Transcript (1) and

Rational Choice
Sociology
Lecture 3
The Measurement of Utility and
Subjective Probability
Why measurement matters?
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Only if there is possible to find out, what p, u and eu (u×p) values are for
specific actors in the specific empirical situations, rational choice theory can be
applied as empirical theory
However, measurement makes sense only if there is something to measure
For an actor choosing under risk, this “something is here”, only if her
preferences and probabilistic beliefs satisfy the set of axioms discussed below
Only if these axioms are satisfied, the expected utility function EU(x) is defined
which ascribes to each choice alternative xi the utility index ui.
As for probability, one (objectivistic) version of EU theory maintains that EU
function is defined only if actors know empirical probabilities (disclosed by
statistical data) of the conditions that co-determine (together with choices) the
outcomes. If one accepts this version, then one doesn’t need to measure
probabilistic expectations (subjective probabilities), but is bound to treat the
situations where probabilistic expectations are not grounded in the statistical
data as those of the choice under uncertainty
Another (subjectivist or Bayesian) version maintains that for the EU function
to be defined, it is sufficient for actors to have consistent subjective
probabilities. This version elaborates the methods for the measurement of
subjective probabilities
Axioms of expected utility theory (or measurability conditions
for utility and subjective probability)
(1) Reflexivity
 (2) Completeness
 (3) Transitivity
These axioms are identical with those for rational choice under
certainty
(4) Continuity condition (modified version):
If a prospect yi includes two outcomes such that one of them is
the worst outcome (xw) and another one the best outcome
(xb), then for all remaining outcomes xij there exist
probabilities p and 1-p such that the actor is indifferent
between the outcome xij and the prospect yk consisting of xb
with probability p and xw with probability 1-p
xij~(pxb; (1-p)xw)
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Continuity axiom for choice under risk:
explanation by example
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Action1
x11
x12=
xw
x13~
(p x15;
1-p
x12)
x14
x15=
xb
x16
Action2
x21
x22
x23
x24
x25
x26
(5) Increasing preference with increasing
probability
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If there are two prospects yi and yk that differ only by the probabilities of the
outcomes that they include, then yi>yk only if probabilities of better outcomes
in yi is greater than probabilities of better outcomes in yk
If r11 > r12, r21 > r22, and r11 ~ r21; r12 ~ r22, then
EU (Action1) > EU (Action2) if and only if p11 > p21
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Action 1; yi
Action 2; yk
r11
p11
r21
p21
C2
>
>
r12
p12
r22
p22
(6) Independence axiom
(addition of supplementary alternatives doesn’t reverse
the order of preferences between initial alternatives)
If xa>xb, then (pxa; (1-p)xc)) > (pxb;
(1-p)xc))
If 1 bottle of beer > 1 apple, then the lottery (prospect) where there is probability 0,7 to
win 1 bottle of bear and probability 0,3 to win 1 pear > lottery (prospect) where
there is probability 0,7 to win 1 apple and probability 0,3 to win 1 pear
Expected utility function
If preferences and probabilistic beliefs satisfy
axioms 1-6 then for all alternatives in the
feasible set the expected utility function EU(x)
is defined that ascribes to all of them utility
indexes that can be found by specific
measurement procedures
Measurement of utility: NeumannMorgenstern procedure
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(1) Let the actor to identify the best and the worst alternatives
(rb; rw) in her feasible set
(2) Assume U(rb)=1; U(rw)=0
(3) Build lottery where there probability p to win rb and
probability 1-p to win rw
(4) Changing the values of p and 1-p, find the point of
indifference between the remaining alternatives ri in the
feasible set
(5) Take the probability value p in the indifference point as
raw utility index for ri
(6) (Optional) Transform raw utility indexes according to
formula u’= a + b×u
Measurement of utility: Neuman-Morgenstern
procedure – an example (1)
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Jonas is asked to say which of the following
3 day tourist tours he considers as the best
and the worst options, if he could get one of
them for free
(1)To Paris;
(2) To Berlin;
(3) To Cracow;
(4) To Moscow
Measurement of utility: Neuman-Morgenstern
procedure – an example (2)
Jonas says his best option is Paris, worst option Cracow
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Build lottery: there is probability 0,6 to win travel to Paris and probability 0,4 to
win travel to Cracow.
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Propose choice: to participate in lottery or take travel to Berlin for sure
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If Jonas chooses Berlin for sure, increase the odds of winning Paris; If Jonas prefers
lottery decrease them. Propose the choice again. Continue till you find the point of
indifference between Berlin for sure and participation in the lottery.
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Assume that Jonas is indifferent between Berlin for sure and lottery with 0,4
probability winning Paris and 0,6 getting Cracow; and Jonas is indifferent between
Moscow for sure and lottery with 0,8 probability of winning Paris, and 0,2 getting
Cracow.
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Raw utility indexes are: Paris 1, Moscow 0,8, Berlin 0,4, Cracow 0.
Reasoning: Because Jonas only slightly prefers Berlin over Cracow, it is enough for
him small probability to win Paris to renounce Berlin for sure (if he doesn’t wins
Paris, he looses little - Berlin is not much more better than Cracow). However,
because he strongly prefers Moscow over Cracow, he must be almost sure to win
Paris to refuse from Moscow for sure.
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Measurement of utility: Neuman-Morgenstern
procedure – an example (3)
Why transform utility indexes?
Sometimes it may be counterintuitive to asume u(rw)=0
Cracow is very nice place, so why u(Cracow)=0?
We can use u’= 10 + 100×u or u’’= 1000 + 20×u or some other linear transformation y=a+bx.
Then u’(Paris)=110, u’(Moscow)=80, u’(Berlin)=50, u’(Cracow)=10
Then u’’(Paris)=1020, u’’(Moscow)=1016, u’’(Berlin)=1008, u’’(Cracow)=1000
These transformations are possible, because/if axioms (1)-(6) satisfied
As far as they are satisfied, utility is variable measurable at the interval scale level;
for all such measurements linear positive transformation y=a+bx is possible
Such transformation is similar to recalculation of temperature according to Fahrenheit
into temperature according Celsius
Celsius to Fahrenheit[°F] = [°C] × 9⁄5 + 32; Fahrenheit to Celsius
[°C] = ([°F] − 32) × 5⁄9
On the Fahrenheit scale, the freezing point of water is 32 degrees Fahrenheit (°F) and the boiling point 212 °F; no unconventional
zero in interval scale
-273 C absolute zero in Kelvin scale
Measurement of utility: Ramsey
procedure
Shortcoming of Neuman-Morgenstern procedure:
the lottery with best and worst outcomes as prizes can be not credible
Ramsey procedure (more flexible):
(1) Let the actor choose: to take prize (some amount of money or some
valuable thing) k for sure or participate in the lottery, where there is
probability p to win the outcome ri, the utility of which we are measuring,
and probability 1-p to get nothing
(2) Change the odds of winning until the indifference point will be found
between taking k for sure and participating in lottery to win ri
(3) Indifference point is described by equation:
u(k)=u{(ri)×p; 0×1-p} or u(k)=u(ri)×p
(4) Assume u(k)=1 util (1u);
(5) Then u(ri)=1u/p
(6) (Optional) Transform raw utility indexes according to formula u’= a + b×u
Measurement of utility: Ramsey
procedure (an example)
Jonas is asked to choose to be kissed by Madonna for sure, or participate in the
lottery where there is probability p to win the 3 days tour to Nida, or get
nothing (probability 1-p)
Find the point of indifference between kiss of Madonna for sure (if utility of the
kiss of Madonna is chosen as measuring rod) and Nida with probability p.
Assume Jonas is indifferent between them at p for Nida= 0,4.
Then u(Nida)=u(kiss of Madonna)/0,4= 1u/0,4=2,5u
Say, Jonas is indifferent between kiss of Madonna for sure and lottery with
probability 0,2 to win an autograph of Vytautas Landsbergis and probability
0,8 to get nothing
u(Vytautas Landsbergis)=u(kiss of Madonna)/0,2 = 1u/0,2=5u
Say John is indifferent between kiss of Madona for sure and lottery with
probability 0,1 to win 1000 Euro and probability 0,9 to get nothing.
u(1000 Euro)= u(kiss of Madonna)/0,1= 1u/0,1=10u
The „measuring rod“ can by any thing or service of value for person, whose
utilities are measured (e.g. sum of money, bottle of wine, autograph of
Landsbergis etc.)
Measurement of utility: Ramsey
procedure (an example)
According our measurement, u(Nida)=2,5; u(autograph of Vytautas
Landsbergis)=5
Does it makes sense to say that the autograph of Vytautas Landsbergis is twice as
valuable for Jonas than travel to Nida?
No, because/if u is variable measurable at the interval scale level.
If u(a)=1000; u(b)=1200; u(c)=2000; u(d)=2200, then u(b)-u(a)=u(d)-u(c), but
u(c)#u(a)×2
Cp. If temperature increased from +20 to +40 C this does not mean that it became
twice as hot as it was
For the same reason (measurability just on the interval level), the utility indexes
measured by the procedures described above are interpersonally
incomparable. Even if we take as our measurement unit not u(kiss of
Madonna), but say u(100 litas), there is no way to find how much u(100 litas)
for Jonas is more or less than u(100 litas) for Petras.
Measurement of subjective
probability (1)
Three concepts of probability:
(1) Empirical (or statistical) probability: relative frequency of an event (say,
P(A)) in the total population or set of events (or limit of the relative
frequency in the infinite sequence of events). Given empirical concept of
probability, it doesn’t make sense to ask about the probability of the
unique events. What is probability that nuclear reactor in the Ignalina
nuclear plant will explode? There was only one (?) event of such type!
What is probability of World War III in the XXI century? Cp: what is
probability of at least 1 snowy day in October in Vilnius?
(2) Logical probability: degree of confirmation of the hypothesis by available
data.
(3) Subjective (personal) probability: degree of belief or confidence Pa(s),
where “a” refers to the person, and “s” refers to the statement
Some decision analysts accept only (1) and (2). So, “expected utility” and
“subjective expected utility” are distinguished, depending by what kind of
probability the utility indexes u are weighted
Measurement of subjective
probability (2)
Conditions of possibility:
preferences of an actor should satisfy axioms 1-6 for choice under risk;
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beliefs of an actor doesn’t violate axioms of the mathematical theory of
probabilities (remember textbook of statistics by Vydas Cekanavicius and
Gediminas Murauskas)
There are several slightly different procedures. This is one of them:
(1)
propose to the actor to make a bet (to pay money too bookmaker who arranges
the bet) on the truth value of the statement s. At first, the researcher bets with n
(best of all, monetary prize or some other divisible valuable good) on the ~s (i.e.
he bets that s is false), and asks to make the actor make her maximal bet m on
the s (or vice versa; is not important who bets on what)
(2)
After actor makes her maximal bet, calculate her Pa=m/n, where n>m, and
n=m+g, where g is the prize.
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Measurement of subjective
probability: examples
(1) Take the statement: it will rain on the October 20th, 2014 (s).
I bet 10 litas that it is false, i.e. that there will be no rain (~s). What is your bet that s is true?
If you bet, e.g. 1 Euro then you will get 10 Euro (net 9 Euro ) if it will rain on October
5th, and loose 1 litas, if there will be no rain.
Why participate in the bets?
Answer: Why do not use the occasion to earn some money, if you really believe that there
will rain!?
If your maximal bet is 2 Euro, then your Pa(s)=2/10=0,2.
(2) I bet 100 litas that Israel will attack Iran in 2015 (s). So, if there will be no attack, you
will receive 100 Euro (net 100-X, where X is your bet) on January 1st, 2016. What is your
bet on (~s)? If your maximum bet is 35 Euro, then your Pa(~s)=0,35, and your Pa(s)=10,35 = 0,65.
Why should be n>m? Why not bet, say 15 Euro against 10 Euro for s (there
will be rain on the October 20th, 2014)?
If you bet 15 Euro on s, you have loss both in the case s is false, and in the case s is true.
If s true, then you will win 10 Euro, but you have paid 15, and your loss is 5 Euro. If s is
false, your loss is 15 Euro.
Attitudes to risk. Expected utility and expected value (1)
Important advantage of the procedures for the utility
measurement as described above is that they take into account
the attitudes of actors towards risk or degree of their
pessimism/optimism. If utility indexes are found out by one of
these methods, they include information, how risk-averse or
risk-loving the actors are.
If they are introduced by equating money and utility, or time lost
and utility, then one needs to correct the utility indexes for the
risk attitudes of risk. We can postulate that u(100 litas) with
probability 0,5 is 50 utiles, only if actor’s attitudes to risk are
neutral. For this case, expected utility (EU) =expected value
(EV).
If EU>EV, then actor is risk-prone or risk-loving (optimist). If
EU<EV, then actor is risk-averse.
Attitudes to risk. Expected utility and expected value
(2)
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Operational definition of the attitudes to risk, including the measurement of
the degree of this attitude
An actor is risk neutral if she is indifferent between taking m for sure and
participating in the lottery where there is probability 0,5 to win 2 m, and
probability 0,5 to get nothing. If the actor in such situation prefers lottery,
she is risk-loving. If she takes m, she is risk averse.
By increasing or decreasing probability of winning 2m it is possible to find
how much the actor is risk-averse or risk loving.
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To remind: if utility indexes are not postulated, but measured according
Neumann-Morgenstern or Ramsey, then one doesn’t needs to bother about
attitudes to risk: information about them is already in them: they reflect both
actor’s order of preferences and her attitudes to risk
Dominance concept
In some situations, to find the best action one can use
the dominance rule instead of EU or SEU
maximization rule
Definition:
Action Vi strongly dominates over action Vj if all
outcomes in the prospect Vi are better than outcomes
in the prospect Vj.
Action Vi weakly dominates over action Vj if at least
one outcome in the prospect of Vi is better than
outcome in the prospect Vj,and remaining outcomes
are as good as those in Vj.
Dominance rule
If actor has in her feasible set an action that dominates (strongly or weakly) over other actions, then she
should choose this action if probabilities of the outcomes are unconditional (same in each column). In
such case, the application of EU maximization rule is redundant. When probabilities are
unconditional, the outcomes of the actions are caused by causes that are not influenced by actions of
an actor (e.g. whether, market conditions e.g). Usually, dominance rule and EU maximization rule
lead to the same conclusion, but sometimes the situations happen when they contradict. See Norkus
Z. (2003):Newcombo problema ir amerikietiškas klausimas”, Problemos, , 63,
19-34 (not in the obligatory readings)
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V1
U11=-3
p11=0,45
u12=10
p12=0,15
u13=90
p13=0,05
u14=11
p14=0,2
U15=9
p15=0,15
V2
u11=-15
p11=0,45
u12=0
u13=55
u14=7
u15=-4
p12=0,15
p13=0,05
p14=0.2
p15=0,15
V3
u11=-20
u12=-8
u13=12
u14=-9
u15=-8
p11=0,45
p12=0,15
p13=0,05
p14=0,2
p15=0,15
Conditions of the applicability of dominance rule
Dominance rule cannot be applied instead of EU maximization rule if probabilities of
outcomes are conditional (not the same in each column)
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V2
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u11=10
p11 =0,5
u12=1
p12=0,5
EU (V1)=5,5
u21=9
p21=0,9
u22= 0
p22=0,1
EU(V2)=8,1
Conditional probability and
conditional expected utility
If the utility of an action is weighted by conditional
probability, then the actor is maximizing conditional
expected utility
 Conditional probability of the outcome is P(rij/Vj)
(probability of the outcome rij given Vi)
 In the case of unconditional probability P(rij/Vji)=Prij;
in the case of conditional probability P(rij/Vij)>Prij or
P(rij/Vij)<Prij,
i.e. the action increases or decreases the probability of
the state of world associated with the action.
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