Choices

1
Chapter 1

• Choices

2
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

3
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

4
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

5
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?

6
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

7
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)

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 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.)

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
• 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.)

10
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)

11
Example

• Let R(Q) 1200Q - 2Q2

• 4Q 1200 Q 300
• max, min, or inflect?
• d2R/dQ2 -4 (a clue?)

12
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

13
Example

• Let R(Q) 1200Q - 2Q2

14
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

15
An application

• If quadratic f(x) w/ maximum at x0 then

• If quadratic f(x) w/ a minimum at x0 then

16
Example

• Let R(Q) 1200Q - 2Q2

• Since f''(Q) lt 0, then maximum

17

• 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,

18
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
utility maximizer (6,3)
6
budget line x2y12
? budget set
12
x
19
How is maximum found?

• Have two functions
• u(x,y) xy (utility function)
• x2y? 12
• u(y) (12-2y)y 12y-2y2
• Need to maximize u
• u(y) 12 -4y 0
• y 3
• x 6

20
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)

21
Figure 1.4want to find value along wire which
maximized utility function
y
?
g(x,y) c
x
22
Lagrange Method

• Maximize u(x,y) under constraint g(x,y)c
• Create the equation
• L(x,y, ? ) u(x,y) ?(c-g(x,y))
• Find maximums by setting all partial derivates
  (with respect to x, y and ?) to zero
• For example, maximize pq under the constraint
  pq1
• Lagrange Method
• Define L(p,q)pq?(pq-1)
• Solve the equations

23
So the solution is

• p ? 0
• q ? 0
• pq 1
• p q ½

24
Consider our example of apples and oranges

• x 2y 12
• u(x,y) xy
• L(x,y, ? ) u(x,y) ?(c-g(x,y)) xy
  ?(12-x-2y)
• y - ? 0 so y ?
• x -2? 0 so x 2 ?
• x2y 12 so 2 ? 2 ? 12 so 4 ? 12 so
  ? 3
• x6, y 3

25
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, ?) vd d ?(24-2v-3d)
• v1 - 3? 0 v 3?-1
• d - 2?0 d 2?
• 2v3d 24 2(3?-1 ) 3(2 ? ) 24
• 6 ? -2 6 ? 24 so 12? 26
• ?13/6
• v 11/2 d 13/3 (utility 28.12)

26
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, ?) 2x2 50000 y2 40000
  ?(1200-xy)
• 4x - ? 0 2y - ? 0 xy 1200
• x ? /4 y ? /2 3? /4 1200
• ?1600
• x 400, y 800

27
Uncertainty and Chance

• In decision making, often you dont 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)

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• 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.

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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
• (Y0) (Tail, Tail)
• (Y1) (Tail, Head), (Head, Tail)
• (Y2) (Head, Head)

30

• 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
• 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

31
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

32
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 Elementary Events
• 2 (1,1)
• 3 (1,2) (2,1)
• 4 (1,3) (2,2) (3,1)
• 5 (1,4) (2,3) (3,2) (4,1)
• 6 (1,5) (2,4) (3,3) (4,2) (5,1)
• 7 (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
• 8 (2,6) (3,5) (4,4) (5,3) (6,2)
• 9 (3,6) (4,5) (5,4) (6,3)
• 10 (4,6) (5,5) (6,4)
• 11 (5,6) (6,5)
• 12 (6,6)

33

• Properties of Probability Mass Functions
  (Discrete probability distributions) for all x
• ?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) 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.

34
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
yF(x)
1/6
1 2 3 4 5 6
x
35
Properties of distribution

• 0? F(x)?1
• F is increasing F(x) ? F(y) if x lty
• As x goes to infinity, F(x) approaches 1
• P(altX ? b) F(b) F(a) if a lt b
• P(Xa) is the jump in the distribution at a

36
Uniform Distribution

• X has a uniform distribution on interval a,b if
  f(x) 1/(b-a) if a ltxltb
• F(x)
• (x-a)/(b-a) if a lt x lt b

37
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

38
Normal Distribution
39
Example - Heights of U.S. Adults

• Female and Male adult heights are well
  approximated by normal distributions
  YFN(63.7,2.5) YMN(69.1,2.6)

Source Statistical Abstract of the U.S. (1992)
40
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)
41
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
mean p and variance 1-p
42
Example 5 (Poisson distribution)
Poisson distribution Discrete probability
distribution for context-independent rare
events Say events occur at some rate ? so that
the expected number of events occuring within
time t is ?t Now break up t into n equal
intervals. Let the probability of an event in a
single interval be p then np ?t The number of
events in interval l is independent of the number
of events in interval l1 The total number of
events in n intervals is described by the
binomial distribution
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44
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

45
Expected Value
The expected value of a random variable X can be
obtained by summation or integration as follows
..Discrete
..Continuous
The expected value is also known as the
distribution mean
46
Variance standard deviation

• Var(X) ? piX(si) E(X)2
• Standard deviation ? sqrt(Var(X))

47
Decision Making Under Uncertainty

• When you buy a car, you dont know whether it
  will be a good one or not.
• We try to capture the goodness of the decision
  with expected utility
• The function u(w) is the utility function over
  wealth or the von Neumann-Morgenstern utility
  function (has to have certain properties)

48

• For our purposes, u(w) is any strictly increasing
  function u0,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.

49
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, Aa1, a2,...,an each of which
  occurs with some known probability pi.

50
Compound Lottery

• With two lotteries (having same set of
  alternatives)
• L1 wi, pi) L2 wi, pi)
• 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

51
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 agents 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 Agents strategy is
the choice of lottery
Risk aversion gt insurance companies
52
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
• Sallyu(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 doesnt care which
  lottery Risk Neutral
• Johnu(a) sqrt(a)
• Lottery 1 ?u(a) p(a) 1sqrt(.5) .7
• Lottery 2 ?u(a) p(a) .5sqrt(0) .5sqrt(1)
  .5
• John prefers lottery 1 Risk averse

53
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 gt 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.

54
Full vs bounded rationality
Bounded rationality How much can I afford to
compute
Full rationality
Descriptive vs. prescriptive theories of bounded
rationality
55
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.

56
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.

57
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
  risk-neutral, 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.

58
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?

59

• 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

60

• If this equals zero, we have px(r-1) (1-p)rx
  0Dividing 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.

61

• 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 pu(w - L - rx x) (1-p)u(w - rx),
• If p r, this means you solve
• max pu(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) pu'(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)

62

• 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 lt L you would
  under-insure. How much you'd underinsure would
  depend on the how much greater r was than p.

63
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)
• The expected utility of gamble 1 is
• ½ 1/2(0) ½ 20 10
• The expected utility of gamble 2
• ½sqrt(136-100) ½ sqrt(225) ½(615) 10.5

64

• 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!

65

• 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).

66

• So u(w) w is risk neutral
• u(w) is risk averse
• u(w) w2 is risk seeking (as large amount of
  money is worth much more than small amounts)

67
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

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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

69
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".

70
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 couldGamble 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 themGamble 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.

71

• 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 relationshipu(1) gt 0.1 u(5)
  0.89 u(1) 0.01 u(0), i.e.0.11 u(1) gt
  0.1 u(5) 0.1 u(0)Adding 0.89 u(0) to
  each side, we get0.11 u(1) 0.89 u(0) gt
  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.

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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.

73
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.

74

• 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.