The Expected Number of Nash Equilibria of a Normal Form Game
Fix finite pure strategy sets <formula format="inline"> <simplemath><sub>S1</sub> <roman>,</roman>… <roman>,</roman><sub>Sn</sub> </simplemath> </formula>, and let <formula format="inline"> <simplemath>S=<sub>S1</sub>×⋯×<sub>Sn</sub> </simplemath> </formula>. In our model of a random game the agents' payoffs are statistically independent, with each agent's payoff uniformly distributed on the unit sphere in <openface>R</openface>-super-S. For given nonempty <formula format="inline"> <simplemath><sub>T1</sub>⊂<sub>S1</sub> <roman>,</roman>… <roman>,</roman><sub>Tn</sub>⊂<sub>Sn</sub> </simplemath> </formula> we give a computationally implementable formula for the mean number of Nash equilibria in which each agent i's mixed strategy has support T<sub>i</sub>. The formula is the product of two expressions. The first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets T<sub>i</sub>. The second may be construed as the "probability" that such an equilibrium remains an equilibrium when the strategies in the sets <formula format="inline"> <simplemath><sub>Si</sub>∖<sub>Ti</sub> </simplemath> </formula> become available. Copyright The Econometric Society 2005.
