Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

# 9.1 Introduction

"What accounts for TIT FOR TAT's robust success is its combination of being nice, retaliatory, forgiving, and clear.  Its niceness prevents it from getting into unnecessary trouble. Its retaliation discourages the other side from persisting when defection is tried.  Its forgiveness helps restore mutual cooperation. And its clarity makes it intelligible to the other player, thereby eliciting long-term cooperation."

-Robert Axelrod

Mathematics has an on-again-off-again relationship with the real world. There are fields of mathematics that exist, more or less, solely "for themselves." Researchers in these fields are primarily motivated by the most abstract of "what-if" conjectures. The field of topology is a prime example of this, as it is known primarily for the beauty of the thinking behind its results rather than for its connections to reality. Much of mathematics, however, has its foundation in the happenings of the world around us. The field of game theory, in some sense, represents the pinnacle of this type of mathematics. It can be thought of as the mathematical study of our human interactions.

Mathematics started off as a way to apply strict, rigorous thinking to the wildly complicated world around us. Throughout its evolution, mathematical thought has oscillated between two alternative paths of thinking. On the one hand, it has often been at the vanguard of quantitative science, helping to shed new light on previously incomprehensible phenomena through the power of logical thinking. On the other hand, it has sometimes been taken to levels of abstraction far beyond what any pragmatic scientist would be interested in. Even in its more applied incarnations, mathematics involves a great many simplifying assumptions. To get anywhere, we must reduce incredibly complicated situations from our everyday reality into problems that can be strictly defined and rigorously analyzed. The subject of game theory provides a great example of this mathematical reduction in action.

Games, to a mathematician, are simplified versions of situations that arise in the course of interactions among people, or any kinds of agents. A game represents an idealized situation that can be analyzed mathematically to shed light on how or why certain outcomes are reached. For example, the Ultimatum Game is a two-player situation in which one player is given an amount of money to share with another in any proportion desired. The receiving player then can choose either to accept or to decline the giver's offer. If the offer is accepted, both players get their share of the money, as determined by the giver. If the receiver declines, no one gets any money. This represents a vastly simplified model of a business negotiation. As we will see later in this unit, this simple game has far-reaching and sometimes surprising implications for how humans judge the importance of justice and equality.

Most games have many simplifying assumptions, as is evident throughout this discussion, but the one that provides the foundation for all others is the assumption that players act rationally. The classic assumption is that a rational player will always act in a way that seems to maximize personal benefit.

This assumption of rationality is what allows the mathematical analysis of games to work. If the people playing a game do not behave rationally, then the results of any game-theory-supported analysis may be less relevant. Nonetheless, the conclusions that are obtained, even under these idealized assumptions, can be useful. Interestingly, an important use of game theory has been to probe the limits of rationality. As we shall see in our study of the Ultimatum Game, there are multiple concepts of what is rational, in addition to the one based on maximizing profit.

One of the most interesting conclusions reached in game theory is that rational actions by both players can result in situations in which both players are worse off. The primary example of this is the Prisoner's Dilemma, which we will see in more detail later on in this unit.

In this unit we will examine the mathematical analysis of games, beginning with a bit of the history and some of the motivations behind the field's development. From there we will examine a few simple games in order to illustrate the basic terminology and concepts. We will then move on to more substantial games, such as the Prisoner's Dilemma and the Hawks and Doves game. We will examine the various types of solutions and equilibria that exist for these games, and we will see how these elements change, depending on whether a game is played once or more than once. In iterated and multi-player games, we will see how the payoff a player can expect from using a particular strategy depends on the strategies that others employ and how frequently they do so. Along the way, we will see how game theory can help us understand the processes of evolution, including the evolution of language. Finally, we will see how analyzing abstract games applies to real-life situations, such as business transactions, language development, and avoiding nuclear war.