Tgk Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/sequentialdecisiOOofek HB31 .M415 Massachusetts Institute of Technology Department of Economics Working Paper Series Sequential Decision Making: How Prior Choices Affect Subsequent Valuations Elie Ofek Muhamet Yildiz Ernan Haruvy Working Paper 02-40 November 2002 Room E52-251 50 Memorial Drive MA Cambridge, 02142 This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssm.com/abstract_id=353421 Sequential Decision Making: How Prior Choices Affect Subsequent Valuations Elie Ofek, Muhamet Yildiz and Ernan Haruvy* November 2002 MASSACHUSETTS INSTITUTE OFTECHNOLOGY LIBRARIES *Elie Ofek: Harvard Business School, Soldiers Field, Boston, MA 02163. Muhamet Yildiz: MA Department of Economics, MIT, Cambridge, 02142. Ernan Haruvy: University of Texas at Dallas School of Management, 2601 North Floyd Road, Richardson, TX 75083-0688. The authors would like to thank Al Roth, Daron Acemoglu, George Wu and Vikram Maheshri for helpful suggestions, and Brian Gibbs for detailed comments on an earlier version. The paper has also benefited from comments ofparticipants at the Behavioral Research Council conference (Great Barrington, MA, July 2002). Abstract This paper develops and tests a model of sequential decision making where a first stage of ranking a set of alternatives is followed by a second stage of determining the value of these same alternatives. The model assumes a boundedly rational Bayesian decision maker who is uncertain about his/her underlying preferences over the relevant attributes, and who has to exert costly cognitive effort toresolve this uncertainty. Compared to when only valuation takes place, the analysis reveals that ranking a set of alternatives prior to determining their value has three primary effects: a) the spread (or dispersion) of valuations between most and least preferred alternatives increases, b) decision makers will, on expectation, exert more effort in the valuation phase, and c) the more each attribute contributes to overall utility the greater the relative impact of ranking is on valuation spread. The analysis also sheds light on how prior ranking impacts the demand for aproduct. Theseresults are then corroborated in aseries ofcontrolled lab experiments with actual prizes. The findings have implications for many real life decision making situations ranging from auctions, where there is a tendency to prioritize items before determining a bid, to the ranking of job candidates prior to determining wages and benefits to be offered. More generally, the results bear on our understanding ofhow past decisions can affect future related decisions. 1 Introduction Past decisions are often used as input to guide future related decisions. This is regarded as beneficial when the information conveyed in a previous decision is expected to shed light on dimensions associated with the decision at hand, even if these decisions are not identical. In the context of such dynamic decision-making, we ask how previous choices would affect the subsequent determination of maximum willingness to pay for a given good. For example, consider an individual who has chosen a job offer with certain wages in a big metropolitan city over an identical job that pays more wages but is located in a small suburban town. Imagine that several months later, this same individual is considering buying a small house downtown or a big house in the suburbs. How would the willingness to pay (and thus bid) for a house located in downtown or suburbia change if the individual is reminded of the fact that she has chosen the lower-paying urban job over the higher paying suburban job? A similar question may arise even within a single decision if the decision maker divides the complex decision problem into multiple sub- decisions. Examples ofsuch procedures are common: an employer might first rank the set of candidates interviewed before determining the details of each offer to be made (Roth 1984), a consumer at an auction site with many similar items might have to repeatedly choose among sellers before determining how much to bid until an item is secured (Peters and Severinov 2001), and amanagement team might first rankorder product development projects before deciding how much R&D resources to devote to each one (Keefer 2001). In an ideal world where individuals know their preferences with certainty or can figure them out effortlessly, Individual's previous choices would not have an informational value for subsequent decisions. In reality, however, individuals seldom know their own preference structure with full confidence (for example, the exact trade-off between two product attributes) or may need to anticipate the likelihood offuture usage/consumption contingencies. Hence, most decision making tasks regarding multiple alternatives entail costly effort, in the form ofcognitive thinking or time-consuming research, and will render past choices potentially useful input. How then should we expect the effort expended in determining valuations to be impacted by the knowledge of a prior choice or rank ordering of the alternatives? How do previous choices impact final valuations, compared to when only valuation takes place? The goal of this paper is to provide an answer to these questions both theoretically and empirically. Social-science literature has examined the implications of various decision tasks on preference elicitation. The focus has been on how performing various tasks, such as choice, rating and matching can yield different outcomes when performed separately (e.g., Tversky et al. 1988, Montgomery et al. 1994, Bazerman et al. 1992, Huber et al. 2002). However, the implications of intertemporally combining a set of tasks on final elicited preferences have not been studied. In this context, our paper focuses on how a previous ranking task, where the output required is an ordinal relationship between the alternatives, affects a subsequent valuation task, where the output required is a measure ofwillingness to pay for each of the alternatives. To examine this prevalent sequence of evaluating alternatives, we construct a model of individual decision-making over a set of two alternatives defined over two attributes that need to be traded-off. The central features of the model that make it relevant for examining the above decision sequence are: a) individuals are uncertain about their preferences (much in the vein of March 1978, and Keeny and Raiffa 1976), b) through costly cognitive effort they can resolve part ofthis uncertainty, and c) though individuals may be forgetful of specific details emerging from cognitive effort during ranking, the outcome ofthis decision phase (i.e., the rank ordering ofalternatives) can be incorporated in the subsequent determination of willingness to pay. As such, our boundedly rational agents use their own previous choices as a source of information about their own utility structure, andmay be perceived to have apreferencefor consistency (similar to the agents in Yariv (2002)). That said, our agents also exhibit other specific patterns of behavior (also observed in our experimental setting) that cannot be explained by such preference for consistency. The analysis of the model reveals three interesting findings. First, it is shown that the spread ofvaluations for the two alternatives is, on expectation, greater when ranking precedes valuation. This is not only because of the extra information embodied in the rankings, but, interestingly, also because rankings may induce the agent to think more when the information is perceived to be more valuable. Second, this increased spread as a result of ranking is more pronounced when the contribution of each attribute to overall utility is higher. Lastly, we find that the effort expended in valuing alternatives is (canonically) expected to be higher when ranking information is present, even though previous effort has obviously already been expended in the ranking stage. We also exam- ine how prior ranking affects the likelihood of purchase (with any given distribution of prices) In particular, we show through a canonical example that prior ranking increases . the probability ofa sale when prices are either very low or very high (but not when they are centered around the mean expected value). A series of experiments designed to allow comparison of valuation and ranking deci- sions (and their combined effect) were carried out. The experiments used actual prizes for future consumption of a familiar product category, namely dining at local restau- rants, and were constructed to induce truth-telling through a Becker-DeGroot-Marschak (1964) mechanism. The empirical results strongly confirm the implications ofthe theory, and were designed to rule out possible alternative explanations (such as learning or task familiarity) In particular, we confirm that the effect of ranking on valuation spread be- . comes more pronounced as the stakes involved in the task increase (i.e., the average value of the prizes is higher) This provides an example for a general possibility that certain . effects that are generated by bounded rationality (Rubenstein 1998) may get even larger as the decisions themselves become more important. The rest of the paper is organized as follows: In the next section we develop theory that allows the modeling of a sequence of decisions regarding the same alternatives, in particular, ranking and subsequent monetary valuation. We formulate the central findings of the model as hypotheses, which we then test through a series of controlled experiments. The paper ends with concluding remarks. A 2 Model of Sequential Decision Making To shed light on our questions of interest, in this section we develop a simple model of utility with uncertain preferences. We then analyze the utility maximizing behavior of an agent confronted with two different alternatives that she either needs to 1) rank in order of preference, 2) provide an exact monetary value for each, or 3) first rank and then value each of the alternatives. Notation: We will use the following standard notation throughout. Given any ran- dom variable x, E [x] is the expected value ofx and Var (x) the variance of x. We write E [x\G] and Var (x\G) for the conditional expected value and the conditional variance of x given event G. We also write Pr(G) for the probability of event G and let E [x G] = : E [x\G] Pr(G). The sets of real and non-negative real numbers are denoted by 1Z and — TZ+, respectively. Given any three-times differentiable function / : TZ > TZ, we write /', /", and /"' for the first, second, and third derivatives, respectively. 2.1 Set-Up Consider a boundedly rational agent who wishes to maximize the expected value of a utility function u, the parameters of which she does not know with certainty. The