Partnerships of Bidders with Constant Relative Risk Aversions

In this paper, we study a dynamic auction for allocating a single indivisible project while different participants have different bid values for the project. When the price rises continuously, the bidders can retreat the auction and obtain the compensation by the difference between the price at retreating time and the previous bid price.*e final successful bidder achieves the project and pays compensations to others. We show that the auction of bidders with constant relative risk aversion (CRRA) has a unique equilibrium. While the relative risk aversion coefficient approaches to zero, the equilibrium with CRRA bidders would approach to the equilibrium with risk-neutral bidders.


Introduction
When we need to distribute an indivisible project or inheritance to multiple people (such as a house to divorced couples or heirs), a financial problem arises. Cramton et al. [1] propose that the results of allocation depend on the bidder's attitude towards risk. When bidders are risk neutral and their personal wealths are independent random variables, the necessary and sufficient conditions for effective allocation among many partners are proved. Moreover, they prove that the only equilibrium partnership is solvable as the number of bidders increases dramatically. Under the framework of homogeneous prospectives, Morgan [2] studies the fairness of dissolving the partnership between two people. Athanassoglou et al. [3] consider how to divide the project so as to minimize the maximum loss of bidders. McAfee [4] introduces a simple mechanism ( [5,6] also study the mechanism) to describe dissolvable partnerships without considering the utility function of bidders or their value distribution. Assuming that the participants are risk averse and that both bidders tend to have constant absolute risk aversion (CARA) coefficients, McAfee studies the dissolvable partnership problem and solves the bidding function under equilibrium. e second best mechanism for given initial ownership is described in [7]. In order to maximize the sum of weighted social surplus and income, the optimal dissolution mechanism for arbitrary initial ownership is demonstrated in [8]. In addition, similar to [9], most existing studies have been established under the assumption that bidders are risk neutral or the number of bidders is only two.
In order to ensure fair distribution, we need to find an appropriate allocation mechanism to allocate indivisible projects or legacies. Matt and John [10] show a dynamic auction. In the auction, the price goes up from 0 to the value that M − 1 bidders withdraw from bidding, and the last bidder who does not withdraw from the auction wins the project. In return, the winner pays the previous bidders a compensation equal to the difference between his/her exiting price and that of the previous bidder. Let p k be the price, at which the kth withdrawer exits the auction. We assume that p 0 � 0. Matt and John [10] describe the necessary and sufficient conditions for the bidding function at the case that equilibrium is symmetric equilibrium and there are M ≥ 2 risk neutral or CARA bidders. ey prove that when absolute risk aversion coefficient tends to zero, CARA equilibrium strategy tends to risk neutral equilibrium strategy.
In the same type of research, the literature studies mostly choose two bidders to study the relationship of dissolution. However, in practical cases, dissolution problems usually occur when there are more bidders. erefore, the balancing strategy of two bidders has great limitations, which cannot solve the complex situation when there are more bidders. In addition, the risk preference of bidders in the relevant literature is mainly risk neutral. is is inconsistent with the diversified risk preferences of bidders in real life. In view of this, this paper synthesizes the two shortcomings above by assuming that bidders with different project value have different risk aversion levels, and the risk aversion level is negatively correlated with the project value of bidders. Furthermore, we mainly consider the impact of the project value of two or more bidders on their risk tolerance. Moreover, this paper chooses exponential utility function with constant relative risk aversion coefficient to discuss the properties of equilibrium strategy, which is different from the absolute risk aversion coefficient selected in [10]. Absolute risk aversion coefficient cannot reflect the impact of bidders' project value on their risk tolerance. When each bidder has the same relative risk aversion coefficient, we obtain their equilibrium strategy in each round and study the relationship between the equilibrium price of multiple bidders and their relative risk aversion coefficient.
In Section 2, we briefly review the models in [10]. Because bidders with different project values usually have different risk aversion degree, we introduce the utility function of relative aversion coefficient to solve equilibrium strategy, which makes the use of equilibrium strategy more widely. In Section 3, we describe the equilibrium bidding function when there are more than two CRRA bidders and prove that when their relative risk aversion coefficient tends to zero, their strategies tend to be risk neutral equilibrium.

The Mathematical Model
and the bidder's instantaneous probability is (see [10]) In an auction, the price rises continuously from 0 to M − 1 bidders retreating from it. e last bidder who stays in the auction wins the project. Let p 0 � 0 and p k be the price of the kth retreating from the auction. In return, the kth round bidder retreating from the auction will receive the compensation which equals to (p k − p k− 1 ) from the winner, for any k ∈ 1, . . . , M − 1 { }. us, the winner will pay p M− 1 for the project.

Equilibrium Strategies
We denote the utility function of bidders by where c (c > 0 and c ≠ 1) is the index of the relative risk aversion coefficient.
As the utility function of bidders is u c (x), the bidder who wins the project will gain the payoff u c ( is his value) and retreating from the auction in the kth round will gain the payoff e term ξ c k (x, p k− 1 ) denotes the equilibrium price retreating from the auction in the round k when their relative risk aversion coefficient is c.
Before we provide and verify a useful proposition, we review two lemmas as follows. It will be used in the consequent proposition.

Lemma 1.
(i) Any increasing and differentiable symmetric equilibrium ξ satisfies the differential equation: and for k ∈ 1, 2, . . . , is a solution to the system of differential equations in (i), then it is an equilibrium.

Lemma 2. When bidders are risk neutral, the unique symmetric equilibrium satisfies
and for k � 1, . . . , M − 2, where Using Lemma 1, we obtain Multiplying both sides of equation (10) by (1 − F(x)), we obtain Substituting equation (11) into equation (9), it follows that z zx In addition, we have that where C is independent of x ∈ [0, x).
From F(x) � 1, the following equality holds: Mathematical Problems in Engineering 3 en, it follows that at is, We have proved that, at the round M − 1, bid function satisfies equation (7). We shall prove that, at the round k < M − 1, bid function satisfies equation (8) when the round k + 1 bid function satisfies equation (8).