On 1-convexity and nucleolus of co-insurance games

Abstract

JEL classification: C71MSC: 91A12 91A40 91B30 a b s t r a c t The insurance situation in which an enormous risk is insured by a number of insurance companies is modeled through a cooperative TU game, the so-called co-insurance game, first introduced in Fragnelli and Marina (2004). In this paper we present certain conditions on the parameters of the model that guarantee the 1-convexity property of co-insurance games which in turn ensures the nonemptiness of the core and the linearity of the nucleolus as a function of the variable premium. Further we reveal conditions when a co-insurance game is representable in the form of a veto-removed game and present an efficient final algorithm for computing the nucleolus of a veto-removed game.

Data Cost Games as an Application of 1-Concavity in Cooperative Game Theory

“…Finally, an alternative proof of the main Theorem 3 is treated in [11] in terms of Dutch soccer teams and their potential fans. Three other applications of oneconcavity or one-convexity, called library game, coinsurance game, and the dual game of the Stackelberg oligopoly game, respectively, can be found in [10,12,13]. The nucleolus for 2convex games is treated in [14].…”

Determining the Nucleolus of Compromise Stable Games

“…In words, the TU game v is said to be 1-convex if its corresponding (nonnegative) gap function g v attains its minimum at the grand coalition. Clearly, the class of compromise stable TU games contains the subclass of 1-convex n-person games [3,4], as well as the 2-convex n-person games [3,5] and the big boss and clan games [2,8,9].…”

Models, theory and applications in cooperative game theory

“…One of these proofs deals with individual data sets that form a partition of the whole data set. Three other applications of one-concavity or one-convexity, called library game, co-insurance game, and the dual game of the Stackelberg oligopoly game respectively, can be found in [10], [13] and [12]. The nucleolus for 2-convex games is treated in [15].…”

“…In summary, the 3-person game v turns out to be 1-convex iff 10.5 ≤ v(N ) ≤ 13 and moreover, to be 2-convex iff v(N ) ≥ 15. Appealing examples of 1-convex games are discovered, like the library game together with a suitably chosen basis [18] as well as the co-insurance game [17]. It is still an outstanding challenge to search for appealing examples of 2-convex games.…”

“…As shown in [?] [17], certain practical problems such as co-insurance situations and library situations can be modeled as a cooperative game in characteristic function form. Formally, a cooperative game on player set N is a characteristic function v : P(N ) → R defined on P(N ) satisfying v(∅) = 0.…”