Endogenous versus Exogenous Fairness Indices in Repeated Ultimatum Games

In ultimatum games, we often observe some participants rejecting offers that may be normally viewed as “fair” while others accept even lower offers that are typically viewed as “unfair”. The objective of this study is to construct and examine an endogenous fairness index that helps explain this phenomenon. To achieve this objective, we construct a repeated ultimatum game environment in which each participant plays the roles of both the sender and the receiver with two different participants. We conjecture that the ratio of the amount that an individual receives divided by the amount the individual sends, captures the benchmark of what constitutes a fair offer for that individual when an offer-acceptance decision has to be made. Our design includes a fixedand random-partners treatment in the repeated ultimatum game as an attempt to identify and isolate the effects of social distance on offer-acceptance decisions. In addition to the inclusion of the fairness indices in the offer-acceptance models, we introduce measures of social value orientations and risk attitudes as control variables in our analyses. We find that our belief-related fairness index is, in some cases, a better explanatory variable for offer-acceptance decisions than the conventional “offer index” and in other cases significantly augments the “offer index”. As well, the offer-acceptance model including the belief-related fairness index can account for likelihoods of accepting less fair offers that can, at times, exceed likelihoods of accepting more fair offers.

Theoretical Economics Letters studying questions related to bargaining power, distribution of resources and fairness concerns. Game theory predicts that a rational self-interested proposer is expected to offer a very small (trivial) portion of an endowment and that a rational self-interested (wealth-maximizing) recipient is expected to accept this offer. Data from controlled ultimatum game experiments do not typically support these predictions. There are various explanations offered in the literature for these anomalous observations. In this study we focus on the fairness of the offer as an important determinant of the recipient's decision. In particular, we propose an endogenous measure of fairness based on the actions of the recipient when in the role of a proposer as an important determinant of the recipient's decision to accept or reject an offer.
The usual measure of fairness in ultimatum game environments is the proportion of the proposer's endowment that is offered to the recipient. An offer that is 50% of the proposer's endowment is viewed as fair. An offer that is less than 50% is viewed as unfair. The primary objective of this study is to develop a fairness measure that will permit us to identify situations in which a recipient may at times reject an offer that may otherwise appear to be fair while in other situations may accept an offer that may appear to be unfair. The standard measure of fairness will never support our objective. Because individuals make such decisions, it would be valuable to identify a measure of fairness that would be consistent with these kinds of choices by rational self-interested recipients. To conclude that people are generally not rational makes it difficult to understand the decisions that they make. In addition, it makes the prediction of what people may do in various situations impossible.

Detailed Background and Literature
The basic ultimatum game involves two players. The first player is endowed with an asset (money) and is asked to share it with an anonymous second player. The incentive to share resources is created by telling the first player that if the second player does not accept the offer, neither player receives a payoff. If the second player accepts the offer, the two players keep the asset allocation proposed by the first player.
Game theory predicts that a rational self-interested proposer is expected to offer a very small (trivial) portion of the endowment and that a rational wealthmaximizing recipient is expected to accept this offer because something is better than nothing. Contrary to these predictions, the common findings in ultimatum game experiments show that proposers often offer substantial amounts (typically varying between 10% and 50% of the endowment), and that the recipients have the tendency to accept offers that are higher than 25% of the endowment. However, it is not uncommon to observe some responders reject offers that are higher than 30% while others under the same environment accept offers as low as 10%. 1 1 See, for example, [1] (Table 4 and Table 5, p. 375) that provide several observations that display this behavior). culture and the size of the stake. 2 In this study, we focus on fairness. We conjecture that the decisions of individuals in simple bargaining environments to accept or reject offers depend, among other things, on their perceptions of the fairness of the allocation of the shared resources. Typically, but not always, an offer equal to half of the sender's endowment is identified as a fair offer. 3 Correspondingly, if an offer is less than half of the sender's endowment, the offer may be described as being unfair. One challenging issue is that the notion of fairness is difficult to quantify as it may be based on a tacit belief that can only be inferred through an individual's behavior. In addition, the perception of fairness is likely to vary among individuals and perhaps even for the same individual from one context to another (see, e.g., [2] [3] [4] [5]).
Azar et al. [4] address the effect on offer acceptances of what individuals believe others will find to be the minimum acceptable offer and on what individuals expect to receive. These beliefs and expectations were obtained for participants and used to evaluate offer-acceptance decisions. The design used in this paper provides an alternative to this mechanism for generating indications of beliefs or expectations that may be less likely to influence the behavior of participants who provide their beliefs or expectations. By asking these questions, the experimenter may be leading the participants to focus on notions of fairness when they are asked to make decisions as senders and the receivers. By asking participants about their expectations and beliefs we may be framing the decisions they must make as senders and responders that will affect the decisions differently than if we had not asked them about these expectations and beliefs. 4 Gomaa et al. [5] compare the effectiveness of an endogenous fairness index to an exogenous fairness index as an explanatory variable that can account for a significant amount of variation in offer-acceptance behavior in a one-shot ultimatum game. Their objective is to identify a fairness index that would permit scenarios in which the likelihood of accepting an unfair offer could exceed the likelihood of accepting a fair offer. The objective in this paper is to extend these results to a repeated ultimatum game environment with either fixed partners or with randomly reassigned partners. In the fixed-partners environment, partners are not reassigned after each decision round. The structure of the repeated game reduces social distance and permits participants to build reputations. In the random-partners environment the one-shot game is extended to evaluate how 2 Güth and Kocher [6] provide a comprehensive review of ultimatum game research over more than thirty years. 3 This notion of what is a fair offer in an ultimatum game is common and is included in recent papers by [4] and [6] (p. 398, fn. 11). 4 Our experiment does not compare the effectiveness of the alternative methods of collecting information about beliefs or expectations and whether our conjecture that asking for beliefs of what is a fair offer or of what individuals expect others will accept will lead to different offer or offer-acceptance behavior than what we find in our sessions. Any differences that may exist between the alternative methods would be worthwhile identifying. The results of this experiment suggest that reducing social distance by maintaining fixed partners leads to convergence to a greater offer than when partners are randomly reassigned after each decision round. However, by the 3 rd round of the repeated game with fixed partners the mean offer is significantly below the conventional "fair" offer of 50 percent of the sender's endowment. With random partners, the offer falls from about 37 percent of the sender's endowment in the one-shot game to 31 percent by the 3 rd round of the repeated game. Offer-acceptance models are estimated for both the one-shot game and the 3 rd round of the repeated game and we find that the inclusion of the endogenous fairness index results in an estimating equation with a better "fit" than does the exogenous fairness index alone.
Therefore, the closer the endogenous fairness index is to unity, the closer is the amount received by the person who must make an offer-acceptance decision to a fair offer. 5 When making an offer-acceptance decision, this endogenous measure will be a better reflection of the receiver's notion of the fairness of the offer than is the offer alone. Finally, the data from this experiment offer some insight regarding risk attitudes and social value orientations with respect to offers made and offers accepted and how these decisions may change once the game is played repeatedly. 6 5 Fairness may be a more important characteristic of an offer than its magnitude. Fehr and Schmidt [7] note "Responders do not behave in a self-interest maximizing manner … the motive indicated for the rejection of positive, yet 'low', offers is that participants view them as unfair." 6 It is important to point out that the role of risk is currently a topic of substantial interest in the Management Science and Operations Research literature that considers the decisions made by a firm to supply output to multiple retailers (see [8] [9] [10]). Uncertainty can arise from retailer costs and market size. For example, in the context of the supply-chain competition, it is argued that high or low risk is associated with the risk faced by the supplier with regard to the decision the supplier must make when choosing the amount to supply to different retail outlets. In this environment, it is the risk associated with the choice made by the supplier that is crucial rather than the risk attitude of the supplier who must make a decision. In our environment we are using the risk attitude of the decision maker (risk averse, risk neutral or risk seeking), to help us understand the decisions that are made by our decision maker given the degree of fairness that characterizes the choice that has to be made by our decision maker. The risk attitude of the decision maker and the riskiness of the situation characterize different contexts in which risk is important.

Variable Selection
To examine the research questions addressed in this study, we conduct a series of sessions in which individuals participate in an ultimatum game. All participants play the roles of sender and receiver. In this environment, each sender is a receiver of an offer from a participant different from the person to whom she sends an offer.

Design Variables
The first design variable is the repeated nature of the game. The game can be played as a one-shot game or as a repeated game. The second design variable is the way partners are assigned in the game. Assignment can be either random or fixed. The one-shot game is necessarily a random-partners game.

Repeated Game with Random Reassignment of Partners
The repeated game provides experience to a participant that is not available through a one-shot random interaction. Knowing that everyone in a session is participating in the repeated interaction may lead participants to consider their interactions as more than simple one-shot encounters; people will learn about the game through repeated interactions with many different people. Introducing the repeated game is an attempt to provide participants with the opportunity to learn more about the game. Maintaining random reassignment is an attempt to avoid confounding the effects of experience with the effects associated with narrowing social distance. This is similar to the repeated design used by [4].

Repeated Game with Fixed Partners
The fixed-partners environment in a repeated game may reduce social distance among individuals and lead to larger allocations than in the random-partners treatment. If you are matched repeatedly with the same sender you may be willing to reject an offer that you believe is too low with the intention of providing an inducement to the sender to respond with a larger offer. This could lead to acceptances being associated with larger offers over time. An alternative, however, is that rejections may lead to lower offers by disappointed senders and ultimately lead to acceptances associated with lower offers over time. The result is not obvious, but it may be affected by the assignment-of-partners treatment as

Risk Attitudes
An individual's risk attitude identifies the individual as being risk neutral, risk averse or risk seeking. Generally, a risk-neutral individual will accept a fair bet (or a bet whose odds favor the individual), a risk-averse individual will only accept a bet whose odds are in her favor while a risk-seeking individual will bet against the odds. Risk attitudes are measured using a BDM mechanism [11] that was extended by [12] [13] [14] [15], in which participants report their minimum selling prices for each of 24 lotteries with the prize of 100 Francs of laboratory currency (LF) but with different probabilities of winning.

Social Value Orientations
An individual's social value orientation identifies the individual as being altruistic, cooperative, individualistic, competitive or aggressive. 7 Social value orientations are elicited using the geometric framework introduced by [16] that was extended to the Ring Game by [17] in which participants select a desired income allocation between themselves and an anonymous other person from each of 24 pairs of income allocations.

Fairness Indices
To define the fairness indices, consider four of nine participants (identified as A through I) in a session of an ultimatum game. Participant A sends an offer to B, B sends an offer to D and G sends an offer to A. With respect to participant B, the first fairness index, identified as the Exogenous Index, is equal to the offer sent by A to the receiver B divided by the A's endowment. This is one measure of fairness that may be used to evaluate the likelihood that a receiver will accept an offer (see examples in [18] and [4]). The second fairness index, again with respect to participant B, identified as the Endogenous Index, is equal to the offer sent by A to the receiver B divided by the offer sent by B to D in the role of a receiver.
The first index is described as an exogenous fairness index because the denominator of the index is exogenous to the decisions made by the sender. It is the maximum amount that a sender is able to transfer to a receiver and is determined by the experimenter not the participant. Typically, as noted earlier, an Exogenous Index of 0.50 is described as characterizing a fair offer.
The second index is described as an endogenous fairness index because both 7 Examples of these categories can be described within the context of how an individual may choose between two different allocations of money between himself and another individual. An altruistic individual will select an allocation that transfers less money to the other person than the individual gives up. Cooperative individuals select allocations that maximize the total received by both people. Individualists select allocations in which their allocation is greatest. Competitors select allocations that give them more than the other individual receives. Aggressors select allocations that minimize the other individual's allocation. Theoretical Economics Letters the numerator and denominator of the index are determined within the experiment. From the perspective of receiver B, the denominator is the amount sent by B to individual D in the role of receiver. The denominator in the Endogenous

Index used to evaluate the likelihood that B accepts A's offer is interpreted as B's
notion of what is a fair share of her endowment to offer to D, and being a fair offer, it is expected to be accepted by C. This is a variant of the receiver's belief variable used by [4]. The receiver's belief about what others would find to be an acceptable offer was used to create an index that was introduced into an offer-rejection model. As the index increased, the likelihood of a rejection was conjectured to rise. Our use of the ultimatum game in which all participants played the role of both sender and receiver permits us to generate a proxy for the beliefs or expectations elicited by [4] without directly asking participants what they believed others would find acceptable. 8 From the perspective of B as a receiver, an Endogenous Index of unity identifies A's offer as a fair offer.

Experimental Design and Conjectures
A total of 86 participants were recruited from undergraduate classes at a medium-sized university. During the recruitment phase, students were told that the experiment involved simple decision-making, and that the details would be given to them during the session. In addition, they were told that they were required to participate in two separate sessions. Each session would be conducted on a different day and each session would last no more than two hours. They were also informed that during the course of the sessions they would earn money that would be paid to them in cash at the conclusion of each session.

Sessions
In the first session, we elicited participants' risk attitudes and social value orientations. The descriptions of the derivation of the risk attitudes and the social value orientation measures are included in [19]. 9  Each participant is endowed with LF500 at the beginning of each period. Beginning with the first round, participants simultaneously play two roles in the ultimatum game. Participants are separated into two groups; the groups are sent to different rooms. In each round each person (in the role of the allocator of resources) makes an offer to another person who is in the other room (who is randomly selected by the experimenter). The person described previously as the allocator, also plays the role of a receiver, and in this role receives an allocation from someone in the other room. 11 Following the first round, a second set of instructions is distributed and, again, read aloud by the experimenter. Participants are informed that the game will be played for three more rounds and the pairing protocol is explained to them. In some sessions, the pairings of participants are retained throughout the course of the second, third and fourth rounds. This is the fixed-partners treatment.
In the random-partners treatment, participants are randomly reassigned after each decision round. Participants always make allocations to and receive alloca- for about 60 minutes. The average payoff was $15.40. 12 The experimental design is summarized in Table 1.

Conjectures
The fixed-partner and random-partner treatments are relevant for the repeated game. The fixed-partner treatment narrows the social distance among individuals and is likely to contribute to a greater tendency to share. This suggests that Conjecture 1. People in a fixed-partners game will offer more than people in a random-partners game and in a one-shot game.
The inability of individuals to send signals and develop reputations in the 11 Suppose there are 6 people participating in a session. The even numbered people are in one room and the odd numbered people are in another room. Participants 1, 3 and 5 are in one room. Participants 2, 4 and 6 are in a different room. Participant 1 sends to participant 2. Participant 2 sends to participant 3. Participant 3 sends to participant 4. Participant 4 sends to participant 5. Participant 5 sends to participant 6. Participant 6 sends to participant 1. In fixed-partners treatments the participant pairings do not change over decision rounds. In random-partners treatments, the pairings are reassigned after each decision round. Participants with even ID numbers (in one room) are always interacting with participants with odd ID numbers (in another room). Theoretical Economics Letters one-shot game and the random-partners repeated game suggests that Conjecture 2. Offers in the random-partners repeated game will not be different from offers in the one-shot game.
The decisions of receivers are likely to be affected by whether they are in a one-shot game, a random-partners game or a fixed-partners game. For individuals in a one-shot game or a random-partners game their rejecting an offer does not send a signal that will necessarily bring a return in a subsequent round. This suggests that Conjecture 3. The likelihood of acceptance ceteris paribus will be greater for an individual in a one-shot game or a random-partners treatment than for an individual in a fixed-partners treatment.
If the offer made by an individual reflects the individual's notion of a fair offer this will be reflected in the decisions the individual makes as a receiver in the ultimatum game. This suggests that Conjecture 4. The greater is the fairness index, the greater is the likelihood that an offer will be accepted.

Data Analyses and Results
The analysis proceeds by first evaluating the data using the design variables of the model. The design variables consist of the game treatments (one-shot versus repeated) and the social-distance and reputation treatments (fixed-partners narrows social distance and can create reputations relative to random-partners).
If the randomization of participants to treatments is fully effective and if there are enough observations there should be no residual effects related to things other than the design variables. Results are first presented for the offer decisions.
These are followed by results for the offer-acceptance decisions.

Offer Decisions
The Offer Model presented in Table 2  The mean offers sent in each of the four rounds are presented in Figure 1.
The fixed-partner treatment leads to greater offers than when individuals are making offers to strangers. These differences are statistically significant (p = 0.041, p = 0.000 and p = 0.000 for rounds 1, 2 and 3 of the repeated games).   The round 1, 2 and 3 mean offers in the random-partners treatment are significantly lower than the mean one-shot game offer (one-sided tests, p = 0.061, 0.000 and 0.000). We can reject the null hypothesis that offers from random-partners games are equal to those of one-shot games in favor of the alternative that offers are lower than those in the one-shot environment. These results do not support Conjecture 2.
Our conjecture was that repeated play among participants in fixed-partners games would reduce social distance between participants relative to what would occur in repeated games with random partners. This would result in increases in offers in the former treatment and relative to the latter treatment. The data confirm this. What was unanticipated was the decay in offers by participants in the repeated game with random partners. Individuals in repeated games tend to contribute about 40 percent of their endowments when with fixed partners and about 30 percent of their endowments when with randomly assigned partners.

Design Variables
The Offer-Acceptance Model presented in Table 3  The mean offer acceptance rates in each of the four rounds are presented in Figure 2. Based on Table 3 regression results, none of these acceptance rates are significantly different from any other rate whether we compare across games, over rounds or across partnering environments. Considering only the design va- riables, it appears as if social distance does not matter and the reputation that 13 The mean estimated z-statistic using the regression coefficients in Table 3 and the 192 observations in the sample is 0.7416. This corresponds to an estimated likelihood of acceptance of 77 percent, which is also the actual percentage of offers accepted.

Behavioral Variables
The behavioral variables that may account for offer-acceptance  Table 2).  The mean Endogenous Index was 1.13 in the one-shot game. This fell to 1.02 and 1.05 for the fixed-partners and random-partners by the 3 rd round of the repeated games. A model identical to that presented in Table 2 for the offer sent, using only the design variables period and partner, does not account for a sig- This sub-section has described the fairness indices we will use to help understand the offer-acceptance behavior of participants in the ultimatum games. The role of social value orientations, risk attitudes, fairness indices and the desirability of the Endogenous Index relative to the Exogenous Index are the objectives of the next sub-section.

Behavioral Models
We are focusing on round 3 of the repeated game because this is the round for which the participants have the most experience and it is more likely that the behavior in this round is more representative of equilibrium behavior than that in earlier rounds. We compare the Offer-Acceptance Models of the one-shot game and the 3 rd round of the repeated game.
Coefficients and robust standard errors for alternative probit regressions of the Offer-Acceptance Model for the one-shot game are presented in Table 4. A comparable set of probit regressions is provided for the repeated game in Table   5. The only design variable included in these regressions is the variable representing the random-partner treatment (equal to 1 if the participant is in a random-partner game and 0 if the participant is in a fixed-partner game) in the regressions for the 3rd round of the repeated game. The constant terms in the regressions capture the estimates of the z-statistic representing the likelihood that an individual who is pro self and risk neutral in the one-shot game or who is pro self, risk neutral and in the fixed-partner game, will accept an offer. Table 4 presents three empirical models and Table 5    1) One-shot games • Model I Model I includes the Exogenous Index and the value orientation and risk attitude variables. The coefficient of Exogenous Index is positive and significant (p = 0.000). As the offer rises, the greater is the likelihood that the offer will be accepted. This provides support for Conjecture 4. The coefficient of Endogenous Index in Model III is positive and significant (p = 0.000). As the offer received rises or as the amount sent by the receiver falls, 15 It is possible that a fairness index may have a different impact on the likelihood that an offer is accepted in the fixed-partners treatment than in the random-partners treatment. If the coefficient of the interaction term were significant, this would identify such an effect. Also, the likelihood that an offer is accepted if it increases from LF175 to LF200 may be different if the Endogenous Index is 1 than if it is 0.8. If the product of the Endogenous Index and the coefficient on the interaction term were not significant, only the change in the offer would matter. If the interaction term were significant, both the change in the offer and the value of the Endogenous Index would matter. 16 "While pseudo R-squared values cannot be interpreted independently or compared across datasets, they are valid and useful in evaluating multiple models predicting the same outcome on the same dataset. In other words, a pseudo R-squared statistic without context has little meaning. A pseudo R-squared only has meaning when compared to another pseudo R-squared of the same type, on the same data, predicting the same outcome. In this situation, the higher pseudo R-squared indicates which model better predicts the outcome." See http://www.ats.ucla.edu/stat/mult_pkg/faq/general/Psuedo_RSquareds.htm Theoretical Economics Letters the greater is the likelihood that the offer will be accepted. This provides support for Conjecture 4.
The only behavioral variable that has a statistically significant impact on the likelihood of acceptance is the social value orientation measure. The Offer-Acceptance model for the one-shot game that includes the Endogenous Index displays a positive effect on offer acceptance as the social value orientation of an individual increases. More pro-social individuals are more likely to accept offers than pro-self individuals (p = 0.085). and as the Exogenous Index (the offer) rises, the greater is the likelihood that the offer will be accepted. This provides support for Conjecture 4.

2) Repeated games
The main effect of the Random Partners treatment is also significant (p = 0.003). However, whether the difference between the likelihood that an offer will be accepted in the fixed-partners or random-partners treatment depends on the size of the Exogenous Index. If the Exogenous Index is 0.5 (offer of LF250), it will be more likely that the offer will be accepted in the fixed-partners treatment than in the random-partners treatment and this difference is significant (p = 0.006). However, if the Exogenous Index falls to 0.36 (an offer of LF180), the difference is still greater for the fixed-partners treatment, but only marginally significant (p = 0.092). When the Exogenous Index falls below 0.355 (an offer of LF177.50) the difference, regardless of whether it is positive or negative, is no longer significantly different from zero. Conjecture 3 is that individuals in fixedpartners treatments will be less likely to accept offers than those in randompartners treatments. The data do not support this conjecture with Model I.
Using Model I, both the social value orientation measure and the risk attitudes of individuals have significant effects on the likelihood that offers are accepted.
More pro-social individuals are less likely to accept offers than pro-self individuals and risk-averse individuals are more likely to accept offers than risk-neutral or risk-seeking individuals.
• Model II The Model II regression in Table 5

Fairness Indices
To evaluate the effect of change in an index on the likelihood that an offer will be accepted in the repeated game requires the consideration of two cases: the fixed-partners case and the random-partners case. From Model II of the repeated game (Table 5) where α is the regression constant, B is the matrix of behavioral variables and β is the matrix of coefficients for these variables, Xi is the Exogenous Index and Ni is the Endogenous Index. We are interested in evaluating how zi(P = 0) and zi(P = 1) change when each index changes. The signs of the derivatives of equations (2) and (3) with respect to Xi and Ni will allow us to evaluate Conjecture 4. Let zi(P = 0)j and zi(P = 1)j represent the derivatives of Equations ( (1) and (2)) where j = (Xi, Ni). ceptance likelihoods 5 times out of 6. This provides qualified support for Conjecture 4 but does not support a conclusion that increases in either of these indices unconditionally results in an increase in the likelihood that an offer is accepted.

Partners Treatments
The Model II regression that includes behavioral variables as well as the Random Partners variable identifies a significant random-partners treatment effect in the 3 rd round of the repeated games that was not identified by the model that accounted for only design variables (see the last paragraph in subsection 5.2.1).
The following describes the effect of random partners (relative to fixed partners) on the estimated z-statistic (that captures the likelihood that an offer is accepted) for the 3 rd period of the repeated game using Model II. zi(P = 0) is the estimated z-statistic given a fixed-partners treatment and a given set of behavioral variables B and the fairness indices Xi and Ni where Xi is the Exogenous Index and Ni is the Endogenous Index. Table 6 presents the mean values for the fairness indices (pooled across partners treatments), their standard deviations and their ranges in the 3 rd round of the repeated games. The change in the z-statistic (and corresponding likelihood of accepting an offer) associated with a move from the fixed-partners treatment to the randompartners treatment is equal to ( ) ( ) zi P 1 zi P 0 zi = − = = ∆ . Based on Model II in If Δzi is evaluated at the sample means (Xi equals 0.38 and Ni equals 1.03) Δzi is greater than 0. This positive difference is statistically significant (χ 2 -test, p = 0.000). The implication is that at the sample means and given a set of behavioral variables, the likelihood that an offer is accepted will be greater for a receiver in the random-partners treatment than for a receiver in the fixed-partners treatment. We have also evaluated Δzi at combinations of the extreme values of the two fairness indices. These combinations, the resulting sign of the differences (Δzi) and the p-values associated with these differences are presented in Table 7.
These    between an individual's social value orientation and the likelihood that an individual will accept an offer (p = 0.394). We conclude that the narrowing of social distance through repeated interaction has dampened the effect that was associated with social value orientation in the one-shot game.

Why Does Someone Accept A While Someone Else Rejects B > A?
Consider the two pairs of players 1 and 2. Each player has an endowment of LF500. For the pair of players, player 1 sends LF200 to player 2 and player 2 sends LF100 to player 3. For the second pair, player 1 sends LF250 to player 2 and player 2 sends LF250 to player 3. Our comparison focuses on the decisions made by the two individuals in the player 2 roles. Who is more likely to accept the offer received from player 1?
Model I in Table 4 19 The difference is significant (p = 0.000). The individual who received the LF200 offer is more likely to accept this offer than the individual who received the LF250 offer.
For the repeated game, both of the fairness indices are included in the offer-acceptance model. Using the regression for Model II in Table 5, for both the fixed-partners and random-partners treatments, the difference between the likelihoods that the offers described above are accepted is determined by the sum of three terms. 20 When these sums are evaluated for the two individuals we have described above, we find that for both the fixed-partners and random-partners treatments the likelihood that the LF200 offer is accepted is greater than the likelihood that the LF250 offer is accepted. 21 Including the Endogenous Index in the Offer-Acceptance model extends the usefulness of the model. With this extension, the model can accommodate situations in with offers that may be characterized as fair may be less likely to be accepted than offers that may appear to be unfair offers. This arises because in addition to the offer, the Endogenous Index includes information about what the receiver sends to others which may be interpreted as what the receiver believes is a fair share of her endowment to allocate to another individual. The amount sent to another by the receiver permits us to generate an endogenous measure of fairness that is not captured by the Exogenous Index made up of the offer and the sender's endowment. This is comparable to obtaining information about the participants' beliefs or expectations, but without alerting the participants to this by explicitly asking about them (which could frame the game in a way that may affect offers). 22

Conclusions
We have used a repeated ultimatum game to examine the role of social distance 19 11.736 is the product of the regression coefficient 5.868 (from Model III in Table 4) and the Endogenous Index 2.00 and 5.868 is the product of the regression coefficient and the Endogenous Index 1.00. 20 For the fixed-partners treatment the three terms are −257.803X − 109.565N + 330.856XN where X and N are the values of the Exogenous and Endogenous Indices. For the random-partners treatment the three terms are −4352.404X − 1501.543 + 6327.811XN. These are from Model III in Table 5. 21 In the fixed-partners treatment the sum of the three terms for the first person is −57.566 and for the second person it is −73.039. In the random-partners treatment the sum of the three terms for the first person is 318.201 and for the second person it is −513.84. In both cases the larger terms are significantly greater than the smaller terms (p = 0.0425 in the fixed-partners treatment and p = 0.000 in the random-partners treatment. 22 Using the offer-acceptance regression (6) from [4] (Table 4, p. 9) we find that if the change in the expectation variable is less than 2.429 times the change in the offer proportion the probability of rejecting the offer falls. Comparing the two cases used in the text above, the change in the offer proportion is -0.1 and the change in the expectation variable is −0.3. −0.3 is less than −2.429. For this example, using the Offer-Acceptance model presented by [4] in regression (6), it is more likely that the higher offer will be rejected than the lower offer. The belief or expectation is an important variable. Theoretical Economics Letters on offers made and on offer-acceptance decisions. The environment used had participants playing the roles of both senders and receivers. This enabled us to create indices that capture different concepts of fairness that we used to evaluate acceptance decisions. The ultimatum game environment enables us to gather information that we believe is a proxy for the beliefs or expectations that receivers have about what constitutes a fair offer. The fairness indices that we use in our offer-acceptance models include the conventional fairness index measured as the amount sent divided by the sender's endowment as well as the ratio of what is sent to the receiver divided by what the receiver sends to another participant in the receiver's role as a sender. In addition to the inclusion of partners treatments to capture social distance and the fairness indices in the offer-acceptance models, we introduce measures of social value orientation as well as risk attitudes as variables in our offer-acceptance models.
A particular objective was to identify a fairness index that would permit scenarios in which the likelihood of accepting an unfair offer could exceed the likelihood of accepting a fair offer. We show that the Endogenous Index successfully accomplishes this. This is an advantage over the conventional fairness index, which is the Exogenous Index in our experiment.
The behavioral variables that we introduce into our analysis help account differences in behavior over time. Pro-social individuals are more likely to accept offers in one-shot games than pro-self individuals but the significant effect of social value orientation disappears by the 3 rd round of the repeated games (with the narrowing of social distance).
Risk attitudes do not have a significant impact on the likelihood that an offer is accepted in the one-shot games, but by the 3 rd round of the repeated game, the likelihood of offer acceptance is greater for risk-averse individuals than risk-neutral individuals and is greater for risk-neutral individuals than risk-seekers.
We find that narrowing social distance among participants in the repeated ultimatum game by using fixed-partners leads to significant increases in offers over those sent when participants are randomly matched period after period in the repeated game. In the random-partners treatment, the mean offer falls to 31 percent of the sender's endowment by the 3 rd period of the repeated game. In the fixed-partners treatment, the mean offer rises to 42 percent of the sender's endowment. These compare with 38 percent in the one-shot game. Even with the narrowing of social distance in the fixed-partners treatment, the offer remains significantly below the conventional fairness measure of 50 percent of the sender's endowment. It may be that the participants believe that the role of sender Our experiment does not compare the effectiveness of the alternative methods of collecting information about beliefs or expectations and whether our conjecture that asking for beliefs of what is a fair offer or asking what individuals expect others will accept will lead to different offer or offer-acceptance behavior than what we find in our sessions. Any differences that may exist between the alternative methods would be worthwhile identifying.
Extrapolating beyond our data, our results that notions of fairness may not be fixed for all parties in a bargaining situation may help us understand some decisions that we observe some economic agents make when they forfeit potentially good opportunities in business interactions such as negotiating mergers and acquisitions or labour and management contracts. Similarly diverse notions of fairness can easily be the source of failures in legislative interactions or treaty negotiations among national governments.