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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Figure 1.
Time series for each player from the dynamics in (7) showing examples of convergence to equilibria and period-2 cycles when
Figure 2.
Bifurcation diagrams for the dynamics from (7) starting with
Figure 3.
Time series plots for each player when we take a symmetric CES utility function giving dynamics (7). We set
Figure 4.
Bifircation diagrams for the dynamics (7), when we have asymmetric CES utility functions for each player. Starting values are
Figure 5.
Time series for both players from the dynamics (10)-(11), when parameters are fixed at
Figure 7.
Time series of the dynamics from reaction curves of type (18) when
Figure 8.
Time series of the dynamics from reaction curves of type (18). Player 1 chooses the + branch every 3 iterations, Player 2 chooses the + branch every 5 iterations. We observe period-15 dynamics. For the parameter values, we take (a)
Figure 10.
Phase portrait
Figure 11.
Time series of the dynamics from reaction curves of type (18) when
Figure 12.
Phase portraits of the dynamics from reaction curves of type (18) with 500 iterations, given parameter values
Figure 13.
Time series (a) and phase portrait of 100 iterations (b) of the dynamics from reaction curves of type (18) when
Figure 15.
Time series of the dynamics from reaction curves of type (18) with asymmetric status choice and parameter values
Figure 16.
Time series (a) and phase portrait (b) of the dynamics from reaction curves of type (18) with asymmetric status choice, given parameter values
Figure 17.
Time series (a) and phase portrait (b) of the dynamics from reaction curves of type (18) with asymmetric status choice, given parameter values
Figure 18.
Bifurcation diagrams showing routes to chaos for the difference equation (41) for (a)
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