**- 1 -**The Satisficer’s Curse Robert E. Marks Australian Graduate School of Management University of New South Wales Sydney, NSW 2052, Australia bobm@agsm.edu.au Ph: +61 2 9931 9271 F: +61 2 9313 7279 ABSTRACT: Following the Winner’s Curse and the Optimizer’s Curse, this paper introduces the Satisficer’s Curse, a systematic overvaluation that occurs when any uncertain prospect is chosen because its estimate exceeds a positive threshold.1 JEL codes: D81, D84, G11, M51 Ke ywords: satisficing; satisficer’s curse; Bayesian models 1. Introduction Capen et al. (1970) named the phenomenon of the Winner’s Curse in auctions, epitomised by the saying, “The good news is you won; the bad news is you paid too much”. Their model was generalised by Harrison and March (1984), and further generalised by Smith and Winkler (2006), who identified the “Optimizer’s Curse,” to argue that any decision that chooses the best prospect from a set of possibilities will fall heir to “post-decision disappointment,” on average. Brown (1974) drew attention to the post-decision disappointment associated with internal capital investment decisions, and Harrison and March (1984, p.27) and Compte (2004, p.9) also commented that selection bias would occur when choosing the best from a set of alternatives, without any recourse to psychological motivation (Lovallo and 1. I wish to thank Dan Lovallo and Danny Kahnemann for the presentation that sparked this paper, even if they disagree with my thesis, and Anna Gunnthorsdottir, Sally Wood, Dennis Turner, and Sharyn Roberts for their helpful comments.- 2 - Kahnemann 2003). This paper formalises the post-decision disappointment flowing from selection bias when the decision-maker is choosing an alternative that satisfies some (positive) hurdle in expected performance, the Satisficer’s2Curse. 2. The Model Assume n independent projects available for internal investment by a company, each of which has an independent value µiand an error term εi. The estimate Viof the ith project’s value is the sum of the true value µiand the error term εi: (1) Vi= µi+ εi. The error terms are independent across projects, and unbiased: (2) E[εi] = 0. Assume that the distributions over values and errors are non-degenerate. Assume that the value µiand the error εieach admit of a density (denoted by fiand girespectively), that the support of each density is an interval, and that each density is positive in its support. There is a single realisation viof the value estimate Vi, and (at most) a single realisation xiof the random variable µi. In a Bayesian world the firm would know the joint distribution over the values µi and the estimates Vi, and would be able to compute, for each realisation viof the estimate Vi, the conditional probability over the value, and hence the conditional expectation of value: (3) i(vi) ≡ E[µi|Vi= vi]. Y* Here, in contrast, assume that the firm is aware that values µiare distributed independently, but that, conditional on Vi, the firm forms an erroneous predictionˆYiof the value µi. Assumption 1: The predicted valueˆYiis given by Vi. 2. Simon (1957) introduced the word “satisfice” as a description of good-enough non-optimizing decision making; satisficing is now institutionalized as a means of making multifarious decisions.

- 3 - Denote the difference between the actual and the correct Bayesian predictions by Hi: (4) Hi(Vi,ˆYi) ≡ˆYi−Y* i(vi). For any giv en realisation of ViandˆYithis can be interpreted as the optimism associated with project i. Under Assumption 1, E(Vi) = E(µi), so that on average there is no optimism associated with any project i, and the prediction errors cancel: E(Hi) = 0. Let ∆idenote the expected difference between predicted and realised values, conditional on being chosen:3 (5) ∆i≡ E[ˆYi− µi|i is chosen] = E[ˆYi] − vi Definition 1: Project i exhibits the Optimizer’s Curse when ∆i> 0. In other words, the Optimizer’s Curse refers to situations in which the value of the chosen project was overestimated. Following Compte (2004), note that ∆ican be rewritten as: (6) ∆i= E[ˆYi−Y* i|i is chosen] = E[Hi|i is chosen], i|i is chosen] = E[µi|i is chosen] by construction. When a firm chooses a single project from a set, with possibly erroneous since E[Y* estimates, the firm will suffer the Optimizer’s Curse, even though each prediction is an unbiased estimate of that project’s value. Only if there was only a single alternative would the firm not experience the Optimizer’s Curse. The act of choosing the project with the highest predicted value induces a selection bias in favour of projects with (overly) optimistic value predictions. Theorem 1: WithˆYi= Vi, if 0 < Pr{i chosen} < 1, then ∆i> 0, that is, if the decision maker uses the naive forecast, and project i could be chosen (i.e. that estimateˆYiis the highest is neither certain nor impossible), then project i exhibits the Optimizer’s Curse. Before proving Theorem 1, consider satisficing: choosing project i when it has a 3. The following formal model is an adaptation from Compte (2004), who treated the Winner’s Curse in auction selection.

- 4 - high value prediction (say,ˆYiabove a positive threshold p), and so appears attractive. We shall prove that in this case, project i exhibits an optimistic prediction of its value, and hence post-decision disappointment. Let Sidenote the expected difference between predicted and realised values, conditional on project i’s predicted performance exceeding a positive hurdle or threshold level p > 0: (7) Si≡ E[ˆYi− µi|ˆYi> p] = E[Hi|ˆYi> p > 0]. Definition 2: Project i exhibits the Satisficer’s Curse when Si> 0. Theorem 2: Consider positive p such that 0 < Pr{ˆYi> p} < 1. That is, the event that the predictionˆYiis above the threshold p is neither certain nor impossible. Then, under Assumption 1, we have Si= E[Hi|ˆYi> p] > 0. That is, for any value realisation, high realisations of the predictionˆYicoincide with high realisations of the error term εiin equation (1), and hence with the over-valuation, so project i exhibits the Satisficer’s Curse. Theorem 1 will follow because project i is only chosen in events where its predicted maxˆYj, the highest prediction across other projects. value is equal to or greater than p = j≠i Proof of Theorem 2: (after Compte 2004). Under Assumption 1,ˆYi= Vi, so the event {ˆYi> p} is equivalent to the event {εi> Zi}, where the random variable Ziis defined: Zi≡ p − µi. Moreover, E[Hi|ˆYi> p] = E[Vi−Y* = E[εi|εi> Zi]. i|εi> Zi] But E[εi] = 0 and εiis independent of values µi, for any realisation zi∈ Zithat falls within the support

- 5 - of εi(which is infinite if the p.d.f. giis Gaussian). Thus we have E[εi|εi> Zi,Zi= zi] > 0. Since Pr{εi≥ Zi} ∈(0,1), the supports of εiand Zimust overlap. This follows because each support is an interval, and because ˆ εiand Ziboth admit of a density that is ev erywhere positive in its support, by definition. Hence E[Hi|ˆYi> p] > 0, and Theorem 2 is proved. Proof of Theorem 1: Define p ≡ maxˆYi j≠i We hav e ∆i= E[Hi|ˆYi> p], given that i is preferred to all others. Since 0 < Pr{i is chosen} < 1, the support of p and ˆYimust overlap, for the same reason as above. Thus, from Theorem 2, the result follows: ∆i> 0, and Theorem 1 is proved (under Assumption 1): the chosen project exhibits the Optimizer’s Curse. 3. Discussion The Satisficer’s Curse is similar to the Peter Principle, which however uses estimates based on past performance alone (Lazear 2004). Our estimates do not require observation of past performance. We could have modelled Assumption 1 as: (8) Vi= µi+ λεi, where λ ∈ [0,1), (Compte 2004). The firm’s predictionˆYiof the value µiis: ˆYi≡ Eλ[µi|Vi], where the superscript λ means that the expectation is taken assuming a joint distribution over Viand µiTheorem 1 proves that a project chosen under the naive Assumption 1 will

- 6 - exhibit the Optimizer’s Curse, and equation (8) shows that only if the full error term εiis acknowledged doesˆYi= Y* i. What about learning? What is our decision-maker to learn? Should he or she ignore the ranking by predicted value because of the error terms εi? To do so would be to throw information away. Raising any return hurdle ˆ p that some projects are predicted to exceed will not obviate the Satisficer’s Curse so long as the error term is ignored (λ = 0) or discounted (λ ∈[0,1)). If the hurdle is an institutional threshold, then an understanding of the Satisficer’s Curse might result in the institution learning to put procedures in place to reduce the prospect of performance reverting to the mean in future. If learning is happening anyway (such as “learning by doing”), then the assumption of stationarity (of distribution fi) is violated. If such learning is for whatever reason not available, then acknowledgement of the Satisficer’s Curse should qualify expectations that future performance will reflect past estimates; on average it will not.

- 7 - References Brown, K.C., 1974, A note on the apparent bias of net revenue estimates for capital investment projects, Journal of Finance, 29(4): 1215−1216. Capen E.C., R.B Clapp and V.M. Campbell, 1971, Competitive bidding in high risk situations, Journal of Petroleum Technology, 23: 641−653. Compte, O., 2004, Prediction errors and the winner’s curse, ENPC-CERAS working paper, www.enpc.fr/ceras/compte/prediction2004.pdf (accessed 2006/05/15) Harrison J.R. and J.G. March, 1984, Decision making and postdecision surprises, Administrative Science Quarterly, 29: 26−42. Lazear, E.P., 2004, The Peter Principle: a theory of decline, Journal of Political Economy, 112(1 pt.2): S141−S163. Lovallo, D. and D. Kahnemann, 2003, Delusions of success: how optimism undermines executives’ decisions, Harvard Business Review, 81(7): 56−, July. Simon, H.A., 1957, Models of Man, (John Wiley, New York). Smith, J.E. and R.L. Winkler, 2006, The optimizer’s curse: skepticism and postdecision surprise in decision analysis, Management Science, 52(3): 311−322.