Disapproval aversion or inflated inequity acceptance? The impact of expressing emotions in ultimatum bargaining

Past experimental research has shown that when rating systems are available, buyers are more generous in accepting unfair offers in ultimatum bargaining. However, it also suggests that, under these conditions, sellers behave more fairly to avoid receiving negative feedback. This paper experimentally investigates which effect is stronger with the use of a rating system: buyers’ inflated inequity acceptance or sellers’ disapproval aversion. We explore this question by varying the information condition on the buyers’ side. Our experiment shows that in a setup where the size of the pie is common knowledge for both buyers and sellers, when a rating system is present, the sellers exhibit disapproval aversion but the buyers do not display greater acceptance of inequity. By contrast, when only sellers are aware of the size of the pie, sellers behave aggressively to exploit buyers and their behavior does not change in the presence of a rating system; however, buyers display greater acceptance of inequity when a rating system is present. We discuss how these results can be explained by a theoretical model that includes sellers’ social disapproval aversion and buyers’ disappointment aversion in addition to the players’ inequality aversion. Electronic supplementary material The online version of this article (10.1007/s10683-017-9554-z) contains supplementary material, which is available to authorized users.


A.1. Standard Theory Predictions for the N-C and R-C Treatments
For an assigned q  [0, 40], a seller maximizes his payoff with respect to the price , given the paired buyer's purchase threshold p b ,: Likewise, the buyer maximizes her payoff with respect to the purchase threshold , given the paired seller's offering price p s : Thus, the best responses of the seller and buyer are calculated as below: These best response correspondences are depicted as below:

A.2. Inequality-Averse Preferences and Best Response Strategies for the N-C and R-C Treatments
Suppose that a seller is inequality-averse as defined in the paper: u sj ( , The term within the first curly bracket is maximized at: = In short, the seller's best response function is summarized as: Here, the acceptance rate of the offering prices in the experiment would be q*/40 in expectation as q is randomly drawn from the uniform distribution between 0 and 40. Note that the intercept in Figure A.2 (the seller's best response price at q = 0) is 1 8 . This is not dependent on p bi .
If < 1 8 , then the seller's best response strategy becomes as follows: For simplicity, we assume that > 1 8 in the rest of this Appendix A. Also, note that * > 0 since * is a point at which y = − 1 2 and y = • (2 − 3 2 ) 2 intersect; both curves shift to the right when p bi increases as shown in the following figure. 2 Thus, we find that the seller's best response strategies shift as below responding to a chance in p bi . 2 * can be greater than 40. In that case, the seller's best response price is less than or equal to for any value q  [0,40]. Suppose also that the buyer is inequality-averse, as assumed in the paper: u bi ( , ) = − • ( − ) 2 . Notice here that the utility weight on inequality μ i is different from that of seller μ j . The buyer maximizes her utility, given p sj : This means that the buyer's best response strategy is: For a given q, the best response correspondences of the buyer and those of the seller described in Conditions (A2) and (A3) characterize the set of Nash equilibria.

A.3. Standard Theory Predictions for the N-IC and R-IC Treatments
For each q  [0, 40], given the buyer's purchase threshold p b , the seller maximizes his payoff with respect to the price : We obtain, from this maximization problem, the best response function of the seller as follows: Likewise, given the seller's strategy ( ), the buyer maximizes her expected payoff with respect to as the value of the commodity is unknown to her. This reduces to the following maximization problem: There exist Bayesian Nash equilibria characterized by the seller's best response specified in condition (A4) and by the following best response strategy of the buyer: Specifically, the following is an example of the equilibria: p s = c for q such that ≤ 2 ; and ( ) = for q such that > 2 , while p b = c. Here, c is any integer that is less than or equal to 20. With this equilibrium, the expected payoff for the seller is: There is no profitable deviation, not only for the seller but also for the buyer. There are many equilibria of this kind.
There is also another kind of equilibrium in which the transaction is not exerted. The following is an example: the seller posts a price that is greater than or equal to 20 always, and the buyer sets her purchase threshold at 0.

A.4. Inequality-Averse Preferences and Best Response Strategies for the N-IC and R-IC Treatments
The best response of the seller is the same as that discussed in Section A.2.
For the best response of the buyer, suppose that the buyer is also inequality-averse like the Then, the buyer's best response strategy is derived by maximizing her expected utility given the seller's strategy = ( ): That is, Here, suppose that ( ) is non-decreasing in q. Then (A8) reduces to the following: Here, −1 ( ) is the upper bound if it has multiple values (correspondence). Since the condition of non-negative utility must be met, we have: Although there are multiple equilibria, there is a common feature in that the buyer obtains a positive material payoff in expectation. This is because condition (A10) implies that: The BNE is characterized by (A2) (or A3), (A8) and (A10).
From Figure A.3, we have the following features of the equilibria: (1) The higher the buyer's equilibrium purchase threshold * , the higher the acceptance rate.
(2) Regardless of which purchase threshold is realized in equilibrium, the seller's equilibrium price is increasing in q and less than * in a region where ≤ 4 3 * − 1 6 .
(3) No trades are closed in the region where q > q*.

A.5. Disapproval Aversion and the Transactions between the Seller and the Buyer in the R-C Treatment
In this analysis, we assume that the seller's payoff function is expressed as in Eq. (4) of the paper (i.e., ′ = + • ( -5)). Then, the utility of seller j is expressed as: Likewise, the utility of buyer i is expressed as: We can solve this situation from the second stage (the rating stage) [backward induction].
The second stage (rating stage): If > , buyer i receives a material payoff of 0 points. In the rating stage, the buyer minimizes the term: • ( • ( -5)) 2 . This means that the buyer's best response rating score is: If ≤ , the buyer tries to minimize the term: in the rating stage. From 2 − 3 2 + • ( -5) = 0, we find: This is the condition of interior solutions. The negative slope (-2/c) shows a negative correlation between r and p sj . The buyer's best response correspondence differs by q because 0 ≤ r ≤ 10. We have the following three cases, considering the size of intercept (5 + The optimality condition suggests a negative correlation between r and p sj .
In the second stage, there are no real decisions to make for the sellers.
The first stage (the transaction between the seller and buyer): The buyer's best response can be quickly derived. Given , if − − • (2 − 3 2 + • ( * -5)) 2 ≥ 0, she submits a purchase threshold that is higher than or equal to . Here, r* is the buyer's best response strategy for rating in the following rating stage. It is clear that because of the rating opportunities, the inequality-averse term in her utility function is smaller in the R-C treatment than in the N-C treatment. Therefore, materially unequal offers by the seller are more likely to be accepted by the buyer. This is consistent with the idea that the buyer substitutes expressing emotions for rejecting offers.
As for the seller, we first consider the interior solution case in the rating stage (see the above . Thus, we see that the seller chooses to offer lower prices in the R-C treatment in order to avoid receiving disapproval points or to enjoy positive psychological gains. When r* = 10 or 0 (corner solution), from ∂ sj /∂p sj = 0, we have: The buyer submits so that ≤ as long as the buyer's utility is positive. Thus, in the corner solution, the seller offers a materially less fair amount (higher price) to the buyer and the buyer selects r* = 0 in the second stage.

IC treatments
In this subsection, we study how theoretical predictions may change if inequality-averse actors also exhibit disappointment aversion due to asymmetric information on q in the incomplete information treatments. For simplicity, we assume that sellers do not exhibit social disapproval aversion in this analysis. That is, we assume that buyers' rating behaviors will not affect the utilities of sellers. We note that calculations become messy if we have both disapproval aversion and disappointment aversion in the model. However, even if we incorporate both disapproval aversion and disappointment aversion into the modeling of subjects' inequality-averse preference (Eqs. (1) to (3)) with the assumption that buyers can cancel out negative emotions from disappointment by releasing the emotions, we obtain the same implications (the degree of inequity acceptance is stronger in the incomplete information than in the complete information settings because of disappointment aversion). 3 In this Appendix, we show a simpler version of the analysis for an illustrative purpose.
We assume that with incomplete information (when is unknown to buyers) buyers select based on the expected value of . In a model with disappointment aversion (e.g., Bonomo, Garcia, Meddahi, and Tédongap 2010, Gul 1991, Routledge and Zin 2010, buyer i will incur a psychological disutility when the realized was less than her expectation (i.e., E(q) = 20). The absolute value of i's psychological loss is assumed to be increasing in . We will incorporate the model of disappointment aversion into our model with inequality aversion (Eqs. (1) to (3) in the paper) as follows. First, we write the payoff function for buyer i in the N-IC treatment as below: where 1 > α > 0.
We further assume that buyers' loss from disappointment would be 0 once it is released (see psychological papers for this argument, such as Xiao and Houser [2005], Campbell-Sills et al. [2006], and Gross and John [2003]). This means that buyers strategically utilize the rating opportunity to deal with their negative emotions from disappointment (realized low ). In other words, buyers do not utilize rating opportunities to verbally punish or reward the behavior of their treatment as below.
Second, we calculate the utilities of buyer i and seller j in the incomplete information treatments based on Eqs.
(1) to (3): Here, the superscripts N and R refer to treatments without and with rating opportunities, respectively.

Proposition 2. When is small enough, <
Sellers' best responses in the R-IC treatment is (A2). By using the same calculation process as in (A2), we can find sellers' best response function in the N-IC treatment.    The Determinants of the Rating Decisions by Buyers (Supplementing Table 2 of the paper) Dependent variable: Rating that buyer i gave to the matched seller j in period t ∈ {1, 2, …, 50} The following are the estimation results with the Heckman's two-stage selection model.

[Second Stage Regression]
Notes: Numbers in parenthesis are robust standard errors. Control variables include buyers' demographic variables: a USA dummy (=1 if sessions were conducted in the USA; 0 otherwise), a female dummy (=1 if female; 0 otherwise), number of economics courses taken, general political orientation (1 = very conservative to 7 = very liberal) and income of the subject's family. We omitted the coefficient estimates of these demographic variables to conserve space as these are not related to the hypotheses in the paper.
*, **, and *** indicate significance at the 0.10 level, at the 0.05 level and at the 0.01 level, respectively.    Please note that the absolute values of coefficient estimates in column (1') are identical to those in column (2') with the sign being opposite. The same holds for columns (3') and (4').
Notes: Numbers in parenthesis are robust standard errors. Control variables include buyers' demographic variables: a USA dummy (=1 if sessions were conducted in the USA; 0 otherwise), a female dummy (=1 if female; 0 otherwise), numbers of economics courses taken, general political orientation (1 = very conservative to 7 = very liberal) and income of the subject's family. We omitted the coefficient estimates of these demographic variables to conserve space as these are not related to the hypotheses in the paper.
*, **, and *** indicate significance at the 0.10 level, at the 0.05 level and at the 0.01 level, respectively.
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