Evolutionary games on graphs and discrete dynamical systems

Figures & data

Table

	Abbreviation	Scenario	Nash equilibria
$a>c>b>d$	FC	Full cooperation	(C,C)
$c>a>b>d$	HD	Hawk and Dove	(C,D), (D,C) and a mixed equilibrium
$a>c>d>b$	SH	Stag hunt	(C,C), (D,D) and a mixed equilibrium
$c>a>d>b$	PD	Prisoner's dilemma	(D,D)

Figure 1 Set of admissible parameters and four game-theoretic scenarios, $a=1$ and $d=0$.

Table 1 $K_n$ – Connection between graph size, different scenarios and attractivity of states $(0,0,\dots ,0)$ and $(1,1,\dots ,1)$.

	PD	HD	SH	FC
$(0,0,\dots ,0)$ attractive	Always	$n<1+\frac{c-d}{b-d}$	Always	$n<1+\frac{c-d}{b-d}$
$(1,1,\dots ,1)$ attractive	Never	Never	$n>1+\frac{a-b}{a-c}$	$n>1+\frac{a-b}{a-c}$

Figure 2 Attractivity regions of $(0,0,\dots ,0)$ (light grey) and $(1,1,\dots ,1)$ (horizontally hatched) ($a=1$ and $d=0$ are fixed, cf. Remark 8). (a) $n=3$ on the left and (b) $n=5$ on the right.

Figure 3 A trajectory of the evolutionary game from Example 13. For possible reproduction we provide its graph 6 notation (see [22]): |WsOPA?OG?[?E@C?o@??@??O?????????s??k?@@_?Cg??KO|.

Figure 4 Illustration of Theorem 27 with $a=1$ and $d=0$. Both $(0,0,\dots ,0)$ and $(1,1,\dots ,1)$ are attractive in the horizontally hatched region, $(0,0,\dots ,0)$ in the light grey region, $(1,1,\dots ,1)$ in the dark grey region. In the vertically hatched region, the attractive cycles (20) and attractors (19) occur. There are no non-trivial attractors in the two dotted regions.

Figure 5 Wheels $W_8$ and $W_{14}$.

B.D. McKay, Graph6 and sparse6 graph formats. Available at http://cs.anu.edu.au/∼bdm/data/formats.html.

 Google Scholar