History-Dependent Risk Aversion and the Reinforcement Effect
Introduction
- Examples of the Reinforcement Effect
- Related Literature
The RE says that people become less risk averse after a good history than after a bad history. The RE says that the agent is less risk averse after a good history (high payoff) than after a bad history (low payoff).
Overview of the Main Result and Its Applications
- Main Result
- Application to Asset Pricing
In Section 1.4, I then apply my model to the classic Lucas tree model of asset pricing and draw the implications of RE on asset price dynamics. Another implication of RE is that in the low state, asset prices fall as agents become more risk averse (point 0 to point 1).
Model
- Basic Setup and Model
- The Reinforcement Effect and Monotonicity
- Time Consistency and Relation to Existing Models
In fact, my model generalizes the following additive version of the Kreps-Porteus model (1.5), where V(x,z) =u(x)+βu(z): the utility of (pi,(xi;Zi)) in= 1 ice. Since the additive Kreps-Porteus model (1.6) is a special case of the history-dependent model (1.3), I can derive the following consequence of Theorem 1 for (1.6).
Application: The Lucas Tree Model with HDDA Agents
- Optimal Consumption Profile
- Market Clearing and Equilibrium Asset Price
- The Dynamics of the Price-Dividend Ratio
- Empirical Regularities: High, Volatile, and Predictable As-
At each date, given (zt−1,st−1), the disappointment parameter δt, and the demand for the asset xt, the agent's continuing value of the asset is V(zt−1,st−1, δt,xt). In HDDA, the price-dividend ratio is 1−ββ times the magnitude of the probability distortion, so.
Behavioral Foundations of the History-Dependent Model
The second half of the regularity imposes the other two properties of the DEU, the discounted utility and the expected utility accumulator. The weak separability between today and tomorrow essentially requires that the utility of a deterministic z-score be independent of history (time-consistency of recall); i.e., the utility of z is unaffected by a lottery X and a deterministic outcome µ. Specifically, i) says that if µ is the security equivalent of a simple lottery X, then µ is still the security equivalent of XE and if the agent will get a deterministic outcome in the second period.
Specifying Risk Preferences: Expected Utility, Disappointment Aver-
- History-Dependent Expected Utility (HDEU)
- History-Dependent Disappointment Aversion (HDDA)
- History-Dependent Rank-Dependent Utility (HDRDU)
- Probability Distortion, Belief Change, and Nontriviality
We now turn to our model in which we specify (2.1) in the following two ways. It turns out that RTI binds t andt' in the following way: t satisfies cancellation if0 satisfies cancellation.
Choosing with the Worst in Mind: A Reference-Dependent Model 35
Model
- Diminishing Sensitivity
- Implications of Diminishing Sensitivity
- Implications of the Compromise and Attraction Effects, and
When can we say that our model is a good model for the compromise and attraction effects? We will now argue that predictions of our model are consistent with the implications of the compromise and attraction effects.
Behavioral Foundation
- Axioms and Representation Theorem
- Characterizing Diminishing Sensitivity
- Sketch of the Proof of Theorem 4
In fact, we now show that WBAE is equivalent to observing the attraction effect in a given neighborhood without adopting our model. Therefore, BAE is equivalent to observing the attraction effect near a triplex,y, and (t1,x2), while not near a triplex,y, and (z1,x2). Since decreasing sensitivity is equivalent to BAE, in our model decreasing sensitivity is equivalent to observing the attraction effect in the neighborhood of x,y, and(l,p), but not in that ofx,y, and(z1,x2) .
Applications
- Intertemporal Choice with Non-Binding Borrowing Constraint 59
In the standard model, the optimal consumption profiles are the same when the constraint is not binding; i.e. y1+b¯ > c∗. The borrowing limit lowers the consumer's maximum consumption level for today and increases the minimum consumption level for tomorrow. Suppose the agent prefers xovery in menu {x,y} (chooses the riskier option) but prefers'overx' in some menu A (chooses the safer option).
Model with General Menu-Dependent References
Finally, we briefly discuss the estimation of risk preference when the agent has a reference-dependent additive preference. However, if we estimate risk preferences using only binary comparisons, then it provides an unambiguous measure of risk level because the effect of f is null (recall Note 1 in Section 2.2). But the behavioral predictions of general models will be weaker compared to using only minimal ones.
Related Literature
More precisely, they have a representation similar to (3.1) in which the time dimension corresponds to the probability dimension in (3.1). One example is "choice overload": the idea that a subject may be inclined to make choices when presented with too many alternatives. Cochrane (1999): "By force of habit: A consumption-based explanation of stock-holding behavior," Journal of Political Economy.
Theory of Decisions by Intra-Dimensional Comparisons
Introduction
- Related Literature
Now we introduce our model for the IDC heuristic in the aforementioned two scenarios and discuss some implications of them. Let us first describe our model in the context of the first scenario: choice over binary lotteries. Next, we define our general model in the context of the second scenario, social choice.
The Basic Model
- Separability and Representation Theorem
- Some Specifications and Properties of Distance-Based Func-
Separability captures the idea of the IDC heuristic and guarantees dimension-specific distance-based functions f1,. It is clear that we cannot get any uniqueness without normalizing distance-based functions and an aggregator. Furthermore, when f is a function of utility difference; i.e. f(x,y)= h(u(x)−u(y)), decreasing sensitivity is equivalent to the strict concavity of u.
Discussion on Changing the Domain
PALM can capture the compromise effect when we assume that it is "between" x and with respect to priority. One is the sequential nature of choice, and the other is the utility of the outside option. Proposition 10 states that the utility of the outside option is the source behind "global" increases in the probability of choosing the outside option.
The Perception-Adjusted Luce Model
Introduction
The final consequence is that adding c causes a larger decrease in the probability of choosing a than in the probability of choosing b. This means that the resulting violation of Luce's IIA takes the form of a decrease in the relative probability of choosing an overb. The risk rate is the probability that you will choose an object, provided that you do not choose any of the objects with a higher perception priority.
Primitives and Luce’s model
- PALM
This pattern of choice cannot be explained by Luce's model; indeed, it cannot be explained by any model of random utility. Here we allow for an external option and use the version of Luce's model where it is not possible to choose in A. Each perceived alternative is chosen with probability described by µ, a function that depends on the utility according to Luce's formula (4.1) .
Axioms
So suppose that a has a higher priority than b and that the relative probability of being chosen over bis is lower when the choice set is A∪ {c} than when the choice set is A. Hazard level IIA means that detection advantage explains the change in relative probabilities. : we need to have a compensatory decrease in the probability of selecting the element that is detected before, relative to the probability of selecting the element that is detected earlier. In other words, the relative probability of choosing a over b decreased, and therefore Luce's IIA was violated, because the probability of choosing the item perceived before the preposition increased relative to the probability of choosing the item perceived earlier.
Theorem
- Discussion of Outside Option
The inequality ˆu(x0) ≥ u(x0) reflects that there are two sources behind choosing the outside option in PALM. For example, if the utility of the external option is zero, the external option will not be chosen in Luce's model. Another advantage of PALM's treatment of the external option is that it allows us to understand forced choices.
Compromise and Attraction Effects
- Compromise Effect– Violation of IIA
- Attraction Effect– A Violation of Regularity
- The Effects of Forced Choice
A famous example of the attraction effect was documented by Simonson and Tversky (1992) using the following experiment. Dhar and Simonson show that introducing the no-choice option weakens the compromise effect and reduces the relative share of an option that is “average” on all dimensions. In PALM, choosing the external option more often is linked to greater utility of the external option.
Correlation between Utility u and Perception Priority %
Related Literature
Before we discuss this, note that we can easily obtain a choice-theoretic version of the ODLM in the following way: for any A∈A,. “Stochastic rationality and revealed stochastic preference,” in Preferences, Uncertainty, and Optimality, Essays in Honor of Leo Hurwicz, Westview Press: Boulder, CO, 161–186. rep., Working Paper, Princeton University. Felfernig (2009): “Minimizing product utility estimation errors in recommendation outcome evaluations,” in Proceedings of the 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology-Volume 01, IEEE Computer Society, 20 –27 .
The Order-Dependent Luce Model
Introduction
In this paper we develop a menu-dependent version of Luce's model in which menu dependence is caused by an underlying linear order of alternatives. In section 5.3 we focus on a case where the underlying linear order of alternatives is observed and provide a characterization theorem. In section 5.4 we focus on a case where the underlying linear order of alternatives is not observed.
Basic
In many cases, the underlying linear sequence is subjective, unobservable, or too expensive for a researcher to obtain. One of the main contributions of the paper is that we can uniquely identify the underlying linear sequence from observed choice probabilities (Theorems 16-17). In this paper, we consider a menu-dependent Luce's model in which menu dependence is determined by some underlying linear order on alternatives.
Axioms and Representation Theorem for given R
- Increasing Order-Dependent Luce Model
In this case, the evaluations of the alternatives increase in shelf order: being on the middle shelf is better than being on the bottom shelf, and being on the top shelf is better than being in the middle. Consider the example of a grocery store where a is at the top, bis is in the middle, and cis is at the bottom. Similar to observation 5 in Section 5.2, eliminating bfrom{a,b,c} helps more than because it moves to the middle shelf while still being on the top shelf.
Identifying Unknown R and Revealed Order
The following result proves that R0 is well-behaved and almost complete under asymmetry and transition. The next result shows that Asymmetry and Transitivity are necessary under the conditions of the utility function, giving us sufficient variety in the choice probabilities. Proposition 16 also shows that R0 is almost complete.2 Then the next result also shows that R0 is in fact consistent with R.
Behavioral Phenomena
- Consistency with Violations of IIA–the Similarity Effect and
- Consistency with Violation of Regularity–Attraction Effect . 110
- Choice Overload
The similarity and compromise effects are defined in the same kind of experimental setup. In the experiment studied by Simonson and Tversky (1992), x is X-370, a very basic model of the Minolta camera; y is MAXXUM 3000i, a more advanced model from the same brand; andzis MAXXUM 7000i, the top model that Minolta offers in this class of cameras. Here we show that adding a new alternative to a menu can lead to a decrease in the agent's satisfaction with his chosen option, even if the added alternative does not decrease the average utility of the menu.
Related Literature
Since Luce's model is a special case of both the ODLM and random utility, the ODLM and random utility intersect. However, the ODLM allows for the attractor effect while random utility models always obey regularity, so the ODLM is not a special case of random utility. In fact, a choice-theoretic account of descending ODLMs is a special case of Masatlioglu et al.
Proofs
- Proofs of Theorem 1 and Corollary 1
- Proofs of Theorems 2-3
Now by Jensen's inequality, the above inequality implies that ux0 is more concave than ux. By Jensen's inequality, to satisfy the above inequality for any T ∈∆(R+), fx,x0 must be concave. Second, let me define a function µ(xi,X) for each history (xi,X). From the weak Separability between today and tomorrow ii), pi,(xi;zi)n.
Behavioral Foundations of HDEU and HDDA
- Characterizing HDEU
- Characterizing HDDA
The second axiom is called Additivity Today, which uses the additive structure of the EU in the first period. The first part of Axiom 27 requires that there be a Gul's Disappointment Aversion view when comparing first-period lotteries. The second part of Axiom 27 requires that there be a Gul's Disappointment Aversion view when comparing second-period lotteries.
Proofs
- Proof of Proposition 2
- Proof of Proposition 3
- Proof of Proposition 4
- Proof of Proposition 5
- Proof of Proposition 6
- Implications of Regularity and INEA
- Useful Lemmas for the proof of Theorem 4
- Proof of Theorem 4
Since the right side does not depend on x2, the left side should not depend on x2 either. Since we have obtained a typical Cauchy functional equation (see Kuczma 2008), there exists α > 0 such thath(t) = αt; that is, u0. Since u1,u2, f, f0 are strictly increasing and continuous, similar to step 1, we obtain a Cauchy functional equation.
Alternatives with n -attributes
Using both Maximums and Minimums
- Attraction Effect and Compromise Effect
- Symmetric Dominance and Two Decoy Effect
Finally, note that the symmetric dominance and the two decoy effects are also equivalent as in section 2.2.3, since the maximum and minimum are identical.
Asymmetry of Two Dimensions and Violations of Transitivity
It turns out that the comparison of two dimensions in terms of decreasing sensitivity is closely related to violations of the transitivity of binary comparisons.1 Violations of transitivity are consistently documented in the experimental literature.2. Therefore, if transitivity is violated in a direction x ∼ y ∼ z x, then the first dimension is more decreasingly sensitive than the second dimension (α < β). Finally, if transitivity is violated in a direction x ∼ y ∼ z ≺ x, the second dimension is more sensitive to decreasing behavior than the first dimension (α > β).
Proof of Theorem 5
Now we will prove a lemma showing that there exist continuous distance-based functions corresponding (in some sense) to . Moreover, by continuity and strong monotonicity, fi is also continuous and strictly increasing in its first argument. Since W is continuously and strictly increasing in all its arguments and every fi is continuous.
Proof of Theorem 6
- Necessity
- Sufficiency
Relation to Manzini and Mariotti
Superficially, this representation is similar to ours, but it is actually very different: it is incompatible with our model, in the sense that the set of stochastic choices satisfying our model is distinct from the set of stochastic choices in Manzini and Mariotti's model. Let ρ have a Manzini and Mariotti (2014) representation as above and let X have at least three elements. In Manzini and Marriott's model, however, there is always at least one violation of Luce's IIA.
Proof of Proposition 10
Proof of Proposition 12
Proof of Proposition 13
Proof of Proposition 14
Finite X
To simplify the explanation, we use the following notation in this proof: a` bifa band er is noc ∈ X witha c b.
A modification without the outside option
Similarly, the following observation which is very similar to Proposition 9 shows that the modified PALM can rationalize the withdrawal effect. The two observations above illustrate that the outside option does not really play a role in explaining the two effects, but the sequential procedure does.
Proof of Theorem 7
Proof of Proposition 15
Proof of Proposition 16
Proof of Proposition 17
Completing revealed order R 0
- Lotteries X and Y and bundles (X ; Z ) and (Y ; Z )
- The Price-Dividend Ratio
- Intertemporal Consumption Lotteries
- The Price-Dividend Ratio and the Disappointment Parameter
- The Dynamics of the Price-Dividend Ratio
- The Price-Dividend Ratio (blue) and the Disappointment Parameter
- Compromise and Attraction Effects
- Diminishing Sensitivity When u 1 (x 1 ) = x
- The Effect of a Shift in Reference Points When f (t) = √
- Compromise Effect, Attraction Effect, and Diminishing Sensitivity
- Two Decoy Effect and Two Compromise Effect
- Symmetric Dominance
- All Possible Third Alternatives That Can Cause a Preference Reversal 50
- Reference Translation Invariance (RTI)
- Cancellation
- The Effect of a Non-Binding Borrowing Constraint
- Average Income and Consumption by Education, Attanasio and We-
- Separability when n = 2
- Difference- i
- Compromise effect in Simonson and Tversky (1992)
- Attraction effect in Simonson and Tversky (1992)
- The Compromise effect in Simonson and Tversky (1992)
- Attraction effect in Simonson and Tversky (1992)
- Choice Probabilities in Tversky (1969)
Ho (1994): "Violations of the betweenness axiom and nonlinearity in probability," Journal of risk and uncertainty. “Dynamic Consistency and Nonexpected Utility Models of Choice Under Uncertainty,” Journal of Economic Literature, 1622–1668. 1988): "Equality and Decision Making under Risk (Is There a Utility-Theoretic Solution to the Allais Paradox?)," Journal of Economic Theory.
Strictly Concave h
Asymmetry of Two Dimensions
