Selfish altruism, fierce cooperation and the predator^*

* Draft of May 10, 2018. Earlier versions of this paper were presented at the IZA brown bag seminar on two occasions (February 2012 and January 2013). Subsequent versions were presented at the 2014 Conference on ‘Norms, Actions, Games’, the 16th Conference on the Foundations of Utility and Risk and the 25th International Conference on Game Theory at Stony Brook University.

Figures & data

Figure 1. Two strikers against one goalie in ‘Xs and Os notation’.

Figure 2. Projected on the xy-plane the set of interior Nash equilibrium points together with four instances selected from the one parameter family of invariant surfaces for selected values of a. Notice how y tends to 1 as a increases.

Figure 3. I run 110,000 rounds of the discrete model given by Equation (17(17a) $x_{i+1}=max\left[0,min\{1,(1-2z_i+c\left)h\right\}\right],$(17a) ) with initial conditions x=.8 (near complete selfishness), y=.1 (near complete defector), and z=.4 with e=1 and c=.1. As in the continuous case, y tends to 1 and x,z to $.5$.

Figure 4. Pairwise dynamics of 110,000 rounds of the discrete model given by Equation (17(17a) $x_{i+1}=max\left[0,min\{1,(1-2z_i+c\left)h\right\}\right],$(17a) ) with initial conditions x=.8, y=.1, and z=.4 with e=1 and c=.1.