Pure Strategy Nash Equilibrium Points and the Lefschetz Fixed Point Theorem

Using the Lefschetz fixed point theorem, a pure strategy Nash equilibrium existence theorem is established for a class of n-person games with possibly nonacyclic strategy sets. It is argued that the Lefschetz approach to fixed point theorems may ultimately prove to be particularly important in economic and game theory due to the generality of spaces that can be considered and the interesting related questions that can be investigated. For example, the Lefschetz approach to fixed point theorems leads naturally to the concept of a "Nielsen Number" of a map f:Y->Y, a homotopy-invariant lower bound for the number of fixed points of f. The Nielsen number provides a lower bound for the number of Nash equilibria in certain n-person games. Annotated pointers to related work can be accessed here: http://www.econ.iastate.edu/tesfatsi/vita.htm#Game