Static Games of Complete Information

(Nash equilibrium)

Nash equilibrium with discrete strategies

The beautiful movie, A Beautiful Mind, depicts John Nash's life. Well worth watching.

Mangasarian and Stone model to compute Nash equilibria in Bertrand problem

There is a nice nonlinear (quadratic) programming formulation due to Mangasarian and Stone (JMAA 1964) which can be used to find a Nash equilibrium of any two-person matrix game. This Maple file [.pdf] solves the Bertrand pricing problem discussed in class. (Note: By varying the initial conditions, it is possible to generate other equilibria, if they exist.)

Bertrand problem with Gambit

The Bertrand problem with three strategies is solved using Gambit here. You may need to download the file before loading in Gambit.

Nash equilibrium with continuous strategies

A simple example

Here is the Maple file [.pdf] of the problem discussed in class. If each player optimizes his and her function by manipulating both x and y, then the optimal solution results in large payoffs. But in game problems, this is not possible, because one's payoff depends on other player's decision also. The Nash equilibrium result is so much more different from the single-decision optimization problem.

Competing newsvendors

Here is the Maple file [.pdf] for the two newsvendors problem as it is presented in Wu and Parlar (2011).