Risk aversion and implicit shortage cost explain the Anchoring and Insufficient Adjustment bias in human newsvendors

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Abstract

The Anchoring and Insufficient Adjustment (AIA) bias has been observed in many newsvendor experiments, although a mathematical explanation for this behavior has previously eluded researchers. We show here that risk aversion coupled with an implicit shortage cost, both of which are well-known components of newsvendor decisions, comprehensively explains this behavior. We construct combinations of a risk-averse utility function and a shortage cost that explain the results from previously reported newsvendor experiments.

Introduction

Laboratory experiments using human newsvendors ([16] first, followed by  [1], [2], [3], [10], [6], [11]) observed that they ordered less (more) than the optimal quantity in the high (low) profit margin condition. [16] showed that this bias cannot be explained exclusively by risk aversion, risk seeking, loss aversion, waste aversion, stockout aversion, or underestimation of opportunity costs. They suggested that prospect theory might be able to account for this order pattern, but this was recently  [12] shown to not be the case.  [16] also suggested that ex-post inventory preference could explain this behavior, but acknowledged that it was not able to account for the observed asymmetry in the low margin and high margin settings. This observed asymmetry also cannot be explained by newsvendor overconfidence as manifested by underestimating the variability of demand, as analyzed by  [14]. Finally,  [16] mentioned in passing that a combination of these preferences, such as risk aversion and stockout aversion, might be able to explain this bias, but did not elaborate on how such a combination could be constructed. We show that risk aversion coupled with shortage cost (sometimes referred to as loss of goodwill, as in  [11]) is the only pair of causes that can comprehensively and reasonably characterize the AIA bias. After analytically establishing this result, we fit the model to data from  [16] and construct the specific forms of the risk aversion function and the shortage cost value that result in their findings.

Section snippets

The newsvendor decision

The basic parameters of a typical newsvendor decision are the demand distribution (ξU[l,u]), selling price (p), and purchase cost (c). While the demand can be from any probability distribution, for ease of analysis and consistency with most of the existing newsvendor experiments, we assume here that the demand follows a uniform distribution. We also consider demand to be continuously distributed, in contrast to much of the existing behavioral literature that assumes it to be discretely

Proposed causes for the human bias

The following seven causes are commonly considered when explaining the AIA behavior in human newsvendor decisions.

  • 1.

    Risk Aversion: Utility is a non-decreasing, concave function of profit, and the newsvendor’s objective is to maximize expected utility.  [5] showed that inclusion of risk aversion results in a lower order quantity for all settings of the problem parameters.

  • 2.

    Waste Aversion (t): Any leftover inventory must be disposed of at an additional cost of t(Qξ)+, implying that the salvage

Newsvendor with risk aversion and an implicit shortage cost

We define the non-decreasing function U(Π) as the utility that a decision maker receives after realizing profit Π. The objective function is then transformed into one of expected utility, Eξ[U(Π(ξ|Q))]. When the utility function is concave, it captures risk aversion. The shortage cost g (g0) quantifies the aversion to stockouts.  [7] demonstrated that human subjects consider loss of goodwill even when it is not explicitly included in the task description. Inclusion of this shortage cost

Explaining AIA observations

We say that AIA exists when the following conditions are satisfied: 0<c̄p/2c¯<p,andQ¯<Q¯oμQ̄o<Q̄. The smallest possible penalty cost, gmin, that explains Q¯o=l+(ul)×(pc¯+gminp+gmin) can be computed as gminc¯(uluQ¯o)p. Further, we compute the following quantities: π̄q=Π(Q̄o|Q̄o)=(pc̄)Q̄oπ̄u=Π(u|Q̄o)=(pc̄)Q̄og(uQ̄o)=π̄qg(uQ̄o)π̄l=Π(l|Q̄o)=plc̄Q̄o=π̄qp(Q̄ol)π¯q=Π(Q¯o|Q¯o)=(pc¯)Q¯oπ¯u=Π(u|Q¯o)=(pc¯)Q¯og(uQ¯o)=π¯qg(uQ¯o)π¯l=Π(l|Q¯o)=plc¯Q¯o=π¯qp(Q¯ol). In constructing

Application to Schweitzer and Cachon data

We apply the above-mentioned approach to explain the results observed in  [16]. Recall that in our continuous adaptation of their setting, l=0, u=300, p=12, c̄=3, c¯=9, Q¯o=134>75, and Q̄o=177<225. To achieve these order quantities, the shortage cost must satisfy: ggmin=9(3000300134)12=4.2651. Fixing g=gmin, we compute the six needed profit levels as follows: π̄q=(pc̄)Q̄o=(123)177=1593π̄u=π̄qg(uQ̄o)=15934.2651(300177)=1068.4π̄l=π̄qp(Q̄ol)=159312(1770)=531π¯q=(pc¯)Q¯o=(129)134=

Robustness of our result

In this section, we evaluate the robustness of our results when (i) Proposition 4 is satisfied but Proposition 6 is violated; (ii) Proposition 4 is not satisfied; and (iii) demand is normally distributed.

Conclusion

The well-documented Anchoring and Insufficient Adjustment bias in newsvendor decision making is consistent only with the combination of implicit risk aversion and shortage aversion. In order to mitigate this bias, managers should educate their decision makers to explicitly account for these aversions.

References (16)

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