Consider a situation where agents have to choose one project among the set of multiple alternatives and at the same time they have to agree with the way of sharing the cost of the project that is actually developed. There are many examples of such situations from a location of public facility such as a public school, a disposal center, and a nuclear-related equipment to Joint venture development or R&D investment agreement among firms.

In the pioneering work of [1] , they formulated the mechanism that determines the project which will be developed and how the agents share the surplus from its development. In their mechanism called a multi-bidding mechanism, each individual is required to reveal payable bids for each project, which can be seen as his relative preference on the project, combined with the project announcement¹. They show that their mechanism has a Nash equilibrium and always chooses the efficient projects. However, their mechanism is slightly artificial because the preference of each agent on the project is embedded in the announcement of his bids, which makes the agents in question to the announcement of his favorite project. Recently, [2] modifies the multi-bidding mechanism to a one where the project announcement is removed. He shows that in his natural mechanism, the existence of Nash equilibrium is no longer guaranteed and the desirability of the mechanism is limited.

The purpose of this paper is to extend their analysis one step forward. We consider a situation where there is a cost to develop the project and agents have to agree with how to share the cost of the project. In [1] and [2] , the cost of the project is implicitly assumed to be zero or if it exists, agents are assumed to agree with the way of sharing the cost of project in advance of participating the mechanism. On the other hand, we consider how the agents share the cost of developing project, in addition to which project will be developed and how the agents share the surplus from its development. The introduction of the cost of projects needs a subtle rule of choosing a project because a project whose cost is not budgeted by the agents’ payments cannot be carried out. In our mechanism, each agent is required to announce his voluntary contribution for each project when the project is actually carried out, in combination with his vote for the projects. A project is chosen from the set of projects whose cost is covered by the agents’ voluntary contributions. From such the set of projects, one project is more likely to be chosen if it is voted by more agents. When there is no project in the set that is voted, one project is chosen from the set of projects whose cost is strictly covered by the agents’ voluntary contribution. We show that our mechanism has a Nash equilibrium, and in any Nash equilibrium of our mechanism, an efficient project is chosen and the cost of its development is exactly covered by the agents’ voluntary contributions.

The reminder of the paper is as follows. In the next section, we present our mechanism, a multi-bidding cost sharing mechanism, and then, we examine the properties of the mechanism. In Section 3, we conclude the paper with some discussion on the relation of our results with the ones obtained from previous studies.

We use a Nash equilibrium as our equilibrium concept to analyze the mechanism.

The next lemma shows that the aggregate net bid is zero for any project at a Nash equilibrium.

Lemma 1. In any Nash equilibrium of the MBCS mechanism, the aggregate net bid of projects is zero. That is, at any Nash equilibrium for any.

Proof. When, we have the desired result because of Equation (1). Thus we consider the case. Then, this implies that there exists some such that. Because of the construction of the score function, project p can be chosen by the mechanism in a positive probability. Therefore, agent i can improve his payoff by reducing by small and increasing the bid for another project in by when is small enough. This is a contradiction. Q.E.D

From Lemma 1, we know that a project may be chosen by some probability at a Nash equilibrium. The next lemma show that the project that is chosen in a positive probability at a Nash equilibrium gives the highest payoff to any agent given the Nash equilibrium bid profile b.

Lemma 2. If project q is chosen by the mechanism at Nash equilibrium in a positive probability, this q must satisfy

Proof. We prove this by the way of contradiction. We assume that there exist agent i and project such that

Let be the maximizer of among the set of projects. Then, agent i can improve his payoff by reducing by small and increasing the bid for project by when is small enough. This contradicts the Nash equilibrium. Q.E.D

The following theorem guarantees the efficiency of the projects that are chosen by the MBCS mechanism.

Theorem 1. If project q is chosen by the mechanism at Nash equilibrium at a positive probability, this project is efficient.

Proof. Let q denote the project with a positive score. By summing the condition of Lemma 2 over, we have

By Lemma 1, this can be reduced to

Thus, q is efficient. Q.E.D

The next theorem shows that the lower bound of the agent’s utility when they participate in the MBCS mechanism is, the utility of agent i when the cost of the projects are shared by the equal sharing rule and all of the projects are chosen in an equal probability:

Theorem 2. In any Nash equilibrium, the final payoff of agent i is greater than or equal to.

Proof. Let q be the project that has a positive score. By summing the condition in Lemma 2 over the set of projects, we have

By the definition of the mechanism,. Thus, the left hand side of the above equation is

Divide the equation by k, and we have the desired result. Q.E.D

The following lemma describes the Nash equilibrium vote profile.

Lemma 3. In any Nash equilibrium, for any,.

Proof. Let q be an inefficient project. When for some i, Lemma 1 means that this inefficient q will be chosen by the mechanism at a positive probability. This contradicts Theorem 1. Q.E.D

The following lemma shows that the indifference between projects p and q which are possibly chosen by the mechanism for any at Nash equilibrium.

Lemma 4. In any Nash equilibrium, for any, the final payoff of agent i is indifferent between any two projects that can be chosen by the mechanism.

Proof. Let p and q be projects with a positive score at Nash equilibrium. The condition of Lemma 2 must be satisfied in equality for any agent. Q.E.D

From the above lemma, we have the important properties of Nash equilibrium of the MBCS mechanism. Even if there are two efficient projects that can be chosen by the mechanism at a Nash equilibrium, the final payoff of each i is indifferent between these two projects.

The following lemma characterizes the Nash equilibrium.

Lemma 5. A strategy profile becomes a Nash equilibrium if and only if the following three conditions are satisfied:

1) for any,

2) Given a bids profile, the final payoff of each i is maximized when an efficient project is chosen, and it is indifferent between any two projects that are efficient, and

3) for each.

Proof. We have already shows the “only if part”. So we will show the “if part” of the theorem. However, the proof of the “if part” is almost same as the proof of Lemma 5 in [1] . Thus, we omit it. Q.E.D

Let us define the set of payoffs P as follows:

Theorem 3. Any payoff vector u in P can be a Nash equilibrium payoff of the MBCS mechanism.

Proof. The proof of this theorem is similar to the proof of Theorem 3 in [1] . Thus, we only explain the outline of the proof and give a small remark on the different point from their proof.

This theorem is proved by the following manner. First, we show that the set of Nash equilibrium and the set of Nash equilibrium payoffs are convex. Then, we show that for any agent i there exists a Nash equilibrium such that all the agents other than i obtain the payoffs of their lower bound, i.e.,. Finally, we end the proof by showing that these payoff vectors are the end points of P.

The different point from the proof of [1] is a bid profile that corresponds to the end point of the set of Nash equilibrium profiles. Take any agent. We consider the following strategy that can be shown to the best for agent j among the set of Nash equilibria:

Each agent’s strategy consists of the above bids profiles and the vote set that is a subset of E. Then, it is shown that the above strategy profile is a Nash equilibrium and becomes the end point of the set of Nash equilibrium bid profiles. Q.E.D

The next theorem says that any Nash equilibrium of the MBCS mechanism is a strong Nash equilibrium.

Theorem 4. Any Nash equilibrium of the MBCS mechanism is a strong Nash equilibrium of the mechanism.

Proof. The proof of this theorem is almost same as the proof of Theorem 4 in [1] . Thus, we omit it. Q.E.D

An important implication obtained from our analysis is that the set of final payoffs that can be supported by Nash equilibrium (or strong Nash equilibrium) of the MBCS mechanism is the same as the set of payoffs that are supported by (strong) Nash equilibrium of the multi-bidding mechanism applied to the situation where the

utility obtained from project is transformed to. This means that if agents can agree with the equal

sharing rule of the cost of the project in advance of playing the multi-bidding mechanism, they can reach the same result as the one in the MBCS mechanism. However, it is uncertain that such an agreement can be formed before the mechanism, and in that case, the MBCS mechanism will be functioned.

What our mechanism leads to if all the projects are efficient should be emphasized. From Theorem 2, we know that in any Nash equilibrium, all the agents obtain the utility greater than or equal to their lower bound, , and under the circumstance where, the summation of their lower bound utilities must be. This implies that when, the Nash equilibrium payoff is uniquely determined, i.e.,. Thus, our mechanism leads to the utility profile which is the same as the expected utility in the random selection and equal sharing mechanism where each project is chosen at an equal probability and the cost of chosen project is shared equally among agents. However, it should be noticed that as in the case with other bidding-based mechanisms, our mechanism achieves the payoff vector as a realized value but not an expected value.

One way to extend our mechanism to the asymmetric case is that burden share of each agent is determined by some exogenous factors but there is some difficulty to reach an agreement on this burden share. In such a case, if we apply the modified version of multi-bidding cost sharing mechanism where now the bids of agent i is

constrained to sum up, where is normalized burden share of agent i to project q, this mechanism

always selects an efficient project at every Nash and strong Nash equilibrium. The difference from the previous mechanism is that in this modified mechanism, the lower bound utility of agent i at Nash equilibrium is

The author thanks Kohei Kamaga for their helpful comments and suggestions.