A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game

Journal of Industrial Engineering and Management, Jun 2018

Purpose: The aim of the paper is to pick up the result of a previously published paper in order to deepen the discussion. We analyze the solution against the background of some well-known concepts and we introduce a newer one. In doing so we would like to inspire the further discussion of supply chain collaborationDesign/methodology/approach: Based on game theoretical knowledge we present and compare seven properties of fair profit sharing.Findings: We show that the nucleolus is a core-solution, which does not fulfil aggregate monotonicity. In contrast the Shapley value is an aggregate monotonic solution but does not belong to the core of every cooperative game. Moreover, we present the Lorenz dominance as an additional fairness criteria.Originality/value: We discuss the very involved procedure of establishing lexicographic orders of excess vectors for games with many players.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.jiem.org/index.php/jiem/article/viewFile/2430/869

A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game

Journal of Industrial Engineering and Management JIEM 2013-0953 A Reply to Ponte et al (2016) Supply Chain Collaboration: Some Comments on the Nucleolus of the Beer Game Brandenburg University of Technology Cottbus-Senftenberg (Germany) Purpose: The aim of the paper is to pick up the result of a previously published paper in order to deepen the discussion. We analyze the solution against the background of some well-known concepts and we introduce a newer one. In doing so we would like to inspire the further discussion of supply chain collaboration. Design/methodology/approach: Based on game theoretical knowledge we present and compare seven properties of fair profit sharing. Findings: We show that the nucleolus is a core-solution, which does not fulfil aggregate monotonicity. In contrast the Shapley value is an aggregate monotonic solution but does not belong to the core of every cooperative game. Moreover, we present the Lorenz dominance as an additional fairness criteria. Originality/value: We discuss the very involved procedure of establishing lexicographic orders of excess vectors for games with many players. beer game; cooperative game theory; profit allocation; Shapley value; nucleolus; core-selection; aggregate monotonicity; Lorenz set 1. Introduction 2.2. Properties of a Game To analyze a solution we have to introduce some classes of games. Cooperation may be successful or not. To concretize the term ?success'', some desirable properties of games may be defined. One goal is the generation of a result which is not worse than the results of isolated actions. This is referred to as superadditivity. A game (N, v) is superadditive if v(R ? S) ? v(R) + v(S) for all S, R ? N with R ? S = ?. In the following we concentrate on situations in which for at least one coalition yields v(R ? S) > v(R) + v(S). In consequence, the grand coalition generates a better result than the sum of all stand-alone coalitions. A game (N, v) is essential if v(N) > ?i?N v({i}). By EN we denote the set of all essential games with the set of players N. In the following essential games only are analyzed. Superadditivity describes the relationship of coalitions of disjoint elements. A similar effect may be claimed for coalitions of conjoint elements. This is called convexity. A game (N, v) is convex (Maschler et al., 2013, p. 718) if v(S ?{i}) ? v(S) ? v(R ? {i}) ? v(R) for all S ? R ? N\{i}. We will denote the set of convex games by CN. 2.3. Properties of a Fair Solution Looking at a game, the question arises of how to share the jointly generated result between the partners, which is equivalent to an allocation of the result. A function f(v) which assigns to a game (N, v) a, possible empty, subset f(v) of is called a solution concept. The function f distributes v(N) and generates a payoff vector x = (x1, x2, x3, ? xn) with x ? . Such a function is referred to as allocation scheme. Definition 1: A solution f is a single-valued solution if | f (v)| = 1 for every v. In this case, f (v) is represented by an element of , i.e. f (v) = x. With the allocation of the jointly generated result, the problem of fairness arises. Several properties of a fair solution have been identified in cooperative game theory in the last decades. The most crucial properties are (Gonz?lez-D?az, Garc?a-Jurado & Fiestras-Janeiro, 2010, p. 226; Calleja, Rafels & Tijs, 2012; Mueller, 2018, p. 406408): ? ? ? ? ? ? Efficiency: A single-valued solution f is efficient if ?i?N fi(v) = v(N). Individual rationality: A single-valued solution f is individual rational if fi(v) ? v({i}) ?i?N. Equal-treatment-property: A single-valued solution f satisfies equal-treatment property if for the players i and j, for which holds: v(S ? {i}) = v(S ? { j }) ?S ? N with i, j ? S yields fi(v) = fj(v). Dummy-player-property: A player i is called a dummy player if v(S ? {i}) = v(S) + v({i}) for all S ? N with i ? S. A single-valued solution f satisfies the dummy-player-property if for a dummy-player i yields: fi(v) = v({i}). Additivity: A single-valued solution f satisfies additivity if for any two games v, w follows (v + w) = f (v) + f (w). Aggregate monotonicity: A single-valued solution f satisfies aggregate monotonicity if for all games v, w with v(N) > w(N) and v(S) = w(S) for all S N follows: fi(v) ? fi(w) ?i?N. There are several other properties (Arin & Katsev, 2018, p. 305) which are not of interest for further discussion. The first two properties are summerized by defining an imputation. Definition 2: The set of imputations I(v) of a game N(v) is defined by Only those imputations are of interest that are not dominated by another imputation. The set of non-dominated imputations forms the core. Definition 3: The Core(v) of a game N(v) is defined by The core of a game contains all solutions which are justified as fair and, therefore, are stable. It may be small, very large, or empty. Without defining the property of balancedness in detail we point out, that th (...truncated)


This is a preview of a remote PDF: http://www.jiem.org/index.php/jiem/article/viewFile/2430/869

David Mueller. A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game, Journal of Industrial Engineering and Management, 2018, pp. 528-534, Volume 11, Issue 3, DOI: 10.3926/jiem.2430