Economic foundations of generalized games with shared constraint: Do binding agreements lead to less Nash equilibria?

A generalized game is a situation in which interaction between agents occurs not only through their objective function but also through their strategy sets; the strategy set of each agent depends upon the decision of the other agents and is called the individual constraint. As opposed to generalized games with exogenous shared constraint literature pioneered by Rosen (1965), we take the individual constraints as the basic premises and derive the shared constraint generated from the individual ones, a set K. For a profile of strategies to be a Nash equilibrium of the game with individual constraints, it must lie in K. But if, given what the others do, each agent agrees to restrict her choice in K, something that we call an endogenous shared constraint, this mutual restraint may generate new Nash equilibria. We show that the set of Nash equilibria in endogenous shared constraint contains the set of Nash equilibria in individual constraints. In particular, when there is no Nash equilibrium in individual constraints, there may still exist a Nash equilibrium in endogenous shared constraint. We also prove a few results for a specific class of generalized games that we call non-classical games. Finally, we give two economic applications of our results to collective action problems: carbon emission and public good problems.