To illustrate the importance of time as a coordination device in collective-risk dilemmas, let us first explore the baseline case where players are not able to interact over multiple rounds. In such a case, only unconditional strategies are available, such as being a defector (who does not contribute, ), a fair-sharer (someone who contributes a proportional share of the target, ), or an altruist (contributing the full endowment, ). In the simplest case of a pairwise game where the target is equal to one player's endowment, , a collective-risk dilemma with these three strategies is represented by the payoff matrix:(1)

In this game, irrespective of the strategy of their co-player, altruists end up with a payoff of zero. Therefore, altruism is a dominated strategy and we may expect that altruistic acts occur at very low frequencies, reducing the collective-risk dilemma to a game between defectors and fair sharers. Individuals strictly prefer defection for all , as the expected loss for missing the target is below the fair share contribution . This prediction is confirmed by replicator dynamics [19], [23], see Figure 1a: irrespective of the initial distribution of strategies in the population, individuals learn to stop contributing. This qualitative behavior changes as the risk of collective loss exceeds . In this case, there are three possible Nash equilibria: all players withholding their contributions, all individuals doing their fair share, and a mixed population of defectors and fair-sharers. In this mixed equilibrium, the fraction of defectors is given by(2)

However, since the mixed equilibrium is not evolutionary stable (Figure 1b), the dynamics either leads to a homogeneous population of defectors or to a homogeneous population of fair sharers. Which of these two possible outcomes is reached, depends on the initial behavior of the individuals: populations with a sufficient initial number of fair sharers eventually succeed in coordinating on the beneficial fair-share equilibrium, whereas populations mostly consisting of defectors end up in the detrimental equilibrium. In general, it depends on the risk of collective loss, whether or not a given initial population succeeds to coordinate on the fair-share equilibrium. To estimate the basins of attraction of each equilibrium, we have recorded the results of the evolutionary dynamics for different randomly chosen initial populations (Figure 1c). According to these simulations, an increasing risk of collective loss stimulates attempts to reach the fair-share equilibrium. Nevertheless, even for high risk values, a substantial proportion of initial populations fails to coordinate on the beneficial equilibrium. For instance, even for , roughly a quarter of all initial states lead to a non-cooperative population of defectors. Increasing the risk of collective loss has therefore a two-fold effect on the achieved welfare: on the one hand, a higher decreases the expected payoff if the target is missed, but on the other hand, high values of make it more likely that the players cooperate. As a consequence, intermediate, and not high values of represent the worst-case scenario for the average payoffs (Figure 1c).

To investigate the impact of time, let us now consider a collective-risk dilemma with two rounds. Again, we assume that each agent has an initial endowment , and that each agent can either contribute or to the common pool in each round. The target is reached if total contributions sum up to a player's endowment, that is . Obviously, this setting allows more than the previous three strategies of defectors, fair sharers, and altruists, as in the two-round game players may condition their behavior in the second round on their co-player's contribution in the first round. We can write the players' strategies as a 3-tuple with . The first variable determines whether the player cooperates in the first round: If , then this player contributes to the common pool, whereas for , the player does not contribute. The second variable determines whether the focal player cooperates in the second round, given that the opponent cooperated in the first round, whereas the third variable corresponds to the focal player's action in the second round, given that the opponent did not cooperate in the first round.

Therefore, this pairwise collective-risk dilemma allows eight possible strategies, which include the previous three strategies of the game without timing: For example, the strategy corresponds to a defector who does not contribute to the common pool, whereas players with strategy are altruists who contribute their full endowment, independent of the opponent's contribution behavior. The two strategies and are fair sharers, unconditionally contributing half of their endowment, either in the first period or in the second period, respectively. However, the collective-risk dilemma with timing allows additional strategies of interest: For instance, one may interpret a player with strategy (0;1,0) as a conditional cooperator, who is cooperative in the second round, given that the co-player was cooperative in the first round. In contrast, a player using applies a wait & see strategy, by awaiting the first round and by cooperating in the second round if there were no contributions in the first round. We can summarize the eight possible strategies' payoffs in a matrix:(3)

Similar to the case without timing, the defector's strategy leads to the highest payoff for all risk values . As the risk of collective loss exceeds , there are three additional pure Nash equilibria, two fair-share strategies and and the wait & see strategy . To estimate the robustness of these equilibria, we have again performed simulations with randomly chosen initial populations (see Figure 2a). As expected, defection is the most abundant strategy for low values of . However, for , the three cooperative equilibrium strategies , and are soon applied by a substantial share of initial populations, leading to complete coordination on a beneficial equilibrium with sufficient contributions as approaches one. Remarkably, for a risk of collective loss of , more than of all initial populations learn to coordinate on an equilibrium with sufficient contributions in this game with timing, while only reach the target in the game without timing. The opportunity to interact and influence one another thus indeed proves as a powerful means to reach cooperation in the collective-risk dilemma.

To analyze the timing of contributions, we recorded the fraction of players who contribute in the first and in the second round, respectively (Figure 2b). Depending on the risk of collective loss, one can roughly distinguish three different parameter regions: for , contributions are are neither made in the first nor in the second round, whereas in the interval , the fraction of contributions increases considerably in both rounds. In this region of intermediate risk, early contributors with strategy make up the majority of the population (Figure 2a), as they benefit from the presence of conditional cooperators . As exceeds approximately , individuals tend to delay their contributions towards the second round. In the limit of full risk, , contributions in the second round are twice as likely as early contributions, which is in line with the experimental observation that subjects tend to procrastinate their contributions towards the second half of the game [6].

The positive effect of time on coordination is reflected in the achieved average payoff (Figure 2c); especially for moderate risk values, the two-round game results in substantially higher payoffs than the one-round game. In particular, the minimum payoff increases by more than if individuals have the option to interact over time. Again, this minimum payoff is not attained at maximum risk, , but rather at an intermediate risk value.

Real-world coordination problems often involve a large group of individuals and multiple interactions over time. It is therefore natural to explore these more general cases. However, games with multiple players and multiple rounds are considerably more complex, and the size of the payoff matrix increases exponentially in both variables. To investigate such large-scale collective-risk dilemmas, we have performed extensive individual-based simulations. Simulations were conducted using the same setup as in [14], which allows for a comparison with previous work. This setup is similar to the two-round case: each of the individuals has an initial endowment and may contribute at maximum per round to the collective pool, in order to reach the group target , which is set to (i.e., the target is reached if all players give their half endowment). For the multi-round case, however, we assume that individuals base their decisions on the collective pool so far, rather than on co-players' individual decisions. That is, for every round a player defines an individual threshold on the total contributions up to round . A player's strategy is then a set , such that the player contributes an amount if the common pool satisfies the threshold , whereas the player contributes if the individual threshold is not satisfied.

To model the evolutionary dynamics, we use a mutation-selection process in a population of finite size . In each generation, individuals participate in several collective-risk dilemmas. Thereafter, the individuals' fitness is calculated as an exponential function of their payoffs, , where the strength of selection parameter measures the importance of a player's payoff for its fitness. Individuals are then selected in proportion to their fitness to give rise to the next generation [24], [25]. Offspring inherits the strategy of the parent with probability ; with the remaining probability a player explores a randomly chosen new strategy. In case of such a mutation event, we assume that changes in the thresholds and in the investments of each round occur independently, and that changes in the thresholds are normally distributed around a mean of with a variance of . We use this evolutionary game setup to explore the impact of group size, , and round number, , on coordination in collective-risk dilemmas.

As one may expect, group size has no effect for low risk values, , where withholding contributions is a weakly dominant strategy (see Figure 3a). However, for , small groups obtain higher payoffs, due to the higher probability to coordinate on a beneficial equilibrium with sufficient contributions. For larger groups, we observe a diffusion of responsibility [26], and it takes higher risks to motivate players to join the collaborative effort. For example, for a risk of collective loss , groups of 12 players typically fail to reach the target (resulting in a low payoff of approximately ), whereas two-player groups almost always reach it (leading to the maximum attainable payoff ). This group effect is quenched when game length increases (see Figure 3b): for , an increase in the number of rounds leads to a higher probability to coordinate on an equilibrium where the target is reached, resulting in a higher average payoff for all players. This positive influence of time is especially pronounced for intermediate risks, such as . However this quenching did not eliminate the group effect; even if the risk of collective loss approaches one and players have 12 rounds to reach the target, there are still instances of collective failure (resulting in an average payoff below the optimum ). Thus, while the inclusion of time in general facilitates coordination, there is no guarantee that subjects reach the target.

The previous simulations also allow us to investigate in more detail how individuals time their contributions in games with multiple rounds. If evolution leads to a contribution scheme comparable to Schelling's rule, then we would expect that individuals signal their willingness to contribute already in early rounds, and they would refrain from further contributions as soon as they realize that their co-players do not follow. If applied by all individuals, such a strategy would lead to overall contributions that are evenly distributed over the rounds. Alternatively, individuals that are rational could also apply backward induction: for , and given that players intend to reach the target, backward induction would suggest that players contribute nothing in the first half of the game, while they would donate the maximum amount in the second half. In this way, late contributions serve as a self-commitment, which allows individuals to signal credibly that they will not contribute more than their fair share. By not contributing in the beginning, they simply forego any possibility to compensate insufficient contributions, but they ensure that others will either contribute or face collective loss.

Our evolutionary simulations suggest that the timing of contributions is somewhere between these two extremes: while average contributions in the last round are typically close to the maximum amount , this does not imply that all contributions are shifted towards the second half of the game. Instead, there is always a baseline level of early contributions, independent of the total number of rounds (see Figure 4a). However, the results rather seem to be in line with the backward induction outcome than with evenly distributed contributions over time. This relative abundance of awaiting strategies does not depend on the maximum contribution per round; nor does it depend on the assumption that players only have the binary option of contributing either or in a given round (see Figure 4b, where subjects could either contribute nothing, or ).

A significant delay of contributions towards the second half of the game was also observed in the experiments of Milinski et. al. [6]. However, a comparison of the experimental data for a high risk of collective loss, with our results reveals that the evolutionary simulations overestimate the extent of procrastination (see Figure 5). There might be several reasons for this discrepancy: first, in our model individuals only consider the total contributions so far. However, the subjects in experiments may also base their actions on the outcome of the previous round, or on individual behaviors. Unfortunately, reasons behind subjects' actions are still wanting and thus we chose to focus on total contributions. Second, the subjects in the experiments were only allowed to play the game once and thus they did not have the opportunity to learn and adapt their strategies, as assumed in our evolutionary simulations. We would therefore expect that experienced subjects exhibit a behavior that is closer to the backward induction solution, as found in other economic interactions [27]. Third, our evolutionary analysis does not include any psychological motives for contributions, such as loss aversion [28], or framing effects suggesting subjects should reach the target by contributing a fair amount each round. The presence of such effects would also explain why subjects contributed a considerable amount of their endowment even in treatments where the risk of collective loss was only , in which case non-contribution would have been the individual and the social optimum. However, if the game is played repeatedly, one might also expect that the impact of these psychological motives decreases [29], and the observed timing of contributions might reveal a stronger tendency to procrastinate.