A Course on Cooperative Game Theory

One of the most important practical optimization problems prevalent in practically all walks of life is the linear programming (LP) problem. It crops up in various engineering, operations research, scheduling and many other different scenarios. Due to its important, the problem has been extensively studied since the middle of the previous century. As a result, a deep literature has developed around the problem and connections have been established to other sciences.

Our reason for considering LP in this book is its connection to several aspects of cooperative game theory. The problem of determining whether the core of a game is empty can be formulated as an LP problem. Similar formulations can be made for the nucleolus. In this chapter, we provide a brief introduction to LP and describe the connections to the core and the nucleolus. The algorithmic complexity of solving LP is discussed in Chapter 11 and in Chapter 13, LP is used to formulate a notion of fairness in the stable matching problem. Our description of LP is minimal and is intended only to familiarise the reader with the basic idea. For a deeper understanding of the area including its algorithmic issues, we refer the reader to Papadimitriou and Steiglitz (1982).

The Diet Problem

LP is best introduced through a practical example. We start by motivating LP with the so-called diet problem. In this problem, a person wishes to obtain a balanced diet at a minimal cost. The basic idea is that there are several types of nutrients (say, proteins, fats, carbohydrates, minerals, etcetera) and for a healthy diet, a person needs to take in a certain minimum amount of each nutrient. The nutrients are not directly available. Instead, what are available are different kinds of foods (say, rice, wheat, meat, etcetera). Each kind of food contains the basic nutrients in varying proportions. We assume that these proportions can be quantified per unit of each kind of food.