Kockesen L., Game Theory Lecture Notes.pdf

Chapter 1

Introduction

1.1 W HAT IS G AME T HEORY ?

We, humans, cannot survive without interacting with other humans, and ironically, it some-

times seems that we have survived despite those interactions. Production and exchange require

cooperation between individuals at some level but the same interactions may also lead to disastrous

confrontations. Human history is as much a history of ﬁghts and wars as it is a history of success-

ful cooperation. Many human interactions carry the potentials of cooperation and harmony as well

as conﬂict and disaster. Examples are abound: relationships among couples, siblings, countries,

management and labor unions, neighbors, students and professors, and so on.

One can argue that the increasingly complex technologies, institutions, and cultural norms that

have existed in human societies have been there in order to facilitate and regulate these interactions.

For example, internet technology greatly facilitates buyer-seller transactions, but also complicates

them further by increasing opportunities for cheating and fraud. Workers and managers have usu-

ally opposing interests when it comes to wages and working conditions, and labor unions as well as

labor law provide channels and rules through which any potential conﬂict between them can be ad-

dressed. Similarly, several cultural and religious norms, such as altruism or reciprocity, bring some

order to potentially dangerous interactions between individuals. All these norms and institutions

constantly evolve as the nature of the underlying interactions keep changing. In this sense, under-

standing human behavior in its social and institutional context requires a proper understanding of

human interaction.

Economics, sociology, psychology, and political science are all devoted to studying human

behavior in different realms of social life. However, in many instances they treat individuals in

isolation, for convenience if not for anything else. In other words, they assume that to understand

Introduction

Game theory studies strategic

interactions

one individual’s behavior it is safe to assume that her behavior does not have a signiﬁcant effect on

other individuals. In some cases, and depending upon the question one is asking, this assumption

may be warranted. For example, what a small farmer in a local market, say in Montana, charges for

wheat is not likely to have an effect on the world wheat prices. Similarly, the probability that my

vote will change the outcome of the U.S. presidential elections is negligibly small. So, if we are

interested in the world wheat price or the result of the presidential elections, we may safely assume

that one individual acts as if her behavior will not affect the outcome.

In many cases, however, this assumption may lead to wrong conclusions. For example, how

much our farmer in Montana charges, compared to the other farmers in Montana, certainly affects

how much she and other farmers make. If our farmer sets a price that is lower than the prices

set by the other farmers in the local market, she would sell more than the others, and vice versa.

Therefore, if we assume that they determine their prices without taking this effect into account,

we are not likely to get anywhere near understanding their behavior. Similarly, the vote of one

individual may radically change the outcome of voting in small committees and assuming that they

vote in ignorance of that fact is likely to be misleading.

The subject matter of game theory is exactly those interactions within a group of individuals (or

governments, ﬁrms, etc.) where the actions of each individual have an effect on the outcome that

is of interest to all. Yet, this is not enough for a situation to be a proper subject of game theory: the

way that individuals act has to be strategic, i.e., they should be aware of the fact that their actions

affect others. The fact that my actions have an effect on the outcome does not necessitate strategic

behavior, if I am not aware of that fact. Therefore, we say that game theory studies strategic

interaction within a group of individuals. By strategic interaction we mean that individuals know

that their actions will have an effect on the outcome and act accordingly.

Having determined the types of situations that game theory deals with, we have to now discuss

how it analyzes these situations. Like any other theory, the objective of game theory is to organize

our knowledge and increase our understanding of the outside world. A scientiﬁc theory tries to

abstract the most essential aspects of a given situation, analyze them using certain assumptions and

procedures, and at the end derive some general principles and predictions that can be applied to

individual instances.

For it to have any predictive power, game theory has to postulate some rules according to which

individuals act. If we do not describe how individuals behave, what their objectives are and how

rules of the game

they try to achieve those objectives we cannot derive any predictions at all in a given situation. For

example, one would get completely different predictions regarding the price of wheat in a local

market if one assumes that farmers simply ﬂip a coin and choose between $1 and $2 a pound

compared to if one assumes they try to make as much money as possible. Therefore, to bring some

1.1. What is Game Theory?

discipline to the analysis one has to introduce some structure in terms of the rules of the game.

The most important, and maybe one of the most controversial, assumption of game theory

which brings about this discipline is that individuals are rational .

We assume that individuals are

rational.

Deﬁnition. An individual is rational if she has well-deﬁned objectives (or preferences)

over the set of possible outcomes and she implements the best available strategy to pursue

them.

Rationality implies that individuals know the strategies available to each individual, have com-

plete and consistent preferences over possible outcomes, and they are aware of those preferences.

Furthermore, they can determine the best strategy for themselves and ﬂawlessly implement it.

If taken literally, the assumption of rationality is certainly an unrealistic one, and if

applied to particular cases it may produce results that are at odds with reality. We should

ﬁrst note that game theorists are aware of the limitations imposed by this assumption

and there is an active research area studying the implications of less demanding forms

of rationality, called bounded rationality . This course, however, is not the appropriate

place to study this area of research. Furthermore, to really appreciate the problems with

rationality assumption one has to ﬁrst see its results. Therefore, without delving into

too much discussion, we will argue that one should treat rationality as a limiting case.

You will have enough opportunity in this book to decide for yourself whether it produces

useful and interesting results. As the saying goes: “the proof of the pudding is in the

eating.”

The term strategic interaction is actually more loaded than it is alluded to above. It is not

enough that I know that my actions, as well as yours, affect the outcome, but I must also know that

you know this fact. Take the example of two wheat farmers. Suppose both farmer A and B know

that their respective choices of prices will affect their proﬁts for the day. But suppose, A does not

know that B knows this. Now, from the perspective of farmer A, farmer B is completely ignorant

of what is going on in the market and hence farmer B might set any price. This makes farmer

A’s decision quite uninteresting itself. To model the situation more realistically, we then have to

assume that they both know that they know that their prices will affect their proﬁts. One actually

has to continue in this fashion and assume that the rules of the game, including how actions affect

the participants and individuals’ rationality, are common knowledge.

A fact X is common knowledge if everybody knows it, if everybody knows that everybody

knows it, if everybody knows that everybody knows that everybody knows it, an so on. This has

Introduction

We assume that the game and

rationality are common

knowledge

some philosophical implications and is subject to a lot of controversy, but for the most part we will

avoid those discussions and take it as given.

In sum, we may deﬁne game theory as follows:

Deﬁnition. Game theory is a systematic study of strategic interaction among rational

individuals.

Its limitations aside, game theory has been fruitfully applied to many situations in the realm of

economics, political science, biology, law, etc. In the rest of this chapter we will illustrate the main

ideas and concepts of game theory and some of its applications using simple examples. In later

chapters we will analyze more realistic and complicated scenarios and discuss how game theory is

applied in the real world. Among those applications are ﬁrm competition in oligopolistic markets,

competition between political parties, auctions, bargaining, and repeated interaction between ﬁrms.

1.2 E XAMPLES

For the sake of comparison, we ﬁrst start with an example in which there is no strategic inter-

action, and hence one does not need game theory to analyze.

Example 1.1 (A Single Person Decision Problem) . Suppose Ali is an investor who can invest his

$100 either in a safe asset, say government bonds, which brings 10% return in one year, or he can

invest it in a risky asset, say a stock issued by a corporation, which either brings 20% return (if the

company performance is good) or zero return (if the company performance is bad).

State

Good Bad

Bonds 10% 10%

Stocks 20%

Clearly, which investment is best for Ali depends on his preferences and the relative likelihoods

of the two states of the world. Let’s denote the probability of the good state occurring p and that of

the bad state 1 − p , and assume that Ali wants to maximize the amount of money he has at the end

of the year. If he invests his $100 on bonds, he will have $110 at the end of the year irrespective

of the state of the world (i.e., with certainty). If he invests on stocks, however, with probability

1.2. Examples

p he will have $120 and with probability 1 − p he will have $100. We can therefore calculate his

average (or expected) money holdings at the end of the year as

p × 120 + ( 1 − p ) × 100 = 100 + 20 × p

If, for example, p = 1 / 2, then he expects to have $110 at the end of the year. In general, if p > 1 / 2,

then he would prefer to invest in stocks, and if p < 1 / 2 he would prefer bonds.

This is just one example of a single person decision making problem , in which the decision

problem of an individual can be analyzed in isolation of the other individuals’ behavior. Any A single person decision

problem has no strategic

interaction

Example 1.2 (An Investment Game) . Now, suppose Ali again has two options for investing his

$100. He may either invest it in bonds, which have a certain return of 10%, or he may invest it in

a risky venture. This venture requires $200 to be a success, in which case the return is 20%, i.e.,

$100 investment yields $120 at the end of the year. If total investment is less than $200, then the

venture is a failure and yields zero return, i.e., $100 investment yields $100. Ali knows that there

is another person, let’s call her Beril, who is exactly in the same situation, and there is no other

potential investor in the venture. Unfortunately, Ali and Beril don’t know each other and cannot

communicate. Therefore, they both have to make the investment decision without knowing the

decisions of each other.

We can summarize the returns on the investments of Ali and Beril as a function of their deci-

sions in the table given in Figure 1.1. The ﬁrst number in each cell represents the return on Ali’s

investment, whereas the second number represents Beril’s return. We assume that both Ali and

Beril know the situation represented in this table, i.e., they know the rules of the game.

Figure 1.1: Investment Game.

Ali

Beril

Bonds Venture

Bonds 110 , 110 110 , 100

Venture 100 , 110

The existence of strategic interaction is apparent in this situation, which should be contrasted

with the one in Example 1.1. The crucial element is that the outcome of Ali’s decision (i.e., the

return on the investment chosen) depends on what Beril does. Investing in the risky option, i.e., the

uncertainty involved in such problems are exogenous in the sense that it is not determined or in-

ﬂuenced in any way by the behavior of the individual in question. In the above example, the only

uncertainty comes from the performance of the stock, which we may safely assume to be inde-

pendent of Ali’s choice of investment. Contrast this with the situation illustrated in the following

example.

120 , 120

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: