Andrea Gaunersdorfer and Josef Hofbauer:
Fictitious play, Shapley polygons, and the replicator equation.
Games Econom. Behav. 11 (1995), 279-303.
This paper compares the limiting properties of the two fundamental dynamic models for noncooperative games in strategic form: (1) the fictitious play dynamics, of which the continuous-time version can be reduced to the best reply dynamics, $dx/dt={\rm BR}(x)-x$, where BR is the best reply correspondence, and (2) the replicator dynamics, $dx\sb i/dt=x\sb i[(AX)\sb i-xAx]$, where $A$ is a payoff matrix. The authors prove that asymptotic behavior of best reply dynamics and time-averaged behavior of best reply dynamics and time-averaged replicator dynamics coincide for three examples: the rock-scissors-paper game, a three-person matching pennies game due to J. S. Jordan [Games Econom. Behav. 5 (1993), no. 3, 368--386; MR 94e:90134], and the well-known $3\times 3$ bimatrix game by L. S. Shapley [in Advances in game theory, 1--28, Princeton Univ. Press, Princeton, NJ, 1964; ]. In all these examples, the best reply dynamics has the limit attractor being a polygon (called Shapley polygon) and moreover the time averages of the solutions of the replicator dynamics "converge" to the same polygon. The rock-scissors-paper game is discussed in great detail, and the paper presents several proofs for asymptotic behavior of the best reply dynamics by using standard techniques of smooth dynamic systems such as Lyapunov functions, Poincare sections and the Bendixson test.