Choice under uncertainty

Choice under uncertainty

Expected utility theory The agent chooses among a set of risky alternatives (lotteries)

Description of risky alternatives (lotteries) a lottery L = a random variable on a set of outcomes C C has finitely many elements n = 1,..., N (to avoid technicalities) The probability π n of n is objectively known (it is not a belief of the agent, a value the agent has to imagine = case of objective probabilities; subjective probabilities are considered later) a lottery L is characterized by (π 1,..., π N ) L is the set of lotteries over C (L is the simplex)

Compound lotteries To define some properties of preferences, we need to define first compound lotteries A compound lottery is a lottery over lotteries (a random variable on L) Call a lottery in L a simple lottery Consider K simple lotteries L k = ( π k 1,..., πk N) and probabilities α k Denote (L 1,..., L k, α 1,..., α K ) or k α kl k the compound lottery The compound lottery is the risky alternative that yields L k with probability α k (first draw k, then draw n according to L k ) The compound lottery induces a lottery over C: the so-called reduced lottery L = (π 1,..., π N ) where: π n = k α k π k n

Description of preferences over lotteries Preferences of the agent are a binary relation on L that satisfies the following assumptions: completeness and transitivity (as any preference relation) continuity: L, L, L L, the sets {α [0, 1] /αl + (1 α) L L } and {α [0, 1] /L αl + (1 α) L } are closed (a small change in probabilities does not change the ordering between lotteries) independence axiom: L, L, L L, α (0, 1), L L αl + (1 α) L αl + (1 α) L

Comments on the axioms: The continuity axiom is the same as the one in the basic theory of preference-based choice (the one you know from a previous course) Recall that completeness, transitivity, continuity implies existence of a continuous function U (defined on L) that represents (that is: L L U (L) U (L )) The new element here is the independence axiom. Independence axiom is then the most important element in this definition (and the most controversial as well). Loosely speaking, this axiom allows U to be written as an expected utility (see below). Independence axiom means that the ordering between 2 lotteries is independent on the mixture with a third lottery

A utility U : L IR has the expected utility form if there is (u 1,..., u N ) IR such that L = (π 1,..., π N ) L, U (L) = n π n u n u n is the utility of outcome n (U (δ n ) = u n where δ n is the lottery assigning probability 1 to outcome n) U (L) is the expected value of the u n When U has the expected utility form, U is called a Von Neumann Morgenstern (VNM) utility function expected utility means that U is linear in probabilities

Technical properties of a VNM utility First property A utility U : L IR has the expected utility form iff it is linear, that is: for every K, for every probability (α 1,..., α K ) and every lotteries L 1,..., L K L ( ) U α k L k = α k U (L k ) k k

Second property Consider a VNM utility U : L IR representing the preference relation. Then, a utility function U 0 : L IR is a VNM utility representing iff there are (α, β) IR ++ IR such that U 0 = αu + β the ordering between utility differences have a meaning in the context of the EU theory: it depends on only (and not on the choice of the utility representing ), if U (L 1 ) U (L 2 ) > U (L 3 ) U (L 4 ), then U 0 (L 1 ) U 0 (L 2 ) > U 0 (L 3 ) U 0 (L 4 ) the levels of utility differences and the utility levels have no meaning (they are utility specific), the ordering between utility levels obviously depends on only (and not on the utility)

The expected utility Theorem the most important result in the theory of choice under uncertainty What assumptions on do imply EU? If a preference relation on L satisfies the 4 axioms (completeness, transitivity, continuity, independence), then admits a utility representation of the expected utility form.

admits a utility representation of the expected utility form This means: is represented by a utility U : L IR and there is (u 1,..., u N ) IR such that L = (π 1,..., π N ) L, U (L) = n π n u n We can write as well: There is (u 1,..., u N ) IR such that for every L = (π 1,..., π N ) and L = (π 1,..., π N ) in L L L n π n u n n π nu n

The converse implication is true (if is represented by a VNM utility, then satisfies the 4 axioms). The proof of this converse implication is (almost) easy.

Proof of the Theorem We skip it

Discussion of the theory of expected utility EU theory is the most analytically convenient theory of choice under uncertainty. This is why it is widely used (non EU theories are hardly used in economics) maximizing EU is the right way to behave if you want your decisions to be consistent with the independence axiom (this is a normative viewpoint: if the 4 axioms must be satisfied, then the decision maker must behave in accordance with EU theory) empirically, the EU theory gives rise to many paradoxes ( = empirical observations that are inconsistent with the theory)

because of these paradoxes, non EU theories have been developed (since the 80 s approximately)... but these alternative theories are less tractable and they give rise to other paradoxes! I guess that many economists would agree on the following statement: decision theory remains a living and interesting research field, but almost every economist believes that EU theory remains the best choice for economic modelling (except in some specific cases). Some alternative theories have proven to be useful. 2 examples are the Epstein - Zin (1989) recursive utility and the multi-prior model à la Gilboa Schmeidler (1989)

We now review a few paradoxes. I begin with the: Saint-Petersburg Paradox This is NOT a paradox concerning EU, but this is a well-known paradox (the oldest one?). So I present it now. Consider the following lottery: I toss a coin. If tails (proba 1 2 ), then you win 2. If heads (proba 1 2 ), then I toss a coin for a second time. If tails, then you win 2 2. If heads, then I toss a coin for a third time.... If tails at the n th tossing, then you win 2 n. If heads, then I toss a coin for a n + 1 th time.

How much are you willing to pay me to play this lottery? One million euros?! No? But your expected profit is +, whatever the price of the lottery is! The explanation is that you don t want to get an infinite expected profit, because this profit is risky. This paradox tells you that you are not a profit maximizer. The common interpretation of this paradox is that you are risk-averse (the utility you assign to the lottery is smaller than the utility you assign to the expected gain of the lottery), see next section.

Allais Paradox N = 3 (monetary) outcomes: 2500000, 500000, 0 A first choice between lotteries L 1 and L 2 L 1 = (0, 1, 0) and L 2 = (.10,.89,.01) A second choice between lotteries L 1 and L 2 L 1 = (0,.11,.89) and L 2 = (.10, 0,.90) Many people choose L 1 and L 2 (so that L 1 L 2 and L 1 L 2 ) These choices are not consistent with EU

Two ways to prove inconsistency with EU check that there is no (u 2500000, u 500000, u 0 ) such that EU L1 > EU L2 and EU L 1 < EU L 2 define the lotteries P = ( 2500000, 0; 10 11, 11) 1 and observe that L1 =.11δ 500000 +.89δ 500000 and L 2 =.11P +.89δ 500000 L 1 =.11δ 500000 +.89δ 0 and L 2 =.11P +.89δ 0 (δx is a lottery that gives x with probability 1) the independence axiom is violated. Indeed, it requires: L 1 L 2 δ 500000 P L 1 L 2

Allais paradox is an example of the effect of common consequence: Consider 4 lotteries, namely P, F, G and δ x (x with probability 1) A first choice is made between L 1 and L 2 L 1 = αδ x + (1 α) F and L 2 = αp + (1 α) F A second choice is made between L 1 and L 2 L 1 = αδ x + (1 α) G and L 2 = αp + (1 α) G The independence axiom implies L 1 L 2 δ x P L 1 L 2 In the case where F SD1 G (see below the section stochastic dominance ) and P gives positive probability to outcomes that are both < x and > x, many people choose L 1 and L 2

4 reactions to Allais paradox Normative: choosing L 1 and L 2 is a mistake and should be corrected (this assumes that decisions must obey the independence axiom) Allais paradox relies on very small probabilities and very large payoffs. It has no practical relevance (common choices don t have such proba and payoffs) Define a regret theory = modify the set of axioms to embody the following idea: an agent chooses L 1 because he does not want to take the risk to have nothing (with proba.01 in L 2 ) and to regret that he could have 500000 with certainty. There is no such problem with the choice L 1 /L 2 (so choosing L 2 is not inconsistent with choosing L 1 ) Define a theory of choice under uncertainty without the independence axiom (you should then replace it with a somewhat weaker axiom - recall that theories need axioms in order to get results - with no result, a theory is uninteresting)

Machina s paradox Another paradox illustrating a possible influence of what might have been on your choices (such influence is not taken into account by EU theory - so, if you want to model these influences, then you need a non EU theory) N = 3 outcomes: n = 1 a trip to Venice n = 2 watching a movie about Venice n = 3 doing nothing Your preferences over the outcomes are: outcome 1 outcome 2 outcome 3

A choice between 2 lotteries L 1 = (99.9,.1, 0) and L 2 = (99.9, 0,.1) EU Theory implies L 1 L 2 But, if you play L 1 and get the outcome 2, then a common reaction is to be so disappointed that you decide not to watch the movie (that is: you finally prefer outcome 3 to outcome 2). Notice that this is disappointment that an outcome did not come up, and not regret over a choice not made (= Allais paradox).

Induced preferences In many examples, the set of outcomes must be properly written Example: you bring a wine bottle to a dinner you decide to bring Red or W hite wine (R or W ) dinner consists either of Meat or F ish (M or F ) you have no preferences on {R, W } nor {M, F }, but you prefer R if M and W if F your preferences must then be defined on the 4 outcomes R if M, R if F, W if M and W if F RM RF, WF WM, RM WF, WM RF

Practical implication: choosing the wine before or after you learn about dinner makes a great difference to you choosing before you know the dinner corresponds to a choice between 2 lotteries (RM, RF, π, 1 π) and (WF, WF, π, 1 π) choosing after you know M (resp F ) corresponds to a choice between 2 certain lotteries RM and WM (resp RF and WF ) your expected utility is larger in the latter case

Risk aversion We define and discuss risk aversion within the context of EU theory (this could be done more generally) We consider monetary outcomes to keep things simple (that is: outcomes in IR + ) But we consider a continuous set of outcomes x IR + (the EU Theorem can be extended to this framework at the cost of some additional technicalities)

A lottery is a random variable that takes non negative real values. It is represented by a cdf (cumulative distribution function) F F : IR + [0, 1], increasing, F (0) = 0, lim x + F = 1 F (x) = the probability that the payoff is smaller or equal than x F (denoted f ) is the density function, whenever it exists F (x) = x f (t) dt F is not necessarily continuous (see next slide) The restriction to IR + of the domain of F is for convenience only

This framework is compatible with a discrete variable If L is a discrete lottery with N outcomes (x 1,..., x N ) and probabilities (π 1,..., π N ), then the associated cdf is F (x) = n/x n x π n (and F (x) = 0 when x < x 1 ) with x 1 <... < x N

Expected utility form We write U (F ) = u (x) df (x) for some measurable function u : IR + IR In Mas-Colell et alii, U is called the VNM utility function and u the Bernouilli utility function Very few people use the name Bernouilli function. VNM utility commonly designates both u and U From now on, we assume that u is continuous and increasing Another common restriction is concavity, it is linked to risk aversion

Risk aversion An agent is risk averse iff, for every lottery F, the agent prefers the degenerate lottery xdf (x) to F An agent is risk neutral iff, for every lottery F, the agent is indifferent between the degenerate lottery xdf (x) and F An agent is strictly risk averse iff, for every (non degenerate) lottery F, the agent strictly prefers the degenerate lottery xdf (x) to F The degenerate lottery x is a lottery that gives x with probability 1

If preferences admit an EU representation, then An agent is risk averse iff ( F, u (x) df (x) u ) xdf (x) An agent is risk neutral iff ( F, u (x) df (x) = u ) xdf (x) An agent is strictly risk averse iff ( F non degenerate, u (x) df (x) < u ) xdf (x)

A very important property: Risk aversion is equivalent to the concavity of u (u 0 if u is C 2 ) because the above inequality is the Jensen s inequality (this is the definition of a concave function) Risk neutrality is equivalent to the linearity of u Strict risk aversion is equivalent to the strict concavity of u (u < 0 except on a set of measure 0 if u is C 2 )

Measurement of risk aversion Consider a given u The certainty equivalent of F is c (F, u) defined by u (c) = u (x) df (x) the agent is indifferent between c and playing F For every payoff x and every ε > 0, the probability premium π (x, ε, u) is defined by ( ) ( ) 1 1 u (x) = 2 π u (x ε) + 2 + π u (x + ε) the expected gain of the lottery is x + 2πε: 2πε is the increase in the expected gain required to make the agent indifferent between x and playing the lottery

Link between c, π and risk aversion The 4 following properties are equivalent: the agent is risk-averse u is concave for every F, c xdf (x) for every x and ε, π 0

The usual concept for measuring risk aversion Consider a C 2 Bernouilli function u The Arrow Pratt coefficient of absolute risk aversion is: r A (x) = u (x) u (x) 0 The Arrow Pratt coefficient of relative risk aversion is: r R (x) = xu (x) u (x) 0 Some motivation for these concepts: measure of the concavity of u r A (x) = 4π (x, 0, u) other applications later

CARA functions: the function with a constant absolute risk aversion a ( x, r A (x) = a) is u (x) = exp ax CRRA functions, the function with a constant relative risk aversion a ( x, r R (x) = a) is u (x) = x 1 a 1 a if a 1 u (x) = ln x if a = 1 (both functions are unique up to an increasing linear transformation, that is: αu (x) + β with α > 0)

Comparison across individuals There are (at least) 5 definitions of the relation agent 1 is unambiguously more risk averse than agent 2, namely x, r 1 A (x) r 2 A (x) F, c ( F, u 1) c ( F, u 2) x, ε, π ( x, ε, u 1) π ( x, ε, u 2) u 1 is more concave than u 2, that is: there is an increasing concave function f such that u 1 = f u 2 x, F, u 1 (x) df (x) u 1 ( x) u 2 (x) df (x) u 2 ( x) Proposition: these 5 definitions are equivalent

Comparison across wealth levels Objective: To give a formal content to the idea that people take more risk when they become wealthier. This is the case of agents with DARA Definition: A Bernouilli function has decreasing - or increasing - absolute risk aversion (DARA - or IARA -) when r A (x) is decreasing - or increasing - in x

Denote for every w 1 and w 2 with w 1 < w 2 u 1 (z) = u (w 1 + z) and u 2 (z) = u (w 2 + z) The 5 following properties are equivalent: u is DARA F, w c w (F ) (= the max price you accept to pay to get rid of the risk z) is decreasing in w where c w (F ) is defined by u (c w (F )) = u (w + z) df (z) π (x, ε, u) is decreasing in x u 1 is more concave than u 2, that is: there is an increasing concave function f such that u 1 = f u 2 F, u (w 1 + z) df (z) u (w 1 ) u (w 2 + z) df (z) u (w 2 )

Comment on this equivalence: absolute risk aversion is adapted to absolute gain or losses (relative risk aversion is adapted to gain or losses that are measured relatively to the initial wealth level)

To state consequences of DRRA, consider lotteries where gain/losses are defined as proportions of the initial wealth w, that is: a lottery is a random variable t on IR + (cdf F ) and the final wealth is tw The 3 following properties are equivalent: u is DRRA (r R (x) is decreasing) for every w 1 < w 2, û 1 (t) is more concave than û 2 (t), that is: there is an increasing concave function f such that û 1 = f û 2 (where û i (t) = u (tw i )) F, w/ĉ w (F ) is decreasing in w where ĉ w (F ) is defined by u (ĉ w (F )) = u (tw) df (t) (or (w ĉ w (F )) /w, that is: the max proportion of w that you accept to lose to give up the risk t)

Empirically, under the assumption of EU, the most plausible utilities satisfy both DARA and CRRA (CRRA functions are DARA, DRRA is stronger than DARA)

Application to insurance 2 events: No Accident with proba 1 π: agent s wealth w Accident with proba π: agent s wealth w d (damage) Insurance contract premium p reimbursement q

Agent chooses (p, q) by maximizing max (1 π) u (w p) + πu (w p + q d) Result 1: If the contract is actuarially fair, that is: if p = πq (expected value of reimbursement) the agent insures completely Proof: q solves (the constraint q 0 is omitted) (1 π) ( π) u (w πq) + π (1 π) u (w πq + q d) = 0 that is q = d

Result 2: If contract is not actuarially fair, and if there is λ 0 such that p = (1 + λ) πq the agent chooses a partial insurance (this means that the risk averse agent chooses to bear risk, because selling this risk is too expensive) Proof: q solves (the constraint q 0 is omitted) u (w p) = 1 π (1 + λ) (1 π) (1 + λ) u (w p + q d) This implies u (w p) < u (w p + q d) (as long as π (1 + λ) 1) that is q d < 0 The case π (1 + λ) 1 requires to consider the constraint q 0 (the agent chooses q = 0)

Application to the demand for a risky asset 2 assets: a risky asset with random return z (cdf F ) a safe asset with known return r zdf (z) > r the investor s problem is to choose between risk and (expected) return The agent chooses how to invest a wealth w in the 2 assets (an amount α in the risky asset, β = w α in the safe asset) by maximizing the EU of the final wealth max α,β 0 u (αz + βr) df (x) s.t. α + β = w

Result 1: α > 0 (the risk-averse agent chooses to bear a (small) amount of risk because this risk is actuarially favorable = Result 2 in the insurance problem) Proof: the Kuhn Tucker FOC is H (α ) = 0 where H (α) = (z r) u (αz + (w α) r) df (x) is decreasing, and H (0) = ( zdf (z) r ) u (wr) > 0

Result 2: if u is CARA, α does not depend on w Proof: max α exp a (wr + (z r) α) df (x) gives the same solution as max exp a (z r) αdf (x) α

Result 3: if u is CRRA, α w does not depend on w Proof in the case r R 1 (the case r R = 1 is similar): max 0 α w gives the same solution as max 0 s 1 (wr + (z r) α) 1 a df (x) 1 a (r + (z r) s) 1 a df (x) 1 a where s = α w is the share of wealth invested in the risky asset

Result 4: if u is DARA, α increases in w Proof: α solves the Kuhn Tucker FOC H (α ) = 0 where H (α) = (z r) u (w α ) df (x) with w α = (z r) α + wr Differentiating the FOC gives check that H α (α) = dα H dw = w (α ) H α (α ) (z r) 2 u (w α ) df (x) < 0

To check that H w (α ) 0, write u (w α ) = r A (w α ) u (w α ) and z r, (z r) r ( r A (w α ) u (w α ) ) (z r) r ( r A (w 0 ) u (w α ) ) this implies r r (z r) r ( r A (w α ) u (w α ) ) df (z) (z r) r ( r A (w 0 ) u (w α ) ) df (z)

Analogously + r + r (z r) r ( r A (w α ) u (w α ) ) df (z) (z r) r ( r A (w 0 ) u (w α ) ) df (z) Summing the 2 inequalities implies H w (α ) r A (w 0 ) r (z r) u (w α ) df (z) = 0 }{{} H(α )

Further results: if u is IARA, α decreases in w if u is DRRA, α w if u is IRRA, α w increases in w decreases in w

Comparison of Payoff Distributions in Terms of Return and Risk Objective: to give a formal content to the idea that: a distribution of returns (or payoffs) F gives unambiguously higher returns than a distribution G: F first-order stochastically dominates G (F SD1 G) a distribution F is unambiguously less risky than a distribution G: F second-order stochastically dominates G (F SD2 G) Technical restriction: for every cdf F in this section, there is x < + such that F (x) = 1

First order stochastic dominance Consider 2 distributions of returns F and G There are (at least) 2 definitions of F gives higher returns than G x, F (x) G (x) (for every x, the probability to get a return higher than x is higher with F than with G) F SD1 G, that is: for every non decreasing function u : IR IR u (x) df (x) u (x) dg (x) (the agent prefers F to G) Proposition: the 2 definitions are equivalent

SD1 and means F SD1 G implies that xdf (x) xdg (x) But the converse implication is not true because the whole distributions F and G matter to establish SD1 (not only their means)

Determining F from G when F SD1 G It is possible to write F as G + an upward shift, that is: first draw x according to a distribution G then draw z according to a distribution H x such that H x (0) = 0 (H x is an upward shift that may depends on x) define F as the compound lottery with final payoff x + z For every F and G such that F SD1 G, it is possible to define F in this way

Second order stochastic dominance We restrict attention to comparison between distributions having the same mean and such that F (0) = G (0) = 0 For any two distributions F and G with the same mean (that is: xdf (x) = xdg (x)), F second order stochastically dominates G if, for every non decreasing concave function u : IR + IR, u (x) df (x) u (x) dg (x) (a risk averse agent prefers F to G) F SD2 G means that F is less risky than G

Mean preserving spreads Consider the following compound lottery G first draw x according to a distribution F then draw z according to a distribution H x such that zdhx (z) = 0 the final payoff is x + z (H x is a white noise added to x, that may depend on x) G has the same mean as F, we say that G is a mean preserving spread of F

Property: G is a mean preserving spread of F iff F SD2 G

Proof: if G is a mean preserving spread of F, then ( ) u (x) dg (x) = u (t + z) dh t (z) df (t) Given that u is concave ( ) u (t + z) dh t (z) u (t + z) dh t (z) ( ) ( ) u (t + z) dh t (z) df (t) u (t + z) dh t (z) df (t) Given that (t + z) dh t (z) = t u (x) dg (x) u (t) df (t) that is: F SD2 G We skip the proof of the converse implication

A last usual property characterizing SD2 Consider 2 distributions F and G with the same mean, F (0) = G (0) = 0 and x such that F ( x) = G ( x) = 1 x, x G (t) dt x F (t) dt x the area between 0 and x below G is larger than the one below F on a graph the total areas below F and G are the same: x 0 F (x) dx = x 0 G (x) dx this means that G puts higher probabilities on extreme values than F

To see why the total areas below F and G are the same, make an integration by parts: x 0 (F (x) G (x)) dx = [x (F (x) G (x))] x 0 }{{} =0 x x (df (x) dg (x)) = 0 0 } {{ } =0 (same mean)

The 3 following properties are equivalent: F SD2 G G is a mean-preserving spread of F for every x, x G (t) dt x F (t) dt

State-dependent Utility There are S states of nature s = 1,..., S. Probability of s is π s (objective probability = everyone assigns the same probability to s) consider the set of random variables on S with non negative real values a random variable is described by (x 1,..., x S ) IR S + (x s the payoff in state s) preferences are a binary relation on this set (that is IR S +) has an extended EU representation if there are S functions u s : IR S + IR such that, for every (x 1,..., x S ) and (x 1,..., x S ) in IR S +, (x 1,..., x S ) ( x 1,..., x S ) π s u s (x s ) s s the utility now depends on s π s u s ( x s )

Existence of an extended EU representation The axioms required on are exactly the same as previously One only needs to define on a set of properly defined lotteries ˆL ˆL is the set of lotteries L that are S-tuples (F 1,..., F S ) of distributions on IR + first draw s (with the objective probabilities (π 1,..., π S )) then draw a monetary payoff x according to F S

Theorem: If a preference relation on ˆL satisfies the 4 axioms (completeness, transitivity, continuity, independence), then admits a utility representation of the extended EU form. That is: there are S functions u s : IR + S IR such that for every L = (F 1,..., F S ) and L = (F 1,..., F S ) in ˆL L L π s u s (x) df s (x) π s u s (x) df s (x) s s The converse implication is true as well

The sure thing axiom There is another axiom implying an extended EU representation of preferences. define directly on IR S + (not on lotteries) satisfies the sure-thing axiom if, for every (event) E S, for every x, x, y, y in IR S + such that s E, x s = x s and y s = y s s / E, x s = y s and x s = y s we have x y x y

Intuition: when you order x and y, only the payoffs on E matters (as the payoffs on S E are common to x and y) if this sure thing ( = what you get if s S E) changes (that is: (x, y) becomes (x, y )), then your ordering between the 2 random payoffs should not be affected this idea is somewhat similar to the independence axiom Assume S 3 (a technical innocuous assumption). If satisfies the 4 axioms (completeness, transitivity, continuity, sure-thing), then it admits an extended EU representation. The converse implication is true as well

Subjective probability theory In many choices under uncertainty, there is nothing like an objective probability that can be assigned to an event. How can we write a theory of choice under uncertainty in these situations with no probability? the situations (tossing a coin, rolling the dice) where the probability of each event can be unambiguously assessed are often referred to as situations with risk the other situations are often referred to as situations with uncertainty this old distinction is due to Knight (1921)

One theoretical question is: can we still represent the choice made by an agent as if the agent first describes the uncertainty he faces by a probability distribution on the set of states of nature (these probabilities are subjective) then makes his decision by maximizing an EU (estimated with the subjective probabilities) (this requires that the agent is able to list the set of states of nature)

Subjective probability theory answers yes : some axioms of consistency of preferences imply that choices under uncertainty are made as if the agent assigns subjective probabilities to the uncertain events and maximizes an EU The classical reference is Savage (1954), the more readable approach is Anscombe Aumann (1963) We follow here the AA approach

The AA approach relies on the set ˆL of lotteries L = (F 1,..., F S ) and the preference relation defined on ˆL recall that L = (F 1,..., F S ) means that, in state s, the value of the (monetary) payoff is randomly drawn according to distribution F s this approach is formally not very different from the above EU theorems. But it is much more abstract the result is : with subjective proba, an EU representation requires one more axiom (state uniformity of preferences) in order to get a Bernouilli utility u independent of s (the 4 usual axioms provide an extended EU representation only)

The first step is to apply the extended EU Theorem How do we apply the Theorem with no proba in this model? A careful examination of the proof shows that the proba on s are not needed The theorem states: If satisfies the 4 axioms, then there are S functions û s : IR + IR such that, for every L and L in ˆL L L û s (x) df s (x) û s (x) df s (x) s s (the u s exhibited in the theorem are û s /π s ) This allows us to derive from the preferences in state s. This is a preference relation s on the set of distributions F on IR + defined by F s F û s (x) df s (x) û s (x) df s (x)

The objective of the first step was to be in a position to define s We need to have a mathematical object like s for every s in order to define the following axiom: Consider a preference relation on ˆL satisfying the 4 axioms. The derived preferences are state-uniform if, for every distributions F and F on IR + s, s, F s F F s F We need this axiom to state a (subjective) EU Theorem

Subjective EU Theorem If the preference relation defined on ˆL satisfies the 4 axioms (completeness, transitivity, continuity, independence) and if the derived preferences are state-uniform Then there are (π 1,..., π S ) 0 and u : IR + IR such that, for every x and x in IR +, S x x π s u (x s ) π s u ( x ) s s s The probabilities are uniquely determined u is unique up to an increasing linear transformation (αu + β with α > 0) A technical remark: x x is consistent with defined on ˆL if x s is seen as a degenerate lottery in state s (that gives x s with proba 1)

Ellsberg paradox (Mas-Colell et alii offers a variant with 2 urns) 1 urn containing 100 balls: 34 red (R) balls 66 black (B) or yellow (Y ) balls A first choice between 2 games: you draw a ball in the urn and you win 1 if game 1: the ball is R game 2: the ball is B A second choice between 2 games: you draw a ball in the urn and you win 1 if game 3: the ball is R or Y game 4: the ball is B or Y A common choice is games 1 and 4 not consistent with SEU theory

To see why, assume that there is a EU representation of preferences: 3 states of nature R, B, Y with subjective proba (π R, π B, π Y ) π R = 34 100 payoffs u 0 and u 1 (normalization assumption u 0 = 0 w.l.o.g.) game 1 game 2 34 100 u 1 π B u 1 game 3 game 4 ( 34 100 + π Y ) u1 (π B + π Y ) u 1 Hence, game 1 game 2 game 3 game 4

A deeper explanation of this paradox is that the sure thing axiom is violated: games 1 and 3 (2 and 4) give the same payoffs on {R, B} games 1 and 2 (3 and 4) give the same payoffs on {Y } the sure thing axiom implies that game1 game2 game3 game4 Interpretation of the choice of games 1 and 4: aversion to ambiguity (you choose the games where you know the probability of winning) Consequence of this interpretation: there may be a meaningful difference between risk and uncertainty (that the SEU theory is unable to account for). Hence, there is a need for an alternative choice theory

The end of the chapter