"An ANC Analytical Payoff Function for 3-Agent-Multistage-Network-Games with Endogenous Coalitions and Communication Structures"
For any three-agent network game, we extend the Aumann-Myerson (1988) (Shapley) solution by allowing non-myopic pairs of players to propose at each stage not only bilateral communication links (necessary for cooperation), but also a pair of individual payoffs. For a link to form the sum of the payoff proposals has to be equal to the sum of both agents' Myerson Values in the prospective graph implied by the added link. For equilibrium selection, we allow agents to use a "two-agent" dynamic extension of the Nash bargaining rule (NBR). This rule selects among multiple credible Nash equilibria of the strategic form of the proposal game. Loosely speaking, the strong Pareto efficiency of the NBR implies in cases of indifference of one agent towards linking with two partners that earlier agreements will be honoured first. Thus, the agent will will end up linked with only the first partner if the latter is as least as better off by the agent doing so. We provide by construction an alnost non-cooperative (ANC) analytical payoff function. The outcome is always efficient. If multiple graph structures exist, they are payoff equivalent. For strictly superadditive games, we only predict two link graphs. A necessary condition for a one link or coalition of two agents to form is that the colluded can achieve what the grand coalition can.