On the Bankruptcy Situations and the Alexia Value

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 [7–10]. 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 v : 2^N → ℝ is a map, assigning to each coalition S ∈ 2^N a real number, such that v(∅) = 0. We identify a game 〈N, v〉 with its characteristic function v and denote by G^N the set of cooperative games with player set N. In this paper we assume that v ∈ G^N is monotonic, that is, v(S) ≥ 0 and v(S₁) ≤ v(S₂) if S₁ ≤ S₂.

An order (permutation) σ is a bijective function σ : N → N = {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 P^σ(i) = {r ∈ N∣σ⁻¹(r) < σ⁻¹(i)} consists of all predecessors of i with respect to the order (permutation) σ.

Let v ∈ G^N and σ ∈ Π(N). The marginal contribution vector with respect to σ and v is denoted by m^σ(v) ∈ ℝⁿ. This vector has the value $$ for each i ∈ N in it’s ith coordinate.

The Shapley value ϕ(v) of a game v ∈ G^N is the average of the marginal vectors of the game, that is, ϕ(v) = (1/n!)∑_σ∈Π(N) m^σ(v).

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.

A game v ∈ G^N is called convex if and only if v(S ∪ T) + v(S∩T) ≥ v(S) + v(T) for each S, T ∈ 2^N.

The following example demonstrates the definitions above.

3. The Model and the Solution

A bankruptcy situation with set of claimants N is a pair (E, d), where E ≥ 0 is the estate to be divided, and $$ is the vector of claims such that ∑_i∈N d_i ≥ E. We assume without loss of generality that d₁ < d₂ < ⋯<d_n.

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

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

4. Solution of the Bankruptcy Situations from the Talmud

For d₁ + d₂ < E the solution is α(v) = (d₁/2,  d₂/2,  E − ((d₁ + d₂)/2)), and for d₁ + d₂ ≥ E, α(v) = (1/6)(E + 2d₁ − d₂, E − 2d₂ − d₁, 4E − d₁ − d₂). 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, 4d₂ − d₁, 6d₃ − d₁ − 2d₂, 12d₄ − d₁ − 2d₂ − 6d₃). 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.