Michał Misiurewicz

One of the main tools used in Economics is Game Theory. In 1994 John Nash won the Nobel prize in Economics, mainly for introducing the notion of what became known as Nash equilibrium, a crucial concept in non-cooperative games. In his original proof of its existence he used a certain map, which can be interpreted as corresponding to the players using a “better response” strategy. That is, the players look at the results of the preceding round of the game and adjust their strategy accordingly, but not trying to change it completely. Strategies involve probabilities, so one should understand a “player” as a group rather than as a single person.

Iterations of the Nash map can be viewed as a dynamical system and analyzed using methods from the theory of Dynamical Systems. We do it for 2 by 2 games, that is, games involving 2 players and 2 pure strategies. The maps are classified, according to the dynamics, as dominant strategy, elliptic or hyperbolic. One of the difficulties that we have to overcome is that the maps are only piecewise smooth. In one of the games, Matching Pennies, we show that even though the game has a unique Nash equilibrium point, the iterative procedure does notlead to convergence to the Nash equilibrium. Instead, the successive plays of the game will follow an orbit around the equilibrium point. This orbit has period eight and it is the only periodic orbit other than the equilibrium point.

For the better response map, one can vary the index of caution, which measures how much the players want to adjust their strategies. We investigate what happens if this parameter goes to zero, and we develop a scaling law.

The lectures are based on my research joint with the economists R. A. Becker and S. K. Chakrabarti, and mathematicians W. Geller and B. Kitchens.