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We 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. We propose a multi-bidding cost sharing mechanism where each agent simultaneously announces his voluntary contribution for each project when the project is actually carried out, in combination with his vote for the projects. We show that a Nash equilibrium exists in this mechanism, and in any Nash equilibrium of this mechanism, the efficient project is always chosen. Moreover, in the Nash equilibrium, the way of sharing the cost of the project is, in a sense, an equal sharing rule.

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 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, [

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 [

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 consider a set of agents ^{2}. The aggregate cost of all the

projects is denoted by

for all^{e} the maximum total pure value that can be obtained by developing the project, that is,

for some

One benchmark mechanism that determines the project that is developed and how the cost of the project is shared is a mechanism that randomly chooses a project and shares the cost of the project equally among the agents. In this random selection and equal sharing mechanism, the expected payoff of agent i is

This simple mechanism has several deficiencies. This mechanism may choose an inefficient project, which means that after the determination of the project by the mechanism, agents may start to bargain because there is a room for all of them to be better off by selecting an efficient project, and this diminishes the credibility of the mechanism. Second, it is uncertain that equal sharing of the cost of project is an appropriate way of sharing the cost among agents when the agents’ preferences show the diversity. The mechanism a priori assumes the desirability of the equal sharing. These points motivate to consider another mechanism that chooses the project and determines the way of sharing the costs of projects.

The multi-bidding mechanism of [

Reference [

The problem considered in this paper involves the stage immediately prior to [

We also consider the natural version of the multi-bidding mechanism in a different way from [

Our mechanism is informally described as follows. Similar to the multi-bidding mechanism, each agent announces k bids, one for each possible project, which is interpreted as his cost burden for the development of the project if it is actually chosen. In addition, agent i votes on the projects. Only the projects whose cost is collected by agents will be chosen as the winner, and among such projects, the project that is voted by more agents is chosen at a higher probability.

A multi-bidding cost sharing (MBCS) mechanism is described as follows. Each i announces k bids, one for each possible project, that are constrained to sum up to per capita total cost

on projects that he wants to be chosen as a winner. Here, i is allowed to vote for all the projects that he likes. Thus, his strategy is a vector

The strategy profile of n agents is

vote profile. Given an announced bid profile b, an aggregate bid of project q is

net bid of project q is

where^{3}. A winner of the mechanism is chosen from

Moreover, there is no case that all the members of

If the mechanism chooses project q, then q is developed and each i receives the payments^{4,5}.

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,

Proof. When

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

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

Let

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

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

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,

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

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

Lemma 4. In any Nash equilibrium, for any

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)

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)

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 [

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 [

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.,

The different point from the proof of [

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 [

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

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,

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

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.