Supplementary Math Handout for Decision Analysis
İrem Demirci∗
Fall 2016
This handout is prepared for CC 501 Decision Analysis course to remind the students of the mathematical tools that they will use in this class. Please do not distribute this handout. I do not take
credits for any mistakes in this handout.
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Ordinal versus Cardinal Functions and Their Transformations
Multi-Attribute Value Theory (MAVT) and Expected Utility Theory (EUT) are two of the most
important topics that we will cover in this class. In both cases, we will use functions to represent
preferences. Under certainty, we will make use of value functions, and if there are uncertainties
involved, utility functions will be employed.
1.1
Ordinal Functions
An ordinal function assigns a real number to every alternative in a way that the preference ranking
over the alternatives is preserved. This is the main requirement for a value function or a utility
function to represent rational preferences, and can be summarized as:
a b iff v(a) ≥ v(b)
which reads that the function assigns a real number to each alternative, such that the value of
alternative a is greater than the value of alternative b if and only if the decision maker prefers
alternative a to alternative b.
The absolute magnitude of the values assigned to the alternatives is not relevant such that only
their order matters. This is an ordinal property of the value function, a property that is preserved
under any positive monotonic transformation. A positive monotonic transformation converts one
set of numbers into another set of numbers such that the rank order of the original set of numbers
∗
University of Mannheim Business School, Area Finance, Mannheim, Germany. e-mail: [email protected].
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does not change. A positive monotonic transformation is simply a strictly increasing function.
An ordinal value function is not unique because any positive monotonic transformation of the original value function preserves the ranking of alternatives, therefore represents the same preferences.
For instance, if v(x) is a value function that represents the preferences, and f (.) is a strictly increasing function, then u(x) = f (v(x)) is also a value function representing the same preference
relation as v(x).
1.2
Cardinal Functions
With ordinal functions, only the ranking of the values of the function matters. For instance, if v(x) is
an ordinal function and a, b, c, and d are four alternatives, then the fact that v(a)−v(b) = v(c)−v(d)
does not necessarily mean that the transition from b to a is valued the same as the transition from
d to c.
A value function is cardinal if it is an ordinal function with the additional property that the
transition from alternative b to alternative a (b → a) is preferred to the transition from alternative
d to alternative c (d → c) if and only if the value difference between a and b is greater than the
value difference between c and d:
b → a d → c iff v(a) − v(b) ≥ v(c) − v(d)
Therefore, cardinal functions not only reflect the ranking of preferences (as ordinal functions do),
but also the ranking of transitions between alternatives.
Cardinal functions are unique up to a positive linear transformation. For instance, if v(x) is a
cardinal value function that represents the preferences, and f (x) = a + bx where b > 0, then
u(x) = f (v(x)) is also a cardinal value function representing the same preference relation as u(x).
To see this, assume that the preferences over x1 , x2 , x3 and x4 satisfy the following:
x2 → x1 x4 → x3 and v(x1 ) − v(x2 ) ≥ v(x3 ) − v(x4 )
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The following holds for any positive linear transformation of v(x) such that u(x) = a + bv(x):
v(x1 ) − v(x2 ) ≥ v(x3 ) − v(x4 ) ⇒ u(x1 ) − u(x2 ) ≥ u(x3 ) − u(x4 )
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Concave, Convex and Linear Functions
This section explains the properties of concave, convex and linear functions which will be especially
useful for studying utility functions and attitudes towards risk.
2.1
Concave Functions
Let f be a function of a single variable defined on the interval I. Then, f is concave if for every x1
and x2 in I, and for every 0 ≤ α ≤ 1
f (αx1 + (1 − α)x2 ) ≥ αf (x1 ) + (1 − α)f (x2 )
Here is an alternative definition for concave functions: A twice-differentiable function f of a single
variable defined on an interval I is concave if and only if for all x in I
f 00 (x) ≤ 0
Therefore, the first derivative of a concave function is monotonically non-increasing.
f (x)
f (x2 )
f (αx1 + (1 − α)x2 )
αf (x1 ) + (1 − α)f (x2 )
f (x1 )
0
x1
αx1 + (1 − α)x2
3
x2
x
2.2
Convex Functions
Let f be a function of a single variable defined on the interval I. Then, f is convex if for every x1
and x2 in I, and for every 0 ≤ α ≤ 1
f (αx1 + (1 − α)x2 ) ≤ αf (x1 ) + (1 − α)f (x2 )
Here is an alternative definition for convex functions: A twice-differentiable function f of a single
variable defined on an interval I is convex if and only if for all x in I
f 00 (x) ≥ 0
Therefore, the first derivative of a convex function is monotonically non-decreasing.
2.3
Linear Functions
A linear function is both concave and convex because its slope is a constant. A linear function has
the form f (x) = mx + c for some constants m and c. The graph is a line with slope m and intercept
c. Note that you can derive any linear function as long as you know two points that satisfy the
function. Let (x1 , y1 ) and (x2 , y2 ) be the two points, then the slope of the function is given by
m=
y2 − y1
x2 − x1
and the intercept is
c = y1 − x 1
3
y2 − y1
x2 − x1
Laws of Logarithmic and Exponential Functions
Logarithmic and exponential utility functions are two very important and common utility functions
that we use in order to represent attitudes towards risk. Therefore, it is important to know the
properties of these functions.
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3.1
Laws of Logarithmic Functions
Assume that a > 0 and x > 0 are both positive real numbers
(i) y = loga (x) ⇒ x = ay
(ii) loga (ax ) = x
(iii) aloga x = x
(iv) loga (x) =
logc (x)
for any c > 0 and c 6= 1
logc (a)
(v) loga (xy) = loga (x) + loga (y) for any positive real number y
x
= loga (x) − loga (y) for any positive real number y
(vi) loga
y
(vii) loga (xk ) = kloga (x) for any real number k
Special case: Laws of natural logarithmic function
(i) ln(x) = y ⇒ x = ey
for any x > 0
(ii) ln(ex ) = x
(iii) eln(x) = x for any x > 0
(iv) loga (x) =
ln(x)
for any x > 0, a > 0 and a 6= 1
ln(a)
(v) ln(xy) = ln(x) + ln(y) for any positive real numbers x and y
x
= ln(x) − ln(y) for any positive real numbers x and y
(vi) ln
y
(vii) ln(xk ) = kln(x) for any real number k
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3.2
Laws of Exponential Functions
Assume that a > 0
(i) a0 = 1
(ii) a−x =
1
ax
(iii) (ax )y = axy
(iv) ax+y = ax · ay
(v) ax−y =
ax
ay
Special case: Laws of natural exponential function
(i) ln(ex ) = xln(e) = x
(ii) eln(x) = x
(iii) ea+b = ea · eb
(iv) ea−b =
ea
eb
(v) e0 = 1
f (x) = ex
f (x) = ln(x)
x
1
1
x
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3.3
Derivatives of the Most Common Functions
(i) Linear Function:
u(x) = a + cx, where a and c are constants, then u0 (x) = c
(ii) Exponential Function:
u(x) = ax , where a > 0, then u0 (x) = ax ln(a)
Special case: Natural exponential function
u(x) = ex , where e = 2.7182818284..., then u0 (x) = ex
(iii) Logarithmic Function:
u(x) = loga x, where a > 0 and a 6= 1, then u0 (x) =
1
xln(a)
Special case: Natural logarithmic function
u(x) = ln(x), then u0 (x) =
1
x
(iv) Power Function:
u(x) = cxa , where c and a are constants, then u0 (x) = caxa−1
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Differentiation
4.1
The Basic Rules of Differentiation
(i) Constant rule:
If f (x) = c where c is a constant, then f 0 (x) = 0
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(ii) Constant multiple rule:
If f (x) = au(x) where a is a constant, then f 0 (x) = au0 (x)
(iii) Multiple rule:
If f (x) = u(ax) where a is a constant, then f 0 (x) = au0 (ax)
(iv) The sum/difference rule:
If f (x) = u(x) ± v(x), then f 0 (x) = u0 (x) ± v 0 (x)
(v) Composite function rule:
If f (x) = u(v(x)), then f 0 (x) = u0 (v(x)) · v 0 (x)
(vi) Product rule:
If f (x) = u(x) · v(x), then f 0 (x) = v
dv
du
+u
dx
dx
(vii) Quotient rule:
If f (x) =
4.2
v du − u dv
u(x)
, then f 0 (x) = dx 2 dx
v(x)
v
Applications of Differentiation
The rules of differentiation are useful for solving an optimization problem which is basically finding
the largest or the smallest value that a function can take, often subject to a constraint. The
constraints are usually described by an equation or an inequality that must hold for any solution.
4.2.1
Finding Absolute Extrema
You can apply the following steps in order to find the absolute extrema of a continuous function
f (x) on the interval [a, b]:
1. Find all the critical points of f (x).
2. Evaluate the function at the critical points and at the end points.
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3. Identify the absolute extrema.
4.2.2
First Derivative Test
Let x = c be a critical point of a continuous and differentiable function f (x) on the interval I.
(a) If f 0 (x) > 0 for all x < c and if f 0 (x) < 0 for all x > c, then f (c) is the absolute maximum
value of f (x) on the interval I.
(b) If f 0 (x) < 0 for all x < c and if f 0 (x) > 0 for all x > c, then f (c) is the absolute minimum
value of f (x) on the interval I.
Note that unlike the previous method, here the endpoints do not need to be finite such that the
function can be defined on unbounded intervals [a, ∞] or on open intervals (a, b).
4.2.3
Second Derivative Test
Let x = c be a critical point of a continuous and differentiable function f (x) on the interval I.
(a) If f 00 (x) > 0 for all x in I, then f (c) is the absolute minimum value of f (x) on the interval I.
(b) If f 00 (x) < 0 for all x in I, then f (c) is the absolute maximum value of f (x) on the interval I.
4.2.4
Constraint optimization with Lagrange multipliers
Let f (x, y) be a function in two variables with continuous second order partial derivatives and g(x, y)
be another differentiable function in two variables. We define L(x, y, λ) := f (x, y) + λ · g(x, y), and
(a,b) is a relative extremum of f subject to g(x, y) = 0. Then, there is some value λ = λ∗ such that
∂L ∂L ∂L =
=
=0
∂x (a,b,λ∗ )
∂y (a,b,λ∗ )
∂λ (a,b,λ∗ )
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EXERCISES
1) Joint and conditional probabilities (EWL 2.4)
There are two events, x and y. Given the joint probabilities p(x ∩ y) = 0.12, p(x ∩ ȳ) = 0.29
and the conditional probability p(y|x̄) = 0.90, calculate the probabilities p(x), p(x̄), p(x̄ ∩ y),
p(x̄ ∩ ȳ), p(y), p(ȳ), p(x|y), p(y|x), p(x|ȳ) and p(x ∪ y).
2) Stochastic independence
a) As an expert in sports you are working in the German National Team. You are supposed
to investigate whether there is a relation between the effort of the team and the weather.
Team effort is measured by football outcomes where “Germany wins” denotes high effort,
while “Germany loses” and “Tie” denotes low effort. Regarding the weather you know that
it could “rain” or be “sunny”. The following matrix contains the joint probabilities for the
events:
Game Outcome
Win
Loss/Tie
Sunny
0.55
0.05
Weather
Rain
0.15
0.25
What is the probability that
I. the sun shines?
II. Germany wins?
III. the sun shines and/or Germany wins?
IV. the sun shines conditional on Germany winning?
V. Germany loses conditional on rainy weather?
b) You are an expert in stock market and your job is to forecast the price movements in the
S&P index. You have the following information about the weather and the direction of the
change in the S&P index tomorrow:
S&P Outcome
Up
Down
Sunny
0.60
0.15
10
Weather
Rain
0.20
0.05
What is the probability that
I.the sun shines?
II. the S&P index goes up?
III. the sun shines and/or the index goes up?
IV. the sun shines conditional on the index going up?
V. the index goes down conditional on rainy weather?
c) Does the weather being rainy or sunny provide extra information regarding (i) the German
team winning or losing, (ii) the S&P index going up or down?
3) Ordinal versus cardinal functions
Due to your new job you have to travel from Mannheim to Zurich. You can choose between four
types of transportation. One important objective is to minimize travel expenses which are given
in the following table:
Plane
Train
Car
Bus
Travel Expenses (in es)
150
90
70
50
a) Assume that the value of each alternative is given by the following value function: v(x) =
a + b/x where x denotes the travel expenses and a and b are constants. If you assign a value
of 1 to “Bus” and a value of 0 to “Plane”, what is a and b?
b) Consider the following monotonic transformation of v(x): f (x) =
p
v(x), and the following
positive linear transformation: g(x) = 5 + 11v(x). Show that the cardinal properties of v(x)
are lost with monotonic transformation and preserved under positive linear transformation.
4) Derivatives
Calculate the first and second derivatives of the following functions:
a) f (x) = x2 + 2ex
b) f (x) = x2 ex
c) f (x) =
ex
x
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d) f (x) = x2 ln(x)
e) f (x) = ex ln(x)
f ) f (x) =
ln(x)
ex
5) Optimization
Big Oil Corp. has to decide on the profit-maximizing quantity of oil to refine. It cannot refine
less than zero or more than 20 barrels. What quantity should it choose if it has the following
profit functions?
a) π(x) = 10x − 0.5x2 − 3
b) π(x) = x3 − 10x2 − 5
6) Constraint optimization with Lagrange multipliers
Find the critical points of the following functions subject to their constraints:
a) f (x, y) = 2y + x ,
s.t. y 2 + xy = 1
b) f (x, y) = x2 − 8x + y 2 − 12y + 48 ,
s.t. x + y − 5 = 3
7) Cumulative distribution function
Consider the probability distribution for the number of raisins in an oatmeal cookie:
p(X = 0) = 0.02, p(X = 1) = 0.05, p(X = 2) = 0.2,
p(X = 3) = 0.4, p(X = 4) = 0.22, p(X = 5) = 0.11.
Find the cumulative probabilities for the number of raisins and plot the CDF.
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