Reciprocity models revisited: intention factors and reference values

Hinz, Janna; Nicklisch, Andreas; Sommer, Mey-Ling Article — Published Version Reciprocity models revisited: intention factors and reference values International Journal of Game Theory Provided in Cooperation with: Springer Nature Suggested Citation: Hinz, Janna; Nicklisch, Andreas; Sommer, Mey-Ling (2024) : Reciprocity models revisited: intention factors and reference values, International Journal of Game Theory, ISSN 1432-1270, Springer, Berlin, Heidelberg, Vol. 53, Iss. 2, pp. 299-324, https://doi.org/10.1007/s00182-024-00898-z This Version is available at: https://hdl.handle.net/10419/314965 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) International Journal of Game Theory (2024) 53:299–324 https://doi.org/10.1007/s00182-024-00898-z 1 3 ORIGINAL PAPER Reciprocity models revisited: intention factors andreference values JannaHinz1· AndreasNicklisch2 · Mey‑LingSommer3 Accepted: 14 April 2024 / Published online: 21 May 2024 © The Author(s) 2024 Abstract We present a test of the two most established reciprocity models, an intention factor model and a reference value model. We test characteristic elements of each model in a series of twelve mini-ultimatum games. Results from online experiments show major differences between actual behavior and predictions of both models: the distance of actual offers to the proposed reference value provides a poor measure for the kindness of offers, while a comparison of offers with extreme offers as suggested by the intention factor model makes offers indiscriminable in richer settings. We discuss possible combinations of both models better describing our observations. Keywords Experiments· Intentions· Mini-ultimatum game· Reference value· Reciprocity models JEL Classification C52· C72· C91 * Andreas Nicklisch [email protected] 1 Department ofEconomics, University ofHamburg, von-Melle-Park 5, 20146Hamburg, Germany 2 Center forEconomic Policy Research, University ofApplied Sciences oftheGrisons, andResearch Unit “Needs based Justice andDistribution Procedures”, Comercialstr. 20, 7000Chur, Switzerland 3 Faculty ofEconomics und Social Sciences, Helmut Schmidt University, Holstenhofweg 85, 22043Hamburg, Germany 300 J.Hinz et al. 1 3 1 Introduction Reciprocity is one fundamental cornerstone of human behavior, and an integral element for other-regarding preferences. The importance of reciprocal behavior for human interactions has been stressed by a large body of economic literature (e.g., Cox and Deck 2003, Andreoni etal. (2003); Falk etal. (2003); Gächter and Thöni (2007)).1 Consequently, behavioral economists have been striving to explain how people digress from self-interested behavior to reward kind actions and punish unkind actions of their opponents. Modeling reciprocity, however, has turned out to be a very complex endeavor. The specific formulation of reciprocal preferences follows predominantly two distinct ways: Rabin (1993) and Dufwenberg and Kirchsteiger (2004) focus on an comparison between a “suggested” payoff and a reference value. In turn, Falk and Fischbacher (2006) rely on a combination of the inequality of the “suggested” payoffs and an intention factor measuring how deliberate the “suggestion” is to capture underlying motivations. Despite the ambiguity of both approaches, their scientific impact is enormous2 leading subsequent studies to rely on one or the other theory (e.g., Stanca etal. (2009), Ambrus and Pathak 2012). In this current study we test key characteristic features of both approaches in a number of mini-ultimatum games similar to the one used in the tradition of Bolton and Zwick (1995) and Falk etal. (2003). Particularly, we focus on two key differences between the two approaches: firstly, the reference value approach measures the extent of (un)kindness according to the distance of the specific offer to the reference value, while the intention factor approach measures the unkindness by the inequity of the specific offer. Secondly, although both models take extreme outcomes into considerations for assessing (un)kindness, the reference value appears more robust against outliers: outliers along other alternative outcomes of the game are converted into a reference value, whereas the intention factor approach measures the intention by a pairwise comparison between an extreme outcome and the specific offer. We show that both approaches have advantages and disadvantages when explaining actual decisions so that a combination of both approaches seems to provide a good description of behavior. The general idea of reciprocal preferences is perhaps best summarized by the Latin principle ‘quid pro quo.’ The overarching non-parametric model by Cox etal. (2008) formalizes these words in the following way: suppose a proposer (player P) has a number of alternatives from which she can choose one. Her choice has consequences in terms of payoffs not only for herself, but also for the responder (player R). Among the alternatives player P can choose from, player R considers P’s choice blue to be more generous than red if blue yields a higher payoff to R than the choice of red, while P’s gain from choosing blue and not red is at most as large as R’s gain from choosing blue and not red. R may or may not accept the proposed alternative. 1 Alternative approaches such as distributional concerns (e.g., Fehr and Schmidt (1999)) or guilt-aversion (e.g., Battigalli and Dufwenberg (2007)) have been shown to explain pro-social behavior partly, but not comprehensively (e.g., Andreoni etal. (2003); Nicklisch and Wolff (2012)). 2 In February 2024, the paper by Falk & Fischbacher is cited more than 3920 times, while Dufwenberg & Kirchsteiger is cited more than 2690 times according to Google Scholar. 301 1 3 Reciprocity models revisited: intention factors andreference… With R holding reciprocal preferences, the likelihood of R’s approval for a suggested alternative increases, the more generous P’s choice is: R gains immaterial utility from P’s material payoffs if P behaves generously, whereas R’s immaterial utility can even be negative, if P chooses a mean alternative. Consequently, R may want to punish P if P behaved unkindly, for instance by destroying all payoffs altogether. Contemporary reciprocity concepts translate the general idea of quid pro quo into well-defined closed preference models. We refer to the first approach as the “reference value model,” first formalized by Rabin (1993).3 In his belief-dependent model, reciprocity is analyzed for two-player, normal-form games. Dufwenberg and Kirchsteiger (2004; herafter D &K) extended Rabin’s model of belief-dependent preferences to extensive n-player games. In both models, R ranks P’s alternatives from the one yielding the lowest payoff for R to the highest payoff for R. Half way between R’s lowest and highest payoff lies R’s equitable payoff dividing P’s alternatives into unkind ones below the equitable payoff and kind ones above (hereafter, we denote the equitable payoff as the reference value). P’s (un)kindness towards R increases in the difference between the payoff corresponding to P’s choice and the reference value. We would like to stress that the reference value model measures the action’s kindness by a “global assessment.” That is, the midpoint of the entire set of alternatives in the game determines the reference value which, in turn, determines the kindness of a specific offer. We test the predictive success of the reference value model according to two characteristics. Firstly, we check for the continuity of the reference value: we analyze whether the likelihood that R accepts an alternative increases in the distance between the equitable payoff and the payoff “normally resulting” from P’s chosen action.4 Secondly, we test for the predictive success of the reference value: varying the game, but keeping the reference value and the distance from the reference value constant, we analyze whether the likelihood that R responds unkindly (i.e., she does not accept) remains constant.5 The second class of reciprocity models, “intention factor models,” contrasts the reference value models in two ways: they decompose P’s (un)kindness towards R into the product of an intention term and an outcome term (e.g., Falk and Fischbacher (2006), hereafter F &F). The first term determines whether R perceives P’s action as intended or not, the second term determines the severeness of P’s perceived (un)kindness. F &F place in R’s immaterial partial utility from reciprocity prior importance on the difference of payoffs between P and R within an option before the difference to the other possible payoffs for R is considered. As mentioned earlier, the intention factor model assesses the action’s kindness in a pairwise comparison with extreme outcomes of the game. This may yield a problem for this approach: one kind alternative can turn all other alternatives inevitable into fully intentional unkind 3 Other models incorporating reciprocity follow the same logic, but apply slightly different techniques (e.g., Cox etal. (2007)). 4 We clarify the meaning of “normally” below. 5 Notice that we do not test the complete approach by D &K. We dismiss some equilibria based on mutual meanness as not plausible in our setting. For further details see Section2. 302 J.Hinz et al. 1 3 alternatives. We will show the consequences of this feature for predictions in games with several alternatives in Section3. The crucial importance of intentions for reciprocal behavior has been shown elsewhere (e.g. Falk etal. (2008)). We test the predictive success of intentions by checking for the continuity of the intention factor: we analyze whether the likelihood that R responds unkindly increases, if the intention factor increases, keeping the inequity of P and R’s payoffs constant. Secondly, we test for consistency of the intention factor: varying the game, but keeping the intention factor and the inequity of payoffs constant, we analyze whether the likelihood that R responds unkindly remains constant. In other words, we test whether an assessment of kindness relying on the pairwise comparison describes P’s perceived kindness properly. For our purpose, we propose a series of twelve mini-ultimatum games. Some of them offer P two alternatives to choose from, some of them offer four alternatives. All of them allow R to reject a proposed alternative and forgo her own income for the sake of punishing P. The games are designed such that they allow us to assess the predictive success of reference value models and intention factor models. We retrieve our data in online experiments with almost 500 participants. As such, our analysis follows Sobel (2005) criticism that existing reciprocity models seem to be fitting for specific situations, but lack a clear characterization of this very situation. Along the same line of arguments, there are some other studies discussing and testing the predictive success of reciprocity models. Firstly, they provide evidence on the importance of intentions for reciprocation: if there is no alternative but to behave unkindly, subjects reciprocate less severely (Falk etal. 2003); the same holds true if an action is taken that is not unambiguously kind, but selfish to some degree (Stanca etal. 2009). Secondly, Dhaene and Bouckaert (2010) elicit first and second order beliefs of participants in a sequential prisoners’ dilemma and a mini-ultimatum game. They show that beliefs and behavior, particularly of second movers, are very consistent with D &K’s reciprocity model. Furthermore, Pelligra (2011) varies systematically the outside options in a trust game, where the first mover’s trusting is either kind or unkind for the second mover. Contrasting the theoretical predictions of the D &K model, the trustworthiness of the second mover remains constant across treatment conditions suggesting that other motives dominate behavior in this setting (cf., Pelligra (2011)). Finally, Nicklisch and Wolff (2012) test an overall characteristic of reference value models and intention factor models: if punishment is sufficiently cheap, reciprocation is modeled as an “all-or-nothing” decision. That is, if P behaves kindly (unkindly), R maximizes her utility by choosing the most kind (unkind) response possible. By means of a modified ultimatum game, the authors show that decisions for a majority of participants in a laboratory experiment do not follow this assumption. Along this partly pessimistic assessment of contemporary reciprocity models, our data shows important shortcomings for both models when predicting behavior. Particularly, the continuity of the reference value model fails to characterize actual behavior: increasing the distance between the reference value and the payoff of the actual offer does not necessarily correspond with increasing rejection rates. Moreover, variation of the game yields differences in the rejection rates although the reference value and the distance from the reference value remain constant. We conclude 303 1 3 Reciprocity models revisited: intention factors andreference… from those findings that the distance to the reference value serves as a poor descriptor for the extent of (un)kindness. On the other hand, experimental results for simple games with two alternatives are nicely predicted by the intention factor model. The likelihood of rejection increases for increasing intention factors. However, there is little consistency between predictions and decisions in the richer games with four alternatives. We conclude from this that the pairwise comparison of alternatives does not characterize behavior adequately, and suggest a combination of both approaches. This combination includes a global assessment for the intention of a choice and the inequity of an alternative for the extent of (un)kindness. The remainder of this article is organized as follows: The following section (re) acquaints with both reciprocity models to be tested with an emphasis on the element we scrutinize. In Section 3 we introduce our experimental design and procedure. Section4 presents results. In Section5 we discuss our findings and suggest potential developments for reciprocity models reflecting our results. Section6 concludes. 2 Reciprocity based utility Let us consider a two-player extensive form game with two stages and common knowledge about the game and decisions in previous stages.6 Formally, suppose player P and player R be proposer and responder in an ultimatum-like game. P chooses an action ap from her set of alternatives Ap . Suppose ap affects P’s and R’s payoffs ( 𝜋p and 𝜋r , respectively). R observes P’s action. Then i’s reciprocity based utility function with i,j∈{P,R} has the following structure: Utility consists of a material payoff 𝜋i and an immaterial utility component. The material part of Ui refers to the payoff assigned to the outcome of a specific choice. The immaterial part of utility is initiated with an individual sensitivity parameter to reciprocal concerns, 𝜓i . If 𝜓i=0 , then utility will equal material payoff as suggested by narrow self-interest. 𝜅ji(.) denotes i’s perceived (un)kindness of j’s action, and 𝜆ij(.) is the (un)kindness of i’s response. The latter two depend on i’s (first-order) expectations concerning j’s behavior in the consecutive game ( b′ ij ), and i’s (secondorder) expectations concerning j’s beliefs of i’s beliefs of the consecutive game ( b′′ iji ). Depending on the firstand second-order beliefs of players, reciprocity models can accommodate generally a very large range of behavior. For instance, a very reciprocal responder may expect to receive a very low offer, perceiving a large offer to be mean. While she assumes that the proposer expects her to accept the offer, she may decide to reject the large offer yielding substantial immaterial utility resulting from the mean rejection of a mean offer. In turn, a very reciprocal proposer7 may (1) U i=𝜋i(ai,aj)+𝜓i𝜅ji(aj,b � ij ,b �� iji )𝜆ij(ai,b � ij ,b �� iji) 6 Of course, both approaches cover much richer settings. Here, we restrict our deliberations to the experimental setting for the notational simplicity. 7 For this argumentation, “very” refers to cases in which the immaterial utility component outweighs the material utility component for both players. 304 J.Hinz et al. 1 3 consider the rejection of offers to be mean. Anticipating that responders consider large offers as mean, the proposer casts a large (mean) offer facing the (anticipated) mean rejection. Consequently, both proposer and responder yield higher utility by a rejection of a large offer than through the acceptance of a small offer.8 Whereas reciprocity allows generally for “masochistic” equilibria that maximize the immaterial utility component of mutual mean actions (see our last footnote), we rule those equilibria out as non-plausible in our dictator games. That is, we assume that proposers do not submit offers for which they seek a rejection in order to maximize their immaterial utility.9 Notice that D&K’s approach allows in principle for those equilibria based on mutual meanness. Hence, we do not test D &K’s full model but dismiss some of its equilibria on plausibility grounds. Thus, we assume in the following that if a proposer casts an offer, she expects its acceptance, and she expects the responder to believe that she aims at the acceptance of the offer. Formally, b� PR = “acceptance” and b�� PRP = “ b� RP = acceptance ”, while b� RP = “acceptance” and b�� RPR = “ b� PR = acceptance ”. As a consequence, the terms for i’s perceived (un)kindness of j’s action and the (un)kindness of i’s response simplify to 𝜅ji(aj) and 𝜆ij(ai) , since proposer’s and responder’s first order and second order beliefs assume the acceptance of the offer. That is, “the normally resulting way” of the ultimatum game assumes the acceptance of the proposed offer. Whether actual behavior aR deviates from the expected way, is in both models a matter of the immaterial partial utility components from reciprocity. In the next sections, we briefly present the key components of both reference value models and intention factor models. Both models provide a solution concept that require an updating of players’ beliefs as the play unfolds.10 We will pay special attention to the differences in the updating between both approaches, and formulate hypotheses based on the theoretical analysis. 2.1 Sequential reciprocity according toDufwenberg & Kirchsteiger With respect to the reference value model by Dufwenberg & Kirchsteiger, R’s immaterial partial utility from reciprocity equals the product of P’s (un)kindness towards R, 𝜅PR(aP) , and R’s (un)kindness towards P by deviating from “the normally resulting way”, 𝜆RP(aR) . 8 Notice that the described behavior creates a logical twist for reciprocal preferences. It appears questionable whether a mutual mean action allowing players to maximize the reciprocal utility is not in fact nice: it creates a higher utility than the one resulting from the acceptance of a small offer. If so, it turns the starting point of this thought experiment – rejecting offers is mean – absurd. 9 If the proposer seeks rejections, this response may not be considered to be mean any longer, while accepting a mean offer creates an equally bad option for the responder. We guess that the coordination on such an unusual equilibrium appears very difficult in an anonymous experimental setting and may be desirable only for a very small number of players. Similar difficulties with distributional fairness preferences (e.g., inequality aversion) are excluded by the assumption that players do not throw away money for the sake of a reduction of income equalities (see, e.g., Fehr and Schmidt (1999),p. 824). 10 In their reference value model, Dufwenberg and Kirchsteiger (2004) refer to it as a “sequential reciprocity equilibrium.” 305 1 3 Reciprocity models revisited: intention factors andreference… R’s reference value separating P’s actions into kind and unkind actions is the value half way between the lowest and highest payoff available at the time when P makes her decision. Formally, R’s equitable payoff 𝜋 eP R (i.e., R’s average payoff following P’s action) is: where 𝜋′ R is the set of payoffs induced by P’s efficient strategies. In turn, not considered are P’s inefficient strategies, that is, strategies for which one finds a Pareto improvement – in terms of R’s and P’s payoffs – among P’s strategies for any strategy choice of R (compare D &K, pp. 275–276). P’s (un)kindness towards R in D &K’s approach is evaluated according to the difference between the payoff resulting from accepting P’s offer and the reference value: After R observes P’s move, it is on her to respond, again influencing the monetary outcomes for both players. That is, ranking R’s efficient alternatives from the one yielding the lowest to the highest payoff for P, the midpoint of the ranking determines R’s reference value for the (un)kindness of her response. In other words, R’s (un)kindness towards P is measured according to the distance between P’s actual payoff to P’s equitable payoff. It is important to stress that P’s equitable payoff 𝜋 eR P in the ultimatum game is the payoff resulting from accepting the offer: recall that P’s equitable payoff averages the maximum and the minimum in the set of payoffs induced by R’s efficient strategies. Since rejections are inefficient, R’s only efficient strategy equals the acceptance of the offer. Therefore, it follows: D &K’s explicit quantification of (un)kindness with the equitable payoff as a reference value allows us to test the predictive success of their model: Hyp D&K : Keeping the offer constant, but decreasing the distance to the equitable payoff across the proposer’s actions implies non-increasing rejection rates for those actions, while keeping the distance constant implies a constant rejection rate. 2.2 Intention‑based reciprocity according toFalk & Fischbacher Within the context of our simple mini-ultimatum games, one can show for Falk & Fischbacher’s approach that 𝜆F&F RP (a R )=𝜆 D&K RP (a R) : according to F &F, the kindness of R’s reciprocation is measured with respect to the degree by which R alters P’s actual from his expected payoff. Since R expects P to propose her an offer for which P seeks R’s acceptance in our mini-ultimatum games, R alters P’s payoff by rejecting the offer implying a negative reciprocation. In turn, the difference between the two approaches is settled in the specific form of 𝜅F&F PR (a P) . F &F separate the perceived (un)kindness into two terms, the outcome (2) 𝜋 e P R =0.5 max(𝜋 � R )+0.5 min(𝜋 � R ) , (3) 𝜅D&K PR (a P )=𝜋 R (a P )−𝜋 eP R (4) 𝜆D&K RP (a R )=𝜋 P (a R )−𝜋 P (acceptance ) 306 J.Hinz et al. 1 3 term ΔP(aP) and the intention factor 𝜗F&F P (a P) . The outcome term is formalized such that the evaluation of kindness is based on the inequity between proposer’s and responder’s payoffs at a specific end node assuming that the responder chooses her efficient strategy: Again, within the context of our mini-ultimatum games, we can simply insert the payoffs following accepted offers into Eq. (5). To derive perceived kindness, the outcome term is multiplied with the intention term, where reciprocal concerns come into play. Here, F &F distinguish between five payoff constellations from which different intentions are derived. More specifically, the intention factor accounts for the proposer’s intentional and unintentional choices depending on how she could have altered payoff constellation with regard to her own in combination with the responder’s payoff: where 𝜋0 R , 𝜋0 P are payoffs resulting from accepting the specific offer in our design. Let  ΠR be the set of payoffs resulting from the acceptance of an alternative offer (but not the specific offer), 𝜋 R be one element in  ΠR , and 𝜖R be an individual parameter with 0 ≤ 𝜖R ≤ 1 . This parameter is denoted as the pure outcome concern parameter. That is, 𝜖R measures the responder’s unease with the inequity between proposer’s and responder’s payoff, although the proposer has no option to avoid the kind or mean offer. The first two cases of (6) refer to intentions of the proposer’s actions favoring the responder money-wise: in the first one, the proposer offers the responder not the smallest payoff possible although it is higher than his own one. This case is considered as fully intentional. In the second case, the proposer offers the responder a higher payoff than her own one, but has no chance to avoid this. In this case, the outcome is nice, but the proposer does not act intentionally so that the intention factor is reduced. The last three cases refer to negative intentions: the final case mirrors the second case into the negative domain; the proposer offers the responder a smaller payoff than her own one, but has no chance to avoid this. In this case, the outcome is mean, but the proposer does not act intentionally so that the intention factor is reduced, whereas the third and fourth case refer to intentionally mean choices. In the fourth one, the proposer offers the responder a smaller payoff than a possible alternative, but the offer is “somehow understandable” in the sense that the alternative yields less for the proposer than for responder. Therefore, the proposer’s intention is discounted according to the proposer’s relative disadvantage under the alternative. In contrast, in the third case, the proposer’s unkindness is fully intentional, since (5) ΔP(aP)=𝜋R−𝜋P (6) 𝜗 F&F P(aP)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 if 𝜋0 R≥𝜋0 Pand ∃𝜋 R∈ ΠR∶𝜋 R<𝜋 0 R, 𝜖Rif 𝜋0 R≥𝜋0 Pand ∀𝜋 R∈ ΠR∶𝜋 R≥𝜋0 R, 1 if 𝜋0 R<𝜋 0 Pand ∃𝜋 R∈ ΠR∶𝜋 R>𝜋 0 Rand 𝜋 R≤𝜋 P max �1−𝜋 R−𝜋 P 𝜋0 P−𝜋0 R ,𝜖R�if 𝜋0 R<𝜋 0 Pand ∃𝜋 R∈ ΠR∶𝜋 R>𝜋 0 Rand 𝜋 R> 𝜋 P 𝜖Rif 𝜋0 R<𝜋 0 Pand ∀𝜋 R∈ ΠR∶𝜋 R≤𝜋0 R, 313 1 3 Reciprocity models revisited: intention factors andreference… Similarly, predictions according to Hyp F&F are misaligned with results due to the lower rejections rate of (8,2) in Γ8 and Γ11 compared to Γ9 , Γ10 and Γ12 . In contrast to our earlier results, which corroborate with F &F’s model, we provide here Fig. 1 First mover decisions (in percent) Fig. 2 The y-axis displays the rejection rates for (8,2) in Γ1 to Γ12 . On the x-axis, the respective rank is plotted according to D&K and F&F with 0 reflecting the lowest and 4 the highest likelihood for rejection 314 J.Hinz et al. 1 3 evidence for a lack of generality (or a limiting specificity) of F &F’s model. More precisely, by changing the structure of the games in a way that we add a number of alternatives, reciprocal behavior cannot be satisfactorily described by F &F’s intention-based model anymore. Interestingly, the comparison across simple and richer games also casts some doubts onto F &F’s model (e.g., Γ4 and Γ12 : p=0.04 ; or Γ2 and Γ8 : p<0.001 ), although in those cases the alternatives of the simple games are subsets of the alternatives in the richer games. We discuss this point in greater detail in the Section5. Provided our observations, it seems that neither D &K’s model nor F &F’s model characterize rejection behavior in the richer games accurately. Particularly, both models fail to predict the occurrence of the two blocks. The results suggest that both models miss to incorporate an important characteristic of reciprocity. 4.2 Individual reciprocity In the following, we test for the consistency of our results on an individual level. For this, we test the personal implications of both models across games. In a first step, we test whether pairwise comparison of acceptances and rejections of two games leads to consistent results. That is, provided the decision in one game, the same subject has to make the same decision in another game associated to the same rank. Moreover, if the subject rejects (accepts) an offer, other offers associated with higher (lower) ranks ought to be rejected (accepted) by the same subject as well. Therefore, we will test the continuity of the two rankings across all games in a second step. That is, we will test whether the rejection of an offer implies the rejection of all Table 3 The first column reports observed rejection rates for (8,2) in ascending order for Γ1 to Γ12 The second and and third column display the rank of rejection according to D&K and F&F with 0 reflecting the lowest and 4 the highest likelihood for rejection along the game’s values and ranks according to the modification of 𝜗CA P (8, 2 ) (for this see below) Game {Alternatives} Rejection rates for 8,2 ΨD&K( 8, 2 ) ΨF&F( 8, 2 ) 𝜗CA P (8, 2 ) ΨCA ( 8, 2 ) Γ3{(9, 1)} 0.32 0 1 𝜖R 1 Γ1{(8, 2)} 0.35 0 1 𝜖R 1 Γ11{(9, 1),(10, 0),(6, 4)} 0.48 0 4 𝜖R 1 Γ6{(3, 7)} 0.48 4 2 𝜎R 2 Γ7{(3, 4)} 0.49 2 3 𝜎R 2 Γ8{(10, 0),(9, 1),(5, 5)} 0.50 1 4 𝜖R 1 Γ5{(4, 3)} 0.54 1 4 1 3 Γ4{(7, 3)} 0.55 1 4 1 3 Γ2{(5, 5)} 0.57 3 4 1 3 Γ12{(9, 1),(7, 3),(6, 4)} 0.59 1 4 1 3 Γ9{(9, 1),(7, 3),(5, 5)} 0.61 2 4 1 3 Γ10{(7, 3),(6, 4),(5, 5)} 0.62 3 4 1 3 315 1 3 Reciprocity models revisited: intention factors andreference… games associated to a higher rank. Likewise, the acceptance of a game implies the acceptance of all games with a lower rank. Let us start with the first step and the pairwise comparison of simple games according to D &K. In line with Hyp D&K , we cannot reject the hypothesis that the same subjects reject Γ4 and Γ5 ( p=.61 ).18 At the same time, we have to reject the hypothesis that at least all responders rejecting (8,2) in Γ4 and Γ5 (203 responders do so) reject (8,2) in Γ2 , Γ6 and Γ7 as well ( p<0.001 ): only 151 responders reject (8,2) in all five games. Likewise, the prediction that responders rejecting (8,2) in Γ7 reject (8,2) in Γ2 and Γ6 is not supported by the data ( p<0.001 ): only 160 responders reject (8,2) in all three games. Finally, the prediction that responders who reject (8,2) in Γ2 do so in Γ6 as well is also not supported by the data ( p<0.001 ): 186 responders reject (8,2) in both games. Turning to F &F, we find that 107 responders reject (8,2) in Γ1 and Γ3 (out of 152/137 rejecting Γ1∕Γ3 ), so that there is weakly significant evidence that not the same subjects reject this offer in both games ( p=0.08 ). Similarly, Hyp F&F is not supported in the sense that from 204 (209) responders rejecting (8,2) in Γ6 ( Γ7 ), 153 (180) reject the same offer in Γ4 , Γ5 and Γ7 ( Γ4 and Γ5 ). Here, we have to reject the hypothesis that the same subjects reject this offer in all four games ( p<0.001 , and p=0.002 , respectively). However, we cannot discard the hypothesis that the same subjects reject (8,2) in Γ4 and Γ5 ( p=0.61 ): from 235 (231) rejecting (8,2) in Γ4 ( Γ5 ), 203 reject the offer in both games. In other words, F &F organizes the data well, if offers are fully intentional referring to their model. For the richer games, D &K predict that responders rejecting (8,2) in Γ8 ( Γ9 ) are expected to reject the same offer in Γ9 and Γ10 ( Γ10 ). There is little evidence for the claims: from 214 (260) responders rejecting (8,2) in Γ8 ( Γ9 ), 194 (236) responders do so in Γ9 and Γ10 ( Γ10 ) as well ( p<0.001 for both comparisons). Likewise, not the same responders reject (8,2) in Γ4 , Γ5 , Γ8 and Γ12 ( p<0.001 ), Γ7 and Γ9 ( p<0.001 ), and Γ2 and Γ10 ( p=0.01 ). Finally, from 167 responders rejecting (8,2) in Γ4 , Γ5 , Γ8 and Γ12 , 137 reject (8,2) in Γ7 , Γ9 , Γ2 , Γ10 and Γ6 as well. Again, the claim that the same responders reject (8,2) across all those games is not supported ( p<0.001 ). We continue our analysis by testing F &F in the context of richer games: according to Hyp F&F , the same responders reject (8,2) in games Γ2 , Γ4 , Γ5 and Γ8 to Γ12 . This claim is not supported by the data ( p<0.001 ). Even if we restrict our analysis to games Γ8 to Γ12 , there is little evidence that the same responders reject (8,2) in those games ( p<0.001 ). However, we cannot reject the claim that the same responders reject (8,2) in Γ9 , Γ10 and Γ12 ( p=0.19 ). Thus there seems to be an important difference between Γ9 , Γ10 and Γ12 on the one hand, and Γ8 and Γ11 on the other that influences reciprocity substantially in our experiment. Now, we turn to the second step of the analysis: we check the continuity of the two rankings across all games. That is, provided that a subject rejects a game, she rejects all games with the same and lower ranks as well. Likewise, provided that 18 Throughout this subsection, we use a two-sided Friedman test for the assessment of statistical significance. 316 J.Hinz et al. 1 3 a subject accepts a game, she accepts all games with the same and higher ranks as well. For instance, suppose a subject accepts the offer (8,2) in Γ7 . Since Γ7 has the rank 2 according to D &K, the subject ought to accept all offers in games with higher or equal ranks, too (i.e., Γ1,Γ3,Γ4,Γ5,Γ8,Γ9,Γ11 , and Γ12 ). In turn, rejecting (8,2) in Γ7 (rank 3 referring to F &F), a subject ought to reject all offers in games with lower or equal rank 3 according to F &F (i.e., Γ2,Γ4,Γ5,Γ8,Γ9,Γ10,Γ11 , and Γ12 ). Altogether, the prediction of both D &K and F &F leads to five patterns for acceptances and rejections each.19 We check for each subject whether the subject’s acceptances and rejections match one of the patterns and, if not, the minimum mismatches from one of the patterns both for D &K and F &F (and CA, see below). Figure3 illustrates the frequency of subjects whose acceptances and rejections of (8,2) match exactly one predicted pattern (denoted as ‘0’), match a predicted pattern except one or two choices at most (‘1’ and ‘2’), respectively (two out of twelve choices is the maximum number of individual mismatches). Overall, only 114 subjects (out of 427) show no mismatch according to the predictions from D &K, while 220 according to the predictions from F &F. Thus, the former model yields significantly fewer no-mismatches than the latter ( p<0.001 , two-sided proportion test comparing both numbers). In line with this observation, we find an average number of 1.47 mismatches for D &K and 0.48 mismatches for F &F. Notice that a large proportion of individual mismatches with D &K’s predictions results from rejections in Γ1,Γ3 , and Γ11 (i.e., D &K predict rank zero for those games). Thus, an advantage of the F &F model and a disadvantage of the D &K model is that the former (latter) model can (cannot) rationalize rejections based on inequity considerations. 5 Discussion Our results indicate for both models weaknesses when predicting perceived kindness. D &K provide with their 𝜅D&K PR (a P) a partition in the degree of kindness which is too detailed. F &F’s kindness measurement yields a partition of kindness which is too general in richer settings due to the pairwise comparison of offers to the extreme alternatives. Consequently, neither model can describe behavior fully. Therefore, we want to discuss a potential combination of both reciprocity models (‘Combined approach’, abbreviated ‘CA’). However, we do not attempt to present a fully elaborated model, but we want to sketch a possible avenue for the further development of research on reciprocity. On the one hand, we want to adjust F &F’s model so that it yields more predictive power in richer environments while keeping the number of mismatches as low as possible (though additional predictive power will lead almost inevitable to more mismatches). On the other hand, we want to incorporate 19 For example, accepting (8,2) in Γ1,Γ3,Γ4,Γ5,Γ7,Γ8,Γ9,Γ11 , and Γ12 , while rejecting (8,2) in Γ2,Γ6 , and Γ10 is one pattern predicted by D &K. Trivially, acceptance of (8,2) in all games is another pattern predicted both by D &K and F &F, whereas rejecting (8,2) in Γ1,Γ3 , and Γ11 cannot be rationalized by D &K. All predicted patterns are described in the Appendix. 317 1 3 Reciprocity models revisited: intention factors andreference… the assessment of alternatives’ kindness according to the reference value approach to provide a sufficiently simple partition of kindness. As our data corroborate in simple games F &F’s distinction between five generic cases of intention (and the distinction between the intention and the extent to which this alternative is considered to be kind or unkind), we do not seek to modify this feature.20 Hence, we propose a 𝜅CA PR (a P) which consists of the product of an intention factor and the outcome term ΔP(aP) according to equation (6). However, the major extension is an assessment of P’s alternatives according to a reference value. Not only do our experimental results support this approach, but we consider it as unrealistic that in more complex situations – and we study in our experiment complexity only to the extent that we test games with four instead of two alternatives – the existence of one clearly (un)kind action determines the choice of another alternative as a fully intentional act of (un)kindness. Thus, similar to D &K’s approach, we model intention relative to some reference value. Specifically, the reference value we propose, the median offer (i.e., the “middle” payoff within the set of corresponding end notes resulting from the consecutive choice of efficient strategies) provides the additional benefit of being robust against outliers in the set of alternatives (D &K discuss the problem of outlying payoffs extensively in their paper). Formally, let us denote with 𝜋M i i’s median payoff among the set of payoffs resulting from j’s choice.21 Then, we define the intention factor of P’s move 𝜗CA P (a P) of offering a specific payoff combination 𝜋0 P ,𝜋 0 R as follows: 012 D&K F&F CA Number of mismatches per person Frequency 0.00.2 0.40.6 0. 8 Fig. 3 Frequency of individual deviations from the prediction patterns according to the theories (for ‘CA’ see below) 20 On a side-note, we consider the fact that there is no significant difference of rejection rates within both blocks – although one game per block yields in sum less than 10 Talers – as evidence that efficiency concerns are less important in this setting. 21 We define 𝜋M i implicitly with respect to the cumulative distribution function F(x) on i’s set of payoffs resulting from i’s and j’s choice of any combination of efficient strategies in a game: 𝜋M i satisfies both inequalities ∫ (−∞,𝜋M i ]dF(x) ≥0.5 and ∫ [𝜋M i ,∞) dF(x) ≥0.5 . 318 J.Hinz et al. 1 3 where  ΠP be the set of proposer’s payoffs resulting from the acceptance of an alternative offer, 𝜋 P be one element in  ΠP , and 𝜎R be another individual parameter with 0 ≤ 𝜖R ≤ 𝜎R ≤ 1 . That is, like F &F’s approach, our intention factor differentiates between five categories of R’s outcomes resulting from P’s action: payoffs implying smaller payoffs for P than for R are considered as fully intentionally kind if they are larger than the reference value, whereas they are accidentally kind if they are smaller or equal to the reference value. In turn, there are accidentally unkind offers which imply larger payoffs for P than for R if they are larger or equal to the reference value. Finally, P’s action leading to R’s payoff being smaller than R’s reference value and P’s payoff is considered to be fully intentional unkind only if P could choose better alternatives for herself. That is, if the unkind offer is “somehow understandable” in the sense that all other alternatives yield less for P than R’s reference value, the offer is still perceived as intentionally unkind but not that much. Only, if there is at least one other alternative which yields for P more than R’s reference value, choosing the specific alternative is fully intentional (and unkind). Notice that the latter two cases translate F &F’s observation that “the perception of the unfair offer depends on how much j has to sacrifice in order to make the more friendly offer” (F &F, 2006, p. 297) into the context of a global assessment of i’s payoffs.22 That is, if making a more friendly offer than (8,2) implies that P earns less than R’s reference value, this offer is still unkind, but with limited intention. Based on our reformulation of 𝜗CA P (a P) , we obtain a new rank order for the likelihood of a rejection for (8,2) which is reported in the last column of Table3: the predictions based on 𝜗CA P (a P) follow qualitatively the one based on 𝜗F&F P (a P) for the simple games. However, using 𝜗CA P (a P) one can predict the two blocks of rejection rates in the richer games. In addition, the rejection rates for (7,3) in Γ9 , Γ10 and Γ12 follow the predicted pattern: in Γ9 and Γ12 where (7,3) is unfavorable but mildly unkind, 101 and 106 responders reject the offer, while 151 do so in Γ10 where this offer is fully intentionally unkind.23 Comparing the number of individual mismatches between decisions and predictions between both models, we obtain a similar performance of the combined approach and the F &F model (see our Fig.3): for 48% of all subjects we compute no mismatch based on the prediction of the combined approach, whereas 52% based (8) 𝜗CA P(aP)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 if 𝜋 0 R>𝜋 M R and 𝜋 0 R>𝜋 0 P , 𝜖Rif 𝜋0 R≤𝜋M Rand 𝜋0 R≥𝜋0 P, 𝜖Rif 𝜋0 R≥𝜋M Rand 𝜋0 R≤𝜋0 P, 1 if 𝜋0 R<𝜋 M R,𝜋0 R<𝜋 0 P, and ∃𝜋 P∈ ΠP:𝜋 P>𝜋 M R 𝜎Rif 𝜋 0 R<𝜋 M R,𝜋 0 R<𝜋 0 P, and ∀ 𝜋 P ∈ Π P:𝜋 P≤𝜋 M R, 22 In this quote, j refers to the proposer and i to the responder in the context of our setting. 23 p < 0.001 for the hypothesis that the rejection rate in Γ10 equals one in the other two games, whereas p=0.466 for the hypothesis that the rejection rates in Γ9 and Γ12 are the same. 319 1 3 Reciprocity models revisited: intention factors andreference… on the prediction of the F &F model. The difference between both frequencies is not significant ( p=0.338 , two-sided proportion test comparing both numbers). Hence it seems that the combined approach is able to sharpen the predictive power while keeping the good performance with regards to the low number of mismatches. Finally, we run a series of linear probability regressions on the individual decisions whether to accept or reject the offer (8,2) (reported in Table4). We control for individual characteristics by including individual dummy variables for all subjects. Independent variables are the ranks resulting from our three models F &F, D &K and CA. We obtain remarkably high R2 values in all regressions along constantly significant coefficients for all three ranks. Most importantly, we assess whether including the ranks of the Combined approach adds to the predictive power of the regressions by running likelihood-ratio tests. Results indicate that in all cases the explanatory power of the regressions increase significantly by adding the ranks of the Combined approach. That is, adding the new ranks to the ranks of the F &F model (i.e., (1) vs. (5)) and to the ranks of the D &K model (i.e., (2) vs. (6)), as well as to a combination of both models (i.e., (4) vs. (7)) contributes significantly to the explanation of variance in the rejections of (8,2). Overall, it seems that 𝜅CA PR (a P) provides a good combination between F &F’s idea of a kindness term which differentiates between the intention of an action and the extend of kindness with D &K’s approach to assess an entire game by means of a reference value. This results in more diverse predictions particularly in the context of richer decision environments. For instance, in an ultimatum-type game with several alternatives, the assumption that players evaluate the kindness of a specific offer by pairwise comparisons between alternatives seems unrealistic to us at least due to the shire computational effort it takes to compare alternatives against each other. Rather, we follow D &K’s idea that players condense alternatives by means of reference values. As such, our approach is located in some sense half way between the models by D &K and F &F: it processes more information than the model by D &K. On the other hand, our approach generalizes over alternatives by a larger extent than F &F by forming reference values. Whether our modifications optimize the tradeoff between the generalisability of the model to various situations and the accurate prediction of specific behavior is an open question and requires future research. 6 Conclusion Although reciprocity is one fundamental cornerstone of human behavior, modeling reciprocity still is a challenge for social scientists. The current study analyzes two of the most established approaches, the reference value model by D &K and the intention factor model by F &F. We point out that there are two major differences between the two approaches: the first model measures perceived kindness of an action in relation to a reference value while the second model distinguishes between the intention of an action and the extent to which the action is perceived as being unkind or kind. The latter element of F &F’s model relies on the inequity of the proposed payoffs whereas the former element results from a pairwise comparison between alternatives. 320 J.Hinz et al. 1 3 We test elements of both models within the context of mini-ultimatum games with two and four alternatives. Results show that F &F’s approach works fine in the games with two alternatives, but has important drawbacks in the games with four alternatives, both with respect to the average numbers but also once we run a within-subject analysis. On the other hand, D &K’s model fails to characterize behavior within both contexts. However, we have to admit that we do not test D &K’s complete model excluding some equilibria based on mutual meanness. This may impair its predictive success in our setting. Despite the shortcoming, we conclude that D &K’s idea to measure perceived kindness in one variable, the distance to the equitable payoff, does not sufficiently capture the nature of perceived (un)kindness. Particularly for games with four alternatives, it seems that the pairwise comparison of alternatives does not predict behavior accurately. Therefore, we present and discuss a potential modification of F &F’s reciprocity model which includes elements of D &K’s approach. Testing the Combined approach’s predictions with our experimental data yields an appropriately low number of mismatches between subjects’ decisions and the predictions. Also the results of the regression analyses indicate a significant improvement of predictive power added by the Combined approach to the existing models. To conclude, more research is needed to model reciprocity in a sufficient way. Perhaps, the question is not whether there is a true model mapping reciprocity, but whether there is a model that adequately balances the need for generalisability across different games with a satisfactory good predictability of behavior within a specific environment. Elsewhere, it has been shown that reciprocity itself encompasses a number of different subtypes of social utility (e.g., Nicklisch and Wolff (2012)). Therefore, we have to ask ourselves whether we want to model the behavior Table 4 Linear probability regressions Significance levels *p<0.1, **p<0.05 and ***p<0.01 Dependent variable Rejecting the offer (8,2) (1) (2) (3) (4) (5) (6) (7) ΨF&F 0.068 ∗∗∗ 0.062 ∗∗∗ 0.046 ∗∗∗ 0.049 ∗∗∗ (0.004) (0.004) (0.005) (0.005) ΨD&K 0.037 ∗∗∗ 0.025 ∗∗∗ 0.009 ∗∗ 0.015 ∗∗∗ (0.003) (0.003) (0.004) (0.004) ΨCA 0.083 ∗∗∗ 0.047 ∗∗∗ 0.077 ∗∗∗ 0.034 ∗∗∗ (0.005) (0.006) (0.006) (0.007) Constant − 0.220 ∗∗ − 0.055 − 0.180 ∗∗ − 0.241 ∗∗∗ − 0.252 ∗∗∗ − 0.179 ∗∗ − 0.255 ∗∗∗ (0.088) (0.090) (0.089) (0.088) (0.088) (0.089) (0.088) Observations 5124 5124 5124 5124 5124 5124 5124 Individual Dummies Yes Yes Yes Yes Yes Yes Yes R 2 0.662 0.645 0.659 0.666 0.666 0.659 0.667 F-statistic 21.50*** 20.01*** 21.24*** 21.82*** 21.89*** 21.22*** 21.94*** LR-test: 𝜒2( 1 ) 70.78*** 204.90*** 27.22*** (1)vs.(5) (2)vs.(6) (4)vs.(7) 321 1 3 Reciprocity models revisited: intention factors andreference… in one specific game which may trigger one specific form of reciprocity, or whether we want to rely on a general model, which, however, has less predicting power in special situations. The answer to this question we cannot provide here. Therefore, we would like to invite future research to follow this avenue, or, maybe, prove it wrong. Appendix: Predictions according tooutcome concerned social utility Inequity aversion Provided inequity averse preferences, we have to claim that responders either accept or reject (8,2) in all games Γ1 to Γ12 . The reason for this is rather obvious: as the same offer (8,2) is considered throughout all games and inequity aversion preferences take only the outcome of a proposal into consideration, there is no difference in the utility resulting from acceptance across games, nor from rejection across games. Levine’s (1998) model In the following we want to derive a prediction for behavior in our games according to Levine’s (1998) model. We choose this model, since it can be characterized as some intermediate step between models based on outcome concerns and reciprocity models. The reason for this is that the proposer partly reveals her taste for altruism through her choice among the alternatives of the game. This information updates the weight for altruism in the responder’s utility function and may lead to rejections if altruism is negative. Formally, we can define the utility function of player i (paired with j) according to Levine (1998) as: where 𝜋i(ai,aj) is i’s monetary payoff of an offer, while 𝛼i is i’s taste for altruism ( −1<𝛼 i<1 ); finally, 𝜚i measures the importance of j’s altruism for i’s utility ( 0≤𝜚i≤1 ). Suppose i is a proposer, while j is a responder. Of course, 𝛼i and 𝜚i are i’s private information. However, by choosing a specific offer in the game, the proposer i partly reveals her taste for altruism to the responder j who updates her utility UL j accordingly. Therefore, j’s utility changes with i’s choice of alternatives in the game. Yet, it turns out that i’s choice of (8,2) is uninformative in Γ1,Γ2 , Γ4 , Γ5 , Γ6 , Γ7 , and Γ10 in the sense that it only reveals ai<1 which is known from the beginning. As an example, consider Γ2 . Here, choosing (8,2) implies 8 + 𝛼 i +𝜚 i 𝛼 j 1 + 𝜚 i 2>5+ 𝛼 i +𝜚 i 𝛼 j 1 + 𝜚 i 5 which is equivalent to 1+𝜚i(1−𝛼j)>𝛼 i . It follows that the maximum of 𝛼i is 1. Thus the choice of a specific offer in our mini-ultimatum games does not result in an update of j’s belief concerning ai . It follows that in all of the previously U L i=𝜋i(ai,aj)+ 𝛼 i +𝜚 i 𝛼 j 1 +𝜚i 𝜋j(ai,aj ) 322 J.Hinz et al. 1 3 mentioned games j rejects the offer of (8, 2) if 0 >2+ 𝛼 j +𝜚 j 𝛼 i 1+𝜚 j 8 . It follows that j rejects if 𝛼j< 𝛼 with 𝛼 ∈ (−1, …, 0.5] depending on j’s specific 𝜚j . That is to say, if a responder rejects (8,2) in one of the games Γ1,Γ2 , Γ4 , Γ5 , Γ6 , Γ7 , or Γ10 , she should reject (8,2) in all seven games. Following the same rational, Levine’s model predicts that i does not offer 8,2 in Γ3 , Γ8 , Γ9 , Γ11 , and Γ12 . Therefore, we are hardly able to form predictions with respect to responder’s behavior. Notice that our deliberations rely on the linear specification of Levine’s model. One may think of a more general interpretation of the model. Let us assume that we can use the frequency by which proposers choose (8,2) as an indicator for the degree of unkindness of this offer. This leads then to an update of R’s belief regarding 𝛼P and, consequently, R’s rejection rate: that is, the more frequent (8,2) is chosen by the proposers, the less mean it is considered by the responders, and, therefore, rejected less frequently (e.g., Utikal and Fischbacher (2014) follow this idea). This leads, for instance, for the richer settings Γ8 to Γ12 to the prediction that the rejection rate for (8,2) in Γ8 is lower than in Γ9 to Γ12 (since (8,2) is offered significantly more frequently in Γ8 than in Γ9 , Γ10 , Γ11 , and Γ12 , see our proposer analysis at the beginning of Section4.1). Although not explaining our data fully – the prediction is violated by comparing Γ8 to Γ11 ( p=0.584 , two-sided proportion test comparing both rejection rates) – we would like to invite researchers to consider this often overlooked model for their future studies. Predicted acceptance andrejection patterns according tothethree approaches See Table5. Table 5 Choice patterns for (8,2) in Γ1 to Γ12 according to the three approaches; ‘a’ indicates predicted acceptances, ‘r’ predicted rejections Game D &K pattern F &F pattern CA pattern (I) (II) (III) (IV) (V) (I) (II) (III) (IV) (V) (I) (II) (III) (IV) Γ1 a a a a a a a a a r a a a r Γ2 a a r r r a r r r r a r r r Γ3 a a a a a a a a a r a a a r Γ4 a a a a r a r r r r a r r r Γ5 a a a a r a r r r r a r r r Γ6 a r r r r a a a r r a a r r Γ7 a a a r r a a r r r a a r r Γ8 a a a a r a r r r r a a a r Γ9 a a a r r a r r r r a r r r Γ10 a a r r r a r r r r a r r r Γ11 a a a a a a r r r r a a a r Γ12 a a a a r a r r r r a r r r