1 Introduction

Promises may foster trust and cooperation. A literature explores why. Charness and Dufwenberg (2006) (C&D) propose and report experimental support for an expectation-based explanation (EBE): A promise feeds a self-fulfilling circle of beliefs about beliefs. Promises are honored because if a person breaks his promise, then he would experience guilt for letting down the co-player’s expectation.Footnote 1 Therefore, the co-player trusts the promisor. Vanberg (2008) proposes an alternative commitment-based explanation (CBE): People like to keep their word.Footnote 2 To experimentally test CBE it is crucial to develop a design that exogenously varies whether a player sends a promise to another. Vanberg ran an experiment that achieved that by relying on an ingenious “partner-switching” feature. His results support CBE.Footnote 3

While the title of Vanberg’s paper includes the question, “Why do people keep their promises?” his approach to CBE is broader as he also refers to obligations “based on agreements or contracts” (p. 1467). His experiment reflects this too. Let us highlight two differences between C&D’s and Vanberg’s designs. First, C&D focus on a binary trust game, where two players move in sequence. Vanberg instead explores a symmetrized dictator game, where only one player is active along any play path and where players initially do not know their role (dictator or recipient). Second, C&D and Vanberg explore different communication protocols. C&D study a single pre-play message that cannot be responded to. Vanberg instead allows subjects to send messages back and forth. If they then reciprocate each other’s promises, their exchange may have the flavor of a conversation that generates an informal agreement.

A summary of Vanberg’s contributions reveals some interesting remaining unchartered research territory: First, Vanberg identified a potential confound to C&D’s result, namely CBE. Second, he also developed a design-tool – partner-switching – that allows testing for CBE. Third, he found support for CBE in a relevant context (with messages back-and-forth). However, he did not run a test of the relevance of CBE in C&D’s context (with unilateral messages). While he identified a potential confound to C&D’s result, he did not test its relevance in C&D’s setting. Since C&D’s study has garnered much interest, and given that the difference between messages back-and-forth and unilateral messages may be psychologically relevant, we propose that running such a test is of interest. In this paper, we report results from a design that accomplishes this.

We did not enter this research exercise with strong prior ideas as to how and why promises and informal agreements might trigger different forms of motivation and behavior. Nevertheless, interest in exploring related issues is enhanced by noting that several papers have documented that sometimes the nature of a communication protocol matters to behavior. For example, Brandts et al. (2019) survey how different communication structures have different impacts. In particular, they highlight dimensions that refer to messages’ order, direction, and frequency—all elements that differ in the communication protocols associated with the two designs described in Fig. 1. The classification of communication contents also discovers different channels through which communication affects choices, potentially driven by different rationales.Footnote 4 To give an example, Krupka et al. (2017), investigating informal agreements, emphasize the social norm channel for their efficacy.Footnote 5 Relatedly, it may seem reasonable that communication that goes back and forth through multiple messages could create personal bonds and support the maintenance of agreements through moral commitment mechanisms, as effectively supported by Vanberg’s results. However, the possibility of creating two-way links is nil with unilateral messages, suggesting that the psychology of promise-keeping mechanisms may be different in such context.

Fig. 1
figure 1

The game trees: C&D’s (to the left) and Vanberg’s (to the right)

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Besides evaluating CBE in C&D’s trust game setting with unilateral promises, we also report additional results regarding EBE. Vanberg’s switching feature, in fact, allows a relevant test, although there are limits to which extent this can be done (as noted by Ederer & Stremitzer, 2017 and Di Bartolomeo et al., 2019b.) We postpone a discussion of details until Sect. 2.

The rest of the paper is organized as follows. Section 2 provides in-depth scientific background: hypotheses, designs (C&D’s, Vanberg’s, ours, and also that of Di Bartolomeo et al.,’s 2019b, which helps add perspective), and other related literature. Section 3 describes our procedures. Section 4 explains what we found. Section 5 concludes.

2 Scientific background

We first recall what C&D and Vanberg did, then explain what we added. Figure 1 depicts C&D’s game (form) to the left and Vanberg’s to the right. Note how they differ, as indicated above.

2.1 EBE

C&D explored experimental treatments with and without pre-play communication. In one treatment, B could send a single pre-play message to A. Suppose that B experiences guilt if he chooses Don’t, and that the guilt increases the more strongly B believes that A believes B will choose Roll.Footnote 6 A promise from B to A may then feed a self-fulfilling circle of beliefs about beliefs that B will Roll, and, therefore, A will choose In. C&D articulated this idea – aka EBE – tested it, and they found support.

2.2 CBE

Vanberg points out that C&D’s story is confounded. Suppose B has an innate preference for keeping his word: If B promises to Roll then he will prefer not to renege. If A anticipates this, he will choose In. This idea – aka CBE – generates the exact prediction as EBE.

2.3 Partner-switching

Vanberg came up with a clever experimental device to test the empirical relevance of CBE, enabling him to draw robust causal inferences regarding the impact of a promise. Namely, he proposed that if subjects i and j formed a chatting pair and then i was chosen to be the dictator, then with 50% probability, j would be “switched” and replaced by another subject k who previously chatted with subject l. Moreover, if there were a switch, i, but not k, would be informed of this. For cases where l sent a similar message to k as i sent to j (note: i could read l’s message to k) EBE suggests that i would behave the same way with or without a switch. CBE, by contrast, implies that i will fulfill the promise if and only if there were no switch.Footnote 7

2.4 Vanberg’s results

Vanberg did not base his test of CBE on C&D’s game but on the game to the right in Fig. 1. That is, if a subject i were selected to be the dictator (by the initial chance move), then subject j, with whom i had initially communicated, would be switched to another subject k who had initially communicated with a fourth subject l. Moreover, instead of using C&D’s single-shot messages from one player to the other, Vanberg allowed the two players to engage in four rounds of back-and-forth messaging. Based on this design, Vanberg reported support for CBE.

Vanberg, furthermore, elicited subjects’ (first- and second-order) beliefs that a dictator would Roll, and he documented that these beliefs increased when subjects that received a promise were involved. Given this data pattern, Vanberg’s design admits the following clean test of EBE: Consider i, j, k, l as described in the previous paragraph, and suppose that i made a promise to Roll to j and is switched. EBE implies that i should be more likely to Roll if l made a promise to Roll to k than if l did not make such a promise. However, Vanberg did not find support for this prediction.

2.5 Unchartered territory

It is natural to wonder whether Vanberg’s key results – support for CBE, not support for EBE – would also be obtained in C&D’s setting. The back-and-forth nature of Vanberg’s communication protocol may generate experiences that look like informal agreements and have a different flavor than one-sided promises. While informal agreements and one-sided promises both evoke issues of keeping-one’s-word, the modes of exchange are conceptually distinct.They may relate differently to how and why trust and cooperation may be induced.

2.6 Our design

We apply Vanberg’s partner-switching feature to C&D’s original game, thus providing new independent tests of CBE and EBE in C&D’s setting. A subject in B’s position can send a single written free-form message to a subject in A’s position. Subsequently, there was a 50% probability that the A-subject would be switched and replaced by another subject (also in the position of A) who previously received a message from yet another subject l (in the position of player B). If there was a switch, only the B-subject was informed.

2.7 Di Bartolomeo et al. (2019b) (DDPP)

To clarify our contribution further, it is helpful to compare it with our previous study DDPP (the extra P is Francesco Passarelli’s). DDPP tweak Vanberg’s design so that the switching probability is not 50% but rather (depending on treatment) 25% or 75%. This induces exogenous variation in whether a player who sent a promise was paired with whoever received that promise (as needed for Vanberg’s test of CBE) and in players’ beliefs.Footnote 8 This new feature allows DDPP to run several new tests of EBE and guilt aversion and CBE. Their main results align with Vanberg’s, supporting CBE but not EBE. Our current design differs in two respects from DDPP’s. First, we work with C&D’s game rather than Vanberg’s. Second, we use Vanberg’s switching probability rather than those of DDPP. In other words, we conduct tests a la Vanberg in C&D’s game, whereas DDPP modify Vanberg’s tests while staying close to his game.

2.8 Less closely related literature

So far, in this section, we referenced the studies that most closely relate to (and directly motivate) us. All this work is part of broader experimental literature on communication in various trust games. We offer brief comments to give readers a richer context and backdrop.

Ellingsen and Johannesson (2004) are pioneers in studies that considered preferences for promise-keeping in trust games. Much of their focus, unlike ours, is on theories that refer to fair distributions of material rewards. They find that promises to behave reasonably mitigate the hold-up problem as investors rely on their trading partner’s promises.

Ederer and Stremitzer (2017) propose a novel design that includes an “unreliable random device,” creating exogenous variation in players’ expectations and allowing clean tests of EBE. Their exercise may be seen as parallel to Vanberg’s, except that the primary focus is on EBE rather than CBE. Their paper relates more closely to DDPP (as discussed above) than our current work. They provide evidence that promisors’ aversion to disappointing others’ expectations leads them to behave more generously.

Ismayilov and Potters (2016, 2017) stress the endogeneity nature of promises in exploring alternative motivations for promise-keeping. They explore “internal consistency” (desired by the party issuing a promise) and “social obligation” (felt because someone received one’s promise). Investigating causality, in the end, they argue that promises do not induce trustworthiness. Instead, cooperators are more likely to send promises than non-cooperators.

As mentioned also above, differences in communication aspects matter (Brandts et al., 2019). In a context like ours, Charness and Dufwenberg (2010) and Di Bartolomeo et al. (2019a) allowed Bs to send or not send a prefabricated bare promise-to-Roll to A. Charness and Dufwenberg (2010) found that these promises did not affect beliefs and Roll rates. Instead, Di Bartolomeo et al. (2019a) focused on silent agents, finding that passive communication (silence) matters as it signals and somehow justifies selfish behavior.

Several recent studies have found that informal agreements may significantly affect subsequent play. For example, Krupka et al. (2017), who elicit social norms in a context with or without them, estimate that honoring an informal agreement in the double dictator game is worth giving up approximately 10% of total earnings. They also compare social norms to guilt aversion and lying aversion. And report that informal agreements affect behavior through their direct effect on social norms as well as through an indirect effect on beliefs. For more work on formal agreements, see Miettinen (2013) and Dufwenberg et al. (2017) for theory and Kessler and Leider (2012), Dufwenberg et al. (2017), and Di Bartolomeo et al. (2023) for experiments.

3 Procedures

Our experiment was conducted at the CIMEO Lab of Sapienza University of Rome in May 2019. It involved 226 undergraduate students (8 sessions) recruited using an online recruitment system.Footnote 9 Upon arrival at the lab, subjects were randomly assigned to isolated computer terminals. Three assistants handed out instructions and checked that participants correctly followed the procedures. Before playing any game, subjects completed a short questionnaire testing their comprehension.

Each session consisted of 10 rounds, with perfect stranger matching. At the end of each session, one of the rounds was randomly chosen for payment. All subjects received a fixed show-up fee of 2.50 tokens, where 1 token = 0.5 euro.

Each round implemented the following sequence of six stages.Footnote 10

  1. 1.

    Role assignment: Player positions B and A are randomly assigned, and pairs are formed.

  2. 2.

    Communication: B can send a free-form message to A (≤ 90 characters).

  3. 3.

    A’s action: A reads B’s message, and then A has to choose In or Out.

  4. 4.

    Switching: Some As were switched with a 50% probability. Only Bs were informed whether or not a switch occurred. Bs with switched As were allowed to read the message previously received by the new A’s pre-switch B.Footnote 11

  5. 5.

    Belief elicitation: This stage has two parts: (a) First-order beliefs: each A was asked to guess if his/her unknown B would choose to Roll or Don’t; (b) Second-order beliefs: each B was asked to guess the guess of the A with whom they would play after the switching stage occurred.Footnote 12

  6. 6.

    B’s action: B chooses between Roll or Don’t. Then all subjects are informed about their payoff in that round. As are neither informed whether they had been switched nor about B’s choice; only payoffs are revealed.Footnote 13

Eliciting first- and second-order beliefs is common in the experimental literature on guilt aversion. Doing so here allows us to compare our findings regarding beliefs to C&D’s. Incentives for beliefs elicitation were provided for all rounds except the one chosen for payment, implying that subjects had no incentive to hedge against bad outcomes and thus to misreport their beliefs.Footnote 14

4 Results

We report our main findings related to CBE and EBE in Sect. 4.2. However, for the relevance of those tests, it is first critical to document that, in our design, promises on balance have the effect of raising subjects’ expectations and Roll rates. We establish that in Sect. 4.1.

4.1 Promises, second-order beliefs, and roll rates

Our sample consists of 1130 pairs of subjects and 1130 messages from B to A. We code these messages according to whether or not they conveyed a promise to Roll (or a similar-in-spirit clear statement of intent to Roll).Footnote 15 This way, we obtained 527 promises out of 1130 messages (47%). The proportion of As who choose In is 61% (691 out of 1130). The proportion of As who choose In after receiving a promise is 76% (398 out of 527). The proportion of As who chose In when Bs did not promise is 49% (293 out of 603). As expected, the proportion of As who choose In after receiving a promise is significantly higher than that of As who do not receive a promise (76% vs. 49%: Z = 2.52, p = 0.012).

Let us now focus on second-order beliefs.Footnote 16 As said before, we observe that As chooses In in 691 cases, so we also have 691 Bs who choose between Roll and Don’t. However, due to the partner-switching feature, these Bs are not necessarily those who sent a message to the As choosing In. Table 1 reports their second-order beliefs (in bold), the Bs’ beliefs about the probability that an A subject believed B would roll the die, and reports standard deviations and the number of observations (in brackets). The first two columns (a and b) refer to the non-switched cases, while the second and third columns (c and d) refer to Bs whose partner was switched. Odd columns (a and c) refer to the case where Bs read a promise, while even ones (b and d) refer to the case where Bs did not. Rows indicate whether Bs made a promise (1) or not (2).

Table 1 Second-order beliefs of B’s
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Looking at Bs with non-switched partners, as in C&D, we find that the second-order beliefs of Bs who made a promise are significantly different from those of Bs who did not send a promise (70% vs. 55%: Z = 2.38, p = 0.017).Footnote 17 It shows the correlation between promises and second-order beliefs, also found by C&D. Among the Bs who made a promise, the average second-order belief of those who read a promise is independent of the switch (70% vs. 67%: Z = 0.00 and p = 1.000), i.e., second-order beliefs of Bs with non-switched partners who made a promise are not significantly different from those of other Bs who made a promise and were re-matched with As who received a promise by someone else. Therefore, like Vanberg, we obtain exogenous variation in promises.

The second-order beliefs of switched promisors who are re-matched with an A who received a promise are higher than those of switched promisors who are re-matched with an A who did not receive a promise by someone else (67% vs. 54%: Z = 2.10, p = 0.036). Similarly, the second-order beliefs of Bs with non-switched partners who made a promise are higher than those of Bs who made a promise and are re-matched with an A who did not receive a promise by someone else (70% vs. 54%: Z = 2.52, p = 0.012).

The second-order beliefs of Bs who did not send a promise (all those in row (2)) are not statistically different.Footnote 18 This is suggestive of heterogeneity between agents who self-select to send promises and those who do not, as reported by Ismayilov and Potters (2016).

4.2 Main results: CBE & EBE

Table 2 reports the average Roll rates of Bs (in bold), standard deviations, and the number of observations (between brackets). The structure is otherwise like that of Table 1. We distinguish the average Roll rates of Bs who promise and Bs who do not promise, by rows. Columns refer to the message they read and indicate whether a switch occurred.

Table 2 B’s Roll rates
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Focusing on columns (a) and (b) of Tables 1 and 2, we observe a correlation between promise-keeping and second-order beliefs, as in C&D. Second-order beliefs of promisors are higher in Table 1 (70% vs. 55%: Z = 2.38 and p = 0.017) as are average Roll rates (74% vs. 29%: Z = 2.52, p = 0.012) in Table 2. Our results show that people are likelier to keep promises, which correlates with high second-order beliefs. However, as argued by Vanberg, correlation does not necessarily imply causation. We need to further investigate the issue by using our exogenous variation in promises.

Among Bs who made a promise, the average Roll rate of Bs with non-switched partners is not statistically different from that of Bs who read a promise made by someone else (74% vs. 70%: Z = 0.14, p = 0.889). Thus, we do not find support for CBE. The behavior of Bs with non-switched partners who keep their own promises is not different from that of Bs who keep promises done by another.

Our result here differs from that of Vanberg, who found support for CBE when he ran an analogous test. Instead, the result is consistent with the idea that people keep promises made by others since those are associated with higher second-order beliefs, as predicted by EBE.

Two direct tests for EBE are obtained by comparing promisor Bs (row (1) and column (a)) who read a promise whether they are switched (row (1) and column (c).) with those who did not (row (1) and column (d)). Remember that they have different second-order beliefs (columns (a) and (c) vs. (d).) Looking at Table 2, the average Roll rate of Bs with non-switched partners and that of other Bs who read a promise made by someone else are both higher than that of Bs with switched partners who did not read a promise made by someone else (74% vs. 59%: Z = 2.52, p = 0.012; 70% vs. 59%: Z = 2.10, p = 0.036). This finding supports EBE. Again, our results differ from Vanberg’s; he found no statistically significant difference and no support for EBE.

5 Conclusions

Humans are social animals who communicate. There is interest in exploring the psychological mechanisms that matter for different forms of communication. We distinguish between unilateral and bilateral messages. The former variety may produce a unilateral promise, while the latter variety may generate an informal agreement. The causal impact of promises or agreements may then be studied using Vanberg’s (2008) partner-switching technique, the central idea being that a switch anullates either form of covenant.

C&D proposed a theory – EBE – that may explain why promises foster trust and cooperation. Vanberg pointed out that C&D’s results are confounded: an alternative explanation – CBE – is conceivable. Using his partner-switching technique, he reported support for CBE and a lack of support for EBE. As Vanberg deviated from C&D’s design as regards the choice of the game and communication protocol, it is natural to wonder if his main result also extends to a setting that uses C&Ds game and communication protocol. This is what we have explored. Our results are the opposite of Vanberg’s, supporting EBE and not supporting CBE.

The findings can be linked to the distinction between promises and informal agreements. C&D’s game is asymmetric. At the root, both players know that player A has to trust player B to Roll, not the other way around. By contrast, Vanberg’s game is symmetric. At the root, both players know that either may have to trust the other to Roll. Moreover, the communication protocols differ, with a one-sided message from B to A in C&D’s case and a conversation-like exchange in Vanberg’s case. The symmetry of Vanberg’s game, and the back-and-forth nature of his communication protocol, invites the reflection that players may be inclined and able to strike a deal of conditional cooperation: “I’ll promise to Roll if you promise to Roll.” And if both players do promise to Roll, their exchange has the semblance of an informal agreement. Vanberg’s results are consistent with and supportive of the idea that players have a belief-independent preference to honor such agreements.

As regards CBE, there is no tension between our results and Vanberg’s. If his study is interpreted as documenting evidence for a preference for honoring informal agreements, then this has no counterpart in our (or C&D’s) design. A preference for keeping a unilateral promise may be a somewhat different animal than a preference for honoring a gentleman’s agreement. Different forms of CBE are considered by Vanberg and by us.

Regarding EBE, it may seem puzzling that this theory is supported in C&D’s setting but not Vanberg’s. Data is what it is though, and, after observing the results, we have the following reflections: Different games may trigger different thinking. Humans are motivated in many ways.Footnote 19 However, perhaps humans cannot consider more than a few such motivations at a time, and perhaps Vanberg’s setting, relative to ours (and C&D’s), triggers other motivations that may crowd out the belief-dependent feelings built into EBE? To mention two such potential motivations, consider the preference for honoring an informal agreement described above. Second, consider reciprocity, such that a player would wish to choose Don’t [Roll] if and only if he or she believed that the other player would have done likewise had he or she been designated to choose whether or not to Roll. In Vanberg’s game, this motivation would be potent (since there is a node where the other player may choose Roll), whereas, in C&D’s game, it would be muted (since the other player has no Roll choice).

We hope that future research may take inspiration from these speculative remarks and develop new designs that may be useful for testing the empirical relevance of our ideas.