Download Chapter 1

Chapter 1
Choices
1
Definition of Game Theory
• Game theory provides a framework in
which to model and analyze conflict and
cooperation among different entities, each
with its own goal
2
Objective Function
• When faced with a decision, we want the
best choice for us.
• Need to maximize an objective function
(which measures our benefit from the
decision)
• Example: buying a house. Want more
space or smaller house in better location
• Example: budget – what activity gets what
money
3
Optimization problem
• Given f:   which assigns a real value
to each alternative in domain 
• We assume a higher value means a better
choice, so we try to maximize f
• Let w* be that value of  which maximizes
the function
4
Example
• Want to buy apples and oranges. Apples cost
$1 per pound and oranges $2 per pound.
• We have $12 total.
• (x,y) represents buying x apples and y oranges.
• Let f(x,y) = xy represent the worth of the choice
(x,y). Which is better (12,0), (6,3), or (5,1)?
• We need to define the domain.  is the set
{(x,y) | x 0, y 0, x + 2y 12}
• How could you find the optimal solution?
5
Relative versus absolute extremum
•
•
•
•
Extrema (c, d, e, f)
Maxima (c, d) minima (e, f)
Relative (c, e) vs. absolute (d, f) extrema
Local (c, e) vs. global (d, f) extrema
D

y
C


E

F
x
6
Critical & stationary values
• The critical value of x is the value x* if f ’(x*) = 0
• A stationary point is a point at which the derivative of
a function f(x) vanishes
• A stationary point may be a minimum, maximum, or
inflection point.
• A stationary value (The value at a stationary point) of
y is f(x*)
• A stationary point is the point with coordinates x*
and f(x*)
7
First-derivative test
• The first-order condition or necessary condition for
extrema is that f '(x*) = 0 and the value of f(x*) is:
• A relative maximum if the derivative f '(x) changes its
sign from positive to negative from the immediate left
of the point x* to its immediate right. (first derivative
test for a max.)
y
A
f '(x*) = 0
x*
8
First-derivative test
• The first-order condition or necessary condition for
extrema is that f '(x*) = 0 and the value of f(x*) is:
• A relative minimum if f '(x*) changes its sign from
negative to positive from the immediate left of x0 to
its immediate right. (first derivative test of min.)
y
B
f '(x*)=0
x
x*
9
First-derivative test
• The first-order condition or necessary condition for
extrema is that f '(x*) = 0 and the value of f(x*) is:
• Neither a relative maxima nor a relative minima if
f '(x) has the same sign on both the immediate left
and right of point x0. (first derivative test for point of
inflection)
y
D

x*
f '(x*) = 0
x
10
Example
• Let R(Q) = 1200Q - 2Q2
dR
 1200  4Q  0
dQ
• 4Q = 1200; Q = 300;
• max, min, or inflect?
• d2R/dQ2 = -4 (a clue?)
11
Derivative of a derivative
• Given y = f(x)
• The first derivative f '(x) or dy/dx is itself a function
of x, it should be differentiable with respect to x,
provided that it is continuous and smooth.
• The result of this differentiation is known as the
second derivative of the function f and is denoted as
f ''(x) or d2y/dx2.
• The second derivative can be differentiated with
respect to x again to produce a third derivative,
f '''(x) and so on to f(n)(x) or dny/dxn
12
Example
• Let R(Q) = 1200Q - 2Q2
dR
 1200  4Q
dQ
d dR d 2 R

 4
2
dQ dQ dQ
2
3
d d R d R

0
2
3
dQ dQ
dQ
13
Interpretation of the second derivative
• f '(x) measures the rate of change of a function
– e.g., whether the slope is increasing or decreasing
• f ''(x) measures the rate of change in the rate of
change of a function
– e.g., whether the slope is increasing or decreasing at
an increasing or decreasing rate
14
An application
• If quadratic f(x) w/ maximum at x0 then
 
f  x  0
0
• If quadratic f(x) w/ a minimum at x0 then
f xo   0
15
Example
• Let R(Q) = 1200Q - 2Q2
dR
 1200  4Q  0
dQ
d dR d 2 R

 4
2
dQ dQ dQ
• Since f''(Q) < 0, then maximum
16
• profit function (on left in red) with 1st derivative
shown in blue.
• on right, 1st deriviative is shown again (on
different scale) and its deriviate (the 2nd
derivative) is shown in red,
17
Figure 1.2
• Shows the set of possible choices for our
problem of apples and oranges.
• Does not show you how the maximum is found.
y
6
utility maximizer: (6,3)
budget line: x+2y=12
 budget set
x
12
18
How is maximum found?
•
•
•
•
•
•
•
•
Have two functions
u(x,y) = xy (utility function)
x+2y 12
u(y) = (12-2y)*y = 12y-2y2
Need to maximize u
u’(y) = 12 -4y = 0
y=3
x=6
19
Optimizing Using Lagrange
• Optimizing when the choice set is an interval is
fairly easy.
• What if the choice set is described by a set of
equations?
• Let g(x,y) be the constraint function.
• Want to maximize u(x,y) given g(x,y)=c
• Geometric meaning is shown in Figure 1.4.
• The “wire” g(x,y)=c show all the solutions in the
choice set which satisfy the constraint function.
• We want to find the point on the wire which
maximizes u(x,y)
20
Figure 1.4
want to find value along “wire” which maximized utility
function
y

g(x,y) = c
x
21
Lagrange Method
•
•
•
•
Maximize u(x,y) under constraint g(x,y)=c
Create the equation
L(x,y, λ )= u(x,y) + λ(c-g(x,y))
Find maximums by setting all partial derivates (with
respect to x, y and λ) to zero
• For example, maximize pq under the constraint: p+q=1
• Lagrange Method:
– Define L(p,q)=pq+λ(p+q-1)
– Solve the equations
L( p, q)
 0,
p
L( p, q)
 0,
q
p  q 1
22
So the solution is
•
•
•
•
p+λ=0
q+λ=0
p+q = 1
p=q=½
23
Consider our example of apples
and oranges
•
•
•
•
•
•
•
x + 2y = 12
u(x,y) = xy
L(x,y, λ )= u(x,y) + λ(c-g(x,y)) = xy + λ(12-x-2y)
y-λ=0
so y= λ
x -2λ = 0 so x = 2 λ
x+2y = 12 so 2 λ + 2 λ = 12 so 4 λ =12 so λ = 3
x=6, y = 3
24
Example
•
•
•
•
•
•
•
•
•
•
•
•
•
I can buy v pounds of vegetables at $ p1 each
I can buy d pounds of dye at $ p2 each
I have $m total
Utility is vd +d
How many of each should I buy if I have $24?
let m= 24, p1 = 2, p2 = 3
L(v,d, λ) = vd +d + λ(24-2v-3d)
v+1 - 3λ = 0
v = 3λ-1
d - 2λ=0
d = 2λ
2v+3d = 24
2(3λ-1 ) + 3(2 λ ) = 24
6 λ -2 + 6 λ = 24 so 12λ =26
λ=13/6
v = 11/2; d = 13/3 (utility 28.12)
25
Example 1.6
• Can manufacture x units of product at factory A
costing 2x2 + 50000
• Can manufacture y units of product at factory B
costing y2 + 40000
• We want to minimize cost but need to produce
1200 units total.
• L(x,y, λ) = 2x2 + 50000 + y2 + 40000 + λ(1200-x+y)
• 4x - λ = 0
2y - λ = 0 x+y = 1200
• x = λ /4
y = λ /2 3λ /4 = 1200
• λ=1600
• x = 400, y = 800
26
Uncertainty and Chance
• In decision making, often you don’t know what
the other player will do, but only have some
guesses of what he will do.
• Thus, we need to deal with our estimates of
what they will do - probability
• A probability space (S,P) where S is a finite set,
called the sample space, and P is a function that
assigns a probability to elements si in S
• pi  0 and  pi = 1 where pi is the probabilty of si
• if A is a subset of S then, P(A) =  pi (when si
A)
27
• subsets of the sample space are called
events
• Events are random outcomes of chance
• Throwing coins has events H (throwing
heads) and T (throwing tails)
• P(H) = P(T) = ½
• A random variable, X, is a function from S
to the Reals. It converts an event like
“throw a head” to a number. Makes it
easier to work with all events in a similar
manner.
• Say X(H) = 1 and X(T) = 2.
28
Example
• Toss a coin twice. Let the random variableY
denote the number of heads.
• Denote (Tail, Tail) to be the elementary event
that the first toss is tail and the second toss is
tail.
• Denote the other elementary events accordingly.
•
•
•
•
Compound Event Elementary Events
(Y=0) (Tail, Tail)
(Y=1) (Tail, Head), (Head, Tail)
(Y=2) (Head, Head)
29
Discrete random variables
Definition: Let X be a random variable that can take only a finite (or
countably infinite) number of values then the function p(x) described
by
p ( x)  P( X  x)
is a probability mass function
Examples of probability mass functions
Example 1 (Uniform probability distribution)
p(x) = 1/n where n = number of possible outcomes of the experiment
e.g. fair dice. p(x) = 1/6
30
Example
• Toss a balanced coin twice. Let Y denote the number of heads. Find the
probability mass function of Y.
• Denote (Tail, Tail) to be the elementary event that the first toss is tail and
the second toss is tail. Denote the other elementary events accordingly.
•
•
•
•
•
•
•
•
•
Number of Heads (y) Elementary Events
0 (Tail, Tail)
1 (Tail, Head) (Head, Tail)
2 (Head, Head)
y f(y)
0 ¼
1 ½
2 ¼
31
Example
•
Toss a pair of dice; win dollars equal to the sum of numbers on the two dice. Let Y denote
the winnings after playing the game once. Find the probability mass function of Y.
Winnings
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12
Elementary Events
(1,1)
(1,2) (2,1)
(1,3) (2,2) (3,1)
(1,4) (2,3) (3,2) (4,1)
(1,5) (2,4) (3,3) (4,2) (5,1)
(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
(2,6) (3,5) (4,4) (5,3) (6,2)
(3,6) (4,5) (5,4) (6,3)
(4,6) (5,5) (6,4)
(5,6) (6,5)
(6,6)
32
Properties of Probability Mass Functions (Discrete probability
distributions) for all x
0  p( x)  1
p(x) = 1
Definition: Distribution Function
The term distribution function is short for cumulative distribution
function and describes the integral of the probability density
function
Let X be a random variable. Then F(x) = P( X  x)
is called the
distribution function of X.
A discrete random variable can be represented as a histogram.
For a discrete random variable, F(x) is just the sum of the area of the
boxes of a histogram below and including x.
33
Example 1.11
•
•
•
•
•
•
Tossing a fair dice
Sample space is {1,2,3,4,5,6}
pi = 1/6 for each i
X(i) = i
F(x) = P(X x) = number of integers less than x/6
F(x) is step function (see figure 1.5)
1
y=F(x)
1/6
1
2
3
4
5
6
x
34
Properties of distribution
•
•
•
•
•
0 F(x)1
F is increasing F(x)  F(y) if x <y
As x goes to infinity, F(x) approaches 1
P(a<X  b) = F(b) –F(a) if a < b
P(X=a) is the jump in the distribution at a
35
Uniform Distribution
• X has a uniform distribution on interval [a,b]
if f(x) = 1/(b-a) if a <x<b
• F(x) =
x
 f (t )dt
-inf
=(x-a)/(b-a) if a < x < b
36
Normal Distribution
• Bell-shaped, symmetric family of distributions
• Classified by 2 parameters: Mean (m) and standard
deviation (s). These represent location and spread
• Random variables that are approximately normal have the
following properties with respect to individual
measurements:
– Approximately half (50%) fall above (and below) mean
– Approximately 68% fall within 1 standard deviation of
mean
– Approximately 95% within 2 standard deviations of mean
– Virtually all fall within 3 standard deviations of mean
• Notation when Y is normally distributed with mean m and
standard deviation s :
Y ~ N (m ,s )
37
Normal Distribution
P(Y  m )  0.50 P( m  s  Y  m  s )  0.68 P( m  2s  Y  m  2s )  0.95
38
Example - Heights of U.S. Adults
• Female and Male adult heights are well approximated by
normal distributions: YF~N(63.7,2.5) YM~N(69.1,2.6)
20
20
18
16
14
12
10
10
8
6
4
Std. Dev = 2.48
Std. Dev = 2.61
2
Mean = 63.7
Mean = 69.1
0
N = 99.68
55.5
57.5
56.5
59.5
58.5
61.5
60.5
63.5
62.5
65.5
64.5
67.5
66.5
INCHESF
69.5
68.5
70.5
N = 99.23
0
59.5 61.5 63.5 65.5 67.5 69.5 71.5 73.5 75.5
60.5 62.5 64.5 66.5 68.5 70.5 72.5 74.5 76.5
INCHESM
Cases weighted by PCTM
Cases weighted by PCTF
39
Source: Statistical Abstract of the U.S. (1992)
Example 2 (Geometric distribution)
If I toss a coin (where p is the probability of tails), how long
do I have to wait until I toss a head? (k is number of throws
before throwing a head)
P(k )  p k (1  p)
40
Example 3 (binomial distribution)
If two distinct outcomes of an experiment are possible, A and
B, and the probability of event A is p, then the probability of k
occurrences of event A from n trials is given by the binomial
distribution
n k
  p (1  p) n  k
k 
mean = p and
variance = 1-p
41
Example 5 (Poisson distribution)
Poisson distribution: Discrete probability distribution for context-independent
‘rare’ events
Say events occur at some rate λ so that the expected number of events
occuring within time t is λt
Now break up t into n equal intervals. Let the probability of an event in a
single interval be p then
np = λt
The number of events in interval l is independent of the number of events in
interval l+1
The total number of events in n intervals is described by the binomial
distribution
42
n
p(k )    p k (1  p) n  k
k 
n!
t 
 t  

  1 

(n  k )! k!  n  
n
k
nk

n!
t  
t 

1 

k
k! 
n
(n  k )! n
k

t k
k!
nk
e  t
43
Two kinds of random variables
• A discrete random variable has a countable
number of possible values.
– X: number of baskets when trying 5 free throws.
A continuous random variable takes all values in an
interval of numbers.
– X: the time it takes for a bulb to burn out.
– The values are not countable.
– has a probabilty density function, rather than
probability mass function
44
Expected Value
The expected value of a random variable X can be obtained by
summation or integration as follows:
E ( X )   xp( x)
…..Discrete
x
E ( X )   xp( x)dx
…..Continuous
x
The expected value is also known as the distribution mean
45
Variance & standard deviation
• Var(X) =  pi[X(si) – E(X)]2
• Standard deviation s = sqrt(Var(X))
46
Decision Making Under Uncertainty
• When you buy a car, you don’t know whether
it will be a good one or not.
• We try to capture the goodness of the
decision with expected utility
•
E (u (d ))   u ( wx) p( x)
x
• The function u(w) is the utility function over
wealth or the von Neumann-Morgenstern
utility function (has to have certain properties)
47
• For our purposes, u(w) is any strictly increasing
function u:[0,inf] 
• Decisions made under uncertainty can be
thought of as choosing a lottery L over
alternative levels of wealth wi where each level
of wealth can be assigned a probability pi
• Lottery L is a collection of pairs {{wi, pi)}
• a lottery or gamble is simply a probability
distribution over a known, finite set of outcomes.
48
Examples:
• For the Derby betting pool, the set of outcomes A =
{Giacomo wins,Closing Argument wins, Afleet Alex wins}
• For the pharmaceutical company, the set of outcomes A
= {Earn $500 million from patent, Earn $200 million from
patent, Earn $0 from patent}
• Each of these outcomes had a probability attached to it,
and so we can define a simple lottery as a set of
outcomes, A={a1, a2,...,an} each of which occurs with
some known probability pi.
49
Compound Lottery
• With two lotteries (having same set of alternatives)
• L1= {{wi, pi)} L2 = {{w’i, p’i)}
• we can combine: pL1 + (1-p)L2 is a compound lottery
• We can then also construct compound lotteries, which
are probability distributions over lotteries - i.e., an
outcome of a lottery may itself be another lottery. As a
concrete example, imagine a Powerball lottery where the
prize is yet another lottery ticket. Let G represent the set
of all lotteries, or gambles, both simple and compound
• Independence Axiom: If L1 is preferred over L2, then
pL1+(1-p)L3 is preferred over pL2+(1-p)L3
50
Goals
• Agent attempts to maximize its expected utility
• Utility function ui of agent i is a mapping from outcomes to reals
– Can be over a multi-dimensional outcome space
– Incorporates agent’s risk attitude (allows quantitative tradeoffs)
Lottery: a process, such as picking a name from a hat,
through which goods are allocated randomly
Lottery 1: $0.5M prob 1
Lottery 2: $1M prob 0.5
$0
prob 0.5
Agent’s strategy is the
choice of lottery
ui
Risk averse
1
Risk neutral
0.5
Risk seeking
0
0
0.5
1
M$
51
Risk aversion => insurance companies
Attitudes towards risk
• Lottery 1: $0.5M prob 1
Lottery 2: $1M prob 0.5
•
$0
prob 0.5
Nick: u(a) = a2
Lottery 1: u(a) p(a) = 1*(.5)2 = .25
Lottery 2: u(a) p(a) = .5*(0)2 + .5(1)2 = .5
Nick with this risk nature prefers lottery 2: Risk Seeking
Sally:u(a) = a
Lottery 1: u(a) p(a) = 1*(.5) = .5
Lottery 2: u(a) p(a) = .5*(0) + .5(1) = .5
Sally with this risk nature doesn’t care which lottery: Risk Neutral
John:u(a) = sqrt(a)
Lottery 1: u(a) p(a) = 1*sqrt(.5) = .7
Lottery 2: u(a) p(a) = .5*sqrt(0) + .5*sqrt(1) = .5
John prefers lottery 1: Risk averse
52
Utility functions are scale-invariant
• Agent i chooses a strategy that maximizes expected utility
•
maxstrategy Soutcome p(outcome | strategy) ui(outcome)
•
p(outcome | strategy) is probability of outcome, given the strategy
• If ui’() = a ui() + b for a > 0 then the agent will choose the same strategy
under utility function ui’ as it would under ui
• Linear relationship between ALL utilities preserves strategies?
• Note that ui has to be finite for each possible outcome
– Otherwise expected utility could be infinite for several strategies, so the
strategies could not be compared.
53
Full vs bounded rationality
Full
rationality
Bounded rationality:
How much can I
afford to compute
Environment
Environment
Perceptions
Actions
Perceptions
Actions
Agent
Agent
Reasoning
machinery
solution quality
Descriptive vs. prescriptive
theories of bounded rationality
worth of solution
time
deliberation cost
54
Expected Utility Theorem
Theorem 1.19 (Expected Utility Theorem) If a
preference relation on the set of lotteries satisfies
independence and continuity, then there is a von
Neumann-Morgenstern utility function u over wealth
such that the induced utility function on lotteries,
for L = {(wi, pi) : i = 1 . . . n }, is compatible with the
preference relation on lotteries.
In other words: we can capture preference using a
numeric function.
55
Utility Over Wealth
• we could use the term Bernoulli Utility
Function to refer to a decision-maker's utility
over wealth - since it was Bernoulli who
originally proposed the idea that people's
internal, subjective value for an amount of
money was not necessarily equal to the physical
value of that money.
• The term von Neumann-Morgenstern Utility
Function, or Expected Utility Function is used
to refer to a decision-maker's utility over
lotteries, or gambles.
56
Risk Aversion and insurance
• risk-averse individuals will always choose to insure
valuable assets, since although the probability of a loss
may be small, the potential loss of the asset itself would
be so large that most people would rather pay small
amounts of money as a premium for certain than risk the
loss.
On the other hand, insurance companies are riskneutral, and earn their profits from the fact that the value
of the premiums they receive is either greater than or
equal to the expected value of the loss.
57
Example
• Our discussion will assume that apart from knowning his
own wealth, an individual making the decision to insure
or not also knows for certain the probability of a loss or
accident.
Say you (a risk-averse consumer) have initial wealth w,
and a von Neumann-Morgenstern utility function u(.).
You own a car of value L, and the probability of an
accident which would total the car is p (we might imagine
p as the current accident rate in the state where you
live).
If x is the amount of insurance you purchase, how much
should x be?
58
• The answer to this question depends, very simply, on the
price of insurance - the premium you'd have to pay. Let's
say this price is r, for $1 worth of insurance, so for $x of
insurance, you'd be paying $rx as a premium.
For insurance to be actuarially fair, the insurance
company should have zero expected profits. We can set
up their problem as:
With probability p, the insurance company must pay $x,
while receiving $rx in premiums. With probability (1-p),
they pay nothing, and continue to receive $rx in
premiums. So their expected profit is:
p(rx - x) + (1-p)rx
59
• If this equals zero, we have: px(r-1) + (1-p)rx = 0
Dividing throughout by x, we get: pr - p + r - pr = 0
i.e. p = r.
So for insurance to be actuarially fair, the premium rate
must equal the probability of an accident.
In actual practice, even if the premium does not equal
the probability of an accident, it certainly depends on it which is why different demographic groups pay widely
differing automobile insurance premiums. Since single
men under the age of 25 have the highest accident risk,
they also pay the highest premiums.
60
• you would want to choose a value of x (the amount you
insure) so as to maximize expected utility, i.e.
Given actuarially fair insurance, where L is car value and
w is total wealth
• maximize p*u(w - L - rx + x) + (1-p)*u(w - rx),
• If p = r, this means you solve:
• max p*u(w - L - px + x) + (1-p)*u(w - px),
Differentiating with respect to x, and setting the result equal
to zero, we get the first-order necessary condition as:
(1-p) p*u'(w - px - L + x) - p(1-p)u'(w - px) = 0,
Note: terms in red/bold are derivatives of insides of u.
which gives us: u'(w - px - L + x) = u'(w - px)
61
• Because utility functions are increasing, the equality of
the marginal utilities of wealth implies equality of the
wealth levels, i.e.
w - px - L + x = w - px,
so we must have x = L.
So, given actuarially fair insurance, you would choose to
fully insure your car. Since you're risk-averse, you'd aim
to equalize your wealth across all circumstances whether or not you have an accident.
However, if p and r are not equal, we will have x < L; you
would under-insure. How much you'd underinsure would
depend on the how much greater r was than p.
62
Example 1.20
• Gamble 1: pay $100 to win $500 with a
probability ½ or win $100 otherwise.
• Gamble 2: pay $100 to win $325 with a
probability of ½ and win $136 otherwise.
• If our u(w) = w
• The expected utility of gamble 1 is
½ 500  100+ 1/2(0) = ½ 20 = 10
The expected utility of gamble 2 =
½*sqrt(136-100) + ½ sqrt(225) = ½(6+15) =10.5
63
• Of course, if the u(w)= w, Gamble 1 is better.
• Individuals have different tolerance for risk.
• An individual who ranks lotteries according to
their expected value (rather than expected
utility) is said to be risk neutral. In other words,
an risk neutral individual who is offered $100
outright or a 50% chance of winning $200 will
value the choices EQUALLY!
64
If the utility function over wealth is linear
u(w) = aw + b
the person is risk neutral
• If the utility function is concave(line between
points is under curve), the individual is risk
averse.
• If the utility function is convex(line between
points is above curve), the individual is risk
seeking. Note, gambling is like staying on the
line as the two endpoints are picked with
probability p or (1-p).
65
• So u(w) = w is risk neutral
• u(w) = w is risk averse
• u(w) = w2 is risk seeking (as large amount
of money is worth much more than small
amounts)
66
Expected Utility Theory
• describes behavior under uncertainty
• If people are risk neutral or risk averse,
they would never play the lottery or
gamble (as return there is usually
negative)
• The expected value of Powerball lottery (if
tickets cost $1 and jackpot is 7 million) is
7000000 * 1/85000000 -1(84999999/85000000) = -.917647
67
But people do play powerball Why?
• Loss is so small, people often ignore it.
• If losses were larger, people may behave
very differently.
• People who buy lottery tickets may behave
in very risk averse manner in other
situation
68
Allais Paradox
• In 1953, Maurice Allais published a paper
regarding a survey he had conducted in 1952,
with a hypothetical game.
• Subjects "with good training in and knowledge of
the theory of probability, so that they could be
considered to behave rationally", routinely
violated the expected utility axioms.
• The game itself and its results have now
become famous as the "Allais Paradox".
69
The most famous structure is the following:
Subjects are asked to choose between the following 2 gambles, i.e.
which one they would like to participate in if they could:
Gamble A: A 100% chance of receiving $1 million.
Gamble B: A 10% chance of receiving $5 million, an 89% chance of
receiving $1 million, and a 1% chance of receiving nothing.
After they have made their choice, they are presented with another 2
gambles and asked to choose between them:
Gamble C: An 11% chance of receiving $1 million, and an 89%
chance of receiving nothing.
Gamble D: A 10% chance of receiving $5 million, and a 90% chance
of receiving nothing.
70
•
This experiment has been conducted many, many times, and most people
invariably prefer A to B, and D to C. So why is this a paradox?
The expected value of A is $1 million, while the expected value of B is $1.39
million. By preferring A to B, people are presumably maximizing expected
utility, not expected value.
By preferring A to B, we have the following expected utility relationship:
u(1) > 0.1 * u(5) + 0.89 * u(1) + 0.01 * u(0), i.e.
0.11 * u(1) > 0.1 * u(5) + 0.1 * u(0)
Adding 0.89 * u(0) to each side, we get:
0.11 * u(1) + 0.89 * u(0) > 0.1 * u(5) + 0.90 * u(0),
implying that an expected utility maximizer must prefer C to D. Of course,
the expected value of C is $110,000, while the expected value of D is
$500,000, so if people were maximizing expected value, they should in fact
prefer D to C. However, their choice in the first stage is inconsistent with
their choice in the second stage, and herein lies the paradox.
From the Von Neumann-Morgenstern axioms, the substitution axiom is the
one that is clearly violated. The probability of receiving $5 million is the
same in both B and D.
71
Ellsberg Paradox
In 1961, Daniel Ellsberg published the
results of a hypothetical experiment he
had conducted, which, to many,
constitutes an even worse violation of the
expected utility axioms than the Allais
Paradox. Ellsberg's subjects in his thought
experiment seemed to run the gamut of
noted economists of the time, from Gerard
Debreu to Paul Samuelson and Howard
Raiffa.
72
The Experiment
•
•
•
•
•
Suppose there are two large pots, each containing black and red balls. The
first pot contains 50 black and 50 red balls. The second pot also contains
100 balls but the mix between red and black balls is unknown.
You win $500 if you draw a red ball. Which pot will you choose? You are
most likely to choose the first pot, as did the people who were part of
Ellsberg's experiment. Why?
You know there is a 50 per cent chance of getting a red ball if you choose
the first pot. The probability of drawing a red ball from the second pot is not
known.
Next, you are offered $500 to draw a black ball. What will you do? Chances
are you will still select the first pot! That is the paradox.
The first time you chose the first pot because you thought the other one had
fewer red balls. Logically, it meant that you thought there were more black
balls in the second pot. So, you should have chosen this pot in the second
experiment —$500 for a black ball.
73
• After a series of such experiments, Daniel
Ellsberg concluded that people behave this way
because they prefer to avoid ambiguity. In the
above case, choosing black or red ball from the
second pot was ambiguous, as the mix was not
known.
• The Ellsberg Paradox essentially states that we
treat ambiguous choices as risky. This has been
cited as one of the reasons for the high returns
in the stock market. Stock price movements are
ambiguous. So we treat the stock market as
risky and demand high returns.
74