1 Introduction
Self-interest often leads to freeloading on the contributions of others in the dynamics associated with common goods and joint enterprises [
22,
41]. As is well known, incentivization, such as rewarding and punishing, is a popular method for harnessing the selfish action and for motivating individuals to behave cooperatively [
5,
7,
8,
15,
17,
38,
39,
45,
47,
53,
54]. Experimental and theoretical studies on joint enterprises under various incentive schemes are growing [
3,
20,
21,
28,
37,
50,
51,
59,
60].
Obviously, whether rewards or penalties, sufficiently large incentives can transform freeloaders into full cooperators, and incentives with small impact do nothing on the outcomes [
50]. However, incentivizing is costly, and such heavy incentives often incur serious costs on those who provide the incentives, whether in a peer-to-peer or institutional manner. Previous game-theoretic studies on the evolution of cooperation with incentives have focused on public good games with compulsory participation, and revealed that the intermediate degrees of punishment lead to a couple of stable equilibria, full defection and full cooperation [
7,
8,
42,
47,
50,
54]. In this bi-stable dynamics, establishing full cooperation requires an initially sufficient fraction of cooperators, or ex ante adjustment to overcome the initial condition [
8,
42]. This situation is a coordination game [
57], which is a model of great interest for analyzing a widespread coordination problem (e.g., in choosing distinct technical standards).
In contrast to a traditional case with compulsory participation, another approach to the evolution of cooperation is an option to opt out of joint enterprises [
1,
6,
10,
18,
24,
25,
32,
35,
40,
49,
52,
62,
65]. The opting-out option can make the freeloader problem relaxed: individuals can exit a joint venture when stuck in a state in which all freeload off one another (“economic stalemate”), and then pursue a stand-alone project; if a joint venture with mutual cooperation is more profitable than in isolation, the individuals once exited will switch to contributing to the venture. This situation, however, will also find defection attractive. Thus, joint enterprises with optional participation can give rise to a rock-paper-scissors cycle [
24,
25,
35,
52].
Recently, Sasaki et al. [
50] revealed that considering optional participation as well as institutional incentives can effect fully cooperative outcomes for the intermediate ranges of incentives. They demonstrated that opting-out combined with rewarding is not very effective at establishing full cooperation, but opting-out combined with punishment is very effective at establishing cooperation. Although there are a series of existing papers on the interplay of punishment and opting-out mechanisms [
9,
13,
16,
26,
55,
56,
61], the main points of these earlier studies comprise solving the puzzling issue of second-order freeloading: the exploitation of the efforts of others to uphold incentives for cooperation [
7,
38,
41,
43,
63]. Sasaki et al. [
50] consider incentives controlled exclusively by a centralized authority (like the empire or state) [
2,
4,
12,
31], and thus, their model is already free from the second-order freeloader problem.
Here we analytically provide a full classification of the replicator dynamics in a public good game with institutional incentives and optional participation. We clarify when and how cooperation can be selected over defection in a bi-stable situation associated with institutional punishment without requiring any ability to communicate among individuals. In particular, assuming that the penalties are large enough to cause bi-stability with both full cooperation and full defection (no matter what the basins of attraction are) in cases of compulsory participation, cooperation will necessarily become selected in the long term, regardless of the initial conditions.
The paper is organized as follows. In Sect.
2, we formalize optional public good games with institutional incentives and determine the average payoffs for the three strategies: cooperation, defection, and non-participation. In Sect.
3, based on analytical results from compulsory games (Sect.
3.1), we explore the interior equilibrium (Sect.
3.2) and in detail classify global dynamics for the three strategies (Sect.
3.3). Finally, in Sect.
4 we provide further discussion and concluding remarks.
4 Discussion
We considered a model for the evolution of cooperation through institutional incentives and analyzed in detail the evolutionary game dynamics. We employed public goods games, which typically assume that there are at least three players. Specifically, based on a public good game with optional participation, we fully analyzed how opting-out impacts on game dynamics; in particular, opting-out can completely overcome a coordination problem associated with punishment for a considerably broader range of parameters than in cases of compulsory participation.
We start from assuming that there is a state-like institution that takes exclusive control of individual-level sanctions in the form of penalties and rewards. In our extended model, nobody is forced to enter a joint enterprise that is protected by the institutional sanctioning. However, whoever is willing to enter, must be charged at the entrance. Further, if one proves unable or unwilling to pay, the sanctioning institution can ban that person from participation in the game. Indeed, joint ventures in real life are mostly protected by enforceable contracts in which members can freely participate, but are then bound by a higher authority. For example, anyone can opt to not participate in a wedding vow, but once it is taken, it is among the strongest enforceable contracts. As far as we know, higher authorities always demand penalties if contracts are broken.
Based on our mathematical analysis, we argue that institutional punishment, rather than institutional rewards, can become a more viable incentivization scheme for cooperation when combined with optional participation. In spite of the fact that the expected payoffs include nonlinear terms, the corresponding replicator dynamics is completely analyzed: in particular, proving that the interior equilibrium for optional participation is unique and a saddle point plays a key role in solving the global dynamics.
We show that combining optional participation with rewards can only marginally improve group welfare (to the same level as the non-participant’s fixed payoff) for a small range of the per capita incentive
δ, with
\(\delta_{-} < \delta< \delta_{\rm r}\) (Fig.
3b). Within this interval, compulsory participation can lead to partial cooperation; however, optional participation eliminates the cooperation and thus drives a population into a state in which all players exit. Hence, freedom of participation is not a particularly effective way of boosting cooperation under a rewards scenario.
Under penalties, the situation changes considerably. Indeed, as soon as
δ>
δ
− (Fig.
2b), the state in which all players cooperate abruptly turns into a global attractor for optional participation. When
δ just exceeds
δ
−, group welfare becomes the maximum (
r−1)
c−
σ. Meanwhile, for compulsory participation, a largest part of the (boundary) state space between cooperation and defection still belongs to the basin of attraction of the state in which all players defect. Because
δ
−=
c/
n, where
n is the group size, and
c is the net contribution cost (a constant), when
n is larger, the minimal sanctioning cost
δ
− to establish full cooperation is smaller.
Collaborating results for compulsory participation have recently been obtained in continuous public good games with institutional incentives by Cressman et al. [
12], who considered the gradual evolution of continuously varying contribution to a public good. The authors show that rewarding and punishing with probabilities depending on the player’s contribution and those of the co-players, can destabilize full defection and stabilize full cooperation, respectively. This model also indicates that combining the best of both incentives would lead the population to full cooperation, irrespective of the initial condition. Looking back at our model, non-participation reflects the common characteristic of destabilizing full defection; thus, it would be fascinating to investigate how efficiently voluntary rewards [
28,
48,
54], instead of voluntary participation, can establish coercion-based cooperation.
In the next two paragraphs, we consider only the penalty scenario and the corresponding coordination situation. There are various approaches to equilibrium selection in
n-person coordination games for binary choices [
19,
29,
34]. A strand of literature uses stochastic evolution models [
14,
33,
64], in which typically, a risk-dominant equilibrium [
23] that has the larger basin of attraction is selected through random fluctuation in the long run. In contrast, considering optional participation, our model typically selects the cooperation equilibrium which provides the higher group welfare, even if the cooperation equilibrium has the smaller basin of attraction when participation is compulsory than has the defection equilibrium. In the sense of favoring the efficient equilibrium, our result is similar to that found in a decentralized partner-changing model proposed by Oechssler [
36], in which players may occasionally change interaction groups.
Higher-order freeloaders are problematic for decentralized peer-to-peer sanctions [
11,
41]. This is not the case, however, for centralized institutional sanctions. In addition, it is clear that sanctioning institutions will stipulate a lesser antisocial punishment targeted at cooperators [
27], which can prevent the evolution of pro-social behaviors ([
44,
46], see also [
18]). Indeed, punishing cooperators essentially promotes defectors, who will reduce the number of participants willing to pay for social institutions. For self-sustainability, thus, sanctioning institutions should dismiss any antisocial schemes that may lead to a future reduction in resources for funding the institution.
Thus, we find that our model restricts the space of possible actions into a very narrow framework of alternative strategies, while increasing complexity. In practice, truly chaotic situations which offer a very long list of possibilities are unfeasible and create inconvenience, as is described by Michael Ende in “The Prison of Freedom” [1992]. Participants in economic experiments usually can make their meaningful choices only from a short and regulated list of options, as is the way in real life. Our result indicates that a third party capable of controlling incentives and membership can play a key role in selecting a cooperation equilibrium without ex ante adjustment. The question of how such a social order can emerge out of a world of chaos is left entirely open.