Secure Implementation in Queueing Problems

This paper studies secure implementability (T. Saijo, T. Sjöström and T. Yamato, “Secure Implementation,” Theoretical Economics, Vol. 2, No. 3, 2007, pp. 203-229) in queueing problems. Our main result shows that the social choice function satisfies strategy-proofness and strong non-bossiness (Z. Ritz, “Restricted Domains, Arrow-Social Welfare Functions and Noncorruptible and Non-Manipulable Social Choice Correspondences: The Case of Private Alternatives,” Mathematical Social Science, Vol. 4, No. 2, 1983, pp. 155-179), both of which are necessary for secure implementation, if and only if it is constant on the domains that satisfy weak indifference introduced in this paper. Weak indifference is weaker than minimal richness (Y. Fujinaka and T. Wakayama, “Secure Implementation in Economies with Indivisible Objects and Money,” Economics Letters, Vol. 100, No. 1, 2008, pp. 91-95). Our main result illustrates that secure implementation is too difficult in queueing problems since many reasonable domains satisfy weak indifference, for example, convex domains.


Introduction
In this paper, we consider queueing problems of allocating positions in a queue to agents, each of whom has a constant unit waiting cost, with monetary transfers.Examples of such problems are the use of large-scaled experimental installations, event sites, and so forth 1 .
Strategy-proofness is a standard property for nonmanipulability: The truthful revelation is a weakly dominant strategy for each agent.However, the strategyproof mechanism might have a Nash equilibrium which induces a non-optimal outcome.This problem is solved by secure implementation (Saijo, et al. [2]), that is, double implementation in dominant strategy equilibria and Nash equilibria 2 .Previous studies illustrate how difficult it is to find desirable and securely implementable social choice functions: Voting environments (Saijo, et al. [2]; Berga and Moreno [4]), public good economies (Saijo,et al. [2]; Nishizaki [5]), pure exchange economies (Mizukami and Wakayama [6]; Nishizaki [7]), the problems of providing a divisible and private good with monetary transfers (Saijo,et al. [2]; Kumar [8]), the problems of allocating indivisible and private goods with monetary transfers (Fujinaka and Wakayama [9]), Shapley-Scarf housing markets (Fujinaka and Waka-yama [10]), and allotment economies with single-peaked preferences (Bochet and Sakai [11]).
This paper is most closely related to the one written by Fujinaka and Wakayama [9].They show a constancy result on secure implementation when the domain satisfies minimal richness (Fujinaka and Wakayama [9]).Our model is a special case of their one and have many reasonable domains which do not satisfy minimal richness.On the basis of this fact, we study the possibility of secure implementation in queueing problems.Unfortunately, our main result shows that only constant social choice functions satisfy strategy-proofness and strong non-bossiness (Ritz [12]), both of which are necessary for secure implementation, on the domains satisfy weak indifference, which is weaker than minimal richness, introduced in this paper.
This paper is organized according to the following sections.In Section 2, we introduce our model, properties of social choice functions, and domain-richness conditions.We show our results in Section 3. Section 4 concludes this paper.


  , 1 For queueing problems, see Suijs [1] and so forth. 2 This concept is considered to be a benchmark for constructing mechanisms which work well in laboratories.See Cason, et al. [3] for experimental results. .
where is a monetary transfer for agent .Let be a profile of monetary transfers and be a profile of consumption bundles, called an allocation.
 be the set of feasible allocations.
For each , let be a unit waiting cost for agent and i C  be a set of unit waiting costs for agent .For each i i   i I  , let i be the utility function for agent such that for each and each be the domain and i be a profile of unit waiting costs.For each be a profile of unit waiting costs for agents other than agent .
be a social choice function.For each , let be the allocation associated with the social choice function f at the profile of unit waiting costs and be the consumption bundle for agent in the allocation .[2] show that strategy-proofness and strong non-bossiness are necessary for secure implementation.
Definition 1 The social choice function f satisfies strategy-proofness if and only if for each ,

Definition 2
The social choice function f satisfies strong non-bossiness if and only if for each , Fujinaka and Wakayama [9] show a constancy result on secure implementation when the domain satisfies minimal richness.
Definition 3 The domain satisfies minimal richness if and only if for each

and each if i i
, then there exists implies that condition 2) in Definition 3 does not old.
Our main result implies a constancy result on secu plementation when the domain satisfies weak indifference which is weaker than minimal richness.
Definition 4 The domain C satisfies wea rence if and only if for each i I  , each , Remark 1 In our model, weak indifference is equiva

Results
lent to convexity 3 .
For simplicity of notation, let   and each i I  .

Preliminary Results
that the social choice func-

onetary transfer de d each
In this subsection, we assume tion f satisfies strategy-proofness.
Le a 1 shows that each agent's m mm pends on her position in the queue given unit waiting costs for other agents.Since the proof is similar to Fujinaka and Wakayama [9], it is omitted.
Lemma 2 shows that if su there exists a unit waiting cost ch that some two different consumption bundles are indifferent in terms of utility level, then the position associated with the unit waiting cost is in between the two positions.In Lemma 2, we use the following notation: . Suppose, radiction, that there e By Lemma 1, this implies , this implies tha We consider the case of j j     .In th Le is case, by mma 2, we have , then, by Equatio nd st onbossiness, we have n (7) a rong n  .This is a contradiction.Therefore, we know .
By applying the above argument to the left inequality repeatedly, we can find , By the same argument stated above, we also have where is a profile of unit waiting costs for agents 1,2 c  an a other th gents 1 and 2. By Equations ( 8) and ( 9), we have By sequentially replacing by j c j c for each 1, 2 j  in this manner, we finally prove . ■ rem Remark 2 The above theo does not depen fin d on the iteness of the number of positions, which is used to prove Claim 3 in Proposition 1 of Fujinaka and Wakayama [9].
Obviously, constant social choice functions are securely implementable.Therefore, by bringing the above theorem together with a characterization of securely implementable social choice functions by Saijo et al. [2], we have the following constancy result on secure implementation.

Corollary 1 Suppose that the domain satisfies weak indifference. The social choice function is securely implementable if and only if it is constant.
Remark 3 In our model, Maskin monotonicity is not st 6 4 Note that the equality does not hold.If it holds, then we have ronger than strategy-proofness .This relationship implies that our main result is established by secure im- plementability but not by Nash implementability.Remark 4 Saijo [14] shows the following constancy re ch uch that  sult on "Nash" implementation: The social choice function satisfies Maskin monotonicity and dual dominance (Saijo [14]) if and only if it satisfies constancy.In line with such domination, Fujinaka and Wakayama [9] show the following constancy result on "secure" implementation: The securely implementable social choice function satisfies non-dominance (Fujinaka and Wakayama [9]) if and only if it satisfies constancy.Note that nondominance is weaker than dual dominance 7 .In our model, similar to the relationship between minimal richness and weak indifference, we have a constancy result on secure implementation by a weaker condition than non-dominance as follows: for each for each each and each i I  ,

Conclusion
secure implementability in queueing by Equations (1) and(2).This implies that f should be a correspondence since we consider the case of i i    .
The following example shows that many reasonable