Andrea Gaunersdorfer and Josef Hofbauer:

** Fictitious play, Shapley polygons, and the replicator equation. **

*Games Econom. Behav.*** 11 ** (1995), 279-303.

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### Math Review item for 96j:90096

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.

**Reviewed** by Akiro Okada

*© Copyright American Mathematical Society 1996, 1997*