Evolution of commitment and level of participation in public goods games

Abstract

Before engaging in a group venture agents may require commitments from other members in the group, and based on the level of acceptance (participation) they can then decide whether it is worthwhile joining the group effort. Here, we show in the context of public goods games and using stochastic evolutionary game theory modelling, which implies imitation and mutation dynamics, that arranging prior commitments while imposing a minimal participation when interacting in groups induces agents to behave cooperatively. Our analytical and numerical results show that if the cost of arranging the commitment is sufficiently small compared to the cost of cooperation, commitment arranging behavior is frequent, leading to a high level of cooperation in the population. Moreover, an optimal participation level emerges depending both on the dilemma at stake and on the cost of arranging the commitment. Namely, the harsher the common good dilemma is, and the costlier it becomes to arrange the commitment, the more participants should explicitly commit to the agreement to ensure the success of the joint venture. Furthermore, considering that commitment deals may last for more than one encounter, we show that commitment proposers can be lenient in case of short-term agreements, yet should be strict in case of long-term interactions.

Appendices

Appendix 1: Some simplifications of the obtained analytical results

Here, using the well-known inequalities [36]

$$\begin{aligned} \log N+ \gamma < F_N = \sum ^{N}_{k=1} \frac{1}{k} \le \log N + 1 \end{aligned}$$

where $\gamma = 0.577215$, we provide some simplifications of the conditions obtained in the main text. First of all, regarding the conditions for risk-dominance of $COMP_F$ against D, FREE and FAKE:

$$\begin{aligned} {\begin{matrix} \epsilon &{} \le \min \left\{ \frac{r+F-N-1}{H_N-H_{F-1}}, \frac{r-1}{H_N} \right\} \times c, \\ \delta &{}\ge \frac{N-r}{N F_{N-1}}c + \frac{F_{N}}{NF_{N-1}}\epsilon . \end{matrix}} \end{aligned}$$

(11)

They can be simplified to

$$\begin{aligned} \begin{aligned} \epsilon&\le \min \left\{ \frac{r+F-N-1}{\log \frac{N}{F-1} + \gamma - 1}, \frac{r-1}{\log N + \gamma } \right\} \times c, \\ \delta&\ge \frac{(N^2-r N) c + \epsilon }{N^2\left( \log (N-1) + 1 \right) } + \frac{\epsilon }{N}. \end{aligned} \end{aligned}$$

(12)

Appendix 2: Risk-dominance conditions when $N \rightarrow \infty$

Note that the risk-dominance condition described in Sect. 3 is valid for the large population size limit ($Z \rightarrow \infty$). We now analyze these conditions for $N \rightarrow \infty$. We assume that the multiplication factor r is a function of N, i.e. r(N).

From (12), if $\log N$ grows much faster than r(N) (or using little-o notation, $r(N) = o(\log N))$, i.e. if

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{r(N)}{\log N} = 0, \end{aligned}$$

then the right-hand side of the first inequality in (12) becomes 0. It implies that when $N \rightarrow \infty$ the cost of arranging the commitment must be infinitely small so that COMP can be risk-dominant against defective strategies (namely, in this case, the FREE strategy). Hence, provided that arranging commitment is costly ($\epsilon > 0$), it must hold that $r(N) \ne o(\log N)$, or, using the big omega notation,

$$\begin{aligned} r(N) = \varOmega (\log N) \text { (as } N \rightarrow \infty ). \end{aligned}$$

(13)

In this case, we have

$$\begin{aligned} \epsilon \le \min \left\{ \lim _{N \rightarrow \infty } \frac{r(N)+F-N-1}{\log \frac{N}{F-1}}, \lim _{N \rightarrow \infty } \frac{r(N)}{\log N} \right\} c \end{aligned}$$

(14)

Hence, the necessary condition for this inequality to hold is that $r(N) = \varOmega (\log N)$ and $\epsilon$ grows at most as fast as $\frac{r(N)}{\log N}$.

Consider now the second risk-dominance inequality

$$\begin{aligned} \begin{aligned} \delta&\ge \frac{(N^2-r N) c + \epsilon }{N^2\left( \log (N-1) + 1 \right) } + \frac{\epsilon }{N}. \end{aligned} \end{aligned}$$

(15)

If $\epsilon$ is a constant, clearly the right hand side becomes 0 as $N \rightarrow \infty$, i.e. $\delta$ can be indefinitely small while still ensuring the condition is satisfied (i.e. COMP is risk-dominant against FAKE). Let’s assume that $\epsilon$ is a function of N. Then, the same conclusion applies if

$$\begin{aligned} \epsilon (N) = o(N). \end{aligned}$$

Otherwise, i.e. if $\epsilon (N) = \varOmega (N)$, then the inequality can be rewritten as

$$\begin{aligned} \begin{aligned} \delta&\ge \lim _{N \rightarrow \infty } \left( \frac{\epsilon (N)}{N^2\left( \log (N-1) + 1 \right) } + \frac{\epsilon (N)}{N} \right) . \end{aligned} \end{aligned}$$

(16)

That is, roughly, $\delta$ needs to grow at least as fast as $\frac{\epsilon (N)}{N}$ when $N \rightarrow \infty$ to guarantee risk-dominance of COMP against FAKE. We also observe that the contribution cost c does not play a role in determining the lower boundary of $\delta$.

In short, as the group size $N \rightarrow \infty$, the risk-dominance conditions are defined in (14) and (16). For these to hold, it is necessary that

• 1. The multiplication factor r(N) satisfies that $r(N) = \varOmega (\log N)$, i.e.

  $$\begin{aligned} \lim _{N \rightarrow \infty } \frac{r(N)}{\log N} > 0 \end{aligned}$$

• 2. The cost of arranging commitment $\epsilon (N)$ grows at most as fast as $\frac{r(N)}{\log N}$ as $N \rightarrow \infty$

• 3. The compensation cost, $\delta (N)$ grows at least as fast as $\frac{\epsilon (N)}{N}$ when $N \rightarrow \infty$

Appendix 3: Results for varying group sizes

Fig. 7

Results in the main text are robust for varying group size N. We plot the total frequency of the commitment proposing strategies as a function of $\epsilon$ and $\delta$. In general, when $\epsilon$ is small enough and $\delta$ is sufficiently high, commitment proposing strategies are frequent, leading to high levels of cooperation in the PGG. Parameters: $Z = 100$; $r=4$; $\beta = 0.25$

In Fig. 7, we plot the total frequency of commitment strategies for varying $\epsilon$ and $\delta$, and for increasing group sizes. The results show that the observations seen in the main text are robust for varying group sizes N as well as compensation cost $\delta$. Furthermore, $\epsilon$ is the essential parameter because as soon as $\delta$ reaches a certain threshold, increasing it does not lead to notable improvement.