extfieldsofstudy: Psychology Medicine

text:
Discounting and Digit Ratio: Low 2D:4D Predicts Patience for a Sample of Females

Inter-temporal trade-offs are ubiquitous in human decision making. We study the relationship between preferences over such trade-offs and the ratio of the second digit to that of the forth (2D:4D), a marker for pre-natal exposure to sex hormones. Specifically, we study whether 2D:4D affects discounting. Our sample consists of 419 female participants of a Guatemalan conditional cash transfer program who take part in an experiment. Their choices in the convex time budget (CTB) experimental task allow us to make inferences regarding their patience (discounting), while controlling for present-biasedness and preference for smoothing consumption (utility curvature). We find that women with lower digit ratios tend to be more patient.

INTRODUCTION
Human decisions involving inter-temporal outcomes are ubiquitous. For example, decisions involving savings and consumption, investments in physical and human capital, and career and health choices all involve trade-offs across time. Economists and other social scientists typically study inter-temporal choices using models which parameterize how an individual weights consumption at different points in time. In particular, discounted utility models assume that individuals place a higher weight on consumption that is sooner; that is, individuals discount the future. Richer models allow for other factors that may also affect inter-temporal choices, such as utility curvature (i.e., the preference to smooth consumption over time), and present biasedness (i.e., higher discounting of the future if choices involve present outcomes) 1 .
Time preferences are heterogeneous among individuals (Harrison et al., 2002;Andreoni et al., 2015). That is, individuals vary in the degree to which they discount the future (their patience), in their preference to smooth consumption, and in their degree of present-biasedness. Given this heterogeneity and that the domain of inter-temporal preferences includes choices over important human capital decisions, it is not surprising that measures of discounting correlate with smoking, alcohol consumption addiction, and drug abuse (Kirby et al., 1999;Mitchell, 1999;Petry, 2001;Chabris et al., 2008;Sutter et al., 2013). In addition, Cadena and Keys (2015) finds that impatient individuals are more likely to make investments that can be classified as dynamically inconsistent and consequently end up with lower income on average. Golsteyn et al. (2014) finds that high discount rates have a negative relationship with school performance, labor supply, health and income. Kirby et al. (2002) also reports evidence of patience being positively correlated with literacy and schooling among the Tsimane' in Bolivia.
Thus, understanding the underlying determinants of intertemporal preferences can help improve our understanding of human behavior over countless domains, as well as the welfare consequences thereof 2 . Indeed, we still know relatively little regarding the underlying determinants of inter-temporal preferences.
In this paper we examine whether a link exists between discounting and second-to-fourth digit length ratios (2D:4D) 3 . 2D:4D is a marker for pre-natal exposure to sex hormones (testosterone and estradiol) in males and females (Manning, 2002;Lutchmaya et al., 2004;Zheng and Cohn, 2011). Evidence suggests that exposure to sex hormones in utero has an organizational effect brain development (Goy and McEwen, 1980;Manning et al., 2001).
If exposure to sex hormones in utero has an effect on the brain, then examining a potential effect on time preferences seems warranted. Several studies find that higher cognitive ability is associated with more patience (Shamosh et al., 2008;Burks et al., 2009;Dohmen et al., 2010;Benjamin et al., 2013). Frederick (2005) introduced the cognitive reflection test (CRT), a simple test designed to capture the cognitive capacity to override an intuitive wrong answer and reflect upon the simple yet non-intuitive correct answer. High scores in this test correlate with higher cognitive abilities (as measured by the Wonderlic Personnel Test, the Need for Cognition Scale, etc.). Furthermore, Frederick finds that individuals with higher CRT scores are generally more patient (using hypothetical choices). In addition, Bosch-Domènech et al. (2014) reports that lower 2D:4D measures are associated with higher scores on the CRT. Collectively, these studies provide a rationale to examine the relationship between 2D:4D and discounting. We use an experimental task, the convex time budget (CTB), to measure time preferences. This method has the advantage of allowing simultaneous structural estimation of discounting, utility curvature, and present-biasedness. The simultaneous estimation is important, as estimating them separately often results in estimates of discounting that are unrealistically high (Andersen et al., 2008).
External validity of time preferences measured via experimental tasks has been documented with different samples. Among school children, experimental measures of impatience are significant predictors of savings decisions, health behavior and school misconduct (Castillo et al., 2011;2 Several papers attempt to explore the covariates of time preferences (Lawrance, 1991;Pender, 1996;Harrison et al., 2002;Tanaka et al., 2010;Cassar et al., 2017). However, establishing a causal effect between the covariates and time preferences has proven to be challenging. For instance, Carvalho et al. (2016) attempts to explore the impact of poverty or lack of liquidity on discounting. 3 The null hypothesis is that no correlation exists. As specified in our registered analysis plan, our alternative hypothesis is that 2D : 4D is negatively correlated with patience; that is, low digit ratio is related to a higher degree of patience. Sutter et al., 2013). Experimentally elicited present-biasedness is correlated with credit card debt among a sample of adults in Massachusetts (Meier and Sprenger, 2010), and predicts payments for environmental services in a sample of Ugandan farmers (Clot and Stanton, 2014). With the experimental task and sample reported here, (Aycinena et al., 2017) shows that preferences for consumption smoothing predict choices among a menu of payment options with large stakes.
There has been limited attention paid to the relationship between 2D:4D and time preferences. Drichoutis and Nayga (2015) uses two experimental tasks involving multiple price list to separately measure risk and time preferences and relates them to 2D:4D. Their evidence is mixed, but suggests that there may be a negative relationship between 2D:4D and discounting. Our paper differs in several important ways: first, they have a final sample of 138 (77 female) university students, while we have a sample size of 419 females who are not students. Second, we use five independent measures of 2D:4D taken from scans of our subjects hands using software designed for this purpose. This is intended to minimize measurement error, and increase the reliability of our measurements. Drichoutis and Nayga (2015) use rulers to measure 2D:4D, and did not scan the hands of their subjects. Third, they used the Holt and Laury (2002) method to measure risk aversion (which is presumed to measure utility curvature). This method involves subjects choosing between lotteries. We employ the CTB task, which does not involve choices over lotteries. Lucas and Koff (2010) analyzes the relationship between 2D:4D and delay discounting, but does not consider other parameters involved in inter-temporal choices (consumption smoothing and present-biasedness). They only find a significant relationship for the right hand for women. They find that a lower 2D:4D ratio is associated with greater delay discounting. Our paper differs significantly from this study in that we use a large sample of non-students, use a different elicitation method and jointly estimate multiple parameters underlying intertemporal preferences.
In addition to contributing to the hormones and economic behavior literature, this study also contributes to the economics literature exploring time preferences on three fronts. First, a robust correlation between time preferences and 2D:4D would provide an exogenous determinant of individual time preferences which could serve as an exogenous instrument to examine causal relations between time preferences and other economic behavior. This could be an important tool to examine causal relationships; for instance, in the growing literature exploring the link between patience and social preferences (Curry et al., 2008;Espín et al., 2012Espín et al., , 2015. Second, most economic theories implicitly or explicitly assume the stability of choice primitives (such as time and risk preferences) and there is empirical evidence of some stability in time preferences at the individual and aggregate levels (Kirby, 2009;Meier and Sprenger, 2015). The link between pre-natal exposure to hormones and time preferences suggests a (partial) mechanism through which time preferences can be heterogeneous across individuals and relatively stable over time. Finally, the third front links to the literature that shows that patience is correlated with higher cognitive ability (Shamosh et al., 2008;Burks et al., 2009;Dohmen et al., 2010;Benjamin et al., 2013). Given that cognitive ability seems to be correlated with 2D:4D (Brañas-Garza and Rustichini, 2011;Bosch-Domènech et al., 2014), our results may suggest a potential mechanism through which 2D:4D affects patience.

MATERIALS AND METHODS
Acuerdo ministerial SP-M-466-2007 (regulating human clinical trials in Guatemala) did not apply to our study and no ethics committee has existed at our (former) institution in Guatemala. Nevertheless, we adhered to standard protocols involving studies that use experimental methods and measures of 2D:4D; specifically, no deception was used in the experiments, we obtained informed consent from participants, and we ensured privacy and security of data and decisions 4 .

Participants
Our sample consists of beneficiaries of Guatemala's Conditional Cash Transfer (CCT) program 5 . Due to CCT program requirements, our sample is 99.1% female and not representative for Guatemala 6 . As might be expected, relative to female respondents on a national representative survey, participants in our experiment are poorer, more likely to be or have been married, live in larger households and their living quarters are more precarious 7 . 4 Given the anticipated low levels of schooling and literacy, assistants read the informed consent sheet to each individual, marked whether subjects gave informed oral consent, and signed the sheet. 5 Mi Bono Seguro (My Security Bonus) is a targeted CCT program overseen by the Ministerio de Desarrollo Social (Ministry of Social Development) of Guatemala. It aims to improve human capital accumulation by promoting investments in health and education for poor households with pregnant women or children under the age of 16. 6 As is conventional among CCT programs, females tend to be the recipients of the funds. This program uses geographic targeting and proxy means testing for eligibility. This program offers two types of conditional transfers: an education transfer and a health transfer. To obtain the health transfer all children under 15, and all pregnant or breastfeeding woman must attend regular medical check-ups. To obtain the education transfer all children between the ages of 6 and 15 must have a school attendance rate of at least 90%. Households may be eligible for both transfers. 7 We compared our sample with the 2011 National Survey of Living Conditions (ENCOVI). ENCOVI is a national representative household survey focused on the measurement of living standards run by the National Institute of Statistics (INE) of Guatemala. To maximize comparability, we restricted attention to female ENCOVI respondents in a comparable age bracket. For detailed results of this comparison, see Aycinena et al. (2015). Not surprisingly, there are limitations with the comparison between our sample and the ENCOVI data. ENCOVI is a national representative survey that was implemented between March and August of 2011, After dropping some observations, the final sample in our analysis consists of 419 individuals 8 . These subjects reside in seven different municipalities across three departments: (El Progreso, Escuintla, and Sacatepéquez) where we ran experimental sessions. Ages range from 20 to 76 (mean 35.9, median 35). All of these women, as a condition for eligibility in the CCT program either have children or were pregnant at the time of the experiment.

Experiment
Participants performed several independent experimental tasks. The first and main task elicits inter-temporal choices using a version of the CTB introduced by Andreoni and Sprenger (2012a,b). The other tasks (which are not used in the current analysis) involve choosing how to spread receipt of financial windfall gains over time when there is no cost associated with receiving funds earlier, eliciting a subject's willingness to forgo funds in order to maintain intra-household control of a financial windfall, and/or a hypothetical CTB which elicited how subjects believed they would behave if questions were asked at a future date.
Participants earn an initial amount of GTQ50 (approximately USD6.4 or PPP$12.3) for taking part in the experiment 9 . In addition, they could earn between GTQ45 -GTQ100 (PPP$11.1 -PPP$24.7) based on their choices in the CTB. To put these amounts in context, CCT's entitled a household to receive GTQ150 (USD19.2 or PPP$37) per month, provided all household members comply with the conditions. Median selfreported household monthly income for the sample was in the range from GTQ500 to GTQ1,000 (PPP$123.5 to PPP$246.9) and 90% of participants report monthly household income below GTQ2,000 (USD256 or PPP$494).

Convex Time Budget (CTB) Task
In the CTB, participants see a series of 24 questions, knowing in advance that one of them will be randomly selected to determine their earnings. Each question presents a choice among six options that involve a combination of money to be obtained at two different times: t and t + k days after the experiment 10 . Implicit in the options was a trade-off between receiving money earlier (at time t) vs. delayed (time t + k): each of these 24 questions allowed subjects to eliminate the delay of partial amounts of money, by "transforming" delayed money (at time t + k) into early money (at time t) at a constant rate (marginal rate of transformation or MRT) that was weakly greater than one.
More specifically, in each question, one option is GTQ100 at time t + k, and GTQ0 at time t (not including the split payments participation fee). Each of the remaining five options involve shifting GTQ20 from time t + k to time t at a constant marginal transformation rate (MRT) or relative price, until only GTQ0 remains at time t + k. Figure 1 illustrates the six options for a question (using MRT = 1.18, t = 0, and k = 35) as presented to participants 11 .
We used two values of t: t = {0, 35}. Each of these, were combined with two different delays: k = {35, 63}. The variation in the delay (k) allows inference regarding discounting of future utility, and the variation in the early period (t = 0 or t > 0) allows inference regarding present-biasedness. For each of the four combinations of t and t + k, participants are presented with six questions, each with a different MRT. As previously mentioned, each question presented six options to choose from. These include two options "at the corners" (all the money delayed or all early) and four options of "interior choices" (involving combinations of both, delayed and early money). The availability of interior choices allows inference regarding preferences for consumption smoothing (Aycinena et al., 2017). Table 1 summarizes the parameters used.
Payments were implemented via post-dated checks made out to the participant. As in Andreoni et al. (2015), to guarantee that the transaction costs associated with obtaining the two associated payments are the same, the GTQ50 participation payment is evenly divided between the payment at time t and the payment at time t + k 12 .
We vary three things between experimental sessions to control for order effects. First, for each pair of t and t + k, we varied the order in which participants see the associated six questions. In some sessions the relative price of money at time t is decreasing over the six questions, and in other sessions it is increasing. We refer to this as the decreasing opportunity cost (DOC) treatment. Second, in some sessions the options within a given question are ordered such that the amount at time t is monotonically decreasing, and in other sessions it is increasing. We refer to this as the decreasing soon amount (DSA) treatment. Third, 11 Since participants have low levels of literacy and numeracy, we presented all choices in the CTB using both numbers, and pictures of the associated quantities of money. Notice that each option specified the amount at time t and the amount at time t + k; as well as the total amount. To further ensure that participants understood the task, assistants asked each participant the questions individually, resolved any questions as they arose and recorded the participant's decision. 12 During the implementation there was a problem with the post-dated check payment mechanism, as some participants were able to cash checks earlier than the dates indicated on them. This would be problematic for our parameter estimates if participants anticipated that this was a possibility, as their effective MRT would then be equal to one in all cases. More specifically, if participants anticipated this, then we would expect that they would choose the option that would allow them to maximize the total amount of money over early and delayed payments. As long as the experimental MRT was greater than one, they would choose the option with the minimum early payment and maximum delayed payment. However, this is not what we observe. Reduced form regressions on early check cashing find no statistically significant correlation between cashing checks early and choosing options that concentrate amounts on delayed payments. Results are available upon request.
in some sessions, the GTQ25 payments for taking part in the experiment which was added to both the payment at time t and time t + k was explicitly shown in each question, and in others it was not. Note that this information was provided to participants prior to the CTB. This treatment simply varies the salience of the participation fee. We refer to this treatment as the included participation fee (IPF) treatment.

Sessions and Protocols
Experimental sessions took place in multipurpose rooms in the municipalities where subjects reside. We ran a total of 23 sessions with 16-24 subjects per session. Each session lasted between 3 and 4 h. All sessions were conducted by a session leader and a team of assistants.
Participants were asked to give informed consent upon arrival. After welcoming participants and giving a general introduction, the session leader projected at the front of the room and read aloud instructions for the CTB 13 . Afterwards, assistants ask each participant to answer several questions to ensure understanding. Then, assistants individually elicit answers for the first six questions (for t = 0 and k = 35, with MRT varying across questions). As noted above, since many participants are illiterate it was important for assistants to provide individual support and show decision sheets (illustrating the available options with pictures of the relevant monetary amounts) for each question. Once all participants have answered the first six questions the session leader explains the changes for the following six questions and assistants individually elicit participant responses. This process continues until all 24 questions of the CTB have been answered.
Once the CTB task is complete, the session leader reads instructions for the remaining tasks and the experiment continues until all experimental tasks are completed. Participants then got a short break where beverages and snacks were provided. A bingo cage was used to determine the question from the CTB task that would be paid. Assistants individually interviewed each participant for a socioeconomic survey. Participants were then called individually to receive their checks and sign receipts. At this time they were asked if we could scan their hands. If they consented to this, their hands were then scanned.

Digit Ratio (2D:4D) Measures
We collected scanned images of the participants' hands 14 . After all images were collected, a research assistant randomly divided the images into five batches 15 . Each batch contained a total of 108 images, including 10 re-inserted images from other batches (so that each rater measured the 2D:4D ratio for a total of 50 subjects twice). These repeated measures serve as the basis for assessing the consistency of measurement for each rater. 13 The supplementary material shows the text of the instructions for both experimental tasks, translated from the original Spanish. 14 Using a digital scanner is a common method for taking digit ratio measures that has been shown to be reliable (Kemper and Schwerdtfeger, 2009). An example of a scan can be seen in Figure 2. 15 We split the measurement of images into batches to break the task into smaller sub-tasks, in an attempt to reduce the effects of fatigue or boredom for research assistants measuring the digit ratios.  Eight raters were instructed and received guidance on using the Autometric software (DeBruine, 2004) designed to measure digit ratios. They then independently measured both hands for each image in all five batches. The order in which each rater received the five batches was randomized. Thus, we collected 8 independent 2D:4D measures for each hand of all participants. In addition, we had 50 randomly selected images measured twice by each rater. The repeated measures for the 50 randomly selected images allowed us to measure intrarater consistency of 2D:4D measures. We drop the measures for three raters with an intraclass correlation coefficient (ICC)  Table 2 shows measures of intra-rater consistency. Table 3 displays the between-rater correlation coefficients. Between rater correlation coefficients range from 0.8663 to 0.9392 for the right hand measures, and from 0.7546 to 0.9668 for the left hand.
We take the average across the five measures 16 . Table 4 shows the summary statistics for the 2D:4D measures. The digit ratios for our sample are lower than those typically found in the literature. For the right hand, mean 2D:4D is 0.9322 (with a standard deviation of 0.0315); for the left hand the mean is 0.9337 (with a standard deviation of 0.0321) 17 . No statistical significant difference is found in variance or mean between hands. Figure 3 illustrates the distribution of the average of all five measures for both hands.
Thus, our final 2D:4D data consists of the average of five (high quality) independent measures for the 419 final sample subjects.

Plan of Analysis
Given the so called "replicability crisis" in scientific findings (see e.g., Ioannidis, 2005;Button et al., 2013;Aarts et al., 2015;Camerer et al., 2016), we attempted to limit the degrees of freedom available to us as researchers 18 . 16 Voracek et al. 18 Studies may give researchers many degrees of freedom, even without explicit fishing (Gelman and Loken, 2014). In 2D:4D research this problem is not absent; if anything it may be exacerbated as there is no consensus regarding which hand  To limit the degrees available to us, we partnered with Anna Dreber to prepare an analysis plan 19 . In the plan, we specify that our main method of analysis will rely on the interval censored Tobit model to structurally estimate time-preference primitives, which allow discounting to vary with 2D:4D. Specifically, we estimate discounting (δ) as a linear function of 2D:4D (among other parameters).
In the analysis plan we also specify three robustness tests. First, we test robustness to changes in the background parameters, since (Andreoni et al., 2015) and (Aycinena et al., 2017) show that the structural estimates may be sensitive to whether or not the participation fee (among other background parameters) is included in the analysis. Thus we perform two robustness checks which modify assumptions about the background parameters.
Second, we examine whether the results are robust at the individual level. To do so, we structurally estimate timepreference primitives at the individual level, and test whether the individual level estimates for δ are correlated with the individual 2D:4D measures. Finally, our third robustness check tests whether our results depend on the method of structural estimation. To do so, we drop the structural estimation approach and test whether 2D:4D measures predict choices of more delayed money using reduced form analysis.
to use, which measures (mean, median, etc.) to use, or the correct specification (linear, quadratic, etc.) to employ. 19 We thank Anna Dreber for her time helping us prepare the analysis plan while she was blind to the data. The plan is posted at the Open Science Framework web platform: https://osf.io/ey67f/register/564d31db8c5e4a7c9694b2be. It should be noted that, technically, this is not a pre-analysis plan, since we developed it after data collection was finished. Nevertheless, we feel that by developing it jointly with a credible third party, it helps to reduce the degrees of freedom of our analysis.

Theoretical and Econometric Framework
To analyze choices, we rely on a model inter-temporal preferences that assumes a time-separable quasi-hyperbolic utility function with constant relative risk aversion. Specifically, denoting the amount of money received by subject i at time t (t + k) as x it (x it+k ), we assume that the following utility function underlies observed choices: Our framework includes three parameters that affect timepreferences: discounting (δ), present biasedness (β) and utility curvature (α). The discount factor, δ, captures the degree to which an individual discounts delays in consumption. A δ = 1 implies that individuals are so patient, that all else equal, they are indifferent to delays in consumption. The lower the value of δ (δ < 1) implies higher discounting of delaying consumption, that is, less patience. Present biasedness, β < 1, captures how much (more) an individual discounts delaying consumption relative to immediate consumption. Note that β = 1 implies a standard discounting model with no present biasedness. Finally α, utility curvature, underlies preferences to inter-temporally smooth consumption. An α = 1 implies that consumption is perfectly substitutable across time, thus no preference to smooth consumption in time. The lower the value of α (α < 1) the higher the preference to smooth consumption. That is, all else equal, the lower α, the more an individual is willing to sacrifice in order to attain a consumption profile that is smoother across time.
Notice that these three parameters are interrelated for timepreferences. That is, it is possible to observe the same choice by two individuals with very different levels of patience (different δ's) if there utility curvature (α) and/or present-biasedness (β) also differ. Given this, it is important to estimate these three parameters jointly (see e.g., Andersen et al., 2008;Andreoni and Sprenger, 2012a).

Main Analysis: Structural Estimation
In our main analysis we employ interval censored tobit regressions 20 . This procedure jointly estimates three parameters: α, β, and δ.
The parameter δ is the aggregate measure of the time preferences in the population (see Andreoni et al., 2015 for a detailed description of the model and the estimation techniques). To test our hypothesis, we allow δ to be a function of the 2D:4D ratio. As specified in our analysis plan, the functional form we assume is as follow: 20 For the structural estimation, the covariance matrix was estimated using sandwhich estimator for robust standard errors. See Aycinena et al. (2014) for a detailed description of the estimation method.
where experimental treatments [included participation fee (IPF) explicitly treatment, decreasing opportunity cost (DOC) treatment, decreasing soon amount (DSA) treatment] are included to control for differences in how the CTB task was presented to subjects.
The first two columns of Table 5, estimated separately, present results of the parameter estimates for the left and right hands of participants. The value for the parameter α shows a strong preference for smoothing consumption over time. The β parameter is higher than one, thus it shows no evidence of present-biasedness 21 . Next we present results in which the parameter of interest, δ, is a function of 2D:4D and treatment controls.
For the parametrization of the discount factor (δ), we see that the coefficient on 2D:4D is negative (−11.899 for the left hand and −15.959 for the right hand) and statistically significant for both hands at the 0.001 level. This implies that lower 2D:4D is correlated with a higher discount factor. That is, individuals with lower 2D:4D (a marker for higher exposure to testosterone in utero) make more patient choices.
Following our analysis plan, we also explore whether there is evidence of a non-linear effect of 2D:4D on discounting. Specifically, we examine whether there is a quadratic relationship by adding 2D:4D 2 as an explanatory variable. Under this specification (not reported but available from the authors upon request), we find that both the linear and squared coefficients Robust standard errors are reported in parenthesis. + p < 0.1, *p < 0.05, **p < 0.01, ***p < 0.001.
Frontiers in Behavioral Neuroscience | www.frontiersin.org are negative, but none are statistically significant at conventional levels.

Robustness to Changes in Background Parameters
It should be noted that the previous parameter estimates may be sensitive to whether or not the participation fee, among other background parameters, is included (e.g., Andreoni et al., 2015;Aycinena et al., 2017). Since all subjects received the participation fee, we included it (Q50, split evenly across two time periods) as a background parameter in the estimates reported in the previous section. For our first set of robustness checks, we test how sensitive our results are to modifying the background parameters. We examine two alternative specifications of the background parameters. Our first examination involves dropping the participation fee from our analysis, so that x it and x it+k do not include the participation fee in our econometric analysis. We report the results for left and right hand in columns 3 and 4 of Table 5 (under the heading "Robustness check 1.1") . For the second, we estimate the parameters with the explicit option displayed to participants, according to the IPF treatment 22 . The last two columns of Table 5 (under the heading "Robustness check 1.2") report the results of such estimates.
As the table shows, estimates of α seem to be quite sensitive to the background parameters used. The estimate of β on the other hand, seems quite robust. Regarding our coefficient of interest, although not quite as sensitive as α, δ does vary with the background parameters employed. Although the impact is not obvious due to the five parameters involved in the estimation of δ, the mean value of δ ranges from 0.6 to 0.85. Nevertheless, the point to note is that the coefficient on 2D:4D is negative and statistically significant (p < 0.001) for both hands across all specifications. Thus, the relationship between 2D:4D and patience reported in the previous section seems robust to the specification of the background parameters.

Individual Level Estimates
The second robustness check involves attempting to estimate time preference primitives at the individual level. We use the interval censored Tobit model with 24 observations per individual (one observation for each of the 24 questions of the CTB) and attempt to jointly estimate α, β, and δ.
Unfortunately, our individual estimates are very imprecise. For our parameter of interest, δ, values range from 0 to 1.4e 191 , and the distribution is very skewed with a mean of 3.4e 188 , and for over half of the observations the estimate of δ < 0.0001. 23 This lack of precision is not surprising given that for each individual, we have 24 observations to estimate eight parameters 24 . To try to overcome this problem, we restrict our analysis to individuals with an (arbitrarily defined) sensible δ parameter: individuals with 0 < δ < 2. This reduces drastically our subsample to 168 individuals.
We use the parameter estimates for the 168 individuals of our restricted sub-sample as a dependent variable and estimate the following reduced form model (separately for left and right hands) using OLS: We present results in the first two columns of Table 6. For the sake of brevity, we only present the results for 2D:4D (point estimate of ρ 1 and its standard error) and the adjusted R 2 . The top row presents the 2D:4D coefficient for the left hand and the 23 The 25th percentile is zero, with a mean of 3.4e 188 and median of .00001. 24 The three parameters which measure preference primitives (α, β, and δ), in addition to the auxiliary parameters (σ , and the five cut-offs λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ). Point estimates for 2D:4D coefficient of the robustness checks. Robust standard errors are reported in parenthesis (clustered at the individual level for robustness checks 3.1 and 3.2) Robustness checks 2 and 3.2 are estimated using OLS; adjusted R 2 for each hand is reported below standard errors. Robustness check 3.1 is estimated using ordered probits; Pseudo R 2 is reported below the standard errors. + p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.
bottom row for the right hand, each estimated independently. None of the coefficients are statistically significant. The signs of the coefficients are consistent with our main analysis, except for the left hand when we include session and surveyor fixed effects. The adjusted R 2 is negative for all four specifications of robustness check two, which indicates that the model is a very poor fit for the data 25 . Overall, this suggests that this approach was not successful in allowing us to test the robustness of the results 26 .

Reduced form Analysis
In our third robustness check, we bypass the structural estimation and directly examine choices with a reduced form approach. The independent variables we employ include our variable of interest (2D:4D), the marginal rate of transformation for the question (MRT j ), the time when the early amount is to be received (t j ), the delay (k j ), and controls for our three treatment variables (DSA, DOC, IPF). Since we have multiple observations per individual, we cluster standard errors at the individual level. In all of our reduced form analysis, we estimate the model for both right and left hand 2D:4D.
Since participants could choose among six discrete ordered options (Y ij ∈ [1, 2, . . . ., 6]), we first examine this using an ordered probit model. Choosing option 1 maximizes the amount received in the early payment; choosing option 6 maximizes the amount received in the delayed payment. Thus, all else equal, a more impatient individual (i.e., with a lower δ) will tend to select lower options than a more patient individual (someone with a with higher δ). If our results are robust, we would again expect a negative coefficient for 2D:4D.
We present the results (of our coefficients of interest) in the middle columns (Robustness check 3.1) of Table 6. Column (1) presents the coefficients for the model described above. We find that for both hands, coefficients are negative and statistically significant (p < 0.01). Again, this supports the findings from the main estimates that lower 2D:4D individuals make more patient choices. Column (2) adds session and surveyor fixed effects. Under this specification, the coefficient for the left hand is no longer statistically significant at conventional levels (p < 0.1).
For our second reduced form approach, we use ordinary least squares and the dependent variable is the early amount chosen (x ijt ) by individual i in question j. We use the same independent variables, with our focus again being on the coefficient of the 2D:4D 27 . Notice the the higher the early amount chosen, the more impatient the individual (given the tradeoffs between early 25 It should be noted that this is not driven by the 2D:4D measure, as a model that excludes 2D:4D as an explanatory variable also has negative adjusted R 2 of similar magnitude. More importantly, the partial R 2 (or coefficient of partial determination) of the 2D:4D coefficient is always positive, suggesting that if anything, it helps the model fit of the data (although clearly not enough). 26 Although our attempt to estimate parameters at the individual level failed, we believe important to stick to our analysis plan and report the attempt despite its failure. 27 It should be noted that the analysis plan specified that the dependent variable for this approach would be the delayed amount chosen. That is a mistake, since the delayed amount is a linear transformation of the dependent variable used in the first approach (Robustness check 3.1). Results are qualitatively and statistically the same if we use delayed amount as our dependent variable. and delayed amounts). Thus, in this approach, we expect a positive correlation between 2D:4D and our dependent variable.
Results for our coefficients of interest are reported in the last two columns (Robustness check 3.2) of Table 6. For the first specification (Column 1), the coefficients for both hands are positive and statistically significant (p < 0.01). In column (2) we add session and surveyor fixed effects. In this case, the coefficient for the left hand is no longer statistically significant at conventional levels (p < 0.1).
Again following our analysis plan, we perform an exploratory analysis of whether the relationship between 2D:4D and discounting is non-linear by adding 2D:4D 2 as an explanatory variable. We do not find any robust evidence for a non-linear relationship between 2D:4D and discounting. Coefficients are not statistically significant either in the ordered probit or the OLS model.
To summarize this last robustness test, we find that results do not depend crucially on the assumption and methods of the structural estimation. Using reduced form analysis, we find evidence that 2D:4D is negatively related to patience for both hands in the first specification, and for the right hand in the second.

DISCUSSION
In this study we investigate the impact of 2D:4D, as a proxy for pre-natal exposure to testosterone, on discounting. We use a large sample (N = 419) of low income females from a wide age range. We rely on 24 choices per individual using the convex-time budget task with large stakes, and the average of five independent measures of 2D:4D.
We follow an analysis plan and jointly estimate time preference parameters and the curvature of the utility function, and allow the discount parameter (δ) to to vary with 2D:4D. We find that, for both hands, 2D:4D is negatively correlated with discount factor (p < 0.001). That is, we find that lower 2D:4D generates more patient choices.
We stick to our analysis plan and perform three robustness tests. First, we examine robustness of our results to varying background parameters; and find that our results are robust. Next, we attempt to estimate time-perference parameters at the individual level and correlate them with 2D:4D using reduced form models. Results of this second robustness check are mixed, since our individual level parameter estimates are very noisy. Our third robustness test involves replacing the parametric estimation method with a direct reduced form analysis. For each hand we run two tests using ordered probits and two using OLS. Given the criteria pre-specified in our analysis plan, our results are mixed. We pre-defined that we would consider a result to be significant if p − value < 0.05 for both hands 28 . Specification (1) of robustness checks 3.1 and 3.2 satisfies this criteria. However, for specification (2), only the result for the right hand is significant at p < 0.05.
Our result are in contrast to those of Lucas and Koff (2010), which reports that lower digit ratios are correlated with greater discounting among women. Our findings also differ from those of Drichoutis and Nayga (2015), which report no effect of digit ratio on (risk or) time preferences. These differences might stem from different samples, methods or protocols used.
However, our finding that lower 2D:4D leads to more patience is consistent with the combined results from other studies that relate 2D:4D, cognitive ability and patience. Bosch-Domènech et al. (2014) find that lower 2D:4D is associated with higher scores in the cognitive reflection test (CRT), and Frederick (2005) finds that higher CRT scores correlate with more patience (in hypothetical choices) and with higher cognitive abilities 29 These results are also consistent with other studies which also find that higher cognitive ability is associated with more patience (Shamosh et al., 2008;Burks et al., 2009;Dohmen et al., 2010;Benjamin et al., 2013).
Why should we care about the relationship between 2D:4D and discounting? Time preferences, and discounting in particular, play an important role in human decision making over countless domains (health, human capital accumulation, labor supply, income, etc.) with important welfare consequences. Our results are thus important, as they point to a potential biological underpinning of time preferences.
On a more methodological note, this finding suggests an exogenous determinant of individual time preferences. This may have broad implications for economic studies on the causal effect of time preferences on different economic behavior. That is, our results could be an important advance in identification strategies for researchers seeking to identify causal relationships between time preferences and other economic behavior, by using 2D:4D as an exogenous instrument.
This study has several peculiarities. First, our sample also differs from typical 2D:4D samples, as we do not rely on a WEIRD (Western Educated Industrialized Rich Democratic) population sample (Henrich et al., 2010a,b). Rather, our sample is particular on different margins: low income non-Caucasian females enrolled in a conditional cash transfer program. In addition, the 2D:4D measures of our sample are lower than those typically found in the literature. As with most findings, our results should be replicated to improve our confidence in the findings (Maniadis et al., 2017). In particular, this work should be replicated with samples of men. One limitation of this study is that our sample is exclusively female. As Frederick (2005) noted, there is a higher correlation of time preferences with CRT for females than males.

AUTHOR CONTRIBUTIONS
DA coordinated the study, designed the experiment, coordinated 2D:4D measurements, conducted statistical analysis and drafted the manuscript. LR coordinated the study, designed the experiment, conducted statistical analysis and drafted the manuscript. All authors gave final approval for publication.

FUNDING
DA greatfully acknowledges financial support from Fundación Capital.


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.