3.1.1 Treatments

Figure 1 summarizes the experimental treatments that aim to change the ambiguity regarding the sharing norms. Firstly, these bargaining environments differ from one another in terms of the existence of symmetry and, among the asymmetric bargaining environments, in terms of the source of the induced bargaining asymmetry (through empowerment, entitlement and information). In the symmetric environment, we expected the ambiguity to be lowest, as the only sensible sharing rule is the 50:50 split. In the asymmetric environments, we expected the ambiguity to be highest, as the 50:50 rule is no longer sensible and there is no other sensible sharing rule. In all of these sessions we provided no information regarding what other participants in previous sessions agreed on. Secondly, we aimed to reduce ambiguity with regard to the available sharing rules for Empowerment and Entitlement, providing subjects with the modal split of the pie in previous sessions. We decided not to carry out additional treatment for the informational asymmetry as it is the lack of information that is the source of the asymmetry, such that providing past agreements may result in canceling out the asymmetry itself.

Fig. 1 Treatments: varying ambiguity in the sharing rules

3.1.2 Time-line of the experiment

All sessions included three different tasks: a real effort task, an alternating-offer bargaining task, and a set of elicitation tasks. The real effort task and the elicitation tasks were identical in all sessions, but we varied the bargaining environment from one treatment to another, as described in Fig. 1. We now provide further details about each of the bargaining environments.

Real effort task: Subjects were presented with a matrix filled with “0”s and “1”s similar to that in Fig. 2 and asked to count the number of ones.Footnote ⁵ Once a number was entered for a matrix and the subject confirmed the input, a new matrix appeared on the screen. Subjects performed this task for 5 minutes and the performance measure was the total number of matrices for which the correct number of “1”s was provided.Footnote ⁶ This task was not directly incentivized but subjects were informed that their performance in this task was important for determining their earnings in the bargaining task.Footnote ⁷ Consistent with previous findings, this task proved to be gender neutral in performance, with regard to the number of matrices attempted, and the precision rate.Footnote ⁸

Fig. 2 Example of a matrix shown to subjects during the real effort task

Subjects’ gender was elicited at the end of this task, just before taking on the bargaining task. In particular, they were presented with two avatars representing the silhouettes of a man and a woman and explicitly asked “Are you a man or a woman?”. As can be seen in Fig. 3, these avatars were chosen to elicit subjects’ gender in the most aseptic and neutral way possible, without giving any further cues such as facial expressions. These avatars were used to make bargainers’ genders common knowledge, as illustrated by Fig. 4.

Fig. 3 Gender avatars

Bargaining task: symmetric. Based on their relative performances in the real effort task, subjects were assigned a score for productivity, which determined the pie to be bargained over. Specifically, the top third of performers were endowed with a productivity of €15 , the middle third with a productivity of €10, and the bottom third with a productivity of €5. Subjects were only given precise details about this protocol after they completed the real effort task, but no information was provided about the actual number of matrices they solved correctly or about their individual productivities.

Each subject was then randomly paired with another subject. One was assigned the role of Participant A (hereafter referred to as the Proposer) and the other that of Participant B (hereafter referred to as the Responder). The Proposer was the paired subject with the higher score in the real effort task, although this protocol was not revealed.Footnote ⁹ Within each pairing, the pie to be bargained over was randomly drawn from the productivity of the proposer and that of the responder with equal probabilities. This means that the pie could be of €5, €10, or €15. Only once the pie size was randomly determined, this information was made public, so the bargaining parties do not know whether they will be bargaining over the pie determined by the proposer’s or the responder’s productivity. Each pairing had 3 minutes to reach a deal on how to split the pie through an alternating-offer bargaining process. During the bargaining, proposers decided on offers to responders while responders decided on demands from proposers. In other words, the whole bargaining process took place in terms of the amount of money that the responder would get. Proposers started the negotiation making the first offer to the responders. During the bargaining, the information available to all subjects consisted of their own avatar and that of the opponent (their gender and that of their paired partner), the size of the pie to be shared, and the bargaining history of offers and demands. See Fig. 4 for an illustration. Importantly, subjects could not see their own productivity or their opponent’s. If they reached a deal within the 3-minute limit, the agreed split was implemented. Otherwise they got 0.

Fig. 4 Screen seen by Proposers during the bargaining task (Symmetric Environment)

The whole bargaining process was repeated for 10 periods in all treatments, with a different paired participant each time.Footnote ¹⁰ Importantly, from one period to the next the role in the bargaining pairing (proposer or responder) and the pie to be split could change. For payment, subjects were informed that the computer would take two periods randomly—one from periods 1–5 and another from periods 6–10—and the resulting outcomes would be implemented.

Bargaining task: empowerment. Everything was the same as for the Symmetric bargaining, except that there was an outside option for the proposer. In particular, if a deal was not reached within the 3-minute limit, the proposer had an outside option while the responder got 0. The outside option available to the proposer was a random amount drawn from a uniform distribution between 50% and 85% of the pie. Both parties knew about the outside option but neither knew its exact value when bargaining.

Bargaining task: empowerment with past agreements. Everything was the same as for Empowerment bargaining, except that we provided subjects with past agreements, i.e., the most frequent amount (mode) agreed for the responder in the sessions with empowerment. These amounts depended on the pie to be shared: 1, 1.5 and 5 euro, when the pies were 5, 10 and 15 euro, respectively. This information was presented to subjects during the negotiation just above the dialog box about the offer/demand.

Bargaining task: entitlement. Everything was the same as for Symmetric bargaining, except that subjects were able to see their own productivity and that of their partners. This was public knowledge. This bargaining environment thus informed subjects of whose productivity determined the size of the pie. This was intended to generate a feeling of entitlement.Footnote ¹¹ In the event of a tie, there is no entitlement effect, meaning that we do not consider those bargaining pairings in the analysis in the rest of the paper (note the lower number of observations in the entitlement treatment).Footnote ¹²

Bargaining task: entitlement with past agreements. Everything was the same as for Entitlement bargaining, except that we provided subjects with past agreements, i.e., the most frequent amount (mode) agreed for the responder in the sessions with entitlement. These amounts depended on the pie to be shared: 2.5, 5 and 7 euros, when the pies were 5, 10 and 15 euros, respectively. This information was presented to subjects during the negotiation just above the dialog box about the offer/demand.

Bargaining task: information. Everything was the same as for the Symmetric environment, except that only the proposer could see the actual size of the pie, while the responder only knew that it could be 5, 10 or 15 euro. This was public knowledge.

Elicitation tasks. After completing the 10 bargaining periods, subjects entered the third and last task of the experiment. We first asked the subjects explicitly: "What do you think the objective of this experiment is?." This answer was not incentivized and they were allowed to provide their answers in free format. One potential concern with the way we made subjects’ genders common knowledge is that this feature of the design could yield some type of experimenter demand effect, which we address in the robustness checks (at the end of Sect. 4.2). Furthermore, in this task we elicited beliefs about self-assessed relative ability both in the real effort task and the bargaining task. As far as the real effort task is concerned, subjects were asked to reveal which quartile of the performance distribution they thought they were in and to state which gender they believed had performed better (or whether there were no gender differences). Similarly, for the bargaining task, subjects were asked to reveal which quartile of the distribution they thought they were in based on the relative surplus obtained during the 10 negotiations and to state which gender on average had obtained a greater share of the pie over the 10 periods (or whether there were no gender differences). Finally, we also elicited risk attitudes following the methodology in Eckel and Grossman (Reference Eckel and Grossman2002) and social preferences via the primary slider measure items described in Murphy et al. (Reference Murphy, Ackermann and Handgraaf2011) and implemented for z-Tree by Crosetto et al. (Reference Crosetto, Weisel and Winter2012). All these measures were incentivized.Footnote ¹³ Table A1 in Online Appendix A shows the mean values for these control variables by gender. The main notable gender differences show up in risk preferences, where women appear to be more risk averse than men, and less confident in both their ability at the real effort task and in their bargaining ability. Figures A1 and A2 in Online Appendix A show subjects’ perceptions about the gender nature of the task and bargaining by gender. Perceptions about the gender nature of the real effort task are split, with slightly more male subjects tending to believe it is a male task, while slightly more female subjects put more weight on the task being a female task. However, both male and female subjects perceive bargaining to be a male task.

3.2.1 Data

We gathered data on 2487 different negotiations from 562 different experimental subjects.Footnote ¹⁴ We focused on three important bargaining outcomes. The first outcome in a negotiation was whether the parties reached an agreement or not. The success rate measured the efficiency of bargaining: only when an agreement was reached could surplus be created. Another important outcome was earnings, measured as the share of the pie. This outcome, however, can be measured in two different ways: overall earnings, not conditional on reaching an agreement, and earnings conditional on reaching an agreement. For example, data from the field on labor markets usually involves the second one, as failed negotiations are rarely observed. However, from an efficiency point of view, the former variable is the most important, for example, when deciding whether to negotiate or in deciding on whom to delegate a negotiation. To sum up, we considered all three variables: probability of reaching an agreement, share of the pie conditional on reaching an agreement, and overall share of the pie or earnings.

3.2.2 Hypotheses

The experimental design consisted of a 2 (Male Proposer, Female Proposer) $\times$ 2 (Male Responder, Female Responder) $\times$ 6 (Symmetric, Empowerment, Empowerment with past feedback, Entitlement, Entitlement with past feedback, Information) factorial design. The first two factors allowed us to test for the existence of gender differences in each of the bargaining roles. Meanwhile, the third factor allowed us to check for the role of gender as an effect modifying factor between symmetric and asymmetric bargaining environments, and between environments with and without past agreements.

Given the experimental design and treatments, we started by testing two different sets of hypotheses. Firstly, we tested for the existence of gender differences in each of the six environments considered.

We hypothesized that gender differences would be non-existent in the symmetric bargaining environment, where the 50:50 norm is prevalent (Hypothesis 1). Our symmetric bargaining setting is closest to the one modeled in Ma and Manove (Reference Ma and Manove1993), where players do not know with certainty whether their offer will be the last one. The reason is that, if they wait for too long, they might not be able to submit the offer and get a response from the other player, while if they send their offer too early, the opponent might send a counteroffer so that their offer is not the last one. In this framework, the expected division of the pie is unique and close to an even split.Footnote ¹⁵

We hypothesized that asymmetric bargaining environments without past agreements may yield gender differences in all three environments: empowerment, entitlement, and information (Hypotheses 2). Note that, by making the proposer the stronger bargaining party, asymmetries break the 50:50 sharing norm but in a way that an alternative clear sharing norm is absent. This lack of clear sharing rule also allows for enough ambiguity and wiggle room for the bargaining parties to show their bargaining abilities. In particular, for the empowerment setting we decided not to provide the exact value of the outside option so as not to make that amount too salient.Footnote ¹⁶ In the entitlement setting, although it was clear the proposer was entitled to a higher share of the pie, because their productivity was higher, it was not clear how much their share of the pie should be, because the pie was determined randomly by the productivity of only one bargaining party. Finally, in the informational asymmetry, bargaining parties might expect the stronger party to try to take advantage of the informational asymmetry.

In asymmetric bargaining environments with past agreements, we intended to maintain the strength of the proposer by breaking with the 50:50 split, but in a way that a new sharing rule arises by providing bargaining parties with past agreements. Under this scenario, we hypothesized that gender differences should be less pronounced and somewhere half way between the symmetric environment and the empowerment and entitlement situations without past agreements (Hypotheses 3).

Secondly, given the fact that we also varied bargaining environments to change the existing ambiguity regarding the available sharing norms, we tested the null hypothesis of whether gender is an effect modifying factor when changing from a symmetric to an asymmetric bargaining environment (Hypothesis 4), and when changing from a bargaining situation without past agreements to a setting with information on past agreements (Hypothesis 5). With this in mind, we compared each of the asymmetric treatments with the Symmetric one, and asymmetric environments without past agreements with those with past agreements.

3.2.3 Identification strategy

In order to test hypotheses 1–3 (whether gender differences exist in different bargaining environments), we started with the following specification:

(1) $\begin{array}{c}{Y}_{\mathrm{ij}}=\alpha +{\beta }_{1}MalePro{p}_i+{\beta }_{2}MaleRes{p}_j+\gamma X_{\mathrm{ij}}+{ϵ}_{\mathrm{ij}}\end{array}$

where $MalePro{p}_i$ ( $MaleRes{p}_j$ ) takes a value of 1 if the Proposer i (Responder j) is a man and 0 for a woman. To control for the characteristics in which the bargaining between Proposer i and Responder j took place, the term $X_{\mathrm{ij}}$ includes session, period, and pie fixed effects. Specification (1) enables us to test whether gender differences in bargaining can be detected, i.e., whether men and women in the role of Proposer/Responder obtain different outcomes from bargaining or whether bargaining with men is different from bargaining with women. In this specification, our coefficients of interest are ${\beta }_1$ and ${\beta }_2$ .Footnote ¹⁷ The estimation results for these tests are shown in Tables 3, 4 and 5.

To test hypothesis 4 (whether introducing asymmetries are gender effect modifying factors), we compared gender differences in each asymmetric environment (without including the treatments with past information) with the symmetric environment by running the following regression:

(2) $\begin{array}{c}\begin{array}{cc}{Y}_{\mathrm{ij}}=& \alpha +{\beta }_{1}MalePro{p}_i+{\beta }_{2}MaleRes{p}_j+{\beta }_{3}Asy{m}_{\mathrm{ij}}+{\beta }_{4}Asy{m}_{\mathrm{ij}}\ast MalePro{p}_i\\ & +{\beta }_{5}Asy{m}_{\mathrm{ij}}\ast MaleRes{p}_j+\gamma X_{\mathrm{ij}}+\theta Asy{m}_{\mathrm{ij}}\ast X_{\mathrm{ij}}+{ϵ}_{\mathrm{ij}}\end{array}\end{array}$

while to test hypothesis 5 (whether providing past information is a gender effect modifying factor), we compared gender differences in each asymmetric environment with the ones under the provision of past information:

(3) $\begin{array}{c}\begin{array}{cc}{Y}_{\mathrm{ij}}=& \alpha +{\beta }_{1}MalePro{p}_i+{\beta }_{2}MaleRes{p}_j+{\beta }_{3}PastAgre{e}_{\mathrm{ij}}+{\beta }_{4}PastAgre{e}_{\mathrm{ij}}\ast MalePro{p}_i\\ & +{\beta }_{5}PastAgre{e}_{\mathrm{ij}}\ast MaleRes{p}_j+\gamma X_{\mathrm{ij}}+\theta PastAgre{e}_{\mathrm{ij}}\ast X_{\mathrm{ij}}+{ϵ}_{\mathrm{ij}}\end{array}\end{array}$

where, as before, $MalePro{p}_i$ ( $MaleRes{p}_j$ ) takes a value of 1 if the Proposer i (Responder j) is a man and 0 for a woman and $X_{\mathrm{ij}}$ incorporates session, period, and pie fixed effects into the analysis to control for the environment in which the bargaining took place. In regression 2, the omitted treatment is the symmetric one, while ${\text{Asym}}_{\mathrm{ij}}$ takes the value of 1 if the pair ij bargained in an asymmetric environment without past agreements. In regression 3, the omitted treatment is the asymmetric one without the past agreements, while ${\text{PastAgree}}_{\mathrm{ij}}$ takes the value of 1 if the pair ij bargained in an environment in which past agreements were provided. In regressions 2 and 3, the coefficients of interest are ${\beta }_4$ and ${\beta }_5$ whose sign and significance show whether the introduction of asymmetries/past agreements indeed modify gender differences with respect to the symmetric/without past agreement environments. The estimation results for these tests are shown in Tables 6 and 7.

Given our interest in understanding gender differences in the three main interdependent outcomes of probability of reaching an agreement, earnings conditional on agreement and overall earnings, the estimation is carried out by using Cragg’s two-part model (Cragg, Reference Cragg1971).Footnote ¹⁸ Cragg’s two-part model relies on the existence of a process that determines whether the outcome is positive or zero –i.e., whether the negotiation ended in agreement or not–, and on a different process that determines the participants’ share of the pie conditional on reaching an agreement.

An interesting feature of two-part models is the interpretation of their coefficients, which provides us with a more comprehensive understanding of the impact of gender on bargaining. Firstly, the model allows us to compute the unconditional semi-elasticity ( $S_z\left(y\right)$ ), i.e., the percentage by which the overall earnings, taking into account failed negotiations, of men differ from that of women in role $z\in \text{Proposer, Responder}$ . Secondly, it allow us to decompose this overall average effect of gender into two different components. On the one hand, we can isolate how the gender of the subject playing in role z impacts the probability of reaching an agreement ( $S_z(P=1)$ ). On the other hand, it allows us to examine how the gender of the person in role z impacts the earnings of the proposer and the responder, conditional on reaching an agreement ( $S_z\left(y\right|y>0)$ ).Footnote ¹⁹ See Table A3 in Online Appendix A.

This decomposition is crucial to understand gender differences in bargaining as having a significant result on overall earnings could be due to different facts: because there are gender differences on the probability of reaching a deal or because there are gender differences in earnings conditional to close a deal. Even more importantly, the absence of gender differences in overall earnings does not necessarily imply the absence of gender differences. It could be the case that one gender in a particular role makes agreement less likely but captures more of the pie when an agreement is reached (for example, because this gender negotiates more aggressively). Since these two gender differences may have opposite effects on overall earnings, it is possible to find a non significant coefficient for the overall earnings despite the existence of gender differences in both parts of the process.

A few final observations are necessary to clarify the subsequent analyses.

First, each of the regression returns 5 closely linked outcomes: probability of reaching a deal, proposer’s earnings when a deal is reached, responder’s earnings when a deal is reached, proposer’s overall earnings and responder’s overall earnings. When talking about earnings, it is necessary to differentiate between proposers’ and responders’ ones. For conditional earnings, this is the case because as we work with the semi-elasticities, the percentage change that the gender of bargainer in role z causes on proposer’s and responder’s earnings will be different when the average earnings of proposers and responders differ (which, as will be shown below, is the case in all asymmetric treatments). Notice however that, despite the magnitude being different, it will always be, by construction, of opposite sign for proposers and responders, as the situation represents a zero-sum game. That is, if one party gets more then the other must get less. For overall earnings, we need to differentiate between proposers’ and responders’ earnings due to the fact that, overall, the game is not a zero-sum game since if the bargaining fail, both parties get zero. This means that for overall earnings, observing a gender in a role affecting the proposers’ overall earnings in certain direction does not imply that it affects the responders’ ones in the opposite direction.

Second, in all specifications for bargaining outcomes, we use a two-way clustering at the subject level, that is, at the proposer and responder level simultaneously (Cameron et al., Reference Cameron, Gelbach and Miller2011; Thompson, Reference Thompson2011; Gu and Yoo, Reference Gu and Yoo2019), such that the number of clusters is the same as the number of different subjects playing the role of proposer and responder respectively.Footnote ²⁰

Finally, given the large number of hypothesis to be tested in the experiment, we also present the p-values corrected through the Romano-Wolf multiple hypothesis correction (Romano and Wolf, Reference Romano and Wolf2005a, Reference Romano and Wolfb, Reference Romano and Wolf2016).Footnote ²¹ Consequently, in Sect. 4, we will only consider that a gender coefficient is significant after taking into account this correction which involves, broadly, those coefficients that are significant at least at the 5% significance level without the correction.