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

To interpret the theorem assume that, unknown to its opponent, firm n "experiments" with a new costing methodology which inflates its relevant costs by some small amount. If all firms adjust their prices adaptively, then firm n's experiment triggers a process of pricing adjustments that leads to higher economic profits for this firm. The theorem asserts that this can be accomplished for a range of values of firm n's distortions independent of the distortions of other firms, the initial equilibrium, and the specific adaptive process of price adjustments.

To provide an intuition for the proof, fix an equilibrium [bar.p] of [GAMMA]([[bar.s].sub.-n], [[s].sub.n] = 0) and consider an arbitrarily small distortion [[??].sub.n]. > 0 that triggers an adaptive adjustment process converging to a new equilibrium price [??]([[s].sub.n]). The question is whether firm n's economic profits increase as a result. There are two possibilities. In the first case, [??]([[s].sub.n]) is bounded away from [??] as [s.sub.n] goes to zero, in which case firm n is easily seen to achieve higher economic profits. Roughly, this happens if [bar.p] is not locally stable, so a slight perturbation leads to a large jump in prices. In the second ease, [??]([[s].sub.n]) is not bounded way from [bar.p] as [[s].sub.n] vanishes. Here, roughly, the net effect on the economic profits of firm n consists of a negative direct effect (due to the distortion of its price, holding the prices of its rivals fixed) and a positive strategic effect (due to its rivals raising their prices). We show that for a small enough distortion of relevant costs, the latter dominates the former.

We note that no assumptions are made about firm n's knowledge about the demand functions, costs, or costing methodologies of its rivals. Nor does any firm need to know, specifically, what new equilibrium would be reached if it distorted its relevant cost.

[] Adjustment of costing methodologies. We now turn to the process through which firms choose costing methodologies. As suggested in the Introduction, new complications arise, namely that firms do not know their rivals' strategies or their own payoff functions. Because firms must learn about their payoffs as well as resolve the strategic uncertainty about their opponents' behavior, the learning model here is of necessity different from the adaptive model used in the pricing game. (17)

We introduce a model of reinforcement learning through trial and error. In this model, motivated by Milgrom and Roberts (1991), firms know their strategy set and observe their realized payoffs. But they do not know their payoff functions or the strategies used by their opponents. Firms experiment with different costing methodologies from time to time and reinforce those that have done well on average in the past.

A dynamic game of experimentation and cost adjustments. We assume firms can adjust prices much more frequently than they can adjust their costing methodologies. To model this, we first restrict firms to choose from a finite grid of relevant cost distortions [[S].sub.n], = {0, [delta], ..., K[delta]), where K is a positive integer and [delta] > 0 ([[S].sub.n] is the same for all firms). Firm n chooses [[s].sub.n]([tau]) [member of] [[S].sub.n] at times [tau] = 1, 2,.... Within each "interval" [[tau], [tau] + 1], we have a sequence of price adjustment sub-periods t = 1, 2, ..., during which firms may adjust prices but not their distortion of relevant costs. (18)

To define the dynamic game of cost adjustments, fix s(0) arbitrarily and let p(0) be any equilibrium of the game [GAMMA](s(0)). (19) At time [tau], firm n is chosen with probability 1 / N, at which point this firm is allowed to reevaluate its costing methodology. With probability [epsilon] > 0, this firm experiments by picking a distortion [[s].sub.n]([tau]) uniformly from [[S].sub.n]. We call these periods "firm n's experimental periods." With probability 1 - [epsilon], firm n picks the distortion that generated the highest average payoff in its past experimental periods. Firms experiment independently from each other and over time. All other firms m [not equal to] n set [[s].sub.m] ([tau]) = [[s].sub.m]([tau] - 1).

Firm n's payoff at [tau] is

[[PI].sub.n](s([tau]), p([tau] - 1)) [equivalent to] [[pi].sup.e.sub.n](p([tau])),

where p([tau]) = [lim.sub.t[right arrow][infinity]] [p.sub.t] and {[p.sub.t]} is any adaptive pricing adjustment sequence for [GAMMA](s([tau])) starting at p([tau] - 1). (20) To motivate this definition, suppose that [tau] > 0 is an experimental period for firm n in which firm n chooses [[s].sub.n]([tau])[not equal to] [s].sub.n]([tau] - 1). This triggers a process of price adjustments starting from p([tau] - 1) and converging to a new equilibrium price p([tau]). Our assumption that prices adjust more rapidly than costing methodologies implies that the limiting price p([tau]) obtains "before" period [tau] + 1 arrives. The payoff function above says that firm n uses the profit from this final price, [[pi].sup.e.sub.n] (p([tau])), in judging its experiment at [tau].

Our model is motivated by Milgrom and Roberts's (1991) study of learning and experimentation in repeated, normal form games. Their analysis is inapplicable in our setting, however, because we are dealing with a dynamic game with history-dependent payoffs. In our model, the payoff structure at time [tau] is a reduced form for an underlying pricing dynamic whose outcome may depend on the initial condition p([tau] - 1).

Learning result.

Theorem 2. There is [delta] > 0 such that for every 0 < [delta] < [DELTA] and any grid [[S].sub.n] = {0, [delta], ..., K[delta]}, with probability one, there is [bar.[tau]] such that for every firm n, in all nonexperimental periods [tau] [greater than or equal to] [bar.[tau]], [s.sub.n]([tau]) > 0.

That is, on almost all paths all firms eventually distort their relevant costs. The strength of the results is in how weak the assumptions are. Finns only know their past distortion choices and, in each case, how well they have done in price competition. This rather coarse information does not allow firms to correlate changes in their payoffs with changes in their rivals' actions, or conduct counterfactuals such as "I would do better by choosing [s.sup.'.sub.n] given the actions of my rivals." Nevertheless, this coarse information is sufficient for firms to eventually reject "rational" costing.

* No-distortion results. Our analysis makes sharp predictions about which environments are unlikely to reinforce the sunk cost bias, and which environments tend to eliminate it.

Monopoly market. The intuition underlying our analysis is that the benefit from distorting relevant costs stems from its favorable strategic effects. This strategic effect is absent in monopoly, so our model implies that no distortion should persist in a monopoly market. Even though a monopolist is just as likely to be predisposed to confound relevant and irrelevant costs, (nonstrategic) learning will eventually lead the firm to price optimally. This conclusion is consistent with the experimental results of Offerman and Potters (2006), who find that the effects of the sunk cost bias disappear in the monopoly treatment of their experiments. (21)

Homogeneous product benchmarks. Firms' behavior also conforms to standard theoretical predictions in two standard benchmarks: perfect competition and price competition in an oligopoly with homogeneous products. Consider, first, perfect competition, and let [p.sup.*] be the equilibrium market price. An individual firm is a price taker, and so its output decision has no effect on the market price and hence on other firms' output decisions. A firm that distorts its relevant costs merely moves its own output away from the profit-maximizing quantity and hence reduces its economic profits.

A no-distortion result can also be established for a Bertrand oligopoly with homogeneous products. Suppose the oligopoly consists of N firms with constant marginal costs ordered so that [[c].sub.1] [less than or equal to] [[c].sub.2] ... [less than or equal to] [[c].sub.N]. Let D(p) be a demand curve which is continuous and strictly decreasing at all p where D(p) > 0 and that there is a "choke price" [bar.p] < [infinity] such that D(p) = 0 for all p [greater than or equal to] [bar.p] and we assume [bar.p] > [c.sub.1]. The demand for firm n is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We also assume that the price firm 1 sets if it is a monopolist is greater than [c.sub.2]. Otherwise, firm 1 is effectively a monopolist even in the presence of competition and hence the analysis is very similar to the monopoly case above.

Proposition 2. There is no incentive to distort relevant costs under price competition in a homogeneous product oligopoly.

The intuition underlying this result is straightforward. In a homogeneous product Bertrand oligopoly, each firm has such a strong incentive to undercut its competitor to capture the entire market that it is impossible to soften price competition via cost distortion.

Although these textbook benchmarks are useful, a more interesting question in practice is what happens in industries that are near-perfectly homogeneous. To explore this, and a number of other questions, the next section presents an example with linear demand that allows us to study how equilibrium distortions change as product differentiation vanishes and as the number of firms increases without bound.

5. Special case: symmetric linear demand