Can prior offers and arbitration outcomes be used to predict the winners of subsequent final-offer arbitration cases?

1. Introduction

Interest arbitration is a dispute resolution procedure in which an independent arbitrator determines the terms of a new contract. The intent of interest arbitration is not to have arbitrators impose settlements on the bargainers but rather to encourage negotiation by the parties. Two main variations of interest arbitration exist. In conventional arbitration (CA), each party submits an offer to the arbitrator, who may select either offer as the settlement or choose one of his or her own. Final-offer arbitration (FOA) requires the arbitrator to select one of the two submitted offers as the award. By forcing the arbitrator to choose one offer over the other, the parties are thought to have an incentive to submit reasonable offers.

In the standard FOA model, the arbitrator formulates a notion of the ideal settlement and then chooses the closest offer as the award. Farber (1980) and Miller (2000) suggest that FOA bargainers strategically align their final offers around the median of the arbitrator's preferred award distribution. The closer the offer is to the median preferred award, the more concessionary it is. Consequently, the submitted offers balance the expected benefits/costs from a more concessionary final offer. Once the offers have been exchanged, the parties may continue to negotiate their own agreement with the knowledge that one of the two offers will be selected by the arbitrator should the bargainers reach an impasse. Presumably, most bargainers would prefer negotiating their own mutually acceptable agreement to the risk of having the opponent's offer imposed on them by the arbitrator.

Given the intent of FOA to encourage negotiated settlements (or at least quality settlements), several factors hinder FOA in practice. First, as the arbitration period begins, each bargainer must file an offer based on some knowledge of the arbitrator's preferred award distribution. Once the final offers are exchanged, the bargainers must also assess the likelihood of winning the case should they remain at impasse. This may be problematic given that between the exchange of offers and the beginning of hearings in the present period, evidence of arbitrator preferences exists only from prior arbitration periods.

Second, prior period FOA outcomes do not directly reveal the preferred award but merely the offer that lies nearest the preferred award. Moreover, the nature of the FOA game suggests that submitted final offers should be fairly "reasonable." Combined, these factors are likely used to draw inferences from historical cases.

Finally, during most (if not all) of this time frame, the bargainers do not know which arbitrator will hear the case should it reach the arbitration stage. Consequently, bargainers must assess the probability of winning the case based on pooled information from prior period arbitration outcomes. (1)

We test the hypothesis that information from past FOA cases can be used to infer the likely outcomes of future cases. In essence, we seek to test the basis of FOA as a viable dispute resolution mechanism--that bargainers can file and exchange offers in an arbitration period and assess, either perfectly or imperfectly, the probability of winning the case before any hearings begin in that period and prior to any knowledge as to which arbitrator will hear the case.

We use data from major league baseball's FOA to test our hypothesis because it has advantages over other FOA settings. First, the factors that may be entered as evidence are clearly specified. Second, abundant baseball performance statistics are convenient proxies for the relevant facts of the cases. Third, baseball represents a "pure" use of FOA as a dispute resolution mechanism. Baseball arbitrators are not permitted to explain their choice of awards; they merely write the award into the Uniform Player's Contract. Consequently, the actual preferred award is never revealed and must be inferred indirectly.

Our analysis provides evidence that, subsequent to the exchange of offers, the outcomes of past FOA cases may be used to infer the likelihood of winning. However, although the outcomes of future cases are correlated with those of past cases, historical outcomes do not perfectly convey the precise position of a pair of offers on the preferred award distribution. This is consistent with the findings of Olson, Dell'omo, and Jarley (1992), who suggest that a fair amount of uncertainty in determining arbitration outcomes exists. This addresses points raised by Farber (1980) and Burgess and Marburger (1993) that FOA drives quality-negotiated settlements while still allowing bargainers the freedom to negotiate their own agreement.

We also find that when FOA arbitrators are confronted with a pair of offers in which one offer is extreme relative to the opponent's offer, they tend to rule in favor of the more "'reasonable" one. Moreover, when the final offers are more or less "equally reasonable," the likelihood of winning one's case takes on the appearance of randomness. Consequently, baseball FOA not only encourages bargainers to submit reasonable offers, it creates an incentive to negotiate a settlement to avoid the risk of an unfavorable arbitrated settlement.

2. Prior FOA Literature

Ashenfelter and Bloom (1984) examined FOA decisions among New Jersey policemen. The governing statute allowed the bargainers to use conventional arbitration to settle their disputes and compelled them to use FOA if they did not agree to use CA. This created a pool of both CA and FOA awards. Assuming that the arbitrator's conventional award was the same as his or her preferred award, Ashenfelter and Bloom (1984) found the same standards used to determine CA awards could be applied to determine FOA winners. In contrast to the arbitration environment used in our study, Ashenfelter and Bloom (1984) had a directly observable proxy for the preferred award, and they used this proxy to determine FOA winners.

Owing to the potential for omitted variables to mask arbitrator decision-making in actual cases, Farber and Bazerman (1986) distributed a set of hypothetical cases to practicing arbitrators and analyzed the decisions arbitrators made. They concluded that arbitrators paid attention to the facts in selecting awards but that there was much variation across arbitrators.

Because the rehire incentive, which is generally assumed to be the objective of arbitrators, is not present in experimental studies, Olson, Dell'omo, and Jarley (1992) compared arbitration decisions in experimental and field settings. (2) Relying on FOA decisions for Wisconsin school teachers, they found that decision rules for cases were essentially the same for both experimental and field cases when the wage was the only disputed issue. Regarding the ability to explain arbitration decisions from pooled data, Olson, Dell'omo, and Jarley (1992) found a substantial amount of uncertainty in determining the preferred award even after controlling for the facts and their respective coefficients. The model improved only slightly when controlling for the identity of the arbitrator. Consequently, although the evidence supported the contention that FOA decisions could be explained through pooled data, the study did not examine the possibility that pooled awards from prior periods could be used to predict awards in the current period. (3)

Efforts to explain FOA decisions in major league baseball have never been particularly fruitful. The inability to predict baseball FOA winners led Scully (1978, 1989) and Erekson, Moser, and Schwartz (1989) to conclude that arbitrators were strategically distributing victories randomly across the two sides to avoid veto. A more likely explanation is that the arbitration process forces the bargainers to submit reasonable offers, thereby creating the illusion that the decisions are unrelated to the quality of the offers (Farber 1980, 1981; Bloom 1986).

3. Theoretical Framework

Crawford (1979) showed that if the arbitrator's preferred award was known with certainty, all settlements would be equal to the preferred award. Moreover, to avoid the costs of arbitration, all settlements would be negotiated. Only in the presence of uncertainty do the parties retain room to negotiate their own settlement.

Farber (1980) and Miller (2000) modeled FOA under the assumption that the arbitrator's preferred award was not known with certainty. Given knowledge of the median and standard deviation of the preferred award distribution, the bargainers strategically align their final offers around the median preferred award. (4) Because the FOA arbitrator must choose one of the two final offers as the award, the arbitrators select the offer that lies nearest the preferred settlement. The probability that management's offer is chosen equals

(1) Pr([W.sub.f] = [W.sub.m]) = Pr[([W.sub.u] + [W.sub.m])/2 > [W.sub.p]],

where [W.sub.f] is the value of the final offer chosen by the arbitrator, [W.sub.m] is management's final offer, [W.sub.u] is the union's final offer, and [W.sub.p] is the arbitrator's preferred award. The parties do not observe [W.sub.p] with certainty. Further, arbitrators have an incentive to conceal their precise preferences because the probability of being selected as an arbitrator decreases as uncertainty about his or her preferred award diminishes. Although [W.sub.p] is not observed with certainty, it is generally assumed to be a function of the facts of the case that are observed by both parties and the arbitrator. Because the facts the arbitrator is allowed to consider are explicitly stated in the arbitration guidelines, the preferred award of the arbitrator can be written as

(2) [W.sub.p] = [beta]X + [epsilon],

where X is the vector of facts observed by both parties, [beta] is the weight assigned to the different facts that determines [W.sub.p], and [epsilon] is the component affecting arbitrator beliefs that is unobserved by the parties.

Substituting Equation 2 into Equation 1 yields

(3) Pr([W.sub.f] = [W.sub.m]) = Pr[([W.sub.u] + [W.sub.m])/2 - X[beta] > [epsilon]].

Previous empirical research of arbitrator decision making has assumed e is normally distributed. Therefore, the probability that management's offer is chosen is

(4) Pr([W.sub.f] = [W.sub.m]) = [PHI][([W.sub.u] + [W.sub.m])/2[sigma] - X[beta]/[sigma])],

where [PHI](*) is the cumulative density function (CDF) for the standard normal distribution. The parameters of Equation 4 can be estimated from a probit model predicting the winning offer as a function of the facts of the case and the mean of the final offers. However, unlike the typical probit model, the standard deviation off([epsilon]) is identified because of variation in the mean final offers across the sample of cases. Because [sigma] is equal to one over the probit coefficient on the mean final offers of the parties, both [sigma] and [beta] are identified and the parameters that determine the preferred awards can be estimated. The parameters obtained from estimating Equation 4 on a sample of FOA cases could then be used to predict the probability that management's offer is selected in an out-of-sample observation using the final offers in the case and the predicted preferred award of the arbitrator:

(5) Pr([W.sub.f] = [W.sub.m]) = [PHI][([W.sub.u] + [W.sub.m])/2[sigma]' ([W'.sub.p]/[sigma]')],

where [sigma]' and [W'.sub.p] are the estimated values of [sigma] and [W.sub.p].

If a large number of out-of-sample observations are available, any link between prior FOA decisions and the outcomes of future cases can be assessed by modeling the actual choices in the out-of-sample observations as a function of the parameter estimates for Equation 5. Specifically, if Z = ([W.sub.u] + [W.sub.m])/2[sigma]' - ([W'.sub.p]/[sigma]'), then in the out-of-sample observations one could estimate

(6) Pr([W.sub.f] = [W.sub.m]) = [PHI][[alpha].sub.1] + [[alpha].sub.1](Z)].

If the outcomes of previous cases provide no insight into future decisions, [[alpha].sub.1] will be equal to zero and [PHI]([[alpha].sub.0]) will equal the CDF for the vector of out-of-sample cases. If, on the other hand, past cases perfectly convey the position of a pair of offers on the preferred award distribution for future cases, [[alpha].sub.1] will be equal to one and [[alpha].sub.0] will be equal to zero.

4. Final-Offer Arbitration in Major League Baseball

We take advantage of data from major league baseball to examine FOA arbitration decisions. Baseball provides an ideal laboratory for empirical tests of the FOA process because the facts of the cases are easily quantifiable. The nature of baseball's FOA process changed markedly from its inception in 1974 until 1981.5 Due to the likelihood of significant arbitrator learning over this early period (Gibbons 1988), we do not analyze any cases prior to 1981. Then, we assume an essentially stable arbitration game for the period 1981-2001.

Baseball arbitration procedures are detailed in Faurot and McAllister (1992). Eligibility for FOA depends on completed years of major league service. (6) The following criteria have been established as admissible evidence in hearings throughout the history of baseball FOA:

(i) the player's contribution during the past season, including overall performance and special qualities of leadership and public appeal;

(ii) length and consistency of career contribution;

(iii) the player's past compensation;

(iv) comparative baseball salaries;

(v) recent club performance; and

(vi) any physical or mental defects in the player.

Management and the players' union each have veto power over the set of eligible arbitrators for cases each year, but the identity of the arbitrator for any particular case is not known to the parties when their final offers are filed.

5. Empirical Results

A total of 345 FOA cases were heard between 1981 and 2001. Of the total, 181 hitters' cases were heard, with management winning 102 (57%) of the cases, as shown in Table 1. The remaining 164 cases were for pitchers. Ninety-one of these cases (55%) were won by management.

We confined our study to hitters' cases. In the hitters' cases, the proxies we used are generally considered to be unambiguously positively correlated with performance. For example, although shortstops typically hit fewer home runs than first basemen, a shortstop who hits 20 home runs is considered to have had a better season than one who hits 5 home runs. (7)

Our sample includes all hitters' FOA cases heard from 1981 through 2001. The purpose of our analysis is to determine if the outcomes of FOA cases are correlated with the decisions in past FOA cases. We proceed by estimating the parameters of Equation 4 over a discrete time frame. Based on the assumption that the parameter estimates can be used to predict the probability that management will win an out-of-sample case, we assess the predictive ability of Equation 5 by accumulating a large number of out-of-sample observations to estimate Equation 6.

Equation 4 is modeled such that the arbitrator's choice of the FOA settlement depends on the player's past performance, his past compensation, and the average level of player salaries. (8) The independent variables used in the analysis are detailed in Table 2. Given the extremely long period (21 years) over which we estimate our models, we used the logarithms of salaries rather than absolute levels in our specifications. We begin by estimating Equation 4 for the 49 hitters' cases heard from 1981 to 1985. The parameters from the equation are then applied to the out-of-sample cases heard in the 1986 season. Next, Equation 4 is estimated for all hitters' cases heard from 1981 to 1986, and the parameters of the equation are applied to the out-of-sample cases heard in the 1987 season. We continue in this fashion through 2001 until we have obtained estimates for our entire sample. (9)

Before proceeding to the next step, two points should be noted. First, by estimating Equation 4 over a discrete time frame and then applying the parameter estimates to the cases heard in the subsequent (out-of-sample) period, we are, in effect, replicating the dilemma confronting bargainers subsequent to the exchange of offers in any given arbitration period. Specifically, until the first decision for that arbitration period is rendered, arbitration outcomes must be inferred from decisions in prior periods. Second, because the identity of the arbitrator is not known to the bargainers at the time the offers are formulated and exchanged (and probably throughout most or all of the negotiation period), Equation 4 is estimated without the use of arbitrator identity dummies.

By combining the estimates of [W'.sub.p] with the actual outcomes, we can estimate the parameters of Equation 6. (10) If the preferred award distribution in the present (out-of-sample) period could be perfectly inferred from the distribution of past (in-sample) FOA decisions, [[alpha].sub.0] would be equal to zero and 0q would be equal to one. On the other hand, if the outcomes of out-of-sample cases are positively correlated with those of past cases, [[alpha].sub.0] would be equal to zero and [[alpha].sub.1] would be less than one.

The results of the estimation are reported in Table 3. A Wald test was used to test the joint hypothesis that [[alpha].sub.0] and the time series dummies are equal to zero and [[alpha].sub.1] = 1. The resulting chi-squared statistic, 51.24, caused the null hypothesis to be rejected at the 1% level. However, as the table also indicates, whereas [[alpha].sub.0] does not significantly differ from zero, 0 [less than or equal to] [[alpha].sub.1] [less than or equal to] 1. This implies that the arbitration decision rules inferred from past decisions are positively, albeit imperfectly correlated, with out-of-sample decisions. Overall, the model successfully predicts 78 of the 132 cases, or 59%.

The imperfect correlation between past and future decisions is consistent with evidence produced in other FOA literature. In contrast to our study, past research has attempted to model, and hence predict, "within-sample" outcomes. Olson, Dell'omo, and Jarley (1992) found substantial uncertainty surrounding a fair award even when the facts of the cases and the weight given to the facts was known. When arbitrator identities were included in the estimation, the fit of the model improved only slightly. Although they could not reject the hypothesis that the appropriate settlement is the same across arbitrators, they did find that some of the differences between arbitrators were, in fact, significant. Similarly, Olson and Jarley (1991) remarked about the relatively high level of uncertainty about arbitrator beliefs in their empirical results, which were not reduced substantially when arbitrator identities were included. Clearly, if within-sample studies reveal a high level of uncertainty, one would expect out-of-sample models to reveal even more uncertainty. (11)

Stevens (1966) theorized FOA as a means of discouraging the submission of extreme offers. Because the "fairest" offer is likely to be selected as the award, bargainers are penalized if their offers are perceived by the arbitrator as extreme. Our evidence allows us to examine the possibility that FOA punishes the submission of extreme offers when they are paired with more reasonable offers. (12)

To investigate this possibility, we rank the 132 out-of-sample cases by fitted values from our estimation of Equation 6. When the fitted value from Equation 6 exceeds the 50% threshold value ([[alpha].sub.0] + [[alpha].sub.1]Z = 0), the player's offer is more extreme relative to management's offer. The reverse is true for fitted values that are below the 50% threshold. In general, the farther the fitted value lies from the 50% threshold, the more extreme one offer is relative to the other.

Accordingly, we group the cases by their distance from the threshold value and report the results in Table 4. (Although our groupings are based on arbitrary dividing lines, we experimented with different groupings, and they all exhibit the same patterns reported in the table.) Group A includes the 64 cases that lie within 1 standard deviation (SD) of the threshold value. In other words, group A consists of the cases in which the final offers are fairly equidistant from the estimated preferred award. Group B consists of the 57 cases for which the fitted value from Equation 6 lies between 1 and 2 SD from the threshold. Finally, group C contains the 11 cases that lie more than 2 SD from the threshold. Of the three groupings, group C includes the cases in which one offer is most extreme relative to the other.

If arbitrators select the most reasonable offer as the award, the more extreme one offer is relative to the other, the more likely the extreme offer will be rejected in favor of the more reasonable one. Table 4 shows that the decision rule from past FOA outcomes correctly predicts the winning offers in 53% of the group A cases and 61% of the group B cases. For the group C cases, which involve the most extreme relative offers, the decision rule correctly predicts the outcomes in nine out of 11 (82%) of the cases. (13)

Note that evidence from Table 4 also suggests that when arbitrators are confronted with a pair of offers that are more or less equally reasonable, the choice of the actual award can appear to be random. This is consistent with arguments raised by Farber (1980, 1981) and Bloom (1986) and also serves to explain the relative difficulties past empirical studies have encountered in predicting baseball arbitration outcomes. Arbitrators usually choose the most reasonable offer when one submission is extreme relative to the other. This forces bargainers to submit reasonable offers. Not surprisingly, therefore, the outcomes from group A, composed of more or less equally reasonable offers representing the largest subset of arbitration cases, are the most difficult to predict.

6. Conclusion

In evaluating FOA as a means of resolving disputes, several aspects of our analysis are important. First, as opposed to simply explaining arbitration outcomes over a discrete period, we found that offers and outcomes in prior periods were positively correlated with outcomes in subsequent periods. Because the arbitrator's preferred award is never directly revealed in a pure FOA setting, these findings are particularly significant. Moreover, because the bargainers do not know which arbitrator will hear the case at the time the offers are filed, the likelihood of winning a case following the exchange of offers must be inferred from pooled outcomes that do not incorporate individual arbitrator preferences.

We also found that when confronted with a pair of offers in which one is clearly more extreme than the other, arbitrators tend to choose the more reasonable offer as the award. Consequently, because FOA can be viewed as a noncooperative game, each bargainer is likely to infer that his or her probability of having his or her offer selected as the award will be higher if a more reasonable offer is submitted. This lends credence to the use of FOA as a viable dispute resolution mechanism with the capacity to drive quality settlements.

Nevertheless, our analysis also reveals that a great deal of uncertainty remains regarding the estimated preferred award distribution. Although the estimated preferred award distribution in the current arbitration period is clearly a function of decisions in prior periods, it cannot be perfectly inferred from those decisions. Moreover, when the two offers are more or less equally reasonable, the outcome of the case is very difficult to predict. Therefore, a strong incentive exists to negotiate a mutually agreeable settlement to avoid the risk of an uncertain arbitration outcome. The effects of this incentive in baseball are clear: during the 1981-2001 period, final offers were exchanged in 1774 cases, but only 345 cases, or 19%, reached the arbitrator. In the remainder of the cases, the parties negotiated a settlement prior to the arbitration hearing.

Table 1. Final-Offer Arbitration Cases by Year (Hitters)

           Number of           % Won
Year     Hitters' Cases    by Management

1981           12               42
1982           10               70
1983           13               61
1984            8               62
1985            6               50
1986           20               65
1987           11               55
1988            7               43
1989            9               44
1990           16               44
1991           12               67
1992           11               64
1993            8               62
1994            7               43
1995            3               67
1996            5               20
1997            1              100
1998            5               80
1999            8               87
2000            4               50
2001            5               20
Total         181               56

Table 2. List of Independent Variables

Variable                             Description

LOG-AVGSALARY     Log of the average player salary in that season
LOG-PREVSALARY    Log of the player's salary in the previous season
LOG-AVGOFFERS     Log of the average of the final offers
NETRUNS           Runs scored minus home runs in previous season
HOMERUNS          Home runs in previous season
NETRBI            Runs batted in minus home runs in previous season
FIELDPCT          Fielding percentage in previous season
CATCHER           Dummy variable equal to 1 if a catcher
FIRST             Dummy variable equal to l if a firstbaseman
SECOND            Dummy variable equal to 1 if a secondbaseman
SS                Dummy variable equal to 1 if a shortstop
THIRD             Dummy variable equal to 1 if a thirdbaseman
Y87-88            Time series dummy variable for 1987 and 1988
Y89-90            Time series dummy variable for 1989 and 1990
Y91-92            Time series dummy variable for 1991 and 1992
Y93-94            Time series dummy variable for 1993 and 1994
Y95-98            Time series dummy variable for 1995 through 1998
Y99-01            Time series dummy variable for 1999 through 2001

Table 3. Estimation of Equation 6

Variable                  Coefficient (t-statistic)

[[alpha].sub.0]                0.211 (0.301)
[[alpha].sub.1]                0.302 (2.20) *
Y87-88                        -0.526 (-1.24)
Y89-90                        -0.590 (-1.52)
Y91-92                         0.031 (0.078)
Y93-94                        -0.175 (-0.396)
Y95-96-97-98                   0.0003 (0.001)
Y99-00-01                     -0.061 (-0.142)
Log likelihood                -86.30
Cases correct                     78
Number of observations           132
Chi-square                    51.24 *

* Significant at 1% level.

Table 4. Summary of Predicting Arbitration Outcomes Based on Past
Decisions

Groupings                           Total Cases    Total Correct

Group A
(less than 1 SD from threshold)          64             34
Group B
(1-2 SD from threshold)                  57             35
Group C
(more than 2 SD from threshold)          11              9
All cases                               132             78

                                     % of Total Cases
Groupings                           Correctly Predicted

Group A
(less than 1 SD from threshold)             53
Group B
(1-2 SD from threshold)                     61
Group C
(more than 2 SD from threshold)             82
All cases                                   59

We benefitted from useful suggestions on earlier versions provided by Orley Ashenfelter, Arthur Blakemore, Scott Burgess, Stuart Low, the participants in the Arizona State University Department of Economics Seminar Series, and an anonymous referee.

(1) Ashenfelter (1987) theorized the arbitrator exchangeability hypothesis, which suggests that arbitrators have an incentive to behave alike. Empirical evidence (Olson, Dell'omo, and Jarley 1992), however, suggests that whereas pooling arbitration decisions across arbitrators may be feasible, differences in arbitrator preferences may exist.

(2) As with most interest arbitration procedures, the list of potential arbitrators are jointly selected by the union and management.

(3) Olson and Rau (1997) investigated the impact of FOA decisions in one round on negotiated settlements in the subsequent round.

(4) Miller's model differs from Farber's in terms of the formulation of final offers by bargainers. In Farber's model, the bargainers select final offers that maximize the expected utility from an arbitrated settlement. Miller's model incorporates the utility from both negotiated settlements and arbitrated outcomes.

(5) Dworkin (1997) describes the evolution of baseball FOA. Helyar (1994) details the landmark Bruce Sutter arbitration case that radically changed FOA in baseball's free agent era.

(6) The number of years required for FOA eligibility vary according to the terms of the collective bargaining contract. From 1974 to 1987, two years of experience were required. The minimum requirement was raised to three years between 1988 and 1990. Finally, from 1991 to the present, players with at least three years of service plus the top 17% of the players with two to three years of experience are eligible for arbitration.

(7) Unlike hillers, the pitchers' roles are more varied, causing distinct differences in the proxies one would use to measure their performance. The number of innings pitched, for example, would be a proxy for the performance of a starting pitcher, but not a relief pitcher. The number of saves earned would be a relevant performance measure for a relief pitcher, but not for a starting pitcher. Moreover, within the subset of relief pitchers, some individuals are "closers" and are likely to accumulate the most saves but will log fewer innings pitched than "setup" relief pitchers, who are likely to pitch more innings but post fewer saves than closers. "Long-relief" pitchers pitch even more innings, but earn fewer saves than the other two relief categories. We attempted to model FOA decisions for pitchers, but found that we could not pool all pitchers into the same sample. Moreover, the sample simply was not large enough to allow us to examine FOA decisions within each pitching role.

(8) A more desirable proxy for comparative salaries is the mean salary of a "comparable" player. Although complete individual salary information has been in the public domain since 1986, only average salary data is available in the 1981-1985 period.

(9) Fourteen separate equations were estimated during this component of the analysis. The equations are not reported in this paper but are available upon request.

(10) Note that [beta] and [sigma] are assumed to be constant for each in-sample subset, but not for the entire Z vector. For example, point estimates for 1986 come from the outcomes of 1981-1985 cases. Point estimates for 1987 come from L981-1986 pooled cases, and so on. Nevertheless. because [beta] and [sigma] may vary in any given year, we include a set of time series dummy variables in the estimation of Equation 6. Because few cases were heard in some years, we combined years such that each time series dummy variable contained at least 15 observations.

(11) We were not able to control for arbitrator identities because the number of cases heard by individual arbitrators is usually quite small. However, in baseball, the parties do not know the identity of the arbitrator who will hear their case throughout most of the negotiating period. Consequently, it is incumbent upon us to model pooled decisions across arbitrators.

(12) We refer to extreme offers in quotations because FOA literature (Gibbons 1988; Curry and Pecorino 1993; Faurot 2001) suggest that extreme offers will not be observed in equilibrium. In contrast, we refer to situations in which one offer is more extreme relative to the other (i.e., the distance between the average of the offers and the arbitrator's estimated preferred award).

(13) This evidence contrasts with the conclusion of Coleman, Jennings, and McLaughlin (1993) that final offers can be unilaterally changed in baseball FOA with minimal impact on the expected arbitration result. Similarly, our empirical results also contrast with Faurot and McAllister (1992), who theorize that final offers will strategically be equidistant from the preferred award.

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Daniel R. Marburger * and Paul L. Burgesst

* Department of Economics, Arkansas State University, Box 4181. State University, AR 72467, USA; E-mail [e-mail]; corresponding author.

([dagger]) Department of Economics, Arizona State University, Economics Department, College of Business, Arizona State University, Tempe, AZ 85287, USA; E mail [e-mail].

Received July 2002; accepted September 2003.