# Ecological public goods game

As an evolutionary game theorist working on cooperation, I sometimes feel like a minimalist engineer. I spend my time thinking about ways to design the simplest mechanisms possible to promote cooperation. One such mechanism that I accidentally noticed (see bottom left graph of results from summer 2009) is the importance of free space, or — more formally — population dynamics. Of course, I was inadvertently reinventing a wheel that Hauert, Holmes, & Doebeli (2006) started building years earlier, except my version was too crooked to drive places.

One of the standard assumptions in analytic treatments of EGT is fixed population size. We either assume that for every birth there is a death (or vice versa) when working with finite populations, or that all relevant effects are captured by the strategy frequency and independent of actual population size (when working with replicator dynamics). This approach only considers evolutionary effects and ignores population dynamics. Hauert, Holmes, & Doebeli (2006) overcome this by building the ecological public goods game.

The authors track three proportions: cooperators ($x$), defector ($y$), and free space ($z$). Since these are still proportions, they must add up to $1 = x + y + z$. Reproduction is modified from standard replicator dynamics, by being restricted to occur only if a free-space is found to reproduce into. An agents fitness is discounted by a multiplicative factor of $z$ — the probability of finding a free space for child placement in an inviscid population. This defined the dynamic system:

\begin{aligned} \dot{x} & = x(zf_C - d) \\ \dot{y} & = y(zf_D - d) \end{aligned}

Where $f_C$ is the average fitness of cooperators, $f_D$ — defectors, and $d$ is a shared death rate. We don’t need to include $\dot{z}$ since we know that $z = 1 - x - y$. Note that if we pick a death rate such that the population density remains constant (by setting $d = z\frac{xf_C + yf_D}{1 - z}$) then we will recover standard replicator dynamics with the constant $z$ as a time-scale parameter.

For interactions, the authors use the public goods game, with max group size $N$ and benefit multiplier $r$. A group is formed to play the game by sampling $N$ times from the distribution $(x,y,z)$ — with probability $x$ a spot is filled by a cooperator, $y$ — defector, and $z = 1 - x - y$ — left empty. This means that the expected number of agents per group is given by a binomial distribution $B(N,z)$. Thus, the average group size is $S = N(1 - z)$.

Each cooperator invests 1 unit of fitness in the public good, and all units invested are multiplied by a constant factor $r$ and uniformly distributed among the agents playing. Defectors invest nothing, but still receive their fraction of the split. Thus, for an agent interacting with $S - 1$ other agents, the expected fitness of being a defector of cooperator are:

\begin{aligned} f_D & = b + \frac{rp(S - 1)}{S} \\ f_C & = b + \frac{r(p(S - 1) + 1)}{S} - 1 \\ & = b + \frac{r(p(S - 1)} + \frac{r}{S} - 1 \\ & = f_D + \frac{r}{S} - 1 \end{aligned}

Where $b$ is the default birth rate and $p = \frac{x}{x + y}$ is the proportion of cooperators among the agents. For the fitness to make sense, we need $f_C > 0$ and so $b > 1 - \frac{r}{S}$, and the strength of selection is given by $\frac{1}{b}$. Note that, unlike the Prisoner’s dilemma (which is dynamically equivalent to PG in the limit of $S \rightarrow \infty$), it becomes rational to cooperate (and irrational to defect) when $r > S$ — this is the regime of weak altruism.

This results in an interesting feedback between the population size ($1 - z$) and the proportion of cooperators. As there are more cooperators in the population, the average fitness becomes higher and the population grows, $\dot{z} < 0$, As the population increased we get $S = (1 - z)N > r$ and then defectors fare better than cooperators, causing the proportion of cooperators (relative to defectors) to decrease $\dot{p} 0$). When the population is small enough ($1 - z > \frac{r}{N}$), then it becomes rational to cooperate ($f_C > f_D$) and the proportion of cooperators starts to grown $\dot{p} > 0$ — restarting the cycle.

Hauert, Holmes, & Doebeli (2006) and later Hauert, Wakano, & Doebeli (2008) carefully analyzed these dynamics. They focus on 4 qualitatively distinct regimes (depending on parameter values). In all settings, if the initial population is too small or has too few cooperators then it will go extinct, however in two of the regimes it is possible to maintain the population leading either to the co-existence of cooperators and defectors or even cooperate dominance.

Four possible phase profiles for the ecological public goods game in groups of at most N = 8. Individuals. The figure plots proportion of cooperation $p = \frac{x}{x + y}$ versus total population density $x + y$. The left hand side is extenction and the top is cooperator dominance. Stable fixed points are in solid black, while unstable fixed points are not coloured in. The four figures differ in their values of (r,d) with (a: 3, 0.5), (b: 5, 1.6), (c: 2.7, 0.5), (d: 2.1, 0.5). Figure 2 in Hauert, Holmes, & Doebeli (2006).

As $r$ is increased, the interaction becomes less competitive and easier for cooperators; Hauert, Wakano, & Doebeli (2008) discuss this dependence on $r$ in detail. The population goes from

1. regime of total extinction (since we assume $b < d$; figure d), to
2. extinction from an unstable fixed point, to
3. oscillations around an unstable focus that lead to extinction (figure c), to
4. Hopf bifurcation resulting in the stable focus, resulting in co-existance of cooperator and defectors through oscillations, to
5. stable fixed point with static distribution of cooperator and defectors co-existing (figure a), to
6. cooperator dominance (figure b)

This means that cooperation in the public goods is possible with just free-space added to the model. Unfortunately, the results do not hold for the inviscid Prisoner’s dilemma. However, Zhang & Hui (2011) showed that in a viscous population, similar dynamics are possible for the Prisoner’s dilemma. The ecological public goods has also been extended to the spatial setting (Wakano, Nowak, & Hauert, 2009; Wakano, & Hauert, 2011), but we will discuss that extension in a future post.

Although cooperation in the ecological public goods game emerges for $N > r$, I don’t think the cooperation can be called strong altruism. The emergence depends on the population density being occasionally low enough that the effective group size $S < r$, which puts us in the weak altruism range. The authors showed the evolution of cooperation driven by weak altruism. The underlying mechanism is similar to Killingback, Bieri, and Flatt (2006) result that we read in EGT Reading group 6, except the ecological variant uses free-space where Killingback et al. uses group structure.

### References

Hauert, C., Holmes, M., & Doebeli, M. (2006). Evolutionary games and population dynamics: maintenance of cooperation in public goods games Proceedings of the Royal Society B: Biological Sciences, 273 (1600), 2565-2571 DOI: 10.1098/rspb.2006.3600

Hauert, C., Wakano, J. Y., & Doebeli, M. (2008). Ecological public goods games: cooperation and bifurcation. Theoretical Population Biology, 73(2), 257.

Killingback, T., Bieri, J., & Flatt, T. (2006). Evolution in group-structured populations can resolve the tragedy of the commons. Proceedings of the Royal Society B: Biological Sciences, 273(1593), 1477-1481.

Wakano, J. Y., Nowak, M. A., & Hauert, C. (2009). Spatial dynamics of ecological public goods. Proceedings of the National Academy of Sciences, 106(19), 7910-7914.

Wakano, J. Y., & Hauert, C. (2011). Pattern formation and chaos in spatial ecological public goods games. Journal of Theoretical Biology, 268(1), 30-38.

Zhang, F., & Hui, C. (2011) Eco-evolutionary feedback and the invasion of cooperation in the prisoner’s dilemma games. PLoS One, 6(11): e27523.