Some results on cooperative interval games

Uncertainty is a daily presence in the real world. It affects our decision-making and may have influence on cooperation. On many occasions, uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e. payoffs lie in some intervals. A suitable game theoretic model to support decision-making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players' payoffs. In this paper, the relations between some set-valued solution concepts using interval payoffs, namely the interval core, the interval dominance core, the square interval dominance core and the interval stable sets for cooperative interval games, are studied. It is shown that the interval core is the unique stable set on the class of convex interval games

İşbirliğine ait aralık oyunları.

Interval uncertainty affects our decision making activities on a daily basis making the data structure of intervals of real numbers more and more popular in theoretical models and related software applications. Natural questions for people or businesses that face interval uncertainty in their data when dealing with cooperation are how to form the coalitions and how to distribute the collective gains or costs. The theory of cooperative interval games is a suitable tool for answering these questions. In this thesis, the classical theory of cooperative games is extended to cooperative interval games. First, basic notions and facts from classical cooperative game theory and interval calculus are given. Then, the model of cooperative interval games is introduced and basic definitions are given. Solution concepts of selection-type and interval-type for cooperative interval games are intensively studied. Further, special classes of cooperative interval games like convex interval games and big boss interval games are introduced and various characterizations are given. Some economic and Operations Research situations such as airport, bankruptcy and sequencing with interval data and related interval games have been also studied. Finally, some algorithmic aspects related with the interval Shapley value and the interval core are considered.Ph.D. - Doctoral Progra

On dominance core and stable sets for cooperative ellipsoidal games

The allocation problem of rewards or costs is a central question for individuals and organizations contemplating cooperation under uncertainty. The involvement of uncertainty in cooperative games is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. The theory of cooperative ellipsoidal games provides a new game theoretical angle and suitable tools for answering this question. In this paper, some solution concepts using ellipsoids, namely the ellipsoidal imputation set, the ellipsoidal dominance core and the ellipsoidal stable sets for cooperative ellipsoidal games, are introduced and studied. The main results contained in the paper are the relations between the ellipsoidal core, the ellipsoidal dominance core and the ellipsoidal stable sets of such a game

HOW TO HANDLE INTERVAL SOLUTIONS FOR COOPERATIVE INTERVAL GAMES

Uncertainty accompanies almost every situation in our lives and it influences our decisions. On many occasions uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our (collaborative) actions, i.e., payoffs lie in some intervals. Cooperative interval games have been proved useful for solving reward/cost sharing problems in situations with interval data in a cooperative environment. In this paper we propose two procedures for cooperative interval games. Both transform an interval allocation, i.e., a payoff vector whose components are compact intervals of real numbers, into a payoff vector (whose components are real numbers) when the value of the grand coalition becomes known (at once or in multiple stages). The research question addressed here is: How to determine for each player his/her/its a payoff generated by cooperation within the grand coalition - in the promised range of payoffs to establish such cooperation - after the uncertainty on the payoff for the grand coalition is resolved? This question is an important one that deserves attention both in the literature and in game practice

Lotniskowe gry przedziałowe i ich wartość Shapleya

This paper deals with the research area of cooperative interval games arising from airport situations with interval data. The major topic of the paper is to present and identify the interval Baker–Thompson rule.Praca dotyczy gier kooperacyjnych opisujących sytuacje występujące na lotniskach, gdzie dane są przedziałowe. Głównym celem pracy jest przedstawienie i identyfikacja przedziałowych zasad Bakera–Thompsona, służących do rozwiązywania problemu opłat lotniskowych dla samolotów lądujących na lotniskach z jednym pasem startowym, gdzie koszt użycia pasa startowego jest nieustalony. Dla takich gier lotniskowych podany jest dowód zgodności pomiędzy alokacją przedziałową Bakera–Thompsona a wartością Shapleya. Pokazano, że zasady przedziałowe Bakera–Thompsona zastosowane do dowolnej sytuacji lotniskowej z danymi przedziałowymi prowadzą do alokacji, należącej do przedziałowego jądra odpowiedniej przedziałowej gry lotniskowej. Zdefiniowano oraz podano niektóre charakterystyki takich wklęsłych gier przedziałowych. Dowiedziono wklęsłości lotniskowych gier przedziałowych oraz użyto tej właściwości do alternatywnego dowodu stabilności przedziałowej alokacji Bakera–Thompsona

Cooperative games under interval uncertainty: on the convexity of the interval undominated cores

Uncertainty is a daily presence in the real world. It affects our decision making and may have influence on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to support decision making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players' payoffs. This paper extends interval-type core solutions for cooperative interval games by discussing the set of undominated core solutions which consists of the interval nondominated core, the square interval dominance core, and the interval dominance core. The interval nondominated core is introduced and it is shown that it coincides with the interval core. A straightforward consequence of this result is the convexity of the interval nondominated core of any cooperative interval game. A necessary and sufficient condition for the convexity of the square interval dominance core of a cooperative interval game is also provided

An Axiomatization of the Interval Shapley Value and on Some Interval Solution Concepts

The Shapley value, one of the most common solution concepts in Operations Research applications of cooperative game theory, is defined and axiomatically characterized in different game-theoretical models. In this paper, we focus on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. In this study, we study the properties of the interval Shapley value on the class of size monotonic interval games, and axiomatically characterize its restriction to a special subclass of cooperative interval games by using fairness property, efficiency and the null player property. Further, we introduce the interval Banzhaf value and the interval egalitarian rule. Finally, the paper ends with a conclusion and an outlook to future studies

Cooperative games under bubbly uncertainty

The allocation problem of rewards/costs is a basic question for players, namely, individuals and companies that are planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world in which noise in observation and experimental design, incomplete information and vagueness in preference structures and decision-making play an important role. In this study, a new class of cooperative games, namely, the cooperative bubbly games, where the worth of each coalition is a bubble instead of a real number, is presented. Furthermore, a new solution concept, the bubbly core, is defined. Finally, the properties and the conditions for the non-emptiness of the bubbly core are given. The paper ends with a conclusion and an outlook to related and future studies

Cost allocations of a grey inventory model with cooperative game theory

Stok yönetimi çalışmalarında birim zamanda ortalama toplam stok maliyetini ve depolanacak ürün miktarını belirlemek firmalar için önemli bir konudur. Ancak gerçek hayatta stok maliyeti parametreleri tam olarak bilinemeyebilir fakat belirli aralık olarak tahmin edilebilir. Ayrıca birden fazla firma ortak hedeflere sahip olan firmalar ile stok ve sipariş maliyetlerinin azaltmak için diğer firmalar ile işbirliği (koalisyon) yoluna giderek giderlerini azaltabilirler. Bu çalışma ile aynı sektörde faaliyet gösteren ve aynı ürünleri sipariş veren firmaların bir araya gelerek ortak sipariş verme durumunda oluşacak toplam maliyetin firmalar arasında nasıl dağıtılacağı konularına katkı sağlanmıştır. Uygulama geliştirmek amacıyla av silahı sektöründe faaliyet gösteren ve aynı ürünleri sipariş eden üç firma incelenmiştir. Ayrıca geliştirilen üç adet maliyet dağıtım kuralı olan gri orantılı kural, gri eşit kayıp dağıtım kuralı, gri kaçırılan alternatif kayıpların dağıtımı ile ilgili adil ve kararlı dağıtımlar karşılaştırılarak incelenmiş firmalar için en uygun dağıtım kuralları önerilmiştir.Inventory management studies on minimizing the avarage total cost per unit time and determines the quantity of stocked materaial which is an important issue for companies. However, in real life inventory cost parameters may not be fully known, but they can be estimated as intervals. Furthermore, the multiple companies can reduce their costs with cooperation (coalition) and with the same target companies to reduce their inventory and ordering costs.In this study, our contribution is how to distribute the total cost of the companies operating the same sector and the same product. In order to make practice, we examine three shotgun firms which order the same products . Also we develop three cost allocation rules which are grey proportional rule, grey equal charge allocation and grey alternative cost avoidiance rule. We propose fair and stable cost distubition rules and compare the best for firms

On cooperative ellipsoidal games

This paper deals with cooperative ellipsoidal games, a class of transferable utility games where the worth of each coalition is an ellipsoid. Ellipsoids are a suitable data structure whenever data are affected by uncertainty and there are some correlations between the items under consideration. The research question addressed here is: How to deal with sharing problems under ellipsoidal uncertainty? This question is an important one in the cooperative game theory literature and its applications. We extend the ellipsoid calculus to introduce a core-like solution concept for cooperative ellipsoidal games. © Izmir University of Economics, Turkey, 2010