Fig 1.
Illustration of meta-incentive game (MIG).
Four individuals are randomly drawn from the population and randomly assigned to one of four roles, recipient, donor, first-order player, and second-order player. In the first stage, the donor decides whether to help the recipient. In the second stage, the first-order player decides whether to provide an incentive for the donor; and in the last stage, the second-order player decides whether to provide an incentive to the first-order player.
Fig 2.
Illustration of replicator dynamics analyses for each type of S-MIG.
This figure illustrates all 24 types of S-MIG. The abbreviations are defined in Table 1. Their vertical layering in the figure reflects the existence condition for the basin of attraction on the point (x, y, z) = (1, 0, 0) related to (μ, δ) under which a cooperative regime emerges. The frames represent the form of local stability at point (x, y, z) = (1, 0, 0): the point is unstable for each type in the top frame which corresponds to (A) in Fig 3, is a non-isolated equilibrium for each type in the bottom right frame which corresponds to (B) in Fig 3, and is asymptotically stable for each type in the bottom left frame which corresponds to (C) and (D) in Fig 3.
Fig 3.
Replicator dynamics analysis of representative S-MIGs on 2-dimensional simplex.
The triangle represents the state space, Δ = {(x, y, z)*** : x, y, z ≥ 0, x+y+z = 1}, where x, y, and z are respectively the frequencies of the cooperative incentive-providers, cooperative incentive-non-providers, and non-cooperative incentive-non-providers. . (A) PR+R, (B) PP, (C) PB+RB(Full), and (D) RB. The abbreviations are defined in Table 1. In (A), (x, y, z) = (1, 0, 0) is unstable, so cooperation is never achieved regardless of the values of (μ, δ). In (B), the whole line z = 0 consists of fixed points, and thus, neutral drift is possible. In (C) and (D), (x, y, z) = (1, 0, 0) is a locally asymptotically stable point depending on the values of (μ, δ), and thus, a cooperative regime can emerge. In (C), the unstable equilibrium in the internal part on z = 0, Kz, is a saddle, and that on y = 0, Ky, is a source. In (D), Kz is a source, while Ky is a saddle.
Table 1.
Types of MIG.
Table 2.
functions on the lines y = 0 and z = 0 for each type of S-MIG. Here, f(x) and g(x) are defined in Eq (5).
Table 3.
Equations and solutions of z* in Eq (6) for each type.