An outside individual option increases optimism and facilitates collaboration when groups form flexibly

Abstract

Voluntary participation is a central yet understudied aspect of collaboration. Here, we model collaboration as people's voluntary choices between joining an uncertain public goods provisioning in groups and pursuing a less profitable but certain individual option. First, we find that voluntariness in collaboration increases the likelihood of group success via two pathways, both contributing to form more optimistic groups: pessimistic defectors are filtered out from groups, and some individuals update their beliefs to become cooperative. Second, we reconcile these findings with existing literature that highlights the detrimental effects of an individual option. We argue that the impact of an outside individual option on collaboration depends on the "externality" of loners - the influence that those leaving the group still exert on group endeavors. Theoretically and experimentally, we show that if collaboration allows for flexible group formation, the negative externality of loners remains limited, and the presence of an individual option robustly aids collaborative success.

Fig. 1. Modeling collaboration under voluntary participation.

A Individuals can either opt in to group collaboration or choose an individual option outside of the group. When opting to collaborate, players are randomly assigned to groups of five members (if the total number of players opting in is not a multiple of five, any remaining players are assigned to the individual option). Group members can choose whether to cooperate (incurring the private cost of 10 points) on the group project. If the number of cooperators is equal to or more than the predetermined threshold value,q (q = 2, 4, or 5), then all the group members earn 30 extra points. When choosing the individual option, players are guaranteed to earn a smaller additional payoff of 10 points irrespective of other players’ choices. Expected payoff (B) and best response (C) as a function of subjective belief about how likely others are to cooperate within groups (i.e., $\gamma$). Each color represents a possible action under mandatory or voluntary participation: orange = cooperate (C), blue = defect (D), and green = leave (L). Recall that under mandatory participation, where leaving is not available, the best response is derived by comparing between the expected payoffs of cooperation and defection.

Fig. 2. Voluntary participation supports collaborative success.

A Breakdown of participants choosing each action in each condition. The pie charts depict the number of participants who selected each action (orange: cooperate [C], blue: defect [D], and green: leave [L]; $n=191$ individuals), with exact numbers in the wedges. B Individual cooperation rate [#C/(#C + #D); Mandatory: $n=191$ individuals in every threshold; Voluntary: Threshold 2: $n=120$; 4: $n=98$, 5: $n=133$], (C) group success rate (same ns as in (B)), and (D) normalized (i.e., ratio against the highest possible payoff for each threshold) average payoff in each condition (Mandatory or Voluntary (including loners): $n=191$ individuals in every threshold; Voluntary (without loners): Threshold 2: $n=120$; 4: $n=98$; 5: $n=133$). In each of the three panels, the x-axis indicates threshold values, and the color coding indicates whether participation in groups is mandatory (red) or voluntary (cyan). Error bars in (B–D) indicate 95% bootstrapped confidence intervals (CIs), with the bar heights showing the exact values from the original experimental data. See Methods for the exact procedures to calculate the CIs. In (D), plain cyan bars show the average payoff under voluntary participation just within groups (excluding loners), and hatched cyan bars show the average payoff for the entire population with loners included.

Fig. 3. Voluntary participation filters out or encourages pessimistic defectors.

A Rates of participants’ defection in the mandatory condition (blue) and leaving in the voluntary condition (green), as a function of their original beliefs about others’ cooperativeness ($\mathrm{\gamma }$ in the mandatory conditions; grouped in 0.1 increments). B Proportions of participants who left groups in the voluntary condition as a function of their actions in the mandatory condition with the same threshold values ($n=191$ individuals for each threshold). Defectors in the mandatory condition (darker green) were more likely to leave groups compared to cooperators (lighter green), across the thresholds under voluntary participation. C Non-loners’ cooperation rates in the two conditions (red: mandatory, cyan: voluntary; Threshold 2: $n=120$ individuals, 4: $n=98$, 5: $n=133$). For the calculation of cooperation rates in the mandatory condition, we included only those participants who stayed in groups in the voluntary condition of the same threshold (i.e., non-loners). Error bars in (B, C) indicate 95% bootstrapped CIs, with the bar heights showing the exact values from the original experimental data. See Methods for the exact procedures to calculate the CIs. D Scatter plots of participants’ subjective beliefs about others’ cooperativeness ($\mathrm{\gamma }$) in the mandatory (x-axis) and voluntary (y-axis) conditions for each threshold (Threshold 2: $n=120$ individuals; 4: $n=98$; 5: $n=133$). Dot colors correspond to the changes in action (orange: positive change where a player changed action from defecting in the mandatory condition to cooperating in the voluntary condition, gray: no change, blue: negative change to defection). As in (C), we display solely the data from non-loners.

Fig. 4. Synthesizing positive and negative effects of the individual option on collaborative success via the degree of loners’ externality.

A Left: When groups are fixed in advance (e.g., the entire population is a group), loners have full externality on the collective outcome and effectively function the same as defectors to the group (e.g., the “village” example in ref. ). Right: When group boundaries are flexible, groups consist only of individuals who voluntarily opt in, and thus loners have no externality on the collective outcome. B Group success rate (computed from participants’ choice data, $r_{\mathrm{C}},r_{\mathrm{D}},\mathrm{and}{r}_{\mathrm{L}}$, in the main experiment) as a function of the degree of loners’ externality, $\mathrm{\rho }$. Blue lines correspond to the situation with the individual option (voluntary participation) and red to the situation without the individual option (mandatory participation). Note that, for the sake of consistency with (A), loners’ externality on the x-axis of graphs is smaller toward the right. C Cooperate, D defect, L leave.

Fig. 5. Results from the additional experiment, manipulating loners’ externality (ρ) on group outcome.

A Number of participants who chose each action (cooperate [C]: orange, defect [D]: blue, or leave [L]: green) in each $\mathrm{\rho }$ condition. Note that ρs are placed inversely from right (0) to left (1) for the sake of consistency with Fig. 4. B Effective cooperation rate ($r_{\mathrm{C}}/\left(r_{\mathrm{C}}+r_{\mathrm{D}}+\rho r_{\mathrm{L}}\right)$; left) and group success rate (right) as a function of $\mathrm{\rho }$. Error bars indicate bootstrapped 95% CIs ($n=182$ individuals), with the midpoints showing the exact values computed from the experimental data. See Methods for the exact procedures to calculate the CIs. C The distributions of participants’ beliefs about (effective) cooperation rate (gray bars) and probability of players choosing each action conditioned on the beliefs (color lines; C: orange, D: blue, or L: green). As participants reported their beliefs about the number of other participants (of 30) choosing each action, beliefs about the effective cooperation rate were computed based on Eq. (4).