Graphon games and an idealized limit of large network games

Graphon games are a class of games with a continuum of agents, introduced to approximate the strategic interactions in large network games. The first result of this study is an equilibrium existence theorem in graphon games, under the same conditions as those in network games. We prove the existence of an equilibrium in a graphon game with an infinite-dimensional strategy space, under the continuity and quasi-concavity of the utility functions. The second result characterizes Nash equilibria in graphon games as the limit points of asymptotic Nash equilibria in large network games. If a sequence of large network games converges to a graphon game, any convergent sequence of asymptotic Nash equilibria in these large network games also converges to a Nash equilibrium of the graphon game. In addition, for any graphon game and its equilibrium, there exists a sequence of large network games that converges to the graphon game and has asymptotic Nash equilibria converging to the equilibrium. These results suggest that the concept of a graphon game is an idealized limit of large network games as the number of players tends to infinity.