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PPE 110

Lecture on preferences over

uncertainty



• Again we observe that there is more to decision-makingof people than is being captured here, but again weproceed 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 isfaced with a pair-wise comparison between twoelements of ∆(X), the decision-maker can say if any onetree is at least as good as another, or is not at least asgood 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 situationsof uncertainty/probability-tree/probabilitydistribution/lottery (the terms are used

interchangeably), while ∆(X) is the set of all suchsituations/probability-trees/probability-distributions/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 strictlybetter than β” or α ≻β by saying this holds if α

≽β and ¬(β ≽ α)• In other words, if individuals can compare pair-wise in order to be able to say “this situation is atleast as good as the other” or this situation is not

at least as good as the other” then that istantamount to giving them the ability to say: “thesituations are similarly good” or “one situation isstrictly better.”



• Again a reminder: always remember whatX, ∆(X), α and β are.

• You are used to comparing objects (thesaying this is like “comparing apples tooranges..”).

• Now we are comparing probabilitydistributions over objects rather thancomparing 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 thecompleteness axiom

• (b) For all α∈

∆(X), α≽

α (this is known asthe reflexivity axiom)

• (c ) For all α, β, γ ∈∆(X), if α ≽ β and β ≽ γthen α ≽ γ (this is known as the transitivityaxiom). This will imply the strong versionof 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 ableto say one is at least as good as the other. Therational person never throws their hands up inthe 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



$3b

=>0.7 0.30.5

0.5

$10$4 $9



• Most people will not know which to prefer, evenweakly 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 wisecomparisons of the “at least as good as sort.”

• The logic is, even if you do not what b is, useyour best possible guess, and then rank the two

situations as better and worse, or equal.



• The reflexivity axiom is relativelyinnocuous, it is imposed for technicalreasons – to complete a mathematical

proof. We will not worry too much aboutthat axiom

• The transitivity axiom, however, is very

powerful. Can you think of a situationwhere 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 andbe denoted by Ck. Assume there are 5001such cups – C

0, C

1, C

2, C

3,…,…C

5000.



• Assume that to be able to distinguish acup of tea as containing more sugar thananother, the cup that is preferred musthave 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 contextoutside decision theory, but is still very useful to

look at. The issues illustrated by this paradoxtranslate to our context as well.

• Suppose there are 3 voters (x,y,z), and 3candidates (A,B,C). Voters rank their choices asbest, middle worst, or as 1,2,3 respectively.

• The way the group makes its decision is bymajority rule: One candidate (say A) is preferred

to another (say B) by the entire group if at least2 people prefer A over B.

• Consider the following situation



Candidates

Voters

A B C

x 1 2 3

y 2 3 1

z 3 1 2



• 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 isthe existence of such choice cycles.

• The last set of axioms involve the substitution

axiom and the Archimedean axiom. They willappear next as III (a) and III (b) respectively.



• III (a) (Substitution Axiom): For any X andcorresponding ∆(X), assume that thepreferences of the individual over elements of ∆(X) obey the axioms I (simplifying axioms) andII (weak-order axioms) stated previously.Further, assume that given α, β, γ ∈∆(X), it istrue that α ≻ β. Then for any number x strictlybetween 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:

α β γ

210 3 2 7 -9 4

0.3 0.7 0.2 0.4 0.4 0.2 0.8



• Then it is true that for any positive fraction x

γ γ

x 1-x x 1-x

α

≻

β



• What this means is that if one situation of uncertainty α is preferred to another β (for whatever reason), then a more complexsituation of uncertainty which results in α withprobability x and any third situation of

uncertainty γ with probability 1-x is to bepreferred to another, more complex situation of uncertainty which gives β with also withprobability 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 whocontributed 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 y

• Since B ≻ A, this means that y≻ x, but then D should be ≻ to C



• III(b) (Archimedean axiom): For any X and

corresponding ∆(X), assume that thepreferences of the individual over elements of ∆(X) obey the axioms I and IIstated previously. Further, assume thatgiven α, β, γ ∈∆(X), it is true that α ≻ β ≻ γ. Then there exists number m and n strictlybetween 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):

γ

210 3 2 7 -9 -8

0.5 0.5 1/3 1/3 1/3 0.2 0.8

β α

≻



• Then there exists numbers m and n (this

property will not hold for all numbers, justsome) such that the following preferenceswill hold true.

1-m 1-nnm

γ γ α α

β ≻



• This implies that if we take a lot of thebest, and a little bit of the worst, then thatis better than the middle. Likewise, If wetake a lot of the worst, and a little bit of thebest, then that is worse than the middle.

• These axioms may sound innocuous andobvious. But they are all we need to statethe following theorem.



• Theorem (informally stated): Given any X and∆(X), and any elements α, β ∈∆(X), where α and

β are probability distributions (or probabilitytrees) over X, then α ≻ β iff the expected utilityof 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 treereplacing 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 isused everywhere: actuarial science, finance, computer

science, economics, indeed, whenever a rational agentis presumed to be in a position to maximize utility, andthe situation can be quantified.

• The theorem states that whenever it is justified to useexpected utility, what we are really assuming is that theagent is acting as if they obey these axioms. Thus, wehave a handle on the implicit cognitive processes andvalue judgments that manifest themselves in suchbehavior. And wherever they do not end up being

expected utility maximizers (such as in a lot of psychology experiments), their failure to use expectedutility is directly attributable to their failure to obey one or more of the axioms we have written down.



• The student might wonder: why use expectedutility instead of expected value?

• To illustrate the necessity of using utility, we lookat 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 headoccurs. If the first time this occurs is on the nthtoss, the person playing is given $2n



1

2

8

2n

4

2

3

n

And so on….



• How much would you be willing to play thisgame? (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 eachreward with the possibility of that rewardoccurring.

• 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 bewilling to pay any amount necessary to be able

to play this game.• Clearly this is absurd. How do we get someinsight of what might be going on in the typicalbehavior of people faced with such a

hypothetical choice, who usually just are willingto pay a very small amount to be able to playthis game?



• If we assume that people value each dollar worth than the previous one, we have a possibleclue to their behavior.

• In other words, every additional dollar still givespeople 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 millionthand oneth, and so on.

• Here is what the utility function would then looklike.



Utility of $

$



• One utility function that has such aproperty is u(x)=√x

• Then the expected utility is what should becalculated, 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 constantmultiplicative term also (1/√2)



• Which yields a value much more consistentwith what people are willing to pay

5.2

)2(1/

11

)2(1/



• There really is no one utility function that is

able to explain human behavior in allcontexts.

• Sometimes it is one kind, sometimesother. Part of the goal of the cognitivesciences is to provide some idea of whenone kind is appropriate, and which typethat is.

• It is a qualified goal, but perhaps better than assuming all kinds are equally likelyat any possible juncture.

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