# Simulating the Cost of Cooperation: A Recipe for Collaborative Problem-Solving
## 3 Results  
According to the effect of the Cost of Cooperation on Problem Simplicity (Figure 1 Left), the simpler the tasks (from \(r=0.9\) to \(r=0.1\)), the lesser the difference on the final agent fitness. Moreover, the difference increases from 15 to 55% in conjunction to the cooperation costs. On the contrary, for a difficult task (\(r=0.01\)), the relationship between the cost of cooperation and the difference in the fitness is almost linear. The main reason for such behavior is that the cost of cooperation influences the reduction of the fitness measure in two ways: (i) directly, where the agent has to pay a cost to cooperate, and (ii) indirectly, where fewer agents want to cooperate because of the direct cost, and, thus, the cooperation is infrequent and the agents have fewer advantages. However, from the point of view of the community size (Figure 2 Right), the cost of cooperation affects the smaller group more than the larger ones. As for the smallest community size (i.e., \(s=1\)), as well as for the smallest problem simplicity (i.e., \(R=0.01\)), the final difference on the fitness is greater than 100%. Such an effect is due to the fact that, especially for very difficult tasks (i.e., \(R=0.01\)), the cost of cooperation is frequently paid without any subsequent payoff, therefore producing a negative final fitness for the agent. For what concerns the maximum group capacity reached by the system at the equilibrium (Figure 1), a general decrease is revealed as related to the cost of cooperation. The effect is caused by the reduction of collectivists' behavior within the system. Nevertheless, its magnitude is largely affected by the two control parameters of the system (i.e., problem simplicity and the size of the group). In particular, as shown in the left plot of Figure 1, the greatest reduction affects the systems facing the hardest problem (i.e., simplicity of the task = 0.01), quite independently to the cost parameters (\(\delta^{c}\)), always reducing the final group capacity at about 40% in comparison to the zero cost condition. On the contrary, the systems facing the easiest problems (i.e., simplicity of the \(task=0.9\) and 0.7) appear not to be affected greatly by the cost, always reaching a reduction of the final group capacity below 5% quite independently from the cost of cooperation. Interestingly, the systems that faced problems of intermediate complexity (i.e., \(R=0.1,0.3,0.5\)) are revealed to be the most sensitive to the cost of cooperation. For instance, a group challenged by a problem simplicity of \(R=0.1\) demonstrated a loss of around 13% when the cost of cooperation was equal to 10% of the expected gain, reaching a loss of 35% for a cooperation cost of 90%. By the same token, the results show an effect contributed by the community size (the right plot of Figure 1). The larger the community, the smaller it appears to be both in the magnitude of the capacity reduction and the sensitivity to the cost of cooperation. In particular, in the extreme case represented by individuals alone, the group capacity reduction ranges between the 2%, for a cooperation cost of 10%, to 60% for a cooperation cost of 90%. In general, when the cost of cooperation is zero, there is an inversely proportional relation between the average probability of cooperation and the size of the group (Figure 3 Left). Moreover, this relation is common despite the different difficulty of the task. Another important aspect is that the optimal equilibrium is reached when there are 2 groups of size 32, respectively. In fact, for smaller groups, more competition is required because an agent has no interest in splitting the gain equally with the others. This aspect is stressed when a cost of cooperation is needed. Actually, the average probability of cooperation is \(10^{-5}\). Figure 1: Percentage differences of the final group cardinality (i.e., the maximum complexity of the problem-solved in the past) in comparison to the cost of cooperation, for each problem simplicity (**left** plot) and for each group size (**right** plot).