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Response Functions⇤ Response functions⇤ Carlos Oyarzun, Adam Sanjurjo, and Hien Nguyen† August 14, 2014 Abstract We analyze a simple adaptive-learning-based model of decision making under uncertainty, and then use it to study subjects’ behavior in an experiment. Subjects choose between two uncertain payoffdistributions, only knowing the support of each. In the first round they choose one alternative and experience a payoff as a result. In the second round they decide whether to choose the same alternative, or to switch. From each subject’s responses we construct a response function,thenuseamixturemodeltoclassifyeachsubject into one of several classes of response function. Further, we explain classes’ differing payoffs in terms of the interaction between the shapes of their response functions and the properties of the payoffdistributions. This constitutes a first step in eliciting and analyzing behavior in plausible environments in which an individual knows almost nothing about alternatives’ outcome distributions, but can learn from a single experience. Key words: Adaptive learning, response functions, experimental economics, stochastic dominance. JEL codes:D81,D83. 1 Introduction Imagine that Bob is a salesman with broad knowledge of green technology, but who only has direct experience in selling LED lighting. Bob is now looking for a higher paying, heavily commission-based sales job, and is choosing between a solar-tech company and a wind-tech company. While it is relatively easy for him to imagine the range of possible earnings he can make in each company, it is difficult for him to have a clear sense of what the probability distributions over possible earnings with each company might look like. On the other hand, Bob will experience a realized level of earnings at the chosen company. If Bob then decides to move to a far-away city and has to again choose between the local solar-tech and wind-tech company, the level of earnings experienced at his first choice may influence his second decision considerably, even if it did little in the way of revealing to him the probability distributions over possible earnings levels in each type of company. Although many important economic decisions share the key features of this example—little opportunity for learning through direct experience, and only minimal knowledge about uncertain payoffdistributions—little is known about how ⇤This paper has benefited from useful discussions with Carlos Alos-Ferrer, Yoram Halevy, Nobuyuki Hanaki, Jonas Hedlund, Ed Hopkins, Nagore Iriberri, Kenan Kalayci, Vadym Lepetyuk, Graham Loomes, Rosario Macera, GeoffMcLachlan, Friederike Mengel, Joshua Miller, Peter Moffatt, Andreas Ortmann, and audiences in seminars at Pontificia Universidad Catolica de Chile, Universidad de Alicante, Universidad de Concepcion, and University of New South Wales. We also acknowledge financial support from the Spanish Ministry of Sciences and Technology (SEJ2007-62656), the Spanish Ministry of Economics and Competition (ECO2012-34928), and the University of Queensland Start-up Grant. All errors remain our responsibility. †Oyarzun is with the School of Economics of the University of Queensland, Sanjurjo is with the department Fundamentos del Análisis Económico, Universidad de Alicante, and Nguyen is with the School of Mathematics and Physics of the University of Queensland. Corresponding author: Carlos Oyarzun, University of Queensland, Brisbane, Queensland 4072, Australia. E.mail: [email protected]. 1 individuals make choices in these settings. In particular, little is known about how the first experienced outcome affects the chances that the individual makes the same choice when given a second opportunity, or how this effect shapes the economic performance of the individual. In this paper we provide an answer to these, and other related questions. The decision setting just described can be made more concrete in the following way: now imagine that Bob walks up to two slot machines. Dropping a coin in either machine will yield him a prize of anywhere from one to five gold bars. However, he knows nothing about the probability distribution over prizes for either machine. Bob has exactly two coins in his pocket. Upon choosing a machine for the first coin it yields him a prize of from one to five gold bars. After experiencing this prize Bob decides to drop his second coin in either the same machine, or switch, at which point the chosen machine yields him a second prize.1 While it would probably be difficult for the decision maker (Bob) to formulate prior beliefs in this setting—such as those hypothesized in Expected Utility Theory or theories of ambiguity aversion—it would perhaps be even more difficult for us, the analysts, to know how to place an operational bound on the range of plausible beliefs for the decision maker. For this reason, when studying experimental subjects’ choices and payoffperformance in this type of a problem, we opt for a relatively less structured approach in which we instead use a simple adaptive-learning-based model of decision making under uncertainty. Our approach is not only well-suited for studying choices and payoff performance, but it also has the advantage that it is agnostic about whether, and how, individuals form priors in such information sparse environments. We observe individuals’ choices in this context—in which they know nothing about the shape of the two probability distributions over outcomes—and analyze how they respond to experienced payoffs. Since the first period choice is made with no information, our model treats it as completely random. On the other hand, the second period choice may depend on the payoffexperienced as a result of the first period choice. Thus, our model represents the second period choice as a Bernoulli distribution in which the probability of choosing the same alternative that was chosen in the first period is determined by the first period payoff. This representation defines a function mapping the experienced payoffto the probability of choosing the same alternative in both periods. We call this mapping the response function. Below, we show that, if each alternative’s payoffdistribution is the same in both periods, the shape of an individual’s response function affects her second period payoffs. In general, the shape of the response function determines the probability of choosing the best alternative with respect to standard criteria for ranking payoffdistributions. In particular, we show that if the response function of an individual is increasing (concave), the individual’s second period choice is more likely to first- (second-) order stochastically dominate the unchosen alternative. The objective of this paper is twofold: 1) to elicit individuals’ response functions, then 2) to study the payoffconsequences that result from the particular shapes of individuals’ response functions. In order to elicit individuals’ response functions, we conduct a controlled laboratory experiment in which the choice problem that subjects face is very similar to the slot machine example given above. In the first round of each task subjects choose one of two alternatives and then experience its corresponding payoff. In the second round, after experiencing the first round payoff, subjects must again choose one of the same two alternatives, but this time they do not observe the payoff. A subject’s payoffs are determined by her choices in both the first round and the second round. Before starting, subjects are informed that, for each alternative in each such task, its distribution of payoffs is identical in the first and second rounds, and that the obtained payoffmay be either 1Notice that this problem is an extremely short-lived multi-armed bandit in which the individual is not endowed with prior beliefs. Thus, her initial choice may be regarded as random and her second (and final) choice as pure exploitation—given that there are no subsequent decisions, there are no incentives to experiment. 2 1, 2, 3, 4, or 5 Euros. In addition, they are told that each alternative’s payoffdistribution changes from one task to the next (in the example above imagine both slot machines being replaced between tasks). Subjects are not told anything else about the distributions. Each subject makes choices in 40 such tasks.2 We estimate the response functions of subjects using the relative frequency of tasks in which subjects choose the same alternative in both rounds, for each experienced payoff.3 Our analysis of subjects’ behavior reveals considerable heterogeneity in response functions. In particular, we use a simple mixture model to identify five different classes of response function. We find that the behavior of more than 60% of our 100 experimental subjects is best fit by an increasing response function. These subjects, in turn, correspond to three different classes. We call two of these classes strongly responsive; Classes 1 and 2 both contain subjects who choose the same alternative with probabilities very close to one if the experienced payoffis greater than 3, and with probabilities very close to zero if the experienced payoffis less than 3. The main difference between Class 1 and Class 2 is their response to a payoffof 3; while subjects in Class 1 choose the same alternative with probability almost one, subjects in Class 2 switch about half of the time. As will be explained below, this seemingly innocuous difference results in non-negligible differences in round two payoffs. Classes 1 and 2 correspond to about 21% and 27% of the subjects, respectively. The remainder of subjects exhibit response functions that are relatively less responsive to experienced payoffs. In particular, in Class 3, which we call weakly responsive, while subjects are still responsive to the obtained payoff, the probability of them choosing the same alternative in both rounds is on average greater than 1/3, even for the lowest experienced payoffs, of 1 and 2. This class represents about 14% of our subjects. Finally there are two additional classes, Classes 4 and 5, of non-responsive subjects.
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