2. CHOICE UNDER UNCERTAINTY. Simple and Compound Lotteries

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1 2 CHOICE UNDER UNCERTAINTY Ref: MWG Chapter 6 Subjective Expected Utility Theory Elements of decision under uncertainty Under uncertainty, the DM is forced, in effect, to gamble A right decision consists in the choice of the best possible bet, not simply in whether it is won or lost after the fact Two essential characteristics: A choice must be made among various possible courses of actions 2 This choice or sequence of choices will ultimately lead to some consequence, but DM cannot be sure in advance what this consequence will be, because it depends not only on his or her choice or choices but on an unpredictable event Simple and Compound Lotteries X = (finite) set of outcomes (what DM cares about) Lset of simple lotteries (prob distributions on X with finite support) A lottery L in L is a fn L : X R, thatsatisfies following 2 properties: L (x) for every x X 2 P x X L (x) = Examples: Take X = {, 9,,,,, 2,, 9, } A fair coin is flipped and the individual wins $ if heads, wins nothing if tails ½ /2 if x {, } L (x) = if x/ {, } 2

2 2 Placing a bet of $ on black on a (European) roulette wheel L 2 (x) = 8/37 if x = 9/37 if x = if x/ {, } 3Apackof52playingcardsisshuffled Win $2 if the top card is an Ace, lose $5 if the top card is the Queen of Spades otherwise no change in wealth L 3 = /3 if x = 2 47/52 if x = /52 if x = 5 if x/ { 5,, 2} 3 4 A balanced die is rolled Win $ if number on top is even & win nothing otherwise L 4 (x) L (x) = ½ /2 if x {, } if x/ {, } A compound lottery is a two-stage lottery in which the outcomes from the first-stage randomization are themselves lotteries Formally, a compound lottery is a fn C : L R, thatsatisfies the following 2properties: C (L) for every L L, with strict inequality for only finitely many lotteries L 2 P L L C (L) = 4

3 Example: A fair coinisflipped and the individual then plays out L 2 if heads and L 3 if tails ½ /2 if L {L2,L C (L) = 3 } if L/ {L 2,L 3 } REDUCTION: Multiply through st -stage prob to reduce a compound lottery to a one-stage lottery Ie, if α,,α n are the prob of the possible 2 nd -stage lotteries L,,L n then the reduction is the lottery α L + α 2 L α n L n Example cont: Reduction of C (L) is lottery R =(/2) L 2 +(/2) L 3, ie, R (x) = /26 if x =2 9/37 if x = 47/4 if x = 9/74 if x = /4 if x = 5 otherwise 5 Consequentialism: assume individual indifferent between any compound lottery and the associated reduced lottery We can also see that the set of lotteries is a mixture space If L & L are lotteries in L then for any α in [, ], αl +( α) L is the lottery L (x) =αl (x)+( α) L (x) To see that L is indeed a lottery, notice that: L (x) =αl (x)+( α) L (x), for every x X 2 P x X L (x) = P x X [αl (x)+( α) L (x)] = α P x X L (x)+( α) P x X L (x) =α +( α) = Further notation: for any x X, let δ x denote the degenerate lottery ½ (x, ) L, ie if y = x δ x (y) = if y 6= x Hence for any lottery L in L we have L = P x X L (x) δ x 6

4 States-of-nature model X = setofoutcomes(whatdmcaresabout) Lset of simple lotteries (probability distributions on X with finite support) S = set of states (uncertain factors beyond the control of the DM) A = set of acts (what the DM controls or chooses) % defined over acts 7 Formally, we will take A to be the set of functions a : S L with finite range Thatis,anyacta maybeexpressedinaform [L,E ; ; L n,e n ] where {E,,E n } forms a finite partition of the state space An act that maps each state to a degenerate lottery may be viewed as a purely subjectively uncertain act [δ x,e ; ; δ xn,e n ] 8

5 Set of acts is also a mixture space For any pair of acts a and a, αa +( α) a is the act a : S L, for which a (s) =αa (s)+( α) a (s) State: complete specification of the past, present and future configuration oftheworld,exceptforthosedetailsthatarepartofthedm sactions Often can analyze situation in terms of a finite partition of the state space {E,,E n } Set of mutually exclusive and exhaustive events 9 Example: Jones faces choice between current employment or doing MBA Jones has the choice between two possible acts : leave and stay Three outcomes x = stay in current employment 2 =incur costs of undertaking MBA, but after graduating only get job similar to the one he had before 3 M = incur costs of undertaking MBA and after graduating land extremely well-paying and exciting job The event E in which Jones obtains the high-paying job if he has chosen to leave leave (s) = ½ δm ½ if s E δ m if s/ E stay (s) = δx if s E if s/ E δ x

6 Whether we have leave % stay or stay % leave would seem to depend on two separate considerations: How good Jones feels the chances of obtaining the high-paying job would have to be to make it worth his while to leave his current employment; 2 How good in his opinion the chances of obtaining the high-paying job actually are Jones s answers to questions of type quantify his personal preference for x relative to and M Jones s answers to questions of type 2 quantify his personal judgement concerning the relative strengths of the factors that favor and oppose certain events If he behaves reasonably then he should choose the solution of the problem which is consistent with his personal preference and his personal judgement How might Jones quantify his preference for x relative to and M? 2 How might Jones quantify his judgement for likelihood of event E? We know δ M  δ x  δ m,sosetu(δ M ):=and U (δ m ):= Let u x be the unique probability for which u x δ M +( u x ) δ m δ x Let π E be the unique probability for which δm on E π E δ M +( π E ) δ m on S E and set Hence we have δ m U (stay) : = u x U (δ M )+( u x ) U (δ m ) U (leave) : = π E U (δ M )+( π E ) U (δ m ) leave % stay π E u x 2

7 Principles of Choice Behavior Axioms for % For ease of exposition, suppose there exists two outcomes M and m, such that δ M  δ m and for all x X, δ M % δ x % δ m Ordering Axiom % is complete and transitive Archimedean Axiom For any three acts, a, a and a,forwhicha  a  a, there exists numbers α and β, bothin(, ) such that αa +( α) a  a  βa +( β) a The Archimedean axiom rules out a lexicographic preference for certainty Thus it plays a similar role to that played by the continuity axiom in decision making under certainty: ruling out discontinuous jumps in the preference relation 3 Example (MWG p7) Suppose M = beautiful & uneventful trip by car x = staying at home m = death by car crash Set a := δ M, a := δ x and a := δ m If a  a  a then there exists sufficiently large α<, such that αa +( α) a  a ie [M,α; m, α]  [x, ] REMARK From people s revealed behavior, axiom is quite sound empirically 4

8 Independence Axiom: For all a, a,a A and λ (, ) we have a % a λa +( λ)a % λa +( λ)a Embodies a substitution principle and a reduction of compound lotteries principle Eg Akbar has free international round-trip ticket and is planning to use it for his winter vacation His preferred destinations, Hawaii and Madrid, are sold out So he makes a reservation for Cancun He can also choose to be wait-listed for Hawaii or Madrid, but not both If he decides to get on the waiting list for Hawaii, then he has a fifty percent chance of ultimately getting a reservation otherwise he will go to Cancun If he decides to get on the waiting list for Madrid, however, the situation is completely different First, the probability of getting a reservation for Madrid is only /4 rather than /2 and secondly, to get on this waiting list, he has to drop his reservation for Cancun If he doesn t get a reservation for Madrid, there is a 2/3 chance he can get back his reservation for Cancun, but there is a /3 chance he will only be able to get a reservation for Toronto 5 X = {C[ancun], H[awii], M[adrid], T[oronto]} Suppose for Akbar, δ M Â δ H Â δ C Â δ T Hence by independence 2 δ M + 2 δ C Â 2 δ H + 2 δ C Furthermore preference between 2 δ H + 2 δ C and 4 δ M is determined by preference between Since 4 δ M + 3 µ δ C + 3 δ T δ H and 2 δ M + 2 δ T µ 2 3 δ C + 3 δ T = 4 δ M + 4 δ T + 2 δ C = µ 2 2 δ M + 2 δ T + 2 δ C 6

9 Further notation: for any act a, anylotteryl and any event E, letl E a denote the act a where ½ a L if s E (s) = a (s) if s/ E State-Independence Axiom: For any pair of lotteries L and L,and any event E, such that (δ M ) E (δ m ) Â δ m, L % L L E a % L Ea The unconditional preference between any pair of lotteries is the same as the preference between those lotteries conditional on any non-null event having obtained 7 A function U : L R is affine, if for all L, L L and α [, ], U (αl +( α) L )=αu (L)+( α) U (L ) Fact: If U : L R is affine, then there exists a function u : X R such that for all L =[x,p ; ; x m,p m ], U (L) = P m i= p iu (x i ) 8

10 The big result: THEOREM: Suppose there exists two outcomes M and m, such that δ M  δ m and for all x X, δ M % δ x % δ m Then the following are equivalent: The preference relation % satisfies the Ordering, Archimedean, Independence and State-Independence Axioms 2 The preference relation % admits a subjective expected utility representation That is, there exists a unique probability measure π and a unique affine function U : L [, ], with U (δ m )=and U (δ M )=, such that for any pair of acts a =[L,E ; ; L n,e n ] & a =[L,E ; ; L n,e n ] a % a nx Xn π (E i ) U (L i ) i= j= π Ej U L j 9 Proof of Theorem: (2) implies () (Exercise) () implies (2) Preliminary Results The axioms imply that properties < exhibits the following Mixture Monotonicity For any a, a A, such that a  a, and any α (, ), a  αa +( α) a  a ProofofMixtureMonotonicity:By independence a = αa +( α) a  αa +( α) a and αa +( α) a  αa +( α) a = a 2

11 Mixture Solvability For any a, a,a A, forwhicha  a  a,there exists a unique α (, ) such that αa +( α) a a Proof of Mixture Solvability: Consider the sets α + = {α [, ] : αa +( α) a  a},and α = {α [, ] : a  αa +( α) a } From Mixture Monotonicity it follows that both α + and α are non-empty, non-intersecting and connected subsets of [, ] Moreover, the greatest lower bound for α + equals the least upper bound for α Denote this number by ᾱ 2 Thus it must be the case that one of the following hold: (i) ᾱ α + and ᾱ/ α, or (ii) ᾱ/ α + and ᾱ α, or (iii) ᾱ/ α + and ᾱ/ α So first suppose ᾱ α + and ᾱ/ α,thatis, ᾱa +( ᾱ) a  a  a Butthenitfollowsthatforanyβ in (, ), wehave a  β (ᾱa +( ᾱ) a )+( β) a = βᾱa +( βᾱ) a (since βᾱ α ) a violation of the Archimedean axiom By similar reasoning we also get a violation of the Archimedean axiom if we assume ᾱ/ α + and ᾱ α Hence we must have ᾱ/ α + and ᾱ/ α, and hence by completeness we have ᾱa +( ᾱ) a a, as required 22

12 We are now in a position to show () implies (2), by explicitly constructing the SEU-representation for % We proceed by first deriving an Expected Utility representation for the preference relation restricted to the set of constant acts That is, we construct the affine real-valued function U defined on L In the second step, we use this U to calibrate the decision weights on events to construct the probability measure π defined on S, that enables us to extend the representation to the entire set of acts 23 Step Constructing the EU-Representation on % restricted to L (the set of constant acts) Set U (δ M ):=and U (δ m ):=Foranyx X set U (δ x ):=β, where, by Mixture Solvability, β is the unique solution to βδ M +( β) δ m δ x For any L = P m i= α iδ xi L we can apply Independence and transitivity of indifference (Ordering) m times to obtain L α (U (δ x ) δ M +( U (δ x )) δ m )+ = mx α i δ xi i=2 mx α i (U (δ xi ) δ M +( U (δ xi )) δ m ) i= Ã X m! Ã Ã m!! X α i U (δ xi ) δ M + α i U (δ xi ) i= i= δ m 24

13 Hence for any pair of constant acts L = P m i= α iδ xi and L = P m j= β jδ xj, transitivity of preference (Ordering) implies L % L iff % à X m! à à m!! X α i U (δ xi ) δ M + α i U (δ xi ) i=i Xm j=i β j U δ xj δ M + i=i δ m Xm β j U δ xj δ m j= But by Mixture Monotonicity this holds if and only if à X m! Xm α i U (δ xi ) i=i j=i β j U δ xj 25 Hence the affine function à m! X U α i δ xi = i= mx α i U (δ xi ) i= represents % restricted to the set of constant acts 26

14 Step 2 Constructing the SEU-Representation for % Fix any a in A and express it in a form [L,E ; ; L n,e n ], where L i % L i+,foralli =,,i For each i =,,n, it follows from Step that there is a unique number U (L i ) [, ], forwhich L i U (L i ) δ M +( U (L i )) δ m For each i =,,n, it follows from Mixture Solvability that there exists a unique π i satisfying [δ M on E E i ; δ m on E i+ E n ] π i δ M +( π i ) δ m From Mixture Monotonicity it follows U (L ) U (L n ) From State-independence it follows π π n 27 By applying State-independence n times we obtain a = L on E U (L ) δ M +( U (L )) δ m on E L n on E n U (L n ) δ M +( U (L n )) δ m on E n =( U (L )) δ m on E δ m on E 2 δ m δ m +(U (L n ) U (L n )) on E n on E n +(U (L ) U (L 2 )) δ M on E δ M on E 2 δ M δ m on E n on E n + U (L n) δ M on E δ m on E 2 δ m on E n δ m on E n δ M on E δ M on E 2 δ M δ M on E n on E n + 28

15 By applying Independence n times we have a is indifferent to: ( U (L )) δ m +(U (L ) U (L 2 )) [π δ M +( π ) δ m ] +(U (L 2 ) U (L 3 )) [π 2 δ M +( π 2 ) δ m ] + +(U (L n ) U (L n )) [π n δ M +( π n ) δ m ]+U (L n ) δ M = " n # X (U (L i ) U (L i+ )) π i + U (L n ) i= δ M " # n X + U (L n ) (U (L i ) U (L i+ )) π i δ m i= 29 Henceifwetakeanygivenpairofacts a = L on E L n on E n and a = L on E L n on En and apply the above methods, it follows from Mixture Monotonicity that a % a if and only if # X (U (L i ) U (L i+ )) π i + U (L n ) " n i= nx U L j U L j+ π j + U (L n ) j= 3

16 Hence, if we set, π ( ) :=, π (S) :=and π i j= E i := πi then we have established that % can be represented by the functional V L on E n X = (U (L i ) U (L i+ )) π i j=e j + U (Ln ) L n on E n i= = U (L ) π (E )+ nx i=2 Just remains to show that π () to be additive That is, for any pair of events A and B, (U (L i )) π i j=e j π i j= E j π (A B) =π (A)+π (B) π (A B) 3 To see that this indeed holds, consider δm on A B + 2 δ m on S (A B) 2 δm on A B δ m on S (A B) 2 (π (A B) δ M +( π (A B)) δ m ) + 2 (π (A B) δ M +( π (A B)) δ m ) (applying Independence twice) = 2 [π (A B)+π (A B)] δ M + µ π (A B)+π (A B) δ m 2 32

17 But δm on A B 2 δ m on S (A B) = δ M on A δ m on B 2 δ m on S (A B) + δm on A B 2 δ m on S (A B) + δ m on A δ M on B 2 δ m on S (A B) 2 (π (A) δ M +( π (A)) δ m )+ 2 (π (B) δ M +( π (B)) δ m ) = 2 (π (A)+π (B)) δ M + 2 ( π (A) π (B)) δ m 33 So by Mixture Monotonicity it follows π (A B)+π (A B) =π (A)+π (B), as required Hence we have established that% can be represented by the functional V L on E L n on E n nx = U (L ) π (E )+ (U (L i )) π i j=e j π i j= E j i=2 = π (E ) U (L )+π (E 2 ) U (L 2 )++ π (E n ) U (L n ) 34

18 Utility Paradoxes and Rationality Violations of the Expected Utility Rule Choice Problem 89 million v million Choice Problem 2 89 million v 9 u () > u () + 89u () + u (5) and u () > u () + u (5) 5 million 5 million 89u () + u () < 9u () + u (5) u () < u () + u (5) 35 Choice Problem ' 89 v 89 / / 5 Choice Problem 2' 89 v 89 / / 5 36

19 Are preferences linear in probabilities? δx δy δz 37 Can Beliefs Always be Represented by Probabilities? Urn contains 9 balls 3 are red The other sixty are black and white balls in unknown proportions a  b Acts EVENT a b a b Red White Black π (Red) U (δ )+( π(red)) U (δ ) > π(white) U (δ )+( π(white)) U (δ ) Ie π (Red) > π(white) 38

20 Can Beliefs Always be Represented by Probabilities? Urn contains 9 balls 3 are red The other sixty are black and white balls in unknown proportions b  a Acts EVENT a b a b Red White Black π (White or Black) U (δ )+( π(white or Black)) U (δ ) > π(red or Black) U (δ )+( π(red or Black)) U (δ ) Ie π (Red) < π(white) 39 Multiple Priors Expected Utility Fix D For any pair of acts a =[L,E ; ; L n,e n ] and a =[L,E; ; L n,e n ] a % a à X n! Xn π (E i ) U (L i ) min π Ej U L j min π D For example with i= π D j= D = {(π R,π B,π W ) : π R =/3, π W {/6, /3, /2}} V (a) = /3 > /6 =V (b) V (b ) = 2/3 > /2 =/3+/6 =V (a ) 4

21 Role of Expected Utility in Economics Economists use EU as an element of models of choice in uncertain environments asset allocation, insurance, saving, investment any alternative needs to be capable of being incorporated into more complicated models of market behavior All models are approximations Challenge is to show departures from maximizing EU have consequences that are not minor for issues under examination eg departures from EU that are not systematic may not bias predictions of market behavior or outcomes 4 Growing body of literature showing some key features of insurance and financial markets can be better explained by models that allow for systematic departures from EU But always easier to obtain better fit by adding parameters to model Unclear, whether out of sample performance is superior to simpler models based on EU For rest of course, we will assume DMs are subjective expected utility maximizers Likely to be better approximation to real world behavior: themoresituationisarepeatedevent the more significant the choice is for individual wealth the more alternative gambles under consideration can be viewed as local deviations from each other 42