scholarly journals The Reflection Effect for Higher Order Risk Preferences

2020 ◽  
pp. 1-45
Author(s):  
Han Bleichrodt ◽  
Paul van Bruggen

Higher order risk preferences are important determinants of economic behaviour. We apply insights from behavioural economics: we measure higher order risk preferences for pure gains and losses. We find a reflection effect not only for second order risk preferences, like Kahneman and Tversky (1979), but also for higher order risk preferences: we find risk aversion, prudence and intemperance for gains, and much more risk loving preferences, imprudence and temperance for losses. These findings are at odds with a universal preference for combining good with bad or good with good, which previous results suggest may underlie higher order risk preferences.

2013 ◽  
Vol 2 (1) ◽  
pp. 1-29
Author(s):  
Philip O'Connor

Exotic bets: exactas, trifectas and superfectas are complicated gambles that depend on the ordering of horse in a race that can be studied by converting them into “synthetic” or “virtual” win bets. Using two ways of constructing synthetic win bets, it is shown that the favorite-longshot bias is a poor description of the returns of the trifecta and superfecta synthetic win bet. Rather, consistent with financial markets, the standard deviation of the payout of the synthetic win bet better describes the different returns of synthetic win bets.It is found that the synthetic win market dislikes standard deviation and kurtosis (and other higher-order even moments) and likes skewness (and other higher order odd moments), implying participants conform to standard utility theory in their choice between win and synthetic win bets and are not risk-loving. A co-efficient of relative risk aversion of about 3 is estimated. Including higher-order moments strongly affects the magnitude of utility function estimates.


2021 ◽  
Author(s):  
◽  
Rana Asgarova

<p>Prospect Theory models behaviour in one-off decisions where outcomes are described. Prospect Theory describes risk aversion when the choice is between gains and risk seeking when the choice is between losses. This asymmetry is known as the reflection effect. In choices about experienced outcomes, individuals show risk seeking for gains and risk aversion for losses. This change in the direction of gain-loss asymmetry is known as the description-experience gap. Across eight experiments, we examined gain-loss asymmetry in two experiential choice procedures. We compared the obtained results with predictions derived from Prospect Theory and the description-experience gap literature.  In Study 1, we evaluated the predictions of the reversed reflection effect in probability discounting. Probability discounting is loss in reinforcer value as a function of uncertainty. In typical tasks measuring discounting, participants choose between smaller, certain amounts and a larger amount at one of several probabilities. In choice from description, most participants show a gain-loss asymmetry consistent with the predictions of the reflection effect, discounting gains more steeply than losses. Across three experiments, we examined whether gain-loss asymmetry also occurred when participants experienced the outcomes they chose, when they chose between two uncertain options, and when these two contexts were combined. Across all of the above contexts, no consistent mean difference in discounting of gains and losses was observed. Rather, in most of the tasks that provided experienced outcomes, the participants showed steeper discounting in the first condition completed, whether it involved choices about gains or losses. Furthermore, subsequent conditions produced shallower discounting, but notably, not shallower than choice based on the expected value of the options. In Studies 2 and 3, we followed-up on this order effect by providing the participants with experience of probabilistic outcomes before the discounting tasks. Participants discounted losses more steeply than gains, consistent with the predictions of a reversed reflection effect.  In Study 2, we examined gain-loss asymmetry in a rapid-acquisition choice procedure using concurrent variable-interval schedules – the Auckland Card Task. Participants repeatedly chose between two decks of cards that varied in the frequency or magnitude of available gains or losses. Participants were more sensitive to changes in gain than loss frequency between the two decks, consistent with the predictions of a reversed reflection effect, while sensitivity to gain and loss magnitude did not show an asymmetry. We found a novel asymmetry in the local effects of gains and losses. In the frequency tasks, gains disrupted the general pattern of responding more than losses. In the magnitude tasks, varying the magnitude of losses had a bigger effect on local-level patterns following outcomes than varying the magnitude of gains.  Across the two tasks we observed patterns of gain-loss asymmetry consistent with the predictions of a reversed reflection effect. We also observed several inconsistencies, particularly when comparing behaviour to choices that would maximize the expected returns. Our research suggested that sufficient exposure to chance outcomes and ensuring delivery of scheduled events are key challenges in further refinement of experiential choice in human operant tasks.</p>


2021 ◽  
Author(s):  
◽  
Rana Asgarova

<p>Prospect Theory models behaviour in one-off decisions where outcomes are described. Prospect Theory describes risk aversion when the choice is between gains and risk seeking when the choice is between losses. This asymmetry is known as the reflection effect. In choices about experienced outcomes, individuals show risk seeking for gains and risk aversion for losses. This change in the direction of gain-loss asymmetry is known as the description-experience gap. Across eight experiments, we examined gain-loss asymmetry in two experiential choice procedures. We compared the obtained results with predictions derived from Prospect Theory and the description-experience gap literature.  In Study 1, we evaluated the predictions of the reversed reflection effect in probability discounting. Probability discounting is loss in reinforcer value as a function of uncertainty. In typical tasks measuring discounting, participants choose between smaller, certain amounts and a larger amount at one of several probabilities. In choice from description, most participants show a gain-loss asymmetry consistent with the predictions of the reflection effect, discounting gains more steeply than losses. Across three experiments, we examined whether gain-loss asymmetry also occurred when participants experienced the outcomes they chose, when they chose between two uncertain options, and when these two contexts were combined. Across all of the above contexts, no consistent mean difference in discounting of gains and losses was observed. Rather, in most of the tasks that provided experienced outcomes, the participants showed steeper discounting in the first condition completed, whether it involved choices about gains or losses. Furthermore, subsequent conditions produced shallower discounting, but notably, not shallower than choice based on the expected value of the options. In Studies 2 and 3, we followed-up on this order effect by providing the participants with experience of probabilistic outcomes before the discounting tasks. Participants discounted losses more steeply than gains, consistent with the predictions of a reversed reflection effect.  In Study 2, we examined gain-loss asymmetry in a rapid-acquisition choice procedure using concurrent variable-interval schedules – the Auckland Card Task. Participants repeatedly chose between two decks of cards that varied in the frequency or magnitude of available gains or losses. Participants were more sensitive to changes in gain than loss frequency between the two decks, consistent with the predictions of a reversed reflection effect, while sensitivity to gain and loss magnitude did not show an asymmetry. We found a novel asymmetry in the local effects of gains and losses. In the frequency tasks, gains disrupted the general pattern of responding more than losses. In the magnitude tasks, varying the magnitude of losses had a bigger effect on local-level patterns following outcomes than varying the magnitude of gains.  Across the two tasks we observed patterns of gain-loss asymmetry consistent with the predictions of a reversed reflection effect. We also observed several inconsistencies, particularly when comparing behaviour to choices that would maximize the expected returns. Our research suggested that sufficient exposure to chance outcomes and ensuring delivery of scheduled events are key challenges in further refinement of experiential choice in human operant tasks.</p>


2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.


Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.


2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.


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