A Boundedly Rational Decision Algorithm Author(S): Xavier Gabaix and David Laibson Source: the American Economic Review, Vol

American Economic Association

A Boundedly Rational Decision Algorithm Author(s): Xavier Gabaix and David Laibson Source: The American Economic Review, Vol. 90, No. 2, Papers and Proceedings of the One Hundred Twelfth Annual Meeting of the American Economic Association (May, 2000), pp. 433- 438 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/117264 . Accessed: 03/10/2011 12:16

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http://www.jstor.org A Boundedly Rational Decision Algorithm

By XAVIER GABAIX AND DAVID LAIBSON*

Cognition requires scarce inputs, including existing literatureon bounded rationality.First, time and concentration. Since cognition is our conceptual approachextends the satisficing costly, sophisticated decision-makers should literature which was pioneered by Herbert use mental short-cuts, or heuristics, to reduce Simon (1955) and subsequentlyformalized with cognitive burdens.We propose and test a model models of deliberation cost.' Second, our ap- motivated by these principles. proach extends the heuristics literature started We base our model on algorithms that sim- by Daniel Kahneman and Amos Tversky plify decision trees. When cognition is costly, (1974). Third,our approachis based on work by the actor will not calculate perfectly, relying Colin Camerer et al. (1994) which shows that instead on a simplification of the tree. The al- decision-makersoften solve problems by look- gorithms are parameterizedto include perfect ing forward, rather than using backward in- rationality(i.e., zero cognition costs) as a spe- duction. Fourth, our emiphasis on empirical cial case. We focus on one algorithmthat sim- evaluationis motivatedby the work of Ido Erev plifies trees by removing branches with low and Alvin Roth (1998) and Camererand Teck- probability. This reduces to a psychologically Hua Ho (1999). Our approach differs from naturalforward simulation rule. theirs since we study first-periodplay in non- We believe that this model achieves four strategic settings instead of learning dynamics goals. First, the model makes quantitativebe- in repeated games. havioralpredictions and, hence, provides a precise alternativeto the rational-actorhypothesis. I. DecisionTrees and an Algorithm Second, the model is psychologically plausible because it is based on the actual decisions al- We propose an algorithm that mimics the gorithms that subjects claim to use. Third, the way that decision-makers evaluate trees. We model is empirically testable; such a test is base our analysis on decision trees since trees provided in this paper. The data overwhelm- can be used to represent a wide class of prob- ingly reject the rational model in favor of our lems. bounded-rationality alternative. Fourth, the Consider the tree in Figure 1. Each starting model is broadly applicable, because it can be box in the left-hand column leads to boxes in used to analyze decision problems that can be the second column. The numbers iniside the representedin tree form. boxes are flow payoffs. 'Probabilisticbranches Our approach combines four strands in the connect boxes. For example, the firstbox in row five contains a flow payoff of 3, and five branches exit from this box, with probabilities * Gabaix: Departmentof Economics, MassachusettsIn- 0.55, 0.1, 0.15, 0.1, and (.1., Startingfrom row stitute of Technology, Cambridge, MA 02142 (e-mail: five, there exist 95 possible outcome paths. The [email protected]); Laibson: Department of Economics, Harvard University, Cambridge, MA 02138, and NBER path that follows the highest-probabilitybranch (e-mail: [email protected]).We are grateful for the at each node is (3, -1, 5, 4, -5). Integrating advice of Gray Becker, Ed Glaeser, Lars Hansen, Larry over all 95 paths, the expected payoff of starting Katz, George Loewenstein, Sendhil Mullainathan,Matthew from row five is 5.04. Rabin, Al Roth, Andrei Shleifer, Richard Thaler, and Richard Zeckhauser. Numerous research assistants helped Imagine that a decision-maker must choose implement and analyze the experiment;we are particularly one of the boxes in the first column of Figure indebted to Meghana Bhatt, David Hwang, Guillermo 1. The decision-makerwill be paid the expected Moloche, and StephenWeinberg. Gabaix acknowledges the value associated with his chosen box. What box hospitality of the University of Chicago. Laibson acknowledges financial supportfrom the National Science Founda- tion, the National Institute of Health, and the MacArthur Foundation. 1 See John Conlisk (1996) for a review. 433 434 AFA PAPERSAND PROCEEDINGS MAY2000

This heuristic describes the FTL value of a 4 1 SD 5 1 starting box bo In our implementation we set p 0.25; perfect rationalitywith zero cognition cost correspondsto p 0. To make FTL precise, call a path a sequence of boxes and associated branch/transitionprob- 14-i 5 4 25 abilities. Abusing notation, let (3, (0.159 2), (0.59 -1), (0.15, -1), (1.0, 4)) represent a path originating in the fifth row of Figure 1. A p- 3 -3 -1 4 2 constrained path (u0, (P1, U1), * 9.. (PN, UN)) has the following properties: (i) all probabili- 3 24-1 -1 4 ties, except possibly the last one, are greater than or equal to p, and (ii) the path cannot be lengthened without violating (i). So, if p 0.25, then the path described above is not a p-constrained path, but (3, (0.15, 2)) is a p- constrained path. Since 0.15 < p the p- constrainedpath does not extend past the branch with probability 0.15. For any path, define a -5 1 5 cumulative probability fl UN1 p, and cumulative payoff U _ '=u Finally,to get the FTL value associatedwith a -5 -4 - 4 1 startingbox b,3 identif the set of p-constrained 5 1 0 -3 paths startingwith box b, the associated set of cumulative probabilities, {IIk}. 1, and cumulative payoffs, {U u}kKk= The FTL value of this box bVpaos is: SU lkUk. Ten p-constrained pathsoriginate in row five of Figure1.4 The EE, value of row five is 4.91, whichis not far fromthe FIGURE 1. DECISION TREE trueexpected value of 5.04. In this example,FrL dramaticallysimplifies the decisiontree. Ninety-five pathsoriginate in the fifth row, but applicationof would you choose?2 We propose a heuristic, FITLreduces consideration to only tn paths,five of dubbed "follow the leaders" (FTL), for evalu- which aretruncated. ating options in such trees. FTL has one param- We chose FTL because it correspondsto our eter, a cutoff probabilityp: intuition about how decision-makers "see" decision trees that do not contain outlier payoffs. From any starting box b follow all Subjectsfollow paths that have high-probability branches that have a probability greater branchesat every node. In the process of deter- than or equal to p. Continue in this way, mining which branches to follow, subjects see moving from left to right across the col- low-probability branches that branch off of umns in the decision tree. If a branchhas high-probability paths. Subjects encode such a probabilityless thanp, consider the box to which the branchleads but do not ad- vance beyond that box. Weight all boxes that are considered with their associated 3For simplicity we describe the algorithm for payoffs cumulativeprobabilities, and calculate the with mean 0. The generalization to nonzero means is weighted sum of the boxes. straightforward. 4 Namely, (3, (0.1, -3)), (3, (0.15, 2)), (3, (0.1, -3)), (3, (0.1, -2)), (3, (0.55, -1), (0.55, 5), (0.15, 4)), (3, (0.55, -1), (0.55, 5), (0.85, 4), (0.55, -5)), (3, (0.55, -1), (0.55, 2 The four most frequentpicks are rows 10 (42 percent),5 5), (0.85, 4), (0.45, 2)), (3, (0.55, - 1), (0.45, - 1), (1.0, 4), (20 percent),3 (16 percent),and 4 (12 percent).The respective (0.33, -5)), (3, (0.55, -1), (0.45, -1), (1.0, 4), (0.57, 2)), expectedvalues are $5.52, $5.04, $3.20, and $5.54. (3, (0.55, -1), (0.45, -1), (1.0, 4), (0.1, 4)). VOL.90 NO. 2 PREFERENCES,BEHAVIOR, AN!) WELFARE 435 low-probability branches, but subjects do not A. Decision Trees follow them deeper into the tree. FTL is only one among many sensible algorithmsthat sub- The experiment was based around 12 ran- jects use to evaluate decision trees. There may domly generated trees, one of which is repro- be close variants that better predict subject duced in Figure 1. We chose large trees, choices. We analyze FTL to demonstratethat because we wanted undergraduatesto use heu- bounded-rationalitymodels can be successfully ristics to "solve" these problems. Half of the empirically implemented. trees have 10 rows and five columns of payoff In the next section, we empiricallycompare boxes; the other half of the trees are 10 X 10. FTL to the rational-actormodel and two other The branchingstructure, probabilities, and pay- choice models, which we call the "column-cutoff offs are independentlyrandomly generated. model" and the "discountingmodel." The ratio- Experimentalinstructions described the con- nal-actormodel with zero cognition cost is the cept of expected value to the subjects. Subjects standardassumption in almostall economicmod- were told to choose one startingrow from each els. This correspondsto the limit case p = 0. For of the 12 trees. Subjects were told that one of presentationalsimplicity, we refer to this bench- the trees would be randomly selected and that markas the rational-actormodel, but we empha- they would be paid the true expected value for size that this standard model assumes both the starting box that they chose in that tree. rationalityand zero cognitioncosts. The column- Subjects were given a maximal time of 40 min- cutoffmodel assumesthat decision-makerscalcu- utes to make their choices on all 12 trees. late perfectlyrationally but only pay attentionto the first Q columnsof the tree, completelyignor- B. Debriefing Form and Expected-Value ing the remaining columns. The discounting Calculations model assumes that decision-makersfollow all paths, but exponentiallydiscount payoffs, using After passing in their completed tree forms, discount factor 8, according to the column in subjects filled out a debriefing formi which which those payoffs arise. asked them to describe their choice algorithm Anticipatingthese tests, we need to close the and their background information. Subjects model by providing a theory that translatesra- were then asked to solve 14 simple expected- tional and quasi-rationalpayoff evaluationsinto value problems (e.g., calculate the expected choices. Assume that there are B possible value of the startingboxes in a 2 X 3 tree). To choices with evaluations { V1, V2, ... , VB} eliminate subjects who did not understandthe given by a candidatemodel. Then the probabil- concept of expected value, our econometric ity of picking box b is analysis excludes subjects who solved fewer than half of these simple problems.5Out of the 251 subjectswho providedanswers for all of the ( exp(I3Vb) 12 decision trees, 230, or 92 percent, answered

B correctly at least half of the simple expected- value questions. Out of this subpopulation,the median score was 12 out of 14 correct. b=l III. Resultsand Analysis We estimatethe parameter,B in our econometric analysis. Had subjects chosen randomly, they would have chosen starting boxes with an average payoff of $1.30.6 Had the subjects chosen the II. ExperimentalDesign starting boxes with the highest payoff, the

We testedthe model using datafrom 259 Har- SThis restrictiondid not affect our results. vard undergraduates.Subjects were guaranteeda 6 In theory this would have been zero, since our games paymentof $7 and earnedmore if they performed are randomly drawn from a mean-zero distribution. Our well; the averagepayment was $20.08. realized value is within the two-standard-errorbands. 436 AEA PAPERSAND PROCEEDINGS MAY2000

average chosen payoff would have been $9.74. Lm(P, 3)-lPm(I3) - p2 In fact, subjects chose starting boxes with an average payoff of $6.72. [Pbg(1) - pbg] We are interestedin comparingfour different b,g classes of models: rationality, FTL, column- cutoff, and discounting. All of these models Lm Min Lm(P, ,3). have a nuisance parameter,,B, which captures pER+ the tendency to pick starting boxes with the highest evaluations [see equation (1)]. For each model, we estimate a different ,3 parameter. Hence, Lm is the empirical (quadratic)distance FTL, column-cutoff, and discounting, also re- between the model and the data. In a companion quire an additionalparameter. paper (Gabaix and Laibson, 1999) we derive FTL requires the probabilitycutoff value p. asymptotic standard errors for Lm and Lm - We exogenously set p = 0.25. We did not Lm, where m m'.i experiment with other values. We choose 0.25 Table 1 reports test statistics for each of our because it corresponds to our intuition about models. The rows of the table represent the which paths could feasibly be considered given different models, m E {rational, FTL, Q- the 40-minute time constraint on the decision column-cutoff, 6-discounting} Column (i) re- process. Because we do not believe that either ports distance metric Lm, which measures the column-cutoff or discounting are psychologi- (square of the) Euclidean distance between the cally plausible models, we do not impose a estimated model and the empirical data. Col single value on either Q or 6. Instead, we eval- umn (ii) reports the difference Lrati-nal Lm uate the models for the full range of Q and 8 When Lrational- Lm > 0 the rational model values. This analysis is providedfor comparison performsrelatively poorly, since the distance of to our preferredmodel, FTL. the rational model from the empirical data (Lrational) is greater than the distance of the competitormodel from the empiricaldata (Lm). Formal Model Testing Column (iii) reportsthe difference LFTL -Lm. Three findings stand out. First, several mod- There are G 12 games, B = 10 starting els statistically significantly outperform ratio- boxes per game, and I = 230 subjects who nality: FTL, Q 8 column-cutoff, Q = 9 satisfy our inclusion criterion. Call P E R column-cutoff, b 0.8 discounting, = 0.9 the empiricaldistribution of choices (i.e., Pbg discounting, and 6 - 0.91 discounting. We re- the fractionof players thatplayed box b in game port = 0.91 discounting, since this value of g). A model m (rationality,FTL, etc.) gives the b minimizes Lratinal - La discounting Second, value Vbg to box b in game g. Recall that we FTL outperforms the rational model by the parameterizethe link between this value and the biggest margin: Lrati-nal LIYL 0.195. We probabilitythat a player will choose b in game reject Lrational LTL 0 with a t statistic of g by 8.75. The next closest contender is Q 8 column-cutoff: Lrati-n - L=8 colmn-cutoff 0.104. Third,we rejectall models in favorof FTL. Vbg) FrL - Pbm(/3) = exp(/3 For the closest competitor, we reject B LQcolumn-cutoff 0 with a t statistic of 6.06. E exp(/3Vb,g) Hence, the experimentalevidence overwhelm- b'=1 ingly supportsFTL.

IV. Extensions for a given ,B.To determine,B, we minimize the Euclidean distance (in RBG) between the FTL exemplifies the general cognitive ten- empirical distribution P and the distribution dency to simulate the future by identifying typ- predicted by the model, given /3. Specifically, ical or representativescenarios. For example, if we take numerousfuture outcomes are equally likely, a VOL.90 NO. 2 PREFERENCES,BEHAVIOR, AND WELFARE 437

TABLE 1-MODEL STATISTICS ing. Consider the well-stu(lied example of a bidder who values a widget 50-percent more (ii) (iii) than a seller. The bidder does not know the (i) L(Rational) L(FTL) m -Model L(m) - L(m) - L(m) underlying value of the widget to the seller. Only the seller knows this value, which is dis- Rational 0.789 -0.195 tributeduniformly from 0 to 100. Replacing this (0.063) (0.022) FTFL 0.594 0.195 distributionof outcomes with the typical out- (0.05 1) (0.022) come (a valuationof 50), leads to a bid of 50 + Column cutoff 1 1.903 -1.114 -1.309 e, far from the optimal bid of 0.7 (0.085) (0.081) (0.083) Column cutoff 2 1.253 -0.464 -0.659 (0.076) (0.074) (0.070) EndogenousAlgorithms Column cutoff 3 0.819 -0.031 -0.225 (0.059) (0.055) (0.047) Column cutoff 4 0.806 -0.017 -0.212 FTL successfully reproducesthe decisions of (0.061) (0.032) (0.026) subjects in our experiment.This result is likely Column cutoff 5 0.852 -0.064 -0.258 (0.062) (0.033) (0.028) to be robust, since our trees are randomly gen- Column cutoff 6 0.831 -0.042 -0.237 erated. However, it is easy to construct special (0.060) (0.030) (0.027) examples, say, with extreme outlier payoffs, in Column cutoff 7 0.824 -0.035 -0.229 which FTL may not do well. We view FTL as a (0.064) (0.023) (0.025) Column cutoff 8 0.685 0.104 -0.091 special case of a richer model that combines (0.059) (0.013) (0.015) several decision algorithms. For decision trees Column cutoff 9 0.728 0.061 --0.133 with extreme outlier payoffs, consider the fol- (0.062) (0.008) (0.019) lowing algorithm: 6 = 0.5 discounting 1.111 -0.322 -0.516 (0.086) (0.064) (0.060) 6 = 0.6 discounting 0.978 -0.189 -0.384 (0.087) (0.056) (0.052) Identify boxes with outlier payoffs (Ipay- 6 = 0.7 discounting 0.864 -0.075 -0.270 offl > 4). Start at these outlier payoffs (0.089) (0.047) (0.042) and apply FTL backward, moving from 6 = 0.8 discounting 0.769 0.019 -0.175 right to left. (0.092) (0.036) (0.031) 6 = 0.9 discounting 0.704 0.085 -0.110 (0.096) (0.020) (0.019) This algorithmhighlights certainpaths, some 6 = 0.91 discounting 0.704 0.085 -0.109 of which intersect with the forward simnulation (0.096) (0.018) (0.019) paths generatedby the standardFTL algorithm. Notes: L(m) is the squaredEuclidean distance between the In general, decision-makers evaluate a given predictionsof model m and the empirical data. Asymptotic starting box by integrating over the union standarderrors appear in parentheses. of forward-FTL paths and the intersecting backward-FTLpaths. This combined algorithm is parameterized by the original probability sensible forecast may focus on the median or threshold,p, and p, a threshold value that de- the mean outcome. Such simplifications are fines the minimal magnitude that a payoff must generally effective strategies for planning in have to attractattention as an extreme outlier. complex environments. Likewise, incomplete Furtherresearch will extend this frameworkin contracts arise because it is too costly to de- two ways. First,new researchwill identify a par- velop contingenciesfor every state of the world. simonious set of parameterizedalgorithrns. Sec- Strategic planning is simplified when one as- ond, new research will provide a theory that sumes that other parties and naturewill behave describeshow the parametersadjust across prob- in typical ways. lems. Natural adjustment candidates include Occasionally, however, these simplifications reinforcement learning and expected-payoff prove to be counterproductive.For example, in adverse selection ("winner's curse") problems, simplifying uncertaintyleads generally to im- 7See Richard Thaler (1994) for a discussion of the portant deviations from rational decision-mak- "winner's curse." 438 AEA PAPERSAND PROCEEDINGS MAY2000 maximizationsubject to constraintson calculation Journal of Economic Literature, June 1996, and memory. We are currentlyexploring these 34(2), pp. 669-700. researchdirections. Erev, Ido and Roth, Alvin. "Predicting How Peo- ple Play Games." American Economic Re- REFERENCES view, September 1998, 88(2), pp. 848-81. Gabaix, Xavier and Laibson, David. "A Positive Camerer,Colin and Ho, Teck-Hua."Experience Theory of Bounded Rationality." Mimeo, Weighted Attraction Learning in Normal- MassachusettsInstitute of Technology, 1999. Form Games." Econometrica, July 1999, Kalhneman, Dan'iel and Tversky, Amos. "Judg- 67(4), pp. 827-74. ment Under Uncertainty: Heuristics and Camerer, Colin; Johnson, E. J.; Rymon, T. and Biases." Science, 27 September 1974, Sen, S. "Cognitionand Framingin Sequential 185(4157), pp. 1124-3 1. Bargaining for Gains and Losses," in Ken Simon, Herbert. "A behavioral model of rational Binmore, Alan Kirman, and P. Tani, eds., choice." Quarterly Joumrnal of Economics, Frontiers of game theory. Cambridge, MA: February 1955, 69(1), pp. 99-118. MIT Press, 1994, pp. 27-47. Thaler, Richard. The winner's curse. New York: Conlhlsk, John. "Why Bounded Rationality?" Russell Sage, 1994.