Nk+1

 un Nk

+ Nk+ Nk+1

¯ − 2: Combining Eqs. (41) and (48) obtains for k = 2, 3, ..., N s + −
Nk+1 Nk−1 + + ∗ Ck = − + u(k) − uk+1 Nk bk − N Nk  2 N− Ck+1 + Nk u(k) −u+ k+1 b∗k = , + Nk+1

(48)

(49)

(50)

which after some simple algebra leads to the following nonhomogeneous linear recursion for the squares of the Ck ’s: (Ck )2 =

+ Nk+1

Nk+

(Ck+1 )2 +

Nk d+ k ¯ − 2. , k = 2, 3, ..., N Nk+ Nk+

(51)

Solving recursion (51) backwards with (previously derived) initial value CN¯ −1 , taking square roots, and recalling the positive sign of the Ck ’s, leads to a simple expression for the Ck ’s: v u ¯ −1 u 1 NX Nl + t ¯ − 1. Ck = d , k = 2, 3, ..., N (52) + Nk l=k Nl+ l Applying Eq. (52) to Eq. (50) yields b∗k

=

Nk− + Nk+1 N

PN¯ −1 +

Nl + l=k+1 N + dl l , + + Nk+1 dk

19

¯ − 1. k = 2, 3, ..., N

(53)

Finally plugging C2 into Eqs. (38)-(39) yields v uN¯ −1 u X Nl + C1 = t + dl , N l l=1 PN¯ −1 Nl + − l=2 N + dl N1 l b∗1 = + . + + + N2 N N2 d1

(54)

(55)

Note that Eqs. (54)-(55) are consistent with the results for Case 1 as given by Eqs. (44)-(45). Thus there is no more need to distinguish between Case 1 and 2. Finally, applying Eqs. (52)-(53)-(54)-(55) to Eq. (35) and performing elementary algebra establishes the result.

Theorem 3.

(i) By strict concavity of V there will be a unique optimal solution. Using the
expression for V of Theorem 2 and the fact that it is differentiable in 0, ¯b (Theorem 1), we may apply first-order conditions on V (b) − c(b). Doing so implies that there will exist a unique ¯ } such that the optimal solution bOP T satisfies: k OP T ∈ {1, 2, ..., N   PN¯ −1 Nl + −   h OP T + dl N l=k OP T N l bOP T = k+ −1 + and bOP T ∈ b+ , b+ kOP T −1 kOP T   4(maxn∈N un )2 NkOP T N

(56)

Looking at the expressions for b+ of Eqs. (9) we see that k OP T will be such that it uniquely satisfies OP T

b

OP T

b

≥

<

b+ kOP T −1 b+ kOP T

 2 ⇒ 4 max un ≤ Nk+OP T d+ kOP T −1

⇒

(57)

n∈N

Nk−OP T −1 Nk+OP T N

PN¯ −1 +

Nl + l=kOP T N + dl l 4(maxn∈N un )2

<

Nk−OP T Nk+OP T +1 N

PN¯ −1 +

Nl + l=kOP T +1 N + dl l .(58) + + NkOP T +1 dkOP T

Algebraic manipulations establish that PN¯ −1 PN¯ −1 Nl + Nl + − N Nk−OP T l=kOP T +1 N + dl l=kOP T N + dl kOP T −1 l l + = + ++ + , + + + + NkOP T +1 N NkOP T +1 dkOP T NkOP T N NkOP T +1 dkOP T so that Eq. (58) is equivalent to 2

 4 max un n∈N

> Nk+OP T +1 d+ kOP T

(59)

Putting Eqs. (58)-(59) together yields: qkOP T < 2 max un ≤ qkOP T −1 . n∈N

The values of V OP T and pOP T follow from the direct application of Theorem 2 and Corollary 1 to bOP T . 20

Corollary 2.

¯ − 1, N ¯ } since Suppose not. Then, by Theorem 3 we must have k OP T ∈ {N

¯ − 1 would automatically imply that at least 3 agents (i.e., the ones with otherwise k OP T < N the three highest valuations) are granted the good with positive probability. ¯ , so that just agents in N ¯ (= N + Suppose first that k OP T = N ¯ ) are assigned positive N N probability. By Theorem 3, we must have qN¯ −1 ≥ 2 max un ⇔ NN¯ (max un − u(N¯ −1) ) ≥ 2 max un ⇒ NN¯ > 2, n∈N n∈N n∈N | {z } >0

leading to a contradiction. ¯ − 1. Again, by Theorem 3, we must have Now, suppose k OP T = N + qN¯ −2 ≥ 2 max un ⇔ NN ¯ −1) ¯ −1 ( u(N n∈N | {z }


leading to a contradiction.

21

+ − u(N¯ −2) ) ≥ 2 max un ⇒ NN ¯ −1 > 2, n∈N | {z } >0

References [1] Alizadeh, F., and Goldfarb, D. (2003). Second-order cone programming. Mathematical Programming B, 95, 3–51. [2] Bertsekas, D. (1999). Nonlinear Programming, Belmont, MA: Athena Scientific. [3] Bertsimas, D., Farias, V. F., and Trichakis, N. (2011). The price of fairness. Operations Research, 59, 17-31. [4] Bertsimas, D., Farias, V. F., and Trichakis, N. (2012). On the efficiency-fairness trade-off. Management Science, 58, 2234-2250. [5] Boyd, S., and Vandenberghe, L. (2008). Convex Optimization. New York, NY: Cambridge University Press. [6] Cowell, F. A. (2000). Measurement of inequality. In Atkinson, A.B. and Bourguignon, F. (ed.) Handbook of Income Distribution, North Holland, Amsterdam, 1, 87-166. [7] Foster, J. E. and Sen, A.K. (1997) On Economic Inequality. After a Quarter Century. Annexe to the Expanded Edition of On Economic Inequality by A.K. Sen., Clarendon Press, Oxford. [8] Iyengar, G. (2005). Robust dynamic programming. Mathematics of Operations Research, 30, 257-280. [9] Moulin, H. (2004). Fair division and collective welfare. MIT press.

22

Online Appendix - not for publication Proof of Proposition 1.

That V is increasing in b follows by definition. Consider the

optimization problems given by the right-hand-side of Eq. (3) for b1 ∈ [0, NN−1 ] and b2 ≥ b1 and denote their optimal solutions by p∗ (b1 ) and p∗ (b2 ) respectively. By feasibility we may note the following: N  X

p∗n (b1 )

n=1

1 − N

2

 N  X 1 2 ∗ ≤ b1 , pn (b2 ) − ≤ b2 . N

(60)

n=1

Consider a convex combination of b1 and b2 given by b(λ) = λb1 + (1 − λ)b2 for some λ ∈ [0, 1] and the optimization problem V (b(λ)) =

max p∈P(b(λ))

N X

pn un .

(61)

n=1

To prove concavity of V in b it suffices to show that V (b(λ)) ≥ λV (b1 ) + (1 − λ)V (b2 ). To this end, consider the probability vector given by p(λ) = λp∗ (b1 ) + (1 − λ)p∗ (b2 ). By feasibility of p∗ (b1 ) and p∗ (b2 ) we immediately deduce that p(λ) ≥ 0 and that

PN

n=1 pn (λ)

=

1. Now we may write N  X n=1

1 pn (λ) − N

2 =

triangle ineq.

≤

  2 N   X 1 1 ∗ ∗ λ pn (b1 ) − + (1 − λ) pn (b2 ) − N N n=1  2 2 ! 12 2 ! 12 N  N  X X 1 1 λ  p∗n (b1 ) − + (1 − λ) p∗n (b2 ) − N N n=1

(60)

≤

n=1

h p i2 p i2 hp λ b1 + (1 − λ) b2 ≤ λb1 + (1 − λ)b2 = b(λ).

(62)

By Eq. (62) and the observations immediately preceding it we can conclude that p(λ) is feasible for optimization problem (61). Thus we may write V (b(λ)) ≥

N X n=1

p(λ)n un = λ

N X

p∗n (b1 )un + (1 − λ)

n=1

N X

p∗n (b2 )un

n=1

= λV (b1 ) + (1 − λ)V (b2 ), where the last equality follows from the assumed optimality of p∗ (b1 ) and p∗ (b2 ). We now proceed to show continuity. By concavity V (b) will be continuous on the open interval (0, NN−1 ) so we need only consider the endpoints 0 and

23

N −1 N .

Since V (b) is increasing in b we must

have limb→( N −1 )− V (b) ≤ V ( NN−1 ). However, if limb→( N −1 )− V (b) < V ( NN−1 ) then we reach a N

N

¯ − 1)/N and other values of b. contradiction if we apply concavity to (N To prove continuity at b = 0 consider an  > 0. Now let δ > 0 and write |V (δ) − V (0)|

=

≤ H¨ older’s ineq.

≤

 N  X 1 ∗ V (δ) − V (0) = pn (δ) − un N n=1 N X ∗ 1 max |un | pn (δ) − N n∈N n=1 "N  2 # 12 X √ 1 ≤ max |un | δ. max |un | p∗n (δ) − n∈N n∈N N n=1

Thus, any choice of 0 < δ <

2 (maxn∈N |un |)2

will ensure that |V (δ) − V (0)| < , completing the

proof.

Proof of Lemma 1.

(i) The function V (b) is bounded above by un for any n ∈ NN¯ . This

upper bound is attained by a probability vector p if and only if it satisfies X

b ⊆N¯ pn = 1, for some N N

b n∈N

b ⊆ N ¯ , with cardinality N b . The convexity of set P(b) implies that the Consider a subset N N b is given by value of b at which it first becomes possible to assign probability 1 to subset N b; N) = b(N

1 1 − . b N N

b ; N ) over N b ⊆ N ¯ is the entire set N ¯ , yielding the desired result. The minimizer of b(N N N Now consider b < ¯b and the optimal solution p∗ (b). As b < ¯b there must exist a j 6= NN¯ such that p∗j (b) > 0. Now consider increasing b by an amount . For δ > 0 small enough the solution p ˜ which is identical to p∗ (b) except that p˜j = p∗j (b) − δ and p˜k = p∗k (b) + δ for some k ∈ NN¯ will be feasible and result in a strictly greater objective value, so that V (b + ) > V (b). (ii) Suppose first that b = ¯b. It is clear here that the unique optimal solution is given by p∗ such that p∗n = 1/NN¯ for all n ∈ NN¯ 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 optimal solution p∗ (b) such that the quadratic inequity constraint is slack. As b < ¯b there must exist an j 6= NN¯ such that p∗j (b) > 0. For  > 0 small enough the solution p ˜ in which p˜j = p∗j (b) −  and p˜k = p∗k (b) +  for some k ∈ NN¯ will be feasible and result in a strictly greater objective value, contradicting p∗ (b)’s optimality. Thus, all optimal solutions must satisfy the quadratic inequity constraint with equality. 24

We now prove uniqueness. Suppose there exist two optimal solutions p∗,1 and p∗,2 . By the preceding argument they must bind the quadratic equity constraint. Consider the set of probability vectors given by their convex combinations p(λ) = λp∗,1 + (1 − λ)p∗,2 , λ ∈ [0, 1]. For λ ∈ (0, 1), p(λ) will satisfy the inequity constraint with strict inequality, since:  N  X 1 2 pn (λ) − N

=

n=1

strict convexity

<

=

  2 N   X 1 1 ∗,2 ∗,1 + (1 − λ) pn − λ pn − N N n=1 "   #   N X 1 2 1 2 ∗,1 ∗,2 λ pn − + (1 − λ) pn − N N n=1   N  N  X X 1 2 1 2 ∗,1 ∗,2 λ pn − + (1 − λ) pn − N N n=1

=

n=1

λb + (1 − λ)b = b.

Thus all solutions p(λ) are feasible. That they are optimal follows trivially by the assumed optimality of p∗,1 , p∗,2 and the linear objective function of (3). But this is a contradiction as all optimal solutions must satisfy the quadratic inequity constraint with equality. The second claim of the Proposition is trivial.

25