SYST 465 / ECON 496 / MATH 465

Pricing in Optimization and Game Theory

George Mason University

Fall 2006

Instructor: Ursula Morris                           Class Room: Enterprise Hall 174

Class Time: Fr. 10:30 - 1:10                        Office Hours:

Email: UMorris1@gmu.edu

Course Description

Finding the adequate mechanism for pricing limited resources, goods and services is one of the main goals of the theoretical analysis of complex systems. Pricing is one of the driving forces for developing numerical methods to find optimal solutions and economic equilibria.

If we assume a market model which consists of producers who supply the market with a fixed amount of goods and of customers with unchanging demand conditions, we observe the following pricing mechanism.

If the goods are offered for a high price P1, supply S is greater than demand  D. A surplus of goods is the consequence. In order to sell the products, the producers have to lower the prices e.g. to P2. This in contrast leads to a shortage of the products as demand is higher than supply. The price has to be changed until total supply equals total demand, all goods are consumed and all money is spent.  This is called a market equilibrium.

The mechanism that leads to the equilibrium can be compared to a two-person game where the producers set the prices and the consumers react to the prices until a point is reached that is best for both parties.

Game theory provides methods to analyze the likely responses of competitors to strategic decisions about prices, expenditures and investments.

The first part of the course will cover the basic ideas and methods in Linear Programming (LP). The fundamental role of pricing in LP will be emphasized: duality, sensitivity analysis and LP decomposition are important topics. The introduction of  matrix games (MG) will show the close relationship between solving the dual pair of linear programs and finding an equilibrium in a two-person matrix game. The lecture will finish with the demonstration of an algorithm for finding an equilibrium in a linear market model using modified barrier functions.

Text: Wayne L. Winston, Operations Research Applications and Algorithms, Fourth Edition, Thomson Learning Inc., Brooks/Cole 2004.

Software: The computational project will use LINDO software which is provided with the book by Wayne L. Winston.

Course Topics

 1 Introduction and real life applications that led to Linear Programming (LP); Gauss-Jordan elimination 2 Simplex method 3 Shadow prices and sensitivity analysis 4 Duality in LP; basic duality theorems and their economic interpretation; Pricing mechanism in LP 5 Two-person matrix games; pure and mixed strategies 6 John von Neumann's theorem for matrix games 7 Matrix games and duality in LP; solving matrix games using LP methods 8 Brown-Robinson iterative method for solving MG; LP decomposition; pricing mechanism for LP using the BR method 9 Equilibrium in a linear exchange market model