Transcript Lecture 2 - Penn Arts & Sciences

Lecture 4
PPE 110
• In most situations when people make choices,
the outcomes are random.
• For example, if you buy a stock, it may go up or
down. If you decide to support a soccer team, it
may finish the season in a number of positions.
If you woo another person, your efforts may lead
to a number of different outcomes.
• We have had a quick acquaintanceship with
probability theory, which is a way of quantifying
uncertainty.
• Now we proceed to modeling choice by adding
in decisions
• It is very convenient to have such situations of uncertainty depicted
by probability trees.
• In the rest of this note, we will write down some rules that we claim
should govern these probability trees from the perspective of a
rational decision-maker.
• At this point in time, we will not bother to investigate how the
probabilities are arrived at – we will assume they are there for the
decision maker. Either someone provided them to her/him (a bookie,
from a lottery, empirical observation) or s/he came up with them on
her/his own.
• The ultimate aim is to present a succinct way to capture rational
human behavior when faced with situations of uncertainty.
• This theory will not be perfect – we will point out many
shortcomings. However, it is the best we have, and in some sense,
incorporates some desirable properties of decision making, such as
consistency.
• Further, in experiments, this theory seems to do better than other
theories. This of course does not mean that it cannot be improved.
• In most real life choices the probabilities are not given,
but rather are a subjective embodiment of the nature of
the uncertainty.
• Thus when you took this class, you did not know what
fraction of students had received what grade, but at the
same time used what information you did have, about
the class and yourself, to come up with something that
rational choice theory claims looks like a probability
distribution -- we will not cover this part of the theory, but
this is known as rational choice with subjective
probabilities.
• Do people always manage to come up with probabilities
that make sense when they are faced with uncertainty?
• The following exercise illustrates this issue in very stark terms.
Subjects were asked about their choices in the following context.
• Part 1: An urn contains 300 colored marbles. 100 of the marbles
are red and the remaining are some mixture of blue and green, but
the number of blue and green marbles cannot be equal. You reach
into the urn and select a marble at random. You will receive $1000 if
the marble you select is of a specific color, which you have specified
before drawing the marble. Would you rather that marble be: (R)
Red (B) Blue
• Part 2 Now, suppose you were to again draw a marble at random
from the same urn, and win $1000 if the marble is not of a specific
color. That color will be announced by you before drawing the
marble. Would you rather that color be: (B) Blue (R) Red
• A majority of people choose red in part 1 and red in part 2.
• Let us analyze their set of choices. Are they consistent with rational
choice? Why or why not? Why do you think they acted the way they
did?
• Since the # of marbles cannot be equal,
there are 2 possibilities, written in
decreasing order: GRB and BGR
• The fact that people chose R in part 1
immediately implies that they believe R>B,
so the order is in effect GRB.
• In that case, in part 2, they want to choose
the color which is least likely to be picked.
So, they should pick B.
•
•
•
•
•
What are these rules? We will first assume that the individual is aware of
the set of outcomes that can occur after the uncertainty resolves (stock
goes up, soccer team wins championship, wooing results in rude rebuffal
etc.)
Then for any situation, we will label the set of outcomes associated with that
outcome X, and the set of all probability trees that end with outcomes in X
as ∆(X). Question: If X has only two elements, how many elements does
∆(X) have?
We will also assume that the individual is a consequentalist. Who is a
consequentalist? It is someone who cares only about the consequences.
This is a well defined term in Philosophy
(http://en.wikipedia.org/wiki/Consequentialism).
Roughly speaking, a consequentalist believes that the end result is what
matters, not the path that got there. The validity/worthwhileness of a path is
to be judged according to its outcome. This may seem amoral – but a
consequentalist would say that if there are particular values attached to the
paths themselves, then the utility of the outcomes should be modified to
reflect the worth of travelling by a particular path.
In other words, describe the set of outcomes, X and then put a probability
distribution over it – if two situations result in the same probability
distribution, then they are identical, no matter in what other aspects they
differ.
• This philosophy may seem very amoral. Loosely
translated, it seems to suggest that the end is what
matters not the means. For instance, should the two
situations depicted in the next slide be seen as the
same, just because the outcomes are the same?
• In that slide an individual has a choice between stealing
and gambling $10,000. The punishment for stealing is a
fine of $10,000, and the potential for loss in gambling is
also $10,000. The probability of ‘success’ and `failure’ is
the same in both situations.
•
You might say: the consequences of being caught stealing and of losing at gambling
are different. Gambling is a private vice, while stealing is a crime. But a
consequentalist would then say: the set of consequences have not been described
properly. One needs to augment the set by describing not just the dollar losses but
also the emotions associated with such an outcome. This is shown next
• In other words, the consequences were
not the same after all.
• But if the consequences are
unambiguously the same, and do not
depend on the interpretation of context,
are people consequentalists?
• The next couple of slides ask you to
choose between a pair of options in
different situations
• Experiment 1 Part 1
• Imagine that the US is preparing for the outbreak
of an unusual Asian disease, which is expected
to kill 600 people. Two alternative programs to
combat the disease have been proposed.
Assume that the exact scientific estimates of the
consequences of the programs are as follows:
• A. If program A is adopted, 200 people will be
saved.
• B. If program B is adopted, there is one-third
probability that 600 people will be saved and a
two-thirds probability that no people will be
saved.
• Experiment 1 part 2
• Imagine that the US is preparing for the outbreak
of an unusual Asian disease, which is expected
to kill 600 people. Two alternative programs to
combat the disease have been proposed.
Assume that the exact scientific estimates of the
consequences of the programs are as follows:
• C. Program C is adopted, 400 people will die.
• D. If program D is adopted, there is a one-third
probability nobody will die and a two-thirds
probability that 600 people will die.
• In the next slide, we see what people chose and
the way this tree can be drawn.
• Here is another example related to
consequentalism:
• Experiment 2, part 1
• Imagine that you have decided to see a
play and paid the admission price of $10
per ticket. As you enter the theater, you
discover that you have lost the ticket. The
seat was not marked and the ticket cannot
be recovered.
• Would you pay $10 for another ticket?
• A: Yes B: No
• Experiment 2, part 2
• Imagine that you have decided to see a
play. As you enter the theater, you
discover that you have lost a $10 bill,
before buying the ticket.
• Would you still pay $10 for a ticket to see
the play?
• A: Yes B: No
• There are problems with assuming consequentalism – yet it is not
clear what those problems are due to.
• Further, if people are pointed out their inconsistencies, will they
continue to behave as before, or will they become more
consequentalist?
• Even if they continue to behave as before, we want the theory we
are developing to serve as a benchmark level of reference, a theory
that embeds some notion of consistency, so that deviations from it
may be clearly apparent.
• We now proceed to some of the more concrete rules.
• We write these rules down one by one. The first category of rules
are what may be called “simplifying axioms.”
• The second category of rules may be called “weak order axioms”
• The third category of rules may be called “Substitution axiom and
Archimedean axiom”
• Simplifying rules
• I (i) Zero probability limbs in a tree may be pruned.
• This rule means that if a state may be conceived of, and
yet be assigned zero probability, then that state may
safely be ignored in the analysis of the decision.
• Example: You are thinking of joining a multi-national
company. Your expected salary is (a) less than <$200,00
with probability 0.8 (b) between $200,000 and $500,000
with probability 0.2 (c ) equal to the salary of the CEO.
• The last state is possible but so unlikely that it may be
assigned probability zero.
0.8
0.2
a
b
<=>
0
c
0.8
a
0.2
b
• This axiom is relatively uncontroversial.
• I(ii): If there is a situation of uncertainty which
leads to another, then the probability of any
outcome that results at the end may be reached
by multiplying the probabilities.
• Example: You are thinking of joining a multinational company. You could join and find your
expected salary is (i) is $200,00 with prob 0.8 (b)
or with probability 0.2 you find your salary
random: with probability 0.5 you may make
$300,000 or with probability 0.5 you may make
$100,000
0.8
=>
0.2
0.8
0.1
0.1
200,000
200000
0.5
300000
0.5
100000
300000
100000
• I (iii) If an outcome occurs more than once
in a situation of uncertainty, then the
probability of that outcome is equal to the
sum of the probabilities associated with
that outcome.
• Example: In the multinational company
you are thinking of joining, you could be
stationed in San Francisco with probability
0.3, New York with probability 0.4, San
Francisco with probability 0.1 and London
with probability 0.2.
• This situation, shown on the left, is
equivalent to the one on the right
0.3
0.4
0.1 0.2
=>
0.4
0.4
0.2
SF
NY
SF
LO
SF
NY
LO
• I(iv) Numerical outcomes that accrue along
different stages of a probability tree may be
added along each path
• Example: Again you find yourself thinking about
joining that MNC. You could join and find your
expected salary is:
• (i) $80,000 with prob 0.8, but if this happens
then with probability 0.4 you will be allowed to try
and get an additional project where you may get
a bonus of $40,000
• Or with probability 0.2 you may make $300,000
with probability 0.5 or $100,000 with probability
0.5
0.8
=>
0.2
0.32
0.48
0.1
0.1
300000
80,000
120000
0.6 0.5
0.4
0.5
0
40000
300000
100000
80000
100000
• Do people behave as these axioms
suggest? Lets do some more experiments
• Experiment 3 part 1
• Which of the following options do you
prefer?
• A: 25% chance to win $30
• B: 20% chance to win $45
• Experiment 3 part 2
• Consider the following 2 stage game. In
the first stage, there is a 75% chance to
end the game without winning anything
and a 25% chance to move into the
second stage. If you move into the second
stage you have a choice between:
• C: A sure win of $30
• D: 80% chance to win $45
•
C
•
Violation of compounding and coalescing as A=C and B=D
• Experiment 4 part 1
• Choose between:
• E: 25% chance to win $240 and 75%
chance to lose $760
• F. 25% chance to win $250 and 75%
chance to lose $750
•
•
•
•
•
Experiment 4, part 2
Two problems:
Choose between
A: A sure gain of $240
B: 25% chance to gain $1000 and 75%
chance to gain nothing
• Also, choose between
• C:A sure loss of $750
• D: 75% chance to lose $1000 and 25%
chance to lose nothing