The impact of entitlements and equity on cooperative bargaining: an experiment.

Bruce, Christopher ; Clark, Jeremy

I. INTRODUCTION

We consider a set of bargaining situations in which the disagreement outcome, or backstop, B, is chosen by a third party. For example, a government agency may signal that it will adopt as policy any agreement that stakeholders reach among themselves; but that otherwise, it will impose a policy of its own choosing. (1) Or an arbitrator may announce the award that will be imposed if a union and employer fail to reach agreement on a new contract.

Of particular interest are those situations in which the status quo allocation of resources, Q, is Pareto inferior to the imposed backstop. Although axiomatic models of cooperative bargaining (Kalai and Smorodinsky 1975; Nash 1950) presume that negotiators in such cases will condition their agreements on B and ignore Q, literatures on entitlements and focal points (Bazerman 1985; Nozick 1974) suggest that "history matters." That is, bargainers may be influenced by the status quo entitlement even when it lies outside the set of outcomes that are Pareto superior to the backstop. In addition, the experimental literature on inequality aversion (e.g., Fehr and Schmidt 1999; Hoffman and Spitzer 1985; Nydegger and Owen 1975) suggests that negotiators may be attracted to outcomes that equalize gains. In this article, we use a laboratory experiment to test the standard cooperative, entitlement, and egalitarian hypotheses in a multidimensional bargaining game.

We begin in Section II by describing a two-party, two-attribute bargaining space, based on the classic Edgeworth box. Drawing from the literature on cooperative bargaining, we initially hypothesize that: (1) the parties will reach an agreement, and that this agreement will be (2) Pareto efficient; (3)Pareto superior to the backstop; and (4) at the Nash bargain. As alternatives to (3) and (4), we consider two predictions from the literatures on entitlements/focal points and inequality aversion. These are that the agreements that parties reach will (5)be conditioned on the status quo allocation rather than the backstop, and (6) equalize the parties' payoffs (when the Nash bargain does not).

In Section III, we describe the design of a laboratory experiment in free-form cooperative bargaining to test these hypotheses. Our design does not use the one-dimensional mechanisms usually employed to analyze bargaining outcomes--such as dictator, "divide the pie," or ultimatum games. These mechanisms offer few opportunities for subjects to choose outcomes that are inefficient or Pareto inferior to the backstop, nor a clear method of differentiating the status quo allocation from the backstop. Instead, we employ an Edgeworth box game involving two subjects with Cobb-Douglas payoff functions over two goods, X and Y. We impose a Pareto inefficient backstop allocation of these goods and allow our subjects to negotiate a reallocation. If they reach an agreement, each party receives the associated value from his or her payoff table; otherwise each receives the payoff associated with the backstop. Each party's payoff table has roughly 200 possible combinations of X and Y over which he or she can bargain.

In Section IV, we present our results. We find that agreement rates are high and that those agreements are mostly in, or "close" to, the Pareto efficient set. The Nash bargain, however, is only supported when it equalizes the value of subjects' payoffs. When the (Pareto efficient) equal payoff outcome differs from the Nash bargain, subjects tend to compromise between the two allocations, some even agreeing to allocations that are Pareto inferior to the backstop. We also find that the tendency of subjects to choose the equal payoff outcome is strengthened when they perceive that the status quo distribution had also been equitable. Section V concludes the article with a discussion of the implications of our findings.

II. HYPOTHESES

We motivate our design with an environmental example drawn from collaborative bargaining over the use of public lands. Assume that two interest groups, environmentalists (Env) and developers (Dev), are in conflict over two dimensions of resource policy: the amount of public land to be protected, A, in acres, and the severity of restrictions to be placed on commercial activity on that land, R. Relative to the status quo allocation, Q, environmentalists prefer more of both A and R, whereas developers prefer less. We illustrate these assumptions in Figure 1, using a conventional Edgeworth box.

[FIGURE 1 OMITTED]

To set policy, the government allows the parties to bargain over the allocation of resources. It commits in advance that if negotiations succeed, it will implement the policy selected by the parties. If negotiations fail, however, it commits to impose a backstop policy, B. In some cases, this backstop will coincide with the status quo, Q--such as at B = Q in Figure 1. In other cases, the government commits to introduce a new policy if negotiations fail, or B [not equal to] Q. (2) As illustrated in Figure 2, the bargaining lens formed by new policy B may exclude Q.

The experimental bargaining literature has primarily focused on two competing hypotheses concerning the outcome that parties will select. The first is that if negotiators are motivated by material self-interest, they will select the Nash bargaining solution (Nash 1950)--the outcome that maximizes the product of their gains relative to the backstop (3) (see, e.g., Nydegger and Owen 1975; Roth and Malouf 1979). As the Nash bargain, N, must be both Pareto superior to B and Pareto efficient, (4) it will lie on the contract curve (CC) within the bargaining lens associated with B, as in Figure 1.

[FIGURE 2 OMITTED]

Alternatively, a number of experiments--notably, Nydegger and Owen (1975), Roth, Malouf, and Murnighan (1981), Hoffman and Spitzer (1985), Shogren (1997), and Bruce and Clark (2010)--have tested whether negotiators behave as if they are egalitarians. With the exception of Shogren, these authors have found that their subjects were drawn toward Pareto efficient outcomes that equalized payoffs--illustrated as E in Figures 1 and 2. Fehr and Schmidt (1999) and Bolton and Ockenfels (2000) have argued that these results are consistent with the assumption that negotiators are motivated both by material self-interest and aversion to inequality.

Neither the Nash bargain nor egalitarian literatures deal explicitly, however, with the effects of a divergence between the status quo and the backstop, particularly where any move from one to the other is Pareto worsening (as in Figure 2). In both models, a nonbinding, Pareto inferior outcome like Q should be irrelevant. Two literatures in economics suggest, however, that Q may affect negotiated outcomes. The first of these concerns "focal points," and the second "entitlements."

The focal point literature developed to explain Schelling's (1960) observation that, for sociocultural reasons, certain outcomes appear to have greater salience than others. Most studies in this literature are concerned with focal outcomes--such as a 50/50 split or the selection of "heads" when parties are asked to choose between heads and tails (Mehta, Starmer, and Sugden 1994). Other studies, however, suggest that negotiators may have focal starting points from which bargaining will proceed. For example, Bazerman (1985), building on the work of Tversky and Kahneman (1974), suggests that parties bargaining over the terms of a labor-management agreement might anchor their negotiations on the terms of the previous contract (see also Binmore, Shaked, and Sutton 1989). This suggests that if the starting allocation differs from the backstop, the negotiated outcome might be drawn away from B towards Q.

A parallel entitlement literature hypothesizes that negotiators' relative bargaining powers may be influenced by their relative perceptions of entitlement to the positions they have taken. A party which believes that its initial position is "fair" or "deserved" might press more vigorously to maintain that allocation than will a party lacking that belief. Parties may feel entitled to the status quo, or historical allocation, if it has been obtained without the use of threat, fraud, or force (Nozick 1974; Zajac 1995); or if a "moral authority" has told them that their "....entitlements are rights" (Hoffman and Spitzer 1985, 266). Roth, Malouf, and Murnighan (1981), Hoffman and Spitzer (1985), Gachter and Riedl (2005), and Falk, Fehr, and Zehnder (2006) have each found some evidence to support this hypothesis. Thus, the focal point and entitlement literatures predict that when the backstop differs from the status quo, negotiators will be drawn from allocations in the bargaining lens conditioned on the backstop to those in the lens conditioned on the status quo.

On the basis of these considerations, we will test three competing hypotheses concerning the agreements collaborative bargainers will reach:

* Nash Bargaining Solution: The parties will negotiate to the Nash bargain, N, that is conditioned on B.

* Egalitarian: The parties will negotiate to the Pareto efficient allocation at which payoffs are equalized, E (which need not be Pareto superior to B).

* Entitlement: The parties will negotiate to a Pareto efficient allocation within the bargaining lens conditioned on Q (which need not be Pareto superior to B).

III. EXPERIMENTAL DESIGN

A. Design Features Across All Treatments

To implement bilateral bargaining over two dimensions of policy, we recruited subjects in groups of ten, and gave each an induced value payoff function over two abstract goods, X and Y. Five subjects were assigned one payoff function, and five another, based on their prior choice of seat in the room. For exposition, we refer to the two preference types induced by these payoff functions as environmentalists and developers, though the neutral labels "you" and "the other person" were used in the experiment. To generate convex indifference curves for each type over the two goods, we used Cobb-Douglas payoff functions:

(1) [P.sub.Env] = [a.sub.Env] [X.sup.[alpha].sub.Env] [Y.sup.1 - [alpha].sub.Env] + [b.sub.Env]

(2) [P.sub.Dev] = [a.sub.Dev] [X.sup.[alpha].sub.Dev] [Y.sup.1 - [alpha].sub.Dev] + [b.sub.Dev].

The use of a common exponent, [alpha], for both types implied that the CC was a diagonal line. The use of constant returns to scale ensured that total payoffs would be constant along the CC, thus controlling for joint payoff efficiency. Each type of individual, i, was endowed with an integer allocation of ([X.sub.i], Q, [Y.sub.i,Q]). We set the total quantity of X and Y at 20 units each, thereby creating 400 potential combinations. Across all treatments, we set a nonsymmetric B at ([X.sub.Env,B], [Y.sub.Env,B]) = (18, 7) and ([X.sub.Dev,B], [Y.sub.Dev,B])= (2, 13), or for brevity, (18,7)/(2,13). This resulted in the portion of the CC within the bargaining lens being located between ([X.sub.Env], [Y.sub.Env]) = (12, 12) and ([X.sub.Env], [Y.sub.Env]) = (14, 14). Because risk preference is thought to influence bargaining outcomes (Murnighan, Roth, and Schoumaker 1988), subjects' risk attitudes were elicited prior to the bargaining instructions using the method of Holt and Laury (2002). (5)

After reading instructions and studying their own payoff tables (and those of their opponents) for as long as any individual wanted, subjects were then placed together in pairs, one environmentalist with one developer. They were then allowed a 3-minute period of unstructured communication in which they could discuss mutually acceptable integer allocations of X and Y. To be accepted as valid, agreements had to be technically feasible, or

(3) [X.sub.Env] + [X.sub.Dev] [less than or equal to] [X.sub.Env,B] + [X.sub.Env,B] = 20

(4) [Y.sub.Env] + [Y.sub.Dev] [less than or equal to] [Y.sub.Env,B] + [Y.sub.Env,B] = 20.

To register a negotiated outcome, one member of the bargaining pair had to describe the allocation on a form, and the other had to tick a box signifying agreement.

To control for the effects of accumulating income on risk preference, only one of the five rounds was implemented at the end of the experiment, chosen by the throw of a die. We prevented subjects from being able to make credible offers of cash side payments after the experiment by (1)ensuring that total earnings were constant along the CC and (2) using a different privately held random draw for each person when being paid to determine which round to count.

Our mixing protocol over the five rounds resulted in each member of one type being paired serially with all five members of the other type. Thus each subject played the same game a total of five times, each time with a different person of the other type. The experiment was conducted manually. Logistically, during the risk elicitation phase, the ten subjects per session were seated at widely spaced individual tables in two rows, with an empty row in between adjacent to the back row. During the bargaining phase, the front row of subjects (all of one type) was turned around and seated at empty tables across from their first set of opponents. There were thus two tables separating each member of the bargaining pair. In subsequent rounds, the two types alternated in having to switch one table to the right. Our design is unusual in that subjects were allowed full, unrestricted communication with their opponents during each 3-minute round. They were warned that threatening or abusive language would not be tolerated, and each pair's conversation was recorded with a microcassette player located midway between them to one side of the tables. Although the unstructured, face-to-face communication introduces "uncontrolled aspects of social interaction" (Roth 1995), it also parallels the in-person, unstructured negotiation used in most forms of bargaining.

B. Design Features of Each Treatment

We ran four treatments, varying the location of the initial allocation and the inequality of payoffs at the Nash bargain in a 2 x 2 design. Sessions were run so as to systematically alternate through the four treatments. Returning to our payoff functions (1) and (2), in all treatments we chose the a's, b's, and [alpha] in such a way as to keep constant the following:

1. the size of the Edgeworth box: [X.sub.Env] + [X.sub.Dev] = 20 and [Y.sub.Env] + [Y.sub.Dev] = 20

2. the size of the bargaining lens (55 cells)

3. the B allocation: ([X.sub.EnvB], [Y.sub.EnvB]) = (18, 7) and ([X.sub.DevB], [Y.sub.DevB]) = (2, 13)

4. the N allocation: ([X.sub.EnvN], [Y.sub.EnvN])= (13, 13) and ([X.sub.DevN], [Y.sub.DevN]) = (7, 7)

5. the sum of payoffs at B: [a.sub.Env] [18.sup.[alpha]][7.sup.1-[alpha]] + [b.sub.Env] + [a.sub.Dev.][2.sup.[alpha]] [13.sup.1-[alpha]] + [b.sub.Dev] = $28.77

6. the sum of all CC payoffs, including at N: [a.sub.Env][l3.sup.[alpha]] [13.sup.1-[alpha]] + [b.sub.Env] + [a.sub.Dev][7.sup.[alpha]][7.sup.1-[alpha]] + [b.sub.Dev] = $45.50.

In addition, we set the parameters to ensure that the total payoffs were substantially higher along the CC (including at N or E) than at Q or B.

To simplify the presentation of payoffs, subjects were provided two colored payoff tables showing the specific earnings they and their opponent would receive for all feasible combinations of X and y. (6) The parameters for all four treatments are reported in Table 1. In treatments where Q and B were identical, they were identified on a payoff table as a single yellow cell. In treatments where they differed, Q and B were identified by green and red cells, respectively.

Treatment I. Treatment I serves as our control treatment, with no divergence between Q and B((18,7)/(2,13)). The payoffs for the environmentalist and developer at B are approximately equal, at $14.67 and $14.10, respectively. In this treatment, N coincides with E at (13,13)/(7,7), with payoffs of $22.75 for each party. Treatment I is represented by the first panel of Figure 3. Here both the Nash bargain and egalitarian hypotheses predict that the parties will agree to N. The entitlement hypothesis predicts only that the parties will settle on the CC within the lens.

Treatment II. In Treatment II, Q is separated from B, but all other parameters are left unchanged from Treatment I. Q is shifted "south-west" from (18,7)/(2,13) to (16,4)/(4,16), yielding initial values for the environmentalist and developer of $0.00 and $27.30, respectively. (7) Thus, payoffs at the status quo are now very unequal in the developer's favor. Q also lies outside the bargaining lens created by B, so that an environmentalist is better off at every point within the bargaining lens associated with B than he or she is at Q, whereas the developer is worse off (except for allocations where the two lenses overlap). In Treatment II, the Nash bargain and egalitarian hypotheses still predict that the parties will agree to N = E. The entitlement hypothesis, however, predicts that agreements will move south-west along the CC to be within the "historical bargaining lens" formed by Q, to reflect the developer's initial advantage.

[FIGURE 3 OMITTED]

Treatments III and IV. Treatments III and IV replicate the Treatment I/II comparison, but now with N separated from E. The physical locations of Q, B, and N remain as in the earlier treatments, but the underlying payoff functions are changed to move the location of E to (10,10)/(10,10). At this allocation, earnings are equalized at $22.75 each, whereas at N the environmentalist and developer now earn $36.40 and $9.10, respectively. Unfortunately, the introduction of an unequal N (i.e., an N at which the parties' payoffs are unequal) also requires the introduction of unequal payoffs at B, to $28.32 and $0.45 for the environmentalist and developer, respectively. Faced with this confound, in Treatment IV where Q diverges from (the unequal) B, we chose to equalize payoffs at Q at $13.65 each. In this way, from Treatments I to II we test whether an unequal Q derails agreements to an equal N conditioned on an equal B; whereas in Treatments III to IV, we test whether an equal Q derails agreements to an unequal N from an unequal B. Treatment IV is represented by panel four of Figure 3.

The Nash bargaining hypothesis for both Treatments III and IV is that the parties will agree to N. The egalitarian hypothesis is that they will agree to E. The entitlement hypothesis is that the parties will agree to a Pareto efficient allocation within the bargaining lens defined by B(= Q) in Treatment III, but by Q in Treatment IV.

IV. RESULTS

Sixteen experiment sessions with ten subjects each were run at the University of Canterbury in April and May of 2008. Four sessions were run per treatment, resulting in 40 people per treatment providing 20 paired bargaining outcomes per round over five bargaining rounds. Each outcome consisted of a physical allocation of X and Y between the environmentalist and developer ([X.sub.Env], [Y.sub.Env])/([X.sub.Dev], [Y.sub.Dev]), and their resulting earnings. Each session took roughly 90 minutes, and subjects earned on average NZ $24.49 (1.00NZ$ = 0.78US$).

We divide our discussion of the results as follows. We begin by comparing agreement rates and proximity to Pareto efficiency across all treatments. We then characterize the location of agreements in each treatment, and finally test whether the Nash, egalitarian, or entitlement hypotheses can explain changes in agreements across treatments. For all statistical tests, we apply the Bonferroni correction to significance levels, requiring a 2.5% significance level for each of our four pair-wise treatment comparisons to achieve an experiment-wide significance level of 10%.

A. Agreement Rates and Proximity to the CC

As our experiment requires subjects to choose from an unusually large number of potential allocations, it provides a strong test of whether subjects can reach agreements in complex negotiations, and whether those agreements are Pareto efficient.

To provide intuition for our results, Figure 3 illustrates all bargaining pair outcomes pooled over the final four rounds for each treatment. Table 2 provides the corresponding descriptive statistics. As shown in Table 2, subjects initially found it harder to reach agreement in Treatment III (N [not equal to] E, Q = B), where both N and Q produced very unequal payoffs, than in the other three treatments. But in all treatments, agreement rates reached 90% or higher by Round 4. Using agreement rates averaged over all five rounds per session as a unit of observation, two-tail Mann-Whitney tests find agreement rates to be significantly lower in Treatment III than in either Treatment I (p = .02) or Treatment IV (p = .02). Looking round by round, however, two-tail Mann-Whitney tests using pair outcomes as the unit of observation found no significant differences between treatments in Rounds 2, 4, or 5. Thus, after only a few rounds of experience, bargainers were generally able to reach agreement in all treatments.

Were these agreements Pareto efficient? Table 2 reports the proportion of agreements that were precisely on the CC. We think, however, that a better indicator comes from measuring the physical or financial deviation of agreements from the CC. This is because allocations immediately adjacent to the CC offered additional options for distributing payoffs with little sacrifice in joint earnings. Beginning with physical deviations, we measure the Euclidean distance of agreements to the nearest Pareto efficient allocation. (8) To illustrate magnitudes, an agreement one or two diagonal units from the CC would be measured to be 1.41 or 2.83 units away from it, respectively, whereas B would be 7.78 units away. As reported in Table 3, we find that agreements were close to or on the CC in all treatments. Average distance ranged from 0.28 to 0.88 units across treatments in Round 1, and from 0 to 0.71 units by Round 5.

Similar support for Pareto efficiency comes from measuring the shortfall in joint earnings of pairs from what was available (NZ$45.50) on the CC. Again to illustrate magnitudes, an agreement one diagonal unit from the CC would reduce joint earnings by $0.46 to $0.51 depending on where it occurred; an agreement two units away would cost $1.84 to $2.03; and having B imposed would cost the pair $16.73. We find in Table 3 that the average joint earnings shortfall ranged from $0.07 to $1.28 in Round 1, narrowing to $0.00 to $0.29 by Round 5. Table 3 also reports p values from Mann-Whitney tests comparing the physical and financial distance of agreements to Pareto efficient outcomes between treatments. We interpret these to confirm that, with limited experience, support for Pareto efficiency is strong across all tour treatments. (9,10)

B. Which Pareto Efficient Allocation?

Table 4 reports three measures of the deviations of agreements from two key allocations: N ((13,13)/(7,7)), which equalizes payoffs in Treatments I and II, and the outcome (10,10)/ (10,10), which equalizes payoffs in Treatments III and IV. Our first two measures of deviation are the Euclidean distance between agreements and the two key allocations, respectively. As before, a one diagonal unit of deviation from a key allocation results in a distance of 1.41 units, and two results in 2.83 units.

Our third measure of deviation relates to the financial distance between agreements and the two key allocations. (11) We measure the relative deviation in the environmentalist's earnings share at actual agreements from what it would have been at the two allocations. This measure takes the absolute value of the difference between the environmentalist's share of earnings at the actual agreement and at (13,13)/(7,7), and subtracts from it the absolute value of the difference between the environmentalist's share at the agreement and at (10,10)/(10,10). This measure can range in value from -0.3, when a pair's division of earnings corresponds exactly to that at (10,10)/(10,10), to +0.3, when it corresponds exactly to that at N ((13,13)/(7,7)). A measure of 0 indicates that the pair's division of earnings was halfway between what it would have been at the two allocations. (12)

Treatment 1. As Table 4 illustrates, the agreements in Treatment I were at or near N even in the first round. By Round 2, mean physical distance from N was 0, and the environmentalists' mean share of earnings measured +0.3. As the Nash, egalitarian, and entitlement models all predict, or are consistent with, this outcome Treatment I provides a reassuring baseline from which to make cross-treatment comparisons.

Treatment II. In Treatment II, recall that Q was moved south-west from the roughly equal payoff allocation B ($14.67, $14.10), to a very unequal one ($0, $27.30). Our results indicate that subjects ignored this unequal Q and any "historical bargaining lens" it might have created. As is seen in Table 4, the agreements in Treatment II appear very similar to those in Treatment I in terms of distance from N and earnings share.

Treatment III. In Treatment III, the payoff functions were altered so that, although the physical locations of Q, B, and N were unchanged, payoffs at B and its corresponding N became unequal. Payoffs became ($28.32, $0.45) at B, ($36.40, $9.10) at N, and ($22.75, $22.75) at E, which we had moved south-west of the bargaining lens formed by B to (10,10)/(10,10). As seen in Figure 3, agreements in Treatment III appear more dispersed than in Treatments I and II. From Table 4 measures, they also appear on average a compromise between E and N, both in physical distance and earnings share. Agreements began relatively close to E in Rounds 1 and 2, and edged slightly closer to N by Rounds 4 and 5. In Round 5, the modal agreement was at (11,11)/(9,9), generating earnings of ($27.30, $18.20). Strikingly, this outcome was (just) outside the bargaining lens, making the environmentalist $1.02 worse off than by forgoing agreement. However most agreements were closer to N than this, and lay within the bargaining lens.

Treatment IV. In Treatment IV, Q diverged south-west from Treatment III's B ($28.32, $0.45), to equalize initial endowment values ($13.65, $13.65) at (16,4)/(4,16). This value of Q defined a "historical bargaining lens" that included E. As illustrated in Table 4, the agreements in Treatment IV again appear on average to be a compromise between E and N, but now much more heavily tilted toward E. Agreements on average were physically closer to E than to N, and the environmentalist's share of earnings was closer to E than to N in all five rounds. Indeed, the modal agreement for all five rounds was (10,10)/(10,10). At this outcome, the environmentalists agreed to leave the bargaining lens defined by B, and earn $5.57 less than they could have by forgoing agreement. This tendency remained as strong in Round 5 as in Round 1.

C. The Three-Way Horse Race

We can now compare the predictive powers of the Nash bargain, egalitarian, and entitlement hypotheses.

The Nash bargain predicts that the parties would select N in all four treatments. In Table 5, we report Mann-Whitney tests for two alternative measures of this prediction: that the geometric distance between agreements and N did not vary among treatments, and that the environmentalist's share of earnings at those agreements did not vary. We conduct tests based on individual pair agreements, round by round, and session average agreements, averaged over all rounds. We see in Table 5 that, consistent with Nash bargaining, support for N did not differ significantly between Treatments I and II (where N = E and Q diverged from an equal B), whether by Euclidean distance or deviation of earning shares. Support for the Nash fell significantly, however, when E was separated from N (I vs. III and II vs. IV) or when Q was separated from a B with unequal payoffs, and N [not equal to] E(III vs. IV). In all these latter comparisons, agreements moved south-west from N towards E.

In contrast, the egalitarian hypothesis successfully predicted most of the cross-treatment effects. Like Nash bargaining, it correctly predicted no significant difference in agreement locations between Treatments I and II, where E remained at N. This is true whether we consider Euclidean distance from the E = N allocation, or the parties' shares of joint earnings. Unlike Nash bargaining, however, the egalitarian hypothesis correctly predicted that agreements would move away from N towards (10,10)/(10,10) when the latter equalized payoffs, whether between Treatments I and III, or Treatments II and IV. These movements were significant whether measured in Euclidean distance or earnings share deviations, and whether using round-by-round pair outcomes or session-based averages. Incorporating aversion to inequality in the manner of Fehr and Schmidt (1999) would easily modify the bargaining lenses as perceived by subjects to explain the agreements reached. In particular, assigning a weight of 0.2 to "disadvantageous" and "advantageous" inequality aversion would create bargaining lenses containing 188 of 191 agreements in Treatments I plus II, and 172 of the 177 agreements in Treatments III plus IV. (13) The only effect not predicted by the egalitarian hypothesis was the significant additional movement of agreements towards E when an equal Q was separated from an unequal B (III to IV).

Finally, the entitlement hypothesis had mixed success. Positively, the small change between Treatments III and IV of shifting Q southwest of an unequal B produced a significant movement of agreements away from the Nash bargain. Negatively, this movement was only observed when the diverging Q was associated with a more equitable division of payoffs than was B--that is, between Treatments III and IV and not between I and II. If it were possible, a more complete test of the entitlement hypothesis would also examine the effect of separating an equal Q from an equal B, and separating an unequal Q from an unequal B.

The positive result is remarkable in that Q was strictly notional (subjects were merely told that this was their initial allocation). Standard bargaining theory would predict that Q would be irrelevant to the negotiation process. The negative result is also notable, in that it suggests that the impact of the initial allocation depends upon its perceived equity: it appears to become focal or an entitlement only when it is equitable. This asymmetry of results suggests that parties may attach greater weight to inequality aversion--in the sense of Fehr and Schmidt (1999)--when the status quo is equitable than when it is inequitable.

V. DISCUSSION AND CONCLUSIONS

In this article, we analyze the case in which opposing parties conduct multidimensional negotiations, subject to the constraint that if they fail to reach agreement, a third party will impose an externally chosen outcome. For example, a government may commit to impose a "backstop" set of regulations if stakeholder groups fail to reach consensus on a new public policy; or a labor arbitrator may announce the award that he or she will impose if a union and employer fail to reach agreement on a new contract. In these cases, it is of interest to determine whether the parties will be able to reach agreement, and what the nature of such an agreement will be.

To answer these questions, we conducted laboratory experiments that implemented a two-person, two-good game modeled on the Edgeworth box. Subjects were asked to negotiate an allocation from among approximately two hundred options, given that a backstop outcome would be imposed if they failed to reach agreement. We tested first whether they would be able to reach agreement at Pareto efficient allocations. Contingent on this, we tested whether parties would be drawn to the Nash bargain or equal payoff outcome when the two were separated, and whether agreements would be conditioned on the backstop or initial allocation when these were separated.

We found that subjects were quickly able to negotiate Pareto efficient (or nearly Pareto efficient) agreements, even when the payoffs imposed at the backstop allocation were very unequal ($28.32, $0.45) or differed substantially from the payoffs at the initial allocation. By the final round of negotiations, treatment agreement rates were never less than 95%, and never averaged less than 94% of the maximum total payoffs available.

Second, we found that our subjects were drawn toward outcomes that equalized payoffs. Under our implementation, this preference was sufficiently strong that when the outcome that equalized payoffs was Pareto inferior to the backstop, subjects negotiated outcomes that were also inferior. In Treatment III, where the initial allocation was unequal, the favored subjects gave up, on average, $1.02 relative to the backstop; and in Treatment IV, where the initial allocation was equal, they gave up $5.57. Third, we found that when separated from the backstop, the initial allocation affected the outcomes negotiated by our subjects if it was equal and the backstop was not, but had no effect if it was unequal and the backstop was (roughly) equal. This finding was all the more striking given the weakness of our implementation. Our subjects did not "earn" or "deserve" their initial allocations, but simply started with them. This suggests both that parties may be able to "negotiate around" a backstop policy that has been poorly chosen, and that a third party may have difficulty using its choice of backstop to induce the parties to accept an outcome that it feels is welfare improving.

Finally, there is an important caveat that needs to be recognized before practical lessons can be drawn from our experiment. Although our subjects had full information about one another's payoffs, negotiators in actual bargaining would have imperfect information at best. Thus our subjects may have found it easier to identify outcomes that equalized overall gains than would real world negotiators. We hope, in future experiments, to avoid this effect by implementing unstructured bargaining with private, unverifiable payoff information.

doi: 10.1111/j.1465-7295.2011.00391.x

ABBREVIATION

CC: Contract Curve

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CHRISTOPHER BRUCE and JEREMY CLARK *

* Funding for this research was provided by the Donner Canadian Foundation and the College of Business and Economics of the University of Canterbury. We would like to thank two anonymous referees for very helpful comments, as well as Michael McKee, C. Brain Cadsby. Kyle Hyndman, Andrea Menclova, and Charles Noussair.

Bruce: Department of Economics, University of Calgary, Calgary, AB T2N 1N4, Canada. Phone 403-220-4093, Fax 403-282-5262, E-mail cjbruce@ucalgary.ca

Clark: Department of Economics and Finance, University of Canterbury, Christchurch 8140, New Zealand. Phone 011-643-364-2308, Fax 011-643-364-2635, E-mail jeremy.clark@canterbury.ac.nz

(1.) This approach, known as collaborative policy making, negotiated rulemaking, deliberative democracy, or consensus-building, is discussed extensively in the environmental policy literature. See, especially: Amy (1985), Coglianese (1997), Harter (1982), Pritzker and Dalton (1995), and Wondolleck and Yaffee (2000).

(2.) This is the effect of the Negotiated Rulemaking Act in the United States (Pritzker and Dalton 1995).

(3.) Other axiomatic models have also been proposed in Raiffa (1953), Kalai and Smorodinsky (1975), and Gachter and Riedl (2005). We restrict our discussion to the Nash bargain, which has been the focus of most of the experimental bargaining literature.

(4.) Some authors have questioned whether, in multidimensional bargaining, negotiators will be able to reach agreement or, if so, reach an efficient outcome. See, for example, Pratt and Zeckhauser (1992) and Binmore et al. (1998).

(5.) The pair average risk aversion as measured by this instrument was not significant in random effects panel regressions predicting whether agreements were (a) in the bargaining lens, (b)Pareto efficient, or (c)at the Nash bargain. Neither were most pair demographic characteristics we elicited. For brevity, we exclude these results in what follows.

(6.) Allocations that yield negative earnings for either party were excluded from consideration, yielding 199 possible allocations in Treatments I and II, and 215 allocations in Treatments III and IV. Calculators were provided for each person.

(7.) If this allocation had been the backstop, the Nash bargain would have occurred at (10,10)/(10,10), with payoffs of $9.10 and $36.40, respectively.

(8.) If the closest allocation on the CC to an agreement is ([X.sub.Env.cc], [Y.sub.Env,cc]), then the Euclidean distance between them is [([([X.sub.Env] - [X.sub.Env.cc]).sup.2] + ([Y.sub.Env] - [Y.sub.Env,cc]).sup.2]).sup.1/2]. If an agreement was equidistant to two cells on the CC, distance was measured to the averaged coordinates.

(9.) Although some round-by-round tests based on pair outcomes find significant treatment differences in physical or financial distance to the CC, the magnitude of the differences is trivial. Significance arises because of perfect adherence to the CC in some rounds of Treatments I and II.

(10). Our finding that parties were able to reach efficient agreements when the payoffs at the backstop were unequal (Treatments III and IV), appears inconsistent with the findings of Binmore, Shaked, and Sutton (1989) and Binmore et al. (1991, 1998).

(11.) We cannot compare joint earnings at the two key allocations with those at actual agreements because joint earnings are identical at the former two.

(12.) This measure does not capture the absolute distance of agreements to either key allocation, but only the relative success of either allocation in predicting earnings shares. Agreements north-east or south-west of the key allocations would yield values capped at -0.3 or +0.3, but this occurred in only 5 of 368 agreements.

(13.) The Fehr and Schmidt (1999) model for two agents is [U.sub.Env] = [Payoff.sub.Env] - [[alpha].sup.*] max{[Payoff.sub.Dev] - [Payoff.sub.Env], 0} -[[[beta].sup.*] max{[Payoff.sub.Env] - [Payoff.sub.Dev], 0}. A weighting of [alpha] = [beta] = .2 narrows the bargaining lens in Treatments I and II, and "pulls" it south-west in Treatments III and IV.

TABLE 1
Parameters Used Across Treatments

 Environmentalist Developer

Treatment I: Status quo = backstop allocation, Nash bargain equalizes
payoffs
 Payoff function [U.sub.Env](X, Y) = [U.sub.Dev](X, Y) =
 4.55[X.sup.1/2] 4.55[X.sup.1/2]
 [Y.sup.1/2]-36.40 [Y.sup.1/2]-9.10
 At Q & B Gets $14.67 from (18,7) Gets $14.10 from (2,13)
 At N & E Gets $22.75 from (13,13) Gets $22.75 from (7,7)

Treatment II: Status quo [not equal to] backstop allocation, Nash
bargain equalizes payoffs
 Payoff function See Treatment I See Treatment I
 At Q Gets $0.00 from (16,4) Gets $27.30 from (4,16)
 At B Gets $14.67 from (18,7) Gets $14.10 from (2,13)
 At N & E Gets $22.75 from (13,13) Gets $22.75 from (7,7)

Treatment III: Statu= backstop, Nash bargain does not equalize payoffs
 Payoff [U.sub.Env](X, Y) = [U.sub.Dev](X, Y) =
 4.55[X.sup.1/2] 4.55[X.sup.1/2]
 [Y.sup.1/2]-22.75 [Y.sup.1/2]-22.75
 At Q & B Gets $28.32 from (18,7) Gets $0.45 from (2,13)
 At N Gets $36.40 from (13,13) Gets $9.10 from (7,7)
 At E Gets $22.75 from (10,10) Gets $22.75 from (10,10)

Treatment IV: Status quo [not equal to] backstop, Nash bargain does not
equalize payoffs
 Payoff function See Treatment III See Treatment III
 At Q Gets $13.65 from (16,4) Gets $13.65 from (4,16)
 At B Gets $28.32 from (18,7) Gets $0.45 from (2,13)
 At N Gets $36.40 from (13,13) Gets $9.10 from (7,7)
 At E Gets $22.75 from (10,10) Gets $22.75 from (10,10)

TABLE 2
Descriptive Statistics of Physical Bargaining Outcomes

 Round

 Pair N 1 2 3
Agreement rates
T I: Q = B, E = N 20 (a) 1.00 .95 1.00
T II: Q [not equal to] B, E = N 20 .85 .95 .95
T III: Q = B, E [not equal to] N 20 .50 .85 .80
T IV: Q [not equal to] B, E [not equal
 to] N 20 .80 .95 1.00

Proportion in bargaining lens
T I: Q = B, E = N 20 (a) 1.00 1.00 1.00
T II: Q [not equal to] B, E = N 20 1.00 1.00 1.00
T III: Q = B, E [not equal to] N 20 .70 .70 .70
T IV: Q [not equal to] B, E [not equal
 to] N 20 .35 .30 .30

Contingent on reaching agreement
Proportion exactly on the contract curve
T I: Q = B, E = N .65 1.00 1.00
T ii: Q [not equal to] B, E = N .65 .89 .84
T III: Q = B, E [not equal to] N .70 .35 .25
T IV: Q [not equal to] B, E [not equal
 to] N .63 .68 .60

Proportion exactly at the Nash bargain (13,13)/(7,7)
T I: Q = B, E = N .65 1.00 1.00
T II: Q [not equal to] B, E = N .59 .84 .84
T III: Q = B, E [not equal to] N .10 .06 .00
T IV: Q [not equal to] B, E [not equal
 to] N .00 .05 .05

Proportion exactly at (10,10)/(10,10) (equalizes earnings in III, IV)
T I: Q = B, E = N .00 .00 .00
T II: Q [not equal to] B, E = N .00 .00 .00
TIII: Q = B, E [not equal to] N .40 .12 .06
T IV: Q [not equal to] B, E [not equal
 to] N .63 .53 .50

 Round

 4 5 Ave.
Agreement rates
T I: Q = B, E = N 1.00 1.00 .99
T II: Q [not equal to] B, E = N .90 1.00 .93
T III: Q = B, E [not equal to] N 1.00 .95 .82
T IV: Q [not equal to] B, E [not equal
 to] N 1.00 1.00 .95

Proportion in bargaining lens
T I: Q = B, E = N 1.00 1.00 1.00
T II: Q [not equal to] B, E = N 1.00 1.00 1.00
T III: Q = B, E [not equal to] N .75 .70 .71
T IV: Q [not equal to] B, E [not equal
 to] N .30 .20 .29

Contingent on reaching agreement
Proportion exactly on the contract curve
T I: Q = B, E = N .90 .95 .90 (b)
T ii: Q [not equal to] B, E = N 1.00 1.00 .88
T III: Q = B, E [not equal to] N .35 .32 .37
T IV: Q [not equal to] B, E [not equal
 to] N .60 .75 .65

Proportion exactly at the Nash bargain (13,13)/(7,7)
T I: Q = B, E = N .90 .95 .90 (b)
T II: Q [not equal to] B, E = N 1.00 1.00 .86
T III: Q = B, E [not equal to] N .05 .00 .04
T IV: Q [not equal to] B, E [not equal
 to] N .00 .05 .03

Proportion exactly at (10,10)/(10,10) (equalizes earnings in III, IV)
T I: Q = B, E = N .00 .00 .01
T II: Q [not equal to] B, E = N .00 .00 .00
TIII: Q = B, E [not equal to] N .05 .00 .10
T IV: Q [not equal to] B, E [not equal
 to] N .55 .60 .56

(a) N = 19 pairs for Round 5 of Treatment I, where a technically
inefficient agreement is omitted.

(b) Average across rounds weighted by the number of agreements per
round.

TABLE 3
Geometric Distance and Loss in Earnings Between Agreements and the
Nearest Point on the CC

 Round

Treatment 1 2

I (Q = B; Mean distance to CC .813 0
N = E) (1.262) (a) (0)
 Mean loss (NZ$) in joint .55 0
 earnings (.96) (0)

II (Q [not Mean distance to CC .666 .558
equal to] B; (1.211) (1.865)
N = E) Mean loss (NZ$) in joint .45 .99
 earnings (1.13) (3.84)

III (Q = B; N Mean distance to CC .283 .749
[not equal to] (.494) (.769)
E) Mean loss (NZ$) in joint .07 .26
 earnings (.15) (.39)

IV (Q [not Mean distance to CC .884 .484
equal to] B; N (2.153) (1.029)
[not equal to] Mean loss (NZ$) in joint 1.28 .30
E) earnings (4.53) (1.00)

 Round

Treatment 3 4

I (Q = B; Mean distance to CC 0 .318
N = E) (0) (.986)
 Mean loss (NZ$) in joint 0 .26
 earnings (0) (.81)

II (Q [not Mean distance to CC .595 0
equal to] B; (1.532) (0)
N = E) Mean loss (NZ$) in joint .67 0
 earnings (1.95) (0)

III (Q = B; N Mean distance to CC .707 .601
[not equal to] (.577) (.770)
E) Mean loss (NZ$) in joint .19 0.24
 earnings (.27) (.73)

IV (Q [not Mean distance to CC .354 .318
equal to] B; N (.487) (.428)
[not equal to] Mean loss (NZ$) in joint .08 .07
E) earnings (.14) (.12)

 Round

Treatment 5

I (Q = B; Mean distance to CC .112
N = E) (.487)
 Mean loss (NZ$) in joint .06
 earnings (.25)

II (Q [not Mean distance to CC 0
equal to] B; (0)
N = E) Mean loss (NZ$) in joint 0
 earnings (0)

III (Q = B; N Mean distance to CC .707
[not equal to] (.816)
E) Mean loss (NZ$) in joint 0.29
 earnings (.75)

IV (Q [not Mean distance to CC .177
equal to] B; N (.314)
[not equal to] Mean loss (NZ$) in joint .03
E) earnings (.05)

Mann-Whitney Test p Values

 Round by Round Overall
 (Obs. = Pair Agreement) (Obs. =
 Session
 1 2 3 4 5 Average)

Mean distance to CC
 I = II? .858 .152 .068 .174 .305 .772
 III = IV? .661 .082 .048 .143 .004# .149
 I = III? .495 .000# .000# .002# .000# .042
 II = IV? .983 .176 .228 .003# .018# .773
Mean loss in joint earnings
 I = II? .844 .152 .068 .174 .305 1.000
 III = IV? .618 .056 .051 .119 .001# .773
 I = III? .112 .001# .000# .319 .011# .772
 II = IV? .725 .880 .518 .004# .021# .773

Notes: Number of pairs reaching agreement in each round is provided in
Table 2. Figures in bold significant at the 2.5%
level.

(a) Standard deviations in parentheses.

Note: Figures in bold significant at the 2.5% level are
indicated with #.

TABLE 4
Mean Distance and Relative Deviation in Environmentalist's Share of
Earnings Between Agreements and Two Key Allocations

 Round

Treatment 1 2 3

I: Distance to Nash (13,13)/(7,7) when N = .84 0 0
E: Q = B (1.28) (0) (0)
 Distance to (10,10)/(10,10) when it [not 4.38 4.24 4.24
 equal to] E: Q = B (.39) (0) (0)
 Relative deviation in Env.'s share of .29 .30 .30
 earnings between (13,13)/(7,7) and (.04) (0) (0)
 (10,10)/(10,10) (a)

II: Distance to Nash (13,13)/(7,7) when N = .87 .63 .66
E: Q [not equal to] B (1.30) (1.88) (1.63)
 Distance to (10, 10)/(10,10) when it [not 4.13 4.44 4.58
 equal to] E: Q [not equal to] B (.70) (1.07) (.81)
 Relative deviation in Env.'s share of .25 .29 .29
 earnings between (13,13)/(7,7) and (.09) (.05) (.05)
 (10,10)/(10,10) (a)

III: Distance to Nash (13,13)/(7,7) when N 3.02 2.43 2.42
[not equal to] E: Q = B (1.53) (1.05) (1.10)
 Distance to (10,10)/(10,10) when it = E: 1.44 2.30 2.35
 Q = B (1.47) (1.17) (1.26)
 Relative deviation in Env.'s share of -.12 .00 .00
 earnings between (13,13)/(7,7) and (.22) (.16) (.18)
 (10,10)/(10,10) (a)

IV: Distance to Nash (13,13)/(7,7) when N 4.13 3.59 3.13
[not equal to] E: Q [not equal to] B (1.77) (1.29) (1.41)
 Distance to (10,10)/(10,10) when it = E: 1.27 1.20 1.30
 Q [not equal to] B (2.28) (1.55) (1.53)
 Relative deviation in Env.'s share of -.23 -.17 -.13
 earnings between (13,13)/(7,7) and (.15) (.19) (.21)
 (10,10)/(10,10) (a)

 Round

Treatment 4 5

I: Distance to Nash (13,13)/(7,7) when N = .32 .12
E: Q = B (1.00) (0.51)
 Distance to (10,10)/(10,10) when it [not 4.32 4.24
 equal to] E: Q = B (.25) (.03)
 Relative deviation in Env.'s share of .30 .30
 earnings between (13,13)/(7,7) and (.02) (.02)
 (10,10)/(10,10) (a)

II: Distance to Nash (13,13)/(7,7) when N = 0 0
E: Q [not equal to] B (0) (0)
 Distance to (10, 10)/(10,10) when it [not 4.24 4.24
 equal to] E: Q [not equal to] B (0) (0)
 Relative deviation in Env.'s share of .30 .30
 earnings between (13,13)/(7,7) and (0) (0)
 (10,10)/(10,10) (a)

III: Distance to Nash (13,13)/(7,7) when N 2.06 2.08
[not equal to] E: Q = B (1.08) (0.91)
 Distance to (10,10)/(10,10) when it = E: 2.63 2.72
 Q = B (1.34) (1.26)
 Relative deviation in Env.'s share of .05 .06
 earnings between (13,13)/(7,7) and (.16) (.15)
 (10,10)/(10,10) (a)

IV: Distance to Nash (13,13)/(7,7) when N 3.34 3.52
[not equal to] E: Q [not equal to] B (1.21) (1.14)
 Distance to (10,10)/(10,10) when it = E: 1.09 .79
 Q [not equal to] B (1.47) (1.18)
 Relative deviation in Env.'s share of -.16 -.20
 earnings between (13,13)/(7,7) and (.20) (.16)
 (10,10)/(10,10) (a)

(a) Ranges from -0.3, indicating the environmentalist's share of
earnings corresponds to that at the allocation (10,10) (10,10), to
+0.3, corresponding to his share at (13,13) (7,7).

TABLE 5
Treatment Comparisons of Distance and Relative Deviation in
Environmentalist's Share of Joint Earnings Between Two Key Allocations

 Round by Round
 (Obs. = Pair Agreement)

 1 2 3
Mann-Whitney two-tail test p values
Mean distance to the Nash (13,13)/(7,7)
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .861 .075 .068
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .206 .004# .074
 I = III? (N = E [right arrow] N [not
 equal to] E; Q = B) .001# .000# .000#
 II = IV? (N = E [right arrow] N [not .000# .000# .000#
 equal to] E; Q [not equal to] B)
Mean distance to the (10,10)/(10,10) allocation
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .563 .553 .068
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .405 .010# .032
 I = III? (N = E [right arrow] N [not
 equal to] E; Q = B) .000# .000# .000#
 II = IV? (N = E [right arrow] N [not equal
 to] E; Q [not equal to] B) .000# .000# .000#
Mean relative deviation in Env.'s share of joint earnings between
(13,13)/(7,7) and (10,10)/(10,10)
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .373 .317 .305
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .171 .005# .054
 I = III? (N = E [right arrow] N [not equal
 to] E; Q = B) .000# .000# .000#
 II = IV? (N = E [right arrow] N [not
 equal to] E; Q [not equal to] B) .000# .000# .000#

 Round by Round
 (Obs. = Pair
 Agreement) Overall
 (Obs. =
 4 5 Session)
Mann-Whitney two-tail test p values
Mean distance to the Nash (13,13)/(7,7)
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .174 .305 .245
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .002# .000# .021
 I = III? (N = E [right arrow] N [not
 equal to] E; Q = B) .000# .000# .020
 II = IV? (N = E [right arrow] N [not .000# .000# .021
 equal to] E; Q [not equal to] B)
Mean distance to the (10,10)/(10,10) allocation
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .174 .305 .772
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .002# .000# .021
 I = III? (N = E [right arrow] N [not
 equal to] E; Q = B) .000# .000# .020
 II = IV? (N = E [right arrow] N [not equal
 to] E; Q [not equal to] B) .000# .000# .021
Mean relative deviation in Env.'s share of joint earnings between
(13,13)/(7,7) and (10,10)/(10,10)
 I = II? (Q = B [right arrow] Q [not equal
 to] B; N = E) .343 .305 .042
 III = IV? (Q = B [right arrow] Q [not
 equal to] B; N [not equal to] E) .002# .000# .021
 I = III? (N = E [right arrow] N [not equal
 to] E; Q = B) .000# .000# .020
 II = IV? (N = E [right arrow] N [not
 equal to] E; Q [not equal to] B) .000# .000# .021

Note: Figures in bold significant at the 2.5% level.

Note: Figures in bold significant at the 2.5% level are
indicated with #.