B OUNDED R ATIONALITY in L ABORATORY B ARGAINING with A SSYMETRIC I NFORMATION Timothy N. Cason and Stanley S. Reynolds Economic Theory, 25, (2005)

Class (Mean)

1 0 p  Continuation probability Buyer value Price percent 54 Price frequency # accepted Acceptance rate

KEY IDEAS: Multi-round monopoly pricing Treatment variable – Pr. of continuing into a second round. Comparison of results (prices and buyers’ purchasing behavior) with equilibrium predictions. Alternative explanation – Bounded Rationality of subjects.

Market Forces: Buyer may have an incentive to wait for a better price offer. Seller may anticipate strategic behavior by buyers when choosing prices. Seller may practice a form of inter-temporal price discrimination. Variables of interest: Changes in information Changes in the discount factor Changes in the time horizon

Related Research Game-theoretic predictions: Patient monopolist – Initial P ▲ with impatience of buyer (P-discrimination) Initial P ▲with time horizon Demand withholding ▼ Impatient monopolist - Initial P ▼ with patience of buyer (Intrapersonal Initial P ▼ with time horizon P-competition) Demand withholding ▲

Güth, Ockenfels and Ritzberger (1995):  Expts with different combination of max # rounds (2/3) and S / B discount factors.  Prices failed to conform to comparative statics predictions and Ps tended to be >> predicted Ps. Rapoport, Erev and Zwick (1995):  1-sided incomplete info.; ∞ time horizon. Price offers tended to decline over time.  Avg. initial P ▲ with impatience of monopolist.  In some cases the avg. initial P > static monopoly P.

Cason and Sharma (2001):  Role of info about buyers’ values (extent of price- discrimination).  Though uncertainty regarding demand vanished in the limit, Ps were much closer to equilibrium predictions of the incomplete info model. Reynolds (2000):  Bargaining expts with one, two and six rounds. Demand withholding results in line with predictions.  Opening Ps ▲ with trading horizon.

Adjustments to take into account risk preferences or preferences for fairness are not adequate to explain the results in these experiments. Experimental Design:  9 sessions with 15 or 13 subjects in each.  z-tree program; “strangers” design*.  Switching roles helps the subjects learn how to backward induct.  Subjects were undergrad students at the Univ. of Arizona and Purdue Univ.  12 lab dollars = $1; Mean earnings = $36 per subject

PBE predictions:  Pr = 0; (50 X 0.4) > 15  Pr = 0.3 or 0.6; Incentive constraint : v – p 1 ≥ Pr.(v – p 2 ) v = 54; p 2 = 15 (Pr, p 1,p 2 ) = {(0,50,15);(0.3,40,15);(0.6,30,15)}  Pr = 0.9; p 1 = 50; Pr(p 2 = 15) ≠ 1

Fairness concerns: Buyers accept “unfair” offers (e.g., price offers of 15 to a v = 18 buyer, and price offers of 45 to a v = 54 buyer) 70 – 85% of times.  Alternating roles could have reduced the importance of fairness concerns – subjects were able to “take turns” taking advantage of the privileged seller position. Risk aversion: A modest level of risk aversion results in fairly small changes in PBE predictions, and what change there is tends to be in a direction away from what is observed.

Bounded Rationality That property of an agent that behaves in a manner that is nearly optimal with respect to its goals as its resources will allow. Used to designate rational choice that takes into account the cognitive limitations of both knowledge and cognitive capacity. Central theme in Behavioral Economics. It is concerned with the ways in which the actual decision-making process influences decisions.

Two models:  Noisy Nash Model (NNM): Play equilibrium strategy with Pr  and randomize (uniformly)* over all strategies with Pr 1- .  Quantal Response Equilibrium (QRE): Agents choose actions that yield higher expected payoffs with higher probability [smoothed-out best response].

NNM – parameter  logit-AQRE – each agents choice probabilities follow a multinomial logit distribution with parameter  logit-AQRE with subject risk aversion. Max. likelihood estimates for these models reveal that the logit-AQRE has a better fit than the NNM model and inclusion of risk aversion has slight improvement. The logit-AQRE model predicts opening offer prices close to observed prices (Table 2) and is also consistent with the wide range of offer prices observed.

Conclusions PBE makes poor predictions for price levels, comparative statics, and buyer purchasing behavior in a simple multi-round pricing game. The logit-AQRE model captures important features of the result. Fairness considerations is not sufficient to explain deviations from equilibrium, and Bounded Rationality and Risk Aversion alone are consistent with a reasonably large proportion of the deviations.