Online Appendix to Optimal Stopping with General Risk Preferences Jetlir Duraj∗ The online appendix is organized as follows. In part 1 we present behavioral foundations of the model considered in the main body of the paper. The main result of part 1 characterizes Expected Utility as the only risk preference model which is consistent with Dynamic Consistency of Preferences in our setting. In part 2 we relate behaviorally weak Risk Aversion to the classical Risk Aversion concept, which is defined as aversion to mean-preserving spreads. Finally, part 3 presents additional applications and examples of Theorems 1 and 2 from the main body of the paper.

1

Behavioral foundations of optimal stopping in continuous time.

In this part we give behavioral foundations in terms of axioms on dynamic choice behavior of the agent for the general optimal stopping problem we have considered in the previous parts. The motivation for this analysis is the following. First, it establishes formally that the assumptions about Naivet´e or Sophistication are indeed needed as soon as  does not have an Expected Utility representation. For dynamic choice over finite lottery trees (a discrete time setting) the following result is well-known: under some weak technical requirements, two axioms of dynamic behavior, ’Consequentialism’ and ’Dynamic Consistency of Preferences’,1 are equivalent to the agent being an Expected Utility-maximizer (see Hammond (1988) for a formal proof and Hammond (1989), Gul, Lantto (1990) but also Machina (1989) for more on the interpretation of this result).2 It follows that to be able to pin down uniquely the dynamic behavior of the agent in finite lottery trees, when the agent is not an Expected Utility-maximizer, additional assumptions on dynamic behavior are needed, besides the existence of the preference relation  on lotteries. Here we formulate Dynamic Consistency of Preferences (DCP) for our setting and extend the classical result to our continuous time diffusion setting where the choice objects are (uniformly integrable) stopping times.3 This means ∗

[email protected] In our setting we don’t need a Consequentialism axiom as it follows from our Consistency axiom, since the prize process, it being a diffusion, is Markovian; see details in the following. 2 The result we prove is not a special case of the classical result about finite lottery trees. Besides the different set up of continuous time, in the classical Theorem as stated in Gul, Lantto (1990) the agent faces finite lottery trees of arbitrary length, but the horizon of the decision problem is fixed once the tree she faces is fixed. Here the horizon of the decision problem is endogenous due to stopping. 3 We haven’t found any similar result for continuous time processes in the literature. Moreover, adapting the proof techniques from the discrete time setting in Hammond (1988) to ours is impossible. 1

1

that DCP has to be relaxed in our setting as well, and that additional rules are needed to fully specify dynamic behavior. Naivet´e and Sophistication are precisely these additional rules.

A behavioral model of optimal stopping. We start explaining the behavioral foundations of the model of this paper by introducing some notation needed to state the axioms. Unless otherwise stated, in the following the index X runs over regular diffusions as defined in Set Up part of the main body of the paper. Fix such a diffusion X and a starting point y0 ∈ (w, b). Denote the elements of the filtration of the diffusion X by FXT (y0 ), T ∈ [0, ∞). Recall, that FXT (y0 ) encodes the prize uncertainty resolved till time T . An arbitrary element n from FXT (y0 ) can be interpreted as a ‘node’ of depth T of the ‘uncertainty tree’ defined by the diffusion X started at y0 . We identify the singleton set 0 FX0 (y0 ) with y0 . We say that n0 ∈ FXT (y0 ) is a continuation of n ∈ FXT (y0 ) if T 0 > T and the occurrence of n0 implies that of n. Denote by FX (y0 ) the union of FXT (y0 ) as T ranges across all positive time periods and by F(y0 ) the union of all FX (y0 ) as X ranges across all regular diffusions. The former encodes all possible histories of prize path realizations when fixing a particular regular diffusion and the latter when considering all regular diffusions started at y0 . FX (y0 ) is a well-defined object because all diffusions X are adapted to the filtration of the underlying Brownian motion W of the diffusion. The choice objects of the agent at each moment in time are given by the set S of (uniformly integrable) stopping times. We recall here the convention from Set Up section of the main body of the paper: all stopping times are assumed to be adapted to the filtration of the underlying Brownian motion which drives the diffusions.4 For each n ∈ F(y0 ) we assume that the agent has a complete and transitive preference relation n over S. An agent can thus be identified with the collection of preference relations A = (n )n∈F (y0 ) : for each diffusion X a collection of preference relations for each node that can be reached by the diffusion. A stopping time can then also be identified with a stopping policy, which is a function from FX (y0 ) to {stop, continue} telling the agent to stop or continue after any event from FX (y0 ). In our model, the preference relations n are related to each other by the existence of a fixed, static risk preference, which evaluates prize lotteries in a history-independent way. To simplify notation in the following we denote for a real-valued random variable Y and n an event from some sigma-algebra σ by FY |n the distribution of the random variable conditional on the event n occurring. The following definition is an adaptation of the similar Definition of Consistency in Gul, Lantto (1990), who consider choice in discrete time problems modeled as finite lottery trees. Definition 1. The agent exhibits Consistency with static preference if there exists a risk preference functional V : ∆([w, b])→R such that for all diffusions X and all events n 4

In particular, this also ensures that all diffusions live in the same filtered probability space.

2

possible under FX (y0 ), it holds n is represented by V (FXτ |n ). Thus, the value of stopping time τ if event n has occurred is given by the static utility of the distribution of Xτ , evaluated conditional on n having occurred. We maintain Consistency with static preference in the following. The next definition gives a formal statement of the well-known Dynamic Consistency axiom in our setting. Definition 2. An agent satisfies Dynamic Consistency of Preferences (DCP) if for all diffusions X, all y0 ∈ (w, b) and three different n, n1 , n2 ∈ FX (y0 ) such that i) n1 , n2 are continuations of n and they are disjoint events ii) the probability that either n1 or n2 happens conditional on n happening, is one, the following implication is true for any four stopping times τ1 , τ10 , τ2 , τ20 from S: ( ( τ , if n τ10 , if n1 1 1 . n τ 0 = if τ1 n1 τ10 and τ2 n2 τ20 then τ = τ20 , if n2 τ2 , if n2 Dynamic Consistency of Preferences says, that if for two mutually exclusive and exhaustive continuation events n1 , n2 of n, stopping time τ1 is preferred to τ10 at event n1 and τ2 is preferred to τ20 at event n2 , it should also hold that the combined stopping time τ is preferred to the stopping time τ 0 after event n. DCP is violated when there is an event n and two stopping times τ, τ 0 from S such that at n it holds τ 0 n τ even though the stopping time τ leads to (weakly) better prospects than τ 0 in all future continuations of event n. DCP puts very strong restrictions on the preference  which the agent uses to evaluate lotteries from ∆([w, b]), as the following Theorem shows. Theorem 1. The static risk preference functional V of an agent has an Expected Utility representation with a strictly increasing, bounded and continuous Bernoulli utility function u : [w, b]→R if and only if the following requirements are met: 1. V satisfies FOSD-monotonicity and is continuous in the topology of convergence in distribution. 2. The agent satisfies Dynamic Consistency of Preferences. Among other things, this result implies that if the agent’s risk preferences are not Expected Utility, the agent is time inconsistent in some optimal stopping problem. This occurs in the case of a naive agent at an event n, because she projects the ‘current’ preference n into all preferences of continuation events: she decides on the stopping time at event n by assuming that n0 = n for all continuation events n0 of n. The sophisticated agent on the other hand, restricts her choice set of stopping times S at event n to the set S(n) of stopping times she knows will not lead to preference reversals, no matter the continuation of the process. 3

Before giving the full proof of Theorem 1, which is technically involved, we sketch the main idea behind the hardest part of the proof: sufficiency of DCP for the risk preference to have an Expected Utility representation, i.e. to satisfy Independence. Suppose the agent prefers distribution F to G and assume that she faces the choice between the mixed lotteries λF +(1−λ)H and λG+(1−λ)H for some other arbitrary lottery H and λ ∈ (0, 1). Suppose agent satisfies DCP and there is a stopping problem where the agent has two stopping strategies τF,H and τG,H which, if implemented, lead to a history h1 where H is realized with probability 1 − λ and otherwise to a history h2 where respectively F or G is realized with probability λ. Under this situation DCP will imply that the agent prefers τF,H to τG,H at the current moment of time, because conditional on either history h1 or h2 the agent prefers τF,H to τG,H . But the distribution induced by, respectively τF,H or τG,H , is λF + (1 − λ)H or λG + (1 − λ)H! Now consistency will imply that the agent prefers λF + (1 − λ)H to λG + (1 − λ)H in a static problem as well. This implies that the risk preference of the agent satisfies Independence. The latter fact and the other assumed technical conditions imply due to classical results, that the risk preference has an Expected Utility representation.5 Proof of Theorem 1. Necessity: Let V be given by V (F ) = EF [u] for some u : [w, b]→R strictly increasing, bounded and continuous. Requirement 1. is then standard. Regarding Requirement 2: take τ1 , τ10 , τ2 , τ20 , n, n1 , n2 as in Definition 2. It follows for τ¯ either τ or τ 0 that E[u(Xτ¯ )|n] = E[u(Xτ¯ ), n1 |n] + E[u(Xτ¯ ), n2 |n] = E[E[u(Xτ¯ )|n1 ]|n] + E[E[u(Xτ¯ )|n2 ]|n], where the last equality follows from the Markov property of diffusion processes. Both summands at the end of the calculation above are weakly higher for τ¯ = τ than τ¯ = τ 0 by hypothesis. This shows necessity of the requirements. Sufficiency: The proof consists of two steps. First, we establish the following important Lemma which states that under the three Requirements, V satisfies the Independence axiom of Expected Utility (cf. the discussion preceding Proposition 1 in the Set Up section of the main body of the paper). Lemma 1. Under the Requirements 1−2, V satisfies Independence, i.e. for all H, G1 , G2 ∈ ∆([w, b]) and α ∈ [0, 1] we have V (G1 ) ≥ V (G2 ) implies V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Proof. Take H as in the statement and assume w.l.o.g. that α ∈ (0, 1). We assume for now additionally that Gi , i = 1, 2 are step functions, i.e. correspond to lotteries of finite support. We relax this assumption at the very end. We divide the proof for this case in several steps. Step 1. Assume first, that {G1 , G2 } is ordered by FOSD-monotonicity. In particular, it holds G1 >F OSD G2 , the other case being excluded by V (G1 ) ≥ V (G2 ). It then follows that αH + (1 − α)G1 >F OSD αH + (1 − α)G2 , 5

See Theorem 3 in Grandmont (1972).

4

and by Requirement 1 that V (αH + (1 − α)G1 ) > V (αH + (1 − α)G2 ). Step 2. Assume now that {G1 , G2 } is not ordered by FOSD-monotonicity and that H has support contained in (w, b). 2,L Recall from the statement and proof of Lemma 2 the set Cinc . Equip it with the metric 2,L given by the maximum norm. Consider the map ψ : Cinc →R given by ψ(S) = EG1 [S] − EG2 [S]. 2,L This map is continuous and since Cinc ([w, b]) is a convex metric space (in particular it is connected), it follows that the image of ψ in R is connected. In particular, it is an interval. We need the following auxiliary Claim. 2,L Claim 1: There exists a S ∈ Cinc ([w, b]) such that ψ(S) = 0. Proof of Claim 1. Assume this is not the case. It then follows that the whole image of ψ, it being connected, consists of either only negative or only positive numbers. It then follows that 2,L either Case 1: EG1 [S] > EG2 [S] for all S ∈ Cinc ([w, b])

or 2,L Case 2. EG1 [S] > EG2 [S] for all S ∈ Cinc ([w, b])

We close the proof of the Claim by showing that this implies that {G1 , G2 } is ordered by FOSD. Focus on Case 1, the other one being analogous. Pick a finite sequence x1 < x2 < · · · < xn in [w, b] which consists of the union of the support of the lotteries corresponding 2,L to G1 and G2 . For a k ∈ {2, . . . , n} pick Sk ∈ Cinc ([w, b]) with Sk (xj ) = 1 − (n − j) for k ≤ j ≤ n and Sk (xj ) = j for all 1 ≤ j < k. This is possible for  > 0 small enough. It follows from Case 1 that n X

(1 − (n − j))(G1 (xj ) − G1 (xj−1 )) + 

>

G1 (xj ) − G1 (xj−1 )

j=1

j=k n X

k−1 X

(1 − (n − j))(G2 (xj ) − G2 (xj−1 )) + 

k−1 X

G2 (xj ) − G2 (xj−1 )

j=1

j=k

for all  > 0 small enough. Letting now  go to zero we recover for all k ∈ {2, . . . , n} that G1 (xj ) ≤ G2 (xj ), j = 1, . . . , n. But this implies that G1 FOSD-dominates G2 .6 End of Proof of Claim 1. 2,L It follows, that there exists some S ∈ Cinc ([w, b]) with EG1 [S] = EG2 [S]. As in the proof of Lemma 2 of the Appendix of the paper it follows that there exists some y2 and some diffusion X, started at y2 ∈ [w, b] with scaling function S : [w, b] × [w, b]→R, so that S(y2 , y2 ) = 0 and Ex∼G1 [S(x, y2 )] = Ex∼G2 [S(x, y2 )] = 0. 6

cf. Proposition 6.D.1. in pg. 195 of Mas-Colell, Whinston, Green (1995).

5

Consider now the function ρ : [w, b]→R given by ρ(z) = Ex∼H [S(x, z)]. This is a continuous function with ρ(b) ≥ 0 and ρ(w) ≤ 0. In particular, it follows, there exists y1 ∈ (w, b) with ρ(y2 ) = 0 (that y1 can be chosen different from w, b follows from the intermediate assumption that supp(H) ⊂ (w, b) ). Due to Proposition 1 in the main body of the paper there exists τH , τG1 and τG2 such that if the diffusion X is started at y1 then FXτH = H and if X is started at y2 then FXτG = Gi , i = 1, 2. i Step 2a. Assume first, that y1 > y2 and consider the stopping time τy2 ,y1 . We show that Independence holds for these kinds of H. The case of y2 > y1 is similar. We know from the proof of Theorem 2 in the main body of the paper and its preceding discussion, that for y ∈ (y1 , y2 ) the probability p(y) that X started at y reaches y1 before it reaches y2 is a strictly increasing continuous function with p(y1 ) = 1, p(y2 ) = 0. There exists thus a y0 ∈ (y1 , y2 ) with p(y0 ) = α. Consider now the stopping times τi for i = 1, 2 given by7 ( τH , if Xτy1 ,y2 = y1 τi = τGi , if Xτy1 ,y2 = y2 . This is again a uniformly integrable stopping time, i.e. an element of S.8 The Markov property of X shows that Xτi has the distribution αH + (1 − α)Gi . Take the ‘root’ event y0 ∈ FX0 (y0 ). It follows that on the event n = {Xτy1 ,y2 = y1 } V (FXτi |n ) = V (FXτH |n ) = V (H), while on the event m = {Xτy1 ,y2 = y2 } V (FXτi |m ) = V (FXτG

i

|m )

= V (Gi ).

Here we have used the fact that diffusions satisfy the strong Markov property which implies that the distribution of Xτ conditional on an event A in the sigma-algebra of another stopping time τˆ, which is finite with probability one, depends on its sigma-algebra σ(ˆ τ ) only through Xτˆ . Note that the union of m and n is the whole sample space and thus has probability one of occurring. This uses continuity of the diffusion process X. Because V (G1 ) ≥ V (G2 ) DCP yields that τ1 y0 τ2 . In all, due to the Definition 1 and DCP, it follows V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Step 2b. We now assume that H with supp(H) ⊂ (w, b) has y1 = y2 instead. In this case, one can find a sequence of distributions Hn , n ∈ N with (1) supp(Hn ) ⊂ (w, b), (2) Hn →H, n→∞ in distribution and (3) so that the y1n in (w, b) defined by Ex∼Hn [S(x, y1n )] = 0 7

Intuitively, we paste together the two distributions Gi and H in such a way so that from the perspective of the agent at time 0, when diffusion is started at y0 the probability that τH is realized is precisely α. 8 One checks easily that the events {τi > t} for t ≥ 0 depend only on the evolution of the process till time t.

6

have y1n > y2 (since S(·, ·) is increasing in the first argument and decreasing in the second, it suffices to perturb H by shifting some probability on higher values within the support of H). The case above gives V (αHn + (1 − α)G1 ) ≥ V (αHn + (1 − α)G2 ) and Continuity of V establishes that V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Step 3. Finally, it remains to consider the case of general H ∈ ∆([w, b]) with support possibly including w or b. In this case, we can again find a sequence Hn , n ∈ N with (1) supp(Hn ) ⊂ (w, b), (2) Hn →H, n→∞ in distribution. The arguments above then give V (αHn + (1 − α)G1 ) ≥ V (αHn + (1 − α)G2 ) and Continuity of V establishes again that V (αH +(1−α)G1 ) ≥ V (αH +(1−α)G2 ). This establishes the proof for the case of step distributions G1 , G2 . We now use the following Claim and continuity to close the proof for the case when G1 , G2 are not necessarily step distributions (i.e. their respective lotteries don’t have finite support). Claim 2 For G a cdf, there exists sequences of step cdf-s Gi,n , n ∈ N, i = 1, 2 so that Gi,n →Gi in distribution with Gi,n V (G1 ) ≥ V (G2 ) > V (G2,n ) and Gi,n →Gi , weakly for n→∞. This, Continuity of V and the fact that Independence holds for the triplets G1,n , G2,n , H finishes the proof of the Lemma. Given the result of Lemma 1 and the Continuity assumption on , Theorem 3 of Grandmont (1972) yields a representation V (F ) = EF [u] with a bounded and continuous u : [w, b]→R. Requirement 1 then establishes that u is also strictly increasing. This finishes the proof of sufficiency. We end this section with a remark which connects Theorem 1 to the results in Duffie, Epstein (1992). Remark 1. Duffie, Epstein (1992) considers recursive preferences in a setting similar to ours of information originating from Brownian diffusions. Their preferences are defined over stochastic consumption streams. They show that recursive preferences are time consistent in a manner similar to Definition 2: if a consumption process c1 is ranked unambiguously better than another c2 after a stopping time and both are indistinguishable before the stopping time, c1 is unambiguously better from the perspective of time zero as well. Recursive preferences don’t necessarily satisfy Consistency in the sense of Definition 1: the evaluation of a stochastic consumption stream depends at different points in time 7

on the stream in consideration. We conjecture that the only recursive preferences which satisfy the natural Consistency definition in the setting of stochastic consumption processes are the standard Discounted Expected Utility preferences, i.e. that an analogous result to Theorem 1 holds in the setting of Duffie, Epstein (1992) as well.

2

Relation between weak Risk Aversion and Risk Aversion

Weak Risk Aversion (wRA) is a crucial concept for the characterization of the optimal stopping behavior of a naive agent. It is an implication of risk aversion, defined as aversion to mean-preserving spreads.9 Here we clarify precisely the behavioral relation between these two properties of preference. The axiom needed to establish the relation is a relaxation of the Independence Axiom from Expected Utility Theory, which we restate here for comparison and reader’s reference. Axiom: Independence For G1 , G2 , F ∈ ∆([w, b]) with G1 G2 and any α ∈ [0, 1] we have that αH + (1 − α)G1 αH + (1 − α)G2 . The following relaxation is needed for the proof of Proposition 1. Axiom: Mixture Monotonicity w.r.t. Certainty (MMC) Let for i = 1, . . . , n be P xi ∈ [w, b], Fi be lotteries with E[Fi ] = xi and αi ≥ 0 with i αi = 1. If Fi δxi for all i = 1, . . . , n, then X X αi F i  α i δx i . i

i

If Fi δxi for all i = 1, . . . , n, then X

αi F i 

X

i

α i δx i ,

i

Here, mixture operator is in the sense of distributions. This axiom says that one can aggregate preference comparisons as long as one (and the same) side of the comparisons concern the certain expected value of the other respective side of the comparisons. Obviously, MMC is implied by Independence. It is well known in the decision theory literature that for Expected Utility preferences risk aversion is equivalent to wRA and that for non-Expected Utility preferences risk aversion is stronger than wRA. The following Proposition states, that in the case of non-Expected Utility risk preferences, MMC is precisely the weakening of Independence needed for which wRA implies risk aversion. Proposition 1. 1) The following are equivalent. (a)  satisfies Risk Aversion in the sense of aversion to mean-preserving spreads 9

See Section 6.D. in Mas-Colell, Whinston, Green (1995) for a definition and discussion of this concept.

8

(b)  satisfies Mixture Monotonicity w.r.t. Certainty and weak Risk Aversion everywhere. 2) The following are equivalent. (a)  satisfies Mixture Monotonicity w.r.t. Certainty and strong not wRA everywhere: for all x ∈ (w, b) and F ∈ ∆([w, b]) with E[F ] = x we have F δE[F ] . (b)  satisfies Risk Loving in the sense of preference for mean-preserving spreads. Proof of Proposition 1. 1) We show that (b) implies (a) first. Assume  satisfies wRA and MMC. Let F be a mean preserving spread of G. Then there exists a probability kernel K : [w, b] × [w, b]→[0, 1] 10 such that  Z Z K(dz, y) dG(y), F (A) = A

and

Z zK(dz, y) = y, for all y ∈ [w, b].

This characterization follows from the arguments in Example 6.D.2 in Chapter 6 of MasColell, Whinston, Green (1995). In particular, we have that the distribution K(·, y) is different from δy for y ∈ [w, b] only by a zero-mean bet. It follows from wRA, that K(·, y)δy , for all y ∈ [w, b].

(1)

Now MMC implies that F G, if G is a step function. For a general distribution G there is a sequence of step functions Gn , which are also probability distributions, such that Gn converges weakly to G. It follows for Z Fn (z) = K(z, y)dGn (y), z ∈ [w, b]. that Fn is a mean preserving spread of Gn and that therefore Fn Gn due to the previous argument. We now use the following Fact 1 If a sequence of distributions Gn , n ≥ 1 converges weakly to G, then for all upper-semicontinuous functions f : [w, b]→R we have Z Z lim sup f (z)dGn (z) ≤ f (z)dG(z). (2) n→∞ 10

I.e. K(z, ·) is measurable for each z ∈ [w, b] and z 7→ K(z, y) is a probability distribution function for all y ∈ [w, b].

9

Proof of Fact 1. For f an indicator of a closed set, the result follows from the socalled Portmanteau Theorem.11 Otherwise, the result is standard, once it is recalled that an upper-semicontinuous function over a compact set has a maximum and thus is bounded from above. A reference is for example Theorem 1.3.4 in Van Der Vaart, Wellner (1996). It follows that due to weak convergence of Gn to G and the fact that K(·, y) is uppersemicontinuous for all y ∈ [w, b], it being a probability distribution, one has due to the Fact just proved, that lim sup Fn (y) = lim sup Ex∼Gn [K(x, y)] ≤ Ex∼G [K(x, y)] = F (y), n→∞

y ∈ [w, b]

(3)

n→∞

Assume by contradiction that F G. Due to continuity, there is a natural number N s.t. F Gn for all n ≥ N . Furthermore, the space ∆([w, b]) being compact w.r.t. convergence in distribution, we have a subsequence Fnk , k ≥ 1 converging weakly to some F¯ ∈ ∆([w, b]). Now we note the following fact. Fact 2 For a distribution F ∈ ∆([w, b]) the set of discontinuities is countable and each open interval of [w, b] has a point where F is continuous. Proof of Fact 2. It suffices to show that the number of discontinuities is countable as the second part follows from the fact that any open, non-empty interval has uncountably many points. Any discontinuity of f is in 1 }. n Since f (b) − f (w) = 1, any of the An sets cannot contain infinitely many elements. ∪n≥1 An := ∪n≥1 {x ∈ [w, b] : f (x+) − f (x−) >

We use this claim on the implication of (3) which implies that F¯ (y) ≤ F (y), for all continuity points y of F¯ .

(4)

Given this, we note that for arbitrary y ∈ [w, b] we have F¯ (y) = lim F¯ (y + n ) ≤ lim F (y + n ) = F (y). n →0+

n →0+

Here, in the first equality we have used Claim 2 to construct a sequence of positive numbers n going to zero, s.t. F¯ is continuous in the points y + n as well as right-continuity of F¯ , whereas the inequality follows from (4) and the last equality again from right-continuity of F . This gives that F¯ FOSD-dominates F . In particular, for all nk ≥ N we have F¯ Gnk . But Fnk converges weakly to F¯ , so that there exists nk , k large enough with Fnk Gnk . This is a contradiction to the assumption. It follows that F G and thus the conclusion for arbitrary F as well. Now we show that (a) implies (b). Risk aversion implies directly wRA. Given this, the case Fxi +i Fxi never occurs in the strict form, so that for MMC we can focus on the case Fxi +i Fxi . But it is easy to see that X αi Fxi +i i 11

See Theorem 4.25 in Kallenberg (2006).

10

is a mean preserving spread of the distribution of the lottery 2) The proof is analogous to the proof of 1).

3

P

i

α i δx i .

Additional Applications of Theorems 1 and 2

S2.1: Applications to Quadratic Preferences Quadratic preferences are a special case of smooth preferences, which were introduced in Machina (1982) as a response to Allais paradox and several other related paradoxes of Expected Utility. Smooth preferences are precisely those risk preferences which can be represented by a functional which is Fr´echet-differentiable.12 This implies that locally, preferences look like Expected Utility, even though globally Independence Axiom may be violated. This allows to explain the Allais paradox. Here we concentrate on the main example of smooth preferences from Machina (1982): quadratic preferences. Given functions R : [w, b]→R and T : [w, b]→R these are given by the preference functional Z Z 1 1 (5) V (F ) = RdF + [ T dF ]2 = EF [R] + EF [T ]2 . 2 2 One says the preference is properly quadratic if T is a non-constant function.13 They were axiomatized in Chew et al. (1991). Another way to represent quadratic preferences used in Chew et al. (1991) is given by Z Z 1 φ(x, y)dF (x)dF (y).14 (6) V (F ) = 2 A sufficient condition in the case of the naive agent for y0 to be in the continuation region of X, is the existence of a stopping time τ such that 1 1 E[R(Xτ )] + [E[T (Xτ )]2 > R(x0 ) + T (x0 )2 . 2 2 Similarly, the naive agent will stop immediately if R, T are such that for every stopping time τ which is strictly positive with positive probability, we have 1 1 E[R(Xτ )] + [E[S(Xτ )]2 < R(x0 ) + S(x0 )2 . 2 2 From this we can derive immediately classical results for the case the agent’s preferences are Expected Utility.15 12

We refer to Machina (1982) for the formal definition and the properties. This definition corresponds to a function being properly quadratic in Chew et al. (1991). 14 To see this rewrite from the definition in equation (6) of Machina (1982) as Z Z 1 V (F ) = (R(x) + R(y) + T (x)T (y)) dF (x)dF (y). 2 13

15 This corresponds to the case that the preferences as represented in (5) are called not properly quadratic.

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A special case: Expected Utility Set T ≡ 0 in (5). From the previous paragraph, the following Proposition is immediate. Proposition 2. An EU-agent with Bernoulli utility u always stops a diffusion Xt with probability one if u(Xt ) is a Supermartingale. A EU agent never stops a diffusion with probability one if u(Xt ) is a Submartingale. This generalizes Proposition 1 from Ebert, Strack (2015), because their assumptions on the Bernoulli utility function and the process Xt are precisely so that R(Xt ) in their setting is a strict Supermartingale. A formal proof of this result, which we omit since it is easy, uses the fact that the Bernoulli utility of an EU agent is bounded, if her risk preference is defined on the set of all possible distributions over [w, b].16 Sufficient conditions for the Supermartingale or Submartingale property are easy to find, for example by employing Ito-s Lemma.17 Corollary 1. Assume that u : [w, b]→R is twice continuously differentiable. Then, u(Xt ) 2 is a Supermartingale if µ(x)u0 (x) + σ 2(x) u00 (x) ≤ 0 and the EU-agent stops with probability 2 one. u(Xt ) is a Submartingale if µ(x)u0 (x) + σ 2(x) u00 (x) ≥ 0 and the EU-agent never stops with probability one. This Corollary has the following nice interpretation in terms of the00 traditional concept (x) of the coefficient of absolute risk aversion, defined as A(x) = − uu0 (x) . Whenever the 1 normalized drift is below 2 A(x) for all x ∈ [w, b] the agent never starts and whenever it is above the agent never stops with probability one. So the preference constraint in the case of an agent with Expected Utility can be described uniquely by the coefficient of absolute risk aversion A. Another special case of quadratic preferences: Choice-Acclimating Personal Equilibrium (CPE) This equilibrium concept was introduced in K¨oszegi, Rabin (2007) to model preferences under risk for an agent who experiences expectation-based loss aversion. They have become very popular in the applied behavioral economic theory due to their tractability compared to other models of stochastic reference dependence. Due to the work in Masatlioglu, Raymond (2016) we know that CPE, taken as a static risk preference, is the intersection of quadratic and rank-dependent preferences. As quadratic preferences they are encoded by the function φ : [w, b] × [w, b]→R given by φ(x, y) =

1 (u(x) + u(y) + (1 − λ)|u(x) − u(y)|) , 2

where u : [w, b]→R is an increasing function and 2 ≥ λ > 1 is the loss aversion parameter.18 This parameter determines the magnitude of the disutility the agent experiences when comparing each possible realized outcome of the lottery with a better outcome which could have been realized. Thus CPE preferences are defined uniquely by the pair (u, λ). Masatlioglu, Raymond (2016) establish that CPE preferences are quasiconvex. 16

See Theorem 3 in Grandmont (1972). See Chapter 3 of Karatzas, Shreve (2012). 18 The restriction 2 ≥ λ > 1 is needed to ensure that the preferences are FOSD-monotonic. 17

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Theorem 1 in the main body of the paper and the results in Masatlioglu, Raymond (2016) give the following characterization for the case of a naive CPE agent. Proposition 3. Consider a CPE agent with loss aversion parameter λ ∈ (1, 2] and utility u : [w, b]→R, who is naive about the dynamic inconsistency she faces. Then the following holds for any stopping problem (X, y0 ) with scaling function S 1. If u ◦ S −1 is concave, then y0 is in the stopping region of X. 2. If u ◦ S −1 is strictly convex at 0, then y0 is in the continuation region of X. We now turn to the sophisticated case general for quadratic preferences. Recall the specification (5) of quadratic preferences. Assume that R, T : [w, b]→R are differentiable and so, that for every F ∈ ∆([w, b]) we have R0 (x) + T 0 (x)EF [T ] > 0.

(7)

This condition ensures, via the characterization of smooth preferences in Machina (1982), that the quadratic preferences we consider fulfill FOSD-monotonicity.19 In line with Axiom 2 from the Set Up section in the main body of the paper, we assume (7) in the following, as it is part of the basic assumptions we have imposed on the preference. We prove the following in the appendix. Proposition 4. 1. If R, T are continuously differentiable it follows that the sophisticated agent always starts some diffusion. 2. A sophisticated agent with monotonic CPE preferences (λ ∈ (1, 2]) and differentiable u always starts some diffusion. Besides the case when R, T are differentiable, one could hope to get the never stopping result if we restrict to quadratic preferences whose representation has non-differentiable R or T . As Proposition 4 shows, this is not true for the subclass of CPE preferences.

S2.2: Proofs for the results on Quadratic Preferences Proof of Corollary 1. Ito-s lemma applied to u(Xt ) gives Z t Z t σ 2 (Xs ) 00 0 u (Xs )ds + σ(Xs )u0 (Xs )dWs . u(Xt ) = u(y0 ) + µ(Xs )u (Xs ) + 2 0 0 The last term in the RHS is a zero-mean Martingale. The result now follows immediately by taking expectations. Proof of Proposition 3. We first prove the following Claim, which holds more generally for quadratic preferences as defined in (6). 19

Details are in the Appendix in the proof of Proposition 4.

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Claim Given a quadratic preference over ∆([w, b]) identified with the properly quadratic function φ, the preference over ∆([S(w, y0 ), S(b, y0 )]) given by VS (G) = V (G ◦ S) is again a proper quadratic preference encoded by φˆ : [S(w, y0 ), S(b, y0 )] × [S(w, y0 ), S(b, y0 )]→R ˆ r) = φ(S −1 (z), S −1 (r)). given by φ(z, Proof of Claim. One writes Z Z Z Z 1 b bˆ 1 b b φ(x, y)dF (x)dF (y) = φ(S(x), S(y))dF ◦ S −1 (S(x))dF ◦ S −1 (S(y)) V (F ) = 2 w w 2 w w Z Z 1 S(b) S(b) ˆ = φ(z, r)dF ◦ S −1 (z)dF ◦ S −1 (r). 2 S(w) S(w)

This claim implies directly, that for CPE preference the preference VS is given by the pair (u ◦ S −1 , λ). 1. According to Proposition 6 in Masatlioglu, Raymond (2016) the CPE agent (u ◦ S −1 , λ) is then risk averse everywhere. 2. If uˆ := u ◦ S −1 is strictly convex at 0 we have for the lottery 1 1 L = δ− + δ+ , 2 2 that 1 1 1 V (L) = (ˆ u(−) + uˆ(+) − (λ − 1)|ˆ u(+) − uˆ(−)|) + uˆ(−) + uˆ(+) > uˆ(0), 4 4 4 for all  > 0 small enough, due to strict convexity and continuity of uˆ (the latter is ensured by convexity). I.e. we have just showed that the agent with (u ◦ S −1 , λ) violates wRA at 0. In particular, part (3) of Theorem 1 in the main body of the paper is applicable.

Proof of Proposition 4. 1. We first establish the following. Claim The quadratic preference given by (5) satisfies FOSD-monotonicity if and only if (7) holds true. Proof of Claim. Given a functional V representing preferences over lotteries, V is called Fr´echet differentiable or smooth if for all F there exists a local Bernoulli utility function u(·, F ), such that for all lotteries G one has V (G) = V (F ) + EG [u(x, F )] − EF [u(x, F )] + o(||F − G||),

F →G

where the norm used in the o-term is Z ||F − G|| =

|F (z) − G(z)|dz.

One calculates that the local utility function for quadratic preferences is x 7→ u(x, F ) = R(x) + T (x)EF [T ]. Theorem 1 in Machina (1982) gives now the claim. 14

Let a < x < c be fixed and let’s calculate the slope of the q-curve. It fulfills the equation 1 (1 − q(y))R(a) + q(y)R(c) + (1 − q(y))2 T (a)2 + (1 − q(y))2 T (c)2 2 1 + 2q(y)(1 − q(y))T (a)T (c)) = R(y) + T (y)2 . 2 Using the Implicit Function Theorem from Real Analysis and letting y→a one gets the equation characterizing q 0 (a): −q 0 (a+)(R(a) − R(c)) − q 0 (a+)T (a)2 − 2q 0 (a+)T (c)2 − 2q 0 (a)T (a)T (c) = R0 (a) + T (a)T 0 (a). Here we see that q 0 (a+) can’t be +∞. We calculate the slope at c in a similar way to find, that it is characterized by −q 0 (c−)(R(a) − R(c)) − 2q 0 (c−)T (a)T (c) = R0 (c) + T (c)T 0 (c). We see that q 0 (c−) = 0 would imply R0 (c) + T (c)T 0 (c) = 0 and we need this to hold for all b > c > x. Plugging this identity into (7) implies though, that we need for all possible F T 0 (c)[EF [T ] − T (c)] > 0. We have a contradiction as soon as we pick F = δc . The result now follows from the general results in Theorem 2 of the main body of the paper. 2. We know from Masatlioglu, Raymond (2016), that a monotonic CPE agent is RDU with the probability weighting function wλ (z) = (2 − λ)z + (λ − 1)z 2 . Calculate from this, that wλ0 (0+) = 2 − λ, wλ0 (1−) = λ and recall that it is assumed λ ∈ (1, 2]. Now Theorem 2 of the main body of the paper completes the proof.

S2.3: Other examples for the naive RDU agent Other examples for RDU naive behavior can be constructed with the help of results from Xu, Zhou (2013), who for example show that, an agent with commitment stops always, 1 if w is convex and U (x) := u(x β ) is concave. The latter is the case if the dynamics of the diffusion Xt are unfavorable (low normalized drift σµ2 ). In particular, there is a whole class of naive RDU agents who are first-order risk averse everywhere in the sense of Segal, Spivak (1990)20 and who stop a whole class of geometric Brownian motions. For a more detailed class of examples in this spirit: it is easy to see, that u concave, µ ∈ (0, 12 σ 2 ) and w convex are sufficient conditions for the naive to stop a whole class of geometric Brownian motions.21 20 21

See Proposition 4 in their paper Calculate 4µ 1 1 1 1 2µ − 2µ2 −1 U 00 (x) = u00 (x β ) 2 x− σ2 − u0 (x β ) x σ . β β σ2

If u is concave and µ ∈ (0, 12 σ 2 ) is concave then U is concave as well.

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S2.4: Applications to Cautious Expected Utility: a result for the sophisticated agent Here we give a sufficient condition for the sophisticated CEU agent to actually start some diffusion. These conditions are rather weak, which implies that extreme sensitivity is generically not satisfied for CEU models. Proposition 5. Assume that the set U is a compact set of strictly increasing functions and that the one-sided derivatives of all u ∈ U are bounded uniformly away from zero and infinity: i.e. the following is fulfilled sup sup u0 (x) < +∞, u∈U x∈[w,b]

inf inf u0 (x) > 0.

u∈U x∈[w,b]

Then for all y0 ∈ [w, b] there are diffusions X such that y0 is in the continuation region of X. Proof. Note first, that to ensure FOSD-monotonicity of V , the minimal set U , in the sense of inclusion, in the representation22 has to contain only strictly increasing functions. Given that U is compact, for each y0 ∈ (w, b) there exists uy some element in the U such that V ((1 − q(y))δw + q(y)δb ) = u−1 y (q(y)) = y. It follows q(y) = uy (y). We know that q(y) is non-decreasing so that it has one-sided derivatives at w, b. It follows 1 − uy (y) 1 q(b) − q(y) = = b−y b−y b−y

Z

b

u0y (s)ds,

y

where we have used that the functions u ∈ U are absolutely continuous due to being non-decreasing. The assumption in the statement implies then that q 0 (b−) > 0. One can show similarly, that q 0 (w−) < +∞. The result follows then from the general results for the sophisticated case.

References Cerreia-Vioglio, S., Dillenberger, D. and Ortoleva, P., 2015. Cautious Expected Utility and the Certainty Effect, Econometrica, 83(2), pp.693-728. Chew, S.H., Epstein, L. G. and Segal, U., 1990. Mixture Symmetry and Quadratic Utility, Econometrica, pp.139-163. Duffie, D. and Epstein, L.G., 1992. Stochastic Differential Utility, Econometrica, pp.353394. Ebert, S. and Strack, P., 2015. Until the Bitter End: On Prospect Theory in a Dynamic Contest, The American Economic Review, 105(4), pp.1618-1633. Grandmont, J-M., 1972. Continuity Properties of Paretian Utility, Journal of Economic Theory, 4(1), pp.45-57. 22

See section 2.5 of and in particular Theorem 2 in Cerreia-Vioglio et al. (2015) for the meaning of ‘minimal set U ’.

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Gul, F. and Lantto, O., 1990. Betweenness Satisfying Preferences and Dynamic Choice, Journal of Economic Theory, 52(1), pp.162-177. Hammond, P.J., 1988. Consequentialist foundations for expected utility, Theory and decision, 25(1), pp.25-78. Hammond, P.J., 1989. Consistent Plans, Consequentialism and Expected Utility, Econometrica: Journal of the Econometric Society, pp.1445-1449. Kallenberg, O., 2006. Foundations of Modern Probability, Springer Science and Business Media. Karatzas, I and Shreve, S., 2012. Brownian motion and stochastic calculus (Vol. 113). Springer Science and Business Media. K¨oszegi, B. and Rabin, M., 2007. Reference Dependent Risk Attitudes, The American Economic Review, 97(4), pp.1047-1073. Machina, M., 1982. Expected Utility Analysis without the Independence Axiom, Econometrica: Journal of the Econometric Society, pp.277-323 Machina, M., 1989. Dynamic Consistency and Non-Expected Utility models of Choice under Uncertainty, Journal of Economic Literature, 27(4), pp.1622-1668. Mas-Colell, A., Whinston, M.D. and Green, J.R., 1995. Microeconomic theory (Vol. 1). New York: Oxford University Press. Masatlioglu, Y. and Raymond, C., 2016. A Behavioral Analysis of Stochastic Reference Dependence, The American Economic Review, 106(9), pp.2760-2782. Segal, U. and Spivak, A., 1990. First Order vs. Second Order Risk Aversion, Journal of Economic Theory, 51(1), pp.111-125. Van der Vaart, A.W. and Wellner, Wellner, J.A., 1996. Weak Convergence and Empirical Processes, Springer Verlag Xu, Z.Q. and Zhou, X.Y., 2013. Optimal Stopping under Probability Distortion, The Annals of Applied Probability, 23(1), pp.251-282.

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