«IZA DP No. 1375 Dividing Justly in Bargaining Problems with Claims: Normative Judgments and Actual Negotiations Simon Gächter Arno Riedl October ...»
specify a bargaining protocol. Bargaining was conducted over a local area network with the help of the experimental software ‘Rabbit’ (Brandel, 2000). The negotiators were allowed to make any (non-negative) proposal as long as the sum of shares was smaller or equal to 2050 points. The negotiators had 15 minutes to reach an agreement. The instructions told the subjects that in case they fail to reach an agreement within the time limit they would earn nothing in this experiment, except their show-up fee. Hence, the ‘threat point’ in this experiment is (0, 0). Random pairing, anonymity, duration and disagreement payoﬀs were common knowledge.
Procedures and payments. The vignettes where administered to 59 ﬁrst year undergraduate students of law, business administration, and computer science of the University of Vienna who were, at the end of class, asked to answer the above problem.
Participation was voluntary and anonymous.6 Each participant only took part either in 1980V or in 1640V. It took the students roughly 10 minutes to answer the question.
The experiments were conducted in the computerized lab of the Institute for Advanced Studies in Vienna. The 80 experimental subjects had the same background as our survey participants. Nobody who participated in the vignette study took part in the experiments. Moreover, each subject only participated in one experiment. The experimental subjects were paid according to their decisions in the experiment. The average earning (including a show-up fee) was approx. Euro 19.-. On average, an experiment lasted 75 minutes, including reading the instructions, performing the experiment, and answering a post-experimental questionnaire.
We ﬁrst analyze the normative judgments and then describe the results of the negotiations. We start our analysis at an aggregate level. In a further important step we will evaluate the predictive power of the four solutions by performing an individual-level analysis.
For convenience, in the remainder of the paper we refer to the subject with the high claim (either 1980 or 1640) as the ‘winner’ and to the subject with the low claim To avoid any experimenter demand eﬀect we did not administer this questionnaire to our own students but instead asked colleagues to administer the questionnaire for us in their class.
(either 510 or 850) as the ‘loser’ of the performance quiz. Moreover, we will adopt the convention to express all allocations in ‘winner shares’, i.e., the share of the total pie of 2050 that goes to the ‘winner’ of the quiz or the better performer.
4.1 Normative judgments What are the normative views our respondents hold about the fair division in the two bargaining with claims environments? Our ﬁrst result records the evidence.
Result 1 The claims strongly aﬀect the normative views of a fair division. If claims are 1640, the average winner share is 65.2 percent, which is closest to PROP. If claims are 1980, the respondents on average think the winner should get 75.4 percent. This comes closest to CEA. Fairness judgments also become more heterogeneous when claims are very asymmetric.
Support. Figure 2 provides a graphical support for our ﬁrst result. It shows a box-plot for each of the two claim levels.7 The medians of the two distributions are far apart The box-plot is a convenient compact way of describing the distribution of outcomes. The rectangle contains 50 percent of the observations. The lower end of the rectangle is the 25th percentile and the upper end the 75th percentile. The horizontal line in the box indicates the median. The vertical line outside the box indicates the adjacent values, and the dots outside the adjacent lines are ‘outliers’.
from each other. With the 1980 claims the whole distribution is shifted upwards. Both a non-parametric Mann-Whitney test and a Kolmogorov-Smirnov test show that the null hypothesis of equal distributions can be rejected at all conventional levels. In the impartial judgments of the vignettes, the average answer in 1980V is on average 10.2 percentage points higher than in 1640V. A further interesting observation is that the variance is considerably higher in 1980V than in 1640V. The standard deviation when claims are strongly asymmetric is 9.7 percent, while it is 6.5 percent in 1640V. Thus, the asymmetry of claims increases the heterogeneity in fairness judgments.
We now look at which rule organizes the data best, i.e., which solution comes closest to the observations on the aggregate level. Table 2 summarizes for each theory and for both claim levels the mean deviation of the observed outcome from a particular theoretical solution. For instance, with claims of (1980, 510) CEL proposes that the winner should get a share of 0.859 of the new pie of 2050. The answers from the survey fall short of the CEL prediction by an average of −.104 and this diﬀerence is signiﬁcant according to a two-tailed t-test.
In 1640V answers come closest to PROP (which predicts a share of 65.9 percent), whereas in the highly asymmetric claims of 1980V the answers are very close to CEA, which predicts 75.1 percent. Note this implies that the normative attractiveness of the diﬀerent rules is not constant across claims. Finally, the equal split EQUA is farthest apart from the observed answers. This holds for both claim points.
4.2 Negotiated agreements While the vignettes elicit the impartial normative views, in actual negotiations many other motives, in addition to the normative views can play a role and inﬂuence the agreements. Result 2 summarizes our main observations from the bargaining experiment.
Result 2 The bargaining agreements are aﬀected by the claims but not by the diﬀerence in the asymmetry of the claims. In 1980E, the average winner share is 58.5 percent, which comes closest to EQUA. In 1640E the winner share is 57.3 percent. This is closest to CEA. The variance in agreements strongly increases in the asymmetry of claims.
Support. Figure 3 depicts the distribution of agreements for the two claim levels.
Recall that we also conducted a control experiment without claims to examine whether asymmetric claims matter at all. In line with previous symmetric free-form bargaining experiments (e.g. Nydegger and Owen, 1975) we ﬁnd that 100% of the agreements implement the equal split. Thus as Figure 3 shows, the presence of asymmetric claims strongly aﬀects the bargaining outcomes. Two further observations are noteworthy.
First, the variance in the agreed winner shares is considerably higher in 1980E than in 1640E. In 1980E the standard deviation is 11.5 percent, which is twice as high than in 1640E, where it is 5.7 percent. Thus, the more asymmetric claims are the more heterogeneous agreements become. Second, the mean agreements are very similar. In 1980E the agreed winner share is 58.5 percent, while it is 57.3 percent in 1640E. Both a non-parametric Mann-Whitney test and a Kolmogorov-Smirnov test show that the null hypothesis of equal distributions cannot be rejected (p-values are 0.825 and 0.389).
Figure 3: Box-plots of distribution of negotiated agreements When we look at which solution comes closest to the observed average agreements we ﬁnd that in 1980E EQUA predicts best, whereas in 1640E CEA does best. However, we observe also that actual winner shares are signiﬁcantly above the share of one half.
Table 3 records the details.
We close this section by comparing the normative judgments and actual agreements.
First, a Kruskal-Wallis test shows that the null hypothesis of equal distributions in all of our four treatments can be rejected at any conventional level (p = 0.0001). Second, a comparison of normative judgments and actual agreements for the 1640-claim (i.e., 1640V and 1640E) shows that the actual agreements are 7.9 percent lower than the normative judgments. This diﬀerence is signiﬁcant according to a two-sided MannWhitney test (p = 0.001). When claims are very asymmetric (i.e., in 1980E and 1980V), the gap between actual agreements and normative judgments amounts to 17 percentage points, which is signiﬁcant at any level. Thus, the asymmetry in claims also increases the divergence of actual agreements and normative judgments.
4.3 Which rule predicts best? An individual-level analysis
as follows. We calculate for each subject in the surveys (each bargaining pair in the experiments) the absolute diﬀerences of his or her answer (reached agreements) from the four solutions. We then rank the solutions according to the smallest diﬀerence of actual and proposed division. For instance, if a subject’s answer comes closest to CEA, and is farthest apart from EQUA, CEA is ranked 1st and EQUA is ranked 4th for this subject. Finally, we calculate, for each treatment, the average rank of each solution.
This provides us with a measure for the relative attractiveness of the diﬀerent solutions at an individual level. We summarize this analysis in the following result.
Result 3 (a) In the normative judgments PROP comes closest to the observed results.
This holds for both claims points. (b) In the negotiations the agreements are closest to CEA for both claims points.
Support. Table 4 displays the distribution of ranks per treatment and per solution. For instance, in 1980V, CEL is 1st -ranked for the answers of 4 subjects, 2nd -ranked for the answers of 2 subjects, 3rd -ranked for the answers of 18 subjects and 4th -ranked for the answers of 5 subjects. Table 4 also reports the mean ranks and the overall ranks for the solutions, based on the mean ranks.
Table 4 clearly shows that in the normative judgments of the vignette treatments, PROP is the ﬁrst ranked solution and EQUA receives the fourth rank. This holds for both claims points. In the negotiation experiments quite a diﬀerent picture emerges.
Here we ﬁnd for both claims points that CEA is the solution than comes closest to the reached agreements and thus receives the 1st rank.
Notes: a numbers indicate the number of observations for which the result (i.e. answer in vignette or agreement in bargaining) is closest, second closest, third closest, and fourth closest, respectively, to the respective rule.
b χ2 -tests compare the distribution of ranks between the diﬀerent claims for each rule.
We know from Result 1 that on average the normative attractiveness of a solution is not independent of the asymmetry of the claims points. Here the question arises, whether there is a similar phenomenon with respect to the rank of the solutions. That is, does the rank of a solution implied by a particular rule depend on the asymmetry of claims? This is an important question because the normative rules are independent of the asymmetry in claims. Here we can report the following ﬁnding.
Result 4 (a) In the experiments, the distribution of ranks is not signiﬁcantly diﬀerent between 1980E and 1640E. This holds for all four solutions. (b) In the normative judgments, however, we ﬁnd signiﬁcant diﬀerences between 1980V and 1640V in the distribution of ranks for CEL and CEA. PROP and EQUA are not signiﬁcantly diﬀerently ranked. This indicates that in the vignettes the preferred rule is not independent of the claims.
Support. In Table 4 we report pairwise χ2 -tests of the distribution of rank orders for each of our solutions and per method of investigation. The results show that in the experiments the distribution of ranks does not diﬀer between claim levels in all four rules (p ≥ 0.12). By contrast, in the normative judgments of the vignette studies, only the ranks of PROP and EQUA are equally distributed between claims. CEL and CEA are not stable since the distribution of ranks is signiﬁcantly aﬀected by the claims points.
At ﬁrst sight Result 4 seems puzzling. We propose the following interpretation. Let us concentrate on the normative judgments ﬁrst. Here we ﬁnd that between claims, only CEL and CEA change ranks; PROP and EQUA are ranked equally across claims.