Games in sequencing situations with externalities

Highlights

• A new model for dealing with sequencing situations with externalities is provided.

• Partition sequencing games satisfy cohesiveness and non-positive externalities.

• The relationships between the Equal Gain Splitting rule and the cores are studied.

• A mechanism for implementing the Equal Gain Splitting rule is presented.

Abstract

In this paper, we consider cooperative games in sequencing situations with externalities, called partition sequencing games, in which the worth of a coalition can be influenced by external players in the queue. We first show that partition sequencing games satisfy cohesiveness and have non-positive externalities. Then by proposing partition rules, which are specifications on how external players should partition themselves, we study the relationships between the allocation given by the Equal Gain Splitting rule (EGS rule) and the cores based on different partition rules. It is shown that the EGS rule always yields a core element no matter which partition rule is applied to the game. Moreover, we propose a mechanism which implements the EGS rule for partition sequencing games.

Introduction

Cooperation in scheduling theory models represents an interesting research field on the frontier between game theory and scheduling problems. The research in this area deals with issues in which a finite number of agents, lined up in front of one or more machines, need to be processed, and these agents are faced with the decision-making problem of reducing their waiting costs by rearranging their orders. In the analysis of a sequencing model in this category, two questions that have aroused wide attention are: how to obtain the optimal order for the agents so as to minimize their total costs; and how the maximal cost savings should be allocated among the agents. Curiel, Pederzoli, and Tijs (1989) first take a game theoretic approach for solving these problems. They introduce sequencing games arising from a single machine sequencing situation in which jobs are waiting in line to be processed in front of a single machine. In their setting, the worth of a coalition is seen as the maximal cost savings the coalition can achieve by admissible rearrangements of its members without jumping over non-members. They also introduce a cost savings allocation rule called the Equal Gain Splitting rule (EGS rule) which generates a core element of the corresponding sequencing game. Since the pioneering work of Curiel et al. (1989), different types of modifications and generalizations about sequencing games have been presented, including ready times (Hamers, Borm, & Tijs, 1995), due dates (Borm, Fiestras-Janeiro, Hamers, Sánchez, & Voorneveld, 2002), multiple machines (Calleja, Borm, Hamers, Klijn, Slikker, 2002, Hamers, Klijn, Suijs, 1999, Slikker, 2005), grouped jobs (Çiftçi, Borm, Hamers, Slikker, 2013, Grundel, Çiftçi, Borm, Hamers, 2013), multistage situations (Curiel, 2010, Curiel, 2015), relaxed sequencing games (Musegaas, Borm, Quant, 2015, Slikker, 2006, Velzen, Hamers, 2003) and sequencing situations without an initial order (Hall, Liu, 2016, Klijn, Sánchez, 2006). A review on sequencing games can be found in Curiel, Hamers, and Klijn (2002).

Up until now, a common feature among the researches of sequencing games and their extended models is that the cooperative game theoretic approach, taken to the scheduling problem, is rooted in characteristic function form (CFF), which inherently assumes that the worth of a coalition is independent of the structure of other players. The underlying assumption behind all the definitions of these sequencing games is: what a group of players in the queue can achieve by cooperating only relies on the maximal cost savings through admissible rearrangements of its own members. This implies that the influences of other coalitions on this achievement is neglected.

In this paper, we take the impacts of external players’ reactions into consideration for determining the worth of a coalition in sequencing situations and related cooperative games. For example, consider the case where a collaborative manufacturing chain is formed by several companies. If some of the members separate from the co-production process, the full cooperation agreement will certainly break and all the remaining companies are free to contract with each other. So now, in order to access its real value, the deviating coalition should focus on not only the cost savings obtained from the rearrangement of its members but also the mutual influences between it and other later-formed coalitions. In this paper, these mutual influences which are regarded as mutual negotiations for trading places of coalitions are the key points of defining the worths of coalitions. That is, the existence of external factors implies that the evaluation of the worth of a coalition will be conducted on two stages: firstly, internal collaboration stage, and secondly, external negotiation stage. Hence, the final surplus for a coalition is mostly conditioned on the structure of external players. This idea happens to coincide with the well known situation of externalities in coalitional game theory, which is first modeled by Thrall and Lucas (1963) as partition function form (PFF) games. In such games, a coalition can have multiple worths depending on how the outside players partition themselves. In the current paper, we first introduce a class of sequencing games in partition function form by defining the worth of a coalition as the sum of surpluses generated from the two-stage process mentioned previously, and we call these games partition sequencing games. We will show that these games are cohesive (the grand coalition generates the largest total surplus) and have non-positive externalities (the merger between two disjoint coalitions does not make other coalitions better off).

For cooperative games, the core is a well-known solution concept and is widely regarded as an effective way to analyze the stability and fairness of coalitional games. There are already many attempts to define a modification of the core for PFF games, e.g., by Funaki and Yamato (1999), Hafalir (2007), Kóczy (2007), Bloch and Van den Nouweland (2014) and Abe and Funaki (2017). The definitions of the core of partition function games proposed in the literature are fundamentally based on pre-specified behavioral assumptions about the reactions of external players. Similarly, we introduce a concept called the partition rule, which is a mapping from a deviating coalition to a partition of the grand coalition containing the deviating coalition itself; namely, a partition rule specifies the reactions of external players to a deviation. Two specific partition rules are given: the m-exogenous rule and s-exogenous rule. We will show that the best-case scenario for a deviating coalition is that external players behave according to the s-exogenous rule, and the worst-case scenario is that external players behave according to the m-exogenous rule. We then define the cores of partition function games based on disparate partition rules and show that the EGS rule always yields a core allocation. Specially, the allocation generated by the EGS rule is the only core element if the s-exogenous rule is applied to a partition sequencing game.

While the EGS rule is a reasonable way of distributing the gains of the cooperation among players in sequencing situations, its noncooperative foundation has not been formally provided in the literature as far as we know. In the last part of this paper, we focus on a noncooperative approach to shape the strategic behavior of individuals for the implementation of the EGS rule. The mechanism we propose in this paper is a noncooperative sequential process in the same spirit of Rubinstein (1982)’s alternating offers bargaining game and its extensions to n players by Chatterjee, Dutta, Ray, and Sengupta (1993). The player in the first position in the queue initiates the game by proposing a coalition containing himself and offers to other players in that coalition. If all the respondents accept the proposal, the coalition is formed. Then the game is played among the remaining players. If at least one respondent rejects the proposal, the proposer is removed from the game and forms a singleton by himself. Then the first rejecter becomes the new proposer in the next period. This dynamic bargaining model is similar to the games proposed by Chatterjee et al. (1993), Bloch (1996) and Ray and Vohra (1999). But the main differences between these games and ours are that first, we do not incorporate the factor of discounting; second, the proposer proposing an unaccepted proposal will certainly be removed from the game in the next round. We will show that the subgame perfect equilibrium outcomes of this mechanism coincide with the payoff vector yielded by the EGS rule.

The plan of the paper is as follows. We continue in Section 2 with some preliminaries on sequencing situations and PFF games. In Section 3 we introduce the partition sequencing games. It is shown that these games are cohesive and have non-positive externalities. In Section 4 the concept of the partition rule is introduced for analyzing the influences of the reactions of outside players on the deviating coalition in sequencing situations. Furthermore, the cores of partition sequencing games are defined based on the partition rules and we present some relationships between the EGS rule and these cores. Section 5 provides a mechanism for implementing the EGS rule and Section 6 concludes.