Table 1.

Each mechanism can be described by a simple 2 × 2 payoff matrix, which specifies the interaction between cooperators and defectors. From these matrices we can directly derive the necessary conditions for evolution of cooperation. The parameters c and b denote, respectively, the cost for the donor and the benefit for the recipient. All conditions can be expressed as the benefit-to-cost ratio exceeding a critical value. The concepts of evolutionarily stable (ESS), risk-dominant (RD) and advantageous strategies (AD) are defined in the text. Further explanations of the underlying calculations can be found in the Supporting Online Material (53).

	Payoff matrix	Cooperation is …
		ESS	RD	AD
Kin selection	$\begin{array}{ccc}& C& D\\ C& \left(b-c\right)\left(1+r\right)& br-c\\ D& b-rc& 0\end{array}$	$\frac{b}{c}>\frac{1}{r}$	$\frac{b}{c}>\frac{1}{r}$	$\frac{b}{c}>\frac{1}{r}$	r…genetic relatedness
Direct reciprocity	$\begin{array}{ccc}& C& D\\ C& \left(b-c\right)/\left(1-w\right)& -c\\ D& b& 0\end{array}$	$\frac{b}{c}>\frac{1}{w}$	$\frac{b}{c}>\frac{2-w}{w}$	$\frac{b}{c}>\frac{3-2w}{w}$	w…probability of next round
Indirect reciprocity	$\begin{array}{ccc}& C& D\\ C& b-c& -c\left(1-q\right)\\ D& b\left(1-q\right)& 0\end{array}$	$\frac{b}{c}>\frac{1}{q}$	$\frac{b}{c}>\frac{2-q}{q}$	$\frac{b}{c}>\frac{3-2q}{q}$	q…social acquaintanceship
Network reciprocity	$\begin{array}{ccc}& C& D\\ C& b-c& H-c\\ D& b-H& 0\end{array}$	$\frac{b}{c}>k$	$\frac{b}{c}>k$	$\frac{b}{c}>k$	k…number of neighbors $H=\frac{\left(b-c\right)k-2c}{\left(k+1\right)\left(k-2\right)}$
Group selection	$\begin{array}{ccc}& C& D\\ C& \left(b-c\right)\left(m+n\right)& \left(b-c\right)m-cn\\ D& bn& 0\end{array}$	$\frac{b}{c}>1+\frac{n}{m}$	$\frac{b}{c}>1+\frac{n}{m}$	$\frac{b}{c}>1+\frac{n}{m}$	n…group size m…number of groups