Mathematical Modeling in the Teaching of Game Theory Ein-Ya Gura The Center for the Study of Rationality The Hebrew University of Jerusalem

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1 Mathematical Modeling in the Teaching of Game Theory Ein-Ya Gura The Center for the Study of Rationality The Hebrew University of Jerusalem Abstract Mathematical modeling is the essence of the teaching of mathematics. The problem is that most of the mathematics taught at school has almost no connection to real-life situations and therefore the mathematical modeling is quite meaningless. In this article, we outline a course in game theory we offered to Israeli high-school students in which we show how mathematics can be used to analyze real-life situations. Game theory is motivated mostly by the social sciences and therefore constructing mathematical models for real-life situations is natural to the theory. Introduction The idea of creating a course in game theory for high-school students originated from an awareness of the general public s limited and narrow conception of mathematics. Mathematics is perceived as primarily technical, computational, and meaningless, and is thus not considered to be intellectually challenging. One can analyze the reasons for this misconception or one can intervene and try to change the situation. We have chosen the latter option. In order to help students sense the spirit of mathematics, one must expose them to as many kinds of mathematics as possible and enable them to see mathematics as related to reality. New curricula and new approaches to mathematics instruction can effect this change. In Israel, a mathematics curriculum consisting of compulsory courses and ninety hours of elective studies was approved in This curriculum change paved the way for an elective in game theory. Unfortunately, this change was never implemented, but we were left with a course in game theory. Game theory in general is a branch of mathematics that is motivated mostly by the social sciences or, better, by human behavior. In game theory one can ensure that students are able to understand the basic problems on which they are working. As game theorist and Nobel laureate Robert Aumann writes, The language of game theory coalitions, payoffs, markets, votes suggests that it is not a branch of mathematics, that it is motivated by and related to the world around us, and that it should be able to tell us something about the world. Yet, despite its tractability, game theory is and remains a branch of mathematics: the resistive medium is the mathematical model with its definitions, axioms, theorems and proofs. Our main goal was to make mathematics a subject that can be discussed and thought about through a basic comprehension of the problem at hand. What we are trying to do in science is to understand our world and therefore the basic aim of scientific activity remains the comprehension itself. The Course The course follows the book Insights into Game Theory: An Alternative Mathematical Experience written by Michael Maschler and Ein-Ya Gura, and published by Cambridge University Press in The choice of topics reflects our purpose: we wanted to present material that does not require mathematical prerequisites and yet involves deep game-theoretic

2 ideas and some mathematical sophistication. Broadly speaking, the topics chosen are all related to the various meanings that can be given to the concept of fair division. The course is a collection of a few topics from the theory that are intended to open a window onto a new and fascinating world of mathematical applications to the social sciences. It selects a small number of topics and studies them in depth. It shows the student how a mathematical model can be constructed for real-life issues. One of the aims of the course is to acquaint the student with a different mathematics, a mathematics that is not buried under complicated formulas, yet contains deep mathematical thinking. Another aim is to show that mathematics can efficiently handle social issues. A third aim is to deepen the mathematical thinking of the students. The book on which the course is based is divided into chapters that are independent so that a teacher and a student can choose one chapter or several and cover them in any order. The first chapter, Mathematical Matching, concerns, among other things, the problem of assigning applicants to institutions of higher learning. Each applicant ranks the universities to which he is applying according to his scale of preferences. The universities, in turn, rank the applicants for admission according to their own scale of preferences. The question is how to effect the matching between the applicants and the universities. This problem leads to unexpected solutions. The second chapter, Social Justice, concerns social decision rules. In a democratic society it is customary to make decisions by a vote. The decision supported by the majority of the voters is adopted. But we show that majority rule does not always yield a clear-cut solution. The attempt to find other voting rules raises unexpected difficulties. The third chapter, The Shapley Value in Cooperative Games, addresses, among other things, the following problem: a group of people come before an arbitrator and inform him of the expected profits of every subgroup, as well as of the whole group, if the groups operate independently. It seems that these data are sufficient for the arbitrator to decide how to divide the profits if all the litigants operate together. The fourth chapter, Analysis of a Bankruptcy Problem from the Talmud, addresses the following problem: several creditors have claims to an estate, but the total amount of the claims exceeds the value of the estate. How should the estate be divided among the creditors? In the chapter several solutions are accepted, two of which are discussed in the Talmud. Every chapter is based on a real-life situation that makes the constructing of a mathematical model relevant and meaningful. To illustrate this, I shall discuss the first two chapters in detail. Mathematical Matching In 1962 a paper by David Gale and Lloyd S. Shapley appeared at the RAND Corporation, whose title, College Admissions and the Stability of Marriage, raised eyebrows. Actually, the paper dealt with a matter of some urgency. According to Gale, the paper owes its origin to an article in the New Yorker, dated September 10, 1960, in which the writer describes the difficulties of undergraduate admissions at Yale University. Then as now, students would apply to several

3 universities and admissions officers had no way of telling which applicants were serious about enrolling. The students, who had every reason to manipulate, would create the impression that each university was their top choice, while the universities would enroll too many students, assuming that many of them would not attend. The whole process became a guessing game. Above all, there was a feeling that actual enrollments were far from optimal. Having read the article, Gale and Shapley collaborated. First, they defined a concept of stable matching, and then proved that stable matching between students and universities always exists. This and further developments are discussed in the first chapter of our book. For simplicity, Gale and Shapley started with the unrealistic case in which there are exactly n universities and n applicants and each university has exactly one vacancy. A more realistic description of this case is a matching between men and women hence the title of their paper. In the book, we start with a community of men and women where the number of men equals the number of women. The objective is to propose a good matching system for the community, where the meaning of good becomes clear. To be able to propose such a system, we need relevant data about the community. Accordingly, we ask every community member to rank members of the opposite sex in accordance with his or her preference for a marriage partner. We assume that no man or woman in the community is indifferent to the choice between two or more members of the opposite sex. This assumption is introduced to simplify our task. Later on we show how to dispense with it. From the specific community we move on to the definition of a stable matching, then to the procedure of finding a stable matching system, and finally to the proof that a stable matching system always exists. The Gale Shapley algorithm is based on three assumptions: 1. The number of men equals the number of women. 2. There is no indifference. 3. Every community member has to rank all members of the opposite sex. These assumptions are needed for the proof that a stable matching system always exists, but once we prove it we show how to dispense with them. Next, we deal with the Gale Shapley algorithm and the assignment problem, specifically the medical school admissions problem. It has been shown that some preference structures yield more than one stable matching system. This raises a few questions. 1. Is there one stable matching system that is everyone s favorite? Assuming that there is no indifference, the answer is no, because if there are two stable matching systems, then at least one man is paired off with a different woman in the second system, and necessarily prefers one system to the other. 2. Is there one stable matching system that is the men s favorite? Surprisingly, the answer is yes. The same goes for the women: there is a stable matching system that is the women s favorite. This leads to a definition of an optimal stable matching system for

4 all men or all women. It is proved that for every preference structure, the matching system obtained by the Gale Shapley algorithm, when the men propose, is optimal for the men. When the women propose they too get an optimal matching system. To sum up, there is a gradual development in this chapter: we start with a real-life problem, and then construct a model that leads to an algorithm for getting a stable matching system and, beyond that, to an optimal matching system for the proposing group. Gale and Shapley were the first to ask whether their algorithm for matching men and women was applicable to the college admissions problem. What they did not know at the time was that the Association of American Medical Colleges had already for ten years been applying the Gale- Shapley algorithm to the task of assigning interns to hospitals in the United States. By a process of trial and error that spanned over half a century, the Association in 1951 adopted the procedure, later rediscovered by Gale and Shapley, that was hospital-optimal. Social Justice In a democratic society, the prevalent method of decision-making is majority rule. This method attempts to aggregate many individual views and opinions into a single social decision. Suppose there is a community of three voters who must make a decision by choosing one of three alternatives (say, disarmament, cold war, or open war). A society that behaves rationally will establish a preference order with regard to the three alternatives on the basis of voter preferences, choosing the alternative that is the first preference. Majority rule is the natural way to make a social decision on the basis of voter preferences. Consider the following example, known as the voting paradox. A certain amount of the municipal budget is unspent and the city council must decide how to invest it. It has three options: investment in education, investment in security, and investment in health (the sum is too small to divide feasibly among the three options). Sitting on the city council are representatives of three parties: the Left party (3 representatives), the Center party (4 representatives), and the Right party (5 representatives). The parties list of preferences is: Center (4) Left (3) Right (5) health education security security health education education security health The preferences are listed in the columns in descending order. It makes little sense to vote on all alternatives together in one vote. Such a vote would result in a decision to invest in security (5 to 3 or 4), whereas there is a clear majority in favor of health over security (7 to 5), because both the Left party and the Center party prefer investment in health to investment in security. Therefore, a proposal is adopted to vote on the different alternatives in pairs. The result of the pairwise voting is: security vs. education: the majority is in favor of security (9 to 3) security vs. health: the majority is in favor of health (7 to 5)

5 In other words, the majority prefers health to security, and it prefers security to education. One way therefore is to conclude that the social preference is: health security education However, one of the council members calls for a vote on health vs. education. Remarkably, it turns out that in the majority opinion, education is preferred to health (8 to 4). In this example, decision by majority leads to the absurd: health security education security education health The voting results shows: health security education health The relation is a cyclic preference relation because for any alternative there exists another alternative that is preferred to it. That is, there is no most-preferred alternative and therefore majority rule provides no clear guidance on how to spend the budget. This paradox has long been known. The French mathematician and philosopher Marquis de Condorcet first noted it in It seems that in this example majority rule, which establishes a social preference on the basis of voter preferences, does not yield rational behavior. The question is whether there is a decisionmaking model for society that can aggregate known personal preferences in a way that will meet our intuitive demands for a just method of decision-making. The American economist Kenneth Arrow tried to answer this question. In this chapter of our book we discuss the result of his research. We start by constructing a mathematical model that presents a formal way of arranging the preference orders of the individuals in the society. This way of listing preferences involves several implicit assumptions, most of them reasonable. The assumptions relate to the preference relation and the relation of indifference. An aggregate of preference orders compiled as described formally is called a preference profile. Our task is to look for a decision rule that assigns to each preference profile a preference order that will represent the social decision. In mathematical terms, we are looking for a function that assigns one preference order to each preference profile. Such a function is called a social choice function. The task we set for ourselves is not only to choose the most-preferred alternative from among all alternatives under discussion, but also to determine the preference order of the society with regard to all the alternatives under discussion. We present a few examples of social decision rules. First we evaluate them against an intuitive standard of justice, and then we present formal conditions for defining the concept of justice. Next we formulate requirements for f, and then we look for a function f that satisfies these requirements. These requirements are called axioms and they are the formal conditions for defining the properties of a just decision rule.

6 The chapter opens with a discussion of the shortcomings of majority rule as a principal method of social decision-making. The question raised in light of these shortcomings is whether it is possible to find another fair way of making a social decision. In the course of the chapter we construct a system of axioms, that is, a system of intuitive requirements for a fair decisionmaking procedure. The question now is whether there is a social decision rule for all possible preference profiles that satisfies this system of axioms. Kenneth Arrow s surprising answer is that there is no social choice function that satisfies all the axioms! This means that every social choice function we can think of will fail to satisfy at least one of the axioms. In other words, there is an internal contradiction in the system of axioms presented in this chapter. Our aim throughout this chapter has been to find a social decision rule that will satisfy our sense of fairness in a democratic society. This aim is not achieved; indeed, it is proved that such a rule does not exist! The remaining two chapters are also based on real-life situations and therefore the study of their mathematical models expands the student s experience in using game theory to understand social phenomena. Conclusion The importance of this course in game theory is the fact that the students can really practice mathematical modeling in connection with real-life issues because this is the essence of game theory. Game theory undertakes to build mathematical models and draw conclusions from these models in connection with interactive decision-making, where a group of people not necessarily sharing the same interests is required to make a decision. References Aumann, R. J. (1985), What Is Game Theory Trying to Accomplish? In The Frontiers of Knowledge, K. Arrow and S. Honkapohja (eds.), Oxford: Basil Blackwell. Gale, D. and L. S. Shapley (1962), College Admissions and the Stability of Marriage, American Mathematical Monthly 69: Gale, D. (2001), The Two-Sided Matching Problem: Origin, Development, and Current Issues, International Game Theory Review 3: Arrow, K. J. (1951), Social Choice and Individual Values, New York: Wiley.