Bargaining with Asymmetric Dispute Costs

The take-it-or-leave-it bargaining structure is stylized and based on the theoretical literature, but there are examples of it being employed in the field. Mnookin et al. (2000) cite the use of this tactic by Ford Motor Company in pretrial negotiations (p. 107) and as its use by General Electric in labor negotiations in the 1950s (p. 215).

A taste for fairness is compatible with rationality in the sense that it may be an argument in the utility function. Here, when we refer to the rational model, we mean a model of narrow rationality in which fairness or other-regarding preferences play no role.

While our concern in this paper is fairness, there have been experiments analyzing a variety of other issues in the law and economics literature. A small sampling includes Coursey and Stanley (1988), Main and Park (2002), and Inglis et al. (2005) who study conditional fee shifting, Stanley and Coursey (1990) who study the Priest and Klein (1984) selection hypothesis, and Pecorino and Van Boening (2004) who study voluntary disclosure. There is also an extensive literature on arbitration, much of which compares the performance of different arbitration procedures. These include final offer arbitration and conventional arbitration, but others as well. A partial listing of papers in this literature includes Ashenfelter et al. (1992), Pecorino and Van Boening (2001), Dickinson (2004, 2005), and Deck and Farmer (2007, 2009).

In the experiment, we avoided using the terms plaintiff or defendant. This contrasts with the approach taken in the literature on self-serving bias in which players are specifically given a role as either a plaintiff or a defendant. For a survey of the self-serving bias literature, see Babcock and Lowenstein (1997).

Among others, see Güth et al. (1982), Hoffman and Spitzer (1982, 1985), Thaler (1988), Slonim and Roth (1998), and Fehr and Schmidt (2000).

For example, players A and B are asked to split $10, with B given the power to make a single offer to A. Player B can make any offer between $0 and $10, and if A rejects it, both receive nothing. The rational model predicts that a self-serving B will offer $0.01 to A (leaving $9.99 for himself) and that A will accept this as it is better than nothing.

See (for example) Table 1 in Slonim and Roth (1998). For offers containing between 30 and 35% of the pie, they find rejection rates of 45.5% in a low stakes ultimatum game, 22.9% in a medium stakes ultimatum game, and 11.1% in a high stakes game. These experiments were conducted in the Slovak Republic and the stakes ranged from 2.5 hours to 62.5 hours of the local wage.

In addition to differences in framing, our embedded game has the feature that it is possible for the proposer to make an offer “outside the pie”, i.e. either one that proposes negative surplus for the responder or one that proposes more than the entire pie for the responder. Neither is possible in the simple ultimatum game. As we discuss in Section 4, we observe 6% of B’s offers in one of these two categories (4% in the first and 2% in the second).

For example, Inglis et al. (2005; Figure 9) show a 10–20% “ex ante inefficiency” for plaintiffs where the last rejected offer is more favorable than the expected court decision; this is comparable our 9–19% A_L rejection rate on positive surplus offers (Section 4.2). Also see the discussion of Slonim and Roth in fn. 7.

For a discussion of differing fairness ideals, see Cappelen et al. (2007), who demonstrate that fairness ideals differ across individuals. Also see the discussion in Konow (2003).

Consider a model where a fair offer of surplus is λ% of the plaintiff’s courts costs with 0<λ≤1. Save-own-cost is a special case of this model with λ=1.

All of our experimental materials are available in “Bargaining with Asymmetric Dispute Costs – Appendix” at http://mycba.ua.edu/~ppecorin/bargaining_with_asymmetric_dispute_costs_-_appendix.pdf.

Subjects were recruited for “approximately 2 hours”; in sessions S3 and S7, period 13 ended after the 2-hour mark, so we did not conduct period 14. Paying and excusing the subjects usually took an additional 10–15 minutes.

The maximum dispute cost for player B is 400+F_B (see step 7). In step 4, we chose 599 – which exceeds 400+F_B in all cases – as a maximum offer to limit the possibility of bankruptcy from unexpectedly high offers while simultaneously not implicitly suggesting to B what offer he might choose. We observed four offers exceeding 400+F_B; all four were in session S3.

In the initial estimations of the regressions in Tables 3 and 5, we included a dummy variable to distinguish between the Treatment 1 baseline and the Treatment 2 baseline. This variable always had p>0.20, so we pool the baseline data in the subsequent analysis.

When B knows what A considers a fair offer, then conditional on that there being a sorting offer, the theoretical predictions are robust to the introduction of risk aversion on the part of either player. When player A makes her accept/reject decision, she faces no risk: she knows her exact payoff if she accepts the offer and her exact payoff if she rejects it. Thus, the predictions on A’s behavior will not change if she is risk averse. The prediction on the amount of the sorting offer is not a function of whether or not B is risk averse: his optimal sorting offer is simply the lowest offer that A_L will accept. If B is sufficiently risk averse, then he may engage in a pooling strategy and hedge against the 1/3 chance of encountering a player A_H (only 7% of offers fall in the pooling range). However, if A’s demand for fairness is not observable, then risk aversion may play a role in explaining our empirical results. We revisit this in Section 4.3.

From Table 8, the sorting offer percentages are 87% (309/357) in the baseline B, 78% (148/189) in T₁, and 87% (159/182) in T₂. The relatively low percentage in T₁ is due almost entirely to the 18 “<0” offers in S3 (see notes in Table 8). From Table 9, the percentages in rounds 4–7 and 11–14 are 88% in B, 87% in T₁, and 91% in T₂.

Offers in the <0, 151–199, and >350 intervals are poor offers from B’s perspective. In the first interval offers are less than A_L’s net dispute payoff and hence are unattractive to both player A types, those in the second offer A_L more than 100% of the surplus from settlement but are too low to be considered viable by A_H, and those in the third are offers that exceed B’s maximum possible dispute cost. We currently do not have parsimonious explanations as to why we observe these offers, but we note that they comprise a relatively small portion of the data.

We also examined dummy variables for site location, session-level effects, sequencing, and so on, but did not find evidence of systematic effects.

The rejections of the rational and equal-split models are due mainly to the constant term. The 95% confidence intervals for the sorting offer regression constant terms are (159.3, 175.0) for all rounds and (157.4, 174.0) for rounds 4–7 and 11–14; these represent about 23–33% of the surplus from settlement. Those models’ comparative static predictions have strong support: the corresponding 95% confidence intervals for T₁ are (41.7, 54.5) all rounds and (40.2, 55.1) rounds 4–7 and 11–14, while for T₂ they are (–54.1, –41.6) and (–56.6, –42.0), respectively.

The 95% C.I.s for the all offers regression constant terms are (167.2, 189.1) all rounds and (170.1, 196.3) rounds 4–7 and 11–14, for T₁ they are (17.8, 44.2) and (8.0, 38.4), respectively, and for T₂ they are (–57.0, –30.1) and (–67.8, –36.8), respectively.

Using the 95% C.I. in fn. 20, α ranges from 23% to 33% of the surplus.

In other words, we can reject the model where a fair offer is λ% of the plaintiff’s courts costs with 0<λ<1. Recall that save-own-cost is a special case of this model with λ=1.

Under the rational model, the recipient of an offer with 0 surplus is indifferent between acceptance and rejection. Player A_L received a total of eight offers with exactly zero surplus; five were rejected and three were accepted. Also, as described in the notes in Table 5, for player A_H surplus 0–25 and >25 refers to the surplus relative to the pooling offer prediction.

Given our parameters, a sorting equilibrium is predicted for all three models under all the distributions of costs we consider. However, in the save-own-cost model, a redistribution of costs toward the defendant could (under alternative parameter values) cause the sorting condition to fail.

Pecorino and Van Boening (2010) do not consider asymmetric dispute costs.

We use the Stata xtlogit postestimation predict command to obtain the probability of rejections, which assumes the random effect is zero.

A similar analysis is performed by (among others) Bolton (1991:1112–1119). Also, we obtain similar ex post optimal offer results when we use interval-level A_L dispute rates and B’s median offer from the respective intervals.

Hoffman et al. (1994) report a posted-offer exchange version of a $10 ultimatum game (B is a seller and A is a buyer) that shifts B’s offers toward the rational prediction of $0. The Contest Exchange treatment, where subjects earn the right to be the seller, has the biggest effect: the mean and median offers (n=24) are $3, and there are no offers above $5. Based on the data in their Figure 3(d), we compute that their average offer contains 30% of the surplus, which is similar to the 20–25% we find here, and that their ex post optimal offer contains 10% of the surplus vs. our finding of 10–17%. We note that their offers are restricted to 10% increments of the pie ($1/$10) while our increments are less than 1% ($0.01/$1.50), their n is smaller than ours, and their subjects played a single round while ours play multiple rounds.

This aspect of the experiment is related to the literature on ultimatum games with outside options. The paper in this literature most closely related to ours is Knez and Camerer (1995). Pecorino and Van Boening (2010) discuss the ways in which the framing in the Knez and Camerer experiments differs from the framing in stylized legal bargaining. In addition, there is no asymmetric information in Knez and Camerer. Schmitt (2003) also considers an outside option. She does so with complete information and incomplete information. Under the incomplete information treatment, players have no information on the value of their bargaining partner’s outside option or on the dollar value of the chips received by their bargaining partner.