Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation

We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good (“cheating”). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or “tax” on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.

Figure 1

Phase portraits for the three-species public goods game: (a) with $r_A=r_B=r_C=1.1$ and $c=2$, and (b) $r_A=1.8,r_B=2.2$ and $r_C=1.5$ and $c=1$, which displaces the fixed point off-center.

Figure 2

Basins of attraction for the three-species public good game when $r_A=r_B=r_C=1.1$ and $c=2$.

Figure 3

(a) Center: setting $\beta =1.5,r_A=r_B=r_C=3$ and $c_A=c_B=c_C=1$ results in a neutrally stable nonlinear cycle. (b) Stable: setting $\beta =2,r_A=r_B=r_C=3$ and $c_A=c_B=c_C=1$ results in an asymptotically stable nonlinear cycle.

Figure 4

(a) A computed control function $\gamma \left(t\right)$ that satisfies the Euler-Lagrange necessary conditions. Here $t_f=6$. (b) The state dynamics under the optimal control function with starting values near the fixed point. Here $t_f=10$.

Figure 5

(a) The state using the quasilinear control function shows slower decay (as expected) than the nonlinear control function. Here $t_f=20$ to illustrate that the quasilinear controller is still driving the system toward the fixed point. (b) The quasilinear control function is shown and compared to the solution to the second order ODE illustrating their equivalence. We also compare the control function computed for the nonlinear control problem to the quasilinearized control. Here $t_f=6$.

Figure 6

The plot of ${\left|\right|\mathbf{\lambda }\left|\right|}^2$ showing that throughout the entire time its value does not exceed $9k$ ($k=1$), thus implying that the identified controller is optimal.