On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule A. Saavedra-Nieves1, P. Saavedra-Nieves2 1 Corresponding author. Departamento de Estatística e Investigación Operativa, Universidade de Vigo. [email protected] 2 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 sit- uations. 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 1 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 farm- ers 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 1 as Ireland and The Netherlands. Although different initiatives were proposed to avoid this, new mea- sures 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 gener- ates 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 Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. 2012 31.85 31.53 30.87 30.00 29.24 28.66 28.41 28.53 29.35 30.48 31.10 31.27 2013 32.65 32.66 32.76 32.84 33.06 33.03 34.44 34.86 35.59 38.57 38.93 39.17 2014 39.24 38.90 38.65 36.05 35.61 35.37 33.72 33.73 33.64 32.24 32.10 31.95 2015 30.52 30.60 30.30 28.80 28.40 27.90 27.60 27.70 28.30 28.70 28.70 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 pro- duction 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 2 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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, Frag- nelli, 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. Car- pente 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 3 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 con- cerning 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 com- putational. 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 expo- nentially 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.