A Matrix Approach to Generalizing Axelrod-Type Tournaments

Abstract

Axelrod's (1980a 1980b) Prisoner's Dilemma computer tournaments have motivated numerous investigations of cooperation, strategy choice, and strategic evolution. By having players adopt various strategies for playing the Prisoner's Dilemma, and then programming a computer to pit the strategies against each other in a round-robin format, Axelrod uncovered important principles about the evolution of cooperation in certain contexts, and stimulated others to extend his basic method into other settings. This article presents a matrix approach for calculating the results of an Axelrod-type tournament. This approach allows one to investigate the impact of changes in the format of the tournament, the nature of the payoff matrix, or the particular strategic choices of the players.

Keywords:

• matrix games
• Prisoner's Dilemma
• Axelrod tournaments
• strategy
• evolution
• dynamics

Notes

Note. Tabled entries are the payoffs to the row player for each of the possible combinations of choices. C = cooperate, D = defect.

Note. C = Cooperation, D = Defection, s ₁ = ALL-C, s ₂ = ALL-D, and s ₃ = TFT. The vertices v ₁ through v ₄ correspond to the vertices in Figure 1.

Note. A is the adjacency matrix, C the choice matrix, and T the tournament matrix. is the adjacency matrix augmented by placing ones on the diagonal. D is the diagonal matrix of degrees. I is the identity matrix, and ⊗ denotes elementwise multiplication. Primes denote transposition. The function f labels the nodes with the strategies adopted.

Note. Strategies are named after their submitters, except for TIT-FOR-TAT, which was submitted by Anatol Rapoport, and RANDOM, which was entered by Axelrod. For strategy definitions, see Axelrod (1980a). The column labeled “Original” contains the rankings from Axelrod's (1980a) round-robin tournament.