Solution of dynamic variational inequalities with application to network equilibrium
There are many important competitive static equilibria network problems. Examples include economic competition in the marketplace, highway and transit traffic, wars, and biological competition problems. Solving competitive static equilibria network problems is difficult by statistical inference, because it depends on historical data, or microsimulation, and because it is generally very labor intensive. In recent years, non-cooperative game theoretic models have been successfully employed to address this type of problem, since they compute static game-theoretic equilibria as a sequence of well-defined mathematical programs. More recently, variational inequalities have become the formation of choice for such network problems, because variational inequalities substantially simplify the study of existence and uniqueness.
A variational inequality can be recast as a fixed point problem. The fixed point problem is generally considerably easier to apply than a static equilibria network problem. My dissertation investigates the algorithm and analyzes its efficiency; it also proves existence of the optimal solution. In particular, we look at the Cournot-Nash oligopoly model and show how (when recast as a fixed point algorithm problem) it is relatively easy to solve with more efficiency.
This suggests that other competitive static equilibria network problems can be solved with similar ease if they are recast as fixed point problems. The method seems especially useful for smaller scale network problems. This algorithm may not work efficiently with larger network problems, given current computer technology. However, with continued increases in computing power, this problem may diminish.