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

https://doi.org/10.1016/j.ejor.2020.02.013Get rights and content

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.

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.

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 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 Fig. 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.

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 Fig. 2). These amounts provide a measure of the maximum capabilities of producing milk of each involved agent. It makes sense to assume that the 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 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 Puerto-Albandoz (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 Fernández-García 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 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.

Section snippets

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 ER+ and cRn are such that, for each agent i ∈ N, ci ≥ 0 with 0 ≤ E ≤ ∑j ∈ Ncj.

The set of bankruptcy problems with set of claimants N is denoted by BN. The value of E is the total amount to be divided, usually referred to as the

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

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

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

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 MTM2016-76969P and MTM2017-089422-P and ERDF.

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