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 forth¹.

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 equilibria². 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 Wakayama [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.

Let be a set of agents. Let be a queue, where, for each, is the position for agent in the queue and for each with For each, let be a consumption bundle for agent, where is a monetary transfer for agent. Let be a profile of monetary transfers and be a profile of consumption bundles, called an allocation. Let

be the set of feasible allocations.

For each, let be a unit waiting cost for agent and be a set of unit waiting costs for agent. For each, let be the utility function for agent such that for each and each,

Let be the domain and be a profile of unit waiting costs. For each, let be a profile of unit waiting costs for agents other than agent.

Let be a social choice function. For each, let be the allocation associated with the social choice function at the profile of unit waiting costs and be the consumption bundle for agent in the allocation.

Saijo et al. [2] show that strategy-proofness and strong non-bossiness are necessary for secure implementation.

Definition 1 The social choice function satisfies strategy-proofness if and only if for each and each,

Definition 2 The social choice function satisfies strong non-bossiness if and only if for each and each, if

then

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, each, each, and each if , then there exists such that 1) and 2) for each.

The following example shows that many reasonable domains do not satisfy minimal richness in our model.

Example 1  Let and. Moreover, let. In this case, we have. Let be such that, that is,. This implies that condition 1) in Definition 3 holds. On the other hand, if, then for. This implies that condition 2) in Definition 3 does not hold.

Our main result implies a constancy result on secure implementation when the domain satisfies weak indifference which is weaker than minimal richness.

Definition 4 The domain satisfies weak indifference if and only if for each, each, each, and each, if, then there exists such that

Remark 1 In our model, weak indifference is equivalent to convexity³.

For simplicity of notation, let, , , , and, , , , for each and each.

This paper studies secure implementability in queueing problems. Fujinaka and Wakayama [9] show a constancy result on secure implementation. Our model is a special case of their one and have many reasonable domains which do not satisfy minimal richness. However, we have the same constancy result under less restrictive domain-richness conditions. On the other hand, it remains to show domain-richness conditions for the existence of non-constant securely implementable social choice functions. However, our main result implies that it is difficult to find such conditions that are reasonable in the economic sense.

This paper is based on my M.A. thesis presented to the Graduate School of Economics, Osaka University. The author would like to thank an anonymous referee, Tatsuyoshi Saijo, Masaki Aoyagi, Yuji Fujinaka, Kazuhiko Hashimoto, Shuhei Morimoto, Shinji Ohseto, Shigehiro Serizawa, Takuma Wakayama, and seminar participants at the 2008 Japanese Economic Association Spring Meeting, Tohoku University, for their helpful comments. The author is especially grateful to Tatsuyoshi Saijo, Yuji Fujinaka, and Takuma Wakayama for their valuable advices. The author acknowledges for the Global COE program of Osaka University for the financial supports. The responsibility for any errors that remain is entirely the author.