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Economics N110,

Game Theory in the Social Sciences

UC Berkeley, Summer 2012

Lecture 1

Instructor: Matt Leister

June 18, 2012

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Table of contents

What is Game Theory?What is Game Theory?What to expect

Mathematical ConceptsSets and NumbersVectors and Vector SpacesFunctions and CorrespondencesOptimization

Sequences and SeriesProbabilityImplication

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What is Game Theory?

the mathematical modeling of agent interaction:

the study of strategy

Examples: competition, cooperation, coordination, bargaining,even communication, and many more aspects of our every daylives can be modeled with game theory.

a basic language to much of microeconomics and oftenapplied in all subfields of economics

the brain child of the late great John Von Neumann and

Oskar Morgenstern Berkeley has a rich history in game theory: John Harsanyi

(1920-2000, Cal professor, and Nobel Laureate) developed thetheory of games with imperfect information.

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What to expect from this course

1. a sound understanding of essential concepts

2. a roadmap of the theorys models and concepts

3. the tools to continue your studies in game theory andmicroeconomics

4. a survey of how game theory has been applied in different

social science fields (e.g. sociology, political science, publicpolicy, psychology, and economics)

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Mathematical Concepts: Sets

A set is a collection of things or elements(e.g. types offruit, people, choices, variables or numbers).

Sets can be finite (e.g. days of the week) or infinite (e.g.time(s) it takes to drive to San Francisco).

finite sets often denoted as {,, . . . ,}

infinite but orderable sets usually denoted as {,, . . .}

this course: 1) sets of choices (actions, strategies)2) sets of states of the world (e,g, {rainy, sunny}) Write x X to denote that the variable x is an element in the

set X.

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Notation

x X... (short: x...), means for all x in X.... x X... (short: x...), means there exists an x in X.... Take X and Y subsets in Z. This is written X Z and

Y Z.

X Y means the set of all elements either in X or in Y X Y means the set of all elements in both X and Y Xc means the set of all elements not in x (but in Z)

X\Y means the set of all elements in X but not in Y(equal to X Yc)

Fact(s): 1) (X Y)c = Xc Yc, 2) (X Y)c = Xc Yc(De Morgans Law)

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Sets of Sets and Sets of Numbers

A set can be composed of sets (each element is a set itself),and we usually denote these types of sets in script (e.g. X)

The set of natural numbers (e.g. 1, 2, . . .) are denoted N

The set of integers (e.g. 5, 0, 3) are denoted Z

The set of real numbers (e.g. 5.23, , 2) are denoted R

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Vectors and Vector Spaces

Taking a set Xcomposed of n R sets X1,...,Xn, a vectorcan be defined as a bundle of n elements, one taken fromeach of the sets in X.

This course: well refer to vectors as profiles.

The set of all such vectors is denoted X1 X2 ... Xn(orni=1 Xi).

If Xi = R, then we denoten

i=1 Xi as Rn.

R

n

is a vector space because we can define component wiseaddition (and also inner products, which are not needed inthis course).Example: x1 = (3.2, 6), x2 = (1.3, 2) (both in R2), withx1 + x2 = (1.9, 8)

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Functions and Correspondences

Functions are mappings from one set to unique elements inanother set (possibly equal to the first set).

Precisely, a function f() maps from a set X (the domain) toa set Y (the range); this is denoted f : X

Y. For each

x X, f gives one, and only one, y Y (denoted f(x)).

This course: the functions well be dealing with are objectivefunctions (next class)

When a mapping gives more than one value for someelements, its referred to as a correspondence.

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Example

A function (left) and a correspondence (right), each fromR

to R, can be graphed:

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Continuity

A function from Rn to R is continuous if for each x X,f(x

) is exceedingly close to f(x) as x

approaches x.

Examples:

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Optimization

Important question: when a function f : X R, then whichx X (possibly multiple x) give the largest f(x)?

The set of all such x is written:

argmaxxXf(x)

The value of f at any such x is written:

maxxXf(x)

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Optimization

Fact: when X is finite, there will always be at least one x Xthat is in argmaxxX f(x)

However, when X is infinite, it may be that argmaxxX f(x)is empty ( denotes the empty set). Example: X =]0, 1[(]a, b[ is set of integers between a and b, not including aand b), with f(x) = x.

Fact: when X is a closed and bounded subset ofRn (i.e. X

includes its boundry and all elements of X have finitecomponents), and f() is continuous, then argmaxxX f(x) isnon-empty (due to Karl Weierstrass).

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Example 1

This function from R2 to R has a maximum:

Figure : multi-modal density function

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Example 2

Parabolas of the form y = f(x) = ax2 + bx + c (with a > 0)

are optimized at x =b

2a :

a > 0, x = b2a gives a minimum;

a < 0, x = b2a gives a maximum.

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Example 2

Fact: for any f(),

argmaxx

f(x) = argminxf(x)

Fact: for any f(),

maxx

f(x) = minxf(x)

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Sequences

A sequence is an ordered infinite set, usually denoted

{x1, x2,...}, or {xi} in short. A geometric sequence is of the form {1, r, r2, r3,...}, r R

(with r0 = 1).

a series is a sequence generated by the cumulative sums of a

sequence. That is, the series of{x1, x2, ...} is the sequence{x1, x1 + x2, x1 + x2 + x3,...}. Fact: the Tth element of the series of a geometric sequence{1, r, r2, r3,...} is:

1 + r + r2 + ... + rT = 1 rT+11 r

Fact: if r (0, 1), then 1 + r + r2 + ... = 11r

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Probability

We can define the set of relevant states of the world anddenote this set , with some probabilility distributioncapturing the llikelihood of each event.

If is finite, this means that we can find Pr() [0, 1], ,with

Pr() = 1. We have Pr() = 0 and Pr() = 1.This course: finite random variables over action sets.

A random variable (in this class) will be a function x from to R. With Pr(

), we write Pr(x = x)

|x(

)=xPr().

A cumulative distribution F of x maps from R to [0, 1], andis defined as F(x)

|x()x Pr() =

xx Pr(x = x).

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Conditional Probability

An event is some subset of .

The probability of an event E is the sum of probabilities ofthe states within E: Pr(E)

E Pr().

The conditional probability of the event E on the event F isthe sum of probabilities of states in both E and F divided bythe probability of F: Pr(E|F) Pr(E and F)Pr(F) .

This course: conditional probabilities will be used to modelposterior beliefs (games with imperfect information)

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Conditional Probability

Example: gene testing

Figure : relative densities

B R l

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Bayes Rule

Using the above definitions, we can write:Pr(E and F) = Pr(E|F)Pr(F) = Pr(F|E)Pr(E)

Rearranging the second equality gives:

Pr(E|F) = Pr(E)Pr(F|E)Pr(F)

=Pr(E)Pr(F|E)

Pr(F) Pr()

=

Pr(E)Pr(F

|E)

Pr(E)Pr(F|E) + Pr(Ec)Pr(F|Ec)

B R l

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Bayes Rule

A partition of is a collection of disjoint subsets of withunion equal to .

Taking finite partition {Ej}nj=1 of , the above expression can

be more generally written:

Pr(Ek|F) = Pr(Ek)Pr(F|Ek)nj=1 Pr(F|Ej)Pr(Ej)

I licatio

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Implication

When a fact, event, or claim A implies another fact, event, orclaim B, we write A B (or B A).

When A B and B A (read A if and only iff B), wewrite A B or A iff B.

Alternative terminology: A B is the same as A issufficient for B and B is necessary for A

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