Complexity of equilibrium in diffusion games on social networks

Abstract

Abstract-In this paper, we consider the competitive diffusion game, and study the existence of its pure-strategy Nash equilibrium when defined over general undirected networks. We first determine the set of pure-strategy Nash equilibria for two special but well-known classes of networks, namely the lattice and the hypercube. Characterizing the utility of the players in terms of graphical distances of their initial seed placements to other nodes in the network, we show that in general networks the decision process on the existence of purestrategy Nash equilibrium is an NP-hard problem. Following this, we provide some necessary conditions for a given profile to be a Nash equilibrium. Furthermore, we study players' utilities in the competitive diffusion game over Erdos-Renyi random graphs and show that as the size of the network grows, the utilities of the players are highly concentrated around their expectation, and are bounded below by some threshold based on the parameters of the network. Finally, we obtain a lower bound for the maximum social welfare of the game with two players, and study sub-modularity of the players' utilities.

Multi-Player Diffusion Games on Graph Classes

“…We studied a competitive diffusion game for three or more players on several classes of graphs, answering-as a main contribution-an open question concerning the existence of a Nash equilibrium for three players on grids [8] negatively. Further, extending previous results on hypercubes [3], we proved that Nash equilibria always exist for four player on d-dimensional hypercubes. With this work, we provide a first systematic study of this game for more than two players.…”

“…Extending results of Roshanbin [8] for two players, we investigate the existence of pure Nash equilibria for at least three players on different classes of graphs including paths, cycles, grid graphs and hypercubes; as a main contribution, we answer an open question proving that there is no Nash equilibrium for three players on m × n grids with min{m, n} ≥ 5. Further, extending results of Etesami and Basar [3] for two players, we prove the existence of pure Nash equilibria for four players on every d-dimensional hypercube.…”

“…Several game-theoretic models have been suggested, including our model of reference, introduced by Alon et al [1]. Some interesting generalizations of this model are the model by Tzoumas et al [12], who considered a more complex underlying diffusion process, and the model studied by Etesami and Basar [3], allowing each player to choose multiple vertices. Dürr and Thang [2] and Mavronicolas et al [7] studied so-called Voronoi games, which are closely related to our model (but not similar; there, players can share vertices).…”

“…HypercubesEtesami and Basar[3] studied diffusion games on hypercubes and proved the existence of a Nash equilibrium for two players on every d-dimensional hypercube. In this section, we extend their result to four players.…”

Multi-player Diffusion Games on Graph Classes

“…Further, extending previous results on hypercubes [3], we proved that Nash equilibria always exist for four player on d-dimensional hypercubes. With this work, we provide a first systematic study of this game for more than two players.…”