On the risk in deviating from Nash equilibrium

Irit Nowik
Shmuel Zamir

The purpose of this work is to offer for any zero-sum game with a unique strictly mixed Nash equilibrium, a measure for the risk when deviating from the Nash equilibrium. We present two approaches regarding the nature of deviations; strategic and erroneous. Accordingly, we define two models. In each model we
define risk measures for the row-player (PI) and the column player (PII), and prove that the risks of PI and PII coincide. This result holds for any norm we use for the size of deviations. We develop explicit expressions for the risk measures in the L1 and L2 norms, and compute it for several games. Although the
results hold for all norms, we show that only the L1 norm is suitable in our context, as it is the only norm which is consistent in the sense that it gives the same size to potentially equivalent deviations. The risk measures defined here enables testing and evaluating predictions on the behavior of players. For example: Do players deviate more in a game with lower risks than in a game with higher risk?

April, 2014