Topics in Cooperative Game Theory

Abstract: Essay 1: A class of cooperative games arising from shortest path problems is defined. These shortest path games are shown to be totally balanced and allow a population-monotonic allocation scheme. Possible methods for obtaining core elements are indicated; first, by relating to the allocation rules in taxation and bankruptcy problems, second, by constructing an explicit rule that takes opportunity costs into account by considering the costs of the second best alternative and that rewards players who are crucial to the construction of the shortest path. Finally, noncooperative games arising from shortest path problems are introduced, in which players make bids or claims on paths. The core allocations of the cooperative shortest path game coincide with the payoff vectors in the strong Nash equilibria of the associated noncooperative shortest path game. Essay 2: In this paper it is shown that the core and the bargaining sets of Davis-Maschler and Zhou coincide in a class of shortest path games. Essay 3: No solution concept is population monotonic on the entire set of transferable utility games with a population monotonic allocation scheme. Therefore, two different criteria, maximality and nonextendibility, are introduced to evaluate the performance of solution concepts as population monotonic allocation devices. These two measures are shown to be equivalent and maximal (or nonextendible) solution concepts are characterized. The aspiration based solution is the maximal solution concept that satisfies an additional fairness property, in the sense that gains or sacrifices are evenly distributed over the players. An axiomatization of this solution concept is provided. Essay 4: A set of necessary and sufficient conditions for convexity of a transferable utility game is provided and shown to be minimal: none of the conditions is redundant. The result is used to provide an axiomatization of the Shapley value on the set of convex games. Essay 5: This paper focuses on cooperative games to model situations in which cautious players share their information with others, thus obtaining more refined information, which creates the potential for better decisions. The set of cautious information sharing games coincides with the collection of games with a population monotonic allocation scheme. A subclass is formed by the cautious information collecting games, in which one uninformed decision maker has to decide on certain actions, while the remaining players can provide additional information about the state of nature, but are themselves not engaged in the decision making process. These games are shown to coincide with the collection of monotonic veto games and form the building blocks of the cautious information sharing games, in the sense that a game is a cautious information sharing game if and only if it can be written as a convex combination of cautious information collecting games. Essay 6: The thieve property for bankruptcy problems entails that if a claimant manages to escape with his claim, the amount allocated to each remaining claimant is not larger than his share in the original problem. An inductive argument yields that a bankruptcy rule yields a population-monotonic allocation scheme in the associated bank\-ruptcy game if and only if the bankruptcy rule is efficient, reasonable (i.e., gives each claimant a nonnegative amount not exceeding his claim), and satisfies the thieve property. Many bankruptcy rules studied in the literature are efficient, reasonable, self- consistent, and monotonic. Rules satisfying these axioms are shown to yield population-monotonic allocation schemes. Essay 7: Total clan games are characterized using monotonicity, veto power of the clan members, and a concavity condition reflecting the decreasing marginal contribution of non-clan members to growing coalitions. This decreasing marginal contribution is incorporated in the notion of a bi- monotonic allocation scheme, where the value of each coalition is divided over its members in such a way that the clan members receive a higher, and the non- clan members a lower share as the coalitions grow larger. Each core element of a total clan game can be extended to both a population monotonic and a bi- monotonic allocation scheme. In total clan games where the clan consists of a single member (the so-called big boss) the use of the nucleolus as an allocation mechanism gives rise to a bi-monotonic allocation scheme.