The Magnitude and “Peanuts” Effects: Searching Implications

Abstract

The framework of this paper is the field of decision-making processes in which people face the choice between probabilistic and dated rewards. Traditionally, the preferences for probabilistic outcomes have been analyzed by the Expected Utility (EU) model whilst the preferences for dated rewards have been studied by the Discounted Utility (DU) model. Nevertheless, recent empirical findings have revealed the existence of several anomalies or paradoxes in both contexts. Specifically, EU and DU models exhibit an anomaly affecting the amount of the reward, viz the "peanuts" and the magnitude effects, respectively, which seem to go in opposite directions. The aim of this paper is to analyze both effects jointly in a wide setting involving choices subject to risk and over a period of time, and thereby identify and consider the implications of one anomaly on the other.

Discounted and Expected Utility from the Probability and Time Trade-Off Model

“…In effect, most existing literature presents the anomalies in intertemporal and probabilistic choices as separate inconsistencies [21] and, indeed, this paper can be used to clarify the equivalence between certain anomalies analyzed in the context of the DU and the EU models. A precedent can be found in the paper by Cruz Rambaud and Sánchez Pérez about the so-called peanuts effects [7]. This methodology could be applied to other anomalies present in both aforementioned models.…”

“…Additionally, we will try to build a PTT model starting from a discounted and an expected utility model able to overcome the limitations pointed out by Baucells and Heukamp.On the other hand, the decision-making under uncertainty involves alternatives whose rewards differ in relation to the probability of being received (the choice between "a smaller reward, to be received with greater probability, and a larger one, but less likely") [3].Both kinds of decisions have been traditionally analyzed by using two main systems of calculation: the Discounted Utility model and the Expected Utility theory which describe the present value of delayed rewards and the actuarial value of risky rewards, respectively. Both models are simple, widely accepted, and with a similar structure, since they use the same theoretical principle: the rewards are assessed by the sum of their utilities [6,7]. In the decision-making process, individuals choose the alternative which maximizes their current utility value (denoted by U 0 ).As seen in Table 1, DU and EU are the basic models employed in the decision-making process which represent the rational choice over time and under risk, respectively.…”

“…Both kinds of decisions have been traditionally analyzed by using two main systems of calculation: the Discounted Utility model and the Expected Utility theory which describe the present value of delayed rewards and the actuarial value of risky rewards, respectively. Both models are simple, widely accepted, and with a similar structure, since they use the same theoretical principle: the rewards are assessed by the sum of their utilities [6,7]. In the decision-making process, individuals choose the alternative which maximizes their current utility value (denoted by U 0 ).…”

“…Another way to conclude that delay and risk choices have non-parallel decision mechanisms has been proved by Cruz Rambaud and Sánchez Pérez [7] and Chapman and Weber [17]. The effect of the reward size on the treatment of delayed and probabilistic outcomes moves in the opposite direction.…”

“…By applying Lemmas 2 and 3, it can be shown that the former definitions are equivalent in the presence of the so-called reverse sub-endurance (see [7]):…”