A modification for prisoner’s dilemma for iterative interactions.

<p>The term <i>w</i> is the probability of a player knowing the strategy of its partner. The tit-for-tat (TFT) strategy cooperates with strangers or with known TFT individuals. It plays defect with known D individuals.</p

Payoff matrix for an iterative prisoner’s dilemma with non-random interactions.

<p>Following many interactions, a portion <i>r</i> occurs with a like-individual, and only the portion 1-<i>r</i> occurs with the randomly selected individual. All-C interacts with another All-C, both <i>r</i> and 1-<i>r</i> of the time the interaction generates a payoff of <i>r</i>(<i>b</i>−<i>c</i>)+(1−<i>r</i>)(<i>b</i>−<i>c</i>), with a total payoff of “<i>r</i>(<i>b</i>−<i>c</i>)+(1−<i>r</i>)(<i>b</i>−<i>c</i>)” for All-C and TFT pair of strategies. Likewise, All-C and All-D obtains <i>r</i>(<i>b</i>−<i>c</i>)+(1−<i>r</i>)( −c), and so forth.</p

A general payoff matrix for games with social dilemmas.

<p>Three conditions hold for social dilemmas: 1) <i>a>d</i>, 2) a>(<i>b+c</i>)/2, and 3) <i>c>a</i>. A mixed strategy is the ESS when <i>b>d</i>, and All-D is a global ESS when <i>d>b</i>.</p

A modification for prisoner’s dilemma for non-random interactions.

<p>The term <i>r</i> is the probability of like interacting with like and (1−<i>r</i>) the probability of random interactions. Payoffs are to the individual playing the row strategy against an opponent playing the column strategy.</p

Payoff matrix for an iterative hawk-dove game with non-random interactions.

<p>Payoff matrix for an iterative hawk-dove game with non-random interactions.</p

Prisoner’s dilemma with varying combinations of <i>r</i> and <i>w</i>.

<p>Below the D ESS isoleg, All-D is ESS. Above the TFT isoleg, TFT is an ESS. Either D or TFT can be the ESS in the region between the D- and TFT-ESS isoleg, leading to alternative stable states. To right of the C-ESS isoleg, C can be an ESS. Parameter values: <i>b</i> = 1.65 and <i>c</i> = 1.</p

The hawk-dove game at varying values of r and w, under low and high cost.

<p>Panel A shows the hawk-dove game under low cost (<i>b</i>/2−<i>c</i>>0; values used were <i>b</i> = 2.4, <i>c</i> = 1). The D and TFT isolegs cross, so there are four regions of outcomes: 1) All-D is a global ESS, 2) All-D or All-TFT are local ESSs (alternate stable states), 3) a mixture of TFT and D are a global ESS, and 4) All-TFT is the global ESS. All-C can be an ESS when <i>r</i>><i>c</i>/<i>b</i>. Panel B shows the hawk-dove game under high cost (<i>b</i>/2−<i>c</i><0; values used were <i>b</i> = 2.4, <i>c</i> = 2) when. At low values of <i>r</i> and <i>w</i>, a mixed strategy of TFT and D is the ESS. At higher values TFT becomes the global ESS. Once <i>r</i> is sufficiently large, then All-C can be an ESS.</p

Payoff matrix for an iterative general game with non-random interactions.

<p>Payoff matrix for an iterative general game with non-random interactions.</p

The snowdrift game with varying combinations of r and w, under low and high costs.

<p>Panel A shows the snowdrift game under low cost (where <i>b</i>/2– <i>c</i> >0; values used were <i>b</i> = 2.4, <i>c</i> = 1). At low <i>r</i> and <i>w</i>, a mixed strategy of TFT and D is the ESS. At high <i>r</i> and <i>w</i>, All-TFT becomes the global ESS, and once <i>r</i> is sufficiently large, All-C can be an ESS. Panel B shows that higher values of <i>r</i> and <i>w</i> are required to promote cooperation when costs of cooperating are higher (where <i>b</i>/2– <i>c</i> <0, values used were <i>b</i> = 2.4, <i>c</i> = 1.5). Because the D and TFT ESS isolegs cross, there are four regions of outcomes: 1) All-D is a global ESS, 2) All-D or All-TFT is local ESS (alternate stable states), 3) a mixture of TFT and D are a global ESS, and 4) All-TFT is the global ESS. All-C can be an ESS when <i>r</i>><i>c</i>/<i>b</i>.</p

Payoff matrix for the snowdrift game.

<p>Digging out of the snowdrift gives each player a benefit of <i>b/2</i>. The cost is born by the digger (C) who splits the cost when both dig, or bears the entire cost as the sole digger. When <i>b/</i>2−<i>c</i>>0, the ESS is a mixture of C and D individuals. When <i>b/</i>2−<i>c</i>/2>0><i>b</i>/2−<i>c</i>, All-D is the sole ESS, even though there is still a social dilemma where All-C yields higher payoffs than All-D.</p