We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional goods and positional goods (the latter entering into "status"). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or non-existence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game
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