Advances in Fuzzy Systems

Volume 2018, Article ID 7031071, 13 pages

https://doi.org/10.1155/2018/7031071

## Simultaneous Resource Accumulation and Payoff Allocation: A Cooperative Fuzzy Game Approach

^{1}Department of Mathematics, Dibrugarh University, Dibrugarh 786004, India^{2}School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Surajit Borkotokey; moc.oohay@robtijarus

Received 29 November 2017; Accepted 17 January 2018; Published 1 April 2018

Academic Editor: Pushpinder Singh

Copyright © 2018 Surajit Borkotokey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We develop a simultaneous resource accumulation and payoff allocation algorithm under the framework of a cooperative fuzzy game that builds on our earlier work on the role of satisfaction in resource accumulation and payoff allocation. The difference between the two models lies in the fact that while focus was more on getting an* exact solution* in our previous model, the negotiation process in the current model accounts more for the role of the intermediate stages. Moreover we characterize our solution using two properties:* asymptotic fairness* and* efficiency*. Our model includes a suitable penalty function to refrain players from unreasonable demands. We focus on real life situations where possibly one or more players compromise on their shares to ensure a binding agreement with the others.

#### 1. Introduction

In the literature, cooperation among self-interested players under binding agreements has been well explained by the theory of crisp cooperative games. In [1–3] it is shown that players’ satisfaction has the ability to set up a solution concept in a cooperative game which is easily computable through an algorithm. It is therefore important to take account of how coalitions are formed* vis-à-vis* and how the worth is allocated among participating players so that they are individually satisfied up to a desired level.

In this paper, we propose a dynamic process of accumulating resources from players and of finding a payoff allocation simultaneously under a cooperative fuzzy game theoretic environment. The preliminary works are done in [3]. Here we enhance the same model incorporating (a) the role of players in the intermediate steps of the negotiation process and (b) the notion of a penalty function that restrains the players from making unreasonable demands.

Consider, for example, the role of Uber and Ola cab services in different Indian cities. Both these companies acquire vehicles from the local vehicle providers of the cities on lease and offer them a portion of the profits accrued from the customers on per day basis. It follows that the cab services accumulate resource (in terms of number of vehicles from different service providers), generate some profit, and finally allocate this profit among the stack holders. It may be the case that the local vehicle providers have their shares in both the companies that results in different levels of satisfactions in terms of their trade relations with these companies. Retention becomes a big concern for both the companies. Therefore they need to satisfy the vehicle providers by providing sufficient opportunity to both generate and share the profit. More precisely let us consider a hypothetical situation where three agents , , and in their respective territorial areas collect monetary resources from the customers and provide this accumulated resource for an investment firm. The firm in turn invests the whole amount in shares of various companies (finitely many) in the market and rewards the agents according to their performances. The agents take notes on how their resources are being invested in different companies. This would determine how much payoff they would get after the firm profits from such investments. Here the manager of the firm acts as the mediator who takes care of all the resources of the agents. The payoffs to the agents are usually measured through their resource collection capacities, whereas, in addition to weighing on such straight performances, many organizations provide extra incentives adopting background corrections. This may include the geographical disadvantages the agents have to comply with in their territorial areas, their prospects towards future expansion of business, team working capabilities, and so on. Such incentives essentially gear up the satisfaction levels of the agents. However it is not found in the literature whether any standard procedure is followed in provision of such incentives. Moreover most of the business organizations follow the principle “if you perform well you get better-off.” Note that performance and satisfactions can never be universally standard. This motivates us to pursue the present study. Similar works can be traced back to [4–6] and so on.

In [7] a dynamic payoff allocation method that converges to a specific allocation under a given set of external restrictions is designed. Similar models of frequency allocations in wireless networks through a game in satisfaction form are found in [8] where payoff to a player accounts only for her satisfactions. However none of these studies uses the notion of fuzzy cooperative games as a tool to address their problems. For similar studies we refer to [4, 9, 10] and so on.

In our previous model [3], the problem of simultaneous resource and payoff allocation among participating players by a mediator is discussed under a dynamic setup. It shows that an exact resource versus payoff allocation matrix evolves as a solution to a minimization problem at each stage of the negotiation process. However we observe that here the emphasis is put on how the* exact solution* (as a part of the limiting process) can be obtained without accounting for the intermediate solutions. In real life situations one needs to consider time and monetary constraints that do not allow the negotiation process to last longer. Therefore, in our present model we focus on the fact that the players get an optimal solution with the least possible time and money. We provide an axiomatic characterization of the* exact solution* which is based on two very natural axioms:* efficiency* and* asymptotic fairness*. We provide an example at the end of the paper to highlight this issue.

Let be the set of players (agents) which is known as the grand coalition. Any subset of is called a crisp coalition. In a crisp coalition, a player can give either full participation (investment in our case) in the coalition with her complete resource or no participation (no resource). The collection of all crisp coalitions of is denoted by . A crisp cooperative game is a pair where is a real valued function known as the characteristic function such that . If the players’ set is fixed, the cooperative game is denoted by the function . For each crisp coalition , the real number is known as the worth of the coalition (or profit incurred from ). On the other hand, if a player needs to participate in more than one coalition simultaneously, with her resources (or power) in hand, it is possible to provide only fractions of her full resource (power) for those coalitions. Such coalitions are called fuzzy coalitions. A fuzzy coalition is nothing but a fuzzy set of , represented by an -tuple, where its th component is the membership degree (or fraction of the power) of player ranging between and . A crisp coalition can be realized as a special type of fuzzy coalition where the degree of participation of any player in it is either or . In the literature Aubin [11], Butnariu [12], and Branzei et al. [13] have well developed the theory of cooperative fuzzy games and justified the fuzzification in terms of the players’ membership degrees in a coalition. In both crisp and fuzzy environment, it is quite necessary to determine how a* fuzzy coalition structure* is formed as well as how a suitable payoff distribution is proposed to the players accordingly. Solution concepts for cooperative games are found in [11–17]. In [1], a new solution concept that evolves as a result of a dynamic negotiation process is defined. It is termed as the payoff allocation. In [2], a process of resource allocation of players is obtained in different cooperative actions with the formation of a* fuzzy coalition structure*. The key objective in [1–3] is to investigate the influence of individual satisfactions upon a payoff and resource allocations (i.e., how it can be used to arrive at a suitable payoff and resource distribution over time). The process of resource allocation (equivalently resource accumulation) is made here synonymous with the formation of a fuzzy coalition. It follows that their model is equivalent to solving an -person cooperative game with distinct fuzzy coalitions.

The goal of our present study is to provide a more developed and systematic treatment of satisfaction levels as a basis for negotiation among rational agents, who are capable of participating in different fuzzy coalitions with possibly varied rate of memberships simultaneously. We introduce the notion of a penalty to restrict the irrational demands of the players. Our present model is seen to be efficient than the one discussed in [3] in situations where the allocation process does not continue for long rather than stopping at some intermediate stage as part of a trade-off. We modify the* variance* function of [3] and show that at an intermediate stage our solution is more efficient than that of the earlier model.

The allocation process (Figure 1) goes exactly in the same way as that in [3]. Initially, the mediator would accumulate budgets/resources from the rational players. The mediator and the players would jointly determine the number of possible coalitions. Then the players would announce their expected total budgets/resources for each coalition. The mediator finds the optimal coalition structure and the optimal total budget allocation in such a way that the sum of all budgets allocated for all coalitions of the optimal coalition structure is equal to the total budget in her hand.