American Institute of Mathematical Sciences

October  2017, 4(4): 335-359. doi: 10.3934/jdg.2017018

Nonlinear dynamics from discrete time two-player status-seeking games

 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom

* Corresponding author: R. A. Van Gorder

Received  January 2017 Revised  July 2017 Published  September 2017

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.

Citation: Borun Shi, Robert A. Van Gorder. Nonlinear dynamics from discrete time two-player status-seeking games. Journal of Dynamics & Games, 2017, 4 (4) : 335-359. doi: 10.3934/jdg.2017018
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Time series for each player from the dynamics in (7) showing examples of convergence to equilibria and period-2 cycles when $y(0) = Y(0)=0.5$, $k = 0.5$, and $\lambda$ varies under the CES utility framework. (A) For $\lambda = 0.6$, we observe convergence to equilibium quantities for both players. (B) For $\lambda = 0.7$, we observe periodic cycles
Bifurcation diagrams for the dynamics from (7) starting with $y(0) = Y(0)=1/2, k = 1/2$, and obtained by varying $\lambda$ first and then a second parameter. (A) One-parameter bifurcation gives the FP-curves (black). The first and only PD occurs at $\lambda \approx 0.6293$. (B) Two-parameter bifurcation gives the PD-curve (red), period-2 arises in the region on and below the PD curve
, although the stability of the fixed points in the second panel precludes the period-2 dynamics seen in Figure 1">Figure 3.  Time series plots for each player when we take a symmetric CES utility function giving dynamics (7). We set $y(0)=1/3, Y(0)=2/3, k = 0.5$, while taking (A) $\lambda=0.6$ or (B) $\lambda=0.7$. Values of fixed points of the system are the same as Figure 1, although the stability of the fixed points in the second panel precludes the period-2 dynamics seen in Figure 1
Bifircation diagrams for the dynamics (7), when we have asymmetric CES utility functions for each player. Starting values are $y(0) = Y(0) = 1/2, k_1=1, k_2=0.5, \lambda_1=\lambda_2=0.7$. There will be one FP-curve and three PD-curves as generated. (A) FP-curve (black) generated by varying $k_1$ first, detecting two PD points. Three PD-curves (blue, red, green) are generated from them. (B) Two-parameter PD bifurcation are generated by varying $k_1, k_2$. (D) Two-parameter PD bifurcations are generated by varying $k_1, \lambda_1$. (D) Two-parameter PD bifurcations are generated by varying $k_1, \lambda_2$
Time series for both players from the dynamics (10)-(11), when parameters are fixed at $y(0)=Y(0)=1/2, k_1=1/2, \lambda_1 = 0.8$ while $k_2$ varies. (a) When $k_2=0.5$, we find gradual convergence to 'period-2 with width-2' dynamics. Since $k_2 < 1$, the corner solution is employed for this case. (b) When $k_2=1.5$, gradual convergence to period-4 dynamics is detected
One-parameter bifurcation diagrams for the dynamcis (10)-(11) starting at $y(0)=Y(0)=1/2, k_2=1.5$. (a) We fix $k_1=0.5$, and find an NS bifurcation at $\lambda_1 = 0.80491$. (b) We fix $\lambda_1=0.8$, and find NS bifurcations at both $k_1 = 0.306389$ and $k_1 = 0.48045$
Time series of the dynamics from reaction curves of type (18) when $r_1 = r_2 = 5, y(0)=Y(0)=0.5$. Player 1 picks the + branch at each iteration. (a) Player 2 picks the + branch every 3 iterations. (b) Player 2 picks the + branch every 4 iterations. (c) Player 2 picks the + branch every 5 iterations. (d) Player 2 picks the + branch every 9 iterations
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) $r_1=5, r_2=5, y(0)=Y(0)=1/2$, (b) $r_1=4, r_2=10, y(0)=Y(0)=1/2$
Time series of the dynamics from reaction curves of type (18) when we take $r_1=r_2=5, y(0)=Y(0)=0.5$. Both players choose between the two optimal branches randomly at each iteration, with the probability of selecting either branch equal to $0.5$
Phase portrait $(y, Y)$ of 1000 iterations of the dynamics from reaction curves of type (18), given $r_1=r_2=5, y(0)=Y(0)=1/2$. Clear structure can be observed. Only a small concentration region (note there are not exactly 4, or 16, or 64 points) of $(y, Y)$ is hit (these are denoted by the red dots). Consumptions fall into those regions immediately, and jump between them frequently (as indicated by the cobweb lines in blue)
Time series of the dynamics from reaction curves of type (18) when $r_1=r_2=5, y(0)=Y(0)=0.5$. Player 1 selects optimal branches deterministically, while Player 2 selects branches randomly with probability $0.5$ for each. (a) Player 1 selects the + branch every 3 iterations, (b) Player 1 selects the + branch every 10 iterations
Phase portraits of the dynamics from reaction curves of type (18) with 500 iterations, given parameter values $r_1=r_2=5, y(0)=Y(0)=0.5$. Player 2 selects branches at random (with probability $0.5$), while (a) Player 1 selects the + branch every 3 iterations, (b) Player 1 selects the + branch every 10 iterations
">Figure 13.  Time series (a) and phase portrait of 100 iterations (b) of the dynamics from reaction curves of type (18) when $r_1=r_2=5, y(0)=Y(0)=0.5$. Player 1 always picks the + branch, while Player 2 chooses branches at random with probability $0.5$ for each iteration. Due to the consistency in Player 1's choices, the concentration clusters (red dots) narrow compared to what was seen in Figure 12
An example of (a) two reaction curves of type (21) for both players and (b) time series showing convergence to the fixed point corresponding to the point of intersection along the reaction curves. Parameter values are $y(0)=Y(0)=1/2, r_1 = 1, r_2 = 1.5$
Time series of the dynamics from reaction curves of type (18) with asymmetric status choice and parameter values $r_1=4, r_2=2, y(0)=Y(0)=1/2$. (a) Player 1 picks the + branch every 5 iterations. (b) Player 1 picks the + branch every 10 iterations. Player 2 always faces a single solution branch (21)
Time series (a) and phase portrait (b) of the dynamics from reaction curves of type (18) with asymmetric status choice, given parameter values $r_1=4, r_2=2, y(0)=Y(0)=1/2$. Player 1 picks branches randomly (with a probability of $0.5$ for each branch) at each iteration, while Player 2 always faces a single solution branch (21)
Time series (a) and phase portrait (b) of the dynamics from reaction curves of type (18) with asymmetric status choice, given parameter values $r_1=10, r_2=1, y(0)=Y(0)=1/2$. Player 1 picks branches randomly (with a probability of $0.5$ for each branch) at each iteration, while Player 2 always faces a single solution branch (21)
]">Figure 18.  Bifurcation diagrams showing routes to chaos for the difference equation (41) for (a) $\sigma =1$, (b) $\sigma =2$, (c) $\sigma =3$, (d) $\sigma =8$, (e) $\sigma = 15$, (f) $\sigma = 40$. Panel (a) corresponds to the dynamics of the system found in Rauscher [21]
Bifurcation diagrams showing routes to chaos for the difference equation (46) for (a) $b = 0.3$, (b) $b = 0.4$, (c) $b = 1.5$, (d) $a = 0.5$, (e) $a = 0.8$, (f) $a = 2.0$
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