Rational Choice Rational Choice SociologySociology

Lecture 3Lecture 3

The Measurement of Utility and The Measurement of Utility and Subjective ProbabilitySubjective Probability

Why measurementWhy measurement matters matters? ? Only if there is possible to find out, what p, u and eu Only if there is possible to find out, what p, u and eu (u(u×p) values are×p) values are for for

specific actors in the specific empirical situationsspecific actors in the specific empirical situations, rational choice theory can be , rational choice theory can be applied as empirical theoryapplied as empirical theory

However, measurement makes sense only if there is something to measureHowever, measurement makes sense only if there is something to measure For an actor choosing under risk, this “something is here”, only if her For an actor choosing under risk, this “something is here”, only if her

preferences and probabilistic beliefs satisfy the set preferences and probabilistic beliefs satisfy the set of of axioms discussed belowaxioms discussed below Only if these axioms are satisfied, the expected utility function EU(x) is defined Only if these axioms are satisfied, the expected utility function EU(x) is defined

which ascribes to each choice alternative xwhich ascribes to each choice alternative xii the utility index u the utility index uii.. As for probability, one (objectivistic) version of EU theory maintains that EU As for probability, one (objectivistic) version of EU theory maintains that EU

function is defined only if actors know empiricfunction is defined only if actors know empiricalal probabilities (dis probabilities (disclosedclosed by by statistical data) of the conditions that co-determine (together with choices) the 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 outcomes. If one accepts this version, then one doesn’t need to measure probabilistic expectations (subjective probabilities), but is bound to treat the probabilistic expectations (subjective probabilities), but is bound to treat the situations where probabilistic expectations are not grounded in the statistical situations where probabilistic expectations are not grounded in the statistical datdataa as those of the choice under uncertainty as those of the choice under uncertainty

Another (subjectivist or Bayesian) version maintains that for the EU function Another (subjectivist or Bayesian) version maintains that for the EU function to be defined, it is sufficient for actors to have consistent subjective to be defined, it is sufficient for actors to have consistent subjective probabilitprobabilities. This version elaborates ies. This version elaborates the methods for the measurement of the methods for the measurement of subjective probabilitiessubjective probabilities

Axioms of expected utility theory (or measurability conditions Axioms of expected utility theory (or measurability conditions for utility and subjective probability)for utility and subjective probability)

(1) Reflexivity (1) Reflexivity (2) Completeness (2) Completeness (3) Transitivity (3) Transitivity

These axioms are identical with those for rational choice under These axioms are identical with those for rational choice under certaintycertainty

(4) Continuity condition(4) Continuity condition (modified version) (modified version)::

If a prospect yIf a prospect yii includes two outcomes such that one of them is includes two outcomes such that one of them is

the worst outcome (xthe worst outcome (xww) and) and another one the best outcome (xanother one the best outcome (xbb), ),

then for all remaining outcomes xthen for all remaining outcomes xijij there exist probabilities p there exist probabilities p

and 1-p such that the actor is indifferent between the outcome and 1-p such that the actor is indifferent between the outcome xxijij and the prospect y and the prospect yk k consisting ofconsisting of xxbb with probability p and x with probability p and xww

with probability 1-pwith probability 1-p

xxijij~(px~(pxbb; (1-p)x; (1-p)xww))

Continuity axiomContinuity axiom for choice under risk for choice under risk: : explanation by exampleexplanation by example

CC11 CC22 CC33 CC44 CC55 CC66

Action1Action1 xx1111 xx1212==

xxww

xx1313~~

(p (p xx1515;;

1-p 1-p xx1212))

xx1414 xx1515==

xxbb

xx1616

Action2Action2 xx2121 xx2222 xx2323 xx2424 xx2525 xx2626

(5) (5) Increasing preference with increasing Increasing preference with increasing probabilityprobability

If there are two prospects yIf there are two prospects yii and y and ykk that differ only by the probabilities of the that differ only by the probabilities of the

outcomes that they include, then youtcomes that they include, then yii>y>ykk only only if probabilities of better outcomif probabilities of better outcomEEs s

in yin yii is greater than probabilities of better outcomes in y is greater than probabilities of better outcomes in ykk

If rIf r1111 > > rr1212, r, r2121 >> r r2222, and r, and r1111 ~~ r r2121; r; r1212 ~~ r r2222, then, then

EU (Action1) EU (Action1) > EU (Action2) if and only if p> EU (Action2) if and only if p1111 > p > p2121

CC11 CC22

Action 1Action 1; y; yii rr1111 >>

pp1111

rr1212

pp1212

Action 2Action 2; y; ykk rr2121 >>

pp2121

rr2222

pp2222

(6) (6) Independence axiomIndependence axiom(addition(addition of supplementary alternatives doesn’t reverse of supplementary alternatives doesn’t reverse

the order of preferencesthe order of preferences between initial alternatives) between initial alternatives)

If xIf xaa>x>xbb, then (px, then (pxaa; (1-p)x; (1-p)xcc)))) > (px > (pxbb; ;

(1-p)x(1-p)xcc))))

If 1 bottle of bear > If 1 bottle of bear > 1 1 apple, then apple, then the the lotterylottery (prospect) (prospect) where where there is probability there is probability 0,70,7 to to win 1win 1 bottle of bear bottle of bear and probability and probability 0,30,3 to win 1 to win 1 pear > lottery pear > lottery (prospect) (prospect) where where there is probability there is probability 0,7 0,7 to win 1 to win 1 apple apple and probabilityand probability 0,3 0,3 to win 1 to win 1 pearpear

Expected utility functionExpected utility function

If preferences and probabilistic beliefs satisfy If preferences and probabilistic beliefs satisfy axioms 1-axioms 1-66 then for all alternatives in the then for all alternatives in the feasible set the expected utility function EU(x) feasible set the expected utility function EU(x) is defined that ascribes to all of them utility is defined that ascribes to all of them utility indexes that can be found by specific indexes that can be found by specific measurement proceduresmeasurement procedures

Measurement of utility: NeumanMeasurement of utility: Neumannn--Morgenstern procedureMorgenstern procedure

(1) Let the actor to identify the best and the worst alternatives (1) Let the actor to identify the best and the worst alternatives (r(rbb; r; rww) in her feasible set) in her feasible set

(2) Assume U((2) Assume U(rrbb)=1; U(r)=1; U(rww)=0)=0 ((33) Build lottery where there probability p to win r) Build lottery where there probability p to win rbb and and

probability 1-p to win rprobability 1-p to win rww

((44) Changing the values of p and 1-p, find the point of ) Changing the values of p and 1-p, find the point of indifference between the remaining alternatives rindifference between the remaining alternatives r ii in the in the feasible setfeasible set

((55) Take the probability value ) Take the probability value pp in the indifference point as in the indifference point as raw utility index forraw utility index for r rii

((66) (Optional) Transform raw utility indexes according to ) (Optional) Transform raw utility indexes according to formula u’formula u’= a + b= a + b×u×u

Measurement of utility: Neuman-Morgenstern Measurement of utility: Neuman-Morgenstern procedureprocedure – an example (1) – an example (1)

Jonas is asked to say which of the following Jonas is asked to say which of the following 3 day tourist tours he considers as the best 3 day tourist tours he considers as the best and the worst options, if he could get one of and the worst options, if he could get one of them for freethem for free

(1)To Paris; (1)To Paris; (2) To Berlin; (2) To Berlin; (3) To Cracow; (3) To Cracow; (4) To Moscow(4) To Moscow

Measurement of utility: Neuman-Morgenstern Measurement of utility: Neuman-Morgenstern procedureprocedure – an example (2) – an example (2)

Jonas says his best option is Paris, worst option CracowJonas 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 Build lottery: there is probability 0,6 to win travel to Paris and probability 0,4 to

win travel towin travel to Cracow Cracow.. Propose choice: to participate in lottery or take travel to Berlin for surePropose 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 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 lottery decrease them. Propose the choice again. Continue till you find the point of indifference between Berlin for sure and participation in the lottery.indifference between Berlin for sure and participation in the lottery.

Assume that Jonas is indifferent between Berlin for sure and lottery with 0,4 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 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 Moscow for sure and lottery with 0,8 probability of winning Paris, and 0,2 getting Cracow.Cracow.

Raw utility indexes are: Paris 1, Moscow 0,8, Berlin 0,4, Cracow 0.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 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 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, 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 because he strongly prefers Moscow over Cracow, he must be almost sure to win Paris to refuse from Moscow for sure.Paris to refuse from Moscow for sure.

Measurement of utility: Neuman-Morgenstern Measurement of utility: Neuman-Morgenstern procedureprocedure – an example (3) – an example (3)

Why transform utility indexes?Why transform utility indexes?Sometimes it may be counterintuitive to asume u(rSometimes it may be counterintuitive to asume u(rww)=0)=0Cracow is very nice place, so why u(Cracow)=0?Cracow is very nice place, so why u(Cracow)=0?We can use We can use u’u’= 10 + 100= 10 + 100×u or ×u or u’u’’= 1000 + 20’= 1000 + 20×u or some other linear transformation y=a+bx.×u or some other linear transformation y=a+bx.Then u’(Paris)=110, u’(Moscow)=80, u’(Berlin)=50, u’(Cracow)=10Then u’(Paris)=110, u’(Moscow)=80, u’(Berlin)=50, u’(Cracow)=10 Then u’’(Paris)=1020, u’’(Moscow)=1016, u’’(Berlin)=1008, u’’(Cracow)=1000Then u’’(Paris)=1020, u’’(Moscow)=1016, u’’(Berlin)=1008, u’’(Cracow)=1000These transformations are possible, because/if axioms (1)-(6)These transformations are possible, because/if axioms (1)-(6) satisfied satisfiedAs far as theyAs far as they are satisfied are satisfied, utility is variable measurable at the interval scale level;, utility is variable measurable at the interval scale level;for all such measurements linear positive transformation y=a+bx is possiblefor all such measurements linear positive transformation y=a+bx is possible

Such transformation is similar to recalculation of temperature according to Fahrenheit Such transformation is similar to recalculation of temperature according to Fahrenheit into temperature according Celsiusinto temperature according Celsius

Celsius to Celsius to FahrenheitFahrenheit[°F] = [°C] × 9⁄5 + 32[°F] = [°C] × 9⁄5 + 32; Fahrenheit to Celsius ; Fahrenheit to Celsius [°C] = ([°F] − 32) × 5⁄9 [°C] = ([°F] − 32) × 5⁄9

On the Fahrenheit scale, the On the Fahrenheit scale, the freezing point of of water is 32 degrees Fahrenheit (°F) and the is 32 degrees Fahrenheit (°F) and the boiling point 212 °F; no unconventional 212 °F; no unconventional zero in interval scale zero in interval scale

-273 C absolute zero-273 C absolute zero in Kelvin scale in Kelvin scale

Measurement of utility: Ramsey Measurement of utility: Ramsey procedure procedure

Shortcoming of Neuman-Morgenstern procedure: Shortcoming of Neuman-Morgenstern procedure:

the lottery with best and worst outcomes as prizes can be not crediblethe lottery with best and worst outcomes as prizes can be not credible

Ramsey procedure (more flexible):Ramsey procedure (more flexible):

(1) Let the actor choose: to take prize (some amount of money or some valuable (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 thing) k for sure or participate in the lottery, where there is probability p to win the outcome rthe outcome rii, the utility of which we are measuring, and probability 1-p to get , the utility of which we are measuring, and probability 1-p to get

nothingnothing

(2) Change the odds of winning until the indifference point will be found(2) Change the odds of winning until the indifference point will be found between between taking k for sure and participating in lottery to win taking k for sure and participating in lottery to win rrii

(3) Indifference point is described by equation:(3) Indifference point is described by equation:

u(k)=u(k)=uu{(r{(rii))×p; 0×1-p} or ×p; 0×1-p} or u(k)=u(ru(k)=u(rii))×p×p

(4) Assume u(k)=1 util (1u);(4) Assume u(k)=1 util (1u);

(5) Then (5) Then u(ru(rii)=1u/p)=1u/p

(6) (6) (Optional) Transform raw utility indexes according to formula u’(Optional) Transform raw utility indexes according to formula u’= a + b= a + b×u×u

Measurement of utility: Ramsey Measurement of utility: Ramsey procedure (an example)procedure (an example)

Jonas is asked to choose to take 100 litasJonas is asked to choose to take 100 litas (or to be kissed by Madonna, or to get an (or to be kissed by Madonna, or to get an autograph of Vytautas Landsbergis or etc.), or participate in the lottery where 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 there is probability p to win the 3 days tour to Nida, or get nothing (probability 1-p)(probability 1-p)

Find the point of indifference between kiss of Madonna for sure Find the point of indifference between kiss of Madonna for sure (if utility of the (if utility of the kiss of Madonna is chosen as measuring rod) kiss of Madonna is chosen as measuring rod) and and Nida Nida with probability p. with probability p. Assume Jonas is indifferent between them at p for Nida= 0,4.Assume Jonas is indifferent between them at p for Nida= 0,4.

Then u(Nida)=u(kiss of Madonna)/0,4= 1u/0,4=2,5uThen u(Nida)=u(kiss of Madonna)/0,4= 1u/0,4=2,5uSay, Say, Jonas is indifferent between kiss of Madonna for sure and lottery with Jonas is indifferent between kiss of Madonna for sure and lottery with

probability 0,2 to win an autograph of Vytautas Landsbergis and probability probability 0,2 to win an autograph of Vytautas Landsbergis and probability 0,8 to get nothing0,8 to get nothing

u(Vytautas Landsbergis)=u(kiss of Madonna)/0,u(Vytautas Landsbergis)=u(kiss of Madonna)/0,2 2 = 1u/0,2=5u = 1u/0,2=5u

Say John is indifferent between kiss of Madona for sure and lottery with 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. 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=10uu(10000 litu)= u(kiss of Madonna)/0,1= 1u/0,1=10u

Measurement of utility: Ramsey Measurement of utility: Ramsey procedure (an example)procedure (an example)

According our measurement, u(According our measurement, u(NidaNida)=2,5; u(autograph of)=2,5; u(autograph of Vytautas Vytautas LandsbergisLandsbergis)=5)=5

Does it makes sense to say that the autograph of Does it makes sense to say that the autograph of Vytautas Landsbergis Vytautas Landsbergis is twice as is twice as valuable for Jonas thanvaluable for Jonas than travel to Nida? travel to Nida?

No, becauseNo, because/if/if u is variable measurable at the interval scale level. 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 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(c)#u(a)×2#u(a)×2

Cp. If temperature increased from +20 to +40 C this does not mean that it Cp. If temperature increased from +20 to +40 C this does not mean that it became twice as hot as it was became twice as hot as it was

For the same reason (measurability just on the interval level), For the same reason (measurability just on the interval level), the utility indexes the utility indexes measured by the procedures described above measured by the procedures described above are interpersonally are interpersonally incomparable. Even if we take as our measurement unit not u(kiss of 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) 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.for Jonas is more or less than u(100 litas) for Petras.

Measurement of subjective Measurement of subjective probability (1)probability (1)

Three concepts of probability:Three concepts of probability:(1)(1) Empirical (or statistical) probability: relative frequency of an event (say, 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 P(A)) in the total population or set of events (or limit of the relative frequency in the infinite sequence of events). frequency in the infinite sequence of events). Given empirical concept of Given empirical concept of probability, it dprobability, it doesn’t make sense to ask about the probability of the oesn’t make sense to ask about the probability of the unique events. What is probability that nuclear reactor in the Ignalina unique events. What is probability that nuclear reactor in the Ignalina nuclear plant will explode? There was only one (?) event of such type! 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 What is probability of World War III in the XXI century? Cp: what is probability of at least probability of at least 1 1 snowy day in October in Vilnius?snowy day in October in Vilnius?

(2) (2) Logical probability: degree of confirmation of the hypothesis by available Logical probability: degree of confirmation of the hypothesis by available data. data.

(3) Subjective (personal) probability: degree of belief or confidence P(3) Subjective (personal) probability: degree of belief or confidence Paa(s), (s), where “a” refers to the person, and “s”where “a” refers to the person, and “s” refers to the statementrefers to the statement

Some decision analysts accept only (1) and (2). So, “expected utility” and Some decision analysts accept only (1) and (2). So, “expected utility” and “subjective expected utility” are distinguished, depending by what kind of “subjective expected utility” are distinguished, depending by what kind of probability the utility indexes u are weightedprobability the utility indexes u are weighted

Measurement of subjective Measurement of subjective probability (2)probability (2)

Conditions of possibility: Conditions of possibility: preferences of an actor should satisfy axioms 1-6 for choice under risk; 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 beliefs of an actor doesn’t violate axioms of the mathematical theory of

probabilities (remember textbook of statistics by Vydas Cekanavicius and probabilities (remember textbook of statistics by Vydas Cekanavicius and Gediminas Murauskas)Gediminas Murauskas)

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

(1)(1) propose to the actor to propose to the actor to makemake a bet (to pay money too bookmaker who arranges 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 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 (best of all, monetary prize or some other divisible valuable good) on the ~~s (i.e. s (i.e. he bets that s is false), and asks to make the actor make her maximal bet m on 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)the s (or vice versa; is not important who bets on what)

(2)(2) After actor makes her maximal bet, calculate her PAfter actor makes her maximal bet, calculate her Paa=m/n, where n=m/n, where n>m, and >m, and

n=m+g, where g is the prize.n=m+g, where g is the prize.

Measurement of subjective Measurement of subjective probability: examplesprobability: examples

(1) Take the statement: it will rain on the October 20(1) Take the statement: it will rain on the October 20 thth, 20, 2014 (s)14 (s)..I bet 10 litas that it is false, i.e. that there will be no rainI bet 10 litas that it is false, i.e. that there will be no rain ( (~~s)s). What is your bet that. What is your bet that s s is true? is true?

If you bet, e.g. 1 litas, then you will get 10 litas, if it will rain on October 5If you bet, e.g. 1 litas, then you will get 10 litas, if it will rain on October 5 thth, and loose 1 , and loose 1 litas, if there will be no rain.litas, if there will be no rain.

Why participate in the bets?Why participate in the bets?Answer: Why do not use the occasion to earn some money, if you really believe that there Answer: Why do not use the occasion to earn some money, if you really believe that there

will rain!?will rain!?

If your maximal bet is 2 litas, then your PIf your maximal bet is 2 litas, then your Paa(s)=2/10=0,2.(s)=2/10=0,2.

(2) I bet 100 litas that Israel will attack Iran in 201(2) I bet 100 litas that Israel will attack Iran in 20155 (s). So, if there will be no attack, you will (s). So, if there will be no attack, you will receive 100 litas on January 1receive 100 litas on January 1stst, 201, 20166. What is your bet on (. What is your bet on (~~s)? If your maximum bet is s)? If your maximum bet is 35 litas, then your P35 litas, then your Paa((~~s)=0,35, and your Ps)=0,35, and your Paa(s)=(s)=1- 1- 0,350,35 = 0,65.= 0,65.

Why should be nWhy should be n>m? Why not bet, say 15 litas against 10 for s (there will be >m? Why not bet, say 15 litas against 10 for s (there will be rain on the October 20th, 20rain on the October 20th, 201414)?)?

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 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, 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.your loss is 15 litas.

Attitudes to risk. Expected utility and expected value (1)Attitudes to risk. Expected utility and expected value (1)

Important advantage of the procedures for the utility Important advantage of the procedures for the utility measurement as demeasurement as desscribed above is that they take into account cribed above is that they take into account the attitudes of actors towards risk or degree of their the attitudes of actors towards risk or degree of their pessimism/optimism. If utility indexes are pessimism/optimism. If utility indexes are found out found out by one of by one of these methods, they include information, how risk-averse or these methods, they include information, how risk-averse or risk-loving the actors are. risk-loving the actors are.

If they are introduced by equating money and utility, or time lost If they are introduced by equating money and utility, or time lost and utility, then one needs to correct the utility indexes for the and utility, then one needs to correct the utility indexes for the risk attitudes of risk. We can postulate that u(100 litas) with 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 probability 0,5 is 50 utiles, only if actor’s attitudes to risk are neutral. For this case, expected utility (EU) =expected value neutral. For this case, expected utility (EU) =expected value (EV).(EV).

If EU>EV, then actor is risk-prone or risk-loving (optimist). If If EU>EV, then actor is risk-prone or risk-loving (optimist). If EU<EV, then actor is risk-averseEU<EV, then actor is risk-averse..

Attitudes to risk. Expected utility and expected value Attitudes to risk. Expected utility and expected value (2)(2)

Operational definition of the attitudes to risk, including the measurement Operational definition of the attitudes to risk, including the measurement of the degree of this attitudeof the degree of this attitude

An actor is risk neutral if she is indifferent between taking m for sure and 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 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, 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. 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 By increasing or decreasing probability of winning 2m it is possible to find

how much the actor is risk-averse or risk lovinghow much the actor is risk-averse or risk loving.. To remind: if utility indexes are not postulated, but measured according To remind: if utility indexes are not postulated, but measured according

Neumann-Morgenstern or Ramsey, then one doesn’t needs to bother about 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 attitudes to risk: information about them is already in them: they reflect both actor’s order of preferences and her attitudes to riskactor’s order of preferences and her attitudes to risk

Dominance conceptDominance concept

In some situations, to find the best action one can use In some situations, to find the best action one can use the dominance rule instead of EU or SEU the dominance rule instead of EU or SEU maximization rulemaximization rule

Definition:Definition:Action VAction Vii strongly dominates over action V strongly dominates over action Vjj if all if all

outcomes in the prospect Voutcomes in the prospect Vii are better than outcomes are better than outcomes in the prospect Vin the prospect Vjj..

Action VAction Vii weakly dominates over action V weakly dominates over action Vj j if at least if at least one outcome in the prospect of Vone outcome in the prospect of Vii is better than is better than outcome in the prospect Voutcome in the prospect Vjj,and remaining outcomes ,and remaining outcomes are as good as those in Vare as good as those in Vjj..

Dominance ruleDominance rule

If actor has in her feasible set an action that dominates (strongly or weakly) over other actions, then she If actor has in her feasible set an action that dominates (strongly or weakly) over other actions, then she should choose this action if should choose this action if probabilities of the outcomes are unconditionalprobabilities of the outcomes are unconditional (same in each column). In (same in each column). In such case, the application of EU maximization rule is redundant. such case, the application of EU maximization rule is redundant. When probabilities are When probabilities are unconditional, the outcomes of the actions are caused by causes that are not influenced by actions of 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.gan actor (e.g. whether, market conditions e.g). Usually, dominance rule and EU maximization rule ). Usually, dominance rule and EU maximization rule lead to the same conclusion, but sometimes the situations happen when they contradict. See lead to the same conclusion, but sometimes the situations happen when they contradict. See NorkusNorkus Z. (2003):Newcombo problema ir Z. (2003):Newcombo problema ir amerikietiškas klausimasamerikietiškas klausimas”, Problemos, , 63, ”, Problemos, , 63, 19-3419-34 (not in the obligatory readings) (not in the obligatory readings)

CC11 CC22 CC33 CC44 CC55

VV11UU1111=-3=-3

pp1111=0,45=0,45

uu1212=10=10

pp1212=0,15=0,15

uu1313=90=90

pp1313=0,05=0,05

uu1414=11=11

pp1414=0,2=0,2

UU1515=9=9

pp1515=0,15=0,15

VV22uu1111=-15=-15

pp1111=0,45=0,45

uu1212=0=0

pp1212=0,15=0,15

uu1313=55=55

pp1313=0,05=0,05

uu1414=7=7

pp1414=0.2=0.2

uu1515=-4=-4

pp1515=0,15=0,15

VV33uu1111=-20=-20

pp1111=0,45=0,45

uu1212=-8=-8

pp1212=0,15=0,15

uu1313=12=12

pp1313=0,05=0,05

uu1414==--99

pp1414=0,2=0,2

uu1515==--88

pp1515=0,15=0,15

Conditions of the applicability of dominance ruleConditions of the applicability of dominance ruleDominance rule cannot be applied instead of EU maximization rule if probabilities of Dominance rule cannot be applied instead of EU maximization rule if probabilities of

outcomes are outcomes are conditional (not the same in each column)conditional (not the same in each column)

C1C1 C2C2

V1V1 uu1111=10=10

pp1111 =0,5 =0,5

uu1212=1=1

pp1212=0,5=0,5 EU (VEU (V11)=5,5)=5,5

V2V2 uu2121=9=9

pp2121=0,9=0,9

uu2222= 0= 0

pp2222=0,1=0,1 EU(VEU(V22)=8,1)=8,1

Conditional probability and Conditional probability and conditional expected utilityconditional expected utility

If the utility of an action is weighted by conditional If the utility of an action is weighted by conditional probability, then the actor is maximizing conditional probability, then the actor is maximizing conditional expected utilityexpected utility

Conditional probability of the outcome is Conditional probability of the outcome is P(rP(rijij/V/Vjj)) (probability of the outcome r(probability of the outcome r ij ij givengiven VVii))

In the case of unconditional probability P(rIn the case of unconditional probability P(r ijij/V/Vjiji)=Pr)=Prijij; ; in the case of conditional probability P(rin the case of conditional probability P(r ijij/V/Vijij)>Pr)>Prijij or or P(rP(rijij/V/Vijij)<Pr)<Prijij,,

i.e. the action increases or decreases the probability of i.e. the action increases or decreases the probability of the state of world associated with the action.the state of world associated with the action.