Rational Choice Sociology

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Rational Choice Sociology. Lecture 3 The Measurement of Utility and Subjective Probability. Why measurement matters ?.

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### RationalChoiceSociology

Lecture 3

The Measurement of Utility and Subjective Probability

Why measurement matters?
• Onlyifthereispossible to findout, what p, u andeu(u×p) values areforspecificactorsinthespecificempiricalsituations, 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. Thisversionelaboratesthe 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 (modifiedversion):

If a prospect yi includes two outcomes such that one of them is the worst outcome (xw) andanother 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 ykconsisting ofxb with probability p and xw with probability 1-p

xij~(pxb; (1-p)xw)

(5) Increasing preference with increasing probability
• 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

(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 bear > 1 apple, then thelottery (prospect) wherethereisprobability0,7 to win 1 bottle of bear andprobability0,3 to win 1 pear > lottery (prospect) where thereisprobability0,7 to win 1 apple andprobability 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: Neumann-Morgenstern procedure
• (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
• 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
• Jonas says his best option is Paris, worst option Cracow
• Build lottery: there is probability 0,6 to win travel to Paris and probability 0,4 to win travel toCracow.
• Propose choice: to participate in lottery or take travel to Berlin for sure
• 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.
• 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.
• 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.

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

OntheFahrenheitscale, thefreezingpoint of wateris 32 degreesFahrenheit (°F) andtheboilingpoint 212 °F; nounconventionalzeroinintervalscale

-273 C absolute zeroinKelvinscale

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 foundbetweentaking k forsureandparticipatinginlottery to winri

(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) Transformrawutilityindexesaccording to formula u’= a + b×u

Measurement of utility: Ramsey procedure (an example)

Jonas is asked to choose to take 100 litas(or to be kissed by Madonna, or to get an autograph of Vytautas Landsbergis or etc.), 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 (ifutilityofthekissofMadonnaischosenasmeasuringrod) and Nida with probability p. Assume Jonas is indifferent between them at p for Nida= 0,4.

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 10000 litas and probability 0,9 to get nothing.

u(10000 litu)= u(kiss of Madonna)/0,1= 1u/0,1=10u

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 thantravel 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. Iftemperatureincreasedfrom +20 to +40 C thisdoesnotmeanthat it becametwiceashotas it was

Forthesamereason (measurability just ontheintervallevel), the utility indexes measuredbytheproceduresdescribedaboveare 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). Givenempiricalconcept 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;
• beliefs of an actor doesn’t violate axioms of the mathematical theory of probabilities (remember textbook of statistics by VydasCekanavicius and Gediminas Murauskas)

There are several slightly different procedures. This is one of them:

• 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)
• After actor makes her maximal bet, calculate her Pa=m/n, where n>m, and n=m+g, where g is the prize.
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 litas, then you will get 10 litas, 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!?

(2) I bet 100 litas that Israel will attack Iran in 2015 (s). So, if there will be no attack, you will receive 100 litas on January 1st, 2016. What is your bet on (~s)? If your maximum bet is 35 litas, then your Pa(~s)=0,35, and your Pa(s)=1- 0,35= 0,65.

Why should be n>m? Why not bet, say 15 litas against 10 for s (there will be rain on the October 20th, 2014)?

If you bet 15 litas 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 litas, but you have paid 15, and your loss is 5 litas. If s is false, your loss is 15 litas.

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 foundoutby 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)
• 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.
• 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)

Conditions of the applicability of dominance ruleDominance rule cannot be applied instead of EU maximization rule if probabilities of outcomes are conditional (not the same in each column)

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 givenVi)
• 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.