Microdynamics for Nash maps (Q977923)

The article deals with the one-parameter family of homeomorphic maps \((x,y) \to (\bar{x},\bar{y})\) \[ \bar{x} = \frac{x + c(1 - 2x) \, \max\, \{0,y\} - c(1 + 2x) \, \max\, \{0,-y\}}{1 + 2c(1 - 2x) \, \max\,\{0,y\} + 2c(1 + 2x) \,\max\,\{0,-y\}}, \] \[ \bar{y} = \frac{y + c(1 - 2y) \, \max\, \{0,-x\} - c(1 + 2y) \, \max\, \{0,x\}}{1 + 2c(1 - 2y) \, \max\,\{0,-x\} + 2c(1 + 2y) \,\max\,\{0,x\}}, \] of the square \(\bigg[-\;D{\frac12},\;D{\frac12}\bigg]^2\) into itself. These maps are Nash maps for the game of Matching Pennies of two players \(X\) and \(Y\) with strategies \((x,1-x)\) and \((y,1-y)\), \(0 \leq x, y \leq 1\) and the payoff matrices \(R_x = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\) and \(P_y = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}\), respectively; a parameter \(c\), \(0 < c < \infty\), is the caution index that indicates how cautious the players are, that is, how much they want to preserve the previous strategy versus how much they want to modify it. The maps are only piecewise smooth; however are quadratic fractional in each quater-squares. As \(c \to 0\) the map \((x,y) \to (\bar{x},\bar{y})\) tends to the identity. The zero is a repeller of all these maps; however, for \(c \to 0\), there exists an attracting simple closed curve that converges to \(0\) as \(c \to 0\) and its form converges to a certain geometrical circle (this fact is also illustrated with graphs of the attraction closed curves for various values of \(c\); these graphs are obtained with computer experiments).