PPE 110
Lecture on preferences over
uncertainty
• Again we observe that there is more to decision-making
of people than is being captured here, but again we
proceed because of the same reasons.
• In addition to these simplifying assumptions, we will
need to impose a little more structure on the set of
probability trees ∆(X)
• We will assume that whenever the decision-maker is
faced with a pair-wise comparison between two
elements of ∆(X), the decision-maker can say if any one
tree is at least as good as another, or is not at least as
good as the other.
• Since this is a mouthful, we will use the
terminology that, given α and β ∈∆(X)
• α ≽β will imply that α is at least as good as β
• ¬(α ≽β) will imply that α is not at least as good
as β
• Again remember that α and β are both situations
of uncertainty/probability-tree/probability
distribution/lottery (the terms are used
interchangeably), while ∆(X) is the set of all such
situations/probability-trees/probabilitydistributions/lotteries
• Note that we may define α ∼β or “α is exactly as
good as β” by saying this holds if α ≽β and β ≽
α
• Similarly, note that we may define “α is strictly
better than β” or α ≻β by saying this holds if α
≽β and ¬(β ≽ α)
• In other words, if individuals can compare pairwise in order to be able to say “this situation is at
least as good as the other” or this situation is not
at least as good as the other” then that is
tantamount to giving them the ability to say: “the
situations are similarly good” or “one situation is
strictly better.”
• Again a reminder: always remember what
X, ∆(X), α and β are.
• You are used to comparing objects (the
saying this is like “comparing apples to
oranges..”).
• Now we are comparing probability
distributions over objects rather than
comparing the objects themselves
• Here then are the weak order axioms:
• II (a) For all α, β ∈∆(X) either α ≽β or β ≽
α or both. This is known as the
completeness axiom
• (b) For all α ∈∆(X), α ≽ α (this is known as
the reflexivity axiom)
• (c ) For all α, β, γ ∈∆(X), if α ≽ β and β ≽ γ
then α ≽ γ (this is known as the transitivity
axiom). This will imply the strong version
of these property as well: if α ≻ β and β ≻
γ then α ≻ γ
• The first axiom says that the decision maker can
always compare two probability trees, and able
to say one is at least as good as the other. The
rational person never throws their hands up in
the air and says the two cannot be compared.
• This assumes you always have some idea of
what the two trees are about.
• Consider the following two probability
distributions. In the picture, let b=the number of
mountains over 10,000 meters in the solar
system
0.7
$10
=>
0.3
$3b
0.5
$4
0.5
$9
• Most people will not know which to prefer, even
weakly because they do not know what b is.
However, rational choice forces that person to
have to be able to be able to make pair wise
comparisons of the “at least as good as sort.”
• The logic is, even if you do not what b is, use
your best possible guess, and then rank the two
situations as better and worse, or equal.
• The reflexivity axiom is relatively
innocuous, it is imposed for technical
reasons – to complete a mathematical
proof. We will not worry too much about
that axiom
• The transitivity axiom, however, is very
powerful. Can you think of a situation
where it may not be obeyed?
• Here is one example. Suppose a person
prefers more sugar to a cup of tea to less,
up to 5000 grains of sugar.
• A sequence of cups of tea are presented
to this person. The sequence is ordered.
Let the kth cup have k grains of sugar and
be denoted by Ck. Assume there are 5001
such cups – C0, C1, C2, C3,…,…C5000.
• Assume that to be able to distinguish a
cup of tea as containing more sugar than
another, the cup that is preferred must
have at least 10 grains of more sugar.
Otherwise, the differences in sweetness
fall below the threshold of distinction for
the person.
• Now note that C0 ≽C1 ≽C2 ≽C3…
• And yet C5000 ≻C0
• Another famous example involves a context
outside decision theory, but is still very useful to
look at. The issues illustrated by this paradox
translate to our context as well.
• Suppose there are 3 voters (x,y,z), and 3
candidates (A,B,C). Voters rank their choices as
best, middle worst, or as 1,2,3 respectively.
• The way the group makes its decision is by
majority rule: One candidate (say A) is preferred
to another (say B) by the entire group if at least
2 people prefer A over B.
• Consider the following situation
Candidates
A
B
C
x
1
2
3
y
2
3
1
z
3
1
2
Voters
• For the entire group, note that A ≻B, and B ≻C,
but C ≻A.
• Thus, this is a failure of transitivity
• What we are ruling out by imposing transitivity is
the existence of such choice cycles.
• The last set of axioms involve the substitution
axiom and the Archimedean axiom. They will
appear next as III (a) and III (b) respectively.
• III (a) (Substitution Axiom): For any X and
corresponding ∆(X), assume that the
preferences of the individual over elements of
∆(X) obey the axioms I (simplifying axioms) and
II (weak-order axioms) stated previously.
Further, assume that given α, β, γ ∈∆(X), it is
true that α ≻ β. Then for any number x strictly
between 0 and 1, it will be true that
• xα + (1-x)γ ≻ xβ+(1-x)γ
• Here is an illustration of the axiom at work.
Suppose X are all dollar amounts.
• Let the trees be as follows:
β
α
0.3
2
0.7
10
0.2
3
γ
0.4 0.4
2
0.2
7
-9
0.8
4
• Then it is true that for any positive fraction x
x
α
≻
1-x
γ
x
β
1-x
γ
• What this means is that if one situation of
uncertainty α is preferred to another β (for
whatever reason), then a more complex
situation of uncertainty which results in α with
probability x and any third situation of
uncertainty γ with probability 1-x is to be
preferred to another, more complex situation of
uncertainty which gives β with also with
probability x and γ with probability 1-x
• In a sense, the γ cancels out in the reckoning.
This property is similar to that of real numbers,
with the ≻ replaced by >.
• We will assume that the reverse is also
true:
• Given any set of outcomes X, any set of
lotteries/probability trees over C denoted
as Δ(X), any p,q,r∈Δ(X) and any α∈(0,1)
• p ≻ q => αp+(1- α)r ≻αq+(1- α)r and
αp+(1- α)r ≻αq+(1- α)r => p ≻ q
•
•
The substitution axiom is not innocuous. In fact, in an experiment, several of the people who
contributed significantly to the theory we are studying violated it.
The experiment they were given is:
•
•
•
•
•
•
•
•
Experiment 5, part 1.
Choose between the following options. All prizes are in dollars.
Lottery A
27,500 with probability 0.33
24,000 with probability 0.66
0 with probability 0.01
Lottery B
24,000 with probability 1
•
•
•
•
•
•
•
•
•
Experiment 5, part 2
Choose between the following options. All prizes are in dollars.
Lottery C
27,500 with probability 0.33
0 with probability 0.67
Lottery D
24,000 with probability 0.34
0 with probability 0.66
A majority chose B in part 1 and C in part 2. Are these answers consistent with the substitution
axiom?
x
•
Since B ≻ A, this means that y ≻ x, but then D should be ≻ to C
y
• III(b) (Archimedean axiom): For any X and
corresponding ∆(X), assume that the
preferences of the individual over
elements of ∆(X) obey the axioms I and II
stated previously. Further, assume that
given α, β, γ ∈∆(X), it is true that α ≻ β ≻ γ
. Then there exists number m and n strictly
between 0 and 1, such that
• mα + (1-m)γ ≻ β ≻ nα+(1-n)γ
• This is illustrated next
• Let the trees be as follows (to begin with, they are arbitrarily
chosen):
β
α
0.5
2
0.5
≻
10
1/3
3
γ
≻
1/3 1/3
2
7
0.2
-9
0.8
-8
• Then there exists numbers m and n (this
property will not hold for all numbers, just
some) such that the following preferences
will hold true.
m
α
1-m
γ
≻
β
≻
n
α
1-n
γ
• This implies that if we take a lot of the
best, and a little bit of the worst, then that
is better than the middle. Likewise, If we
take a lot of the worst, and a little bit of the
best, then that is worse than the middle.
• These axioms may sound innocuous and
obvious. But they are all we need to state
the following theorem.
• Theorem (informally stated): Given any X and
∆(X), and any elements α, β ∈∆(X), where α and
β are probability distributions (or probability
trees) over X, then α ≻ β iff the expected utility of
α greater than the expected utility of β. And α ∼
β iff the expected utility of α equals the expected
utility of β.
• What is the expected utility of α or β? It is simply
the expected value of the probability tree
replacing the outcomes with the utility of the
outcomes.
• You might say, this is obvious – why bother with
writing down the axioms? Here is why:
• Expected utility is the cornerstone of decision theory. It is
used everywhere: actuarial science, finance, computer
science, economics, indeed, whenever a rational agent
is presumed to be in a position to maximize utility, and
the situation can be quantified.
• The theorem states that whenever it is justified to use
expected utility, what we are really assuming is that the
agent is acting as if they obey these axioms. Thus, we
have a handle on the implicit cognitive processes and
value judgments that manifest themselves in such
behavior. And wherever they do not end up being
expected utility maximizers (such as in a lot of
psychology experiments), their failure to use expected
utility is directly attributable to their failure to obey one or
more of the axioms we have written down.
• The student might wonder: why use expected
utility instead of expected value?
• To illustrate the necessity of using utility, we look
at a particular bar game.
• A popular bar is charging customers for the right
to play a particular game.
• In a computer simulation, a fair coin is tossed for
the person playing the game.
• The tossing continues until the first time a head
occurs. If the first time this occurs is on the nth
toss, the person playing is given $2n
1
2
2
3
4
8
n
2n
And so on….
• How much would you be willing to play this
game? (What is the maximum possible
cover charge you would be willing to pay?)
• Small winnings are possible with high
probability. Large winnings are possible
with low probability.
• A rational way to add up the worth of
playing this game should be to weigh each
reward with the possibility of that reward
occurring.
• In other words, use expected value.
•
•
•
•
And yet, the expected value is:
1(1/2)+2(1/4)+….+(1/ 2n)(2n )…..
=1+1+1…..=∞
Thus, expected value tells us we/you should be
willing to pay any amount necessary to be able
to play this game.
• Clearly this is absurd. How do we get some
insight of what might be going on in the typical
behavior of people faced with such a
hypothetical choice, who usually just are willing
to pay a very small amount to be able to play
this game?
• If we assume that people value each dollar
worth than the previous one, we have a possible
clue to their behavior.
• In other words, every additional dollar still gives
people an “utility”, but this utility is less than what
the previous dollar contributed.
• The 100th dollar is worth more than the 101st, the
millionth dollar is worth more than the millionth
and oneth, and so on.
• Here is what the utility function would then look
like.
Utility of $
$
• One utility function that has such a
property is u(x)=√x
• Then the expected utility is what should be
calculated, not expected value.
• This expected utility =
(1/2)(√2)+(1/4)(√4)+(1/8)(√8)+(1/ 2n)(√ 2n)
= (1/√2)+(1/√4)+(1/√8)+…(1/√ 2n)
Which is the same as a geometric series
with first term (1/√2) and constant
multiplicative term also (1/√2)
• Which yields a value much more consistent
with what people are willing to pay
(1/ 2 )
 2. 5
1
1
(1/ 2 )
• There really is no one utility function that is
able to explain human behavior in all
contexts.
• Sometimes it is one kind, sometimes
other. Part of the goal of the cognitive
sciences is to provide some idea of when
one kind is appropriate, and which type
that is.
• It is a qualified goal, but perhaps better
than assuming all kinds are equally likely
at any possible juncture.