On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule

On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule

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On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule A. Saavedra-Nieves, P. Saavedra-Nieves PII: DOI: Reference:

S0377-2217(20)30131-4 https://doi.org/10.1016/j.ejor.2020.02.013 EOR 16333

To appear in:

European Journal of Operational Research

Received date: Accepted date:

4 August 2019 7 February 2020

Please cite this article as: A. Saavedra-Nieves, P. Saavedra-Nieves, On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.013

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Highlights • We study the milk conflict in Galicia after suppressing the European quotas. • We use bankruptcy rules for determining new systems of quotas. • The random arrival rule is difficult to compute in large-scale problems. • We propose a sampling method to estimate it and that solves this drawback. • We analyse the statistical properties before its innovative application.

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On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule A. Saavedra-Nieves1 , P. Saavedra-Nieves2 1 2

Corresponding author. Departamento de Estatística e Investigación Operativa, Universidade de Vigo. [email protected]

Departamento de Estatística, Análise Matemático e Optimización, Universidade de Santiago de Compostela. [email protected]

Abstract This paper addresses a sampling procedure for estimating the random arrival rule in bankruptcy situations. It is based on simple random sampling with replacement and it adapts an estimation method of the Shapley value for transferable utility games, especially useful when dealing with large-scale problems. Its performance is analysed through the establishment of theoretical statistical properties and bounds for the incurred error. Furthermore, this tool is evaluated on two well-studied examples in literature where this allocation rule can be exactly calculated. Finally, we apply this sampling method to provide a new quota for the milk market in Galicia (Spain). After the abolition of the milk European quota system in April 2015, this region represents an example where dairy farmers suffered a massive economic impact after investing heavily in modernising their farms. The resulting quota estimator is compared with two classical rules in bankruptcy literature.

Keywords: Multi-agent systems, bankruptcy, random arrival rule, sampling techniques, milk quotas

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Introduction

The milk quota regime in the European Union (EU) was initially imposed in 1984 with the only purpose of overcoming the surpluses obtained when the production of milk far outstripped the demand. Before this date, EU dairy farmers had been guaranteed a price for their milk (considerably higher than on the main markets) regardless of market demand. The system substantially influenced the prices, as the EU frequently subsidized exports to the world market. In this sense, a tax was imposed for those farmers that exceed the quantities of milk were above the defined thresholds. Those changes in the Common Agriculture Policy of the EU were devoted to help those dairy farmers from specially vulnerable areas. A main goal consisted of providing EU producers more flexibility in order to respond to an increasing demand of milk, by balancing price and production of milk. The end of the milk quotas regime in April 2015 indicates an increase in production which derives into new problems in the dairy sector. The major consequence was that the milk price per litre of milk reduced. Mainly, because the EU was unable to predict the brutal production increase of some countries such

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as Ireland and The Netherlands. Although different initiatives were proposed to avoid this, new measures after the abolition of these quotas (as a new Community policy in this field or the establishment of a new upper-bound of the milk production, for instance) have not yet been addressed to solve the problems of milk prices. A realistic example that illustrates situations as the ones mentioned is the conflict of the milk sector in Galicia, a region in the Northwest of Spain shown in Figure 1. According to the Consellería de Medio Rural of Xunta de Galicia (https://mediorural.xunta.gal/), this region is the leading dairy power in Spain (more than 50% of farms and about 40% of production). In fact, the dairy sector in Galicia generates 1.5% of gross domestic product and it manages 35% of the farmland. Although the regulation of milk quotas a priori sought the sustainability of the conditions of milk producers, through the increase in the price of the litre of milk, the truth is that with its suppression the effect has been the opposite. In particular, Table 1 depicts the evolution of the prices of the milk in Galicia along four years, showing its reduction because of the abolition of the system of quotas. This region is characterized by a high volume and quality of its production of milk, although it also stands out for the absence of an efficient common policy for the cooperatives of farms. Furthermore, the agricultural and livestock sectors in this area have not improved their structures as other European regions have done in recent years, even though they are considered key to the regional economy.

Figure 1: Geographical location of Galicia, see red rectangle.

Year 2012 2013 2014 2015

Jan. 31.85 32.65 39.24 30.52

Feb. 31.53 32.66 38.90 30.60

Mar. 30.87 32.76 38.65 30.30

Apr. 30.00 32.84 36.05 28.80

May. 29.24 33.06 35.61 28.40

Jun. 28.66 33.03 35.37 27.90

Jul. 28.41 34.44 33.72 27.60

Aug. 28.53 34.86 33.73 27.70

Sep. 29.35 35.59 33.64 28.30

Oct. 30.48 38.57 32.24 28.70

Nov. 31.10 38.93 32.10 28.70

Dec. 31.27 39.17 31.95 28.80

Table 1: Averaged prices of the milk in Galicia (in euros per 100 litres) per month in the period 2012-2015.

Under these conditions, one of the measures to be taken by the institutions for solving the milk conflict in Galicia (Spain) after the abolition of the milk quotas may consist of bounding the milk production for the region. An alternative may be reducing this amount with respect to the ones globally imposed in 2014-2015 (the last under regulation of the EU milk quotas). Surely, this decision would increase the prices of a litre of milk after the decreasing arisen with the ending of milk quotas. Table 15 in Appendix A refers to the milk quotas, given in thousands of kilos, imposed for the councils that divide Galicia in the period 2014-2015 (see Figure 2). These amounts provide a measure of the maximum capabilities of producing milk of each involved agent. It makes sense to assume that the

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Figure 2: Map of Galicia and the involved councils (in grey). In dark grey, the size of the bullet of each council indicates the proportion of its milk quota. aggregate of the milk quotas for Galicia has to decrease. Under the considered approach, our main goal consists of searching a new distribution of the milk quotas for the 190 councils under a low-production scenario (with respect to April 2015). The situation described can be seen as a particular bankruptcy problem when the maximum of tons of milk in 2014-2015 imposed for Galicia must be reduced. Bankruptcy problems have taken relevance over the years because of their multiple applications in the real world. Their name is due to a very common problem in economics as the bankruptcy of a company. In particular, the existence of several creditors that claim a portion of a total estate is assumed. The main goal consists of determining how we must divide the resource among all those agents who have claims on it. In a more general setting, this is also the case for the milk problem in Galicia. Bankruptcy problems are initially introduced in O’Neill (1982) and Aumann and Maschler (1985). For a complete survey on this topic, see Moulin (2002) or Thomson (2015). Furthemore, bankruptcy problems have been also considered from a game theoretic perspective in Curiel, Maschler, and Tijs (1987). In fact, any multi-agent allocation problem under cooperation can be analysed as a bankruptcy case. It is only necessary the existence of an estate to be allocated where each agent claims a portion of the total. Many references have illustrated these situations. For instance, Casas-Méndez, Fragnelli, and García-Jurado (2011) analyse the museum pass problem under a bankruptcy approach and Estévez-Fernández (2012) uses bankruptcy to manage those situations with delays in projects. Carpente et al. (2013) illustrates how to divide a cake according to the metabolism of the diners. From an eco-approach, Gutiérrez, Llorca, Sánchez-Soriano, and Mosquera (2018) use bankruptcy to limit the greenhouse gas emissions in production problems. For settings of systems of quotas, Gallástegui, Iñarra, and Prellezo (2002) analyse the policies followed by the European Union (EU) for obtaining the

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fishing quotas of the members under this approach. The definition of rules for distributing the available resources becomes an open problem in the analysis of bankruptcy situations. Different approaches are taken into account in literature. Several alternatives of allocation procedures in bankruptcy were recently characterized in Thomson (2015). Three of the most usual proposals are the proportional rule, that assigns proportionally to the claims, the Talmud rule introduced by Aumann and Maschler (1985), and the random arrival rule (O’Neill, 1982). Although the random arrival rule has been analysed from a theoretical point of view in several papers (see Hwang, 2015, for example), its computation is still a hard task. The main drawbacks concerning the random arrival rule for bankruptcy problems, similar to the ones described for obtaining the Shapley value (Shapley, 1953) of transferable utility games (in what follows, TU-games), are computational. A key assumption in this work refers to the fact of obtaining this rule as the Shapley value of a bankruptcy game (cf. O’Neill, 1982). For large-scale problems as the one we deal, determining the random arrival rule substantially complicates. This is due to the fact of that the complexity exponentially increases in the computations with the number of agents. In fact, Aziz (2013) provides an algorithm to determine the allocation that this rule proposes, but only for a special case of bankruptcy (that in which claims are given by integer numbers). Nevertheless, it does not reduce the computational problem for the calculation in practice when considering large sets of real-valued claimants. Due to the wide applications of bankruptcy in real world, where exact solutions are often not possible in practice, sampling techniques (Cochran, 2007) becomes an alternative tool to solve this kind of problems. Multiple papers deal with the problem of estimating the Shapley value for TU-games. Mann and Shapley (1960) firstly use sampling techniques in estimating the power of the members of electoral systems. However, a polynomial estimation procedure for the Shapley value of general TU-games, based on simple random sampling with replacement, is introduced in Fernández-García and PuertoAlbandoz (2006) and Castro, Gómez, and Tejada (2009). Alternative methods as stratified sampling are recently considered in Maleki (2015) or Castro, Gómez, Molina, and Tejada (2017) as a guarantee to reduce the variance of the estimators obtained by simple random sampling. Benati, López-Blázquez, and Puerto (2019) provide an estimation method for the Shapley value based on the stochastic approximation of deterministic games and sampling techniques. However, the use of these approximation methodologies is not reduced to the estimation of the Shapley value. For example, Perea and Puerto (2019) provide a heuristic procedure for computing the nucleolus for TU-games (cf. Schmeidler, 1969). The main goal of this work is to propose a method to estimate the random arrival rule for bankruptcy problems based on simple random sampling with replacement. Concretely, the ideas of FernándezGarcía and Puerto-Albandoz (2006) and Castro et al. (2009) for the estimation of the Shapley value for TU-games are adapted to the bankruptcy setting. Therefore, an innovative way to face large-scale real problems such as estimating a new system of milk quotas in Galicia after 2015 will be provided. This paper is organized as follows. Section 2 introduces the formal notation for the understanding of bankruptcy situations and the random arrival rule. The sampling procedure to estimate this allocation vector is described in Section 3. Moreover, its statistical properties are studied and some theoretical results on bounding the error are provided. Then, the performance of this proposal is also evaluated on two well-known real examples in literature where the allocation rule can be exactly determined. Section 4 contains the approximation of the random arrival rule for each council in Galicia when the milk quota conflict is considered as a bankruptcy problem. Some concluding remarks are deferred to Section 5. Finally, three appendices are included. Appendix A shows the milk quotas for 190 councils in Galicia

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in the period 2014-2015. Appendix B contains two proposals of new systems of milk quotas by using two well-known rules in bankruptcy. Finally, Appendix C depicts the R code used for approximating the random arrival rule for bankruptcy.

2

Preliminaries in bankruptcy

A bankruptcy problem is a multi-agent situation in which agents claim a portion of a good larger than the one available. Formally, these situations were initially analysed in O’Neill (1982). A bankruptcy problem with a set of n claimants N is given by ( N, c, E), where E ∈ R+ and c ∈ Rn are such that, for each agent i ∈ N, ci ≥ 0 with 0 ≤ E ≤ ∑ c j . j∈ N

The set of bankruptcy problems with set of claimants N is denoted by B N . The value of E is the total amount to be divided, usually referred to as the estate, and ci is the claim of agent i, for each i ∈ N.

The condition E ≤ ∑ c j ensures the existence of a real bankruptcy problem, since that the estate is not j∈ N

sufficient to satisfy all the claimants. This case is the only one of interest, since otherwise we face with a trivial bankruptcy problem in which each agent receives its claim. The definition of procedures to allocate the estate among the claimants becomes fundamental. A division rule f for a bankruptcy problem ( N, c, E) ∈ B N is formally given by a function which assigns a vector f ( N, c, E) ∈ Rn satisfying that: (a) ∑i∈ N f i ( N, c, E) = E, and (b) 0 ≤ f i ( N, c, E) ≤ ci , for each i ∈ N. In this sense, several division rules have been introduced in bankruptcy literature. Some particular proposals are the proportional rule, the Talmud rule (Aumann and Maschler, 1985) and the random arrival rule. A more general overview of the different allocation rules applicable to this class of problems will be given at the end of the paper on a real case. Hereafter, we will focus on the random arrival rule and the computational problems that arise for its calculation. Let ( N, c, E) ∈ B N be a bankruptcy problem and let Π( N ) be the set of all permutations

of N. For each π ∈ Π( N ), Piπ is the set of predecessors of i under π. The random arrival rule (O’Neill, 1982) assigns to ( N, c, E) an allocation vector RA = ( RAi )i∈ N such that, for every agent i ∈ N,    1 RAi ( N, c, E) = min ci , max 0, E − ∑ c j . ∑ n! σ∈Π( N ) j∈ Pσ

(1)

i

The definition of this rule is based on the following idea. We assume that agents claim at their arrival time. When an agent arrives, it receives the minimum between its claim and the remainder, depending the final division on the claimants’ arrival rule. The fact of that this rule for an agent i is obtained as the average of the marginal contributions (all equally likely) provides to the allocation some fairness. A bankruptcy problem can be also analysed from a game theoretic approach. For instance, O’Neill (1982) assigns to each bankruptcy problem a TU-game. Recall that a TU-game is formally given by a pair

( N, v), where N denotes the finite set of agents and v : 2 N −→ R is a map satisfying v(∅) = 0 (more

details, in González-Díaz, García-Jurado, and Fiestras-Janeiro, 2010). A bankruptcy game is a TU-game

( N, v) associated to each ( N, c, E) ∈ B N and given, for each S ⊆ N, by   v(S) = max 0, E − ∑ ci . i∈ /S

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(2)

In practice, v(S) corresponds to the estate that remains when the agents in N \ S received their claims

for each possible coalition S ⊆ N. For a TU-game ( N, v) and for each agent i ∈ N, the Shapley value (Shapley, 1953) of ( N, v) is given by Φi ( N, v) =

∑

S⊆ N \{i }

|S|! (| N | − |S| − 1)! (v(S ∪ {i }) − v(S)). | N |!

Besides, O’Neill (1982) also proves that the Shapley value for the associated bankruptcy game coincides with the allocation vector given by the random arrival rule. It is often stated that the Shapley value satisfies a certain criteria of fairness, through a certain compromise for the set of agents involved. This is consequently extended to the random arrival rule in bankruptcy, justifying its choice as an allocation mechanism in this setting.

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Estimating the random arrival rule

The computation of the random arrival rule for bankruptcy situations becomes a difficult task when the number of involved agents increases. In what follows, we analyse a general procedure for estimating this rule by using simple random sampling with replacement. It is an application of the proposal introduced by Fernández-García and Puerto-Albandoz (2006) and Castro et al. (2009) for approximating the Shapley value for general TU-games to bankruptcy problems (avoiding the use of the corresponding bankruptcy game).

3.1

Numerical implementation of the sampling algorithm

The procedure for approximating the random arrival rule for bankruptcy situations based on sampling techniques is formally described next. Let ( N, c, E) ∈ B N be a bankruptcy problem. The steps of the sampling procedure that we propose

are described below.

1. The population of the sampling procedure is the set of permutations of N, i.e. Π( N ). 2. The vector of parameters to be estimated is RA = ( RAi )i∈ N , where RAi denotes RAi ( N, c, E) for all i ∈ N.

3. The characteristic to be studied in each sampling unit σ ∈ Π( N ) is the vector ( x (σ)i )i∈ N , where    x (σ)i = min ci , max 0, E − ∑ c j , for all i ∈ N. (3) j∈ Piσ

4. The sampling procedure takes each permutation σ ∈ Π( N ) with the same probability. By construction, we obtain a sample with replacement S = {σ1 , . . . , σ` } of ` orders of N.

5. The estimation of the ith -component of RA, RAi , is the mean of the marginal contributions vectors over the sample of permutations S . Formally, we obtain a vector RA = ( RAi )i∈ N where, for each i ∈ N, RAi = 1` ∑σ∈S x (σ)i approximates RAi being ` the sampling size.

Algorithm 3.1 presents the pseudocode of this sampling procedure. Algorithm 3.1. Let ( N, c, E) ∈ B N be a bankruptcy problem. Set `. Do cont = 0 and RAi = 0, for all i ∈ N.

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while cont < ` do Take σ ∈ Π( N ) with probability

1 n! .

for i ∈ N do

  Calculate Piσ and x (σ)i = min ci , max 0, E − ∑ c j . j∈ Piσ

Do RAi = RAi + x (σ)i .

end for cont = cont + 1. end while i Finally, RAi = RA ` for each i ∈ N.

The problem of estimating the random arrival rule for i ∈ N corresponds to the approximation of a

population parameter since it is the mean of the marginal contributions x (σ)i . Therefore, the approximation of this rule is seen as a very common task in Statistics, for which simple random sampling becomes a useful tool. Below, we analyse some properties of our sampling procedure from a statistical point of view. Given an agent i ∈ N, the estimator RAi is clearly unbiased since that E( RAi ) = E



1 x ( σ )i ` σ∑ ∈S

Besides, Var( RAi ) = Var





= E( x (σ)i ) = RAi .

1 x ( σ )i ` σ∑ ∈S



=

θ2 `

where θ 2 denotes the variance of the marginal contributions with respect to the theoretical mean RAi . Hence, taking into account that MSE( RAi ) = E( RAi − RAi )2 = (E( RAi ) − RAi )2 + Var( RAi ), and the unbiased character of RAi , MSE( RAi ) = Var( RAi ), that goes to zero when ` increases.

3.2

Error analysis of sampling estimation

A fundamental issue in a problem as the one dealt with focuses on bounding the absolute error in the estimation. Since this error is often not possible to be measured, a probabilistic bound on its value is theoretically provided instead. This means that the error is guaranteed to be not within a bound ε with a certain probability α as maximum. Formally, this is equivalent to P(| RAi − RAi | ≥ ε) ≤ α. It is easy to check that the estimated value usually is a good approximation of the random arrival rule for bankruptcy when sampling sizes is sufficiently large. In what follows, we state a collection of statistical results that can be useful for determining the required sample size. The first result we provide, based on Tchebyshev’s inequality, helps with bounding the incurred estimation error in the approximation of the random arrival rule. Proposition 3.1 uses the variance of the marginal contributions to this aim even though, in most cases, its value is unknown (or not be explicitly calculated). This fact would limit its direct use. To solve this drawback, Popoviciu’s inequality on variances (Popoviciu, 1935) provides an upper bound on its value based on the range of the random

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variable. Formally, it ensures that, given any bounded random variable X, Var( X ) always satisfies that 1 ( K − k )2 , 4

Var( X ) ≤

(4)

being K and k the upper and the lower bounds on the values of X, respectively. Thus, determining the range of the marginal contributions in bankruptcy plays a fundamental role in the analysis of the error. Its value for a fixed i ∈ N, denoted by wi , is given by wi =

max

σ,σ0 ∈Π( N )



x ( σ )i − x ( σ 0 )i .

Below, we introduce some basic comments about x (σ)i that may result of interest before the statistical analysis. By using Expression (3), the value of x (σ)i is particularly written for bankruptcy problems as

x ( σ )i =

    ci   

E ≥ ∑ j∈ Piσ c j + ci ;

if

E − ∑ j∈ Piσ c j

0

0 ≤ E − ∑ j∈ Piσ c j ≤ ci ;

if

E ≤ ∑ j∈ Piσ c j .

if

Hence, the marginal contributions always satisfies that 0 ≤ x (σ)i ≤ ci for a fixed agent i, in such way that it can naturally tolerated wi = ci . Consequently, we write two usual results to bound the incurred error in the estimation of the random arrival rule directly in terms of the value of ci . They can be considered as specific theoretical results for bankruptcy. Proposition 3.1. Let ( N, c, E) ∈ B N be a bankruptcy problem and take ε > 0, α ∈ (0, 1). Then,

`≥

c2i implies that P(| RAi − RAi | ≥ ε) ≤ α. 4αε2

Proof. Clearly, we have that

q ε Var( RAi )  P(| RAi − RAi | ≥ ε) = P(| RAi − E( RAi )| ≥ ε) = P | RAi − E( RAi )| ≥ q . Var( RAi ) 

Applying Tchebyshev’s inequality, it holds that q  ε Var( RAi )  Var( RAi ) θ2 ≤ = 2, P | RAi − E( RAi )| ≥ q 2 ε `ε Var( RAi )

where θ 2 denotes the variance of the x (σ)i . Since that Popoviciu’s inequality ensures that θ 2 can be bounded by wi2 /4 and wi = ci , taking α such that wi2 c2i θ2 ≤ = ≤ α. ` ε2 4` ε2 4` ε2

This concludes the proof. Let us introduce a second statement based on Hoeffding’s concentration inequality. It helps in bounding the error in the estimation of the random arrival rule through the direct use of the range of the marginal contributions. Hoeffding’s inequality, introduced in Hoeffding (1963), states that if ∑kj=1 X j denotes the sum of k observations X1 , . . . , Xk drawn with replacement, with a j ≤ X j ≤ b j for all j ∈ {1, . . . , k}, then

k

k

j =1

j =1

P(| ∑ X j − E( ∑ X j )| ≥ t) ≤ 2 exp

8



 −2t2 , for all t ≥ 0. ∑kj=1 (b j − a j )2

(5)

Proposition 3.2. Let ( N, c, E) ∈ B N be a bankruptcy problem, take ε > 0 and α ∈ (0, 1). Then,

`≥

ln(2/α)c2i implies that P(| RAi − RAi | ≥ ε) ≤ α. 2ε2

Proof. Clearly, since that RAi = 1` ∑ x (σ)i for a sample of ` elements, σ∈S

P(| RAi − RAi | ≥ ε) = P(| RAi − E( RAi )| ≥ ε) = P(| ∑ x (σ)i − E( ∑ x (σ)i )| ≥ ε`). σ ∈S

σ ∈S

Applying Hoeffding’s inequality (5) and the previous comments on wi , it holds that
P ∑ x (σ)i − E( ∑ x (σ)i ) ≥ ε` ≤ 2 exp σ∈S

σ∈S



−2ε2 ` wi2



= 2 exp



−2ε2 ` c2i



≤ α,

concluding the proof.

The following corollary combines Propositions 3.1 and 3.2. It establishes a very general result in terms of the claims for determining the sample size. Corollary 3.1. Let ( N, c, E) ∈ B N be a bankruptcy problem and take ε > 0 and α ∈ (0, 1). Then,

` ≥ min



 1 ln(2/α) 2 , ci implies that P(| RAi − RAi | ≥ ε) ≤ α. 4αε2 2ε2

Below, we introduce some final comments that are helpful in bounding the error in the estimation of the random arrival rule for bankruptcy problems. • For the usual values of α (α = 0.1, α = 0.05 or α = 0.01), Maleki (2015) proves that Hoeffding’s inequality requires a smaller sampling size than Chebyshev’s inequality. In our setting, min



 1 ln(2/α) 2 ln(2/α) 2 , ci . ci = 4αε2 2ε2 2ε2

Even more, the previous equality only holds for those values of α such that α ≤ 0.23 (p. 46 of Maleki, 2015).

• If ci is obtained in a polynomial time for all i ∈ N, Bachrach et al. (2010) state that the number of required samples to approximate the random arrival value is O( p(n)ln(1/α)) for a given value ε = p(1n) with p(n) a polynomial, and a given confidence level of 1 − α. When α is exponentially small, the required amount of samples is polynomial.

3.3

Algorithm performance on two numerical examples

We illustrate the performance of our sampling procedure in two well-known examples in literature where the exact allocation vector can be determined. However, in both situations, the computation of the random arrival rule is not an easy task using Expression (1) since the set of agents is large enough. O’Neill (1982) ensures that this rule can be obtained as the Shapley value of bankruptcy games. The needing of the procedure proposed is justified by the lack of examples in bankruptcy literature with large set of players because until now, they were not be treated under the approach of the random arrival rule.

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The results included in next sections have been performed computing our sampling proposal into the statistical software R (R Core Team, 2019) on a personal computer with Intel(R) Core(TM) i5-7400 and 8 GB of memory and with a single 3.00GHz CPU processor.

3.3.1

The Pacific Gas and Electric Company

This example of bankruptcy is extracted from Borm, Carpente, Casas-Méndez, and Hendrickx (2005). It describes the situation that arises when the American company Pacific Gas and Electric Company declares bankruptcy. Although that work emphasizes in the analysis of bankruptcy rules for situations with a priori unions of claimants, we use this example to evaluate the performance of our estimating proposal. First, we determine the random arrival rule as the Shapley value of the bankruptcy game in (2). i 1 2 3 4 5 6 7 8 9 10

ci 2207.2500 1966.0000 1302.1000 1228.8000 938.4610 310.0000 57.9284 49.4526 48.4006 45.7064

RAi 186.5294 186.5294 186.5294 186.5294 174.4384 52.6898 10.3302 8.8338 8.6475 8.1700

i 11 12 13 14 15 16 17 18 19 20

ci 40.1472 40.1221 32.8679 29.5235 28.2106 24.7183 23.8495 22.5765 21.5061 19.8002

RAi 7.1829 7.1785 5.8877 5.2914 5.0571 4.4334 4.2782 4.0505 3.8590 3.5538

Table 2: Claims and random arrival rule for the bankruptcy problem ( N, c, E). Table 2 displays the claims (in millions of dollars) for the set of the different agents N = {1, . . . , 20}

that are involved. The estate to be allocated among the creditors is equal to E = 1060 millions of dollars. Table 2 also displays the vector RA = ( RA1 , . . . , RA20 ) provided by the random arrival rule for this bankruptcy situation (of dimension 20). i α α α i α α α i α α α i α α α

= 0.1 = 0.05 = 0.01 = 0.1 = 0.05 = 0.01 = 0.1 = 0.05 = 0.01 = 0.1 = 0.05 = 0.01

1 2919013106 3594409142 5162630175 6 57577975 70900264 101833660 11 965706 1189149 1707967 16 366077 450779 647451

2 2315794515 2851618912 4095764632 7 2010555 2475753 3555911 12 964495 1187658 1705826 17 340793 419645 602733

3 1015831491 1250872766 1796621706 8 1465250 1804276 2591472 13 647257 797018 1144752 18 305385 376044 540110

4 904680854 1114004294 1600038269 9 1403570 1728326 2482384 14 522240 643075 923644 19 277113 341231 490107

5 527673705 649765903 933255210 10 1251661 1541268 2213715 15 476822 587149 843318 20 234896 289245 415441

Table 3: Sampling sizes for estimating the random arrival rule with wi = ci for all i ∈ N. We use our proposal of sampling procedure to estimate the random arrival rule. In view of the exact allocation, we consider that bounds of the absolute error of ε = 0.1, 0.05 or 0.01 (which correspond to 1000 thousands, 50 thousands or a thousand of dollars, respectively) can be naturally tolerated. For instance, the minimum sampling sizes to guarantee that absolute errors smaller than or equal to ε = 0.05 with probability at least 1 − α are the ones in Table 3.

10

First, we compare the estimations of the random arrival rule obtained by using several sampling sizes. Table 4 displays the absolute errors obtained when we take ` = 103 , ` = 106 and ` = 109 . Besides, it is easy to check that when the sizes of the samples increase, the absolute error and the estimated variance associated reduce. As a direct consequence, MSE( RAi ) also decreases when ` enlarges. i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Abs. error 5.0019 1.9747 9.1442 3.7237 9.8521 2.5480 0.6930 0.7409 0.7575 0.5922 0.2079 0.1260 0.2788 0.3450 0.0149 0.0139 0.1628 0.3480 0.0031 0.2980

` = 103 Err. th., α = 0.01 113.6071 101.1900 67.0191 63.2463 48.3026 15.9557 2.9816 2.5453 2.4912 2.3525 2.0664 2.0651 1.6917 1.5196 1.4520 1.2723 1.2275 1.1620 1.1069 1.0191

Est. variance

Abs. error

140.5611 141.3841 137.1674 141.9141 130.8937 13.9170 0.5093 0.3797 0.3172 0.2858 0.2405 0.2390 0.1526 0.1336 0.1164 0.0896 0.0858 0.0699 0.0677 0.0537

0.0676 0.0963 0.0823 0.0950 0.1740 0.07823 0.0078 0.0206 0.0040 0.0093 0.0209 0.0012 0.0043 0.0091 0.0138 0.0116 0.0076 0.0039 0.0051 0.0050

` = 106 Err. th., α = 0.01 3.5926 3.1999 2.1193 2.0000 1.5275 0.5046 0.0943 0.0805 0.0788 0.0744 0.0654 0.0653 0.0535 0.0481 0.0459 0.0402 0.0388 0.0368 0.0350 0.0322

Est. variance

Abs. error

1.4253 · 10−1 1.4270 · 10−1 1.4262 · 10−1 1.4259 · 10−1 1.2485 · 10−1 1.3347 · 10−2 4.8833 · 10−4 3.5599 · 10−4 3.4168 · 10−4 3.0527 · 10−4 2.3618 · 10−4 2.3530 · 10−4 1.5834 · 10−4 1.2762 · 10−4 1.1695 · 10−4 8.9858 · 10−5 8.3626 · 10−5 7.4911 · 10−5 6.8006 · 10−5 5.7675 · 10−5

0.0057 0.0085 0.0017 0.0042 0.0005 0.0056 0.0001 2.6000 · 10−5 0.0005 1.1000 · 10−5 0.0001 0.0003 0.0002 0.0003 0.0002 0.0001 0.0002 0.0002 0.0002 0.0001

` = 109 Err. th., α = 0.01 0.1136 0.1012 0.0670 0.0633 0.0483 0.0160 0.0030 0.0026 0.0025 0.0024 0.0021 0.0021 0.0017 0.0015 0.0015 0.0013 0.0012 0.0012 0.0011 0.0010

Est. variance 1.4260 · 10−4 1.4260 · 10−4 1.4260 · 10−4 1.4259 · 10−4 1.2499 · 10−4 1.3329 · 10−5 4.8806 · 10−7 3.5665 · 10−7 3.4172 · 10−7 3.0498 · 10−7 2.3566 · 10−7 2.3535 · 10−7 1.5825 · 10−7 1.2778 · 10−7 1.1672 · 10−7 8.9683 · 10−8 8.3505 · 10−8 7.4857 · 10−8 6.7942 · 10−8 5.7610 · 10−8

Table 4: Estimation of the random arrival rule for ( N, c, E) for several values of `.

From Table 4, we conclude that our proposal satisfactorily accurates the random arrival rule in this example. When comparing the theoretical absolute errors and the ones depicted in Table 4, the absolute errors obtained in practice are significantly smaller than the ones provided by Corollary 3.1. To check this conjecture, we do a small simulation study. In particular, Table 5 shows the theoretical errors for α = 0.1 as well as the minimum, maximum and mean observed absolute errors in 1000 estimations by using ` = 107 . Notice that the absolute errors are smaller than the theoretical ones. However, there exist some components in which the maximum absolute error slightly exceeds the above-mentioned bounds (less than an α). i Theoretical, α Maximum Average Minimum i Theoretical, α Maximum Average Minimum i Theoretical, α Maximum Average Minimum i Theoretical, α Maximum Average Minimum

= 0.1

= 0.1

= 0.1

= 0.1

1 0.8543 3.972 · 10−1 9.250 · 10−2 2.710 · 10−4 6 0.1200 1.217 · 10−1 2.885 · 10−2 7.400 · 10−5 11 0.0155 1.666 · 10−2 3.693 · 10−3 1.000 · 10−6 16 0.0096 9.896 · 10−3 2.428 · 10−3 3.000 · 10−6

2 0.7609 3.851 · 10−1 9.601 · 10−2 2.400 · 10−5 7 0.0224 1.978 · 10−2 5.757 · 10−3 3.000 · 10−5 12 0.0155 1.819 · 10−2 3.707 · 10−3 2.000 · 10−6 17 0.0092 8.772 · 10−3 2.279 · 10−3 4.000 · 10−6

3 0.5039 3.752 · 10−1 9.216 · 10−2 7.000 · 10−6 8 0.0191 1.994 · 10−2 4.783 · 10−3 5.000 · 10−6 13 0.0127 1.391 · 10−2 3.232 · 10−3 1.000 · 10−6 18 0.0087 8.468 · 10−3 2.189 · 10−3 3.000 · 10−6

4 0.4756 3.622 · 10−1 9.556 · 10−2 3.920 · 10−4 9 0.0187 1.708 · 10−2 4.501 · 10−3 1.000 · 10−6 14 0.0114 1.260 · 10−2 2.992 · 10−3 1.000 · 10−6 19 0.0083 8.718 · 10−3 2.178 · 10−3 3.000 · 10−6

5 0.3632 4.612 · 10−1 9.015 · 10−2 1.750 · 10−4 10 0.0177 1.887 · 10−2 4.293 · 10−3 8.000 · 10−6 15 0.0109 9.845 · 10−3 2.702 · 10−3 4.000 · 10−6 20 0.0077 7.396 · 10−3 1.936 · 10−3 3.000 · 10−6

Table 5: Summary of the absolute errors in the 1000 simulations.

11

3.3.2

A conflictive situation in university management

The situation of allocating resources in a Spanish university extracted from Pulido, Sánchez-Soriano, and Llorca (2002) is considered next. The aim of this example consists of distributing the available money among the entities to finance the purchase of equipment to be used in teaching. This problem is also modelled from a bankruptcy approach. i 1 2 3 4 5 6 7 8 9

ci 15720.66 25532.200 32960.440 13664.610 8173.760 3904.170 14869.040 289753.130 250962.130

RAi 3052.382 4943.097 6368.163 2654.940 1590.825 760.817 2887.800 50564.210 44711.150

i 10 11 12 13 14 15 16 17 18

ci 126857.130 248338.50 227091.640 63069.720 15915.980 10059.720 530070.440 121229.150 233163.45

RAi 23837.810 44306.360 40983.460 12084.870 3090.118 1956.776 79348.800 22807.450 41940.310

i 19 20 21 22 23 24 25 26 27

ci 248008.450 169534.830 240404.840 250845.440 70752.960 140679.050 227684.440 234125.140 264726.580

RAi 44255.370 31455.250 43075.050 44693.170 13518.360 26346.780 41077.080 42091.090 46813.040

Table 6: Claims and random arrival rule for the bankruptcy problem ( N, c, E).

The elements that describe the problem are indicated below. There exists a collection of claims (in euros, in Table 6) for the set of the 27 agents, i.e. N = {1, . . . , 27}. The estate is E = 721214.5

euros and the random arrival rule proposes for this bankruptcy situation is given by the vector RA =

( RA1 , . . . , R27 ) in Table 6. The random arrival rule is exactly obtained as the Shapley value of the associated bankruptcy game. We evaluate how our sampling proposal performs to estimate the random arrival rule also for this example. We choose the sampling size and again, the question that arises is what the appropriate sampling size is. Previously, we manage a bearable value of the incurred error. In these contexts, we assume as natural a value of ε = 10 euros. Table 7 provides the required sampling sizes that ensure an absolute error smaller or equal than 10 with probability at least 1 − α. i α α α i α α α i α α α

= 0.1 = 0.05 = 0.01 = 0.1 = 0.05 = 0.01 = 0.1 = 0.05 = 0.01

1 3701814 4558333 6547109 10 241047575 296820732 426321993 19 921310369 1134481516 1629449590

2 9764489 12023778 17269687 11 923764163 1137503064 1633789424 20 430517563 530129947 761422741

3 16272677 20037820 28780212 12 772458749 951188873 1366187372 21 865684052 1065984481 1531067674

4 2796840 3443967 4946551 13 59581964 73367932 105377959 22 942508822 1160584829 1666941636

5 1000730 1232277 1769913 14 3794371 4672306 6710807 23 74982900 92332309 132616390

6 228313 281140 403800 15 1515811 1866536 2680895 24 296436623 365025599 524284269

7 3311608 4077842 5856982 16 4208624456 5182408462 7443464905 25 776496865 956161322 1373329272

8 1257561621 1548533981 2224150880 17 220133999 271068211 389333787 26 821049050 1011021912 1452125238

9 943385911 1161664858 1668492876 18 814317838 1002733244 1440220269 27 1049707018 1292586352 1856534702

Table 7: Sampling sizes for estimating the random arrival rule with wi = ci for all i ∈ N. Table 8 illustrates the cases of estimating the random arrival rule with ` = 103 , ` = 106 and ` = 109 . We check that the absolute error, the estimated variances and, as a consequence, the mean square error of the estimator reduce when sampling size enlarges. In view of Table 8. the random arrival rule is correctly approximated also in this example. However, the theoretical values of the error provided by Corollary 3.1 are usually larger than the ones observed in the estimations (the amount of absolute errors that exceed the theoretical values is, as maximum, only α per cent of the total). We do a small simulation to check this conjecture. Table 9 displays the

12

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Abs. error 260.9850 516.2350 145.8130 58.6190 3.6250 50.2579 132.7960 629.1200 2196.9100 3708.4500 6223.8900 3054.7900 633.7700 190.0520 115.5260 3314.8500 1000.3700 161.1900 1955.2000 417.5300 4359.7900 1392.6100 286.3300 1465.8300 6262.1300 3365.4000 3583.1500

` = 103 Err. th., α = 0.01 809.1420 1314.1418 1696.4731 703.3172 420.7033 200.9475 765.3092 14913.5874 12917.0154 6529.3337 12781.9772 11688.4018 3246.1971 819.1951 517.7736 27282.7141 6239.6617 12000.9178 12764.9896 8725.9540 12373.6319 12911.0094 3641.6533 7240.7477 11718.9132 12050.4159 13625.4714

Est. variance

Abs. error

41065.500 108435.400 165042.200 29569.760 10486.070 2269.249 35681.980 11188237.000 8121076.000 2656939.000 9271984.000 6692133.000 585676.000 37569.550 16552.342 7884021.000 2097994.000 7464195.000 8073612.000 4095551.000 7203787.000 8520330.000 753941.000 3046781.000 7903876.000 8107100.000 8765723.000

5.2420 6.4590 3.2470 7.4280 5.9420 0.2149 2.4480 125.5300 50.3400 17.4400 34.8000 57.3800 8.0000 5.3780 0.6940 55.5200 65.3200 58.0800 60.7800 67.1500 15.1700 2.3300 22.6300 25.7000 6.4700 64.0500 9.9700

` = 106 Err. th., α = 0.01 25.5873 41.5568 53.6472 22.2408 13.3038 6.3545 24.2012 471.6090 408.4719 206.4757 404.2016 369.6197 102.6538 25.9052 16.3734 862.7552 197.3154 379.5023 403.6644 275.9389 391.2886 408.2820 115.1592 228.9725 370.5846 381.0676 430.8752

Est. variance

Abs. error

38.5847 101.1176 168.2288 29.2014 10.4243 2.3901 34.5004 11081.4900 8567.1990 2390.6900 8396.9070 7144.8320 607.3063 39.4421 15.8160 9021.0900 2190.9600 7508.8610 8379.1700 4163.8070 7927.7780 8559.0500 762.3654 2917.3440 7187.5180 7549.9990 9420.8310

0.0030 0.1290 0.0090 0.1100 0.0790 0.0127 0.2230 6.1600 1.9000 1.4300 5.9700 1.4500 0.0000 0.2170 0.0560 1.6000 0.4700 1.0700 1.7100 2.3400 5.1000 1.5400 0.0300 1.2800 3.2200 0.1600 7.4300

` = 109 Err. th., α = 0.01 0.8091 1.3141 1.6965 0.7033 0.4207 0.2009 0.7653 14.9136 12.9170 6.5293 12.7820 11.6884 3.2462 0.8192 0.5178 27.2827 6.2397 12.0009 12.7650 8.7260 12.3736 12.9110 3.6417 7.2407 11.7189 12.0504 13.6255

Est. variance 0.0385 0.1012 0.1682 0.0291 0.0105 0.0024 0.0345 11.0591 8.5646 2.3885 8.4034 7.1545 0.6076 0.0395 0.0158 9.0500 2.1859 7.5033 8.3847 4.1704 7.9292 8.5569 0.7612 2.9193 7.1885 7.5590 9.4195

Table 8: Estimation of the random arrival rule for ( N, c, E) for several values of `.

minimum, maximum and averaged absolute errors for 1000 estimations of the random arrival rule by using ` = 107 in this example. i Theoretical, α = 0.1 Maximum Average Minimum i Theoretical, α = 0.1 Maximum Average Minimum i Theoretical, α = 0.1 Maximum Average Minimum

1 6.0843 6.668 1.550 0.001 10 49.0966 62.960 12.744 0.010 19 95.9849 85.900 23.542 0.070

2 9.8815 13.509 2.528 0.001 11 96.1127 102.560 23.137 0.060 20 65.6138 73.300 15.948 0.090

3 12.7564 13.682 3.332 0.001 12 87.8896 96.850 21.837 0.040 21 93.0421 105.230 22.217 0.070

4 5.2885 7.146 1.304 0.000 13 24.4094 30.770 6.103 0.000 22 97.0829 88.140 23.322 0.010

5 3.1634 3.743 0.810 0.001 14 6.1599 6.948 1.599 0.005 23 27.3830 27.890 6.862 0.010

6 1.5110 1.635 0.393 0.000 15 3.8933 4.660 0.998 0.000 24 54.4460 56.130 13.361 0.060

7 5.7547 5.656 1.479 0.001 16 205.1493 200.140 43.988 0.090 25 88.1191 84.020 21.323 0.170

8 112.1411 123.240 27.119 0.040 17 46.9184 53.770 11.872 0.010 26 90.6118 86.880 21.702 0.000

9 97.1281 96.600 22.727 0.090 18 90.2396 98.800 21.558 0.070 27 102.4552 106.640 24.329 0.030

Table 9: Summary of the absolute errors in the 1000 simulations.

4

An application: the dairy sector in Galicia

The sampling procedure considered in this work will be applied on the real bankruptcy situation which arises from the end of the milk quotas regime imposed by the European Union (EU) to regulate the milk market. Notice that the analysis of this class of situations can be modelled as a bankruptcy problem when the maximum of tons of milk in 2014-2015 imposed for Galicia reduces by ρ% of the total, with ρ ∈ (0, 100]. From Table 15 in Appendix A, the set of involved agents is given by the 190 councils of

Galicia indicated above, that is, N = {1, . . . , 190}.

13

Council A Baña A Capela A Coruña A Laracha Abegondo Ames Aranga Arteixo Arzúa As Pontes As Somozas Bergondo Betanzos Boimorto Boqueixón Brión Cabana de Bergantiños Cabanas Cambre Carballo Carnota Carral Cee Cerceda Cerdido Cesuras Coirós Coristanco Culleredo Curtis Dumbría Fene Frades Irixoa Laxe Lousame Malpica de Bergantiños Mañón Mazaricos Melide Mesía Miño Moeche Monfero Mugardos Muxía Narón Neda Negreira Noia Ferrol O Pino Oleiros Ordes Oroso Ortigueira Outes Oza dos Ríos Paderne Padrón Ponteceso Pontedeume Rianxo Ribeira Rois San Sadurniño Santa Comba Santiago de Compostela Santiso Sobrado Teo Toques Tordoia Touro Trazo Val do Dubra Valdoviño Vedra Vilarmaior Vilasantar Vimianzo Zas A Fonsagrada A Pastoriza A Pobra de Brollón A Pontenova Abadín Alfoz Antas de Ulla As Nogais Baleira Baralla Barreiros Becerreá Begonte

ρ = 40% 11531.6000 5417.1210 363.2557 8327.6310 1769.9210 1171.9090 11158.8100 1845.1650 30524.8800 1323.5130 580.2840 215.7458 119.4559 10750.7700 8098.4790 7768.3190 5692.7240 466.1630 1384.6440 6381.0260 23.1046 1317.4720 325.7019 6913.5900 697.4539 5257.6970 31.6774 6570.8510 520.4388 16277.0700 9750.5620 217.6453 29396.9900 5269.7750 354.7188 2943.0940 1448.7900 178.4471 36701.8500 9334.8630 25858.8600 983.3110 1571.1770 12659.9900 197.9694 10002.7800 651.0677 221.4089 15926.8000 57.1563 282.4139 6693.3010 108.3295 15990.1300 5114.8610 1876.8090 1075.2080 1815.1300 2029.8490 828.1204 8106.6410 296.9162 12.7543 146.6077 8000.5860 4641.1250 39973.4800 7502.0780 10507.5600 7107.6680 1178.3510 3843.9140 22535.4300 13997.2000 21689.4700 13567.4800 567.3625 2058.7940 1359.7190 6628.8350 13666.1800 10433.4500 845.7969 52481.5400 4777.4240 3063.3700 7384.5610 728.7218 2836.3770 160.8673 6162.3120 3267.4730 10629.2200 882.1060 3755.0420

ρ = 10% 17267.4000 8099.3200 542.3041 12460.1100 2643.5450 1750.3730 16704.6300 2757.6750 45873.0500 1977.7620 866.9816 322.1830 178.3510 16097.7000 12114.9100 11616.2800 8513.0140 696.2620 2067.4180 9541.6740 34.5039 1968.1560 486.5507 10340.8200 1041.5410 7862.4080 47.3176 9823.1400 777.1865 24391.7800 14588.3200 325.2654 44167.0700 7876.2040 529.6529 4398.6690 2164.3870 266.5277 55227.9900 13965.2600 38812.6900 1469.1400 2345.7400 18953.7100 295.5340 14965.3200 972.3772 330.5858 23855.4400 85.3480 421.6757 10007.3400 161.7344 23959.2200 7646.7010 2803.2130 1606.0250 2711.8630 3032.5960 1236.6070 12128.3400 443.5647 19.0509 218.7929 11960.2600 6938.0720 60195.0700 11220.2900 15726.2900 10628.8700 1759.4100 5741.7960 33807.0200 20958.4300 32543.0900 20312.9700 847.2032 3074.9620 2030.3710 9915.3780 20463.9300 15613.2800 1263.5530 79234.2400 7140.5840 4578.3320 11042.4200 1088.3740 4239.5500 240.3023 9214.0920 4883.0550 15902.6300 1317.4200 5611.9550

ρ = 5% 18222.4700 8547.0650 572.0433 13147.0800 2789.7360 1846.7140 17627.3800 2909.3920 48429.1500 2086.6130 914.7122 339.9142 188.1838 16987.2600 12784.8300 12258.9200 8982.7910 734.6168 2181.5220 10068.7000 36.4038 2076.3000 513.3582 10912.4700 1098.8760 8296.0430 49.9307 10365.4100 819.9246 25743.6200 15394.4200 343.1988 46632.6300 8309.7230 558.8793 4641.1260 2283.5240 281.1681 58311.7500 14736.0500 40972.5200 1550.1140 2474.7720 20003.9500 311.7936 15793.8100 1025.9590 348.7634 25173.8700 90.0524 444.8918 10560.4800 170.6231 25287.7700 8069.0780 2957.2840 1694.4280 2861.0410 3199.9150 1304.6930 12796.9800 467.9681 20.0992 230.8559 12619.7600 7319.8620 63563.4700 11840.8400 16595.4100 11214.8400 1856.4650 6058.7230 35688.8300 22118.7100 34353.5600 21435.1400 893.8259 3244.6190 2141.9020 10462.7800 21595.7400 16478.1700 1333.1780 83694.4100 7534.1030 4830.7010 11653.0200 1148.3560 4472.9380 253.5094 9723.3690 5152.2790 16783.0000 1389.9710 5921.7600

Council Bóveda Carballedo Castro de Rei Castroverde Chantada Cospeito Foz Friol Guitiriz Guntín Láncara Lourenzá Lugo Meira Mondoñedo Monforte de Lemos Monterroso O Corgo O Incio O Páramo O Saviñao O Valadouro Ourol Outeiro de Rei Palas de Rei Pantón Paradela Pedrafita do Cebreiro Pol Portomarín Ribadeo Ribeira de Piquín Riotorto Samos Sarria Sober Taboada Trabada Triacastela Vilalba Xermade A Bola A Gudiña A Merca A Mezquita A Teixeira Allariz Baltar Barbadas Cartelle Celanova Coles Lobios Maceda Manzaneda O Carballiño O Irixo Piñor Rairiz de Veiga Ramirás Río Ríos San Amaro San Cristobo de Cea Sandiás Sarreaus Taboadela Trasmiras Viana do Bolo Vilariño de Conso Xinzo de Limia Xunqueira de Ambia Xunqueira de Espadanedo A Estrada A Golada Caldas de Reis Campo Lameiro Cerdedo Cotobade Cuntis Dozón Forcarei Lalín Meis Moraña O Grove Poio Pontecesures Pontevedra Portas Ribadumia Rodeiro Silleda Valga Vila de Cruces

ρ = 40% 2555.9690 14240.9400 43154.7300 22061.1100 32706.7400 32626.9300 2235.4110 16240.3600 22464.8100 25653.7300 9940.5520 2636.5110 27199.3300 6841.9430 2786.7910 3518.3930 11016.2300 9598.9370 2737.1650 12073.6300 14155.1300 1571.8250 13.5460 8749.5040 15914.7600 710.4362 16522.6300 1.4718 25674.7400 14866.0900 16530.4000 1054.6520 4935.1300 6440.8260 32054.6200 597.9441 24445.0000 10338.5200 902.0951 14606.0200 11750.5400 7.9266 158.6295 18.9051 481.8933 194.4081 1043.0100 78.7605 31.4843 278.2216 264.5972 124.5448 40.4932 2205.3550 71.1075 38.6345 171.2959 631.0525 20.8091 31.0982 172.6113 670.2090 93.4178 330.7577 116.7777 923.0562 108.5349 107.8688 800.6951 90.5826 1704.8260 517.4539 42.8836 11671.1100 6015.1940 3.5531 30.2840 40.0274 131.6613 501.4282 9422.7210 7465.7150 49602.2200 17.6661 24.3372 16.0321 11.6189 4.8334 183.9782 53.7867 79.0249 28936.9900 31302.4600 203.8511 9305.9700

ρ = 10% 3819.7830 21330.4700 65038.6900 33093.7100 49173.6900 49081.5300 3341.3530 24338.5000 33695.5100 38511.6100 14876.5200 3939.1940 40846.0700 10232.0500 4163.8580 5257.6370 16488.6500 14364.5800 4092.0650 18072.1600 21188.4600 2348.6640 20.2255 13083.3600 23839.0900 1061.7550 24742.9400 2.1984 38546.3500 22276.8800 24765.0900 1575.2920 7375.6660 9631.6520 48183.2600 892.9664 36681.9700 15467.8900 1347.5870 21874.0300 17584.2900 11.8415 236.7992 28.2302 719.8033 290.2887 1558.2260 117.6245 47.0310 415.5066 395.0777 185.9922 60.4744 3295.7610 106.1976 57.7216 255.7892 942.8653 31.0727 46.4235 257.8572 1000.7630 139.5357 494.1651 174.3335 1378.5490 162.0779 161.0809 1195.6240 135.2569 2546.5290 773.2849 64.0441 17465.6500 8993.7280 5.3061 45.2223 59.7849 196.6076 748.8389 14096.9100 11168.6400 74842.2300 26.3758 36.3548 23.9402 17.3550 7.2170 274.6791 80.3168 118.0362 43456.1600 47054.1600 304.4061 13925.6300

Table 10: Estimated random arrival rule for 190 councils in Galicia.

14

ρ = 5% 4030.4790 22512.1700 68689.1500 34935.1100 51920.4300 51823.4200 3525.8600 25688.2500 35568.2500 40650.8200 15699.3400 4155.6290 43114.1300 10796.6700 4393.1370 5547.5890 17399.9200 15159.6000 4317.4400 19073.4800 22361.4100 2478.0400 21.3426 13805.9500 25162.5300 1120.0690 26116.3700 2.3193 40692.1300 23512.2500 26134.9600 1662.1080 7781.8790 10163.8500 50871.4400 942.1605 38725.0900 16322.1800 1421.7990 23087.4700 18559.1500 12.4932 249.8258 29.7850 759.3857 306.3173 1644.0690 124.1030 49.6190 438.3916 416.8071 196.2302 63.8022 3477.2570 112.0551 60.9118 269.8679 994.7491 32.7827 48.9724 272.0695 1055.8310 147.2227 521.3146 183.9148 1454.4450 171.0030 169.9560 1261.6040 142.6903 2686.8870 815.7465 67.5714 18430.3700 9490.2410 5.5985 47.7087 63.0750 207.4030 790.1454 14877.6000 11784.7500 79045.5000 27.8289 38.3527 25.2599 18.3114 7.6143 289.8145 84.7369 124.5270 45877.0700 49676.3100 321.1784 14695.4500

Thus, the elements of the bankruptcy problem are formally defined as follows. The resources to be allocated among the different councils (the estate) corresponds to the (100 − ρ)% of the aggregate milk

quota for the region in 2015 that is, of 2229811.281 tons of milk. We take as claims of the councils the maximum bound of milk (in tons) given by the individual quotas in the period 2014-2015. A natural solution may consist in reducing the individual milk quotas in the same proportion. However, the random arrival rule for bankruptcy problems provides a new milk quota for each council under the fairness criteria that also bases the Shapley value for TU-games. The main difficulty for its obtaining is that the large set of players that are involved, implicitly increases the computational complexity for the exact computation. For this reason, sampling techniques are useful. We approximate the random arrival rule of each council in Galicia by using simple random sampling with replacement, with ` = 107 . In practice, we are only taking a small fraction of the total of the population of permutations to estimate the random arrival rule. Table 10 details an estimation in the three different scenarios that we consider. In particular, we choose ρ = 5%, ρ = 10% and ρ = 40%, in

such way that the portion of the regional milk quota for 2015 we take as estate is 95%, 90% and 60% of the total, respectively. In view of these results, we do some comments about the performance of the used method. Table 11 illustrates the list of those 10 councils in Galicia with the largest milk quotas. These values do not respect the ranking of the councils in terms of the number of farms that they have. For instance, Lalín or Santa Comba have the largest number of farms in the ranking in spite of they do not have the largest milk quotas. Position Council Farms Position Council Farms

1 A Pastoriza 313 6 Chantada 311

2 Lalín 478 7 Cospeito 255

3 Castro de Rei 259 8 Sarria 234

4 Santa Comba 389 9 Silleda 361

5 Mazaricos 307 10 Arzúa 250

Table 11: List of councils with the largest portion of the estimated milk quota for Galicia.

To conclude the analysis, we check the variability of the estimations of the random arrival rule. Just to simplify, we do a small simulation study only restricted to those components with the largest estimations of the random arrival rule (see Table 11). Table 12 summarizes such results for ρ = 40%. We only observe small variations in the estimations for the considered councils. The largest difference between the maximum and the minimum estimations is, as maximum, equal to 24 tons of milk for the indicated cases. Analogous conclusions can be obtained for ρ = 10% and ρ = 5%. Council A Pastoriza Lalín Castro de Rei Santa Comba Mazaricos Chantada Cospeito Sarria Silleda Arzúa

Maximum 52499.22 49633.35 43188.07 40002.56 36723.09 32731.26 32668.26 32068.42 31319.75 30539.21

Average 52472.49 49593.97 43161.96 39973.33 36696.91 32708.51 32645.75 32050.23 31300.44 30519.81

Minimum 52425.39 49561.72 43135.33 39942.24 36668.44 32681.45 32626.93 32030.53 31281.85 30502.95

Table 12: Summary of 100 estimations of the milk quotas for the councils with ρ = 40%.

15

4.1

Computation of milk quota systems using alternative bankruptcy rules

The previous section is mainly devoted to the estimation of the random arrival rule in different scenarios for the bankruptcy problem that arises in the milk conflict of Galicia. As we have mentioned, the use of specific allocation rules for the problem of bankruptcy allows the establishment of new milk quota systems. The random arrival rule that underlies this work is not the only one applicable in these settings. As we have previously mentioned, there are multiple procedures in the bankruptcy literature for this purpose and that are based on different criteria. In particular, the Talmud rule or the proportional rule are two of the most representative examples in this context that can be exactly obtained in a reasonable time. Next, we compare the different milk quotas that the above-mentioned rules provide. The Talmud, that collects a set of rabbinic discussions related to law, ethics, customs, and history under a Jewish criteria, includes several alternatives of division in bankruptcy. Aumann and Maschler (1985) rationally analyse these proposals. Let ( N, c, E) ∈ B N be a bankruptcy problem. The Talmud rule T = ( Ti )i∈ N assigns to each claimant i ∈ N,

 c   min 2i , λ , Ti ( N, c, E) =    ci − min c2i , λ ,

if

E≤ ∑

cl 2,

if

E≥ ∑

cl 2,

l∈N

l∈N

being λ such that ∑ Tl ( N, c, E) = E. Notice that this bankruptcy rule has interesting properties from l∈N

a game-theoretic point of view. Aumann and Maschler (1985) proves that the Talmud rule coincides with the nucleolus (Schmeidler, 1969) of the corresponding bankruptcy game. We can say that the nucleolus is that allocation that minimizes the dissatisfaction of the players belonging to the less favoured coalitions. Another option for sharing the estate among claimants is the proportional rule, denoted by P =

( Pi )i∈ N . This is an easily obtained allocation rule that assigns to each agent i in the bankruptcy problem ( N, c, E) ∈ B N , the portion of E corresponding to the weight given by its claim. Formally, it is given by Pi ( N, c, E) =

ci E, for each claimant i ∈ N. ∑ cl

l∈N

Unlike the calculation of the random arrival rule when the set of claimants is large enough, the computation of the two rules just described is relatively simple because they can be obtained in a polynomial time. For this reason they have been chosen to make a small comparative study of the milk quotas they provide under the perspective of bankruptcy in the example of Galicia. Table 16 in Appendix B depicts the distributions of new systems of milk quotas obtained with both bankruptcy rules for the ρ = 40% case, although the remaining cases, with ρ = 10% and with ρ = 5%, can be supplied on request. Tables 13 and 14 detail the list of the 10 councils with the largest claims in 2014-2015 and of the 10 ones with the shortest claims in that period. Moreover, the allocations proposed by the estimation of the random arrival rule, the Talmud rule and the proportional rule are depicted for the different bankruptcy problems that arise when reducing the total demand in Galicia ρ = 40%, 10% and 5%, respectively. It is worth mentioning that as an approximation of the random arrival rule we use the average of the 100 estimations used in the simulation study of the previous section. The Central Limit Theorem statistically ensures an improvement in the accuracy for the approximation to the real value of the corresponding allocation.

16

Council i A Pastoriza Lalín Castro de Rei Santa Comba Mazaricos Chantada Cospeito Sarria Silleda Arzúa

ci 87103.41 82356.54 71736.13 66465.18 61046.06 54438.74 54336.02 53345.57 52106.51 50811.05

RAi 52472.49 49593.97 43161.96 39973.33 36696.91 32708.51 32645.75 32050.23 31300.44 30519.81

ρ = 40% Ti 71688.74 66941.87 56321.46 51050.51 45631.39 39024.07 38921.35 37930.89 36691.84 35396.38

Pi 52262.05 49413.93 43041.68 39879.11 36627.64 32663.25 32601.61 32007.34 31263.91 30486.63

RAi 79238.94 74841.90 65040.18 60193.21 55222.70 49177.28 49084.05 48178.44 47046.42 45865.32

ρ = 10% Ti 85172.40 80425.53 69805.12 64534.17 59115.05 52507.73 52405.01 51414.55 50175.50 48880.04

Pi 78393.07 74120.89 64562.52 59818.66 54941.46 48994.87 48902.42 48011.01 46895.86 45729.95

RAi 48423.62 49671.73 50866.95 51823.27 51922.52 58309.75 63563.41 68687.52 79048.09 83695.25

ρ = 5% Ti 49981.25 51276.71 52515.76 53506.22 53608.94 60216.26 65635.38 70906.33 81526.74 86273.61

Pi 48270.50 49501.18 50678.29 51619.22 51716.81 57993.76 63141.92 68149.33 78238.72 82748.24

Table 13: List of the 10 councils with the largest milk quotas in 2015 and the allocations given by the estimated random arrival rule, the Talmud rule and the proportional rule.

In view of Table 13, the (estimated) random arrival rule and the proportional rule provide similar allocations for those 10 councils from Galicia with the largest milk quota. More specifically, we check that the milk quotas obtained under the proportional rule are slightly shorter than the ones given by the estimation of the random arrival rule in the three considered scenarios for the indicated councils. However, the differences between the estimated milk quota and the one under the Talmud rule are clearly more pronounced. In particular, the Talmud rule provides much larger milk quotas. Council i A Merca Meis O Grove Ourol Rianxo Poio A Bola Pontecesures Caldas de Reis Pedr. do Cebreiro

ci 31.59 29.52 26.79 22.64 21.32 19.42 13.25 8.08 5.94 2.46

RAi 18.91 17.66 16.03 13.55 12.76 11.62 7.93 4.83 3.55 1.47

ρ = 40% Ti 15.80 14.76 13.40 11.32 10.66 9.71 6.63 4.04 2.97 1.23

Pi 18.96 17.71 16.08 13.58 12.79 11.65 7.95 4.85 3.56 1.48

RAi 28.23 26.38 23.94 20.23 19.05 17.36 11.84 7.22 5.31 2.20

ρ = 10% Ti 15.80 14.76 13.40 11.32 10.66 9.71 6.63 4.04 2.97 1.23

Pi 28.43 26.57 24.11 20.37 19.19 17.48 11.93 7.27 5.34 2.21

RAi 29.78 27.83 25.26 21.34 20.10 18.31 12.49 7.61 5.60 2.32

ρ = 5% Ti 15.80 14.76 13.40 11.32 10.66 9.71 6.63 4.04 2.97 1.23

Pi 30.01 28.04 25.45 21.51 20.25 18.45 12.59 7.67 5.64 2.34

Table 14: List of the 10 councils with the shortest milk quotas in 2015 and the allocations given by the estimated random arrival rule, the Talmud rule and the proportional rule.

In the case of the 10 councils with the shortest milk quota, conclusions are analogous (see the results included in Table 14). The estimations of the random arrival rule and the proportional rule also assign similar portions of the estate although, in this case, councils obtain slightly larger milk quotas when using the proportional procedure. Contrary to what was happening for the councils with larger milk production in 2014-2015, the system of milk quotas obtained under the Talmud rule ensures milk production considerably smaller than those given by the other two procedures proposed.

5

Concluding remarks

In this work, we have described a procedure to estimate the random arrival rule for bankruptcy problems based on simple random sampling with replacement, as Fernández-García and Puerto-Albandoz (2006) and Castro et al. (2009) do for approximating the Shapley value. It results specially useful when dealing with those bankruptcy situations with a large set of agents. In such scenarios, the size of the set of permutations enlarges enough to make difficult the exact computation of the rule. We have provided some theoretical results to ensure that our proposal correctly approximates the real value and, consequently, to determine the adequate sampling size in the estimation. The performance of our sampling proposal has been evaluated in two real examples taken from the game-theoretic literature.

17

It is worth mentioning the application of this proposal of sampling on a practical case. It corresponds to the real-world situation with large set of involved players that motivates this paper and that is modelled under a bankruptcy approach. Specifically, this situation refers to the milk problem arisen in Galicia (Spain) after the suppression of the European milk quotas in April 2015 and that led to a social conflict in the region. In this setting, the random arrival rule in bankruptcy is introduced as a mechanism of determining a new system of milk quotas based on a fairness criteria as solution. Besides, we compare this new system of quotas with the ones obtained under other two well-known rules in bankruptcy literature, as the Talmud rule and the proportional rule. As in the estimation of coalition values for TU-games, the use of sampling methodologies to estimate allocation rules as the one that considered in this work is due to the fact of its definition in terms of a population mean. However, it is noteworthy that sampling methodologies may be also applied in approximating another allocations. For instance, Perea and Puerto (2019) make use of them to obtain through heuristic methods the nucleolus of general TU-games. Finally, we include several open tasks to be tackled in the future from the two different directions. • An interesting issue may consist in extending the estimation procedure which uses stratified sam-

pling for those scenarios of approximating the random arrival rule for bankruptcy. This idea would be analogous to the one provided by Maleki (2015) and Castro et al. (2017) for the estimation of the Shapley value using the mentioned sampling technique.

• Moreover, some future research is needed for extending the use of sampling techniques to esti-

mate another allocation rules in bankruptcy as the ones proposed, for instance, by Borm et al. (2005). In that work, the existence a priori unions restricts the affinities for the collaboration of agents. The use of alternative sampling techniques in this case may result of interest.

From a computational perspective, it is important to emphasize that the proposed procedure can be easily computed in parallel. This proposal considerably reduces the required time and the complexity with respect to the exact computation of the random arrival rule for bankruptcy problems.

Acknowledgments A. Saavedra-Nieves acknowledges the financial support of Ministerio de Economía y Competitividad of the Spanish government under grant MTM2017-87197-C3-2-P and of Xunta de Galicia through the ERDF (Grupos de Referencia Competitiva) ED431C 2016-040. P. Saavedra-Nieves acknowledges the financial support of Ministerio de Economía y Competitividad of the Spanish government under grants MTM201676969P and MTM2017-089422-P.

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Hwang, Y.-A. (2015). Two characterizations of the random arrival rule. Economic Theory Bulletin, 3(1), 43–52. Maleki, S. (2015). Addressing the computational issues of the Shapley value with applications in the smart grid (Unpublished doctoral dissertation). University of Southampton. Mann, I., and Shapley, L. S. (1960). Values of large games, IV: Evaluating the electoral college by Montecarlo techniques. The RAND Corporation: Santa Monica. Moulin, H. (2002). Axiomatic cost and surplus sharing. Handbook of social choice and welfare, 1, 289–357. O’Neill, B. (1982). A problem of rights arbitration from the Talmud. Mathematical Social Sciences, 2(4), 345–371. Perea, F., and Puerto, J. (2019). A heuristic procedure for computing the nucleolus. Computers & Operations Research, 112. Popoviciu, T. (1935). Sur les équations algébriques ayant toutes leurs racines réelles. Mathematica, 9, 129–145. Pulido, M., Sánchez-Soriano, J., and Llorca, N. (2002). Game theory techniques for university management: an extended bankruptcy model. Annals of Operations Research, 109(1-4), 129– 142. R Core Team. (2019). R: A Language and Environment for Statistical Computing [Computer software manual]. Vienna, Austria. Retrieved from https://www.R-project.org/ Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on applied mathematics, 17(6), 1163–1170. Shapley, L. S. (1953). A value for n-person games. Contributions to the Theory of Games, 2(28), 307–317. Thomson, W. (2015). Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update. Mathematical Social Sciences, 74, 41–59.

20

Appendix A. Milk quotas for 190 councils in Galicia in the period 2014-2015 Table 15 shows the milk quotas for 190 councils in Galicia in the period 2014-2015. Data were extracted from the website of Consellería de Medio Rural of Xunta de Galicia (https://mediorural.xunta.gal/). Council A Baña A Capela A Coruña A Laracha Abegondo Ames Aranga Arteixo Arzúa As Pontes As Somozas Bergondo Betanzos Boimorto Boqueixón Brión Cabana de Bergantiños Cabanas Cambre Carballo Carnota Carral Cee Cerceda Cerdido Cesuras Coirós Coristanco Culleredo Curtis Dumbría Fene Frades Irixoa Laxe Lousame Malpica de Bergantiños Mañón Mazaricos Melide Mesía Miño Moeche Monfero Mugardos Muxía Narón Neda Negreira Noia Ferrol O Pino Oleiros Ordes Oroso Ortigueira Outes Oza dos Ríos Paderne Padrón Ponteceso Pontedeume Rianxo Ribeira

Farms 105 56 1 105 27 16 94 15 250 22 11 3 2 102 61 57 37 7 16 73 1 17 5 70 6 71 1 65 7 122 77 4 183 63 7 23 15 2 307 114 198 8 13 159 1 86 8 3 140 1 1 78 1 170 55 10 21 26 15 3 54 4 1 4

Quota (ci ) 19253.147 9049.543 606.673 13905.151 2957.506 1958.122 18626.978 3084.636 50811.051 2212.419 970.062 360.532 199.583 17949.237 13523.385 12969.246 9509.237 779.033 2313.154 10655.94 38.603 2201.589 544.511 11547.136 1165.265 8784.328 52.966 10970.444 869.505 27152.543 16275.765 363.962 48950.157 8799.154 592.689 4918.423 2421.117 298.197 61046.064 15579.976 43066.524 1643.428 2623.408 21125.185 330.711 16694.545 1087.991 369.915 26551.744 95.521 471.831 11176.461 180.976 26674.213 8543.421 3135.175 1796.511 3032.885 3391.447 1383.377 13537.976 496.303 21.32 244.889

Council Rois San Sadurniño Santa Comba Santiago de Compostela Santiso Sobrado Teo Toques Tordoia Touro Trazo Val do Dubra Valdoviño Vedra Vilarmaior Vilasantar Vimianzo Zas A Fonsagrada A Pastoriza A Pobra de Brollón A Pontenova Abadín Alfoz Antas de Ulla As Nogais Baleira Baralla Barreiros Becerreá Begonte Bóveda Carballedo Castro de Rei Castroverde Chantada Cospeito Foz Friol Guitiriz Guntín Láncara Lourenzá Lugo Meira Mondoñedo Monforte de Lemos Monterroso O Corgo O Incio O Páramo O Saviñao O Valadouro Ourol Outeiro de Rei Palas de Rei Pantón Paradela Pedrafita do Cebreiro Pol Portomarín Ribadeo Ribeira de Piquín Riotorto

Farms 68 35 389 60 115 91 14 66 202 111 136 117 9 36 21 65 129 136 6 313 62 25 119 14 33 4 52 32 57 15 47 27 148 259 173 311 255 14 165 185 174 85 33 233 56 47 34 84 81 31 89 129 11 2 82 117 12 107 1 141 98 96 19 43

Quota (ci ) 13350.368 7752.548 66465.183 12528.073 17539.229 11865.545 1968.363 6417.981 37560.049 23348.929 36161.907 22630.365 947.869 3439.555 2270.729 11073.675 22799.899 17415.502 1413.767 87103.413 7978.205 5118.89 12328.51 1217.884 4740.799 268.871 10292.19 5458.796 17735.873 1473.789 6273.036 4271.243 23757.953 71736.134 36769.831 54438.744 54336.024 3736.9 27096.989 37429.143 42735.528 16596.042 4404.102 45298.976 11425.526 4655.363 5877.472 18387.254 16027.173 4575.134 20147.93 23605.507 2626.861 22.638 14598.522 26541.407 1187.648 27541.073 2.46 42774.089 24811.809 27561.883 1762.278 8239.8

Council Samos Sarria Sober Taboada Trabada Triacastela Vilalba Xermade A Bola A Gudiña A Merca A Mezquita A Teixeira Allariz Baltar Barbadas Cartelle Celanova Coles Lobios Maceda Manzaneda O Carballiño O Irixo Piñor Rairiz de Veiga Ramiras Río Ríos San Amaro San Cristovo de Cea Sandiás Sarreaus Taboadela Trasmiras Viana do Bolo Vilariño de Conso Xinzo de Limia Xunqueira de Ambia Xunqueira de Espadanedo A Estrada A Golada Caldas de Reis Campo Lameiro Cerdedo Cotobade Cuntis Dozón Forcarei Lalín Meis Moraña O Grove Poio Pontecesures Pontevedra Portas Ribadumia Rodeiro Silleda Valga Vila de Cruces TOTAL Galicia

Table 15: Milk quotas for 190 councils in Galicia in the period 2014-2015.

21

Farms 58 234 4 212 55 6 169 80 2 4 1 6 1 4 2 1 11 7 1 1 6 1 1 9 8 2 2 5 7 3 9 1 6 2 1 9 1 2 3 1 216 100 1 2 4 1 31 87 121 478 1 2 1 1 1 1 1 2 275 361 1 147 11974

Quota (ci ) 10757.825 53345.565 999.119 40720.833 17250.763 1507.664 24364.386 19607.536 13.253 264.956 31.593 805.467 324.91 1743.131 131.653 52.63 464.957 442.096 208.133 67.674 3685.43 118.854 64.606 286.276 1054.85 34.773 51.947 288.59 1119.825 156.146 552.869 195.067 1542.098 181.373 180.27 1337.802 151.339 2848.472 864.995 71.678 19471.968 10045.584 5.938 50.603 66.908 220.001 837.933 15730.244 12467.846 82356.544 29.518 40.681 26.793 19.424 8.076 307.417 89.872 132.076 48166.205 52106.51 340.646 15537.076 2229811.281

Appendix B. New systems of milk quotas for 190 councils in Galicia in the period 2014-2015 Table 16 shows two new systems of milk quotas for the set of 190 councils in Galicia in the period 2014-2015 when we impose a reduction of production equal to ρ = 40%, that is, when the estate is E = 1337887. They were obtained by using the Talmud rule and the proportional rule for the corresponding bankruptcy situation, respectively. The cases ρ = 10% and ρ = 5% can be supplied on request. Council A Baña A Capela A Coruña A Laracha Abegondo Ames Aranga Arteixo Arzúa As Pontes As Somozas Bergondo Betanzos Boimorto Boqueixón Brión Cabana de Bergantiños Cabanas Cambre Carballo Carnota Carral Cee Cerceda Cerdido Cesuras Coirós Coristanco Culleredo Curtis Dumbría Fene Frades Irixoa Laxe Lousame Malpica de Bergantiños Mañón Mazaricos Melide Mesía Miño Moeche Monfero Mugardos Muxía Narón Neda Negreira Noia Ferrol O Pino Oleiros Ordes Oroso Ortigueira Outes Oza dos Ríos Paderne Padrón Ponteceso Pontedeume Rianxo Ribeira

Talmud rule 9626.574 4524.772 303.337 6952.576 1478.753 979.061 9313.489 1542.318 35396.379 1106.210 485.031 180.266 99.792 8974.619 6761.693 6484.623 4754.619 389.517 1156.577 5327.970 19.302 1100.795 272.256 5773.568 582.633 4392.164 26.483 5485.222 434.753 13576.272 8137.883 181.981 33535.485 4399.577 296.345 2459.212 1210.559 149.099 45631.392 7789.988 27651.852 821.714 1311.704 10562.593 165.356 8347.273 543.996 184.958 13275.870 47.761 235.916 5588.231 90.488 13337.107 4271.711 1567.588 898.2555 1516.4425 1695.7235 691.6885 6768.9880 248.1515 10.6600 122.4445

Proport. rule 11551.888 5429.726 364.004 8343.091 1774.504 1174.873 11176.187 1850.782 30486.631 1327.451 582.037 216.319 119.750 10769.542 8114.031 7781.548 5705.5422 467.420 1387.892 6393.564 23.162 1320.953 326.707 6928.282 699.159 5270.597 31.780 6582.266 521.703 16291.526 9765.459 218.377 29370.094 5279.492 355.613 2951.054 1452.670 178.918 36627.638 9347.986 25839.914 986.057 1574.045 12675.111 198.427 10016.727 652.795 221.949 15931.046 57.313 283.099 6705.877 108.586 16004.528 5126.053 1881.105 1077.907 1819.731 2034.868 830.026 8122.786 297.782 12.792 146.934

Council

Talmud rule 6675.184 3876.274 51050.511 6264.037 8769.615 5932.773 984.182 3208.991 22145.377 11674.465 20747.235 11315.183 473.935 1719.778 1135.365 5536.838 11399.950 8707.751 706.884 71688.741 3989.103 2559.445 6164.255 608.942 2370.400 134.436 5146.095 2729.398 8867.937 736.895 3136.518 2135.622 11878.977 56321.462 21355.159 39024.072 38921.352 1868.450 13548.495 22014.471 27320.856 8298.021 2202.051 29884.304 5712.763 2327.682 2938.736 9193.627 8013.587 2287.567 10073.965 11802.754 1313.431 11.319 7299.261 13270.704 593.824 13770.537 1.230 27359.417 12405.905 13780.942 881.139 4119.900

Rois San Sadurniño Santa Comba Santiago de Compostela Santiso Sobrado Teo Toques Tordoia Touro Trazo Val do Dubra Valdoviño Vedra Vilarmaior Vilasantar Vimianzo Zas A Fonsagrada A Pastoriza A Pobra de Brollón A Pontenova Abadín Alfoz Antas de Ulla As Nogais Baleira Baralla Barreiros Becerreá Begonte Bóveda Carballedo Castro de Rei Castroverde Chantada Cospeito Foz Friol Guitiriz Guntín Láncara Lourenzá Lugo Meira Mondoñedo Monforte de Lemos Monterroso O Corgo O Incio O Páramo O Saviñao O Valadouro Ourol Outeiro de Rei Palas de Rei Pantón Paradela Pedrafita do Cebreiro Pol Portomarín Ribadeo Ribeira de Piquín Riotorto

Proport. rule 8010.221 4651.529 39879.110 7516.844 10523.537 7119.327 1181.018 3850.789 22536.029 14009.357 21697.144 13578.219 568.721 2063.733 1362.437 6644.205 13679.939 10449.301 848.260 52262.048 4786.923 3071.334 7397.106 730.730 2844.479 161.323 6175.314 3275.278 10641.524 884.273 3763.822 2562.746 14254.772 43041.680 22061.899 32663.246 32601.614 2242.140 16258.193 22457.486 25641.317 9957.625 2642.461 27179.386 6855.316 2793.218 3526.483 11032.352 9616.304 2745.080 12088.758 14163.304 1576.117 13.583 8759.113 15924.844 712.589 16524.644 1.476 25664.453 14887.085 16537.130 1057.367 4943.880

Council Samos Sarria Sober Taboada Trabada Triacastela Vilalba Xermade A Bola A Gudiña A Merca A Mezquita A Teixeira Allariz Baltar Barbadas Cartelle Celanova Coles Lobios Maceda Manzaneda O Carballiño O Irixo Piñor Rairiz de Veiga Ramiras Río Ríos San Amaro San Cristovo de Cea Sandiás Sarreaus Taboadela Trasmiras Viana do Bolo Vilariño de Conso Xinzo de Limia Xunqueira de Ambia Xunqueira de Espadanedo A Estrada A Golada Caldas de Reis Campo Lameiro Cerdedo Cotobade Cuntis Dozón Forcarei Lalín Meis Moraña O Grove Poio Pontecesures Pontevedra Portas Ribadumia Rodeiro Silleda Valga Vila de Cruces TOTAL Galicia

Talmud rule 5378.913 37930.893 499.560 25306.161 8625.382 753.832 12182.193 9803.768 6.627 132.478 15.797 402.734 162.455 871.566 65.827 26.315 232.479 221.048 104.067 33.837 1842.715 59.427 32.303 143.138 527.425 17.387 25.974 144.295 559.913 78.073 276.435 97.534 771.049 90.687 90.135 668.901 75.670 1424.236 432.498 35.839 9735.984 5022.792 2.969 25.302 33.454 110.001 418.967 7865.122 6233.923 66941.872 14.759 20.341 13.397 9.712 4.038 153.709 44.936 66.038 32751.533 36691.838 170.323 7768.538 1337887

Proport. rule 6454.695 32007.339 599.471 24432.500 10350.458 904.598 14618.632 11764.522 7.952 158.974 18.956 483.280 194.946 1045.879 78.992 31.578 278.974 265.258 124.880 40.604 2211.258 71.312 38.764 171.766 632.910 20.864 31.168 173.154 671.895 93.688 331.721 117.040 925.259 108.824 108.162 802.681 90.803 1709.083 518.997 43.007 11683.181 6027.350 3.563 30.362 40.145 132.001 502.760 9438.146 7480.708 49413.926 17.711 24.409 16.076 11.654 4.846 184.450 53.923 79.246 28899.723 31263.906 204.388 9322.246 1337887

Table 16: New systems of milk quotas for 190 councils in Galicia under the Talmud rule and the proportional rule for the case ρ = 40%.

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Appendix C. R code The R procedure (R Core Team, 2019) required to estimate the random arrival rule for bankruptcy problems is displayed below. In particular, we have built a function that returns the approximation when the following arguments are considered: • myseed. A seed specifies the start point when a computer generates a random collection of per-

mutations. That is, we can replicate the results when using the function in any computer and with the same value of the seed for a given set of parameters.

• nitmax. A natural number which indicates the number of samples imposed for the estimation. • E. This argument corresponds to the value of the estate in the bankruptcy problem. It is a real and non-negative number.

• claims. A vector that contains the values of the claims for the different agents involved in the bankruptcy situation.

estimating _ RA < - function ( myseed , nitmax ,E , claims ){ set . seed ( myseed ) n < - length ( claims ) RA < - rep ( 0 ,n ) it = 0 while ( it < nitmax ){ it < - it + 1 sigma < - sample ( 1 :n ,n , replace = F ) for ( l in 1 : n ){ pred < - c () if (l > 1 ){ pred < - sigma [ 1 :( l - 1 )]} xi < - min ( claims [ sigma [ l ]] , max ( 0 ,E - sum ( claims [ pred ]))) RA [ sigma [ l ]] < - RA [ sigma [ l ]]+ xi } } return ( RA / nitmax ) }

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