Market forces meet behavioral biases: cost misallocation and irrational pricing.

Related ideas appear in the literature on the evolution of preferences. Samuelson (2001) offers an excellent overview of this literature. (26) Closer to our model is the recent work of Heifetz, Shannon, and Spiegel (2004). They offer an interesting model of the evolution of optimism, pessimism, and interdependent preferences in dominance-solvable games. Like strategic delegation, the evolution of preferences literature underscores the value of commitment, but this time using distortions of players' perceptions of their payoffs as a commitment device. The unobservability of firms' costing practices again separates our model from the evolution of preferences approach. For instance, as Dekel, Ely, and Yilankaya (2007) point out, this approach has little bite when players cannot observe their opponents' preferences. And as in the delegation paradigm, sunk cost plays no role in evolutionary arguments; these arguments work equally well in settings where no sunk cost is present.

7. Concluding remarks

* There is extensive evidence that real-world decision makers violate the predictions of standard economic theory. Among these violations, the sunk bias in pricing decisions stands out on a number of grounds. First, its impact is not limited to small-stakes decisions: pricing is one of the most critical decisions a company can make. Second, unlike other cognitive biases that disappear once the underlying fallacy is explained, distorted pricing seems to thrive despite the relentless efforts of economics and business educators to stamp it out. Third, although many cognitive biases disappear through learning and training, the survey evidence we report provides no indication that this bias is disappearing over time.

In this article, we provided a theory of why confusion of relevant and irrelevant costs persists in pricing practice. (27) We showed that under conditions rooted in actual cost accounting practices, price competition with product differentiation reinforces managers' innate predisposition to confound relevant and irrelevant costs. And although there is no reason to suspect this predisposition to systematically vary across industries, no similar forces appear in monopoly, perfect competition, or price competition in a homogeneous product oligopoly. Thus, our analysis makes predictions that can be empirically and experimentally tested.

Our theory builds on recent advances in learning in games played by naive players who know little about their environments. As such, this article illustrates how learning theory can be a valuable tool in understanding the nature of behavioral biases in economics.

Appendix

* The proof of Theorem I relies on Proposition 1 and the following result.

Proposition 9. There exists [DELTA] > 0 such that for any firm n, any vector of distortions [s.sub.-n] [greater than or equal to] 0, any [s.sub.n] [less than or equal to] [DELTA], and any equilibrium price vector p of [GAMMA] ([s.sub.-n], 0) with corresponding quantity vector q > > 0, if [p.sup.+] is an equilibrium of [GAMMA] ([s.sub.-n], [s.sub.n]) with [p.sup.+] > p,

[[pi].sup.e.sub.n]([p.sup.+]) > [[pi].sup.e.sub.n](p).

Proof of Theorem 1. Let [DELTA] be as in Proposition 9 and choose any 0 < [[??].sub.n], [less than or equal to] [DELTA]. Fix any [[bar.s].sub.-n] and p as in the statement of the theorem. By Proposition 1, any adaptive price adjustment {[p.sub.t]} starting from [bar.p] for [GAMMA] ([[bar.s].sub.-n], [[??].sub.n]) converges to p = inf{z [member of] E ([[bar.s].sub.-n], [[??].sub.n]), z [greater than or equal to] [bar.p]}. It is easy to see that in fact p >> [bar.p]. The result now follows from Proposition 9.

Proof of Proposition 9. Suppose, by way of contradiction, that for every positive integer k, there is [s.sub.n] < 1/k, a vector of distortions [s.sup.k.sub.-n], an equilibrium [[bar.p].sup.k] of [GAMMA] ([[bar.s].sup.k.sub.-n]], 0), and an equilibrium [[??].sup.k] of [GAMMA]([s.sup.k.sub.-n], [s.sub.n]) with [[??].sup.k] > [[bar.p].sup.k] yet [[pi].sup.e.sub.n]([[??].sup.k]) [less than or equal to] [[pi].sup.e.sub.n]([[bar.p].sup.k]). Passing to subsequences if necessary, we may assume that [[s.sup.k.sub.-n] [right arrow] [s.sub.-n], [[bar.p].sup.k] [right arrow] [bar.p], and [[??].sup.k] [right arrow] [??]. Clearly, [??] and [[pi].sup.e.sub.n]([??]) [less than or equal to] [[pi].sup.e.sub.n]([bar.p]). We note that, by Lemma 2 below, [??] is an equilibrium of [GAMMA]([s.sub.-n], 0). We consider two cases.

Case 1. [??] > [bar.p]. In this case, the fact that best responses are strictly increasing implies [??] >> [bar.p]. Next we show that [[pi].sup.e.sub.n]([??]) > [[pi].sup.e.sub.n]([bar.p]). From our assumptions on demand and the fact that q > > 0.

[[pi].sub.n]([[??].sub.-n], [[bar.p].sub.n], [s.sub.n] = 0) > [[pi].sub.n]([bar.p], [s.sub.n] = 0) = [[pi].sup.e.sub.n]([bar.p]). (A1)

That is, relative to [bar.p], firm n achieves higher profit if its opponents strictly increase their prices while all other prices remain unchanged. Because [??] is an equilibrium, we must also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which combining with (A1) implies [[pi].sup.e.sub.n]([??]) > [[pi].sup.e.sub.n]([bar.p]). By the continuity of [[pi].sup.e.sub.n], for k large enough, [[pi].sup.e.sub.n]([[??].sup.k]) > [[pi].sup.e.sub.n]([[bar.p].sup.k]), a contradiction.

Case 2. [??] = [bar.p]. For each k, define

[z.sup.k] = [[??].sup.k] - [[bar.p].sup.k]/[parallel][[??].sup.k] - [[bar.p].sup.k][parallel].

Let [S.sup.N] denote the unit sphere and, given [alpha] > 0, define [B.sub.[alpha]] = [S.sup.N] [intersection] {[x.sup.N] [member of] [R.sup.N.sub.+], [x.sub.n] [less than or equal to] 1 - [alpha]}. We first need the following lemma.

Lemma 1. There is [alpha] > 0 such that for all large-enough k, [z.sup.k] [member of] [B.sub.[alpha]].

From the definition of derivatives, for every [epsilon] > 0, there is [bar.h] such that for any 0 < h < h and any z [member of] [S.sup.N],

[absolute value of ([[pi].sup.e.sub.n]([[bar.p].sup.k] + hz) - [[pi].sup.e.sub.n]([[bar.p].sup.k] - [[partial derivatives][[pi].sup.e.sub.n]([[bar.p].sup.k])]x)] < [epsilon],

where [partial derivative][[pi].sup.e.sub.n]([[bar.p].sup.k]) is a row vector of partial derivatives of [[pi].sup.e.sub.n]. evaluated at [[bar.p].sup.k]. Notice that [partial derivative][[pi].sup.e.sub.n]/[partial derivative][p.sub.n] ([[bar.p].sup.k] = 0 as [[bar.p].sup.k] is an equilibrium and [partial derivative][[pi].sup.e.sub.n]/[partial derivative][p.sub.m]([[bar.p.sup.k) > 0 for m [not equal to] 0 and [q.sup.k.sub.m] > 0 for all k large enough. Note that [[partial derivative][[pi].sup.e.sub.n]([bar.p)]z > 0 for all z [member of] [B.sub.[alpha]].

Next we note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To see this, write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

For any pair of subsequences {[[??].sup.l]}, {[[??].sup.l]} converging to [??] and [??], respectively, we have [??] [member of] [B.sub.[alpha]] and [??] [member of] ([s.sub.-n], 0) (the latter holds by Lemma 2). From the continuity of [[pi].sup.e.sub.n] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Fix [alpha] > 0 and set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, given any z [member of] [B.sub.[alpha]] and [[bar.p].sup.k] [member of] ([s.sup.k.sub.-n], 0), for all k large enough we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [[pi].sup.e.sub.n]([[??].sup.k]) - [[pi].sup.e.sub.n]([[bar.p].sup.k]) > 0 for all k large enough. A contradiction.

Proof of Lemma 2. Given k, for m [not equal to] n

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [[??].sup.k] [right arrow] [bar.p], [[bar.p].sup.k] [right arrow] [bar.p], and [s.sup.k.sub.-n] [right arrow] [s.sub.-n], because [partial derivative][r.sub.m]/[partial derivative][p.sub.n] is continuous, for every [member of] > 0, there is a [bar.k] such that for all k > [bar.k]

[r.sub.m]([[??].sup.k.sub.-n], [[bar.p].sup.k.sub.n] + ([[??].sup.k.sub.n] - [[bar.p].sup.k.sub.n]), [s.sup.k.sub.m]) - [[bar.p].sup.k.sub.m] > [partial derivative][r.sub.m]/[partial derivative][p.sub.n]([bar.p], [s.sub.m]) - [epsilon] (A2)

for all m [not equal to] n.

Note further that

[[??].sup.k.sub.m] = [r.sub.m] ([[??].sup.k.sub.-n], [[??].sup.k.sub.n], [s.sup.k.sub.m]) = [r.sub.m]([[??].sup.k.sub.- n], [[bar.p].sup.k.sub.n] + ([[??].sup.k.sub.n] - [[bar.p].sup.k.sub.n]), [s.sub.k.sub.m])

and hence from (A2), for every [epsilon] > 0, for all k large enough and all m [not equal to] n,

[[??].sup.k.sub.m] - [[bar.p].sup.k.sub.m]/[[??].sup.k.sub.n] - [[bar.p].sup.k.sub.n] > [partial derivative][r.sub.m]/[partial derivative][p.sub.n] ([bar.p], [s.sub.m]) - [epsilon] > 0

if [epsilon] is small enough.