The intertemporal choice experiments reviewed in this introduction have been of great value in understanding temporal discounting and impulsive choice, however, there is potentially a major confound in their interpretation – a confound which a large part of this thesis is devoted to.
The root of this problem lies in the way that the choices differ in both delay and magnitude. Take, for example, an abstaining smoker who is offered a cigarette. His choice is between a small, immediate reward (the cigarette) and a large, delayed reward (better health in the future, spend less etc.). If he acts impulsively where another does not, it could be because he is less influenced by outcomes that are delayed considerably (more temporally impulsive), or it could be because he does not perceive the larger reward to be as valuable (in relation to the smaller reward) as his self-controlled counterpart does. In the latter scenario, the smoker acts impulsively because there is not enough utility to be gained in waiting for the larger option, given the normal effects of delay.
To fully understand this problem, consider the relationship between the size or magnitude of a reward, and its subjective value or utility. This relationship is not a straightforward linear one, but is more likely to be concave (Figure 7). This can be
easily demonstrated by a simple thought experiment – for the average person, winning £2000 in a lottery would make you much happier than winning £1000, but winning £1,002,000 would not really make you much happier than winning
£1,001,000. The idea that every unit (marginal) increase in a good brings us successively less and less (marginal) utility has a long history in economic theory and goes to the very heart of value.
Adam Smith (1776) explored the paradox of value when considering the disparity between the ‘value in exchange’ of water, whose ‘value in use’ is fundamental, and diamonds whose value in use is trivial. Whereas Smith was led to the conclusion that value lies in the labor required to extract a good, a subjective theory of value addressed this perplexity by positing that the value of a good is not determined by its maximal utility, rather by the increase in utility obtained by consuming one extra unit of that good, i.e., its marginal utility. A salient feature of marginal utility is that it diminishes as the quantity of a good increases – hence the utility provided by a fixed amount of £10, is greater when added to an option worth £50 than to one worth £500. Since water is so plentiful, its marginal utility (and hence value in exchange) is much smaller than that of scarce diamonds. The concept (often referred to as a law) remains integral to economic theory, most notably in the microeconomic concept of the indifference curve which explains preferences between different bundles of goods, consumer theory and laws of supply and demand (Pindyck and Rubinfeld, 2004), as well as in modern analyses of decision under risk and uncertainty (Kahneman and Tversky, 1979; von Neumann and Morgenstern, 1947). In fact diminishing marginal utility was first proposed by Bernoulli (1738) as a solution to the famous St Petersburg paradox.
It is therefore likely that two features of preference, the discounting of time and also the discounting of magnitude (diminishing marginal utility), contribute to choice outcome in intertemporal choice. This is because the rate at which the
marginal utility of an individual diminishes will determine the perceived increase in (subjective) utility of the larger reward relative to the sooner (independently of temporal discounting). Returning to our smoker, it is now possible to understand why if he had a greater rate of magnitude discounting than his self-controlled counterpart, he could have chosen to smoke – even though they had the same rate of temporal discounting.
Figure 7. The non-linearity of (instantaneous) utility. Utility functions describe the relationship between increasing reward magnitudes and subjective value, or utility, derived from them. The most salient feature is that the function is concave for gains – this equates to diminishing marginal utility.
Because the majority of discounting experiments rely on the standard hyperbolic model and assume a linear relationship between utility and reward magnitude, choice outcome is deemed to be solely a product of the discount rate – this leads to two potential confounds. The first is that if diminishing marginal utility does play a role in intertemporal choice, it is likely that many studies overestimate the temporal discount rate because diminishing marginal utility has the effect of decreasing the ratio of the (instantaneous) subjective value of the later reward relative to the sooner, and thus shifting preference to the smaller-sooner option. The second and more serious problem is that we now have an interpretational difficulty when comparing differences in intertemporal choice behaviour across experimental groups, be it in lesion, pharmacological manipulation, population comparison or imaging studies. This arises because we cannot determine whether a difference in impulsive choice – arising from for example a lesion, or being a drug addict – is caused by a change in the temporal discount rate or a change in the rate of diminishing marginal utility of the chooser, or both. Since all previous studies assume that the discount rate/temporal discounting is the only determinant of choice outcome, behavioural changes are automatically associated with changes in K – this casts a shadow of doubt over the conclusions of many of the studies reviewed above.
Moreover, the assumed relationship between K and impulsivity in choice (e.g.
Ainslie, 2001) should break down if K is not the sole determinant of choice outcome. It would therefore be problematic to equate impulsivity in choice directly with this parameter. Accordingly, studies of temporal discounting are confounded because they assume that behavioural changes reflect a change in the discount rate, and studies of impulsivity are confounded because they assume that impulsivity is a sole product of the temporal discount rate – one needs to know a variable’s affect
on both determinants to be able to know how impulsivity will be affected in different choice conditions.
The idea that diminishing marginal utility may play a role in intertemporal choice is rarely discussed although it has been noted by economists (Andersen et al., 2008; Frederick et al., 2002; Kirby and Santiesteban, 2003; Loewenstein and Prelec 1992; Read, 2003), mainly with regard to the possible overestimation of discount rates in economic studies, or in explaining various intertemporal anomalies. Remarkably, no one has empirically tested this idea in humans.
How could we potentially dissociate these two influences on choice and obtain a reliable quantitative measure of each? In original rodent work by the Bradsahw and Szabadi group (see Ho et al., 1999) this has been achieved by way of an integration of temporal discount functions with magnitude discount functions (or utility functions, in economic terms) which mathematically model the relationship between objective and subjective value for magnitude. For example, a common utility function in economic literature – the power law – has the formulation where U is the utility or subjective value of a reward of magnitude M.
Here r is a free parameter which determines the rate at which marginal utility diminishes. If r is less than 1 the utility function will be concave (Figure 7), i.e.
exhibiting diminishing marginal utility. The lower the value of r here, the more concave the utility function and the faster the rate of diminishing marginal utility.
If r is also a determinant of intertemporal choice, the lower the value of r the more impulsive the individual should be, as the less valuable the larger reward is in relation to the smaller and vice versa.
Ho et al. (1999) have developed a two parameter model and a method for estimation of each parameter individually (described in Chapter 2). This has been successfully used in animal studies and has proved to be an accurate description of rodent intertemporal choice behaviour. In some cases, both parameters have been
observed to change in response to a manipulation, occasionally in opposite directions, exerting opposing influences on impulsive choice, and possibly explaining previous discrepancies in the literature.
There would be numerous advantages to implementing such a model in human studies. First, it would potentially allow for a more accurate and less confounded measure of temporal discount rates, and a better overall description of choice behaviour, revealing fundamental truths about human valuation and decision-making systems. It would also allow us to remove a major confound in empirical experiments of intertemporal choice and determine whether changes in behaviour have as their underlying cause a change in the rate of temporal discounting or magnitude discounting. This would facilitate a better understanding of personality characteristics such as impulsivity.