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{\Large
\textbf{Supporting Information: 
Evolutionary dynamics of strategic behavior in collective-risk dilemmas}
}
% Insert Author names, affiliations and corresponding author email.
\\
Maria Abou Chakra and %$^{1}$, 
Arne Traulsen%$^{1,\ast}$ 
\\
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\section*{Evolutionary Game Dynamics in the Collective-Risk Dilemma}

A collective-risk game is played among $M$ players selected at random from a well mixed population of size $N$. 
In each round, players simultaneously invest $I_r$ units into a common pool towards a group target $T=M R$, where $R$ is the total number of rounds played in a game.
$C=\sum_{r=1}^R I_r$ is the total investment of that player.
An individual player commences each game with an initial endowment of $2R$. 
If as a group they fail to meet the target, each player's payoff is zero with probability $p_{\mathrm {risk}}$.
Players keep what they have not invested with probability $1-p_{\mathrm {risk}}$ if the target investment is missed, and with probability $1$ if the players have collectively reached the target investment.
Thus, a player obtains an average payoff of  $2R-C$ when the target is reached and 
$(1-p_{\mathrm {risk}})(2R-C)$ when the target is missed.

\subsection*{Evolutionary Game}
Evolutionary game dynamics were simulated using a mutation-selection process in a population of finite size. 
In each generation, $G$ games are played ($G \gg N $). 
An individual $i$ on average plays $G \cdot M/N$ games and the individual's payoff $\pi_i$ is calculated as the average payoff of the games played. 
The payoff $\pi_{i}$, is translated into a fitness value $f_i=\exp[\beta \pi_i]$, where $\beta$ measures the intensity of selection \cite{nowak:2004pw,traulsen:2008aa}. 
Higher payoffs increase an individual's reproductive potential towards the next generation. 
To simulate the evolutionary dynamics, we assume a frequency-dependent Wright-Fisher process, 
where $N$ individuals are selected at random in proportion to their fitness to reproduce and give rise to the next generation. 
Offspring inherits the strategy of the parent at the end of a generation ($G$ games). 
However, we also incorporate errors in this process.
With probability $\mu$, errors in the threshold values add Gaussian noise with standard deviation $\sigma_E$ to them. 
Independently, the contribution amounts change randomly to a different value with the same probability $\mu$.
As a result, in our dynamical setup, `evolution' operates at the level of strategies while `selection' operates at the behavioral level.


\section*{Variations of the Collective-Risk dilemma}

\subsection*{Group Size} 
A game is played among $M$ individuals selected at random from a well mixed population of size $N$. 
The whole group collectively has to invest a target sum $T= M R$ by the end of the game after $R$ rounds. 
If they fail to do so, their individual payoff is zero with probability $p_{\mathrm {risk}}$.
Therefore, the individual payoff is dependent on the contributions of the $M-1$ co-players.
We were interested in group size effect, and thus, we varied the parameter $M$ while all other parameters were kept constant. 
Simulation results are shown in Fig.\ S1. 
When only few players have to coordinate their actions, a smaller strategy space has to be explored. Furthermore, simulations show that when players are in smaller groups, contributions start at a lower risk value, compared to larger groups: 
For $M=2$, contributions started for $p_{\mathrm {risk}} \geq 0.5$,  and at $p_{\mathrm {risk}}\approx 0.6$, more than half of the games meet the target. For $M=4$, contributions started for $p_{\mathrm {risk}} \geq 0.65$,  and at $p_{\mathrm {risk}}\approx 0.7$, more than half of the games meet the target.  
For $M=6$, contributions started for $p_{\mathrm {risk}} \geq 0.75$,  and at $p_{\mathrm {risk}}\approx 0.8$, more than half of the games meet the target. 
And for $M=8$, contributions started for $p_{\mathrm {risk}} \geq 0.8$,  and at $p_{\mathrm {risk}}\approx 0.85$, more than half of the games meet the target.

\subsection*{Interest} 

Because earlier contributions could be more valuable for time sensitive collective actions, we 
explored the effect of an interest $\delta$ on the common account.
This means that in every round, the common account increases by a factor $1+\delta$. 
This leads to a situation in which all contributions occur early, in contrast to all other variations of the
collective-risk dilemma that we have analyzed.
Simulation results are shown in Fig.\ SI; with interest $\delta=0.1$, investment mainly occur in the first half of the game with players investing up to 30\% of $E$,  this was consistent for different group sizes.
For $M=2$, contributions started for $p_{\mathrm {risk}} \geq 0.35$, and at $p_{\mathrm {risk}}\approx 0.45$, more than half of the games meet the target. 
For $M=4$, contributions started for $p_{\mathrm {risk}} \geq 0.45$, and at $p_{\mathrm {risk}}\approx 0.55$, more than half of the games meet the target.  
For $M=6$, contributions started for $p_{\mathrm {risk}} \geq 0.55$, and at $p_{\mathrm {risk}}\approx 0.70$, more than half of the games meet the target. 
And for $M=8$, contributions started for $p_{\mathrm {risk}} \geq 0.6$, and at $p_{\mathrm {risk}}\approx 0.85$, more than half of the games meet the target.


\subsection*{Target Uncertainty} 
The whole group collectively has to invest a target sum $T= M R$ by the end of the game. 
However what happens if the target varied from one game to the other?  To explore how uncertainty affects behavior we included parameter $\sigma_T$ which can change the target (within a $\pm \sigma_T$ range) for each game.   $\sigma_T$ set to zero indicates that the target will not vary (as in previous simulations), however any value above zero changes the target based on a Gaussian noise with standard deviation $\sigma_T$. 
For simulation results, see Fig.\ S1. With target uncertainty ($\sigma_{T}=2$) the target success frequency dropped to $\approx 80\%$ and contributions start at a lower risk value, compared to larger groups; For $M=2$, contributions started for $p_{\mathrm {risk}} \geq 0.65$, and at $p_{\mathrm {risk}}\approx 0.7$, more than half of the games meet the target. For $M=4$, contributions started for $p_{\mathrm {risk}} \geq 0.7$, and at $p_{\mathrm {risk}}\approx 0.75$, more than half of the games meet the target. 
For $M=6$, contributions started for $p_{\mathrm {risk}} \geq 0.75$, and at $p_{\mathrm {risk}}\approx 0.8$, more than half of the games meet the target. 
And for $M=8$, contributions started for $p_{\mathrm {risk}} \geq 0.8$, and at $p_{\mathrm {risk}}\approx 0.85$, more than half of the games meet the target. 


\subsection*{Risk Curve} 
We also analyzed what happens when the risk continuously decreases with increasing investments.
To this end, we employed
a Fermi function, $p=1/(1+\exp[\gamma(T-\Gamma)])$, instead of the Heaviside step function,
\begin{equation}
p= \begin{cases}
0 & \Gamma\geq T\\
p_{risk}. \\
\end{cases}
\end{equation}
When the risk curve was smooth instead of the step function, we find that the general picture does not change - late contributions are favored for sufficiently high risk.

\subsection*{Maximum Contribution} 
In each round $r$, players simultaneously invest $I_{r}$ units into a common pool. 
In our analysis, we focused on a six player game in which  players can invest 0, 1 or 2 units for ten rounds, as in \cite{milinski:2008lr}.  
However we were also interested whether there is an effect  if the maximum contribution allowed changed.
Thus, to explore the maximum contribution allowed we varied individual investment $I$ between 1 and endowment, $E=2R$, for instance instead of 3 possible contributions players can invest 0, 1, 2, 3, 4, and 5 units. 
We find that in all cases, contributions start as late as possible. For instance, given the same 10 round game contributions began in the sixth round when the maximum contribution was 2 units per round  whereas contributions began in the ninth round when the maximum was 5 units per round, see Fig.\ \ref{max}. 

\section*{Collective-Risk Game versus Other Sequential Games}
Collective-Risk dilemmas games involve several rounds. 
As multiround games they fall into the category of sequential games (games with time), however without timing ($R =1$) they fall into the category of simultaneous games. 
In sequential games such as the ones explored by \cite{erev:1990,varian:1994, coats:2009,goren:2004} individuals play in sequence.
In contrast, in the collective-risk game explored herein individuals play simultaneously over a sequence of rounds \cite{milinski:2008lr, tavoni:2011fk, milinski:2011aa}) .  
Sequential game experiments \cite{Rapoport:1985uq,Rapoport:1987fl,Rapoport:1987jl,Rapoport:1989wl,erev:1990,dorsey:1992,goren:2004,potters:JPE:2005,coats:2009} reveal that individuals in the later part of the sequence become pivotal for the efficiency of the provision. 
The pattern is similar to the collective-risk `fair rational' behavior we observe.
However, in our case the player's position in a sequence is not defined.
Instead, every player's behavior becomes pivotal in the later part of the game. 


To explore the differences between the sequential game and the collective-risk game we expanded the scope of our computational model by introducing sequence allotment for players. Now individual strategies evolve based on the position in the sequence.  As before, each individual has a strategy composed of a threshold, $\tau_{s}$ and the contributions above and below the threshold, but now it is for each position in sequence, $s$ instead of round number, $r$.
The investment is thus determined by a player's strategy as well as the collective contributions by the individuals ahead in sequence so far.  
An individual player commences each game with an initial endowment of $E$ and target of $M\cdot E /2$. 
As before, a player obtains an average payoff of  $E-C$ when the target is reached and 
$(1-p_{\mathrm {risk}})(E-C)$ when the target is missed. 

We compared a sequential game with 10 players to a 10 round collective-risk game under different risk probabilities, see Fig.\ \ref{sequential}. For the collective-risk game late contribution dominated for $p_{\mathrm {risk}} >0.9$. While in the sequential game it was for $p_{\mathrm {risk}} >0.95$ when individuals later in sequence contributed more than individuals ahead in sequence.  As a result of the sequence assignment, there was a change in behaviors frequencies (compared to the collective risk game), $C=0$ behavior occurs at high frequencies for $p_{\mathrm {risk}} \leq 0.5$.  Now the $C<R$ behavior dominates for  $p_{\mathrm {risk}} >0.5$. Behaviors where $C > R$ start to increase by $p_{\mathrm {risk}} >0.9$ and $C=R$ are rare. 
Our simulations reveal that a 10 round collective-risk game has an increase in efficiency compared to a sequential game with 10 players for $p_{\mathrm {risk}}>0.7$.

%We also compared a sequential game where individual did not evolve based on their position, in this case it was equivalent to the collective-risk game only this time individuals played in a sequence and were aware of the previous contributions. The outcomes from such simulations were similar to the collective-risk.}

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