Dynamics of Cooperation in a Task Completion Social Dilemma

We study the situation where the members of a community have the choice to participate in the completion of a common task. The process of completing the task involves only costs and no benefits to the individuals that participate in this process. However, completing the task results in changes that significantly benefit the community and that exceed the participation efforts. A task completion social dilemma arises when the short-term participation costs dissipate any interest in the community members to contribute to the task completion process and therefore to obtain the benefits that result from completing the task. In this work, we model the task completion problem using a dynamical system that characterizes the participation dynamics in the community and the task completion process. We show how this model naturally allows for the incorporation of several mechanisms that facilitate the emergence of cooperation and that have been studied in previous research on social dilemmas, including communication across a network, and indirect reciprocity through relative reputation. We provide mathematical analyses and computer simulations to study the qualitative properties of the participation dynamics in the community for different scenarios.

1 Proofs Theorems and Corollary

Proof Corollary 1.1
Let p * the fixed participation load taken by the individuals. Form the previous proof, we have that Theorem 3], we have then thatẑ converges exponentially to zero. From Eq (S1), given the initial conditionẑ(0), the trajectory ofẑ(t) is given bŷ Substitutingẑ byz − z, we obtain Eq (5).

Proof Theorem 2
This proof is based on Lagrange multiplier theory and the Karush-Kuhn-Tucker necessary and sufficient conditions for constrained optimization problems [3,Ch 3]. We show how the assumptions on the marginal gain functions and the formulation of the problem allows us to state the theorem.
Since U is a strictly concave function, and n i=1 p i = P is in the feasible region, we know that p * satisfies n i=1 p i = P since every increment in p will generate an increment in U . From the Karush-Kuhn-Tucker necessary conditions, there exist µ * , λ * i ≥ 0, i = 1, . . . , n, such that where µ * is the Lagrange multiplier associated with the constraint P i=1 p i ≤ P , and λ * i is the Lagrange multiplier associated with the constraint −p i ≤ 0. Since g i (p i ) > 0 for p i ∈ [0, P ], the Lagrange multipliers satisfy µ * −λ * i > 0. If −p * i > 0, that is, the constraint is inactive, then λ * i = 0, and g(p * i ) = µ * . Therefore, g(p * i ) = g j (p * j ) will be satisfied for every i, j = 1, . . . , n for which p * i , p * j > 0. From the previous result and the assumptions on g i as a function of p i , we have that p * satisfies the sufficient conditions to be a strict maximum of U over the feasible region [3, Proposition 3.2.2]. Since this is a strict concave optimization problem, this maximum is unique.

Proof Theorem 3
We show that the load distribution algorithm satisfies the conditions presented in [4] under which a load is distributed across the nodes in a network so that the marginal gain of any pair of nodes, quantified in this case by g i (p i (t)), is equalized. In this proof, we refer to the marginal gain g i (p i ) as the suitability of node i given p i to follow the notation in [4].
In [4,Theorem 3.4], it is shown that if some conditions are satisfied, then the rule in Eq (14) will lead to a distribution of the total available load P so that the nodes' suitability in the network is equalized. In our theorem, we have the assumptions that the network is connected, and the marginal gain is positive, and decreasing and differentiable with respect to p i . These assumptions, along with lines 6 to 9 in the algorithm, satisfy the conditions (a), (b), (c) in Assumption 1 in [4]. Condition (d) states that the total participation load P must be greater that some bound that can be unnecessarily conservative. In our problem, since we have shown the relationship between the optimization problem in Theorem 2 and the distribution algorithm, it is enough to have a value of P so that the optimization problem has an optimal distribution whose entries are (strictly) greater than zero.
From [4,Theorem 3.4], these four conditions guarantee that the transference of load between nodes leads to a state that is invariant where the suitability between every pair of nodes associated with the community members is the same, converging at an exponential rate. If the optimization problem in Theorem 2 has a maximum with zero entries, then the algorithm will equalize the suitability of those nodes associated with the positive entries.

Proof Theorem 4
This proof is a straightforward result from Theorem 1. Since at least one individual satisfies g i (p i , t) > 0 for p i ∈ [0, P ], the distribution algorithm will assign positive amounts of participation load based on the current marginal gains at each instant of time. Let η = min t≥0 {h(p(t))}, where we know that η > 0. Then, we obtain The right-hand side of this equation is a strictly increasing function of |z − z(t)|. Then, from Theorem 1, we have that z converges to z(t).
2 Implementation Details