Whenever we have a one-sided hypothesis, we use one-sided test statistics. When comparing means across treatments, we use Mann–Whitney-U (MW) tests, and in comparing distributions of variables, we use Kolmogorov–Smirnov (KS) tests. When comparing ratios (or percentages), we use both Pearson’s Chi square and Fisher’s exact tests. For reasons of convenience, we label high performing department heads as “winners” and low performing department heads as “losers”. Fairness judgments, first proposals, and agreements are all given in shares to winners.Footnote 7 We defer non-parametric tests related to differences in bargaining behavior (or outcomes) across different stake sizes to the Online Appendix (see Online Appendix A.3, Table A.3.5). Those results show that the main outcome variables of bargaining are not systematically influenced by the variation in the stake size. Hence, in conducting the non-parametric tests below, we pool data from the two stake sizes.Footnote 8
For every regression analysis, we present four specifications: (1) only the main variables of interest as independent variables (Specification 1), (2) Specification 1 + control variables (Specification 2), (3) Specification 1 + interaction variables (Specification 3), and (4) Specification 3 + control variables (Specification 4). Control variables are subjects’ justice centrality scores, agreeableness scores, risk attitudes, gender composition of the pair, and the budget level dummy (1 for the higher budget). In the following, we provide the estimates for our main variables of interest. Detailed regression tables are provided in Online Appendix A.3, and we report marginal treatment effects from our regressions in Table A.3.6.Footnote 9
Frequency of (dis)agreements
Table 2 shows the distribution of disagreements in our treatments. We observe that 4 pairs (out of 89; 4.5%) disagreed in LTP, and 22 pairs (out of 70; 31.4%) disagreed in HTP. Both tests show that the difference is highly significant (for both, p < 0.001), strongly in line with Hypothesis 1.
Table 3 presents the results of (robust) probit regressions investigating the effects of time pressure (T_pressure), tension in first proposals (the difference in first proposals; Diff_first), and the interaction between the two variables on the probability of disagreements.Footnote 10 In all specifications, the time pressure coefficient has the expected positive sign, and it is significant, except for Specification 3. The interaction of time pressure and the tension in first proposals has the expected positive sign. We interpret this result as follows: for those pairs with high levels of tension in first proposals, the effect of time pressure on the likelihood of a disagreement is seems to be stronger, but the evidence is not fully conclusive.
Notice that there are only four disagreements in LTP (out of 89 pairs), and probit models can become unstable if the binary dependent variable takes one of the two values only very infrequently. Therefore, as a further sensitivity analysis, we also run an exact logistic regression with time pressure, a binary variable describing the tension in first proposals, and an interaction of these two as explanatory variables (see Table A.2.3 in the Online Appendix).Footnote 11 An exact logistic regression is recommended when the sample size is very small and/or when some of the cells formed by the dependent or independent categorical variables contain very few observations.Footnote 12 Estimation results reported in Table A.2.3 provide further support for our conclusions.
Time pressure increases the frequency and likelihood of disagreements.
The effect of time pressure seems to be stronger for those pairs with high levels of tension between players’ first proposals.
Content of agreements
Figure A.1.1 in the Online Appendix shows that high performers’ subjective entitlements are (on average) 0.66 (in both treatments), directly coinciding with the induced reference point. In contrast, low performers’ subjective entitlements (on average) are around 0.58, the midpoint between 1/2-1/2 and 2/3-1/3. Figure A.1.2 shows that high-performers’ first offers bunch around 2/3-1/3 (mean = 0.69, median = 0.67) and low-performers’ first offers bunch around 1/2-1/2 (mean = 0.50, median = 0.53). Moreover, 28% of a total of 363 offers in HTP are equal to either exactly 2/3-1/3 or to the closest prominent numbers around 2/3-1/3 (i.e. the 17,000-10,000 division of the 27,000 budget), and 17% are equal either exactly to 1/2-1/2 or to the closest prominent/round numbers around 1/2-1/2 (i.e., the 8,000–7000 division of 15,000 and the 14,000–13,000 division of 27,000). These percentages are 26 and 13% respectively for LTP (of a total of 1032 offers). Subjects’ fairness judgments (or entitlement perceptions), first proposals, and the rest of the bargaining process were all influenced by the two reference points.
Given these observations, our next look is at bargaining outcomes. The average agreed share of winners (conditional on agreement) is 0.59 in LTP and 0.60 in HTP (one-sided, MW, p = 0.44; KS, p = 0.49; see Fig. 1).Footnote 13 Understandably, these numbers constitute compromises between players’ first proposals (i.e., 0.50 and 0.69), replicating findings from studies with similar designs. In fact, the shares happen to be at the midpoint of the range of first offers. Moreover, 103 out of 133 (77%) agreements were in this range (0.50–0.69). On average, time pressure does not have a level effect on agreement terms. Even a less ambitious test, where we check whether agreements close to 2/3-1/3 are more frequent in HTP does not give a statistically significant affirmative answer: 73% (65%) of all agreements in HTP (LTP) are closer to 2/3-1/3, but Fisher’s exact and Pearson’s Chi square tests both reject the notion that the difference is significant (one-sided, p = 0.219 and 0.331).
Conditional on agreements, neither the average winner agreed share nor the distribution of winner agreed shares differ between LTP and HTP.
Table 4 presents the results of (robust) seemingly unrelated bi-probit regressions, investigating the effects of time pressure, winner entitlements, loser entitlements, interaction variables, and control variables on the probability of reaching 2/3-1/3 agreements and 1/2-1/2 agreements (conditional on agreement), whereas Table 5 presents a (robust) OLS regression on winners’ agreed shares.Footnote 14 In the tables, T_pressure is a dummy variable taking a value 0 for LTP and 1 for HTP; W_fair and L_fair denote winner and loser entitlements, respectively (in the share going to the winner) – the two are taken from the question regarding the allocation choice of a fair and neutral arbitrator.
In Table 4 (upper panel), Model 1, where the dependent variable is a dummy (i.e., Statusquo) showing whether the agreement reached is the 2/3-1/3 division or not, the coefficient of T_pressure is positive in all specifications, but significant on convention levels only in Specifications 3 and 4. This implies that HTP per se does not increase the probability of reaching 2/3-1/3 agreements. In Model 2 (lower panel of Table 4), where the dependent variable is a dummy (i.e., Equal) showing whether the agreement reached is the 1/2-1/2 division, we see that the coefficient of T_pressure is significant and negative in all specifications. The results imply that HTP decreased the probability of reaching 1/2-1/2 agreements.
Conditional on agreements, time pressure decreases the likelihood of reaching agreements on the implicit reference point, i.e., the 1/2-1/2 division. However, it does not systematically relate to the likelihood of 2/3-1/3 agreements.
Table 5 presents the results of (robust) OLS regressions investigating the effects of time pressure, winner entitlements, loser entitlements, and interaction variables on winners’ agreed shares, conditional on agreements. We observe that the coefficients of both winners’ and losers’ fairness judgments have the expected positive sign in all four specifications. W_fair is highly significant in all specifications, whereas L_fair is only significant in Specifications 3 and 4. The coefficient of time pressure is positive in all specifications but, again, only significant in Specifications 3 and 4. Thus, winners’ agreed shares appear to be unaffected by the variation in time pressure, as already indicated by our non-parametric analysis.
Conditional on agreements, time pressure does not influence the share winners receive in agreements. The influence of subjects’ fairness judgments (or perceived entitlements) on agreements is weaker under time pressure.
The bargaining process and the timing of agreements
Since our participants know the deadline before they make their first proposals, the latter might have been influenced by the variation in the bargaining deadline, affecting the tension in first proposals. However, the distributions of tension in first proposals (measured in percentage points; i.e., 0.17 means that the sum of what bargainers demand in their first proposals corresponds to 117% of the pie) in LTP and HTP do not present evidence for a difference in the tension between first proposals (0.19 for LTP and 0.17 for HTP; two-sided MW, p = 0.40; KS, p = 0.61).
First proposals are also potentially informative since impasse may start with a specific constellation of initial positions. Figure A.2 in the Online Appendix depicts the distribution of winners’ and losers’ first proposals across treatments. It shows that the induced reference point outcome, that is the 2/3-1/3 division, is utilized by high performers, whereas the implicit reference point outcome, that is the 1/2-1/2 division, is utilized by low performers in making a first proposal. This is in line with Roth’s (1985) argument: “[The] bargainers sought to identify initial bargaining positions that had some special reason for being credible, and that these credible bargaining positions then served as focal points that influenced the subsequent conduct of negotiations [italics in original].” Since 2/3-1/3 and 1/2-1/2 are possibly the two most salient (or credible) reference/focal points in our environment, it is natural to observe that they influence subjects’ initial bargaining positions.
It is worthwhile re-emphasizing that the differences that we observe in disagreement rates (reported in Sect. 4.1) between LTP and HTP are not due to a prohibitively high level of time pressure. As we reported above, the tension in first proposals (a variable describing an important layer of bargaining) is identical across treatments. The frequency of winners (losers) making the opening proposal in a pair (i.e., the very first offer in a pair) and the frequency of winners (losers) accepting an offer in a pair is also the same in the two treatments (Fisher’s exact tests; p-values are 0.71 and 0.56 for winners, and 0.78 and 0.56 for losers). Moreover, the average number of proposals made in a pair is 11.5 in LTP and 5.2 in HTP. Despite the 85% reduction in allotted time (i.e. from 600 to 90 s) the number of proposals dropped by only 54%. In other words, bargainers in our HTP treatment still found time to exchange several proposals.
A look at the distribution of concessions shows that concession behavior looks very similar in LTP and HTP. MW tests show that average concessions, defined as the difference between what a subject has asked for herself in her first proposal and what she agreed to receive at the end (excluding disagreements), are identical across LTP and HTP (two-sided, p = 0.29 for winners, p = 0.14 for losers). KS tests deliver the same conclusion (two-sided, p = 0.57 for winners, p = 0.54 for losers).
Timing of agreements
The average agreement time in LTP (395 s) is 65.8% of the allotted time (600 s), whereas the average agreement time in HTP (78 s) is 86.7% of the allotted time (90 s). Table 6 displays the distribution of agreement times (in relation to the total time allotted) in LTP and HTP.Footnote 15 Numbers in the top row refer to the time intervals in percentiles (e.g., first 10% of the allotted time and so on), and the numbers in the second and third rows refer to the percentage of agreements that occurred in the corresponding time interval. These numbers show that the distribution of (relative) agreement times in HTP first-order stochastically dominates the ones in LTP. 41% of all agreements in LTP are reached in the last 10% of the allotted time (i.e., between seconds 541 and 600), whereas 72.90% of all agreements in HTP are reached in the last 10% of the allotted time (i.e., between seconds 81 and 90).
Table 7 shows the distribution of last-moment agreements (defined as agreements reached in the last 5 s before the deadline), both conditional and unconditional on reaching an agreement, across treatments. In the latter, a disagreement is coded as 0 (i.e., not a last-moment agreement). We observe that 26 pairs (out of 85 that reached an agreement; 30.5%) in LTP reach the agreement in the last five seconds, and 24 pairs (out of 48 that reached an agreement; 50%) in HTP reach the agreement in the last 5 s. Both Pearson’s Chi square and Fisher’s exact tests show that the difference is significant, in line with Hypothesis 4.
That said, there is another possible definition of last-moment agreements. As it can also be seen in Table 7, if one does not take disagreements into account, then the percentage of last-moment agreements in LTP is 29.2 and in HTP it is 34.3. The difference is in the same direction, but not significant (Pearson’s Chi square test, p = 0.49, one-sided Fisher’s Exact test, p = 0.30).
Conditional on agreements, the frequency of last-moment agreements is higher in HTP than in LTP. Including disagreements in the analysis does not change the direction of this result, but renders it statistically insignificant.
Combined with our findings on first proposals and initial tensions, the similarity of concession behavior across treatments hints at the increased disagreement rate and last-moment agreement frequency under HTP: if average initial conflict and concession behavior are very similar across two situations, but the negotiators have only 90 s to strike a deal instead of 10 min, naturally it is likely that they end up in stalemates (or reach an agreement in the very last moment).
Table 8 presents the results of (robust) probit regressions, investigating the effects of time pressure, tension in first proposals, and the interaction between the two on the probability of last-moment agreements (i.e. agreements reached in the last 5 s), conditional on agreements.Footnote 16 In all specifications, the coefficients of time pressure and the tension in first proposals are highly significant and have the expected positive signs.Footnote 17
Conditional on agreements, time pressure increases the likelihood of last-moment agreements.