1 Introduction

Time preferences are arguably the most important factor of sequential-bargaining models. A patient player has more bargaining power than her less patient opponent because her cost of delay in case of disagreement is lower.

There are many ways of implementing discounting in an experiment. One of the canonical methodologies is the shrinking-pie procedure in which the value the players bargain over (the pie) is reduced between rounds. A newer alternative is the effective-discounting procedure implemented in an infinite-time setting by Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023), where disagreement leads to payment being delayed into the future. Both the shrinking-pie and the effective-discounting procedures are attempts at implementing the real-time costs of delaying bargaining in an experimental setting. An alternative is to let substantial time pass between bargaining rounds in the experiment. We call this novel approach the bargaining-delay procedure.

There is currently a dearth of comparisons between discounting methodologies in bargaining experiments, although (Fréchette & Yuksel, Reference Fréchette and Yuksel2017) directly compare implementations in cooperation games. Comparisons of methodologies serve several useful purposes. Social preferences may be activated to different degrees with different methodologies and documenting differences in outcomes may aid comparisons of results between studies. Related, some types of discounting may be more compelling than others to subjects in the lab, resulting in different outcomes. Furthermore, as Fréchette and Yuksel (Reference Fréchette and Yuksel2017) discuss, different contexts may motivate different discounting procedures, so understanding potential practical and empirical differences are crucial.

This paper tests and compares different discounting methodologies in bargaining games. We report the impact of different discounting implementations in alternating-offer bargaining experiments. We conduct treatments with three different discounting implementations; the shrinking-pie procedure, the effective-discounting procedure and the bargaining-delay procedure.Footnote ¹ We follow the canonical Ochs and Roth framework closely. We chose this framework because it serves as a simple benchmark for the comparison of discounting methodologies and ensures some comparability with the complete information framework of Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023).Footnote ²

There is a clear trade-off between realism and control in the three discounting procedures we test. In the shrinking-pie procedure the discount factor is tightly controlled, but is the most removed from the real life delay it represents. The effective-discounting procedure introduces payment delay into bargainers payoffs, but although the length of delay is controlled and is common knowledge, control over individual patience and discount factors and bargainers beliefs is lost. The bargaining-delay procedure is an additional step towards realism, as actual delay is incorporated into the bargaining process itself, rather than just the payoffs. This comes at the cost of necessitating an online experiment with a less controlled environment.

The literature has found no sensitivity to discount factors and number of periods in classic sequential-bargaining models in the lab, finding instead behavioral responses where the modal response is 50–50 and lower offers are rejected with positive frequency (Binmore et al., Reference Binmore, Swierzbinski and Tomlinson2007; Camerer et al., Reference Camerer, Johnson, Rymon and Sen1993; Ochs & Roth, Reference Ochs and Roth1989; Rapoport et al., Reference Rapoport, Weg and Felsenthal1990; Weg et al., Reference Weg, Rapoport and Felsenthal1990; Zwick et al., Reference Zwick, Rapoport and Howard1992). In particular, Ochs and Roth (Reference Ochs and Roth1989) reject predictions of the finite-time alternating-offers model, finding no sensitivity to time preferences (discount factors) or the number of periods in the game. In line with Güth et al. (Reference Güth, Schmittberger and Schwarze1982), these findings set the stage for a large literature on ”Other Regarding Preferences” (ORP) in order to explain the behavioral responses in such lab experiments (Andreoni, Reference Andreoni1990; Fehr & Schmidt, Reference Fehr and Schmidt1999; Fehr & Fischbacher, Reference Fehr and Fischbacher2002).Footnote ³ Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023) find comparative statics that broadly support the theoretical predictions of Rubinstein (Reference Rubinstein1982) when using the effective-discounting procedure.

While discount factors has largely produced negative results, the modern bargaining literature has been more positive, with recent frameworks often focusing on unstructured real-time bargaining, see Camerer et al. (Reference Camerer, Nave and Smith2019), Karagözoğlu and Kocher (Reference Karagözoğlu and Kocher2019) and Bochet et al. (Reference Bochet, Khanna and Siegenthaler2020). Using field data from an online marketplace, Backus et al. (Reference Backus, Blake, Larsen and Tadelis2020) document bargaining behavior consistent with the theoretical predictions of an incomplete information bargaining model, but they also find evidence of bargaining patterns consistent with behavioral factors such as reciprocity in addition to supporting the split-the-difference behavioral norm. Keniston et al. (Reference Keniston, Larsen, Li, Prescott, Silveira and Yu2021) gather data from a range of real-world sources and document the split-the-difference norm's prevalence. Fanning and Kloosterman (Reference Fanning and Kloosterman2022) find strong support for theoretical predictions of the Coase conjecture in an infinite-horizon environment.

We start out by revisiting Ochs and Roth's (1989) original study where discounting is implemented by the shrinking-pie procedure. Our main treatment measure is the first-period offer, which is tested with four empirical approaches. Our primary test is the Wilcoxon rank sum test, which is complimented by regression analysis, Kolmogorov–Smirnov (KS) distributional tests and Barrett and Donald (Reference Barrett and Donald2003) (BD) first-order stochastic dominance tests. First, we vary the number of periods played in a game, keeping discount factors constant. The model predicts that going from two to three periods should decrease the first-period proposer's offer because she is also the proposer in the last period in the three-period game (as opposed to in the two-period game). As in Ochs and Roth (Reference Ochs and Roth1989), this prediction fails to materialize in our data. Second, we increase the discount factor of the first-period responder. The model also predicts that making the first-period responder more patient should increase the first-period offer because the proposer has to offer more to make her opponent indifferent between accepting the offer or going to the next round. We partially replicate Ochs and Roth's findings in this matter as we do not find a clear treatment difference of this change in the discount factor with the rank sum test, and while this is supported by regression analysis, both the KS and BD tests show a significant difference in the direction predicted by the model. Average first-period offers in these treatments are in the range of 42–45 percent of the pie, close to a 50–50 split, and rejection rates are in the range of 0.12–0.23.

We then repeat the same treatments as above with the only difference that we implement discounting using the effective-discounting procedure. That is, if an offer is rejected and bargaining goes on to a new round, payments are deferred by 1 week or 1 month. One week is meant to represent a high discount factor while 1 month is meant to represent a low discount factor. Again, we do not find any treatment differences when changing the number of bargaining rounds from two to three. On the other hand, we do find a treatment difference when changing payment deferrals for the first-period responder from 1 week to 1 month in the rank sum test. This is supported by the BD test which finds a significant difference between treatments. However, the regression analysis and the KS distributional test do not. As with the shrinking-pie procedure, first-period offers in treatments with effective discounting are in the range of 45–50 percent of the pie, again close to a 50–50 split. Furthermore, rejection rates are quite high, ranging from 0.28 to 0.36. Overall we find that the effective-discounting procedure performs similarly to the shrinking-pie procedure in our finite-time bargaining environment.

We then implement the bargaining-delay procedure in an online experiment. We use an online experiment rather than a lab experiment for this procedure because we believe it better addresses the issues of attrition and transaction costs between bargaining rounds. In the online experiment, we delay bargaining between rounds by periods of 1 week and vary the number of periods in a game. We find no treatment effect, and in line with the other treatments, first-period offers are in the 43–45 percent range. We also find high rejection rates, with results in the 0.29–0.36 range. That is, the bargaining-delay procedure seems to induce bargaining behavior similar to that of the shrinking-pie and the effective-discounting procedures.

2 Model

There are two players, $i\in \{1,2\}$ and two or three periods (bargaining rounds). In the first period, player 1 chooses an offer $x\in [0,1]$ where the total amount bargained over (the pie) is normalized to 1. Player 2 may accept or reject this offer. If player 2 accepts, then the game ends and players receive $(1-x,x)$ . If player 2 rejects, the game proceeds to the next round. In the next round, the pie is discounted according to each player's individual discount factor ${\delta }_i$ , and player 2 makes an offer $y\in [0,1]$ which player 1 may accept or reject. If player 1 accepts, the game ends and players receive $({\delta }_{1}y,{\delta }_2(1-y))$ . If player 1 rejects and there are only two periods, the game ends and players receive (0, 0). If player 1 rejects and there are three periods, the game moves to the last round where player 1 again makes an offer $z\in [0,1]$ . If the offer is accepted, then the game ends and players receive $({\delta }_1^2(1-z),{\delta }_2^{2}z),$ while if player 2 rejects, the game ends and players receive (0, 0).

We here give a brief intuition of the subgame perfect Nash equilibrium of the game. Subgame perfection implies that the optimal offer in a round makes the receiving player indifferent between accepting that round's offer or moving to the next period and getting her anticipated payoff net of discounting. In the last period, the game turns into an Ultimatum game, where the equilibrium offer is 0, which is accepted by the responder. The equilibrium offers are calculated as follows (see Table 1).

Table 1

Equilibrium offers

Offer	2-period game	3-period game
1st period	${\delta }_2$	$(1-{\delta }_1){\delta }_2$
2nd period	0	${\delta }_1$
3rd period	N/A	0

In the experiment, we focus on the following predictions of the model:

1. 1. Increasing the periods from two to three decreases the first-period offer;

2. 2. Increasing Player 2's discount factor increases the first-period offer;

3 Design and procedures

In our experiment, we implement an alternating-offers bargaining protocol to match the model presented above. The aim of our design is to examine whether different procedures of implementing discounting affect experimental results from the bargaining game. In this section, we lay out the design and procedures for the main treatments, where we employ the traditional shrinking-pie procedure and the new effective-discounting procedure. The design (and results) of treatments using the bargaining-delay procedure is deferred until Sect. 5.

We conduct six main treatments, S3A, S2A, S2S, E3A, E2A, and E2S. The first three treatments are pure replications of Ochs and Roth (Reference Ochs and Roth1989) (treatments given by Cell 2, Cell 3 and Cell 6) where we have followed their methodology as closely as possible. Treatments E3A, E2A and E2S follow the same procedure except we replace the shrinking-pie procedure with the effective-discounting procedure from Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023) and include a time-preference elicitation exercise. In the effective discount procedure, if agreement is not reached, then actual payments from the experiment are deferred to actual future dates.Footnote ⁵ Treatments are named after their method, shrinking-pie or effective-discounting, the number of periods, two or three, and whether discount factors are symmetric or asymmetric. Our treatments are summarized in Table 2.

Table 2

Main treatments

Treatment	Periods	Proposer $\delta$	Responder $\delta$	Equilibrium offer
S3A	3	.6	.4	16%
S2A	2	.6	.4	40%
S2S	2	.6	.6	60%
E3A	3	1 week	1 month	${\delta }_{\mathrm{month}}(1-{\delta }_{\mathrm{week}})$ %
E2A	2	1 week	1 month	${\delta }_{\mathrm{month}}$ %
E2S	2	1 week	1 week	${\delta }_{\mathrm{week}}$ %

Equilibrium offers are reported as a percentage of the pie

Because E3A, E2A and E2S all rely on the real value of money in 1 week or 1 month, we need to perform an elicitation task to calculate equilibrium predictions. We used the simplest and most common elicitation task, the multiple price list (MPL) (Cohen et al., Reference Cohen, Ericson, Laibson and White2020; Andersen et al., Reference Andersen, Harrison, Lau and Rutström2006; Freeman et al., Reference Freeman, Manzini, Mariotti and Mittone2016). Elicitations of time preferences can vary based on method and population, and elicitations over a week and a month will almost certainly be higher than the .6 and .4 used in our replication treatments (Cohen et al., Reference Cohen, Ericson, Laibson and White2020; Frederick et al., Reference Frederick, Loewenstein and O’donoghue2002; Matousek et al., Reference Matousek, Havranek and Irsova2022). However, as can be seen from Table 2, equilibrium predictions are that there should be a treatment effect of changing the number of periods or the payment delay of Player 2. One of the weaknesses of the effective-discounting procedure is that we lose precise control of the discount factors. We are able to create public knowledge of the relative delays, but not the discount factors implied by these delay periods. See Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023) for a discussion of this point. In contrast to Kim et al. (Reference Kim, Lim and Schweighofer-Kodritsch2023), we do not include random termination probability, because we have a finite number of periods in this setup. Additionally, we do not include front-end delay for payments in any treatments.

Sessions were conducted in the Research Lab at BI Norwegian Business School in Oslo and at the LEE lab at the University of Copenhagen in the period 4/10/19 to 22/11/21.Footnote ⁶ The study was preregistered on 25/2/20 after an initial pilot was run.Footnote ⁷ Subjects were recruited from the general student populations of BI Norwegian Business School and the University of Copenhagen, respectively. Each subject participated in 10 bargaining games and we collected 5 blocks of 10 subjects per treatment, meaning a total of 50 subjects and 250 bargaining games per treatment. Each session contained between 10 and 30 subjects and no subject participated in more than one session. A total of 300 subjects participated in a total of 1500 bargaining games played. We implemented randomized matching within blocks. At the conclusion of a session, a random game was selected for each individual and earnings in an experimental currency unit (ECU) were converted to NOK/DKK at a publicly announced exchange rate. On average, subjects in the Norwegian sessions earned 31.4 EUR while subjects in the Danish sessions earned 20.0 EUR.Footnote ⁸ z-Tree was used to program and conduct the experiment (Fischbacher, Reference Fischbacher2007).

As in Ochs and Roth (Reference Ochs and Roth1989), we implemented fixed initial roles. The size of the pie was fixed at 100 points per player, and the exchange rate of point to ECU varied in treatments with the shrinking-pie procedure. When implementing the effective-discounting procedure, the exchange rate was constant but the date of payment varied. See Supplementary Information (SI) section B for more procedural details.

5 Bargaining-delay procedure

There are of course many ways to approach the real-time costs of delayed bargaining in experiments. In this section, we explore a bargaining-delay procedure in which substantial time passes between bargaining rounds in the experiment. That is, in an online experiment, we delay bargaining between rounds by periods of one week in case of rejections.Footnote ¹² The aim is to test whether bargaining outcomes differ when introducing a real-time delay between bargaining rounds compared with bargaining immediately, but delaying payments into the future. This procedure focuses on prediction 1 of the model, that increasing the number of periods from two to three decreases the first period offer.Footnote ¹³

5.1 Design and procedures

Apart from the discounting procedure, the bargaining game in the online experiment is implemented as closely to the lab experiment as possible. We conducted two treatments, B3S and B2S, in the online experiment, summarized in Table 3.

Table 3

Additional treatments

Treatment	Periods	Proposer delay	Responder delay	Equilibrium offer
B3S	3	1 week	1 week	${\delta }_{\mathrm{week}}(1-{\delta }_{\mathrm{week}})\%$
B2S	2	1 week	1 week	${\delta }_{\mathrm{week}}\%$

Equilibrium offers are reported as a percentage of the pie

This online study had some notable changes from our main treatments. First, subjects were paid for each game they played, rather than randomly selecting a single game. This change was necessary because we cannot randomize across payments that occur at different times. In addition, all games were played concurrently, i.e., subjects played the first period of all games during session 1 and played all second (third) periods during session 2 (3). This is the only practical way to gather multiple observations for each subject. We also reduced the number of games from 10 to 5 for each subject. This reduction increases the incentive for returning for a single match and reduces potential attrition.

Block size remains constant with 10 subjects and fixed roles which means that each pair matched exactly once. We collected 5 blocks for each treatment. These treatments were programmed using oTree (Chen et al., Reference Chen, Schonger and Wickens2016). In the case of attrition, we automated responses based on the behavior of subjects in E2S and E3A, the most similar treatments. Treatments were conducted between 9/4/22 and 26/5/22 on Prolific. One hundred subjects participated in a total of 250 bargaining games. Each bargaining game had a total pie of $5 USD and the average payment per subject was $16.47 USD.Footnote ¹⁴ Each session took subjects approximately 6 min to complete and was implemented asynchronously. As is common practice in online experiments, we included three comprehension questions in the instructions, which subjects answered with a high degree of accuracy. Most matches were completed in the first round without automation (68.8 percent). Attrition was 42.4 percent and 52.8 percent in B3S and B2S, respectively.

Because B3S and B2S occurred over a series of weeks, we could not collect time elicitations on the day of the treatments without contaminating the incentives of subjects for the remaining rounds. Instead, we performed a follow-up study targeted directly at those who participated in B3S and B2S. This study had a 75 percent response rate and was used to calculate the equilibrium offers for these treatments. See SI section C for additional details regarding technical implementation, automation of responses in the case of attrition, timeline of experiments, comprehension tests and the elicitation follow up study.

5.2 Results for the bargaining-delay procedure

Our third main result is that the bargaining-delay procedure yields similar results as the other two procedures for implementing discounting. This result can be seen in Fig. 3, which shows average first-round offers and equilibrium predictions of the two treatments using the bargaining-delay procedure (B3S and B2S).

Fig. 3

Average first-period offers for B3S and B2S

We make several observations. First, as in both the shrinking-pie procedure and the effective-discounting procedure, we find no treatment effect when comparing first-period offers between the 2-period and 3-period games with the non-parametric test ( $p_n$ = 1.000), and this is supported by all three complimentary tests ( $p_p$ = 0.762; $p_{\mathrm{KS}}$ = 0.729; $p_{\mathrm{BD}}$ = 0.333). Second, first-period offers are 44.4 and 45.6 for B3S and B2S, respectively, which are also closer to a 50-50 split than they are to the model's predictions of 9.5 and 89.3, respectively. Third, rejection rates are in the range of.29–.36, which is similar to what was found in the effective-discounting procedure. The bargaining-delay procedure is also less efficient than the shrinking-pie procedure and is similar in efficiency to the effective-discounting procedure.