The emergence of variance-sensitivity with successful decision rules

Behavioral Ecology doi:10.1093/beheco/arq026 Advance Access publication 15 March 2010 The emergence of variance-sensitivity with successful decision rules Eva Maria Buchkremer and Klaus Reinhold Department of Evolutionary Biology, University Bielefeld, Morgenbreede 45, 33615 Bielefeld, Germany Experiments testing for variance-sensitivity (also called risk-sensitivity) usually offer 2 options delivering identical expected payoffs, with one option providing a constant and the other one a variable reward or delay. Animals often show a preference for the constant option when variance is in amount (variance-aversion) and a preference for the variable option when variance is in delay (varianceproneness). Variance-sensitivity is a taxonomically widespread phenomenon. Variance-sensitive foraging preferences contradict predictions derived from evolutionarily motivated models that emphasize long-term energetic benefits. We discuss a new approach of explaining variance-sensitive preferences. We hypothesize that variance-sensitivity results from decision mechanisms that are adjusted to ensure close to optimal responses to the environment. This paper demonstrates that simple decision rules ensure long-term rate maximization and exhibit variance-sensitive behavior when tested in a classical risk-sensitivity situation. We also show that behavioral patterns observed in experiments like preferences for constant reward amounts and variable time delays are produced by the decision rules. The decision rules presented here are a first step toward a decision mechanism that is psychologically plausible, is advantageous in natural foraging situations, and explains irrational behavior-like variance-sensitivity. Key words: decision rule, long-term maximization, model, optimal foraging, risk-sensitivity. [Behav Ecol 21:576–583 (2010)] Variance-sensitivity oraging animals continuously have to decide where to feed. For many small animals, making the right foraging decision on a minute-to-minute basis can literally make the difference between life and death. For others, a wrong decision might not be life threatening, but the maximization of energy intake per time unit is nevertheless important because the less time is spent foraging, the more time is left for other activities, such as reproduction. The optimality approach to understanding adaptation requires us to identify the currency that foraging animals are maximizing (Stephens and Krebs 1986). The first generation of optimal foraging models assumed long-term rate of energy intake to be the currency that is maximized by foragers. Long-term energy intake is equal to the expected amount of energy obtained from a foraging option divided by the expected total length of time spent obtaining this energy. Long-term rate maximization had some notable successes in explaining foraging decisions in animals (Pyke et al. 1977; Kacelnik 1984; Stephens and Krebs 1986). However, results from experiments on risk-sensitivity have been interpreted such that animal decision making is more complex. Risksensitive behavior is not predicted by long-term rate maximization and suggests that animals base their foraging decisions not only on the net energy gain of foraging options but also on their variance. Risk-sensitivity is sometimes also called ‘‘variance-sensitivity’’ (such as in Stephens et al. 2007). We will resume using this notation for the following reason. The term ‘‘decision under risk’’ is unambiguously defined as a situation in which the decision maker knows the possible outcomes of a decision and their associated probabilities. Because we argue that risk-sensitive behavior is a product of the underlying cognitive mechanism that is based on a continuous learning F Address correspondence to K. Reinhold. E-mail: klaus.reinhold @uni-bielefeld.de. Received 3 August 2009; revised 10 February 2010; accepted 10 February 2010.
The Author 2010. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology. All rights reserved. For permissions, please e-mail: process, and not a direct reaction to well-known environment statistics, we prefer the usage of the term ‘‘variance-sensitivity.’’ Experiments testing for variance-sensitivity usually offer 2 options delivering equivalent mean reward rates, with one option providing a constant and the other one a variable reward or delay. If an animal is insensitive to variance, it should be indifferent between the 2 options with equal mean, whereas a variance-averse animal will show a preference for the constant option and a variance-proneanimal will showa preference for thevariable option. The large body of empirical data suggests that variance-sensitivity is a common and taxonomically widespread phenomenon. Variance-sensitivepreferences were observedininsects, fish,birds, and mammals (Kacelnik and Bateson 1996). Some patterns have emerged from the extensive literature. First, there is clear evidence that the direction of variance-sensitive preferences is affected by whether the variance is in amount of food or in time delay (Kacelnik and Bateson 1997; Bateson 2002). When variance is in amount of food, animals are usually variance-averse. Only few studies on monkeys report variance-prone behavior toward reward amounts (Heilbronner et al. 2008; Hayden and Platt 2009). When variance is in time delay, animals are almost universally variance-prone. Second, some studies have shown that the direction of variance-sensitive preferences may be influenced by the energetic status (energy budget) of the forager (Caraco et al. 1980, 1990; Caraco 1981; Cartar and Dill 1990; Cartar 1991). Animals on a positive energy budget tend to be variance-averse, and animals on a negative energy budget tend to be varianceprone. Third, Shafir (2000) and Weber et al. (2004) showed that the strength of preference is influenced by the variability of the variable option (measured as the coefficient of variation [CV]) when variance is in amount. Their meta-analysis of data from different taxa shows that the strength of preference increases with increasing variability of the variable option. Models explaining variance-sensitivity Ultimate explanation: risk-sensitive theory Several models in biology try to resolve the problem as to why animals show variance-sensitive foraging preferences. Here, we Buchkremer and Reinhold • Variance-sensitivity and decision rules will mention the most influential models. Risk-sensitive theory includes a set of models that address how animals should respond to variance. Stephens (1981) introduced the energy budget rule, which links variance-sensitive behavior to its fitness consequences. The energy budget rule assumes a nonlinear relationship between energy intake and fitness. How an animal should respond to variance depends on its energetic status. When on a positive budget, fitness is assumed to be a decelerating function of resource intake (concave fitness function), for which (...truncated)