Mathematics and the Real World (40 page)

The public mood in Israel in 1976 brought about the establishment of a new political movement, the Democratic Movement for Change. In the course of its establishment, those active in setting it up had to decide on the method for selecting the list of candidates that the movement would put up in the forthcoming election for the Knesset (Israel's parliament). They turned to mathematicians and physicists for help in choosing the best system, and these advisors came up with a sophisticated system. We will not describe the details of the proposed system, but we will observe that even a quick look showed that it would be easy for an organized group within the party to obtain far greater representation on the list of candidates than their relative size warranted. When this was brought to the attention of the system's architects, they dismissed it offhandedly, repeating the claim mentioned above that there is no need to relate to Arrow's result, because if it were adopted, it would result in the list of candidates being chosen by a dictator, in total opposition to the founders’ stated aim of creating a democratic movement. They added that they had carried out simulations of possible outcomes, and those did not indicate any minority group taking control. When the actual internal elections took place, two groups took advantage of the system and obtained representation far in excess of their relative size within the movement. This eventually led to the disintegration of the movement, although in the general election to parliament it was quite successful in the number of its candidates that were elected to the Knesset. One of the movement's founders summarized the events in a book and wrote more or less that the organized groups did not vote as expected. In effect, he and his colleagues who performed the simulations did not show much understanding of ways of voting.

As mentioned, Arrow's mathematical theorem can be interpreted also in the framework in which an individual has to decide on his priorities in complex situations. One way of dealing with a list of possibilities that we
must grade, say a list of places to visit that we are checking before the next holiday, is to draw up a list of criteria against which we rate the options. The criteria will probably include such features as the cost, the amount of pleasure it will give us, the physical effort required to get there, and so on. Each criterion will have the list of possibilities as we have graded them, and we must decide on a combined rating based on those for the separate criteria. Arrow's requirements will then look as follows:

1. Independence of irrelevant alternatives
   : If one of the choices is ruled out without affecting the ratings of any of the criteria of the remaining options, the grading that the system gives to the other options also remains unchanged.
   
2. Unanimity
   : If one of the options, say A, is rated higher than option B in all the criteria, then the system will rate A above B.
   
3. No dominance
   : No single criterion is dominant, that is, such that the final rating will always be the rating in that criterion, regardless of the ratings in the other criteria.
   

The interpretation of the three requirements is similar to that given in the case of choices. In the current instance the third requirement has no social aspect or implication. Indeed, if one of the criteria, say the price, is so dominant that it determines the overall rating, the rating problem becomes simple. Arrow's theorem says that it is impossible to rate the options in such a way that the three criteria are satisfied at the same time. How then can alternatives be rated? The discussion above, regarding choices, applies here too. For example, we can adopt Borda's approach of assigning points to each of the criteria and combining the points of the different rankings to obtain a group ranking. That is the method commonly used in most combined ratings we usually encounter, even though in that method not all of Arrow's conditions are met.

47. THE MATHEMATICS OF CONFRONTATION

In this section we will discuss the mathematical analysis of methods of decision making by humans as individuals. What is special in a social framework is that the decisions of any particular person may clash with the activities of others, and the person making the decisions must take into consideration what his colleagues or opponents will do. At this point we can distinguish between two different scenarios. In one, the impact of the actions of the individual decision maker on say the market situation will be negligible. In the other, the actions of each one of the individual decision makers could influence the final outcome.

In the first case the decision maker assesses the circumstances he faces, and in light of those circumstances he assesses the results of the different decisions he can make and chooses what seems to him the best. The function of mathematics in this process is to construct a mathematical model that can be analyzed quantitatively, and to propose methods of reaching the optimal decision. The mathematical subject is called optimization. We will not expand on it here, because the mathematical methods it uses are not essentially different from those we have encountered previously. We will nonetheless mention the outstanding example of such a mathematical development that had a significant impact on decisions in the capital market. We are referring to the Black-Scholes model and the Black-Scholes formula, which analyzes the risks and chances incurred in investing in the capital market.

The economists Fisher Black and Myron Scholes published a paper in 1973 in which they put forward a mathematical framework that enabled them to analyze the risks involved in investing in options in the capital market. Following their paper, Robert Merton published a paper in which he extended the system and developed the mathematics to the level that it became a tool in the hands of investors in the stock market. The mathematical tool employed in the model is differential equations, which we came across in our discussions of models of natural phenomena, with the difference that the variables are not location, energy, and speed, but prices and rates of interest, and so on. The model was put to routine use by investors
in the stock market, and in 1997 Merton and Scholes were awarded the Nobel Prize in Economics. Black died in 1995.

In the second of the two scenarios outlined above, the decision maker takes into consideration the reactions of other individual decision makers to his own decisions. They too act in light of their assessments of the actions of others. In this situation the use of the word
optimal
may sometimes mislead. For example, take the situation in which several people make their individual decisions, and the outcome is determined by their decisions combined. What is optimization in this case? Optimal for one participant may be terrible for another. If one of the group members knows or assesses the decisions of the others, he will be faced with the optimization problem described above. But then each of the others can estimate the choice of that same individual and can change his own choice accordingly. In which case, the first can alter his decision in light of the new situation, and so on. It is then not clear what the optimal decision is. The mathematical field that analyzes such situations is
game theory
. The rather simplistic name is likely to be misleading because this branch of mathematics is dealing with a very serious subject, the analysis of confrontation. On the other hand, the name
game theory
has become so accepted by the public that it is commonly assumed that the term
game
in this context means confrontation.

The situation that we have described is relevant also to conflicts between decision makers in daily life as well as to social games such as chess. A mathematical analysis of the theoretical possibilities in a game of chess was carried out in 1913 by the German mathematician Ernst Zermelo (1871–1953), famous for his contribution to the foundations of mathematics, whose paper titled “On an Application of Set Theory to the Theory of the Game of Chess” introduced new concepts to that area of research and provided the source of the name
game theory
. Other well-known mathematicians continued to develop that field, including the French mathematician Émile Borel (1871–1956), who in 1921 introduced the mixed-strategy concept, and John von Neumann (1903–1957) who in 1928 proved the minimax theory. We will meet these two concepts again later in this section. Zermelo's paper dealing with social games led to
the development of the subject, which analyzes conflicts between individual decision makers, be they private individuals, company directors, or military and political leaders. The insights reached by game theory have been used since then to analyze situations of people and companies and to understand conflicts of interest between animals. With animals we cannot identify conscious decision making, but the process itself takes place as if someone were making deliberate decisions. In particular, viewing the evolutionary struggle itself as a conflict between species helps in the analysis of the evolutionary process.

Just as in other areas of specialization that mathematics uses to describe and explain phenomena, before using them we must specify precisely what framework of mathematics we are working in. One of the models in game theory is called
games in strategic form
(we shall allude to another model, namely cooperative games, later in the section). This is a game between several players, each one of whom must choose one of a given number of possibilities, called strategies. The choice is made simultaneously, and when each participant makes his decision, he does not know what the other players are choosing. The game ends when the decisions of all players have been received. Next, each player receives a “payoff” that depends on the combination of all the decisions taken. All the players know the payoff and its dependence on the strategies in advance. The payoff can be monetary, but it can also be in another form, assuming that the player has a full rating of preferences for the rewards he is entitled to receive. Clearly the strategic-form model does not cover all conflict situations between players. We will limit the mathematical analysis by imposing the condition that the number of strategies facing each player is finite (for discussion purposes only; the professional literature also analyzes situations with infinite possibilities). Our purpose at this point is to arrive at an understanding of what mathematics can offer at the conceptual level, and that can be achieved with a simple model. If we wish to use the results to analyze day-to-day situations we will need to check the extent to which the circumstances match the mathematical model.

Even at this level of presenting the definition of a game, we can propose
“obvious” properties of the players’ possible decisions. For instance, assume that a player recognizes the following property of one of the strategies facing him, say strategy A: for every possible move by the other players, strategy A yields him the highest payoff. It is then proper for us to refer to the decision to adopt strategy A as the optimal decision. In game theory, such a strategy is called a
dominant strategy
. It stands to reason that a player who identifies a dominant strategy among the options confronting him will adopt that strategy. If each of the players has a dominant strategy, we have “solved” the game. However, players do not always have a dominant strategy. In those cases the best response to one's opponents’ moves is achieved via different strategies.

Another possibility that a player can choose is to look for a maximum-minimum strategy. In other words, the player can calculate the lowest payoff that every strategy might yield him and choose the best of those low payoffs. This concept describes behavior that settles for minimizing possible losses or that reflects concern over the worst-case scenario. In many situations the use of these strategies does not yield reasonable outcomes.

A crucial step in the analysis of possibilities in a game in strategic form was made by John Nash of Princeton University, New Jersey. Nash is known by the general public mainly because of the book and subsequent film,
A Beautiful Mind
. The book, written by Sylvia Nasar, recounts Nash's life history, from his graduate studies at Princeton University in 1948, his illness, his disappearance from the world of research, and his being awarded the Nobel Prize in Economics in 1994. Nash, who also made significant contributions in many other areas of mathematics, proposed the following definition.

Assume that every player chooses one of the strategies open to him. This collection of strategies is in equilibrium if no player can by himself earn a higher payoff by changing his strategy while the other players keep to their original ones
.

The rationale underlying his definition was that if the players agreed on a choice of strategies that are in equilibrium, or if they discovered somehow
that the other players would choose strategies in equilibrium, there is no incentive for any of them to switch from that strategy unilaterally.

Nash did not put forward his definition in a vacuum. Exactly the same concept had been proposed more than a hundred years earlier by the French mathematician and economist Antoine Augustin Cournot (1801–1877), who made valuable contributions to economic theory in several areas. Cournot defined the concept of equilibrium between two firms that constitute a duopoly, that is, the two firms control the market. As Cournot formulated the concept in a relatively complex model, his definition lacks the simplicity and clarity of Nash's definition, but nevertheless many show their respect for him by referring to the concept as Cournot-Nash equilibrium. Others, after Cournot, also used the same concept in various forms, but it was Nash's precise formulation that led to the recognition and widespread use of the concept. Nash went further and proved the existence of equilibrium in a framework we will describe a little later, but first we will give three examples of the concept of equilibrium that are standard in any textbook on game theory.

The first example is known as the
prisoner's dilemma
. Despite its eye-catching name, the game reflects situations we encounter frequently in trade, economics, our social lives, and so on. The dilemma is between cooperation that would to some extent benefit both players, on the one hand, and noncooperation that would be advantageous to one player at the expense of the other. That is indeed a common situation, but in our case each participant must make the decision without any communication with the other one. The mathematical version of the tale is of two suspects who committed a crime, but the police have insufficient evidence for a conviction without one of the suspects incriminating the other. The police therefore make an offer that if one of the suspects will testify against the other and the other continues to claim he is not guilty, the cooperative suspect will go free and the other one will be sentenced to four years in prison, the punishment prescribed by law. If neither agrees to testify against the other, it will be impossible to prove the more serious charge against them, but they can both be convicted of a lesser offense that carries a one-year prison
sentence. If they both accept the offer and each agrees to testify against the other, they will both be convicted but their testimony will earn them a reduced sentence of three years. Each suspect is faced with two possibilities, or in the terminology of game theory, two strategies. One option is to testify, and the other is to refuse the offer, that is, to deny the charges. Each must make the decision without knowing what the other will do. It is customary to summarize the possibilities in the following table.