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Incomplete information and multiple machine queueing problems
European Journal of Operational Research 165 (2005) 251–266 www.elsevier.com/locate/dsw
Stochastics and Statistics
Incomplete information and multiple machine queueing problems Manipushpak Mitra
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Economic Research Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India Received 16 June 2000; accepted 6 October 2003 Available online 25 March 2004
Abstract In mechanism design problems under incomplete information, it is generally difficult to find decision problems that are first best implementable. A decision problem under incomplete information is first best implementable if there exists a mechanism that extracts the private information and achieves efficiency with a transfer scheme that adds up to zero in every state. One can find queueing problem with one machine that are first best implementable under certain cost conditions. In this paper we identify the conditions on cost structure for which queueing problems with multiple machines are first best implementable. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Queueing problems; First best implementability
1. Introduction In a queueing problem with multiple machines, there is a server (for example, a computer server), with more than one identical machines (computers) which has to process a finite number of jobs for a set of individuals. The machines are identical in the sense that a given job takes the same length of time for completion. We assume that it takes one unit of time to complete one job. Each individual has one job to be processed. The server can serve one individual in one machine, that is, it takes one unit of time to process one job in a machine. If the number of jobs, to be processed, is more than the number of machines then individuals will have to wait in a queue. Waiting in a queue is costly for each individual. The server’s objective is to order the individuals in a queue efficiently so as to minimize the aggregate waiting cost. If the cost of waiting in the queue is private information then an individual, if asked, will announce her cost strategically so as to get her job done as early as possible. Therefore, in the queueing scenario described above, the server’s role is that of a planner who has to solve an incentive problem under incomplete information. More precisely, we have a mechanism design problem of a social planner (server in the
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Tel.: +91-33-25752608; fax: +91-33-25778893. E-mail address: [email protected] (M. Mitra).
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.10.050
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queueing problem) whose objective is to extract the privately held information (true waiting cost) of each individual and select the efficient decision (to order the individuals in a queue so as to minimize the aggregate waiting cost) in each state. One of the most significant achievements in the planner’s mechanism design problems under incomplete information has been the existence of a class of mechanisms called Groves–Clarke mechanisms (see Clarke, 1971; Groves, 1973). These mechanisms achieve the twin objectives of truthful revelation of private information and efficiency of decisions provided the agents have quasi-linear preferences. Moreover, for a very broad class of preference structures, in a quasi-linear set up, Groves–Clarke mechanisms are the only class of mechanisms that achieve these objectives (see Holmstr€ om, 1979). However, the drawback of such mechanisms is that they are, in general, not Pareto-optimal. This means that there are preference realizations where the sum of Groves–Clarke transfers are non-zero. In the pure public goods problem, Hurwicz (1975), Green and Laffont (1979) and Walker (1980) proved the budget imbalance of a Groves– Clarke scheme. Hurwicz and Walker (1990) proved the impossibility result in the context of pure exchange economies (economies in which there are no production, no public goods and other externalities). The damaging nature of budget imbalance, in the public goods context, was pointed out by Groves and Ledyard (1977). They proposed, using a very simple model, that an alternative procedure based on majority rule voting may lead to an allocation of resources which is Pareto superior to the one produced by Groves mechanism. However, there are certain decision problems where first best or Pareto optimality can be achieved. In the public goods problem, Groves and Loeb (1975) have proved that if preferences are quadratic then we can find balanced Groves transfer. This result was generalized by Tian (1996) and Liu and Tian (1999). In a single server (one machine) queueing problem with linear cost, Mitra and Sen (1998) showed the existence of first best mechanisms. A problem similar to the queueing problem with linear costs in Mitra and Sen (1998) is the sequencing problem in Suijs (1996). Unlike the queueing problem, where it takes one unit of time to serve one individual, in a sequencing problem the servicing time can differ from one individual to another. Therefore, while the linear cost queueing problem is a discrete time problem, the sequencing problem in Suijs (1996) is a continuous time problem. By assuming servicing time to be common knowledge, Suijs proved the existence of first best mechanisms for the sequencing problem. The existence result in Mitra and Sen (1998) was further generalized by Mitra (2001) for a broader class of cost structures. It was proved that the class of cost structures under which a ‘one machine queueing problem’ is first best implementable is ‘fairly’ large. In this paper, we deal with the question of first best implementability of queueing problems with multiple machine. Therefore, this paper is a generalization of the one machine queueing framework of Mitra (2001) to a multiple machine framework. A multiple machine queueing problem is first best implementable if there exists a mechanism that can extract the private information with a vector of transfers that add up to zero. This allows the server to order the jobs in a way that minimizes the aggregate cost. The most important implication of first best implementability is that the server can extract the private information costlessly. If a queueing problem is first best implementable then there is no welfare loss as the transfers used to extract the private information adds up to zero in all states. A multiple machine queueing problem resembles some of the sequencing problems that are analyzed in the operations research literature. Papers relating to sequencing n jobs in m machines by Dudek and Teuton Jr. (1964), flow shop sequencing problems with ordered processing time by Dudek et al. (1975) and flow shop problems with dominant machines by Krabbenborg et al. (1992) deal with finding algorithms to order (or queue) the n jobs in m machines in a way that minimizes the total elapsed time. However, in all these models, unlike multiple machine queueing problems, machines are not identical. The processing time for the same job can be different in different machines. Moreover, unlike a multiple machine queueing problem where cost parameter is private information, the cost structures in all the above mentioned sequencing problems are common knowledge. In the incomplete information set up, sequencing problems were analyzed by Hamers et al. (1999). They analyzed a multiple identical machine sequencing (or scheduling) problem with linear time cost. Therefore,
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their problem is a continuous time version of the multiple machine queueing problem with linear cost. Hamers et al. (1999) look at the n jobs and m identical machines sequencing situation both in a co-operative and non-cooperative environments. In a co-operative set up, a sequencing problem is called a sequencing game. Curiel et al. (1989) analyzed the sequencing game in a one machine framework. Hamers et al. (1999), by extending the sequencing game of Curiel et al. (1989) to a multiple identical machine framework, addressed the issues of balancedness and non-emptiness of core in m-sequencing games. In a non-cooperative set up, they address the issue of first best implementability by assuming job processing time to be equal to one. Thus, in the non-cooperative set up, the sequencing situation analyzed by Hamers et al. (1999) is identical to the multiple machine queueing problem described in this paper with the restriction that the cost is linear over time. The main objective of this paper is to achieve first best implementability in multiple machine queueing problems. More precisely, this paper identifies the conditions on the cost structure that lead to first best implementability. Therefore, in this paper, we also generalize the non-cooperative sequencing situation, analyzed by Hamers et al. (1999), by allowing for a very general time cost. The results that we get suggest that first best implementability depends critically on the number of machines and the number of jobs to be processed on those machines. If the number of machines (that is m) is even or if the number of jobs to be processed is strictly greater than the number of machines but less than or equal to twice the number of machines, then a multiple machine queueing problem is not first best implementable. For all other multiple machine queueing problems there exists a non-trivial cost structure under which it is first best implementable. Finding an algorithm to obtain the cost minimizing queue is not a very important issue since the conditions on the cost structure under which a multiple machine queueing problem is first best implementable are such that the algorithm for finding the cost minimizing queue is transparent. Therefore, obtaining a state contingent transfer scheme that extracts the private information while adding up to zero is of prime importance. The paper is organized in the following way: in Section 2 we develop the general class of problems, in Section 3 we derive some characterization results, in Section 4 we deal with separable cost multiple machine queueing problems and finally in the concluding remarks of Section 5 we summarize the results obtained in this paper. 2. The general class of problems Let N ¼ f1; 2; . . . ; ng be the set of individuals and mð> 1Þ be the set of identical machines. Define ½xþ to be the lowest integral value not less than x. For example, ½2:005þ ¼ 3 and ½3þ ¼ 3. Given n and m, the total number of queue positions are M ¼ ½n=mþ . Here the number of individuals (and hence the number of jobs) n is strictly greater than the number of machines m in order to have a meaningful queueing problem. hj ðkÞ measures the cost of waiting k periods in the queue for individual j where k 2 f1; . . . ; Mg. Let Rþ be the non-negative orthant of the real line R. A cost vector or type of an individual j 2 N is a vector of the form M M hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 RM þ . An -neighborhood around any vector hj 2 Rþ is the set N ðhj Þ ¼ fx 2 Rþ : khj xk < g. 1 A typical domain of cost vectors of an individual j 2 N is H RM þ . Observe that the do_ A cost main H is assumed to be common for all individuals. For a domain H, we denote its interior by H. _ _ vector hj 2 H, if 9 > 0 such that N ðhj Þ H. We assume that the interior H (of the domain H) is nonempty. Let Dhj ¼ ðDhj ð1Þ; . . . ; Dhj ðM 1ÞÞ represent the vector of first differences generated from the vector hj . Here Dhj ðkÞ ¼ hj ðk þ 1Þ hj ðkÞ for all k 2 f1; . . . ; M 1g. We say that Dhj < Dh0j if and only if Dhj ðkÞ 6 Dh0j ðkÞ for all k 2 f1; . . . ; M 1g and there exists at least one k 0 2 f1; . . . ; M 1g such that 1
Here khj xk denotes the Euclidean distance between the two cost vectors hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ and x ¼ ðxð1Þ; . . . ; xðMÞÞ, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM 2 k¼1 ðhj ðkÞ xðkÞÞ .
khj xk ¼
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Dhj ðk 0 Þ < Dh0j ðk 0 Þ. The three main assumptions we make on the common domain of preference H are as follows. Assumption 1. For all j 2 N and for all cost vector hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H, 0 6 hj ð1Þ 6 hj ð2Þ 6 6 hj ðMÞ. Assumption 2. H is convex, that is, if hj 2 H and h0j 2 H, then khj þ ð1 kÞh0j 2 H for all k 2 ½0; 1. _ there exists ða; bÞ 2 H H such that Da < Dh < Db. Assumption 3. H is sufficiently open if for all hj 2 H, j Moreover, Dhj ðkÞ < DbðkÞ for all k 2 f1; . . . ; M 1g and DaðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. 2 Assumption 1 simply says that the individuals are impatient. Assumption 2 is the standard convexity requirement. Finally, Assumption 3 is an openness assumption on the cost domain which says that for any type hj , in the interior of the domain, there exists one type b in the domain that, in terms of difference, strictly dominates hj and there exists another type a with null first difference vector in the domain which is dominated, in terms of difference, by hj . Let H be the class of domains satisfying Assumptions 1–3. Let H be any domain belonging to H satisfying Assumptions 1–3. The utility of each individual j is assumed to be quasi-linear and is of the form: Uj ðk; tj ; hj Þ ¼ vj hj ðkÞ þ tj where vj ð> 0Þ is the gross benefit derived by individual j from the service, hj ðkÞ is the cost of individual j at the kth queue position and tj is the transfer that individual j receives. The server’s aim is to achieve efficiency or minimize the aggregate cost. To define what we mean by efficiency in a queueing problem with m machines, we need to develop the concept of a multi-set. A multi-set is a set where all elements may not be distinct. For example, X ¼ f1; 1; 1; 3; 6; 6; 9g is a multi-set. Given a queueing problem with n individuals, m machines and hence M ¼ ½n=mþ queue positions, consider the multi-set Xn;m of the form Xn;m ¼ f1; . . . ; 1; 2; . . . ; 2; . . . ; M 1; . . . ; M 1 ; M; . . . ; M g. Let P ðXn;m Þ be the |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} m
m
m
nðM1Þm
set of all possible permutations of the multi-set Xn;m . In this problem, a queue r is a one-to-one correspondence between the set of jobs N ¼ f1; . . . ; ng and the queue positions P ðXn;m Þ, that is, r ¼ ðr1 ; . . . ; rn Þ : N ! P ðXn;m Þ. Thus, rj ¼ k indicates that individual j has the kth position in the queue. Given a queue r ¼ ðr1 ; . . . ; rn Þð2 P ðXn;m ÞÞ, the cost of an individual j 2 N is hj ðrj Þ. A state of the world is n h ¼ ðh1 ; . . . ; hn Þ 2 H where hj is a 1 M vector. n
Definition 2.1. Given a state h 2 H , a queue r 2 P ðXn;m Þ is said to be efficient if r 2 arg minr2P ðXn;m Þ P j2N hj ðrj Þ. An efficient queue r is an assignment that gives each individual exactly one queue position and each of the first M 1 queue positions to exactly m individuals and the Mth queue position to the remaining n ðM 1Þm individuals in such a way that the aggregate cost is minimized. Observe, that there can be states with more than one efficient queue. So we have an efficiency correspondence. An efficient rule is a single valued selection from the efficiency correspondence. Note that efficiency of a queue r is a concept independent of transfers and gross benefits of all individuals. If the server knows the true state h ¼ ðh1 ; . . . ; hn Þ then she can calculate an efficient queue. However, if hj is private information for individual j, the server’s problem then is to design a mechanism that will elicit this information truthfully. We refer to such a problem as a multiple machine queueing problem under incomplete information and is written as C ¼ hN; m; Hi where H 2 H. Note that we are assuming that the domain
2
This means that the vector a is a constant vector.
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H is common knowledge and that the cost vector of an individual is private information. Therefore, each individual, if asked, will announce a cost vector from the domain H. Formally, a mechanism M is a pair n n hr; ti where r : H ! P ðXn;m Þ and t ðt1 ; . . . ; tn Þ : H ! Rn . Thus, a mechanism M is a direct revelation mechanism where each individual j 2 N announces a cost vector hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ and based on the announcements of all individuals (that is, h ¼ ðh1 ; . . . ; hn Þ), the planner (or server) specifies a queue r and a vector of transfers t ¼ ðt1 ; . . . ; tn Þ. Under M ¼ hr; ti, given all others’ announcement hj , the utility of individual j of type hj when her announcement is h0j is given by Uj ðrj ðh0j ; hj Þ; tj ðh0j ; hj Þ; hj Þ ¼ vj hj ðrj ðh0j ; hj ÞÞ þ tj ðh0j ; hj Þ. Definition 2.2. A multiple machine queueing problem C ¼ hN; m; Hi is implementable if there exists an n efficient rule r : H ! P ðXn;m Þ and a mechanism M ¼ hr ; ti such that for all j 2 N, for all pairs of n1 announcement vectors ðhj ; h0j Þ 2 H H and for all announced hj 2 H , Uj ðrj ðhÞ; tj ðhÞ; hj Þ P 0 0 Uj ðrj ðhj ; hj Þ; tj ðhj ; hj Þ; hj Þ. This definition says that C ¼ hN; m; Hi is implementable if there exists a direct mechanism, with an efficient queueing rule r and a vector of transfers, that induces each individual to tell the truth independent of others’ report. Definition 2.3. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable, if there n exists a mechanism M ¼ hr ; ti such that (1) M implements C and (2) for all announcements h 2 H , P j2N tj ðhÞ ¼ 0. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable if it can be implemented with a budget balancing transfer. Thus, if C is first best implementable then incomplete information does not impose any welfare loss.
3. Characterization results n
Definition 3.4. A mechanism M ¼ hr; ti is a Groves–Clarke mechanism if for all j 2 N and for all h 2 H , the transfer is of the form X tj ðhÞ ¼ hl ðrl ðhÞÞ þ cj ðhj Þ: ð3:1Þ l6¼j
In a Groves–Clarke mechanism, the transfer of any P individual j 2 N in any state Ph is the negative of aggregate cost plus the cost of individual j (that is l2N hl ðrl ðhÞÞ þ hj ðrj ðhÞÞ ¼ l6¼j hl ðrl ðhÞÞ), plus a constant cj ðhj Þ. The utility of individual j with a Groves–Clarke transfer is her gross benefit vj less the aggregate cost in state h plus the constant. We now proceed to verify that given a mechanism M ¼ hr ; ti where the queue r is efficient and the transfer satisfies condition (3.1), truth-telling is a dominant strategy. Suppose, it were not the case. Then there exists an individual j with true cost hj and there exists a report hj such that individual j strictly benefits by misreporting her cost to be some h0j ð6¼ hj Þ. That is, Uj ðrj ðhÞ; tj ðhÞ; hj Þ < Uj ðrj ðh0j ; hj Þ; tj ðh0j ; hj Þ; hj Þ. Simplifying this inequality after substituting the Groves– P P 0 Clarke transfer, we get j2N hj ðrj ðhÞÞ > j2N hj ðrj ðhj ; hj ÞÞ. This contradicts efficiency of decision (or aggregate cost minimization) in state h. Hence, the Groves–Clarke transfer leads to truth-telling in dominant strategies. According to a well known result of Holmstr€ om (see Holmstr€ om, 1979), decision problems with ‘‘convex’’ domains are implementable if and only if the mechanism is a Groves–Clarke mechanism (see Theorem (2) in Holmstr€ om (1979)). Since by Assumption 1, any domain H 2 H is convex, all multiple
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machine queueing problems C ¼ hN; m; Hi with H 2 H are implementable if and only if the mechanism is a Groves–Clarke mechanism. Therefore, the question of first best implementability of a multiple machine queueing problem reduces to finding domain restrictions under which we can find a balanced Groves– Clarke transfer. P Let Cðr ðh0 Þ; hÞ ¼ j2N hj ðrj ðh0 ÞÞ where, as stated earlier, r ðh0 Þ is an efficient queue for the announced 0 state h . Thus, Cðr ðh0 Þ; hÞ is the minimum aggregate cost with respect to the announced state h0 when the actual state is h. For notational simplicity we define CðhÞ Cðr ðhÞ; hÞ to be the minimum aggregate cost with respect to the actual state h when the announced state is also h. Remark 3.1. From the definition of efficiency of the queue r it follows that for all h and h0 , CðhÞ 6 Cðr ðh0 Þ; hÞ. When is a multiple machine queueing problem first best implementable? In our first theorem we show that the General Combinatorial Property, defined below, is necessary for first best implementability of a multiple machine queueing problem. Definition 3.5. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the General Combinatorial Property (or GCP) if for all hj 2 H M X aðk; n; mÞhj ðkÞ ¼ 0; ð3:2Þ k¼1 M
where aðn; mÞ ¼ faðk; n; mÞgk¼1 and ( m1 X ðk1Þm l aðk; n; mÞ ¼ ð1Þ ð 1Þ
) n1 if k 2 f1; . . . ; M 1g ðk 1Þm þ l l¼0 ( nðM1Þm1 ) X n1 ðM1Þm l aðk; n; mÞ ¼ ð1Þ ð 1Þ if k ¼ M: ðM 1Þm þ l l¼0
and
The following example illustrates the GCP. b ¼ hN ¼ f1; . . . ; 10g; m ¼ 3; Hi. Here the number of queue positions is M ¼ Example 3.1. Consider C b ½10=3þ ¼ 4 and að10; 3Þ ¼ ðað1; 10; 3Þ ¼ 28; að2; 10; 3Þ ¼ 84; að3; 10; 3Þ ¼ 57; að4; 10; 3Þ ¼ 1Þ. Thus, C satisfies GCP if 8j 2 f1; . . . ; 10g, hj ¼ ðhj ð1Þ; hj ð2Þ; hj ð3Þ; hj ð4ÞÞ 2 H is such that 28hj ð1Þ P 84hj ð2Þ þ M¼4 57hj ð3Þ hj ð4Þ ¼ 0. Note that the coefficient vector að10; 3Þ ¼ ð28; 84; 57; 1Þ is such that k¼1 aðk; 10; 3Þ ¼0. From Definition 3.5 it is quite obvious that for a multiple machine queueing problem C, the coefficient vector aðn; mÞ ¼ faðk; n; mÞgM k¼1 is such that M n X X p1 n 1 ¼ 0: ð3:3Þ aðk; n; mÞ ¼ ð1Þ p1 k¼1 p¼1 The first-order difference at queue position k (that is, Dhj ðkÞ ¼ hj ðk þ 1Þ hj ðkÞ) represents the increase in queueing cost for individual j if he is moved from kth queue position to ðk þ 1Þth queue position. By simplifying Eq. (3.2) using Dhj ðkÞ we get M 1 X k¼1
zðk; n; mÞDhj ðkÞ ¼ 0;
ð3:4Þ
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P where the partial sum coefficient vector zðn; mÞ ¼ fzðk; n; mÞgM1 that zðk; n; mÞ ¼ kr¼1 aðr; n; mÞ for k¼1 is such Pr q a r a1 ¼ ð1Þ , it follows that all k 2 f1; . . . ; M 1g. From the mathematical identity q¼0 ð1Þ q r n2 km1 for all k 2 f1; . . . ; M 1g (see Tomescu, 1985). zðk; n; mÞ ¼ ð1Þ km 1 Theorem 3.1. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable only if it satisfies the GCP. Before proving Theorem 3.1, a lemma due to Walker (1980) is stated below. Consider two profiles h ¼ ðh1 ; . . . ; hn Þ and h0 ¼ ðh01 ; . . . ; h0n Þ. Define for S N, a type hj ðSÞ ¼ hj if j 62 S and hj ðSÞ ¼ h0j if j 2 S. Thus for each S N, we have a state hðSÞ ¼ ðh1 ðSÞ; . . . ; hn ðSÞÞ. Lemma 3.1. A multiple P machine queueing problem C ¼ hN; m; Hi is first best implementable only if for all n n jSj pairs ðh; h0 Þ 2 H H , SN ð1Þ CðhðSÞÞ ¼ 0. 3 1ÞCðhÞ ¼ P By adding the Groves–Clarke transfer of all individuals and setting it to zero we get ðn n n om (1977) for a more general result). Thus, for all ðh; h0 Þ 2 H H , j2N cj ðhj Þ (see Holmstr€ P P P jSj jSj 1 SN ð1Þ CðhðSÞÞ ¼ ðn1Þ j2N SN ð1Þ cj ðhj ðSÞÞ ¼ 0 (see Walker, 1980). Proof of Theorem 3.1. Consider a multiple machine queueing problem C ¼ hN; m; Hi. We start with a given type h1 for individual 1 in the interior of the domain and construct h1 and h0 . Then we apply Lemma 3.1 due to Walker (1980) to derive the result. Consider individual 1 and any announcement h1 ¼ ðh1 ð1Þ; . . . ; _ Given h , we consider two states h ¼ ðh ; h ; . . . ; h Þ and h0 ¼ ðh0 ; . . . ; h0 Þ of the h ðkÞ; . . . ; h ðMÞÞ 2 H. 1
1
1
1
2
n
1
n
following type: for all k ¼ 1; . . . ; M 1, Dh1 ðkÞ < Dhj ðkÞ < Dhjþ1 ðkÞ for all j 2 f2; . . . ; n 1g and h0j ðkÞ ¼ g, for all j 2 N. Therefore, h0j ¼ ðg; g; . . . ; gÞ for all j 2 N which implies that Dh0j ðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. Note that this sort of construction is possible due to Assumptions 1 and 3. Consider any two queue positions k and k þ 1 and any two individuals j and j þ 1 with types hj and hjþ1 , respectively. Note that from the construction of h, on the one hand, it follows that if individual j gets the kth position and ðj þ 1Þth individual gets the ðk þ 1Þth position, then the costs for these two positions add up to fhj ðkÞ þ hjþ1 ðk þ 1Þg. If, on the other hand, the positions of j and ðj þ 1Þ are interchanged then the costs add up to fhj ðk þ 1Þ þ hjþ1 ðkÞg. The former cost strictly exceeds the latter for all k ¼ 1; . . . ; M 1 since, by construction, Dhj ðkÞ < Dhjþ1 ðkÞ for all j ¼ 1; . . . n 1. Thus, the queue that minimizes the aggregate cost requires that, rjþ1 ðhÞ 6 rj ðhÞ for all j ¼ 1; . . . ; n 1. In other words, rj ðhÞ ¼ ½ðn þ 1 jÞ=mþ for all j 2 N. Now consider profiles hðSÞ ¼ ðh1 ðSÞ; . . . ; hn ðSÞÞ where hj ðSÞ ¼ hj if j 62 S and hj ðSÞ ¼ h0j if j 2 S. Observe that from the arguments applied to find the efficient queue in state h it follows that if j; l 62 S and j < l, then rl ðhðSÞÞ 6 rj ðhðSÞÞ. Again, given any S N and S 6¼ /, rj ðhðSÞÞ 6 rs ðhðSÞÞ for all ðs; jÞ 2 S N S. This is because, for any given queue position k 2 f1; . . . ; M 1g, the incremental loss of any individual j 62 S (that is, Dhj ðkÞ) is strictly more than that of any individual s 2 S. Note that for all queues r ðhðSÞÞ, satisfying (1) rj ðhðSÞÞ 6 rs ðhðSÞÞ for all ðs; jÞ 2 S N S and (2) rl ðhðSÞÞ 6 rj ðhðSÞÞ for all ðj; lÞ 2 N S N S, j < l, are efficientPsince the cost of all individuals s 2 S are identical. jSj We now consider the sum SN ð1Þ CðhðSÞÞ. We break this sum in three parts in the following way: (i) P P P P P jSj jb Sj je Sj b ð1Þ h1 ðr1 ðh1 ; ^ ~ SN;j2 S^ ð1Þ hj ðrj ðhð S ÞÞÞ and (iii) SNf1g j6¼1 j2N SN;j62S ð1Þ hj ðrj ðhðSÞÞÞ, (ii) h1 ð Se ÞÞÞ. We consider each of these parts separately in the next three paragraphs.
3
Here jX j denotes the cardinality of X .
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We first consider part (i). Consider an individual j 2 N with j 6¼ 1. Let Pj ¼ fp : p > jg be the set of individuals with the higher ranking index than j. Consider all sets S such that j 62 S and there are x number of individuals from the set Pj and jSj x number of individuals from the set N fPj [ jg. The queue position of individual j for all such S is rj ðhðSÞÞ ¼ ½ðn þ 1 j xÞ=mþ . By collecting all such sets S, that is, P jSj by considering the coefficient of the term hj ð½ðn þ 1 j xÞ=mþ Þ, in the sum SN ð1Þ CðhðSÞÞ we get P x nj jT j ð1Þ T NfPj [jg ð1Þ . Note that x j1 X X jT j r j1 j1 ¼ ð1 þ ð1ÞÞ ¼ 0 since j 6¼ 1: ð1Þ ¼ ð1Þ r r¼0 T NfPj [jg
Therefore, of a term hj ð½ðn þ 1 j xÞ=mþ Þ is zero for all jð6¼ 1Þ and for all xð 6 jPj jÞ. Thus, P the coefficient P jSj ð1Þ h ðr j j ðhðSÞÞÞ ¼ 0. j6¼1 SN;j62S We now consider part (ii). Observe that by adding the cost of an individual j 2 N for all b S ð NÞ such that j 2 b S , we get n X X ^ n1 g ¼ ð1 þ ð1ÞÞn1 g ¼ 0: ð1ÞjSj g ¼ ð1Þs s 1 ^ s¼1 j2SN
P P ^ jSj b Therefore, j2N SN;j2 ^ S^ ð1Þ hj ðrj ðhð S ÞÞÞ ¼ 0. P jSj Since the sums in (i) and (ii) are both zero, it follows that the sum SN ð1Þ CðhðSÞÞ is equal to the P P ~ jSj jSj ð1Þ h1 ðr1 ðh1 ; h1 ð e S ÞÞÞ. For individual 1, with type h1 , sum in (iii), that is SN ð1Þ CðhðSÞÞ ¼ SNf1g ~ e e e we get r1 ðh1 ; h1 ð S ÞÞ ¼ ½ðn j S jÞ=mþ for all S N f1g. Therefore, 3 2 M X X X ~ n 1 jSj x 5h1 ðkÞ: 4 ð1Þ h1 ðr1 ðh1 ; h1 ð e S ÞÞÞ ¼ ð 1Þ ð3:5Þ x ~ k¼1 SNf1g
x:½ðnxÞ=mþ ¼k
P Simplifying condition (3.5) and then by applying Lemma 3.1 we get M k¼1 aðk; n; mÞh1 ðkÞ ¼ 0. Since the selection of individual 1, for the above construction, was arbitrary, the result follows. h Note that the GCP is an additional restriction on the class of domains H satisfying Assumptions 1–3. Let b Hð HÞ be the class of domains satisfying the GCP and Assumptions 1–3. Proposition 3.1. If the number of machines m is even or if the number of queue positions M ¼ ½n=mþ ¼ 2, then b there is no first best implementable multiple machine queueing problem C ¼ hN; m; Hi such that H 2 H. Proof. If the number of machines m is even then for all k 2 f1; . . . ; M 1g, ð1Þkm1 ¼ 1 because km is also even. From Assumption 1, it follows that for all hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H, Dhj ðkÞ P 0 for all km1 k 2 f1; . . . ; M 1g. By substituting ð1Þ ¼ 1 and Dhj ðkÞ P 0 in (3.4) we get Dhj ðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. Hence, the GCP implies that hj ð1Þ ¼ ¼ hj ðMÞ for all hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H and we have a violation of Assumption 3. If M ¼ 2, then GCP implies að1; n; mÞhj ð1Þ þ að2; n; mÞhj ð2Þ ¼ 0 for all hj ¼ ðhj ð1Þ; hj ð2ÞÞ 2 H and condition (3.3) implies að1; n; mÞ þ að2; n; mÞ ¼ 0. Therefore, the GCP implies that hj ð1Þ ¼ hj ð2Þ for all hj ¼ ðhj ð1Þ; hj ð2ÞÞ 2 H. This again is a violation of Assumption 3. h So far we have imposed restrictions on individual preferences. The next property is a restriction on group preferences.
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Definition 3.6. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the General Independence Property (or GIP) if for all pairs ðj; lÞ 2 N N, j 6¼ l, all pairs of cost vectors ðhj ; hl Þ 2 H H are such that one of the following two conditions holds: 1. Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; 2; . . . ; M 1g 2. Dhj ðkÞ 6 Dhl ðkÞ for all k 2 f1; 2; . . . ; M 1g. The GIP for a multiple machine queueing problem C implies that if for any pair of individuals ðj; lÞ 2 N N, j 6¼ l, the respective cost vectors hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H and hl ¼ ðhl ð1Þ; . . . ; hl ðMÞÞ 2 H are such that there exists a k 2 f1; . . . ; M 1g such that Dhj ðkÞ > Dhl ðkÞ, then Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g. The relationship between the GCP and the GIP is captured in the next proposition. Proposition 3.2. For C ¼ hN; m; Hi with M ¼ 3, GCP ) GIP. Proof. Consider any multiple machine queueing problem C ¼ hN; m; Hi such that M ¼ 3 and H satisfies the GCP. Consider hj ¼ ðhj ð1Þ; hj ð2Þ; hj ð3ÞÞ and hl ¼ ðhl ð1Þ; hl ð2Þ; hl ð3ÞÞ for individuals j and l respectively. Since H satisfies the GCP, from condition (3.4) we know that zð1; n; mÞDhi ð1Þ þ zð2; n; mÞDhi ð2Þ ¼ 0 for all zð2;n;mÞ i 2 fj; lg. Therefore, for all i 2 fj; lg, Dhi ð1Þ ¼ P . Dhi ð2Þ where P ¼ zð1;n;mÞ > 0. Thus, Dhj ð1Þ < ð>ÞDhl ð1Þ if and only if Dhj ð2Þ < ð>ÞDhl ð2Þ. h b ¼ hN ¼ f1; . . . ; 10g; m ¼ 3; Hi of Example 3.1. Here the number of queue positions M ¼ 4 Consider C h ¼ and að10; 3Þ ¼ ð28; 84; 57; 1Þ. Consider individuals j and l P4l with costshj ¼ ð1; 3; 4; 4Þ 2 H and ð1; 2; 3; 31Þ 2 H respectively. Observe that for i 2 fj; lg, aðk; 10; 3Þ h ðkÞ ¼ 0. However, D h ð1Þ ¼ i j k¼1 b 2 > D hl ð1Þ ¼ 1 and D hj ð3Þ ¼ 0 < D hl ð3Þ ¼ 28. Therefore, the GIP is not satisfied. Hence for C, GCP ; GIP. Let C represent the class of multiple machine queueing problems with odd number of machines and with at least three queue positions and satisfying the GCP, the GIP and Assumptions 1–3. Therefore, a multiple e belongs to C if Hð e b is a domain satisfying the GCP, the GIP machine queueing problem C ¼ hN; m; Hi HÞ and Assumptions 1–3. We now derive the efficient rule for any multiple machine queueing problem e 2 C. Before doing that we give some more relevant notations and definitions. ConC ¼ hN; m; Hi e For a state h 2 H e n , define Qj ðhÞ ¼ ½l 2 N fjg such that either 9k 2 f1; . . . ; M 1g sider C ¼ hN; m; HÞ. and fDhl ðkÞ > Dhj ðkÞg or 8k 2 f1; 2; . . . ; M 1g, fDhl ðkÞ ¼ Dhj ðkÞ and l < jg. Let Rj ðhÞ ¼ 1 þ jQj ðhÞjð2 f1; . . . ; ngÞ be the rank of individual j in state h. Observe that the way we have specified the ranking, there is no possibility of a tie in the ranking of different individuals in any given state. Therefore, Rj ðhÞ measures the rank of individual j in state h ¼ ðh1 ; . . . ; hn Þ. Using this definition of ranking we state and prove an efficient e 2 C. rule (that is, a single-valued selection from the efficiency correspondence) for C ¼ hN; m; Hi e 2 C. For all h 2 H e n , let r ðhÞ be the queue such that r ðhÞ ¼ Proposition 3.3. Consider C ¼ hN; m; Hi j ½Rj ðhÞ=mþ for all j 2 N. The queue r ðhÞ ¼ ðr1 ðhÞ; . . . ; rn ðhÞÞ is efficient. e n , the queue r ðhÞ ¼ ðr ðhÞ; . . . ; r ðhÞÞ, defined in Proposition 3.3, is For any state h ¼ ðh1 ; . . . ; hn Þ 2 H 1 n obtained in the following way: the individuals having rank 1 to m get the first position, the individuals having rank m þ 1 to 2m get the second position and this goes on till the individuals having rank mðM 2Þ þ 1 to mðM 1Þ get the ðM 1Þth position and finally the remaining individual(s) having rank higher than mðM 1Þ get the Mth position.
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e n and any queue r. We define a sequence sðrÞ Proof of Proposition 3.3. Consider a state h ¼ ðh1 ; . . . ; hn Þ 2 H as an ordering of the jobs obtained from r by placing the jobs in the first position up to first m slots, the jobs in the second position in the following m slots and so on. 4 For a sequence sðrÞ, let j; l 2 N be neighbors with l in front of j, that is sr ðjÞ ¼ sr ðlÞ þ 1. If l and j switches their position (by keeping the position of all other jobs in the sequence unchanged), then the total cost changes by the amount Dðr; r0 ; hÞ Cðr; hÞ Cðr0 ; hÞ. We can have two types of switch-one that leaves the queue positions unaltered and one where the queue position of only the neighbors in question (that is, l and j) change. Therefore, 0 if r ¼ r0 ði:e: rj ¼ rl ¼ k for some k 2 f1; . . . ; MgÞ; Dðr; r0 ; hÞ ¼ Dhj ðkÞ Dhl ðkÞ if r 6¼ r0 ði:e: rj ¼ rl þ 1 and rl ¼ k for some k 6¼ MÞ: e satisfies the GIP, from Dðr; r0 ; hÞ it follows that switching l and j is weakly cost Given that the domain H reducing if and only if Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g. Observe that the queue r ðhÞ in state h is such that if Rj ðhÞ < Rl ðhÞ then (a) Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g and (b) rj ðhÞ 6 rl ðhÞ. Hence for sðr ðhÞÞ, it is not possible to find neighbors for whom switching is cost reducing. This means that r ðhÞ e was arbitrary, the result follows. h is efficient in state h. Since the selection of h 2 H Proposition 3.3 shows that finding an efficient queue is quite transparent if C 2 C. Observe that for C 2 C, the relative ranking of any two individuals ðj; lÞ, for some given costs hj and hl respectively, is independent of the costs announced by all other individuals. Formally, if in state h ¼ ðh1 ; . . . ; hn Þ, e n2 . Rj ðhÞ > Rl ðhÞ for some ðj; lÞ 2 N N, j 6¼ l, then Rj ðhj ; hl ; h0jl Þ > Rl ðhj ; hl ; h0jl Þ for all h0jl 2 H Hence, what determines the efficient queue is the ranking that each individual gets in a given state. We now argue that if one individual is eliminated from the queue then the relative ranking of all other individuals remain unchanged. Before doing that we introduce some more relevant notations and definitions that captures the idea of elimination of an individual from the queue in any given state. Define M 0 ½n1 to be m þ the number of queue positions that remains in a multiple machine queueing problem with n jobs and m machines after one individual (and hence one job) is eliminated from the queue. Observe that M 0 ¼ M 1 if n ¼ rm þ 1 where r ¼ 2; 3; . . . and M 0 ¼ M otherwise. Using the idea of ranking of individuals for C 2 C, we define Qj ðhl Þ ¼ ½i 2 N fj; lg such that either 9k 2 f1; . . . ; M 0 1g and fDhi ðkÞ > Dhj ðkÞg or 8k 2 f1; 2; . . . ; M 0 1g, fDhi ðkÞ ¼ Dhj ðkÞ and i < jg. Let Rj ðhl Þ ¼ 1 þ jQj ðhl Þjð2 f1; . . . ; n 1gÞ. Therefore, in a state h, Rj ðhl Þ measures the rank of individual j in state h by eliminating the cost vector hl of individual lð6¼ jÞ. Remark 3.2. Consider any multiple machine queueing problem C satisfying the GIP but not the GCP such that M 0 ¼ M 1. Consider a state h ¼ ðh1 ; . . . ; hn Þ such that Dh1 ðkÞ ¼ Dh2 ðkÞ for all k 2 f1; . . . ; M 2g and Dh1 ðM 1Þ < Dh2 ðM 1Þ. Moreover, assume that R1 ðhÞ ¼ n and R2 ðhÞ ¼ n 1. Hence R1 ðhÞ > R2 ðhÞ. Observe, that for all i 2 N f1; 2g, R1 ðhi Þ ¼ n 2 and R2 ðhi Þ ¼ n 1 because Dh1 ðkÞ ¼ Dh2 ðkÞ for all k 2 f1; . . . ; M 2g and 1 < 2. Therefore, R1 ðhÞ > R2 ðhÞ and for all i 2 N f1; 2g, R1 ðhi Þ < R2 ðhi Þ. Therefore, for any multiple machine queueing problem C satisfying the GIP but not the GCP, if M 0 ¼ M 1, then the above construction shows that there exist cost vectors for which the relative ranking of a pair of individuals can change if an individual outside the pair under consideration is eliminated from the queue. It is not hard to verify that the construction specified above is the only type of construction that can lead to such a rank reversal in a multiple machine queueing problem C satisfying the GIP when an individual is eliminated. Moreover, such a rank reversal can take place only if M 0 ¼ M 1 and C fails to
4 For example, if there are only 2 machines and 4 jobs and jobs 1 and 2 are in front of machine 1 and jobs 3 and 4 are in front of machine 2 (i.e., if r ¼ ðr1 ¼ 1; r2 ¼ 2; r3 ¼ 1; r4 ¼ 2Þ), then sðrÞ ¼ ð1; 3; 2; 4Þ.
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satisfy the GCP. If, a multiple machine queueing problem C satisfies the GIP and the GCP and if Dhj ðkÞ ¼ Dhl ðkÞ for all k 2 f1; . . . ; M 2g then the GCP implies that Dhj ðM 1Þ ¼ Dhl ðM 1Þ. Therefore, the construction that led to rank reversal is not possible for a multiple machine queueing problem that satisfies the GCP. Hence, if a multiple machine queueing problem C satisfies both the GCP and the GIP, then the relative ranking of any pair of individuals with any given pair of cost vectors remain unchanged if some other individual is eliminated from the queue. More formally, if C 2 C, then we obtain the following e n, relationship between Rj ðhÞ and Rj ðhl Þ. For allðj; lÞ 2 N N, j 6¼ l and for all h 2 H Rj ðhÞ if Rj ðhÞ < Rl ðhÞ; Rj ðhl Þ ¼ Rj ðhÞ 1 if Rj ðhÞ > Rl ðhÞ: Using Remark 3.2 we derive the sufficiency condition under which a multiple machine queueing problem C 2 C is first best implementable. Theorem 3.2. A multiple machine queueing problem C 2 C is first best implementable. We first state and prove a lemma that will be used in proving Theorem 3.2. Lemma 3.2. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the GCP, if and only if for all cost vector hj 2 H, there exists a unique 1 ðn 1Þ vector Hj ¼ fhj ð1Þ; . . . ; hj ðn 1Þg such that for all p 2 f1; . . . ; ng, hj ð½p=mþ Þ ¼ ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ: Proof. Consider a hj 2 H for individual j 2 N that satisfies Hj ¼ fhj ð1Þ; . . . ; hj ðn 1Þg such that for all p 2 f1; . . . ; n 1g, hj ðpÞ ¼
p X
ð1Þpr
r¼1
ð3:6Þ PM
k¼1
aðk; n; mÞhj ðkÞ ¼ 0. Define a vector
ðp 1Þ!ðn p 1Þ! hj ð½r=mþ Þ: ðr 1Þ!ðn rÞ!
ð3:7Þ
We prove Lemma 3.2 in two steps. In the first step it is proved, using (3.7), that for all p 2 f1; . . . ; n 1g, condition (3.6) holds. In the next step we prove that for p ¼ n, condition (3.6) holds only if C satisfies the GCP. ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ ¼ ðn pÞ
p X
ð1Þ
r¼1
þ ðp 1Þ
p1 X r¼1
¼
pr
ðp 1Þ!ðn p 1Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
ð1Þpr1
ðp 2Þ!ðn pÞ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
p1 n o X pr pr1 ðp 1Þ!ðn pÞ! hj ð½r=mþ Þ þ hj ð½p=mþ Þ ð 1Þ þ ð 1Þ ðr 1Þ!ðn rÞ! r¼1
¼ hj ð½p=mþ Þ
ðbecause ð1Þpr þ ð1Þpr1 ¼ 0Þ:
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For p ¼ n, ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ ¼ ðn 1Þhj ðn 1Þ ¼ ðn 1Þ
n1 X
n1r
ð1Þ
r¼1 n1 X
ðn 2Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
ðn 1Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ! r¼1 n1 X n1r n 1 ¼ ð1Þ hj ð½r=mþ Þ r1 r¼1 ¼
ð1Þ
n2
¼ ð1Þ
n1r
M X
aðk; n; mÞhj ðkÞ þ hj ðMÞ
ðfrom the GCPÞ ¼ hj ðMÞ:
k¼1
Therefore, the last step not only proves the necessity of the GCP but also guarantees that for hj , the 1 ðn 1Þ vector Hj is unique. PM We now prove the other part of Lemma 3.2. Observe that the sum k¼1 aðk; n; mÞhj ðkÞ ¼ Pn p1 n 1 h ð½p=mþ Þ. Therefore, p¼1 ð1Þ p1 j n X p1 n 1 h ð½p=mþ Þ ð1Þ p1 j p¼1 n X p1 n 1 ¼ ð1Þ fðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þg p1 p¼1 ( ) n1 n X X p1 n 2 p1 n 2 h ðpÞ þ h ðp 1Þ ¼ 0: ð 1Þ ð 1Þ ¼ ðn 1Þ p1 j p2 j p¼1 p¼2 Lemma 3.2 gives rise to a particular type of separability (as given by condition (3.6)) that will be used in e 2 C. deriving the explicit form of the transfer that first best implements any C ¼ hN; m; Hi P e n for individual j 2 N. From the Proof of Theorem 3.2. Consider the sum l6¼j hj ðRj ðhl ÞÞ in state h 2 H GIP and Remark 3.2 we get X hj ðRj ðhl ÞÞ ¼ ðn Rj ðhÞÞhj ðRj ðhÞÞ þ ðRj ðhÞ 1Þhj ðRj ðhÞ 1Þ l6¼j
¼ hj ð½Rj ðhÞ=mþ Þ ðfrom condition ð3:6Þ in Lemma 3:2Þ: We consider a Groves–Clarke mechanism c M ¼ hr ; ^ti where the term independent of j’s announcement P is ^cj ðhj Þ ¼ ðn 1Þ l6¼j hl ðRl ðhj ÞÞ. Then it follows that ( ) X XX X X X ^cj ðhj Þ ¼ ðn 1Þ hl ðRl ðhj ÞÞ ¼ ðn 1Þ hj ðRj ðhl ÞÞ ¼ ðn 1Þ hj ð½Rj ðhÞ=mþ Þ j2N
j2N
l6¼j
j2N
l6¼j
j2N
¼ ðn 1ÞCðhÞ: Observe that the last step follows from the efficiency rule of Proposition 3.3. The last step implies that for all e n , the sum of transfers P ^tj ðhÞ ¼ ðn 1ÞCðhÞ þ P ^cj ðhj Þ ¼ 0. h h2H j2N j2N Observe that from Theorems 3.1 and 3.2 and from Proposition 3.2 it follows that a multiple machine queueing problem satisfying A1–A3 and with three machines is first best implementable if and only if it satisfies the GCP.
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4. Separable cost multiple machine queueing problems In this section we first define a class of multiple machine queueing problems with separable cost and then verify under what conditions these problems are first best implementable. The following conditions describe a typical separable cost multiple machine queueing problem. 1. Define a real number h > 0. Given h > 0, the cost parameter or type of an individual belongs to the interval ½0; h. Moreover, there exists a function f : f1; . . . ; Mg ! R such that f ðkÞ P f ðk 1Þ for all k 2 f2; . . . ; Mg and f ðk Þ > f ðk 1Þ for at least one k 2 f2; . . . ; Mg. 2. hj ðkÞ ¼ f ðkÞhj for all j 2 N, for all k 2 f1; 2; . . . ; Mg and for all hj 2 ½0; h. For a separable cost multiple machine queueing problem, the cost of each individual for each position is multiplicatively separated into two parts. The first part is a function f that depends on the queue position. Observe that the functional form f is assumed to be identical for all individuals j 2 N. Moreover, we assume that f is common knowledge. The second part which is a non-negative number hj represents the type (or cost parameter) of an individual that belong to the interval ½0; h. In this set up a type vector of individual j 2 N is given by hj ¼ ðhj ð1Þ ¼ f ð1Þhj ; . . . ; hj ðMÞ ¼ f ðMÞhj Þ. Therefore, from now on we will write hj as the cost parameter or type of an individual. The cost parameter hj 2 ½0; h for all j 2 N is private information. Finally, h ¼ ðh1 ; . . . ; hn Þ 2 ½0; hn represents a state of the world or a profile. It is important to observe that the domain H ðf ; ½0; hÞ of any separable cost multiple machine queueing problem satisfies Assumptions 1–3. In this framework the set of individuals N, mð> 1Þ number of machines and H ðf ; ½0; hÞ define the b ¼ hN; m; Hi. We will completely characterize the class multiple machine separable cost queueing problem C of first best implementable multiple machine separable cost queueing problems. We start by showing that these problems satisfy the GIP. b ¼ hN; m; Hi satisfies the GIP. Proposition 4.4. A multiple machine separable cost queueing problem C Proof. Consider any pair ðj; lÞ 2 N N, j 6¼ l, with cost parameters hj and hl respectively. It is obvious that either hj P hl or hj 6 hl . Since f ðk þ 1Þ P f ðkÞ for all k 2 f1; . . . ; M 1g, it is also obvious that if hj P hl , then ff ðk þ 1Þ f ðkÞghj P ff ðk þ 1Þ f ðkÞghl for all k 2 f1; . . . ; M 1g. Therefore, Dhj ðkÞ ¼ ff ðk þ 1Þ f ðkÞghj P Dhl ðkÞ ¼ ff ðk þ 1Þ f ðkÞghl for all k 2 f1; . . . ; M 1g. Similarly, if hj 6 hl , then Dhj ðkÞ 6 Dhl ðkÞ b ¼ hN; m; Hi satisfies the GIP. h for all k 2 f1; . . . ; M 1g. Thus, it follows that C The next two remarks follow trivially from the discussion of the GCP in the previous section. b ¼ hN; m; Hi satisfies the GCP if Remark 4.3. A multiple machine separable cost queueing problem C M X aðk; n; mÞf ðkÞ ¼ 0: ð4:8Þ k¼1
Using Df ðkÞ ¼ f ðk þ 1Þ f ðkÞ and simplifying equation (4.8) we get M 1 X zðk; n; mÞDf ðkÞ ¼ 0; k¼1 Pk n2 km1 where zðk; n; mÞ ¼ r¼1 aðr; n; mÞ ¼ ð1Þ . km 1
ð4:9Þ
b ¼ hN; m; Hi satisfies the GCP, if and only if there exists Remark 4.4. From condition (3.6) it follows that C a unique vector H ¼ fhð1Þ; . . . ; hðn 1Þg such that for all p 2 f1; . . . ; ng,
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f ð½p=mþ Þ ¼ ðn pÞhðpÞ þ ðp 1Þhðp 1Þ; where hðpÞ ¼
Pp
r¼1
ð1Þ
ð4:10Þ
pr ðp1Þ!ðnp1Þ! f ð½r=mþ Þ. ðr1Þ!ðnrÞ!
The next result completely characterizes the class of first best implementable multiple machine separable cost queueing problems. b ¼ hN; m; Hi is first best impleProposition 4.5. A multiple machine separable cost queueing problem C mentable if and only if the cost function satisfies the GCP. The necessity part of Proposition 3.1 is similar to that of Theorem 3.1 and the sufficiency part is similar to Theorem 3.2. Therefore, we omit the proof of this proposition. It is easy to verify that Proposition 3.1 is also true for separable cost queueing problems. Therefore, all multiple machine separable cost queueing problems with either (1) even number of machines or (2) two queue positions are not first best implementable. Let CðSÞð CÞ be the class of multiple machine separable cost queueing problems where m is odd and n > 2m and let C ð CðSÞÞ be the class of first best implementable multiple machine separable cost queueing problems. The next proposition proves the existence of C . Proposition 4.6. There exists C 2 CðSÞ such that C 2 C . b 0 ¼ hN; m; Ho ðf 0 ; ½0; hÞi 2 CðSÞ with odd number of queue positions M and with f 0 of Proof. Consider C n2 0 0 0 0 the following form: f ð1Þ ¼ c P 0 and Df ðkÞ ¼ f ðk þ 1Þ f ðkÞ ¼ 1 for all k 2 f1; . . . ; km 1 0 b ¼ hN; m; ðf 0 ; ½0; hÞi 2 C by showing that f 0 satisfies condition (4.9). ObM 1g. We will prove that C km1 for all k 2 f1; . . . ; M 1g. Therefore, by substituting serve first that zðk; n; mÞDf 0 ðkÞ ¼ ð1Þ PM1 km1 km1 in the left-hand side of condition (4.9) we get k¼1 ð1Þ . Since both m and zðk; n; mÞDf 0 ðkÞ ¼ ð1Þ PM1 km1 0 0 b ¼ 0. Thus, C ¼ hN; m; ðf ; ½0; hÞi 2 C . M are odd, it is obvious that, k¼1 ð1Þ b e ¼ hN; m; He ðf e ; ½0; hÞi 2 CðSÞ with even number of queue positions M and Similarly, consider C n2 for all k 2 with f e of the following form: f e ð1Þ ¼ c P 0 and Df e ðkÞ ¼ f e ðk þ 1Þ f e ðkÞ ¼ 1 km 1 km1 f1; . . . ; M 2g and f e ðM 1Þ ¼ f e ðMÞ (that is, Df e ðM 1Þ ¼ 0). Observe that zðk; n; mÞDf e ðkÞ ¼ ð1Þ for all k 2 f1; . . . ; M 2g and zðM 1; n; mÞDf e ðM 1Þ ¼ 0. Therefore, by substituting zðk; n; mÞDf e ðkÞ for PM2 km1 all k 2 f1; . . . ; M 1g in the left hand side of condition (4.9) we get . Here k¼1 ð1Þ PM2 km1 b e 2 C . h ¼ 0 because m is odd and M is even. Thus, C k¼1 ð1Þ Observe that given C CðSÞ C, it follows from Proposition 4.6 that there exist first best implementable multiple machine queueing problems in C. We conclude this section with an important observation. Observation 1. Given the co-efficient vector aðn; mÞ, it follows that if m is odd, M ¼ 2q þ 1 and n ¼ m ð2q þ 1Þ (where q 2 f1; 2; . . .g) then aðk; P n; mÞ ¼ að2q þ 2 k; n; mÞ for all k 2 f1; . . . ; qg. Using q this result and by substituting aðq þ 1; n; mÞ ¼ 2 k¼1 aðk; n; mÞ in Eq. (3.2) we get q X
aðk; n; mÞfhj ðkÞ þ hj ð2q þ 2 kÞ 2hj ðq þ 1Þg ¼ 0:
ð4:11Þ
k¼1
Observe that if hj ðkÞ ¼ khj for all k 2 f1; . . . ; Mg, then condition (4.11) holds. Thus, if m is odd, n ¼ mM, M is also odd and f l ðkÞ ¼ k for all k, then Cl ¼ hN; m; Hl ðf l ; ½0; hÞi 2 C .
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5. Concluding remarks We have obtained the following results regarding the first best implementability of the class of multiple machine queueing problems with domains satisfying Assumptions 1–3. 1. A multiple machine queueing problem is first best implementable only if it satisfies the GCP. 2. If the number of machines is even or if there are only two queue positions, then a non-trivial multiple machine queueing problem fails to satisfy the GCP and hence is not first best implementable. 3. If the number of machines m is odd and M ¼ ½n=mþ ¼ 3 then a multiple machine queueing problem is first best implementable if and only if it satisfies the GCP. 4. If the number of machines m is odd and M ¼ ½n=mþ P 4 then a multiple machine queueing problem is first best implementable if it satisfies the GCP and the GIP. 5. For all n > 2m such that m is odd, there exists a non-trivial cost function for which a multiple machine queueing problem is first best implementable. 6. Finally, if m is odd, the number of queue positions M ¼ ½n=mþ > 2 is also odd and n ¼ m M then a multiple machine queueing problem with linear cost function is first best implementable. Thus, first best implementability of a multiple machine queueing problem depends heavily on the number of machines and on the number of jobs.
Acknowledgements The author is grateful to Arunava Sen, Jeroen Suijs and two anonymous referees for their invaluable advice. The author is thankful to Parikshit Ghosh and Suryapratim Banerjee for helpful discussions. The author is also thankful to the seminar participants at the European Summer Symposium in Economic Theory 2003 (held in Gerzensee, Switzerland). The author gratefully acknowledges the financial support from the Indian Statistical Institute and from the Deutsche Forschungsgemeinschaft Graduiertenkolleg 629 at the University of Bonn. The author is solely responsible for the errors that may still remain.
References Clarke, E.H., 1971. Multi-part pricing of public goods. Public Choice 11, 17–33. Curiel, I., Pederzoli, G., Tijs, S., 1989. Sequencing games. European Journal of Operational Research 40, 344–351. Dudek, R.A., Panwalkar, S., Smith, M., 1975. Flow-shop sequencing problem with ordered processing time matrices. Management Science 21, 544–549. Dudek, R.A., Teuton Jr., O.F., 1964. Development of M-stage decision rule for scheduling n jobs through M machines. Operations Research 12, 471–497. Green, J., Laffont, J.J., 1979. Incentives in Public Decision Making. North Holland Publication, Amsterdam. Groves, T., 1973. Incentives in teams. Econometrica 41, 617–631. Groves, T., Ledyard, J.O., 1977. Some limitations of demand revealing processes. Public Choice 29, 107–124. Groves, T., Loeb, M., 1975. Incentives and public inputs. Journal of Public Economics 4, 211–226. Hamers, H., Klijn, F., Suijs, J., 1999. On the balancedness of multiple machine sequencing games. European Journal of Operational Research 119, 678–691. Holmstr€ om, B., 1977. On incentives and control in organisation. Ph.D. Dissertation, Stanford Graduate School of Business. Holmstr€ om, B., 1979. Groves’ schemes on restricted domains. Econometrica 47, 1137–1144. Hurwicz, L., 1975. On the Existence of Allocation Systems Whose Manipulative Nash Equilibria are Pareto Optimal, Mimeo., University of Minnesota. Hurwicz, L., Walker, M., 1990. On the generic non-optimality of dominant strategy allocation mechanisms: A general theorem that includes pure exchange economies. Econometrica 58, 683–704.
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Krabbenborg, M., Potters, J., van den Nouweland, A., 1992. Flow-shops with a dominant machine. European Journal of Operational Research 62, 38–46. Liu, L., Tian, G., 1999. A characterization of the existence of optimal dominant strategy mechanisms. Review of Economic Design 4, 205–218. Mitra, M., 2001. Mechanism design in queueing problems. Economic Theory 17, 277–305. Mitra, M., Sen, A., 1998. Dominant Strategy Implementation of First Best Public Decisions, Mimeo., Indian Statistical Institute, New Delhi. Suijs, J., 1996. On incentive compatibility and budget balancedness in public decision making. Economic Design 2, 193–209. Tian, G., 1996. On the existence of optimal truth-dominant mechanisms. Economics Letters 53, 17–24. Tomescu, I., 1985. Problems in Combinatorics and Graph Theory (Melter, R.A., Trans.) Wiley-Interscience, New York. Walker, M., 1980. On the non-existence of dominant strategy mechanisms for making optimal public decisions. Econometrica 48, 1521–1540.
Stochastics and Statistics
Incomplete information and multiple machine queueing problems Manipushpak Mitra
*
Economic Research Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700108, India Received 16 June 2000; accepted 6 October 2003 Available online 25 March 2004
Abstract In mechanism design problems under incomplete information, it is generally difficult to find decision problems that are first best implementable. A decision problem under incomplete information is first best implementable if there exists a mechanism that extracts the private information and achieves efficiency with a transfer scheme that adds up to zero in every state. One can find queueing problem with one machine that are first best implementable under certain cost conditions. In this paper we identify the conditions on cost structure for which queueing problems with multiple machines are first best implementable. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Queueing problems; First best implementability
1. Introduction In a queueing problem with multiple machines, there is a server (for example, a computer server), with more than one identical machines (computers) which has to process a finite number of jobs for a set of individuals. The machines are identical in the sense that a given job takes the same length of time for completion. We assume that it takes one unit of time to complete one job. Each individual has one job to be processed. The server can serve one individual in one machine, that is, it takes one unit of time to process one job in a machine. If the number of jobs, to be processed, is more than the number of machines then individuals will have to wait in a queue. Waiting in a queue is costly for each individual. The server’s objective is to order the individuals in a queue efficiently so as to minimize the aggregate waiting cost. If the cost of waiting in the queue is private information then an individual, if asked, will announce her cost strategically so as to get her job done as early as possible. Therefore, in the queueing scenario described above, the server’s role is that of a planner who has to solve an incentive problem under incomplete information. More precisely, we have a mechanism design problem of a social planner (server in the
*
Tel.: +91-33-25752608; fax: +91-33-25778893. E-mail address: [email protected] (M. Mitra).
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.10.050
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queueing problem) whose objective is to extract the privately held information (true waiting cost) of each individual and select the efficient decision (to order the individuals in a queue so as to minimize the aggregate waiting cost) in each state. One of the most significant achievements in the planner’s mechanism design problems under incomplete information has been the existence of a class of mechanisms called Groves–Clarke mechanisms (see Clarke, 1971; Groves, 1973). These mechanisms achieve the twin objectives of truthful revelation of private information and efficiency of decisions provided the agents have quasi-linear preferences. Moreover, for a very broad class of preference structures, in a quasi-linear set up, Groves–Clarke mechanisms are the only class of mechanisms that achieve these objectives (see Holmstr€ om, 1979). However, the drawback of such mechanisms is that they are, in general, not Pareto-optimal. This means that there are preference realizations where the sum of Groves–Clarke transfers are non-zero. In the pure public goods problem, Hurwicz (1975), Green and Laffont (1979) and Walker (1980) proved the budget imbalance of a Groves– Clarke scheme. Hurwicz and Walker (1990) proved the impossibility result in the context of pure exchange economies (economies in which there are no production, no public goods and other externalities). The damaging nature of budget imbalance, in the public goods context, was pointed out by Groves and Ledyard (1977). They proposed, using a very simple model, that an alternative procedure based on majority rule voting may lead to an allocation of resources which is Pareto superior to the one produced by Groves mechanism. However, there are certain decision problems where first best or Pareto optimality can be achieved. In the public goods problem, Groves and Loeb (1975) have proved that if preferences are quadratic then we can find balanced Groves transfer. This result was generalized by Tian (1996) and Liu and Tian (1999). In a single server (one machine) queueing problem with linear cost, Mitra and Sen (1998) showed the existence of first best mechanisms. A problem similar to the queueing problem with linear costs in Mitra and Sen (1998) is the sequencing problem in Suijs (1996). Unlike the queueing problem, where it takes one unit of time to serve one individual, in a sequencing problem the servicing time can differ from one individual to another. Therefore, while the linear cost queueing problem is a discrete time problem, the sequencing problem in Suijs (1996) is a continuous time problem. By assuming servicing time to be common knowledge, Suijs proved the existence of first best mechanisms for the sequencing problem. The existence result in Mitra and Sen (1998) was further generalized by Mitra (2001) for a broader class of cost structures. It was proved that the class of cost structures under which a ‘one machine queueing problem’ is first best implementable is ‘fairly’ large. In this paper, we deal with the question of first best implementability of queueing problems with multiple machine. Therefore, this paper is a generalization of the one machine queueing framework of Mitra (2001) to a multiple machine framework. A multiple machine queueing problem is first best implementable if there exists a mechanism that can extract the private information with a vector of transfers that add up to zero. This allows the server to order the jobs in a way that minimizes the aggregate cost. The most important implication of first best implementability is that the server can extract the private information costlessly. If a queueing problem is first best implementable then there is no welfare loss as the transfers used to extract the private information adds up to zero in all states. A multiple machine queueing problem resembles some of the sequencing problems that are analyzed in the operations research literature. Papers relating to sequencing n jobs in m machines by Dudek and Teuton Jr. (1964), flow shop sequencing problems with ordered processing time by Dudek et al. (1975) and flow shop problems with dominant machines by Krabbenborg et al. (1992) deal with finding algorithms to order (or queue) the n jobs in m machines in a way that minimizes the total elapsed time. However, in all these models, unlike multiple machine queueing problems, machines are not identical. The processing time for the same job can be different in different machines. Moreover, unlike a multiple machine queueing problem where cost parameter is private information, the cost structures in all the above mentioned sequencing problems are common knowledge. In the incomplete information set up, sequencing problems were analyzed by Hamers et al. (1999). They analyzed a multiple identical machine sequencing (or scheduling) problem with linear time cost. Therefore,
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their problem is a continuous time version of the multiple machine queueing problem with linear cost. Hamers et al. (1999) look at the n jobs and m identical machines sequencing situation both in a co-operative and non-cooperative environments. In a co-operative set up, a sequencing problem is called a sequencing game. Curiel et al. (1989) analyzed the sequencing game in a one machine framework. Hamers et al. (1999), by extending the sequencing game of Curiel et al. (1989) to a multiple identical machine framework, addressed the issues of balancedness and non-emptiness of core in m-sequencing games. In a non-cooperative set up, they address the issue of first best implementability by assuming job processing time to be equal to one. Thus, in the non-cooperative set up, the sequencing situation analyzed by Hamers et al. (1999) is identical to the multiple machine queueing problem described in this paper with the restriction that the cost is linear over time. The main objective of this paper is to achieve first best implementability in multiple machine queueing problems. More precisely, this paper identifies the conditions on the cost structure that lead to first best implementability. Therefore, in this paper, we also generalize the non-cooperative sequencing situation, analyzed by Hamers et al. (1999), by allowing for a very general time cost. The results that we get suggest that first best implementability depends critically on the number of machines and the number of jobs to be processed on those machines. If the number of machines (that is m) is even or if the number of jobs to be processed is strictly greater than the number of machines but less than or equal to twice the number of machines, then a multiple machine queueing problem is not first best implementable. For all other multiple machine queueing problems there exists a non-trivial cost structure under which it is first best implementable. Finding an algorithm to obtain the cost minimizing queue is not a very important issue since the conditions on the cost structure under which a multiple machine queueing problem is first best implementable are such that the algorithm for finding the cost minimizing queue is transparent. Therefore, obtaining a state contingent transfer scheme that extracts the private information while adding up to zero is of prime importance. The paper is organized in the following way: in Section 2 we develop the general class of problems, in Section 3 we derive some characterization results, in Section 4 we deal with separable cost multiple machine queueing problems and finally in the concluding remarks of Section 5 we summarize the results obtained in this paper. 2. The general class of problems Let N ¼ f1; 2; . . . ; ng be the set of individuals and mð> 1Þ be the set of identical machines. Define ½xþ to be the lowest integral value not less than x. For example, ½2:005þ ¼ 3 and ½3þ ¼ 3. Given n and m, the total number of queue positions are M ¼ ½n=mþ . Here the number of individuals (and hence the number of jobs) n is strictly greater than the number of machines m in order to have a meaningful queueing problem. hj ðkÞ measures the cost of waiting k periods in the queue for individual j where k 2 f1; . . . ; Mg. Let Rþ be the non-negative orthant of the real line R. A cost vector or type of an individual j 2 N is a vector of the form M M hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 RM þ . An -neighborhood around any vector hj 2 Rþ is the set N ðhj Þ ¼ fx 2 Rþ : khj xk < g. 1 A typical domain of cost vectors of an individual j 2 N is H RM þ . Observe that the do_ A cost main H is assumed to be common for all individuals. For a domain H, we denote its interior by H. _ _ vector hj 2 H, if 9 > 0 such that N ðhj Þ H. We assume that the interior H (of the domain H) is nonempty. Let Dhj ¼ ðDhj ð1Þ; . . . ; Dhj ðM 1ÞÞ represent the vector of first differences generated from the vector hj . Here Dhj ðkÞ ¼ hj ðk þ 1Þ hj ðkÞ for all k 2 f1; . . . ; M 1g. We say that Dhj < Dh0j if and only if Dhj ðkÞ 6 Dh0j ðkÞ for all k 2 f1; . . . ; M 1g and there exists at least one k 0 2 f1; . . . ; M 1g such that 1
Here khj xk denotes the Euclidean distance between the two cost vectors hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ and x ¼ ðxð1Þ; . . . ; xðMÞÞ, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM 2 k¼1 ðhj ðkÞ xðkÞÞ .
khj xk ¼
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Dhj ðk 0 Þ < Dh0j ðk 0 Þ. The three main assumptions we make on the common domain of preference H are as follows. Assumption 1. For all j 2 N and for all cost vector hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H, 0 6 hj ð1Þ 6 hj ð2Þ 6 6 hj ðMÞ. Assumption 2. H is convex, that is, if hj 2 H and h0j 2 H, then khj þ ð1 kÞh0j 2 H for all k 2 ½0; 1. _ there exists ða; bÞ 2 H H such that Da < Dh < Db. Assumption 3. H is sufficiently open if for all hj 2 H, j Moreover, Dhj ðkÞ < DbðkÞ for all k 2 f1; . . . ; M 1g and DaðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. 2 Assumption 1 simply says that the individuals are impatient. Assumption 2 is the standard convexity requirement. Finally, Assumption 3 is an openness assumption on the cost domain which says that for any type hj , in the interior of the domain, there exists one type b in the domain that, in terms of difference, strictly dominates hj and there exists another type a with null first difference vector in the domain which is dominated, in terms of difference, by hj . Let H be the class of domains satisfying Assumptions 1–3. Let H be any domain belonging to H satisfying Assumptions 1–3. The utility of each individual j is assumed to be quasi-linear and is of the form: Uj ðk; tj ; hj Þ ¼ vj hj ðkÞ þ tj where vj ð> 0Þ is the gross benefit derived by individual j from the service, hj ðkÞ is the cost of individual j at the kth queue position and tj is the transfer that individual j receives. The server’s aim is to achieve efficiency or minimize the aggregate cost. To define what we mean by efficiency in a queueing problem with m machines, we need to develop the concept of a multi-set. A multi-set is a set where all elements may not be distinct. For example, X ¼ f1; 1; 1; 3; 6; 6; 9g is a multi-set. Given a queueing problem with n individuals, m machines and hence M ¼ ½n=mþ queue positions, consider the multi-set Xn;m of the form Xn;m ¼ f1; . . . ; 1; 2; . . . ; 2; . . . ; M 1; . . . ; M 1 ; M; . . . ; M g. Let P ðXn;m Þ be the |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} m
m
m
nðM1Þm
set of all possible permutations of the multi-set Xn;m . In this problem, a queue r is a one-to-one correspondence between the set of jobs N ¼ f1; . . . ; ng and the queue positions P ðXn;m Þ, that is, r ¼ ðr1 ; . . . ; rn Þ : N ! P ðXn;m Þ. Thus, rj ¼ k indicates that individual j has the kth position in the queue. Given a queue r ¼ ðr1 ; . . . ; rn Þð2 P ðXn;m ÞÞ, the cost of an individual j 2 N is hj ðrj Þ. A state of the world is n h ¼ ðh1 ; . . . ; hn Þ 2 H where hj is a 1 M vector. n
Definition 2.1. Given a state h 2 H , a queue r 2 P ðXn;m Þ is said to be efficient if r 2 arg minr2P ðXn;m Þ P j2N hj ðrj Þ. An efficient queue r is an assignment that gives each individual exactly one queue position and each of the first M 1 queue positions to exactly m individuals and the Mth queue position to the remaining n ðM 1Þm individuals in such a way that the aggregate cost is minimized. Observe, that there can be states with more than one efficient queue. So we have an efficiency correspondence. An efficient rule is a single valued selection from the efficiency correspondence. Note that efficiency of a queue r is a concept independent of transfers and gross benefits of all individuals. If the server knows the true state h ¼ ðh1 ; . . . ; hn Þ then she can calculate an efficient queue. However, if hj is private information for individual j, the server’s problem then is to design a mechanism that will elicit this information truthfully. We refer to such a problem as a multiple machine queueing problem under incomplete information and is written as C ¼ hN; m; Hi where H 2 H. Note that we are assuming that the domain
2
This means that the vector a is a constant vector.
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H is common knowledge and that the cost vector of an individual is private information. Therefore, each individual, if asked, will announce a cost vector from the domain H. Formally, a mechanism M is a pair n n hr; ti where r : H ! P ðXn;m Þ and t ðt1 ; . . . ; tn Þ : H ! Rn . Thus, a mechanism M is a direct revelation mechanism where each individual j 2 N announces a cost vector hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ and based on the announcements of all individuals (that is, h ¼ ðh1 ; . . . ; hn Þ), the planner (or server) specifies a queue r and a vector of transfers t ¼ ðt1 ; . . . ; tn Þ. Under M ¼ hr; ti, given all others’ announcement hj , the utility of individual j of type hj when her announcement is h0j is given by Uj ðrj ðh0j ; hj Þ; tj ðh0j ; hj Þ; hj Þ ¼ vj hj ðrj ðh0j ; hj ÞÞ þ tj ðh0j ; hj Þ. Definition 2.2. A multiple machine queueing problem C ¼ hN; m; Hi is implementable if there exists an n efficient rule r : H ! P ðXn;m Þ and a mechanism M ¼ hr ; ti such that for all j 2 N, for all pairs of n1 announcement vectors ðhj ; h0j Þ 2 H H and for all announced hj 2 H , Uj ðrj ðhÞ; tj ðhÞ; hj Þ P 0 0 Uj ðrj ðhj ; hj Þ; tj ðhj ; hj Þ; hj Þ. This definition says that C ¼ hN; m; Hi is implementable if there exists a direct mechanism, with an efficient queueing rule r and a vector of transfers, that induces each individual to tell the truth independent of others’ report. Definition 2.3. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable, if there n exists a mechanism M ¼ hr ; ti such that (1) M implements C and (2) for all announcements h 2 H , P j2N tj ðhÞ ¼ 0. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable if it can be implemented with a budget balancing transfer. Thus, if C is first best implementable then incomplete information does not impose any welfare loss.
3. Characterization results n
Definition 3.4. A mechanism M ¼ hr; ti is a Groves–Clarke mechanism if for all j 2 N and for all h 2 H , the transfer is of the form X tj ðhÞ ¼ hl ðrl ðhÞÞ þ cj ðhj Þ: ð3:1Þ l6¼j
In a Groves–Clarke mechanism, the transfer of any P individual j 2 N in any state Ph is the negative of aggregate cost plus the cost of individual j (that is l2N hl ðrl ðhÞÞ þ hj ðrj ðhÞÞ ¼ l6¼j hl ðrl ðhÞÞ), plus a constant cj ðhj Þ. The utility of individual j with a Groves–Clarke transfer is her gross benefit vj less the aggregate cost in state h plus the constant. We now proceed to verify that given a mechanism M ¼ hr ; ti where the queue r is efficient and the transfer satisfies condition (3.1), truth-telling is a dominant strategy. Suppose, it were not the case. Then there exists an individual j with true cost hj and there exists a report hj such that individual j strictly benefits by misreporting her cost to be some h0j ð6¼ hj Þ. That is, Uj ðrj ðhÞ; tj ðhÞ; hj Þ < Uj ðrj ðh0j ; hj Þ; tj ðh0j ; hj Þ; hj Þ. Simplifying this inequality after substituting the Groves– P P 0 Clarke transfer, we get j2N hj ðrj ðhÞÞ > j2N hj ðrj ðhj ; hj ÞÞ. This contradicts efficiency of decision (or aggregate cost minimization) in state h. Hence, the Groves–Clarke transfer leads to truth-telling in dominant strategies. According to a well known result of Holmstr€ om (see Holmstr€ om, 1979), decision problems with ‘‘convex’’ domains are implementable if and only if the mechanism is a Groves–Clarke mechanism (see Theorem (2) in Holmstr€ om (1979)). Since by Assumption 1, any domain H 2 H is convex, all multiple
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machine queueing problems C ¼ hN; m; Hi with H 2 H are implementable if and only if the mechanism is a Groves–Clarke mechanism. Therefore, the question of first best implementability of a multiple machine queueing problem reduces to finding domain restrictions under which we can find a balanced Groves– Clarke transfer. P Let Cðr ðh0 Þ; hÞ ¼ j2N hj ðrj ðh0 ÞÞ where, as stated earlier, r ðh0 Þ is an efficient queue for the announced 0 state h . Thus, Cðr ðh0 Þ; hÞ is the minimum aggregate cost with respect to the announced state h0 when the actual state is h. For notational simplicity we define CðhÞ Cðr ðhÞ; hÞ to be the minimum aggregate cost with respect to the actual state h when the announced state is also h. Remark 3.1. From the definition of efficiency of the queue r it follows that for all h and h0 , CðhÞ 6 Cðr ðh0 Þ; hÞ. When is a multiple machine queueing problem first best implementable? In our first theorem we show that the General Combinatorial Property, defined below, is necessary for first best implementability of a multiple machine queueing problem. Definition 3.5. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the General Combinatorial Property (or GCP) if for all hj 2 H M X aðk; n; mÞhj ðkÞ ¼ 0; ð3:2Þ k¼1 M
where aðn; mÞ ¼ faðk; n; mÞgk¼1 and ( m1 X ðk1Þm l aðk; n; mÞ ¼ ð1Þ ð 1Þ
) n1 if k 2 f1; . . . ; M 1g ðk 1Þm þ l l¼0 ( nðM1Þm1 ) X n1 ðM1Þm l aðk; n; mÞ ¼ ð1Þ ð 1Þ if k ¼ M: ðM 1Þm þ l l¼0
and
The following example illustrates the GCP. b ¼ hN ¼ f1; . . . ; 10g; m ¼ 3; Hi. Here the number of queue positions is M ¼ Example 3.1. Consider C b ½10=3þ ¼ 4 and að10; 3Þ ¼ ðað1; 10; 3Þ ¼ 28; að2; 10; 3Þ ¼ 84; að3; 10; 3Þ ¼ 57; að4; 10; 3Þ ¼ 1Þ. Thus, C satisfies GCP if 8j 2 f1; . . . ; 10g, hj ¼ ðhj ð1Þ; hj ð2Þ; hj ð3Þ; hj ð4ÞÞ 2 H is such that 28hj ð1Þ P 84hj ð2Þ þ M¼4 57hj ð3Þ hj ð4Þ ¼ 0. Note that the coefficient vector að10; 3Þ ¼ ð28; 84; 57; 1Þ is such that k¼1 aðk; 10; 3Þ ¼0. From Definition 3.5 it is quite obvious that for a multiple machine queueing problem C, the coefficient vector aðn; mÞ ¼ faðk; n; mÞgM k¼1 is such that M n X X p1 n 1 ¼ 0: ð3:3Þ aðk; n; mÞ ¼ ð1Þ p1 k¼1 p¼1 The first-order difference at queue position k (that is, Dhj ðkÞ ¼ hj ðk þ 1Þ hj ðkÞ) represents the increase in queueing cost for individual j if he is moved from kth queue position to ðk þ 1Þth queue position. By simplifying Eq. (3.2) using Dhj ðkÞ we get M 1 X k¼1
zðk; n; mÞDhj ðkÞ ¼ 0;
ð3:4Þ
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P where the partial sum coefficient vector zðn; mÞ ¼ fzðk; n; mÞgM1 that zðk; n; mÞ ¼ kr¼1 aðr; n; mÞ for k¼1 is such Pr q a r a1 ¼ ð1Þ , it follows that all k 2 f1; . . . ; M 1g. From the mathematical identity q¼0 ð1Þ q r n2 km1 for all k 2 f1; . . . ; M 1g (see Tomescu, 1985). zðk; n; mÞ ¼ ð1Þ km 1 Theorem 3.1. A multiple machine queueing problem C ¼ hN; m; Hi is first best implementable only if it satisfies the GCP. Before proving Theorem 3.1, a lemma due to Walker (1980) is stated below. Consider two profiles h ¼ ðh1 ; . . . ; hn Þ and h0 ¼ ðh01 ; . . . ; h0n Þ. Define for S N, a type hj ðSÞ ¼ hj if j 62 S and hj ðSÞ ¼ h0j if j 2 S. Thus for each S N, we have a state hðSÞ ¼ ðh1 ðSÞ; . . . ; hn ðSÞÞ. Lemma 3.1. A multiple P machine queueing problem C ¼ hN; m; Hi is first best implementable only if for all n n jSj pairs ðh; h0 Þ 2 H H , SN ð1Þ CðhðSÞÞ ¼ 0. 3 1ÞCðhÞ ¼ P By adding the Groves–Clarke transfer of all individuals and setting it to zero we get ðn n n om (1977) for a more general result). Thus, for all ðh; h0 Þ 2 H H , j2N cj ðhj Þ (see Holmstr€ P P P jSj jSj 1 SN ð1Þ CðhðSÞÞ ¼ ðn1Þ j2N SN ð1Þ cj ðhj ðSÞÞ ¼ 0 (see Walker, 1980). Proof of Theorem 3.1. Consider a multiple machine queueing problem C ¼ hN; m; Hi. We start with a given type h1 for individual 1 in the interior of the domain and construct h1 and h0 . Then we apply Lemma 3.1 due to Walker (1980) to derive the result. Consider individual 1 and any announcement h1 ¼ ðh1 ð1Þ; . . . ; _ Given h , we consider two states h ¼ ðh ; h ; . . . ; h Þ and h0 ¼ ðh0 ; . . . ; h0 Þ of the h ðkÞ; . . . ; h ðMÞÞ 2 H. 1
1
1
1
2
n
1
n
following type: for all k ¼ 1; . . . ; M 1, Dh1 ðkÞ < Dhj ðkÞ < Dhjþ1 ðkÞ for all j 2 f2; . . . ; n 1g and h0j ðkÞ ¼ g, for all j 2 N. Therefore, h0j ¼ ðg; g; . . . ; gÞ for all j 2 N which implies that Dh0j ðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. Note that this sort of construction is possible due to Assumptions 1 and 3. Consider any two queue positions k and k þ 1 and any two individuals j and j þ 1 with types hj and hjþ1 , respectively. Note that from the construction of h, on the one hand, it follows that if individual j gets the kth position and ðj þ 1Þth individual gets the ðk þ 1Þth position, then the costs for these two positions add up to fhj ðkÞ þ hjþ1 ðk þ 1Þg. If, on the other hand, the positions of j and ðj þ 1Þ are interchanged then the costs add up to fhj ðk þ 1Þ þ hjþ1 ðkÞg. The former cost strictly exceeds the latter for all k ¼ 1; . . . ; M 1 since, by construction, Dhj ðkÞ < Dhjþ1 ðkÞ for all j ¼ 1; . . . n 1. Thus, the queue that minimizes the aggregate cost requires that, rjþ1 ðhÞ 6 rj ðhÞ for all j ¼ 1; . . . ; n 1. In other words, rj ðhÞ ¼ ½ðn þ 1 jÞ=mþ for all j 2 N. Now consider profiles hðSÞ ¼ ðh1 ðSÞ; . . . ; hn ðSÞÞ where hj ðSÞ ¼ hj if j 62 S and hj ðSÞ ¼ h0j if j 2 S. Observe that from the arguments applied to find the efficient queue in state h it follows that if j; l 62 S and j < l, then rl ðhðSÞÞ 6 rj ðhðSÞÞ. Again, given any S N and S 6¼ /, rj ðhðSÞÞ 6 rs ðhðSÞÞ for all ðs; jÞ 2 S N S. This is because, for any given queue position k 2 f1; . . . ; M 1g, the incremental loss of any individual j 62 S (that is, Dhj ðkÞ) is strictly more than that of any individual s 2 S. Note that for all queues r ðhðSÞÞ, satisfying (1) rj ðhðSÞÞ 6 rs ðhðSÞÞ for all ðs; jÞ 2 S N S and (2) rl ðhðSÞÞ 6 rj ðhðSÞÞ for all ðj; lÞ 2 N S N S, j < l, are efficientPsince the cost of all individuals s 2 S are identical. jSj We now consider the sum SN ð1Þ CðhðSÞÞ. We break this sum in three parts in the following way: (i) P P P P P jSj jb Sj je Sj b ð1Þ h1 ðr1 ðh1 ; ^ ~ SN;j2 S^ ð1Þ hj ðrj ðhð S ÞÞÞ and (iii) SNf1g j6¼1 j2N SN;j62S ð1Þ hj ðrj ðhðSÞÞÞ, (ii) h1 ð Se ÞÞÞ. We consider each of these parts separately in the next three paragraphs.
3
Here jX j denotes the cardinality of X .
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We first consider part (i). Consider an individual j 2 N with j 6¼ 1. Let Pj ¼ fp : p > jg be the set of individuals with the higher ranking index than j. Consider all sets S such that j 62 S and there are x number of individuals from the set Pj and jSj x number of individuals from the set N fPj [ jg. The queue position of individual j for all such S is rj ðhðSÞÞ ¼ ½ðn þ 1 j xÞ=mþ . By collecting all such sets S, that is, P jSj by considering the coefficient of the term hj ð½ðn þ 1 j xÞ=mþ Þ, in the sum SN ð1Þ CðhðSÞÞ we get P x nj jT j ð1Þ T NfPj [jg ð1Þ . Note that x j1 X X jT j r j1 j1 ¼ ð1 þ ð1ÞÞ ¼ 0 since j 6¼ 1: ð1Þ ¼ ð1Þ r r¼0 T NfPj [jg
Therefore, of a term hj ð½ðn þ 1 j xÞ=mþ Þ is zero for all jð6¼ 1Þ and for all xð 6 jPj jÞ. Thus, P the coefficient P jSj ð1Þ h ðr j j ðhðSÞÞÞ ¼ 0. j6¼1 SN;j62S We now consider part (ii). Observe that by adding the cost of an individual j 2 N for all b S ð NÞ such that j 2 b S , we get n X X ^ n1 g ¼ ð1 þ ð1ÞÞn1 g ¼ 0: ð1ÞjSj g ¼ ð1Þs s 1 ^ s¼1 j2SN
P P ^ jSj b Therefore, j2N SN;j2 ^ S^ ð1Þ hj ðrj ðhð S ÞÞÞ ¼ 0. P jSj Since the sums in (i) and (ii) are both zero, it follows that the sum SN ð1Þ CðhðSÞÞ is equal to the P P ~ jSj jSj ð1Þ h1 ðr1 ðh1 ; h1 ð e S ÞÞÞ. For individual 1, with type h1 , sum in (iii), that is SN ð1Þ CðhðSÞÞ ¼ SNf1g ~ e e e we get r1 ðh1 ; h1 ð S ÞÞ ¼ ½ðn j S jÞ=mþ for all S N f1g. Therefore, 3 2 M X X X ~ n 1 jSj x 5h1 ðkÞ: 4 ð1Þ h1 ðr1 ðh1 ; h1 ð e S ÞÞÞ ¼ ð 1Þ ð3:5Þ x ~ k¼1 SNf1g
x:½ðnxÞ=mþ ¼k
P Simplifying condition (3.5) and then by applying Lemma 3.1 we get M k¼1 aðk; n; mÞh1 ðkÞ ¼ 0. Since the selection of individual 1, for the above construction, was arbitrary, the result follows. h Note that the GCP is an additional restriction on the class of domains H satisfying Assumptions 1–3. Let b Hð HÞ be the class of domains satisfying the GCP and Assumptions 1–3. Proposition 3.1. If the number of machines m is even or if the number of queue positions M ¼ ½n=mþ ¼ 2, then b there is no first best implementable multiple machine queueing problem C ¼ hN; m; Hi such that H 2 H. Proof. If the number of machines m is even then for all k 2 f1; . . . ; M 1g, ð1Þkm1 ¼ 1 because km is also even. From Assumption 1, it follows that for all hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H, Dhj ðkÞ P 0 for all km1 k 2 f1; . . . ; M 1g. By substituting ð1Þ ¼ 1 and Dhj ðkÞ P 0 in (3.4) we get Dhj ðkÞ ¼ 0 for all k 2 f1; . . . ; M 1g. Hence, the GCP implies that hj ð1Þ ¼ ¼ hj ðMÞ for all hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H and we have a violation of Assumption 3. If M ¼ 2, then GCP implies að1; n; mÞhj ð1Þ þ að2; n; mÞhj ð2Þ ¼ 0 for all hj ¼ ðhj ð1Þ; hj ð2ÞÞ 2 H and condition (3.3) implies að1; n; mÞ þ að2; n; mÞ ¼ 0. Therefore, the GCP implies that hj ð1Þ ¼ hj ð2Þ for all hj ¼ ðhj ð1Þ; hj ð2ÞÞ 2 H. This again is a violation of Assumption 3. h So far we have imposed restrictions on individual preferences. The next property is a restriction on group preferences.
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Definition 3.6. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the General Independence Property (or GIP) if for all pairs ðj; lÞ 2 N N, j 6¼ l, all pairs of cost vectors ðhj ; hl Þ 2 H H are such that one of the following two conditions holds: 1. Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; 2; . . . ; M 1g 2. Dhj ðkÞ 6 Dhl ðkÞ for all k 2 f1; 2; . . . ; M 1g. The GIP for a multiple machine queueing problem C implies that if for any pair of individuals ðj; lÞ 2 N N, j 6¼ l, the respective cost vectors hj ¼ ðhj ð1Þ; . . . ; hj ðMÞÞ 2 H and hl ¼ ðhl ð1Þ; . . . ; hl ðMÞÞ 2 H are such that there exists a k 2 f1; . . . ; M 1g such that Dhj ðkÞ > Dhl ðkÞ, then Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g. The relationship between the GCP and the GIP is captured in the next proposition. Proposition 3.2. For C ¼ hN; m; Hi with M ¼ 3, GCP ) GIP. Proof. Consider any multiple machine queueing problem C ¼ hN; m; Hi such that M ¼ 3 and H satisfies the GCP. Consider hj ¼ ðhj ð1Þ; hj ð2Þ; hj ð3ÞÞ and hl ¼ ðhl ð1Þ; hl ð2Þ; hl ð3ÞÞ for individuals j and l respectively. Since H satisfies the GCP, from condition (3.4) we know that zð1; n; mÞDhi ð1Þ þ zð2; n; mÞDhi ð2Þ ¼ 0 for all zð2;n;mÞ i 2 fj; lg. Therefore, for all i 2 fj; lg, Dhi ð1Þ ¼ P . Dhi ð2Þ where P ¼ zð1;n;mÞ > 0. Thus, Dhj ð1Þ < ð>ÞDhl ð1Þ if and only if Dhj ð2Þ < ð>ÞDhl ð2Þ. h b ¼ hN ¼ f1; . . . ; 10g; m ¼ 3; Hi of Example 3.1. Here the number of queue positions M ¼ 4 Consider C h ¼ and að10; 3Þ ¼ ð28; 84; 57; 1Þ. Consider individuals j and l P4l with costshj ¼ ð1; 3; 4; 4Þ 2 H and ð1; 2; 3; 31Þ 2 H respectively. Observe that for i 2 fj; lg, aðk; 10; 3Þ h ðkÞ ¼ 0. However, D h ð1Þ ¼ i j k¼1 b 2 > D hl ð1Þ ¼ 1 and D hj ð3Þ ¼ 0 < D hl ð3Þ ¼ 28. Therefore, the GIP is not satisfied. Hence for C, GCP ; GIP. Let C represent the class of multiple machine queueing problems with odd number of machines and with at least three queue positions and satisfying the GCP, the GIP and Assumptions 1–3. Therefore, a multiple e belongs to C if Hð e b is a domain satisfying the GCP, the GIP machine queueing problem C ¼ hN; m; Hi HÞ and Assumptions 1–3. We now derive the efficient rule for any multiple machine queueing problem e 2 C. Before doing that we give some more relevant notations and definitions. ConC ¼ hN; m; Hi e For a state h 2 H e n , define Qj ðhÞ ¼ ½l 2 N fjg such that either 9k 2 f1; . . . ; M 1g sider C ¼ hN; m; HÞ. and fDhl ðkÞ > Dhj ðkÞg or 8k 2 f1; 2; . . . ; M 1g, fDhl ðkÞ ¼ Dhj ðkÞ and l < jg. Let Rj ðhÞ ¼ 1 þ jQj ðhÞjð2 f1; . . . ; ngÞ be the rank of individual j in state h. Observe that the way we have specified the ranking, there is no possibility of a tie in the ranking of different individuals in any given state. Therefore, Rj ðhÞ measures the rank of individual j in state h ¼ ðh1 ; . . . ; hn Þ. Using this definition of ranking we state and prove an efficient e 2 C. rule (that is, a single-valued selection from the efficiency correspondence) for C ¼ hN; m; Hi e 2 C. For all h 2 H e n , let r ðhÞ be the queue such that r ðhÞ ¼ Proposition 3.3. Consider C ¼ hN; m; Hi j ½Rj ðhÞ=mþ for all j 2 N. The queue r ðhÞ ¼ ðr1 ðhÞ; . . . ; rn ðhÞÞ is efficient. e n , the queue r ðhÞ ¼ ðr ðhÞ; . . . ; r ðhÞÞ, defined in Proposition 3.3, is For any state h ¼ ðh1 ; . . . ; hn Þ 2 H 1 n obtained in the following way: the individuals having rank 1 to m get the first position, the individuals having rank m þ 1 to 2m get the second position and this goes on till the individuals having rank mðM 2Þ þ 1 to mðM 1Þ get the ðM 1Þth position and finally the remaining individual(s) having rank higher than mðM 1Þ get the Mth position.
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e n and any queue r. We define a sequence sðrÞ Proof of Proposition 3.3. Consider a state h ¼ ðh1 ; . . . ; hn Þ 2 H as an ordering of the jobs obtained from r by placing the jobs in the first position up to first m slots, the jobs in the second position in the following m slots and so on. 4 For a sequence sðrÞ, let j; l 2 N be neighbors with l in front of j, that is sr ðjÞ ¼ sr ðlÞ þ 1. If l and j switches their position (by keeping the position of all other jobs in the sequence unchanged), then the total cost changes by the amount Dðr; r0 ; hÞ Cðr; hÞ Cðr0 ; hÞ. We can have two types of switch-one that leaves the queue positions unaltered and one where the queue position of only the neighbors in question (that is, l and j) change. Therefore, 0 if r ¼ r0 ði:e: rj ¼ rl ¼ k for some k 2 f1; . . . ; MgÞ; Dðr; r0 ; hÞ ¼ Dhj ðkÞ Dhl ðkÞ if r 6¼ r0 ði:e: rj ¼ rl þ 1 and rl ¼ k for some k 6¼ MÞ: e satisfies the GIP, from Dðr; r0 ; hÞ it follows that switching l and j is weakly cost Given that the domain H reducing if and only if Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g. Observe that the queue r ðhÞ in state h is such that if Rj ðhÞ < Rl ðhÞ then (a) Dhj ðkÞ P Dhl ðkÞ for all k 2 f1; . . . ; M 1g and (b) rj ðhÞ 6 rl ðhÞ. Hence for sðr ðhÞÞ, it is not possible to find neighbors for whom switching is cost reducing. This means that r ðhÞ e was arbitrary, the result follows. h is efficient in state h. Since the selection of h 2 H Proposition 3.3 shows that finding an efficient queue is quite transparent if C 2 C. Observe that for C 2 C, the relative ranking of any two individuals ðj; lÞ, for some given costs hj and hl respectively, is independent of the costs announced by all other individuals. Formally, if in state h ¼ ðh1 ; . . . ; hn Þ, e n2 . Rj ðhÞ > Rl ðhÞ for some ðj; lÞ 2 N N, j 6¼ l, then Rj ðhj ; hl ; h0jl Þ > Rl ðhj ; hl ; h0jl Þ for all h0jl 2 H Hence, what determines the efficient queue is the ranking that each individual gets in a given state. We now argue that if one individual is eliminated from the queue then the relative ranking of all other individuals remain unchanged. Before doing that we introduce some more relevant notations and definitions that captures the idea of elimination of an individual from the queue in any given state. Define M 0 ½n1 to be m þ the number of queue positions that remains in a multiple machine queueing problem with n jobs and m machines after one individual (and hence one job) is eliminated from the queue. Observe that M 0 ¼ M 1 if n ¼ rm þ 1 where r ¼ 2; 3; . . . and M 0 ¼ M otherwise. Using the idea of ranking of individuals for C 2 C, we define Qj ðhl Þ ¼ ½i 2 N fj; lg such that either 9k 2 f1; . . . ; M 0 1g and fDhi ðkÞ > Dhj ðkÞg or 8k 2 f1; 2; . . . ; M 0 1g, fDhi ðkÞ ¼ Dhj ðkÞ and i < jg. Let Rj ðhl Þ ¼ 1 þ jQj ðhl Þjð2 f1; . . . ; n 1gÞ. Therefore, in a state h, Rj ðhl Þ measures the rank of individual j in state h by eliminating the cost vector hl of individual lð6¼ jÞ. Remark 3.2. Consider any multiple machine queueing problem C satisfying the GIP but not the GCP such that M 0 ¼ M 1. Consider a state h ¼ ðh1 ; . . . ; hn Þ such that Dh1 ðkÞ ¼ Dh2 ðkÞ for all k 2 f1; . . . ; M 2g and Dh1 ðM 1Þ < Dh2 ðM 1Þ. Moreover, assume that R1 ðhÞ ¼ n and R2 ðhÞ ¼ n 1. Hence R1 ðhÞ > R2 ðhÞ. Observe, that for all i 2 N f1; 2g, R1 ðhi Þ ¼ n 2 and R2 ðhi Þ ¼ n 1 because Dh1 ðkÞ ¼ Dh2 ðkÞ for all k 2 f1; . . . ; M 2g and 1 < 2. Therefore, R1 ðhÞ > R2 ðhÞ and for all i 2 N f1; 2g, R1 ðhi Þ < R2 ðhi Þ. Therefore, for any multiple machine queueing problem C satisfying the GIP but not the GCP, if M 0 ¼ M 1, then the above construction shows that there exist cost vectors for which the relative ranking of a pair of individuals can change if an individual outside the pair under consideration is eliminated from the queue. It is not hard to verify that the construction specified above is the only type of construction that can lead to such a rank reversal in a multiple machine queueing problem C satisfying the GIP when an individual is eliminated. Moreover, such a rank reversal can take place only if M 0 ¼ M 1 and C fails to
4 For example, if there are only 2 machines and 4 jobs and jobs 1 and 2 are in front of machine 1 and jobs 3 and 4 are in front of machine 2 (i.e., if r ¼ ðr1 ¼ 1; r2 ¼ 2; r3 ¼ 1; r4 ¼ 2Þ), then sðrÞ ¼ ð1; 3; 2; 4Þ.
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satisfy the GCP. If, a multiple machine queueing problem C satisfies the GIP and the GCP and if Dhj ðkÞ ¼ Dhl ðkÞ for all k 2 f1; . . . ; M 2g then the GCP implies that Dhj ðM 1Þ ¼ Dhl ðM 1Þ. Therefore, the construction that led to rank reversal is not possible for a multiple machine queueing problem that satisfies the GCP. Hence, if a multiple machine queueing problem C satisfies both the GCP and the GIP, then the relative ranking of any pair of individuals with any given pair of cost vectors remain unchanged if some other individual is eliminated from the queue. More formally, if C 2 C, then we obtain the following e n, relationship between Rj ðhÞ and Rj ðhl Þ. For allðj; lÞ 2 N N, j 6¼ l and for all h 2 H Rj ðhÞ if Rj ðhÞ < Rl ðhÞ; Rj ðhl Þ ¼ Rj ðhÞ 1 if Rj ðhÞ > Rl ðhÞ: Using Remark 3.2 we derive the sufficiency condition under which a multiple machine queueing problem C 2 C is first best implementable. Theorem 3.2. A multiple machine queueing problem C 2 C is first best implementable. We first state and prove a lemma that will be used in proving Theorem 3.2. Lemma 3.2. A multiple machine queueing problem C ¼ hN; m; Hi satisfies the GCP, if and only if for all cost vector hj 2 H, there exists a unique 1 ðn 1Þ vector Hj ¼ fhj ð1Þ; . . . ; hj ðn 1Þg such that for all p 2 f1; . . . ; ng, hj ð½p=mþ Þ ¼ ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ: Proof. Consider a hj 2 H for individual j 2 N that satisfies Hj ¼ fhj ð1Þ; . . . ; hj ðn 1Þg such that for all p 2 f1; . . . ; n 1g, hj ðpÞ ¼
p X
ð1Þpr
r¼1
ð3:6Þ PM
k¼1
aðk; n; mÞhj ðkÞ ¼ 0. Define a vector
ðp 1Þ!ðn p 1Þ! hj ð½r=mþ Þ: ðr 1Þ!ðn rÞ!
ð3:7Þ
We prove Lemma 3.2 in two steps. In the first step it is proved, using (3.7), that for all p 2 f1; . . . ; n 1g, condition (3.6) holds. In the next step we prove that for p ¼ n, condition (3.6) holds only if C satisfies the GCP. ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ ¼ ðn pÞ
p X
ð1Þ
r¼1
þ ðp 1Þ
p1 X r¼1
¼
pr
ðp 1Þ!ðn p 1Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
ð1Þpr1
ðp 2Þ!ðn pÞ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
p1 n o X pr pr1 ðp 1Þ!ðn pÞ! hj ð½r=mþ Þ þ hj ð½p=mþ Þ ð 1Þ þ ð 1Þ ðr 1Þ!ðn rÞ! r¼1
¼ hj ð½p=mþ Þ
ðbecause ð1Þpr þ ð1Þpr1 ¼ 0Þ:
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For p ¼ n, ðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þ ¼ ðn 1Þhj ðn 1Þ ¼ ðn 1Þ
n1 X
n1r
ð1Þ
r¼1 n1 X
ðn 2Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ!
ðn 1Þ! hj ð½r=mþ Þ ðr 1Þ!ðn rÞ! r¼1 n1 X n1r n 1 ¼ ð1Þ hj ð½r=mþ Þ r1 r¼1 ¼
ð1Þ
n2
¼ ð1Þ
n1r
M X
aðk; n; mÞhj ðkÞ þ hj ðMÞ
ðfrom the GCPÞ ¼ hj ðMÞ:
k¼1
Therefore, the last step not only proves the necessity of the GCP but also guarantees that for hj , the 1 ðn 1Þ vector Hj is unique. PM We now prove the other part of Lemma 3.2. Observe that the sum k¼1 aðk; n; mÞhj ðkÞ ¼ Pn p1 n 1 h ð½p=mþ Þ. Therefore, p¼1 ð1Þ p1 j n X p1 n 1 h ð½p=mþ Þ ð1Þ p1 j p¼1 n X p1 n 1 ¼ ð1Þ fðn pÞhj ðpÞ þ ðp 1Þhj ðp 1Þg p1 p¼1 ( ) n1 n X X p1 n 2 p1 n 2 h ðpÞ þ h ðp 1Þ ¼ 0: ð 1Þ ð 1Þ ¼ ðn 1Þ p1 j p2 j p¼1 p¼2 Lemma 3.2 gives rise to a particular type of separability (as given by condition (3.6)) that will be used in e 2 C. deriving the explicit form of the transfer that first best implements any C ¼ hN; m; Hi P e n for individual j 2 N. From the Proof of Theorem 3.2. Consider the sum l6¼j hj ðRj ðhl ÞÞ in state h 2 H GIP and Remark 3.2 we get X hj ðRj ðhl ÞÞ ¼ ðn Rj ðhÞÞhj ðRj ðhÞÞ þ ðRj ðhÞ 1Þhj ðRj ðhÞ 1Þ l6¼j
¼ hj ð½Rj ðhÞ=mþ Þ ðfrom condition ð3:6Þ in Lemma 3:2Þ: We consider a Groves–Clarke mechanism c M ¼ hr ; ^ti where the term independent of j’s announcement P is ^cj ðhj Þ ¼ ðn 1Þ l6¼j hl ðRl ðhj ÞÞ. Then it follows that ( ) X XX X X X ^cj ðhj Þ ¼ ðn 1Þ hl ðRl ðhj ÞÞ ¼ ðn 1Þ hj ðRj ðhl ÞÞ ¼ ðn 1Þ hj ð½Rj ðhÞ=mþ Þ j2N
j2N
l6¼j
j2N
l6¼j
j2N
¼ ðn 1ÞCðhÞ: Observe that the last step follows from the efficiency rule of Proposition 3.3. The last step implies that for all e n , the sum of transfers P ^tj ðhÞ ¼ ðn 1ÞCðhÞ þ P ^cj ðhj Þ ¼ 0. h h2H j2N j2N Observe that from Theorems 3.1 and 3.2 and from Proposition 3.2 it follows that a multiple machine queueing problem satisfying A1–A3 and with three machines is first best implementable if and only if it satisfies the GCP.
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4. Separable cost multiple machine queueing problems In this section we first define a class of multiple machine queueing problems with separable cost and then verify under what conditions these problems are first best implementable. The following conditions describe a typical separable cost multiple machine queueing problem. 1. Define a real number h > 0. Given h > 0, the cost parameter or type of an individual belongs to the interval ½0; h. Moreover, there exists a function f : f1; . . . ; Mg ! R such that f ðkÞ P f ðk 1Þ for all k 2 f2; . . . ; Mg and f ðk Þ > f ðk 1Þ for at least one k 2 f2; . . . ; Mg. 2. hj ðkÞ ¼ f ðkÞhj for all j 2 N, for all k 2 f1; 2; . . . ; Mg and for all hj 2 ½0; h. For a separable cost multiple machine queueing problem, the cost of each individual for each position is multiplicatively separated into two parts. The first part is a function f that depends on the queue position. Observe that the functional form f is assumed to be identical for all individuals j 2 N. Moreover, we assume that f is common knowledge. The second part which is a non-negative number hj represents the type (or cost parameter) of an individual that belong to the interval ½0; h. In this set up a type vector of individual j 2 N is given by hj ¼ ðhj ð1Þ ¼ f ð1Þhj ; . . . ; hj ðMÞ ¼ f ðMÞhj Þ. Therefore, from now on we will write hj as the cost parameter or type of an individual. The cost parameter hj 2 ½0; h for all j 2 N is private information. Finally, h ¼ ðh1 ; . . . ; hn Þ 2 ½0; hn represents a state of the world or a profile. It is important to observe that the domain H ðf ; ½0; hÞ of any separable cost multiple machine queueing problem satisfies Assumptions 1–3. In this framework the set of individuals N, mð> 1Þ number of machines and H ðf ; ½0; hÞ define the b ¼ hN; m; Hi. We will completely characterize the class multiple machine separable cost queueing problem C of first best implementable multiple machine separable cost queueing problems. We start by showing that these problems satisfy the GIP. b ¼ hN; m; Hi satisfies the GIP. Proposition 4.4. A multiple machine separable cost queueing problem C Proof. Consider any pair ðj; lÞ 2 N N, j 6¼ l, with cost parameters hj and hl respectively. It is obvious that either hj P hl or hj 6 hl . Since f ðk þ 1Þ P f ðkÞ for all k 2 f1; . . . ; M 1g, it is also obvious that if hj P hl , then ff ðk þ 1Þ f ðkÞghj P ff ðk þ 1Þ f ðkÞghl for all k 2 f1; . . . ; M 1g. Therefore, Dhj ðkÞ ¼ ff ðk þ 1Þ f ðkÞghj P Dhl ðkÞ ¼ ff ðk þ 1Þ f ðkÞghl for all k 2 f1; . . . ; M 1g. Similarly, if hj 6 hl , then Dhj ðkÞ 6 Dhl ðkÞ b ¼ hN; m; Hi satisfies the GIP. h for all k 2 f1; . . . ; M 1g. Thus, it follows that C The next two remarks follow trivially from the discussion of the GCP in the previous section. b ¼ hN; m; Hi satisfies the GCP if Remark 4.3. A multiple machine separable cost queueing problem C M X aðk; n; mÞf ðkÞ ¼ 0: ð4:8Þ k¼1
Using Df ðkÞ ¼ f ðk þ 1Þ f ðkÞ and simplifying equation (4.8) we get M 1 X zðk; n; mÞDf ðkÞ ¼ 0; k¼1 Pk n2 km1 where zðk; n; mÞ ¼ r¼1 aðr; n; mÞ ¼ ð1Þ . km 1
ð4:9Þ
b ¼ hN; m; Hi satisfies the GCP, if and only if there exists Remark 4.4. From condition (3.6) it follows that C a unique vector H ¼ fhð1Þ; . . . ; hðn 1Þg such that for all p 2 f1; . . . ; ng,
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f ð½p=mþ Þ ¼ ðn pÞhðpÞ þ ðp 1Þhðp 1Þ; where hðpÞ ¼
Pp
r¼1
ð1Þ
ð4:10Þ
pr ðp1Þ!ðnp1Þ! f ð½r=mþ Þ. ðr1Þ!ðnrÞ!
The next result completely characterizes the class of first best implementable multiple machine separable cost queueing problems. b ¼ hN; m; Hi is first best impleProposition 4.5. A multiple machine separable cost queueing problem C mentable if and only if the cost function satisfies the GCP. The necessity part of Proposition 3.1 is similar to that of Theorem 3.1 and the sufficiency part is similar to Theorem 3.2. Therefore, we omit the proof of this proposition. It is easy to verify that Proposition 3.1 is also true for separable cost queueing problems. Therefore, all multiple machine separable cost queueing problems with either (1) even number of machines or (2) two queue positions are not first best implementable. Let CðSÞð CÞ be the class of multiple machine separable cost queueing problems where m is odd and n > 2m and let C ð CðSÞÞ be the class of first best implementable multiple machine separable cost queueing problems. The next proposition proves the existence of C . Proposition 4.6. There exists C 2 CðSÞ such that C 2 C . b 0 ¼ hN; m; Ho ðf 0 ; ½0; hÞi 2 CðSÞ with odd number of queue positions M and with f 0 of Proof. Consider C n2 0 0 0 0 the following form: f ð1Þ ¼ c P 0 and Df ðkÞ ¼ f ðk þ 1Þ f ðkÞ ¼ 1 for all k 2 f1; . . . ; km 1 0 b ¼ hN; m; ðf 0 ; ½0; hÞi 2 C by showing that f 0 satisfies condition (4.9). ObM 1g. We will prove that C km1 for all k 2 f1; . . . ; M 1g. Therefore, by substituting serve first that zðk; n; mÞDf 0 ðkÞ ¼ ð1Þ PM1 km1 km1 in the left-hand side of condition (4.9) we get k¼1 ð1Þ . Since both m and zðk; n; mÞDf 0 ðkÞ ¼ ð1Þ PM1 km1 0 0 b ¼ 0. Thus, C ¼ hN; m; ðf ; ½0; hÞi 2 C . M are odd, it is obvious that, k¼1 ð1Þ b e ¼ hN; m; He ðf e ; ½0; hÞi 2 CðSÞ with even number of queue positions M and Similarly, consider C n2 for all k 2 with f e of the following form: f e ð1Þ ¼ c P 0 and Df e ðkÞ ¼ f e ðk þ 1Þ f e ðkÞ ¼ 1 km 1 km1 f1; . . . ; M 2g and f e ðM 1Þ ¼ f e ðMÞ (that is, Df e ðM 1Þ ¼ 0). Observe that zðk; n; mÞDf e ðkÞ ¼ ð1Þ for all k 2 f1; . . . ; M 2g and zðM 1; n; mÞDf e ðM 1Þ ¼ 0. Therefore, by substituting zðk; n; mÞDf e ðkÞ for PM2 km1 all k 2 f1; . . . ; M 1g in the left hand side of condition (4.9) we get . Here k¼1 ð1Þ PM2 km1 b e 2 C . h ¼ 0 because m is odd and M is even. Thus, C k¼1 ð1Þ Observe that given C CðSÞ C, it follows from Proposition 4.6 that there exist first best implementable multiple machine queueing problems in C. We conclude this section with an important observation. Observation 1. Given the co-efficient vector aðn; mÞ, it follows that if m is odd, M ¼ 2q þ 1 and n ¼ m ð2q þ 1Þ (where q 2 f1; 2; . . .g) then aðk; P n; mÞ ¼ að2q þ 2 k; n; mÞ for all k 2 f1; . . . ; qg. Using q this result and by substituting aðq þ 1; n; mÞ ¼ 2 k¼1 aðk; n; mÞ in Eq. (3.2) we get q X
aðk; n; mÞfhj ðkÞ þ hj ð2q þ 2 kÞ 2hj ðq þ 1Þg ¼ 0:
ð4:11Þ
k¼1
Observe that if hj ðkÞ ¼ khj for all k 2 f1; . . . ; Mg, then condition (4.11) holds. Thus, if m is odd, n ¼ mM, M is also odd and f l ðkÞ ¼ k for all k, then Cl ¼ hN; m; Hl ðf l ; ½0; hÞi 2 C .
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5. Concluding remarks We have obtained the following results regarding the first best implementability of the class of multiple machine queueing problems with domains satisfying Assumptions 1–3. 1. A multiple machine queueing problem is first best implementable only if it satisfies the GCP. 2. If the number of machines is even or if there are only two queue positions, then a non-trivial multiple machine queueing problem fails to satisfy the GCP and hence is not first best implementable. 3. If the number of machines m is odd and M ¼ ½n=mþ ¼ 3 then a multiple machine queueing problem is first best implementable if and only if it satisfies the GCP. 4. If the number of machines m is odd and M ¼ ½n=mþ P 4 then a multiple machine queueing problem is first best implementable if it satisfies the GCP and the GIP. 5. For all n > 2m such that m is odd, there exists a non-trivial cost function for which a multiple machine queueing problem is first best implementable. 6. Finally, if m is odd, the number of queue positions M ¼ ½n=mþ > 2 is also odd and n ¼ m M then a multiple machine queueing problem with linear cost function is first best implementable. Thus, first best implementability of a multiple machine queueing problem depends heavily on the number of machines and on the number of jobs.
Acknowledgements The author is grateful to Arunava Sen, Jeroen Suijs and two anonymous referees for their invaluable advice. The author is thankful to Parikshit Ghosh and Suryapratim Banerjee for helpful discussions. The author is also thankful to the seminar participants at the European Summer Symposium in Economic Theory 2003 (held in Gerzensee, Switzerland). The author gratefully acknowledges the financial support from the Indian Statistical Institute and from the Deutsche Forschungsgemeinschaft Graduiertenkolleg 629 at the University of Bonn. The author is solely responsible for the errors that may still remain.
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