# Schedule - Parallel Session 4 - Intertemporal Choice 2

WMG IMC Boardroom 2 - 11:00 - 12:30#### About Delay Aversion: the Topological Approach

Lorenzo Bastianello

#### Abstract

One of the standard assumptions made in most economic models is that agents have preferences for advancing the time of future satisfaction. The classical way of describing impatient preferences is to use a discounted sum of utilities. The willingness of the Decision Maker (DM) to anticipate future consumption is expressed through a discount function. Instead of working utility (and discount) functions, this paper takes as a starting point a behavioral definition that represent the notion of “delay aversion”. Suppose that a DM is to choose between two extra amounts of income, the smaller one paid at an earlier period. Then she is delay averse if she chooses the smaller and earlier extra amount whenever the bigger one is delivered sufficiently far in the future. It turns out that the discounted sum of utilities entails very strong notions of preferences for advancing the time of future satisfaction. If a DM behaves as if she is discounting utilities, then she is, for instance, myopic (see Brown and Lewis (1981)). As a consequence, such a model does not allow to distinguish among different nuances of impatience. In order to study delay averse preferences, a more general framework is needed: this paper uses a topological approach. New topologies over the space of bounded sequences are presented. These topologies “discount” the future consistently with the notion of delay aversion. More formally, I defined a new topology, T_{DA}, over the space of streams streams of income. The key idea is simple. A suitable topology should make a cash flow which pays one unit of income in the n-th period very close to the cash flow paying zero at all periods, provided that n is big enough. Such a property could be rephrased as ‘the far future is negligible”. There are two main mathematical results. First, the delay averse topology, T_{DA}, is weaker than the sup-norm topology, and stronger than the Mackey topology. Second, the dual of T_{DA} is the set of bounded charges, ba (following the standard mathematical notation). These results have, in turns, interesting economic implications. – Delay aversion is a weak notion of preferences for advancing the time of future satisfaction. A delay averse DM is “in between” a myopic agent (see Brown and Lewis (1981)) and a patient one. – If agents do not discount the future strongly enough (namely as a myopic DM), a market equilibrium may fail to exist. This refines a famous result in general equilibrium proven in Araujo (1985). – Usually it is thought that price-bubbles cannot occurs when agents are impatient (see Gilles and LeRoy (1992)). We show that if the DM are delay averse, then this is no longer true. – I show that delay aversion is the correct notion derived from the paper of Benoît and Ok (2007), where the authors studied the concept of more delay aversion but not delay aversion itself.

**Lorenzo Bastianello**

#### Pooled Data Do Not Tell Much About Individuals - an Illustration with Intertemporal Choice Paradigm

Muye Chen; Michel Regenwetter

#### Abstract

Pooled data are frequently and widely used to infer properties of individuals. Combining individuals’ opinions and preferences, however, can be misleading. Pooled data usually are not representative enough for a population because they do not capture all information regarding preferences and opinions conveyed by individuals. Additionally, aggregating data can engender conclusions that contradict the behavior of a population. These issues have been manifested by many paradoxes in social choice theory. Yet, problems of pooled data are still oftentimes overlooked today. Using intertemporal choice paradigms, we show that pooled data do not tell much about individuals due to well-known voting paradoxes. Decisions involving trade-offs among benefits and costs at different time points are referred to as intertemporal choices (Frederick, Loewenstein, and O’Donoghue, 2002). To illustrate issues with pooled data, we consider five intertemporal choice paradigms: the exponential discounting (ED) model (Samuelson, 1937), the hyperbolic discounting (HD) model (Ainslie, 1975), the quasi-hyperbolic discounting (QHD) model (Laibson, 1997), the DRIFT model (Read, Frederick, and Scholten, 2013), and the intertemporal choice heuristics (ITCH) model (Ericson et al., 2015). Using simulation-based examples, we show that pooled data can induce the Condorcet paradox and the paradox of multiple elections within intertemporal choice paradigms. According to the Condorcet paradox (Condorcet, 1785), individual transitive preferences predicted by ITCH can collectively satisfy the logistic specification of an intransitive preference predicted by DRIFT. Furthermore, individual transitive preferences violating all models can collectively satisfy the logistic specification of an intransitive preference predicted by ITCH. Even if data are fit perfectly and repeatedly by a logistic specification of a theory, every individual might violate that theory. Additionally, according to the paradox of multiple elections (Brams et al., 1998), individuals satisfying QHD can collectively satisfy a mismatching QHD preference pattern, and also an HD only preference pattern. We tested aforementioned intertemporal choice models with individual and pooled data collected from laboratory experiments. Using state-of-the-art Bayesian methods (Regenwetter et al. 2014) and assuming each decision maker had fixed preference with choice variability caused by response errors, we disclosed that combining individuals’ decisions can be misleading. The analysis results suggest that HD can best describe and predict individuals’ behavior. Treating all individuals as a whole, however, led to either mismatching HD predictions or chaos that no model can explain. Scholars should take care in inferring properties of individuals from pooled data, and rely more on individual data when they are accessible.

**Michel Regenwetter**

#### A Decision-Theoretic Diagnosis of Attitude Toward Debt

Marc Scholten; Daniel Read; Daniel Walters; Carsten Erner; Craig Fox

#### Abstract

People greatly differ in their attitude toward debt, or their feelings about acquiring debt, and about prolonging debt once acquired. We define debt as an obligation to pay an amount of money in the future, and develop a diagnostic device, the debt battery, to assess people’s attitude toward debt. The debt battery facilitates the emergence of four debt-attitude groups: (1) the debt prone, who discount delayed payments, and both acquire debt (prefer to make a payment in the future rather than now) and prolong debt (prefer a payment later in the future rather than sooner in the future); (2) the debt holders, who combine discounting with a future bias, so that they so not acquire debt, but prolong debt once acquired; (3) the debt averse, who amplify delayed payments, and neither acquire nor prolong debt; and (4) the debt takers, who combine amplification of delayed payments with a present bias, so that they acquire debt, but do not prolong it once acquired. Our focus is on a contradiction. One of the presumably most robust phenomena in intertemporal choice is the magnitude effect: Implied discount rates decrease with the magnitude of the outcomes. Recently, Hardisty et al. (2013) confirmed the magnitude effect for earnings, but found a reverse magnitude effect for payments: Implied discount rates increase, from negative (debt aversion) to positive (debt proneness) with the magnitude of payments. The authors ascribed their results to biases operating in addition to discounting: Present bias for earnings, and future bias for payments. The contradiction is that, in repeated applications of the debt battery, the debt holders, who combine discounting with future bias, either do not emerge, or, as in the experiment that we report, emerge only as a small group. To resolve the contradiction, we propose a framework of arithmetic discounting and amplification. We show how arithmetic discounting yields the magnitude effect for earnings and payments, and how arithmetic amplification yields a reverse magnitude effect for payments. In an experiment on decisions about whether to acquire debt, we obtain the same aggregate results as Hardisty et al. (2013). However, the reverse magnitude effect is largely carried by a small group of debt holders, who combine discounting with future bias, and by a large group of debt averse, who, according to our framework, engage in arithmetic amplification of delayed payments. In both groups, negative implied discount rates (debt aversion) become less negative as payments become larger. Among the debt prone and the takers, positive implied discount rates (debt proneness) only show a dip for very small payments, which might be a peanuts effect. Overall, the debt battery accurately captures heterogeneity in debt attitude, and the framework of arithmetic discounting and amplification improves our understanding of intertemporal decisions about payments as well as earnings.

**Marc Scholten**

#### A Comprehensive Comparison of Intertemporal Choice Models

Lisheng He; Daniel Read; Nick Chater

#### Abstract

Backgrounds There are three types of static intertemporal choice models. Delay discounting models assume that options are evaluated independently by assigning a value to their outcomes, discounting those values as a function of the delays to the outcomes, and choosing the option with the highest discounted value overall (alternative-based choice). Tradeoff models, instead, assume that options are directly compared along the time and outcome attributes, and the option favored by these comparisons is chosen (attribute-based choice). Interval discounting models adopt hybrid evaluation rules. They differ from delay discounting models by assuming direct comparisons along the time attribute, so that the discount functions for options are interdependent, but that the evaluation is otherwise alternative-based. Problems with the few existing studies that compared different types of intertemporal choice models include: (1) Always missing some prominent models, (2) rarely comparing different forms of tradeoff models, (3) model misrepresentations, (4) sometimes unjustifiably pooling data with heterogeneity and (5) neglect of model selection’s sensitivity to the structure of stimuli. These problems not only challenge the reliability and the validity of existing studies, but also render between-study comparisons practically impossible, and thus leave each study as a discrete piece of evidence. Methods The present study made a comprehensive comparison among the three types of intertemporal choice models. To address the issues above, we collected or formulated a list of seven delay discounting models, two interval discounting models and four tradeoff models and collected 260 intertemporal preference datasets from 99 studies, most of which were not originally designed for model selection. First, we presented a qualitative analysis of each model’s explanatory or predictive power across different empirical phenomena. Second, we used Bayesian model selection to quantitatively compare models across datasets, calculating pairwise Bayes factors for every pair of models separately for every given dataset. We further aggregated evidence from different datasets in a no-pooling manner. Results Results consistently suggest that attribute-based tradeoff models fit data better than other models. First, according to [overall] pairwise Bayes factors, every tradeoff model outperforms any of the delay or interval discounting models. Especially, the power tradeoff model, with a power utility function for outcomes and a power weight function for delays, is the best among all. Second, based on our qualitative analysis, a difference between a tradeoff model and a discounting model is that the tradeoff model can always accommodate the magnitude effect while the discounting model cannot. Thus we further compared models based on the subset of the datasets whose structure cannot elicit the magnitude effect. Results show that any of the tradeoff models still outperform all discounting models.

**Lisheng He**