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Preface...................................................................xv 1 Probability Theory....................................................1 2 Bayesian Decision Theory..............................................18 3 Game Theory: Basic Concepts...........................................32 4 Eliminating Dominated Strategies......................................52 5 Pure-Strategy Nash Equilibria.........................................80 6 Mixed-Strategy Nash Equilibria........................................116 7 Principal-Agent Models................................................162 8 Signaling Games.......................................................179 9 Repeated Games........................................................201 10 Evolutionarily Stable Strategies......................................229 11 Dynamical Systems.....................................................247 12 Evolutionary Dynamics.................................................270 13 Markov Economies and Stochastic Dynamical Systems.....................297 14 Table of Symbols......................................................319 15 Answers...............................................................321 Sources for Problems......................................................373 References................................................................375 Index.....................................................................385

Doubt is disagreeable, but certainty is ridiculous.

Voltaire

**1.1 Basic Set Theory and Mathematical Notation**

A *set* is a collection of objects. We can represent a set by enumerating its
objects. Thus,

*A* = {1, 3, 5, 7, 9, 34}

is the set of single digit odd numbers plus the number 34. We can also represent the same set by a formula. For instance,

*A* = {*x|x* [member of] **N** [conjunction] (*x* < 10 [conjunction] *x* is odd)
[disjunction] (*x* = 34)}.

In interpreting this formula, **N** is the set of natural numbers (positive integers),
"|" means "such that," "[member of]" means "is a element of," [conjunction] is the logical
symbol for "and," and [disjunction] is the logical symbol for "or." See the table of
symbols in chapter 14 if you forget the meaning of a mathematical symbol.

The subset of objects in set *X* that satisfy property *p* can be written as

{*x* [member of] *X|p(x)*}.

The *union* of two sets *A, B* [subset] *X* is the subset of *X* consisting of elements
of *X* that are in *either A* or *B*:

*A* [union] *B* = {*x|x* [member of] *A* [disjunction] *x* [member of] *B*}.

The *intersection* of two sets *A, B* [subset] *X* is the subset of *X* consisting of
elements of *X* that are in *both A* or **B**:

*A* [intersection] *B* = {*x|x* [member of] *A* [conjunction] *x* [member of]
*B*}.

If *a* [member of] *A* and *b* [member of] *B*, the *ordered pair (a,b)* is an entity
such that if *(a, b) = (c, d)*, then *a = c* and *b = d*. The set {*(a, b)|a* [member of]
*A* [conjunction] *b* [member of] *B*} is called the *product* of *A* and *B* and
is written *A* x *B*. For instance, if *A = B* = **R**, where **R** is the set of real
numbers, then *A* x *B* is the real plane, or the real two-dimensional vector space. We also write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A *function f* can be thought of as a set of ordered pairs (*x, f(x)*). For
instance, the function *f(x)* = [chi square] is the set

{*(x, y)|(x, y* [member of] **R**) [conjunction] (*y* = [chi square])}

The set of arguments for which *f* is defined is called the *domain* of *f* and is
written dom (*f*). The set of values that *f* takes is called the *range* of *f* and
is written range (*f*). The function *f* is thus a subset of dom(*f*) x range(*f*).
If *f* is a function defined on set *A* with values in set *B*, we write *f* :
*A* -> *B*.

**1.2 Probability Spaces**

We assume a finite *universe* or *sample space* [OMEGA] and a set *X* of subsets
*A, B, C,* ... of [OMEGA], called *events*. We assume *X* is closed under finite
unions (if [*A*.sub.1], [*A*.sub.2], ... *[A.sub.n]* are events, so is [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]), finite intersections (if [*A*.sub.1], ..., *[A.sub.n]* are events, so is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), and complementation (if *A* is an event
so is the set of elements of [OMEGA] that are not in *A*, which we write *[A.sup.c]*). If *A*
and *B* are events, we interpret *A* [intersection] *B = AB* as the event "*A* and *B*
both occur," *A* [intersection] *B* as the event "*A* or *B* occurs," and *[A.sup.c]*
as the event "*A* does not occur."

For instance, suppose we flip a coin twice, the outcome being *HH*
(heads on both), *HT* (heads on first and tails on second), *TH* (tails on
first and heads on second), and *TT* (tails on both). The sample space is
then [OMEGA] = {*HH, TH, HT, TT*}. Some events are {*HH, HT*} (the coin
comes up heads on the first toss), {*TT*} (the coin comes up tails twice), and
{*HH, HT, TH*} (the coin comes up heads at least once).

The *probability* of an event *A* [member of] *X* is a real number P[*A*] such that 0 [less
than or equal to] P[*A*] [less than or equal to] 1. We assume that P[[OMEGA]] = 1, which says that with
probability 1 *some* outcome occurs, and we also assume that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII], where *[A.sub.i]* [member of] *X* and the {*[A.sub.i]*} are disjoint (that is,
*[A.sub.i]* [intersection] *[A.sub.j]* = 0 for all *i* [not equal to] *j*), then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which says that probabilities are additive over finite
disjoint unions.

**1.3 De Morgan's Laws**

Show that for any two events *A* and *B*, we have

[(*A* [intersection] *B*).sup.c] = *[A.sup.c]* [intersection] *[B.sup.c]*

and

[(*A* [intersection] *B*).sup.c] = *[A.sup.c]* [intersection] *[B.sup.c]*.

These are called *De Morgan's laws*. Express the meaning of these formulas
in words.

Show that if we write *p* for proposition "event *A* occurs" and *q* for "event
*B* occurs," then

not (*p* or *q*) [??] (not *p* and not *q*);

not (*p* and *q*) [??] (not *p* or not *q*).

The formulas are also De Morgan's laws. Give examples of both rules.

**1.4 Interocitors**

An interocitor consists of two kramels and three trums. Let *[A.sub.k]* be the event
"the *k*th kramel is in working condition," and *[B.sub.j]* is the event "the *j*th trum
is in working condition." An interocitor is in working condition if at least
one of its kramels and two of its trums are in working condition. Let ITLITL be
the event "the interocitor is in working condition." Write ITLITL in terms of the
*[A.sub.k]* and the *[B.sub.j]*.

**1.5 The Direct Evaluation of Probabilities**

Theorem 1.1 *Given [a.sub.1], ..., [a.sub.n] and [b.sub.1], ..., [b.sub.m], all distinct, there are n x m
distinct ways of choosing one of the [a.sub.i] and one of the [b.sub.j]. If we also
have [c.sub.1], ..., [c.sub.r], distinct from each other, the [a.sub.i] and the [b.sub.j], then there are
n x m x r distinct ways of choosing one of the [a.sub.i], one of the [b.sub.j], and one of
the [c.sub.k].*

Apply this theorem to determine how many different elements there are in the sample space of

a. the double coin flip

b. the triple coin flip

c. rolling a pair of dice

Generalize the theorem.

**1.6 Probability as Frequency**

Suppose the sample space [OMEGA] consists of a finite number *n* of equally probable
elements. Suppose the event *A* contains *m* of these elements. Then the
*probability of the event A* is *m/n*.

A second definition: Suppose an experiment has *n* distinct outcomes, all
of which are equally likely. Let *A* be a subset of the outcomes, and *n(A)* the
number of elements of *A*. We define the *probability* of *A* as P[*A*] = *n(A)/n*.

For example, in throwing a pair of dice, there are 6 x 6 = 36 mutually
exclusive, equally likely events, each represented by an ordered pair (*a, b*),
where *a* is the number of spots showing on the first die and *b* the number
on the second. Let *A* be the event that both dice show the same number of
spots. Then *n(A)* = 6 and P[*A*] = 6/36 = 1/6.

A third definition: Suppose an experiment can be repeated any number
of times, each outcome being independent of the ones before and after it.
Let *A* be an event that either does or does not occur for each outcome. Let
*[n.sub.t](A)* be the number of times *A* occurred on all the tries up to and including
the *[t.sup.th]* try. We define the *relative frequency* of *A* as *[n.sub.t](A)/t*, and we
define the *probability of A* as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We say two events *A* and *B* are *independent* if P[*A*] does not depend on
whether *B* occurs or not and, conversely, P[*B*] does not depend on whether
*A* occurs or not. If events *A* and *B* are independent, the probability that
both occur is the product of the probabilities that either occurs: that is,

P[*A* and *B*] = P[*A*] x P[*B*].

For example, in flipping coins, let *A* be the event "the first ten flips are
heads." Let *B* be the event "the eleventh flip is heads." Then the two events
are independent.

For another example, suppose there are two urns, one containing 100
white balls and 1 red ball, and the other containing 100 red balls and 1
white ball. You do not know which is which. You choose 2 balls from the
first urn. Let *A* be the event "The first ball is white," and let *B* be the event
"The second ball is white." These events are not independent, because if
you draw a white ball the first time, you are more likely to be drawing from
the urn with 100 white balls than the urn with 1 white ball.

Determine the following probabilities. Assume all coins and dice are "fair" in the sense that H and T are equiprobable for a coin, and 1, ..., 6 are equiprobable for a die.

a. At least one head occurs in a double coin toss.

b. Exactly two tails occur in a triple coin toss.

c. The sum of the two dice equals 7 or 11 in rolling a pair of dice.

d. All six dice show the same number when six dice are thrown.

e. A coin is tossed seven times. The string of outcomes is HHHHHHH.

f. A coin is tossed seven times. The string of outcomes is HTHHTTH.

**1.7 Craps**

A roller plays against the casino. The roller throws the dice and wins if the sum is 7 or 11, but loses if the sum is 2, 3, or 12. If the sum is any other number (4, 5, 6, 8, 9, or 10), the roller throws the dice repeatedly until either winning by matching the first number rolled or losing if the sum is 2, 7, or 12 ("crapping out"). What is the probability of winning?

**1.8 A Marksman Contest**

In a head-to-head contest Alice can beat Bonnie with probability *p* and can
beat Carole with probability *q*. Carole is a better marksman than Bonnie,
so *p* > *q*. To win the contest Alice must win at least two in a row out
of three head-to-heads with Bonnie and Carole and cannot play the same
person twice in a row (that is, she can play Bonnie-Carole-Bonnie or Carole-Bonnie-Carole).
Show that Alice maximizes her probability of winning the
contest playing the better marksman, Carole, twice.

**1.9 Sampling**

The mutually exclusive outcomes of a random action are called *sample
points*. The set of sample points is called the *sample space*. An *event A*
is a subset of a sample space [OMEGA]. The event *A* is *certain* if *A* = [OMEGA] and
*impossible* if *A* = 0 (that is, A has no elements). The *probability* of an
event *A* is P[*A*] = *n(A)/n*([OMEGA]), if we assume [OMEGA] is finite and all [omega] [member of]
[OMEGA] are equally likely.

a. Suppose six dice are thrown. What is the probability all six die show the same number?

b. Suppose we choose *r* object in succession from a set of *n* distinct objects
[*a*.sub.1], ..., *[a.sub.n]*, each time recording the choice and returning the object
to the set before making the next choice. This gives an ordered sample
of the form ([*b*.sub.1], ..., *[b.sub.r]*), where each *[b.sub.j]* is some *[a.sub.i]*. We
call this *sampling with replacement*. Show that, in sampling *r* times with replacement
from a set of *n* objects, there are *[n.sub.r]* distinct ordered samples.

c. Suppose we choose *r* objects in succession from a set of *n* distinct
objects [*a*.sub.1], ..., *[a.sub.n]*, without returning the object to the set. This gives an
ordered sample of the form ([*b*.sub.1], ..., *[b.sub.r]*), where each *[b.sub.j]* is some unique
*[a.sub.i]*. We call this *sampling without replacement*. Show that in sampling
*r* times without replacement from a set of *n* objects, there are

*n(n* - 1) ... (*n - r* + 1) = *n*!/(*n - r*)!

distinct ordered samples, where *n*! = *n* x (*n* - 1) x ... x 2 x 1.

**1.10 Aces Up**

A deck of 52 cards has 4 aces. A player draws 2 cards randomly from the deck. What is the probability that both are aces?

**1.11 Permutations**

A linear ordering of a set of *n* distinct objects is called a *permutation* of the
objects. It is easy to see that the number of distinct permutations of *n* > 0
distinct objects is *n*! = *n x (n* - 1) x ... x 2 x 1. Suppose we have a deck
of cards numbered from 1 to *n* > 1. Shuffle the cards so their new order
is a random permutation of the cards. What is the average number of cards
that appear in the "correct" order (that is, the *k*th card is in the *k*th position)
in the shuffled deck?

**1.12 Combinations and Sampling**

The number of *combinations* of *n* distinct objects taken *r* at a time is the
number of subsets of size *r*, taken from the *n* things without replacement.
We write this as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, we do not care about the order of
the choices. For instance, consider the set of numbers (1,2,3,4}. The number of
samples of size two without replacement = 4!/2! = 12. These are precisely
{12,13,14,21,23,24,31,32,34,41,42,43}. The combinations of the four numbers
of size two (that is, taken two at a time) are {12,13,14,23,24,34}, or
six in number. Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A set of *n* elements has
*n*!/*r*!(*n - r*)! distinct subsets of size *r*. Thus, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

**1.13 Mechanical Defects**

A shipment of seven machines has two defective machines. An inspector checks two machines randomly drawn from the shipment, and accepts the shipment if neither is defective. What is the probability the shipment is accepted?

**1.14 Mass Defection**

A batch of 100 manufactured items is checked by an inspector, who examines 10 items at random. If none is defective, she accepts the whole batch. What is the probability that a batch containing 10 defective items will be accepted?

**1.15 House Rules**

Suppose you are playing the following game against the house in Las Vegas.
You pick a number between one and six. The house rolls three dice, and
pays you $1,000 if your number comes up on one die, $2,000 if your number
comes up on two dice, and $3,000 if your number comes up on all three dice.
If your number does not show up at all, you pay the house $1,000. At first
glance, this looks like a *fair game* (that is, a game in which the expected
payoff is zero), but in fact it is not. How much can you expect to win (or
lose)?

**1.16 The Addition Rule for Probabilities**

Let *A* and *B* be two events. Then 0 [less than or equal to] P[*A*] [less than or equal to] 1 and

P[*A* [union] *B*] = P[*A*] + P[*B*] - P[*AB*].

If *A* and *B* are disjoint (that is, the events are mutually exclusive), then

P[*A* [union] *B*] = P[*A*] + P[*B*].

*(Continues...)*

Excerpted fromGame Theory EvolvingbyHerbert GintisCopyright © 2009 by Princeton University Press. Excerpted by permission.

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