Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 813060 | https://doi.org/10.1155/2012/813060

S. Z. Alparslan Gök, A. Sarıarslan, "On the Bankruptcy Situations and the Alexia Value", Journal of Applied Mathematics, vol. 2012, Article ID 813060, 7 pages, 2012. https://doi.org/10.1155/2012/813060

On the Bankruptcy Situations and the Alexia Value

Academic Editor: Maurizio Porfiri
Received27 Mar 2012
Revised22 Jun 2012
Accepted15 Jul 2012
Published16 Aug 2012

Abstract

The main result of this paper is to show that the three ancient bankruptcy situations from the 2000-year-old Babylonian Talmud can be solved by using the average lexicographic value (Alexia) from cooperative game theory.

1. Introduction

This paper is based on the results of Guiasu [1] who solves the three ancient problems, namely, the bankruptcy problem, the contested garment problem, and the rights arbitration problem, from the 2000-year-old Babylonian Talmud by using the Shapley value [2] from cooperative game theory. We also refer to such games as a TU (transferable utility) game. The main objective of Guiasu [1] is to show that the Shapley value can be used for justifying the ancient solutions given to all these three problems viewed as cooperative 𝑛-person games.

Recently, Tijs et al. [3] introduce the average lexicographic (Alexia) value, a value which averages the lexicographic maxima of the core [4], for games with a nonempty core and show that the Alexia value coincides with the Shapley value for the class of convex games. Further, Curiel et al. [5] show that bankruptcy games are convex.

We notice that two solution concepts are dominant in game theory. One is the Nash equilibrium set [6], and the other is the core. There are remarkable similarities if one looks at the role of the Nash equilibrium set in noncooperative game theory and the role of the core in cooperative game theory. Noncooperative games without Nash equilibria as well as cooperative games with an empty core are not very attractive for a game theorist and also not in practice.

In the sequel we show that the Alexia value can also be an alternative to the Shapley value for the solution of these problems, and note that in general the most popular one-point solution concepts such as the Shapley value for general TU-games do not provide a core element as a solution, but the Alexia value provides a core element as a solution for all games with a nonempty core.

Bankruptcy situations and the problems from the Talmud have been intensively studied in literature [710]. In this study we aim to give a uniform method by using the Alexia value for solving these problems which is done by Guiasu [1] by using the Shapley value.

We first consider the bankruptcy problem which is known as Talmudic problem of three wives. In the story a man married with three women and promised them in their marriage contract the sum of 𝑑1=100, 𝑑2=200, and 𝑑3=300 units of money after his death. But, the estate 𝐸 was less than 600 units. The division of the estate among the three wives is as follows: for 𝐸=100, the wives get 33(1/3), and 33(1/3),33(1/3), for 𝐸=200 the wives get 50, 75, and 75, and for 𝐸=300 the wives get 50, 100, and 150, respectively (cf. [11]).

The second story is called the contested garment problem. The two hold a garment and one of them wants all and the other half of it. Then the division of the garment is three quarters for the former one and one quarter for the latter one (cf. [7]).

Finally, we look at the right arbitration problem. The story is based on a father and his four sons. The father Jacob willed different units of money from his estate to his four sons. So after his death, the four sons produced different deeds, and all of them bear the same date. The son Reuben produced a deed duly witnessed that Jacob willed to him the entire estate on his death, the son Simeon produced a deed that his father willed to him half of the estate, the son Levi produced a deed giving him one third, and the son Judah produced a deed giving him one forth. Assuming that the estate is 𝐸=120 the division between the sons is 7(1/2),10(5/6),20(5/6), and 80(5/6), respectively (cf. [7]).

Following the steps of Guiasu [1], we show that the division of the estate for these three ancient problems may be obtained by using the Alexia value if the bankruptcy problem is viewed as being a cumulative game and the rights arbitration problem is viewed as being a maximal game. In a cumulative TU game the members are not willing to reach a compromise and share their claims. So the values of the characteristic function for the coalitions are calculated additively. On the other hand in a maximal TU game the members of the coalition are willing to reach a compromise and share their claims. For this reason the values of the characteristic function for the proper coalitions are calculated by taking into account the maximum claim.

The paper is organized as follows. In Section 2, we give some preliminaries from cooperative game theory which are necessary in the following sections. The model and the general solution is given in Section 3. Section 4 gives solutions to the three Talmudic problems by using the Alexia value. Finally, we conclude in Section 5.

2. Preliminaries

A cooperative game in coalitional form is an ordered pair 𝑁,𝑣, where 𝑁={1,2,,𝑛} is the set of players, and 𝑣2𝑁 is a map, assigning to each coalition 𝑆2𝑁 a real number, such that 𝑣()=0. We identify a game 𝑁,𝑣 with its characteristic function 𝑣 and denote by 𝐺𝑁 the set of cooperative games with player set 𝑁. In this paper we assume that 𝑣𝐺𝑁 is monotonic, that is, 𝑣(𝑆)0 and 𝑣(𝑆1)𝑣(𝑆2) if 𝑆1𝑆2.

A map 𝜆2𝑁{}+ is called a balanced map if 𝑆2𝑁{}𝜆(𝑆)𝑒𝑆=𝑒𝑁. Here, 𝑒𝑆 is the characteristic vector for coaliton 𝑆 with 𝑒𝑆𝑖=1,if𝑖𝑆0,if𝑖𝑁𝑆.(2.1) An 𝑛-person game 𝑁,𝑣 is called a balanced game if for each balanced map 𝜆2𝑁{}+ we have 𝑆2𝑁{}𝜆(𝑆)𝑣(𝑆)𝑣(𝑁).

The core of a game 𝑁,𝑣 is the set 𝐶(𝑣)=𝑥𝑁𝑖𝑁𝑥𝑖=𝑣(𝑁),𝑖𝑆𝑥𝑖𝑣(𝑆)𝑆2𝑁.{}(2.2) A game is called balanced if its core is nonempty [12, 13].

An order (permutation) 𝜎 is a bijective function 𝜎𝑁𝑁={1,2,,𝑛}, where 𝜎(𝑘) denotes the player at position 𝑘{1,2,,𝑛} in the order (permutation) 𝜎. The set of all orders (permutations) of 𝑁 is denoted with Π(𝑁).

The set 𝑃𝜎(𝑖)={𝑟𝑁𝜎1(𝑟)<𝜎1(𝑖)} consists of all predecessors of 𝑖 with respect to the order (permutation) 𝜎.

Let 𝑣𝐺𝑁 and 𝜎Π(𝑁). The marginal contribution vector with respect to 𝜎 and 𝑣 is denoted by 𝑚𝜎(𝑣)𝑛. This vector has the value 𝑚𝜎𝑖(𝑣)=𝑣(𝑃𝜎(𝑖){𝑖})𝑣(𝑃𝜎(𝑖)) for each 𝑖𝑁 in it’s 𝑖th coordinate.

The Shapley value 𝜙(𝑣) of a game 𝑣𝐺𝑁 is the average of the marginal vectors of the game, that is, 𝜙(𝑣)=(1/𝑛!)𝜎Π(𝑁)𝑚𝜎(𝑣).

For a balanced game 𝑁,𝑣 and an order 𝜎Π(𝑁), the lexinal 𝜆𝜎(𝑣)𝑁 is defined as the lexicographic maximum on 𝐶(𝑣) with respect to 𝜎, that is, 𝜆𝜎𝜎(𝑘)𝑥(𝑣)=max𝜎(𝑘)𝑥𝐶(𝑣),𝑥𝜎(1)=𝜆𝜎𝜎(𝑙)(𝑣)𝑙{1,2,,𝑘1},(2.3) for all 𝑘{1,2,,𝑛}.

The lexinal is recursively defined such that every player gets the maximum he can obtain inside the core under the restriction that the players before him in the corresponding order obtain their restricted maxima.

For a balanced game 𝑁,𝑣, the Alexia value 𝛼(𝑣) is defined as the average over the lexinals as follows: 1𝛼(𝑣)=𝑛!𝜎Π(𝑁)𝜆𝜎(𝑣).(2.4)

A game 𝑣𝐺𝑁 is called convex if and only if 𝑣(𝑆𝑇)+𝑣(𝑆𝑇)𝑣(𝑆)+𝑣(𝑇) for each 𝑆,𝑇2𝑁.

The following example demonstrates the definitions above.

Example 2.1 (See [14]). Let 𝑣𝐺𝑁 be a two-person-balanced game. Then, 𝑣(1,2)𝑣(1)+𝑣(2) and 𝐶(𝑣)=conv{𝑣(𝑁)𝑣(2),𝑣(2)}.

3. The Model and the Solution

A bankruptcy situation with set of claimants 𝑁 is a pair (𝐸,𝑑), where 𝐸0 is the estate to be divided, and 𝑑𝑁+ is the vector of claims such that 𝑖𝑁𝑑𝑖𝐸. We assume without loss of generality that 𝑑1<𝑑2<<𝑑𝑛.

If the values (𝑣({1}),,𝑣({𝑛})) of 𝑣 are given, then for each subset of players {𝑖1,𝑖2,,𝑖𝑘} the characteristic function 𝑣 of the corresponding cooperative cumulative game is defined by 𝑣𝑖1,𝑖2,,𝑖𝑘𝑣𝑖=min1𝑖+𝑣2𝑖++𝑣𝑘,𝐸,(3.1) for all 𝑘=2,3,,𝑛.

If the values (𝑣({1}),,𝑣({𝑛})) of 𝑣 are given, then for each subset of players {𝑖1,𝑖2,,𝑖𝑘} the characteristic function 𝑣 of the corresponding cooperative maximal game is defined by 𝑣𝑖1,𝑖2,,𝑖𝑘𝑣𝑖=minmax1𝑖,𝑣2𝑖,,𝑣𝑘,𝐸,(3.2) for each 𝑘=2,3,,𝑛.

To each bankruptcy situation (𝐸,𝑑) one can associate a bankruptcy game 𝑣 defined by 𝑣({𝑖})=min{𝑑𝑖,𝐸} for each 𝑖=1,2,,𝑛 which can be considered as an arbitrary cumulative or maximal TU game.

Then the Alexia value for each player is 𝛼𝑖(𝑣)=𝐸/𝑛 for each 𝑖=1,2,,𝑛. Accordingly for a cumulative game 𝑣𝑑({𝑖})=min𝑖𝑣𝑖,𝐸=𝐸foreach𝑖=1,2,,𝑛,1,𝑖2,,𝑖𝑘𝑣𝑖=min1𝑖+𝑣2𝑖++𝑣𝑘,𝐸=𝐸,(3.3) for each 𝑘=2,3,,𝑛.

On the other hand for a maximal game 𝑣𝑑({𝑖})=min𝑖𝑣𝑖,𝐸=𝐸foreach𝑖=1,2,,𝑛,1,𝑖2,,𝑖𝑘𝑣𝑖=minmax1𝑖,𝑣2𝑖,,𝑣𝑘,𝐸=𝐸,(3.4) for each 𝑘=2,3,,𝑛.

Then the Alexia value is also 𝛼𝑖(𝑣)=𝐸/𝑛 for each 𝑖=1,2,,𝑛.

Remark 3.1. As it is stated more detailed in Guiasu [1] the division of the estate in the bankruptcy problem can be interpreted as the judge looking at the game as a cumulative game with the wives that does not want to compromise and share parts of their claims. The division of the estate in the rights arbitration problem can be interpreted as the judge looking at the game as a maximal game supposing that the four brothers want to reach a compromise and share parts of their claims when they form a coalition. Conversely in the contested garment problem since there are no proper coalitions in the total set 𝑁={1,2}, the two claims 𝑑1, 𝑑2, and the estate 𝐸 completely determine the game.

Remark 3.2. Guiasu [1] show that the three bankruptcy situations from the Talmud can be modeled by using the cumulative or maximal TU games, and their solution can be found by using the Shapley value. Further, Tijs et al. [3] show that if a game is convex, then 𝜙(𝑣)=𝛼(𝑣), and Curiel et al. [5] states that bankruptcy games are convex. In view of these results we conclude that the solution of the three TU games that we have constructed can also be calculated by using the Alexia value.

Before closing this section, we note that we also have to consider the generalized situation 𝑑1<𝑑2<<𝑑𝑚1<𝐸𝑑𝑚<<𝑑𝑛. Then the game is constructed as follows: 𝑣()=0,𝑣({𝑖})=𝑑𝑖,foreach𝑖=1,2,,𝑚1,𝑣({𝑖})=𝑑𝑚1+𝐸𝑑𝑚1(𝑛𝑚+1),foreach𝑖=𝑚,,𝑛.(3.5) As an interpretation the claim of the first 𝑚1 players whose claims are less than the estate 𝐸 can claim 𝑑𝑖, but the other 𝑛𝑚+1 players whose claims are more than the estate 𝐸 cannot claim more than 𝑑𝑚1 plus the excess of the remaining estate 𝐸𝑑𝑚1 equally divided between them.

4. Solution of the Bankruptcy Situations from the Talmud

We first consider the solution of the Talmudic problem with three wives. In this situation the estate will be divided between the three wives so 𝑁={1,2,3}. We assume that the game to be constructed is cumulative. We give solutions according to the following cases.(i)For 𝑛=3, 𝑚=1 we have the situation 𝐸𝑑1<𝑑2<𝑑3. The game is constructed as follows: 𝑣()=0, 𝑣(𝑆)=𝑑1 otherwise, and the solution is 𝛼(𝑣)=(1/3)(𝐸,𝐸,𝐸). So for 𝐸=100, 𝑑1=100, 𝑑2=200, and 𝑑3=300, the solution is 𝛼(𝑣)=(33(1/3),33(1/3),33(1/3)).(ii)For 𝑛=3, 𝑚=2 we have the situation 𝑑1<𝐸𝑑2<𝑑3. Then the game is constructed as follows: 𝑣()=0,𝑣({1})=𝑑1,𝑣({2})=𝑑1+𝐸𝑑12,𝑣(𝑆)=𝐸otherwise,(4.1) and 𝛼(𝑣)=(1/12)(2(𝐸+𝑑1),5𝐸𝑑1,5𝐸𝑑1). So for 𝐸=200, 𝑑1=100, 𝑑2=200, and 𝑑3=300, the solution is 𝛼(𝑣)=(50,75,75).(iii)For 𝑛=3, 𝑚=3 we have the situation 𝑑1<𝑑2<𝐸𝑑3. Then the game is constructed as follows: 𝑣()=0, 𝑣({1})=𝑑1, 𝑣({2})=𝑑2, 𝑑𝑣({1,2})=min1+𝑑2=𝑑,𝐸1+𝑑2,𝑑1+𝑑2<𝐸𝐸,𝑑1+𝑑2𝐸,(4.2)𝑣(𝑆)=𝐸 otherwise.

For 𝑑1+𝑑2<𝐸 the solution is 𝛼(𝑣)=(𝑑1/2,𝑑2/2,𝐸((𝑑1+𝑑2)/2)), and for 𝑑1+𝑑2𝐸, 𝛼(𝑣)=(1/6)(𝐸+2𝑑1𝑑2,𝐸2𝑑2𝑑1,4𝐸𝑑1𝑑2). So for 𝐸=300, 𝑑1=100, 𝑑2=200, and 𝑑3=300, the solution is 𝛼(𝑣)=(50,100,150).

Next we consider the solution of the congested garment problem. In this situation 𝑁={1,2} and 𝑑1<𝐸=𝑑2, that is, 𝑛=𝑚=2. We have a two-person game, so the game is same if we take either maximal or cumulative. Then the game is constructed as follows: 𝑣()=0, 𝑣({1})=𝑑1, 𝑣({2})=𝐸, and 𝑣(𝑁)=𝐸. The Alexia value 𝛼(𝑣) is (𝑑1/2,𝐸(𝑑1/2)). So for 𝐸=1, 𝑑1=1/2, 𝑑2=1, the solution is 𝛼(𝑣)=(1/4,3/4).

Finally, we look at the solution of the rights arbitration problem. Here the estate will be divided between the four sons, so 𝑁={1,2,3,4}. The claims of the sons can be written as 𝑑1<𝑑2<𝑑3<𝐸=𝑑4, that is, 𝑛=𝑚=4. We assume that the game is maximal. Then the game is constructed as follows: 𝑣()=0, 𝑣({1})=𝑑1, 𝑣({2})=𝑑2, 𝑣({3})=𝑑3, 𝑣({4})=𝑑4, 𝑣({1,2})=𝑑2, 𝑣({1,3})=𝑣({2,3})=𝑣({1,2,3})=𝑑3, 𝑣(𝑆)=𝐸, otherwise.

The solution is 𝛼(𝑣)=(1/12)(3𝑑1,4𝑑2𝑑1,6𝑑3𝑑12𝑑2,12𝑑4𝑑12𝑑26𝑑3). So for 𝐸=120, 𝑑1=30, 𝑑2=40, 𝑑3=60, 𝑑4=120, 𝛼(𝑣)=(7(1/2), 10(5/6), 20(5/6), 80(5/6)).

In the sequel we show that the three ancient problems from Babylonian Talmud can be solved by using the Alexia value from cooperative game theory. The calculations for the Talmudic problems above are done by using the Alexia value, and it is seen that they are equal to the Shapley value. We also notice that each situation is modeled by using bankruptcy games. We know that each bankruptcy game is convex (see Curiel et al. [5]), and for each convex game the Shapley value and the Alexia value are equal (see Tijs et al. [3]). Finally, we notice that the iterative solutions of the three ancient problems by using the Alexia value are straightforward by following the procedure given by Guiasu [1].

5. Concluding Remarks

In this paper we give an alternative solution, the Alexia value from cooperative game theory, to solve the three ancient bankruptcy situations from the 2000-year-old Babylonian Talmud. Our intuition and results are based on Guiasu [1] who used the Shapley value for the solution of these problems. Since each situation is modeled by using bankruptcy games, we used the result of Curiel et al. [5] that each bankruptcy game is convex and the result of Tijs et al. [3] that for each convex game the Shapley value and the Alexia value are equal.

Notice that the Alexia value is an interesting alternative to the Shapley value because the Alexia value provides a core element as a solution for all games with a nonempty core. Moreover it can be seen as a run-to-the-core rule for games with a nonempty core. To be more precise every lexinal player is running to the core according to a certain order where every player takes the maximum he can obtain within the subset of the core that remains after the players before him have made their respective choices. Further, the Alexia value combines two often applied arguments with respect to choosing an allocation: using orderings of the players, and at the same time respecting the fairness criterion of the core. Hence, the Alexia value combines the attractive properties of the Shapley value.

Acknowledgment

The authors gratefully acknowledge two anonymous referees.

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Copyright © 2012 S. Z. Alparslan Gök and A. Sarıarslan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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