Cooperative Functions

51 Pages Posted: 11 Feb 2019

See all articles by Zhigang Cao

Zhigang Cao

Beijing Jiaotong University - School of Economics and Management

Date Written: February 7, 2019


We study a common scenario in industry where returns to scale are nondecreasing and thus full cooperation via pooling all the resources together among related firms is usually the most efficient way of production. This scenario is often modelled as a class of cooperative games, referred to as resource pooling games. We argue that resource pooling games could be better understood via directly analyzing the underlying functions that are referred to as the cooperative functions than via analyzing the induced cooperative games. By combining the pioneering works of Sharkey and Telser (1978) and Aubin (1981), we provide a framework for analyzing cooperative functions. We focus on cooperative functions that are supportable in that nonemptiness of the core is guaranteed for all related resource pooling games, and argue that Aubin core can be adapted to study cooperative functions and has several remarkable advantages over the core. We characterize concave supportable functions and convex supportable function. We find that a cooperative function always derives a convex game if and only if it is ultramodular (i.e., supermodular and coordinate-wise convex). Various related solution concepts, including unnormalized Aubin core and PMAS, are studied. We also provide several applications of this framework, including linear production games, EOQ games, and newsvendor games.

Keywords: Cooperative games, core, Aubin core, supportable functions, marginal pricing, Aumann-Shapley pricing, PMAS

JEL Classification: C71, D51

Suggested Citation

Cao, Zhigang, Cooperative Functions (February 7, 2019). Available at SSRN: or

Zhigang Cao (Contact Author)

Beijing Jiaotong University - School of Economics and Management ( email )


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