On the Allocation of Multiple Divisible Assets to Players with Different Utilities

Abstract

When there is a dispute between players on how to divide multiple divisible assets, how should it be resolved? In this paper we introduce a multi-asset game model that enables cooperation between multiple agents who bargain on sharing K assets, when each player has a different value for each asset. It thus extends the sequential discrete Raiffa solution and the Talmud rule solution to multi-asset cases.

Appendices

Appendix

The Linear Programming Solution for 2 Players

For the two player case the linear programming problem can be dramatically simplified, and we provide an $O(K \log _2 K)$ complexity algorithm (K is the number of assets). We show that the two players share at most a single commodity, regardless of the ratio between the users. To that end let, $\alpha _{1k}=\alpha _k$, and $\alpha _{2k}=1-\alpha _k$.

We want to solve the following optimization problem:

$$\begin{aligned} L(\varvec{\alpha }, \varvec{\delta }, \varvec{\mu }, \varvec{\lambda })= & {} -\displaystyle \sum _{k=1}^K(1-\alpha _{k})u_{2k}-\sum _{k=1}^K\mu _{k}\alpha _{k}\nonumber \\&+\,\lambda \left( \sum _{k=1}^K\alpha _{k}u_{1k}-u_1\right) . \end{aligned}$$

(13)

To better understand the problem, we first derive the KKT conditions (Boyd and Vandenberghe 2004). Taking the derivative with respect to $\alpha _{k}$, we obtain

$$\begin{aligned} \begin{array}{c} u_{2k}+\lambda u_{1k}- \mu _{k}=0.\ \end{array} \end{aligned}$$

(14)

with the complementarity conditions:

$$\begin{aligned} \begin{array}{cl} 1.&{} \sum _{k=1}^K\alpha _{1k} u_{1k}=u_1,\\ 2.&{} \mu _{k}\alpha _{k}=0, \ \mu \ge 0. \end{array}. \end{aligned}$$

(15)

Based on (14–15), we can easily see that the Lagrange multipliers in (15) satisfy the following conclusions :

• 1. $\mu _{nk}= 0$, if $\alpha _k>0, \ \forall \ k$ (see (15.2)).

• 2. If $0<\alpha _{1k}^j<1$, then the players share an asset p if $\frac{u_{2p}}{u_{1p}}=-\lambda$ (see (14.2)).

• 3. Asset p is assigned to player 2 if $\frac{u_{2p}}{u_{1p}}>-\lambda$.

• 4. Asset p is assigned to player 1 if $\frac{u_{2p}}{u_{1p}}<-\lambda$.

• 5. $\sum \alpha _{1k}u_{1k}=u_1$.

Assuming that a feasible solution exists and that the assets are ranked in decreasing order according to the ratio $L\left( k\right) =\frac{u_{1k}}{u_{2k}}$, it follows from the KKT conditions that the allocation is made according to the following rules

• 1. The ideal point of player 1 is $I_1$ given by

  $$\begin{aligned} \begin{array}{lcl} I_1(u_2)= & {} \sum _{k=1}^{p-1}u_{1k}+\alpha _pu_{1p}, \end{array} \end{aligned}$$

  (16)

  where p and $\alpha _p$ are set such that

  $$\begin{aligned} u_{2}=\sum _{k=p}^{K}u_{2k}-\alpha _p u_{2p}. \end{aligned}$$

  (17)

• 2. Similarly, the ideal point of player 2 is $I_2(u_1)$ is given by

  $$\begin{aligned} \begin{array}{lcl} I_2(u_1)= & {} \sum _{k=p}^{K}u_{2k}-\alpha _pu_{2p}, \end{array} \end{aligned}$$

  (18)

  where p and $\alpha _p$ are set such that

  $$\begin{aligned} u_{1}=\sum _{k=1}^{p-1}u_{1k}+\alpha _pu_{1p}. \end{aligned}$$

  (19)

Therefore, no more than one asset can be shared by the two players. The algorithm for computing the ideal point of player 1 is as follows. Let $L_k=\frac{u_{1k}}{u_{2k}}$ be the ratio between the utilities of asset k. We can sort the assets in decreasing order according to $L_K$. If all the values of $L_{k}$ are distinct, there is at most a single asset that has to be shared between the two players. Since only one asset satisfies Eq. (19), we denote this asset $k_s$, and all the assets $1\le k<k_s$ will be allocated to player 1, while all the assets $k_s < k \le K$ will be be allocated to player 2. Asset $k_s$ must be shared accordingly between the players. The complexity of this algorithm is at most $O(K\log K)$, due to the sorting operation. For the Raiffa bargaining solution the sorting operation only has to be done once at the beginning. The complexity of computing the next disagreement point is on the order of O(K).