x. functions on the lines <i>y</i> = 0 and <i>z</i> = 0 for each type of S-MIG. Here, <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are defined in Eq (5).

<p></p><p></p><p></p><p><mi>x</mi><mo>.</mo></p><p></p><p></p> functions on the lines <i>y</i> = 0 and <i>z</i> = 0 for each type of S-MIG. Here, <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.e016" target="_blank">Eq (5)</a>.<p></p

Illustration of meta-incentive game (MIG).

<p>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.</p

Replicator dynamics analysis of representative S-MIGs on 2-dimensional simplex.

<p>The triangle represents the state space, Δ = {(<i>x</i>, <i>y</i>, <i>z</i>)*** : <i>x</i>, <i>y</i>, <i>z</i> ≥ 0, <i>x</i>+<i>y</i>+<i>z</i> = 1}, where <i>x</i>, <i>y</i>, and <i>z</i> are respectively the frequencies of the cooperative incentive-providers, cooperative incentive-non-providers, and non-cooperative incentive-non-providers. </p><p></p><p><mo stretchy="false">(</mo><mi>μ</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo></p><p><mn>1</mn><mn>2</mn></p><mo stretchy="false">)</mo><p></p><p></p>. (A) PR+R, (B) PP, (C) PB+RB(Full), and (D) RB. The abbreviations are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.t001" target="_blank">Table 1</a>. In (A), (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0) is unstable, so cooperation is never achieved regardless of the values of (<i>μ</i>, <i>δ</i>). In (B), the whole line <i>z</i> = 0 consists of fixed points, and thus, neutral drift is possible. In (C) and (D), (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0) is a locally asymptotically stable point depending on the values of (<i>μ</i>, <i>δ</i>), and thus, a cooperative regime can emerge. In (C), the unstable equilibrium in the internal part on <i>z</i> = 0, <i>K</i>_<i>z</i>, is a saddle, and that on <i>y</i> = 0, <i>K</i>_<i>y</i>, is a source. In (D), <i>K</i>_<i>z</i> is a source, while <i>K</i>_<i>y</i> is a saddle.<p></p

Types of MIG.

<p>Types of MIG.</p

x. functions on the lines <i>y</i> = 0 and <i>z</i> = 0 for each type of S-MIG. Here, <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are defined in Eq (5).

<p></p><p></p><p></p><p><mi>x</mi><mo>.</mo></p><p></p><p></p> functions on the lines <i>y</i> = 0 and <i>z</i> = 0 for each type of S-MIG. Here, <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.e016" target="_blank">Eq (5)</a>.<p></p

Illustration of replicator dynamics analyses for each type of S-MIG.

<p>This figure illustrates all 24 types of S-MIG. The abbreviations are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.t001" target="_blank">Table 1</a>. Their vertical layering in the figure reflects the existence condition for the basin of attraction on the point (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0) related to (<i>μ</i>, <i>δ</i>) under which a cooperative regime emerges. The frames represent the form of local stability at point (<i>x</i>, <i>y</i>, <i>z</i>) = (1, 0, 0): the point is unstable for each type in the top frame which corresponds to (A) in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.g003" target="_blank">Fig 3</a>, is a non-isolated equilibrium for each type in the bottom right frame which corresponds to (B) in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.g003" target="_blank">Fig 3</a>, and is asymptotically stable for each type in the bottom left frame which corresponds to (C) and (D) in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.g003" target="_blank">Fig 3</a>.</p

Equations and solutions of <i>z</i>* in Eq (6) for each type.

<p>Equations and solutions of <i>z</i>* in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004232#pcbi.1004232.e060" target="_blank">Eq (6)</a> for each type.</p