Location-Price Game in a Dual-Circle Market with Different Demand Levels

This paper researches a location-price game in a dual-circle market system, where two circular markets are interconnected with different demand levels. Based on the Bertrand and Salop models, a double intersecting circle model is established for a dual-circle market system in which two players (firms) develop a spatial game under price competition. By a two-stage (location-then-price) structure and backward induction approach, the existence of price and location equilibrium outcomes is obtained for the location game. Furthermore, by Ferrari method for quartic equation, the location equilibrium is presented by algebraic expression, which directly reflects the relationship between the equilibrium position and the proportion factor of demand levels. Finally, an algorithm is designed to simulate the game process of two players in the dual-circle market and simulation results show that two players almost reach the equilibrium positions obtained by theory, wherever their initial positions are.

Spatial evolutionary games with weak selection

Recently, a rigorous mathematical theory has been developed for spatial games with weak selection, i.e., when the payoff differences between strategies are small. The key to the analysis is that when space and time are suitably rescaled, the spatial model converges to the solution of a partial differential equation (PDE). This approach can be used to analyze all 2×2 games, but there are a number of 3×3 games for which the behavior of the limiting PDE is not known. In this paper, we give rules for determining the behavior of a large class of 3×3 games and check their validity using simulation. In words, the effect of space is equivalent to making changes in the payoff matrix, and once this is done, the behavior of the spatial game can be predicted from the behavior of the replicator equation for the modified game. We say predicted here because in some cases the behavior of the spatial game is different from that of the replicator equation for the modified game. For example, if a rock–paper–scissors game has a replicator equation that spirals out to the boundary, space stabilizes the system and produces an equilibrium.