Some observations about the network core and convexity
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In this study, we analyze the relationships between the value function - allocation rule setting and the TU game setting induced by value functions. As several different value functions may induce the same TU game, there is some information lost when passing to the TU game setting. We inquire in this study the impact of this lost information upon the preservation of the nonemptiness of the core when we pass from the network to the TU game setting. We pass from a value function to a TU game by associating with each coalition the maximal value of the graphs this coalition can form under the given value function. Conversely, we may associate with each TU game one of the value functions that induce the given TU game. Keeping this fixed, we define the network core as the collection of graphs where no coalition has an incentive to change the cooperation structure in itself, assuming that the rest of the society consists of isolated agents. Besides, we define convexity in the value function setting in an analogous fashion to convexity for the TU games. As convexity implies the nonemptiness of the core in both settings (a well-known result in the TU setting, and a trivial one in the value function setting), we inquire if convexity is preserved in passing from one setting to the other. We find that convexity of the value function is equivalent to a stronger type of convexity of the induced game.