Proportional scheduling, split-proofness, and merge-proofness

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Games and Economic Behavior 63 (2008) 567–587 www.elsevier.com/locate/geb

Proportional scheduling, split-proofness, and merge-proofness ✩ Hervé Moulin Department of Economics-MS22, Rice University, P.O. Box 1892, Houston, TX 77005-1892, USA Received 28 March 2005 Available online 12 December 2006

Abstract If shortest (respectively longest) jobs are served first, splitting a job into smaller jobs (respectively merging several jobs) can reduce the actual wait. Any deterministic protocol is vulnerable to strategic splitting and/or merging. This is not true if scheduling is random, and users care only about expected wait. The Proportional rule draws the job served last with probabilities proportional to size, then repeats among the remaining jobs. It is immune to splitting and merging. Among split-proof protocols constructed in this recursive way, it is characterized by either one of three properties: job sizes and delays are co-monotonic; total delay is at most twice optimal delay; the worst (expected) delay of any job is at most twice the smallest feasible worst delay. A similar result holds within the family of separable rules, © 2006 Elsevier Inc. All rights reserved. JEL classification: C71; D62 Keywords: Probabilistic scheduling; Merging; Splitting; Proportional rule

1. Introduction Processing jobs of different lengths that share a single server raises multiple issues of incentive-compatibility and fairness when agents are impatient. Here we focus on two related maneuvers by which the users may be able to “game” the scheduling discipline, namely splitting one’s job into several smaller jobs requested under different aliases (so that the server believes they come from different users), or merging the jobs of several users into one large job presented to the server by a single agent. ✩

This work is supported by the NSF under Grant SES-0414543. E-mail address: [email protected].

0899-8256/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2006.09.002

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Such maneuvers are feasible when the server cannot monitor the identity of the real beneficiaries of the jobs it processes. In large networks such as the Internet, aliases are easy to generate and not easy to track, raising concerns about the vulnerability of Peer-To-Peer systems (Douceur, 2002; Cheng and Friedman, 2005). In such contexts, preventing merging and splitting is simply not feasible. Moreover the inability to detect the real identity of users may be a design constraint of the system, protecting the users’ privacy when the jobs involve confidential information. Think of users sharing a medical or financial data base. In line with most of the scheduling literature (e.g., Lawler et al., 1993), we assume that partially completed jobs are useless, so that an efficient server processes jobs whole. Splitting is a non-cooperative manipulation where agent A who needs a job of size x, creates several aliases A , A , . . . and requests on their behalf jobs of sizes a, a  , a  , . . . such that a + a  + a  + · · · = x. If the job completed last among A, A , A , . . . , is done strictly earlier than job A in the initial problem, the maneuver is profitable. Merging is the cooperative move where agents A, B, C, . . . , who need jobs of sizes a, b, c, . . . , choose one of them to request a single job of size x = a + b + c + · · · . The merged agents schedule their true jobs as they please during the time where job x is processed, and can enjoy the benefit of their own job as soon as it is completed. Merging is profitable if the waiting time of at least one merged agent can thus be reduced, without increasing that of any other. Many deterministic service disciplines are either “merge-proof” or “split-proof” but not both. Shortest Job First is the benchmark scheduling rule that minimizes total waiting time across all users. It is merge-proof because a merged job is never served earlier than any of its component jobs, but is badly vulnerable to splitting. Longest Job First (that maximizes total waiting time) is similarly “split-proof” but not merge-proof. Our first result (Proposition 1 in Section 3) is that no deterministic scheduling rule is both merge-proof and split-proof. Randomizing the service ordering of jobs is the easiest way to restore fairness when efficiency compels to process them whole. We submit another advantage of randomization: among riskneutral participants—minimizing the expected wait until the completion of their job-, we can construct probabilistic scheduling rules that are simultaneously split-proof and merge-proof. The simplest example is the Uniform scheduling rule ignoring all differences in job lengths, namely choosing each ordering of the n participants with equal probability 1/n!. The expected wait 1 of job i with size xi is yi = xi + 2 j =i xj , because there is a 50% chance that any other job j precedes job i. When a coalition of users other than i merge their jobs, or when user j , j = i, splits her job, the sum j =i xj is unaffected, and so is user i’s expected wait. By Pareto optimality it follows that neither the splitting agent nor the merging coalition can benefit.1 There are many other split-proof and merge-proof scheduling rules (Proposition 2 in Section 6). We argue that the most interesting among these is the Proportional rule. Given n jobs of sizes x1 , . . . , xn , this rule orders them randomly as follows: choose first the job scheduled last, xi ; then select in the same way the job scheduled next job i being selected with probability x1 +···+x n to last among the remaining jobs, and so on. In particular for any two jobs i, j the probability xi , hence the name of this rule. that j is scheduled before i is xi +x j Our main results, Theorem 1 and 2 in Section 9, are two alternative characterizations of Proportional scheduling based on split-proofness. In addition we require one of two natural invariance properties, and one of three normative requirements. 1 This argument is due to Ozsoy (2005).

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A rule is separable if the relative ranking of any subset of jobs is independent of the number and length of other jobs. It is recursive if it is generated by n independent successive draws, choosing first the agent served last, then the agent served next to last in the reduced problem with n − 1 agents, and so on. If w is a positive weight function, the w-quasi-proportional rule schedw(xi ) , and proceeds recursively as above. Proposition 3 ules job i last with probability w(x1 )+···+w(x n) in Section 7 essentially characterizes the class of quasi-proportional rules by the combination of Separability and Recursivity. The three normative requirements (Section 8) include one equity test (Ranking) and two performance indices (excess delay and liability). Ranking simply requires that the longer the job, the longer its expected delay: xi < xj ⇒ yi − xi  yj − xj . Our performance indices are inspired by two conflicting normative goals playing a central role in the management of queues: to minimize the total expected waiting time of users; and to equalize slowdown across users, namely the ratio of waiting time to job size. See Friedman and Henderson (2003), Wierman and Harchol-Balter (2003), Cheng and Friedman (2005), and references in Moulin (2005). The (relative) excess delay of a rule is the worst ratio of actual to optimal expected total delay, where the maximization bears on all scheduling problems. The excess delay of Shortest Job First is 1, that of the Proportional rule is 2, that of the Uniform and Longest Job First rules is infinite. The liability of a job is the worst expected slowdown of that job, when other jobs are arbitrarily large. The liability of a rule is the worst liability of any job under that rule. The Proportional and Shortest Job First rules both have a liability of n, the number of other jobs in the queue. This is about twice the smallest feasible liability n+1 2 (Section 8). Among split-proof scheduling rules that are either separable (Theorem 1), or recursive (Theorem 2), the Proportional one is characterized by any one of (a) Ranking, (b) excess delay no larger than 2, or (c) liability not larger than n. 2. Related literature The role of splitting and merging maneuvers is a familiar theme in the fair division literature. In the rationing problem (Banker, 1981; Moulin, 1987; De Frutos, 1999; Ju, 2003), mergeproofness or split-proofness characterize at once the Proportional rule. In the quasi-linear social choice model, the same properties lead to the egalitarian division of surplus (Moulin, 1985; Chun, 2000), and in the cost sharing problem with variable demands, to the Aumann–Shapley rule (Sprumont, 2005). Most of these results are surveyed in Ju et al. (2005). A problem closely related to this paper is that of assigning randomly and sequentially homogeneous indivisible units according to individual claims (Moulin, 2002; Moulin and Stong, 2002, 2003). It can be viewed as a variant of the current model under “processor sharing.” Each user requests a job of size xi , where xi is an integer, and the server processes successive unit jobs, typically alternating between the different users (instead of serving jobs whole). This is not inefficient if partially completed jobs bring positive utility. Another Proportional rule emerges in that model, in which xi balls of color i are thrown in an  urn, and the server empties the urn without replacement to determine the order in which the xi unit jobs are processed. In Section 6 we show that the Proportional rule in this paper is similarly generated by throwing xi balls of color i in an urn, then serving the jobs (whole) in the order in which each color “vanishes” from the urn. The key axioms in the processor sharing model are different, yet the invulnerability to transfers plays a role (see Theorem 1 in Moulin and Stong, 2003). The companion paper Moulin (2004) discusses merging and splitting maneuvers in the popular scheduling model where monetary transfers are feasible, and users’ preferences are linear in

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waiting costs and money. The results there are less encouraging than in our probabilistic model, in the sense that no efficient mechanism can be both split-proof and merge-proof. Like here, merge-proofness proves much easier to meet than split-proofness. Unlike here, the normative consequences of split-proofness are dire: it is incompatible with either Ranking, a finite individual liability, or the monotonicity of the net waiting cost in one’s job size. In the model with cash transfers just alluded to, the truthful revelation of waiting costs by means of Vickrey–Clarke–Groves mechanisms has been extensively discussed: Dolan (1978), Suijs (1996), Mitra (2001, 2002). Closer to our model, Hain and Mitra (2004), Kittsteiner and Moldovanu (2003, 2005) consider the truthful revelation of job sizes, assuming that agents cannot be punished ex post for misreporting. In our model the system manager can punish users who report ex ante a smaller job, and they have no incentive to report a larger one either. The follow up paper Ozsoy (2005) explores the robustness of our results to partial transfers of jobs and to a coalitional version of the splitting maneuver. More on this paper in Section 10.2. 3. The model A scheduling problem is a pair (N, x) where N is a finite set of users or agents, and x ∈ RN ++ is a profile of strictly positive job sizes, xi > 0 for all i ∈ N . The problem is to choose a scheduling order for those jobs. We write Φ(N ) for the set of orderings of N , with generic element σ : N → {1, . . . , |N |}; σ (i) < σ (j ) means that job i is served/scheduled before job j . Given the finite or infinite set N ∗ of potential users, a deterministic scheduling rule r selects an ordering σ ∈ Φ(N) for any scheduling problem (N, x) where N ⊆ N ∗ . Proposition 1. Assume |N ∗ |  3. A deterministic scheduling rule r is vulnerable to either merging or splitting maneuvers. Proof. Set N = {A, B, C} and pick r, a merge-proof and split-proof rule. Without loss of generality, assume r(N, (2, 2, 2)) = σ , with σ (A) < σ (B) < σ (C). Then r({A, B}, (2, 4)) = σ  must be σ  (A) < σ  (B), otherwise B and C both reduce their wait in (N, (2, 2, 2)) by merging. Next r({B, C}, (4, 2)) = σ  must be σ  (B) < σ  (C), otherwise B shortens his wait by splitting into {A, B}, (2, 2). Next consider r(N, (1, 4, 1)) = σ ∗ . If σ ∗ serves B last, then in ({B, C}, (4, 2)), C can effectively be served first by splitting to {A, C}, (1, 1) and this reduces his wait. If σ ∗ serves B second, consider in (N, (1, 4, 1)) the merging of A, C into A, resulting in the problem {A, B}, (2, 4) where A is served first by σ  : A and C can use this slot of length 2 to serve first whomever is served first in σ ∗ , and reduce the wait of the other user. Similarly if σ ∗ serves B first, both A and C strictly reduce their wait by merging into A. This contradiction ends the proof. 2 From now on, we allow the server to schedule jobs randomly. A random ordering is a probability distribution p on Φ(N ); the set of such distributions is denoted Δ[Φ(N )]. If Ψ is a subset of Φ(N ), and p ∈ Δ[Φ(N )], we use the notation pΨ = σ ∈Ψ pσ . Examples used repeatedly below include pi≺j for Ψ = {σ | σ (i) < σ (j )}, and if S ⊂ N ∗ does not contain i, pS≺i for Ψ = {σ | σ (j ) < σ (i) for all j ∈ S}, pi≺S for Ψ = {σ | σ (i) < σ (j ) for all j ∈ S} and pi≺last(S) for Ψ = {σ | σ (i) < σ (j ) for some j ∈ S}. Merging and splitting maneuvers are only feasible if the server cannot monitor the identity of the users; if the scheduling rule introduces a bias based on the unverifiable name of the users,

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it cannot deal with a situation where several agents claim the same name, therefore the relevant scheduling rules must be anonymous. Thus the following definition, where n∗ is the largest feasible number of users the server can handle. Note that n∗ can be finite or infinite. Definition 1. Given n∗ , a scheduling rule is a mapping ρ associating to each problem (N, x), |N|  n∗ , x ∈ RN ++ , a random ordering ρ(N, x) = p ∈ Δ[Φ(N )]. The rule is anonymous if for any two problems of the same size (N, x), (N  , x  ), with ρ(N, x) = p, ρ(N  , x  ) = p  , and any bijection τ : N → N       xτ (i) = xi for all i ∈ N ⇒ pσ  ◦τ = pσ  for all σ  ∈ Φ(N  ) . As we restrict attention to anonymous rules, a set of users can always be written as N = {1, . . . , n}. Denote Φ(n) = Φ({1, . . . , n}). An (anonymous) scheduling problem is (n, x), where n  n∗ and x ∈ Rn++ , and an anonymous rule maps any such problem into ρ(n, x) = p ∈ Δ(Φ(n)). We turn to the users’ preferences. User i wishes to minimize the expected completion time yi of her own job. Given a problem (n, x) and p ∈ Δ(Φ(n)), her disutility is  yi = xi + pj ≺i · xj . (1) j ∈N i

Given (n, x), it will be important to describe in closed form the set F (n, x) of feasible disutility profiles y ∈ Rn++ , namely such that (1) holds for some lottery p. Define for all S ⊆ {1, . . . , n},   ) of non-ordered v(S, x) = S xi2 + S(2) xi ·xj , where S(2) is the set (with cardinality |S|·(|S|−1) 2 pairs from S. Note that the mapping S → v(S, x) is supermodular. Lemma 1. (Queyranne, 1993) Given x ∈ Rn++ , the profile y is feasible at (n, x), y ∈ F (n, x), if and only if     xi · yi = v {1, . . . , n}, x and xi · yi  v(S, x) for all S ⊆ {1, . . . , n}. N

S

Corollary. Any y ∈ F (n, x) is efficient: for any y  ∈ F (n, x){y   y} ⇒ {y  = y}. Proof. Fix a problem (n, x), x ∈ Rn++ . Using the notation x · y for the coordinate-wise multiplication of vectors, Lemma 1 can be phrased as follows: y ∈ F (n, x) if and only if x · y belongs to the core C({1, . . . , n}, x) of the n-person cooperative game S → v(S, x), S ⊆ {1, . . . , n}. σ σ  For all σ ∈ Φ(n) let y be the profile of (deterministic) waits under σ namely yi σ= P (i;σ ) xj , where P (i; σ ) = {j | σ (j )  σ (i)}. Note that F (n, x) is the convex hull of {y | σ ∈ Φ(n)}. Set P− (i; σ ) = P (i; σ ){i}. Routine computation shows xi · yiσ = v(P (i; σ ), x) − v(P− (i; σ ), x), namely x · y σ is the vector of marginal contributions in the game v(·, x) for the ordering σ of the agents. As v(·, x) is supermodular, a classic result (Shapley, 1971) says that C(N, x) is the convex hull of x · y σ , σ ∈ Φ(N). All coordinates of x are positive, thus the vector z is in the convex hull of {y σ | σ ∈ Φ(n)} if and only if x · z is in the convex hull of {x · y σ | σ ∈ Φ(n)}.  The corollary follows because the weighted sum N xi · yi is independent of y in F (n, x), and xi > 0 for all i. 2 In many results below, we are only interested in the disutility profile y achieved by our scheduling rule, the choice of the lottery p to implement it (typically not unique) does not matter.

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Definition 2. An anonymous scheduling method is a mapping μ associating to each problem (n, x), n  n∗ , x ∈ Rn++ , a profile μ(n, x) = y ∈ F (n, x). In the rest of the paper when we speak of a rule or a method, we always mean that it is anonymous. 4. Split-proofness and merge-proofness Throughout Sections 4–8, the cap n∗ on the size of a problem plays no role, and is omitted. We say that the scheduling problem (n , x  ) obtains from the problem (n, x) by a k-split of job n, where k  2, if 

n = n + k − 1;

xi = xi

for all i = 1, . . . , n − 1;

n 

xi = xn ;

n

a similar definition applies to the k-split of any other job i. Definition 3. We say that the scheduling rule ρ is split-proof if whenever (n , x  ) obtains from (n, x) by a k-split of job n, we have yn  x n +

n−1  j =1

pj ≺last({n,...,n }) · xj ,

where y = ρ(n, x) and p  = ρ(n , x  ).

(2)

After user n splits his job, his real job is done only when all “sub-jobs” of lengths xn , . . . , xn  are completed, hence the delay he experiences from job j, j  n − 1, is determined by the probability pj ≺last({n,...,n }) that this job is scheduled before at least one of n’s sub-jobs. Thus the right-hand side of inequality (2) is the (true) expected wait of user n after the split. Because ρ is anonymous, if the above inequality holds for splits of job n, it holds for splits of other jobs as well. We say that the scheduling problem (n , x  ) obtains from the problem (n, x) by a merging of jobs 1, . . . , k if 

n = n − k + 1;

x1

=

k 

xi ;

xi = xi+k−1

for i = 2, . . . , n − k + 1.

1

A similar definition applies to the merging of any other k jobs. After users 1, . . . , k merge their jobs, they can choose freely the scheduling order, possibly randomized, of their “true” jobs in the time interval of length x1 allocated to the merged job. Let y1 be the expected wait of the latter job, then y1 − x1 is its expected delay generated by the other jobs 2, . . . , n − k + 1. This delay is borne by all users 1, . . . , k upon merging. The other part of their true wait comes from the processing order of jobs 1, . . . , k within the time slot of the merged job: by Lemma 1 they can implement any z ∈ F (k, (x1 , . . . , xk )). Thus we see that users 1, . . . , k can achieve after merging the profile  y ∈ Rk++ of (real) expected waits if and only yi = y1 − x1 + zi for all i = 1, . . . , k. If y is if they can find z ∈ F (k, (x1 , . . . , xk )) such that  the profile of waits before merging, this move is profitable if and only if there exists  y such that yi  yi with at least one strict inequality. Notice that the description of profitable merging relies only on the vectors y, y  of disutility profiles before and after the move, not on the actual lotteries implementing them. Therefore

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merge-proofness is defined directly for a scheduling method. This simplifies greatly the analysis of this property. Definition 4. We say that the scheduling method μ is vulnerable to merging if we can find (n , x  ) obtaining from (n, x) by a merging of jobs 1, . . . , k, and a profile z ∈ F (k, (x1 , . . . , xk )) such that y1

−

k 

xj + zi  yi

for i = 1, . . . , k, with at least one strict inequality

(3)

j =1

where y = μ(n, x) and y  = μ(n , x  ). The method μ is called merge-proof if it is not vulnerable to merging. The scheduling rule ρ is merge-proof if the associated method is. As above, anonymity of μ implies the similar system of inequalities for merges involving any other k users. To illustrate Definitions 3 and 4, we check that the Uniform rule, selecting the uniform lottery on Φ(n) for any problem (n, x),is both split-proof and merge-proof. Recall that the Uniform method is given by yi = xi + 12 j =i xj . For split-proofness, note that pj ≺last({n,...,n }) = k > 12 , implying (2) at once. For k+1  1 j =i;j =1,...,k xj , so the efficiency of 2

merge-proofness, inequality (3) reduces to zi  xi + the Uniform rule is enough.

5. Separable rules and methods We introduce a key invariance property of scheduling rules and methods, under which splitproofness and merge-proofness take a very simple form. In the next section we describe a natural family of separable rules, containing all benchmark scheduling rules (Uniform, Shortest Job First, and Longest Job First) as well as the Proportional rule that we promote. Separability is key to one of our two axiomatizations of the latter (Theorem 1). Notation: given n, S ⊆ {1, . . . , n} and σ ∈ Φ(n), we write σ [S] ∈ Φ(S) for the ordering of S induced by σ . Definition 5. The scheduling rule ρ is separable if for any problem (n, x) and any k < n, setting p = ρ(n, x) and p ∗ = ρ(k, (x1 , . . . , xk )) we have  for all σ ∗ ∈ Φ(k): pσ (N, x) = pσ∗ ∗ . σ ∈Φ(n): σ [{1,...,k}]=σ ∗

This means that the (random) relative ranking of the jobs 1, . . . , k follows a probability distribution independent of the number and size of other jobs. Applying this property for k = 2, we see that p2≺1 only depends upon (x1 , x2 ): p2≺1 = θ (x1 , x2 ). By anonymity the function θ is the same for any pair of users i, j : θ (a, b) is the probability that a job of length b is scheduled before one of length a. Therefore the method μ defined by ρ takes the form  θ (xi , xj ) · xj for all n, x. (4) μi (n, x) = yi = xi + j ∈N i

Note that by construction θ (a, b) + θ (b, a) = 1.

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Lemma 2. The separable scheduling rule ρ with corresponding function θ is split-proof if and only if for all problem (n, x) we have θ x1 ,

n 

xi  p{2,...,n}≺1 ,

2



or equivalently θ

n 

(5)

xi , x1  p1≺last({2,...,n}) ,

2

where as above we set p = ρ(n, x). Proof. Statement if. Fix (n , x  ) obtaining n−1form (n, x) by a k-split of job n (Definition 3). By Separability of ρ, we have yn − xn = 1 θ (xn , xj ) · xj . For any j ∈ {1, . . . , n − 1}, the k-split of job n in problem (2, (xj , xn )) yields the problem (k +1, (xj , xn , . . . , xn  )), so the lower version   of inequality (5) gives θ ( nn xi , xj ) = θ (xn , xj )  pj ≺last({n,...,n }) . The desired inequality (2) follows. Statementonly if. Choose (n, x) as in the premises of (5). Note that (n, x) obtains from ({1, 2}, (x1 , n2 xi )) by a (n − 1)-split of job 2. Apply (2):

n n   y2  xi + p1≺last({2,...,n}) · x1 where y = ρ {1, 2}, x1 , xi . 2

2

On the other hand, Eq. (4) gives n

n   y2 = xi + θ xi , x1 · x1 2

2

which completes the proof.

2

Next we write merge-proofness for a separable rule. Recall that this property bears only on the corresponding method, which in this case is given by Eq. (4). Instead of restricting attention to those methods associated with separable rules, it is natural to allow for a richer class of methods by taking Eq. (4) to be the very definition of separable methods. Definition 6. The scheduling method μ is separable if there exists a function θ from R2++ into (0, 1], such that θ (a, b) + θ (b, a) = 1 for all a, b  0, and such that Eq. (4) holds for all (n, x). We speak of the θ -separable method μ. For this definition to make sense, we must check that for any choice of the function θ as above, Eq. (4) defines a feasible profile of expected waits, hence an (anonymous) scheduling method in the sense of Definition 2. Compute   xi · yi = v(S, x) + θ (xi , xj ) · xi · xj  v(S, x) S

i∈S,j ∈N S

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with equality for S = N . This proves the claim by Lemma 1. Note however that for an arbitrary function θ , the method defined by Eq. (4) may not be implementable by a separable scheduling rule.2 Lemma 3. The θ -separable method is merge-proof if the mapping a → θ (a, b) · a is superadditive on R++ ; the latter holds in particular if θ (a, b) is non-decreasing in a. Proof. We show first a preliminary result. The scheduling method μ is merge-proof if for every k, n, n , and every (n , x  ) obtained from (n, x) by a merging of jobs 1, . . . , k, setting y = μ(n, x) and y  = μ(n , x  ), we have k

k    xi · yi + xi · xj  xi · y1 (6) {i,j }∈K(2)

i=1

1

where K(2) is the set of non-ordered pairs in {1, . . . , k}. Fix any z ∈ F (k, (x1 , . . . , xk )), and use Lemma 1 and the definition of v(·, x) to compute



k k k k       xj · y1 − xi + zj = xj · y1 − xi + v {1, . . . , k}, x j =1

j =1

i=1

=

k 

xj

i=1

· y1 −

j =1



xi · xj .

K(2)

   Thus (6) is equivalent to kj =1 xj · yj  kj =1 xj · (y1 − ki=1 xi + zj ), and implies that the system of inequalities (3) is impossible. We pick now a function θ and prove (6) for the θ -separable method. We develop this inequality in several steps. The term ki=1 xi · yi in the LHS is k 

xi · yi =

i=1

k 

 xi · xi +

=

i=1

θ (xi , xj ) · xj

j ∈{1,...,n}{i}

i=1 k 





xi2 +

 K(2)

xi · xj +



θ (xi , xj ) · xi · xj .

i∈{1,...,k},j ∈{k+1,...,n}

  Setting x1···k = ki=1 xi , the LHS of (6) is now (x1···k )2 + {1,...,k}×{k+1,...,n} θ (xi , xj ) · xi · xj .   The RHS is (x1···k ) · [x1···k + nj=k+1 θ ( ki=1 xi , xj ) · xj ], therefore (6) amounts to 

     xj · θ (xi , xj ) · xi  xj · θ (x1···k , xj ) · x1···k . j ∈{k+1,...,n}

i∈{1,...,k}

j ∈{k+1,...,n}

This holds if a → θ (a, b) · a is superadditive. The latter is true if θ (a, b) is non-decreasing in a because θ  0. 2 2 For a given matrix [θ i,j (x , x )] = [t ] with non-negative entries such that t + t i j i,j i,j j,i = 1, the existence of a lottery p ∈ Δ(Φ(n)) such that prob{σ (j ) < σ (i)} = ti,j for all i, j is not guaranteed. On this difficult existence problem, see

Fishburn (1992).

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Lemma 3 shows that many separable methods are merge-proof. By contrast Lemma 2 does not point to a specific set of rules meeting inequality (5). We turn to a subset of separable rules for which (5) becomes transparent. 6. Parametric and quasi-proportional rules We construct a large family of separable scheduling rules encompassing all the practical rules discussed in this paper. Choose for each a > 0 a cumulative distribution function Fa on [0, +∞[ with no mass at 0. Thus Fa is any non-decreasing and right-continuous function on [0, +∞[ such that Fa (0) = 0 and Fa (∞) = 1. Definition 7. Given a scheduling problem (n, x), x ∈ Rn++ , the parametric rule associated with the family {Fa , a > 0} draws for each i = 1, . . . , n a number Zxi with cdf Fxi , and these draws are stochastically independent. The jobs are ordered according to the realization of these variables, i.e., job i precedes job j if Zxi (ω) < Zxj (ω) and ties are broken with uniform probability. Lemma 4. A parametric rule is separable, and the corresponding function θ is 1 θ (a, b) = prob{Zb < Za } + prob{Zb = Za } 2

for all a, b.

(7)

Proof. Anonymity is clear: the (random) ordering of jobs only depends upon their length. The relative ordering of any subset of jobs depends only upon the draws for this subset of jobs, hence the rule is separable as well. 2 Definition 8. Let w be a strictly positive function on R++ . The w-quasi-proportional rule is the parametric rule associated with Fa (z) = zw(a) for all 0  z  1. w(a) . Lemma 5. For this rule we have θ (a, b) = w(b)+w(a) At problem (n, x) the probability of the ordering σ : 1 ≺ 2 ≺ · · · ≺ n is

w(xn−1 ) w(x2 ) w(xn ) . · n−1 ··· pσ = n w(x ) + w(x ) w(x ) 1 2 k w(x ) k=1 k k=1

(8)

Proof. The variables Zxi are stochastically independent,and the support of the distribution Fa is non-atomic: therefore the probability of a tie Zxi = Zxj is null, and Eq. (7) reduces to 1 θ (a, b) = prob{Zb < Za } =

Fb (t) dFa (t) = 0

=

1 t w(b) w(a)t w(a)−1 dt 0

w(a) . w(b) + w(a)

This proves (8) for n = 2. Assume next it holds for n − 1. The event {σ } is the intersection of [1 ≺ · · · ≺ n − 1] and [{1, . . . , n − 1} ≺ n], and these two events are independent. The c.d.f. n−1 of maxi=1,...,n−1 Zxi is Fx1 (z) · · · Fxn−1 (z) = z 1 w(xk ) , therefore the probability of [{1, . . . , n) n − 1} ≺ n] is w(x , and (8) follows by the inductive assumption. 2 n w(x ) 1

k

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The Uniform scheduling rule is the w-quasi-proportional rule for which w(a) = 1, all a. 1 . Shortest Job First is the parametric rule for which the distribution Formula (8) is then pσ = n! of Fa is concentrated at a so that job i precedes job j if xi < xj . Similarly, Longest Job First is the parametric rule where the distribution of Fa is concentrated at a1 . The Proportional scheduling rule, to which our two characterization results are devoted, is the w-quasi-proportional rule for which w(a) = a, all a. A natural choice of the weight function w is the power function w(a) = a α for some real number α. When α goes to +∞ (respectively −∞), the corresponding rule converges to Shortest (respectively Longest) Job First in the “pointwise” sense: for any (n, x) and any σ , the corresponding probabilities of σ converges. There is an alternative, perhaps more intuitive definition of quasi-proportional rules, for those problems (n, x) where the numbers w(xi ) are all rationals, namely w(xi ) = bdi for some integers bi , d. For each agent i put bi balls of color i in an urn and empty the urn by successive draws with uniform probability and without replacement; schedule the jobs in the order in which each color vanishes in the urn (when the last ball of this color is drawn). This generates precisely the random ordering described in Lemma 5. Our next result shows that for parametric or quasi-proportional rules, our two manipulability requirements are easy to describe. Proposition 2. (i) If the cdfs {Fa , a > 0} satisfy Fa ·Fb  Fa+b for all a, b > 0, the corresponding parametric rule is split-proof. (ii) If the cdfs {Fa , a > 0} are (stochastic-dominance) monotonic in a (a  b ⇒ Fb (z)  Fa (z) for all z), the corresponding parametric rule is merge-proof. Corollary. (i) The w-quasi-proportional rule is split-proof if and only if w is subadditive. (ii) The w-quasi-proportional rule is merge–proof if w is non-decreasing. (iii) The rule corresponding to w(a) = a α is split-proof iff α  1, and merge-proof iff α  0. Proof. We assume for simplicity that all distributions Fa are atomless (i.e., Fa is continuous), so that ties occur with probability zero, and θ (a, b) = prob{Zb < Za }. The proof can be adapted to the case of distributions Fa with atoms. This non-essential argument is available upon request from the author.  Statement (i). Fix a parametric rule {Fa , a > 0} and a problem (n, x); set b = n2 xi . The left-hand inequality in (5) is prob{Zb < Zx1 }  prob{ max Zxj < Zx1 }. 2,...,n

For all a, Fa is differentiable almost everywhere hence the probability distribution of Za has a density fa (z) with respect to the Lebesgue measure. Compute   prob{Zb  Zx1 } =

fb (w)fx1 (z) dw dz wz

∞ z =



∞

fb (w) dw fx1 (z) dz = 0

0

Fb (z)fx1 (z) dz. 0

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The same computation where max2,...,n Zxj replaces Zb , and its c.d.f. gives prob{ max Zxj < Zx1 } = 2,...,n

∞ 



i=2,...,n Fxi

replaces Fb ,

Fxi (z)fx1 (z) dz

0 i=2,...,n

 Therefore the left-hand inequality in (5) holds if Fb (z)  i=2,...,n Fxi (z) for all z, and the proof of statement (i) is complete. ∞ Statement (ii). We have θ (a, b) = 0 (1 − Fa (z)) dFb (z). Thus the monotonicity of Fa (z) w.r.t. a implies that θ (a, b) is non-decreasing in a, and the desired conclusion follows by Lemma 3. Corollary. For the w-quasi-proportional rule, the cdfs Fa are atomless. If w is non-decreasing, a → Fa is non-increasing, hence statement (ii). For statement (i) observe that inequality Fa (z) · Fb (z)  Fa+b (z) amounts to w(a) + w(b)  w(a + b). This proves the “if” statement. Conversely, suppose n the w-rule is split-proof. By Lemma 2, inequalities (5) must be true. As  n i=2 w(xi ) , the lower inequality in (5) reads i=2 Fxi (z) = z θ x1 ,

n 

xi

2

w(x1 )   = w(x1 ) + w( n2 xi )

1 z

n

i=2 w(xi )

w(x1 )zw(x1 )−1 dz

0

w(x1 )  = w(x1 ) + ni=2 w(xi ) implying the subadditivity of w. Statement (iii) adds to (i) and (ii) the observation that merge-proofness holds only if α  0. We omit the easy proof. 2 The Uniform and Proportional rules are the endpoints (for α = 0 and 1) of the interval of merge-proof and split-proof rules identified by the Corollary. Proposition 2 implies that there are many other merge-proof and split-proof parametric scheduling rules. It suffices that the cdfs satisfy Fa · Fb  Fa+b  Fa for all a, b > 0. Two examples are the cdfs Fa (z) = min{ (1+a)z a+z , 1}, a2

and Fa (z) = e− z(a+z) . To generate more merge-proof and split-proof rules that are separable but not parametric, consider a fair lottery over two quasi-proportional rules, with weight functions w1 (a) w2 (a) + 12 w2 (b)+w is non-decreasing in a, this rule is mergew1 , w2 . As θ (a, b) = 12 w1 (b)+w 1 (a) 2 (a) proof (Lemma 3); moreover, split-proofness is preserved by lotteries over rules. Remark 1. For a general w-quasi-proportional rule, merge-proofness does not require w to be w(a) ·a non-decreasing. If a → a · w(a) is superadditive, it is easy to check that a → w(b)+w(a) is superadditive as well, thus by Lemma 3 the w-rule is merge-proof. We can choose such a function w that is not monotonic. 7. Recursive rules We introduce another invariance property of scheduling rules satisfied by quasi-proportional rules and many other rules as well. This property is the key to our characterization of the Proportional rule (Theorem 2).

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Consider Eq. (8) in Lemma 5: pσ is computed recursively by finding first the probability that user n is ranked last in problem (n, x), multiplying it by the probability that user n − 1 is ranked last in the reduced problem (n − 1, (x1 , . . . , xn−1 )), and so on. In other words the rule is entirely determined by the family of probability distributions, denoted π(n, x) below, of the job served last. Notation. For x ∈ Rn++ and i ∈ {1, . . . , n}, x−i ∈ Rn−1 ++ obtains from x by deleting the ith coordinate; for σ ∈ Φ(n), σ−i is the restriction of σ to {1, . . . , n}{i}. The set of probability distributions on {1, . . . , n} is Δ(n). n) w(x n w(x k) 1

Definition 9. The scheduling rule ρ is recursive if there exists for all (n, x) a distribution π(n, x) ∈ Δ(n), such that for all σ ∈ Φ(n) ranking i last 

 p = ρ(n, x) and p −i = ρ(n − 1, x−i )

⇒

pσ = πi (n, x) · pσ−i−i .

By the anonymity of ρ, π(n, x) is symmetric in all variables xi . The quasi-proportional rules are recursive by Lemma 5. Our next result implies, essentially, that no other parametric rule, and more generally no other separable rule, is recursive. Proposition 3. A scheduling rule is quasi-proportional if and only if it is separable, recursive, and meets the Positivity property: for any (n, x) and any i, j p = ρ(n, x)

⇒

pi≺j > 0.

The proof is in Appendix A. Note that Positivity cannot be relaxed: Shortest (or Longest) Jobs First meets the other three properties. Parallel to Lemma 2, we have a sufficient condition for split-proofness among recursive rules. The recursive scheduling rule ρ is split-proof if whenever (n + 1, x  ) obtains from (n, x) by a 2-split of job n, we have: πk (n, x)  πk (n + 1, x  )

for all k = 1, . . . , n − 1.

This says that when agent i splits his job in two (or more) pieces, the probability that another job is scheduled last does not become larger. Loosely speaking, each function πk (n, x) is subadditive upon splitting. The straightforward proof is omitted. Remark 2. An ‘forward-looking’ definition of Recursivity works by drawing the user scheduled first in (n, x), then repeating in the reduced problem and so on. Denote λ(n, x) the probabili) : the resulting rule is ity distribution of the job served first and choose λi (n, x) = 1/w(x n 1/w(x ) 1

w(xi ) w(xj )+w(xi ) ,

j

separable and the probability that job j precedes job i is just like with the w-quasiproportional rule of Definition 8. These two rules are different with three or more users, but they implement the same method. However the forward-looking version of Recursivity does not yield interesting split-proof rules. For instance the corresponding Proportional rule (scheduling i first with probability x1i ) is not split-proof.

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8. Ranking, excess delay and liability 8.1. The ranking axiom Definition 10. The scheduling method μ meets Ranking if for all (n, x), y = μ(n, x) we have xi  xj

⇒

yi − xi  yj − xj

for all i, j ∈ {1, . . . , n}.

The interpretation of Ranking is standard (see e.g., Moulin and Sprumont, 2005): a larger job causes more delay externalities, hence it is fair that it experiences a larger delay. Note that the Uniform rule fails Ranking: the delay of a shorter job is always larger than that of a longer job. Ranking is easy to express for separable—and parametric—methods. Lemma 6. (i) The θ -separable method meets Ranking if θ (a, b) is non-decreasing in a and for all i, j, all a, b > 0:

ab

⇒

θ (a, b) 

a . a+b

(ii) If n∗ = ∞, the converse statement holds true. Proof. The “if” statement follows by developing the inequality yi − xi  yj − xj with the help of (4). To prove “only if” when n∗ = ∞, apply first Ranking to the problem (2, (x1 , x2 )), to get the upper bound on θ (x1 , x2 ). Next show by contradiction that θ (a, b) is non-decreasing in a. Suppose for some a, a  , b, we have a < a  and θ (a, b) > θ (a  , b). Consider the problem (n, (a, a  , b, . . . , b)): for n large enough (4) implies y1 − x1 > y2 − x2 , in contradiction of Ranking. 2 In particular, the w-quasi-proportional method meets Ranking if and only if w(a)/a is nondecreasing. The latter property implies that w is superadditive. Compare with the Corollary to Proposition 2: split-proofness is equivalent to subadditivity of w. For w(a) = a α , Ranking amounts to α  1 and split-proofness to α  1. Recall that α = 1 is the Proportional method. The results of Section 9 show that this tension occurs in the much larger classes of separable or recursive rules. 8.2. Excess delay and limited liability We write ∇(n, x) for the smallest feasible total delay at problem (n, x), obtained by schedul ing shortest jobs first. Thus ∇(n, x) = N (2) min{xi , xj }, where N (2) is the set of non-ordered pairs of {1, . . . , n}. Definition 11. Fix the size n and a scheduling method μ. Setting y = μ(n, x), define n 1 (yi − xi ) the excess delay δ(n) = sup , ∇(n, x) x∈Rn++ the liability λ(n) = sup

x∈Rn++

y1 . x1

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The liability of Shortest Job First is λ(n) = n, because the worst that could happen to job i is that all other jobs are barely shorter than xi . But the optimal (smallest feasible) liability is about twice smaller. Call the w-quasi-proportional rule with w(z) = z2 the Quadratic rule. Lemma 7. (i) The Quadratic method has the liability λ∗ (n) = n+1 2 . (ii) The liability of any method μ is not smaller than λ∗ : λ(n)  λ∗ (n). Proof. Fix μ consider the problem (n, (a, . . . , a)). Feasibility of y = μ(n, x) (Lemma 1) and n implies a · ( 1 yi ) = v(n, x) = n(n+1) · a 2 , thus by anonymity yi = n+1 2 2 · a for all i, implying n+1 λ(n)  2 . For statement i, fix x1 and compute the largest possible expected wait y1 for a general θ -separable method: sup y1 = x1 + x−1

n  j =2

sup θ (x1 , xj ) · xj = x1 + (n − 1) sup θ (x1 , b) · b. xj

b

Thus μ has liability λ∗ if and only if supb>0 θ (a, b) · b = a2 , which one checks easily for the Quadratic method. 2 The above proof gives the simple form of the liability λ for the θ -separable method: b λ(n) = 1 + (n − 1) sup θ (a, b) · . a a,b>0

(9)

Note that there are other separable methods with the optimal liability λ∗ , namely other choices of θ for which supb>0 θ (a, b) · b = a2 . But among quasi-proportional rules, only the Quadratic has the optimal liability λ∗ .3 Lemma 8. For the θ -separable method μ the excess delay δ(n) = δ is independent of n δ = 1 + sup θ (a, b) · ba>0

b−a . a

Proof. We set β = supba>0 θ (a, b) · n 

(yi − xi ) =

1

b−a a ,

possibly infinite. Fix a problem (n, x) and compute

  θ (xi , xj ) · xj + θ (xj , xi ) · xi N (2)

=

(10)

 {i,j }∈N (2):xi xj



xj − xi xi · 1 + θ (xi , xj ) · xi

 (1 + β) · ∇(n, x).

Thus δ(n)  1 + β for all n. To prove the opposite inequality we fix n, pick β  such that β  < β 1 and x1 , x2 such that x1  x2 and θ (x1 , x2 ) · x2x−x  β  . For any ε such that 0 < ε < x1 , consider 1 the problem (n, (x1 , x2 , ε, . . . , ε)). Compute w(a) w(a) a a 3 The w-method guarantees λ∗ iff w(a)+w(b)  2b ⇔ w(b) + 1  2 b for all a, b > 0. For a = b + ε, this gives w(b+ε)−w(b) w(a+ε)−w(a)  2 w(b)  2 w(a) ˙ = 2 w(a) ε ε a−ε . Hence w is differentiable and w(a) a . b ; for b = a + ε,

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(n − 2)(n + 1) · ε, 2  n  x2 − x1  x1 · (1 + β  ). (yi − xi )  x1 · 1 + θ (x1 , x2 ) · x1 ∇(n, x) = x1 +

1

Letting ε go to zero, we get δ(n)  1 + β  and the desired inequality when β  goes to β.

2

Comparing Eqs. (9) and (10) we see that δ is infinite if and only if λ(n) is infinite for all n. We apply now Lemma 8 to compute our two indices for the most important w-quasiproportional methods. We have

 b w(a) a r(b) −1 b · = inf + sup θ (a, b) · = sup . a,b>0 b a a,b>0 w(a) + w(b) a r(a) a,b>0 r(b) r(∞) a Suppose first that r(z) = w(z) z is non-increasing. Then infa,b>0 b + r(a) = r(0) . Therefore the excess delay and liability are both finite if and only if r is bounded away from 0 and from +∞. For instance w(z) = zα with α < 1 has infinite excess delay and liability. r(b) Next suppose that r(z) is non-decreasing. For any a, b we have then ab + r(a)  1, hence

supa,b>0 θ (a, b)· ab  1 so liability and excess delay are both finite. Straightforward computations for w(z) = zα with α  1 give x −1 δ = 1 + sup α ; x1 x + 1

1

(α − 1)1− α λ(n) = 1 + (n − 1) α

for 1  α  +∞.

Liability decreases with α, from λ(n) = n for α = 1 (Proportional method) to λ(n) = n+1 2 for α = 2 (Quadratic method); it increases with α beyond 2, and its limit is λ(n) = n at infinity (Shortest Job First). √ 1+ 2 The excess delay decreases for all α, from δ = 2 for the Proportional method, to δ = 2  1.21 for the Quadratic rule, and δ = 1 at infinity. 9. Characterizing the Proportional method The first result concerns separable rules, the second one recursive rules. They rely on almost identical additional requirements about the excess delay, liability, or the Ranking property. Recall that n∗ caps the size of all problems. In Theorems 1, 2, n∗  3. Theorem 1. (i) If the separable rule ρ is split-proof, then λ(n)  n for all n; (ii) If the separable method μ meets Ranking, then λ(n)  n for all n; (iii) If the separable rule ρ is split-proof and meets at least one of   Ranking, or {δ  2}, or λ(n)  n for some n then its method is Proportional. Theorem 2. The Proportional rule is the only recursive and split-proof rule that meets at least one of     Ranking, or δ(2)  2 , or λ(2)  2 .

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Both results are proven in Appendix A. Theorem 1 is weaker than Theorem 2: it characterizes the set of rules of which the associated method is Proportional, not that rule itself. I conjecture that for any n∗  3 it is possible to construct a separable and split-proof rule that meets any one of the three additional requirements and is not the Proportional one. Is Recursivity a necessary assumption in Theorem 2? I.e., can we find split-proof rules meeting at least one of Ranking, {δ(2)  2}, or {λ(2)  2}, and different from the Proportional one? I conjecture that such rules exist. 10. Concluding comments 10.1. Split- and merge-invariance The Uniform method meets a stronger property than merge-proofness, namely it is mergeinvariant. When a coalition merges, it can achieve after merging precisely the same profile of expected wait as before merging, and no better. Moreover users outside the coalition are not affected by the merge. Formally merge-invariance requires that for any problem (n , x  ) obtaining from (n, x) by a merging of jobs 1, . . . , k, zi = yi − y1

+

k 

xj

for i = 1, . . . , k

  defines z ∈ F k, (x1 , . . . , xk ) .

j =1

It turns out that the Uniform method is the only merge-invariant scheduling method. Thus mergeinvariance is not compatible with even a modicum of responsiveness, or a finite cap on individual liability or excess delay. On the other hand, the Proportional rule is split-invariant, namely splitting one’s job is a matter of indifference. To check this, observe that the cdfs Fa (z) = za satisfy Fa · Fb = Fa+b for all a, b > 0, therefore inequality (5) is actually an equality for the Proportional rule, and the same holds true for inequality (2), expressing that the expected wait of agent 1 is the same before and after the split. However one checks easily that the split increases the expected wait of every user other than 1. While merge-invariance single-handedly characterizes the Uniform rule, the Proportional rule is not the only split-invariant rule. Yet it is characterized by split-invariance within our two benchmark families. Mimicking the proof of Theorems 1, 2, one shows (i) If the separable rule ρ is split-invariant, its method is Proportional; (ii) The Proportional is the only recursive and split-invariant scheduling rule. 10.2. Transfers and coordinated splitting A strategic maneuver related to merging and splitting is the partial transfer of jobs among users. Two (or more) agents choose to reallocate their jobs of sizes x1 and x2 as x1 and x2 , with x1 + x2 = x1 + x2 . Ozsoy (2005) shows that the Proportional rule is not vulnerable to pairwise transfers (involving only two agents). The same is true for all quasi-proportional rules with w(a) = a α and 0  α  1. However, transfers among three agents can be profitable under all quasi-proportional rules, with the single exception of the Uniform rule. Another strategic maneuver threatening the

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Proportional rule is the cooperative version of splitting: several agents split each job into smaller pieces, and decide on the allocation of these slots conditional upon the actual service ordering. Among separable scheduling rules, immunity to this sort of moves points again to the Uniform rule. In the quasi-linear model of the companion paper Moulin (2004), no efficient rules is invulnerable to transfers involving three or more agents, but pairwise transfer-proofness points to a one-dimensional line of rules borne by two appealing rules, one split-proof and the other mergeproof. These solutions are not built on any proportionality idea, but apply instead the Shapley value to an appropriate cooperative game. They can also be derived from invariance or consistency properties (Maniquet, 2003; Chun, 2004, 2006). Acknowledgments Thanks to R.J. Aumann, Hatice Ozsoy, Harris Schlesinger, Jay Sethuraman and two anonymous referees for helpful comments on earlier versions. I am grateful for their hospitality in the Spring 2004 to the Universities of Auckland, New Zealand, and of Namur, Belgium, and to the New School of Economics in Moscow. Appendix A. Remaining proofs A.1. Proof of Proposition 3 Only the “if” statement requires a proof. Let ρ be a rule with the three stated properties, and π(n, x) be the associated probability distribution of the last scheduled agent. By anonymity of ρ, it is symmetric in all variables xi . Fix n, x and partition the event {1 ≺ 2 ≺ · · · ≺ n − 1} according to the ranking of agent n. Setting p = ρ(n, x) this gives  p1≺···≺i−1≺n≺i≺···≺n−1 . (A.1) p1≺2≺···≺n−1 = p1≺2≺···≺n + 1in−1 −(n−1),n

By Separability and Recursivity, the summation in the RHS is πn−1 (n, x) · p1≺2≺···≺n−2 , where p −(n−1),n = ρ(n − 2, x−(n−1),n ). On the other hand, {1 ≺ 2 ≺ · · · ≺ n − 1}{1 ≺ 2 ≺ · · · ≺ n} is the intersection of {1 ≺ 2 ≺ · · · ≺ n − 1} and {n ≺ last({1, . . . , n − 1})}, two independent events by Separability. Setting p −n = ρ(n − 1, x−n ), Eq. (A.1) now writes   −n −(n−1),n 1 − πn (n, x) · p1≺2≺···≺n−1 = πn−1 (n, x) · p1≺2≺···≺n−2 . (A.2) Apply (A.2) first for n = 3, x ∈ R3++ , with the notation θ (b, a) = π2 (2, (a, b)) for the probability that a job of size a precedes one of size b. Writing πi (3, x) simply as πi (x), we get (1 − π3 (x)) · θ (x2 , x1 ) = π2 (x). By Positivity, θ (x2 , x1 ) > 0, so π2 (x) = 0 implies π1 (x) = 0. Exchanging the role of agents 1 and 3, we see that π2 (x) = 0 implies π3 (x) = 0 as well, contradiction. Thus πi (x) > 0 for i = 1, 2, 3, and we have θ (x2 , x1 ) = so that the ratio

π2 (x) π1 + π2 π1 π2 (x)

⇔

π1 1 (x) = −1 π2 θ (x2 , x1 )

only depends on x1 , x2 . From

three real positive functions wi such that and w1 (x1 ) =

π1 π3 (x1 , 1),

w2 (x2 ) =

πi πj

(xi , xj ) =

π2 π3 (x2 , 1):

then

π1 π2

·

π1 π2 π3 π2 · π3 · π1 = 1, we claim there exist wi (xi ) wj (xj ) for all i, j in {1, 2, 3}. Set x3 = 1 w1 (x1 ) π2 π3 π1 π3 · π1 = 1 implies π2 (x1 , x2 ) = w2 (x2 ) .

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π2 Next set x1 = 1: now ππ12 · ππ23 · ππ31 = 1 implies π3 (x2 , x3 ) = ww21(x(1)2 ) ππ13 (x3 , 1), and w3 (x3 ) = w1 (1) ππ31 (x3 , 1) establishes the claim. Anonymity gives wi = w for all i, therefore πi (x) = w(xi ) w(x1 )+w(x2 )+w(x3 )

for all i = 1, 2, 3 and all x ∈ R3++ .

i) Finally we show πi (n, x) = w(x for all (n, x), by induction on n. Assume this equality n w(x j) 1 holds up to n − 1. By Recursivity

−(n−1),n

−n p1≺2≺···≺n−1 = πn−1 (n − 1, x−n ) · p1≺2≺···≺n−2 −(n−1),n and by the inductive assumption p1≺2≺···≺n−2 > 0. Therefore (A.2) becomes   1 − πn (n, x) · πn−1 (n − 1, x−n ) = πn−1 (n, x).

As the choice of agents n, n − 1 was arbitrary, the above equation implies first πi (n, x) > 0 for all i, then the desired quasi-proportional formula. A.2. Proof of Theorems 1 and 2 For any scheduling rule ρ, not necessarily separable or recursive, write θ 2 (x1 , x2 ) = p2≺1 , where p = ρ(2, (x1 , x2 )), for the probability that a job of size x2 precedes a job of size x1 in a two-agent problem. By anonymity the function θ 2 is well-defined. Step 1. If ρ is split-proof, for all c > 0 the mapping a → θ 2 (a, c − a) is subadditive on ]0, c[; moreover supb θ 2 (a, b) · b  a for all a, namely λ(2)  2. Proof. Write for simplicity f (a, c) = θ 2 (a, c − a). Fix x ∈ R3++ and consider the split of user 2 in problem (2, (x1 , x2 + x3 )) to users 2, 3 in (3, x). Setting p = ρ(3, x), split-proofness implies f (x1 , x1 + x2 + x3 ) = θ 2 (x1 , x2 + x3 )  p{2,3}≺1 . Exchanging the role of the agents gives f (xi , x1 + x2 + x3 )  p{j,k}≺i for {i, j, k} = {1, 2, 3}. Summing the three inequalities and using 1 − f (x3 , x1 + x2 + x3 ) = f (x1 + x2 , x1 + x2 + x3 ) (from θ 2 (a, b) + θ 2 (b, a) = 1) gives the subadditivity of f in a as announced. Clearly f (b, 2b) = 12 for all b. Then subadditivity of f in a implies for all a > 0, m ∈ N: 1 1 f (a, 2m · a)  m1 f (m · a, 2m · a) = 2m ⇔ θ 2 (a, (2m − 1) · a)  2m , from which supb θ 2 (a, b) · b  a follows. Step 2: Statements (i) and (ii) in Theorem 1. Statement (i) follows from Step 1 and Eq. (9) for separable methods. For statement (ii), fix a separable method ρ and apply Ranking to the problem (2, (x1 , x2 )) where x1  x2 . We get x1 x1  θ 2 (x1 , x2 ) · x2  θ 2 (x2 , x1 ) · x1 ⇒ θ 2 (x1 , x2 )  x1 + x2 x2 and λ(n)  n then follows from Eq. (9). Step 3. Let ρ be a split-proof rule. If one of Ranking, or {δ(2)  2}, or {λ(2)  2} holds, then a a+b for all a, b. Assume Ranking. Applying it to the problem (2, (x1 , x2 )) where x1  x2 , we get as in 1 . Recalling the definition of f in Step 1, this implies f (a, c)  ac Step 2 θ 2 (x1 , x2 )  x1x+x 2 for 0 < a < c. By subadditivity of f in a, we get θ 2 (a, b) =

1 = f (c, c)  f (a, c) + f (c − a, c) 

a c−a + =1 c c

(A.3)

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H. Moulin / Games and Economic Behavior 63 (2008) 567–587

hence f (a, c) = ac and the claim follows. Assume λ(2)  2. From λ(2) = 1 + supa,b>0 θ 2 (a, b) · ab , which does not require Separability, a we get θ 2 (a, b)  ab for all a, b. Equivalently, f (a, c)  c−a for 0 < a < c. Now apply k times the subadditivity of f :   a a a k f (a, c)  2f ,c  ···  2 f k ,c  2 2 c − 2ak implying f (a, c)  ac . We can repeat the argument around inequality (A.3) and the claim follows. Assume δ(2)  2. For any method, separable or not, δ(2) = 1 + supba>0 θ 2 (a, b) · b−a a . On the other hand a c b−a  1 ⇔ f (a, c)  for 0 < a < . sup θ 2 (a, b) · a c − 2a 2 ba>0 Applying k times the subadditivity of f as above, we get f (a, c) 

a

c−

a 2k−1

, hence f (a, c) 

a c

whenever a < 2c . As f ( 2c , c) = 12 , f must be linear in a on ]0, 2c [, and once more subadditivity on ] 2c , c[ gives f (a, c)  ac everywhere and we are done. Step 4. End of proof. If the rule ρ is separable, Eq. (9) implies that the properties λ(n)  n and λ(n )  n are logically equivalent for any n, n  2 . Assume ρ is split-proof, separable and meets one of Ranking, or {δ(2)  2}, or {λ(2)  2}. a By Separability θ 2 defines the method associated with ρ (Eq. (4)), and by Step 3 θ 2 (a, b) = a+b . This proves statement (iii) in Theorem 1. Now suppose ρ is split-proof, recursive and meets one of Ranking, or {δ(2) 2}, or {λ(2)  2}. Fix a problem (n, x) and consider the split of 2 in problem(2, (x1 , n2 xi )) to {2, 3, . . . , n} in (n, x). Split-proofness and Step 2 give π1 (n, x)  θ 2 (x1 , n2 xi ) = xn1x . Re1 i peat the argument for all i: as π(n, x) is a probability distribution, it must be the proportional one, hence by Recursivity ρ is the Proportional rule. 2 References Banker, R., 1981. Equity consideration in traditional full-cost allocation practices: An axiomatic perspective. In: Moriarty, S. (Ed.), Joint Cost Allocations. University of Oklahoma, Norman. Cheng, A., Friedman, E., 2005. Sybilproof reputation mechanisms. In: SIGCOMM’05 Workshops, August 22–26, Philadelphia, USA. Chun, Y., 2000. Agreement, separability, and other axioms for quasi-linear social choice problems. Soc. Choice Welfare 17, 507–521. Chun, Y., 2004. Consistency and monotonicity in sequencing problems. Mimeo. Seoul National University. Chun, Y., 2006. A pessimistic approach to the queuing problem. Math. Soc. Sci. 51 (2), 171–181. De Frutos, M.A., 1999. Coalitional manipulations in a bankruptcy problem. Rev. Econ. Design 4, 255–272. Dolan, R., 1978. Incentive mechanisms for priority queuing problems. Bell J. Econ. 9, 421–436. Douceur, J., 2002. The Sybil Attack. In: Proceedings of the 1st International Workshop on Peer-to-Peer Systems. IPTPS 2002. LNCS 2429, pp. 251–260. Fishburn, P., 1992. Induced binary probabilities and the linear ordering polytope: A status report. Math. Soc. Sci. 23, 67–80. Friedman, E.J., Henderson, S.G., 2003. Fairness and efficiency in web server protocols. In: Proceedings of the 2003 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems. ACM Press, pp. 229–237. Hain, R., Mitra, M., 2004. Simple sequencing problems with interdependent costs. Games Econ. Behav. 48, 271–291. Ju, B.G., 2003. Manipulations via merging and splitting in claim problems. Rev. Econ. Design 8, 205–215.

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