Experimental treatments
The baseline treatment (T1) was based on the one-shot version of the risky investment game first used by Gneezy et al. (2009). Respondents were told that they would play a real game with money. In this game, they could choose to keep or invest the whole or part of an initial endowment \(X=30\) ETB.Footnote 1 They could invest a share x/X (multiples of 5 ETB) in a 50–50 lottery with the outcome 3x or 0. In case of loss, the respondent only received \(X-x\). The lucky winners obtained \(X-x+3x=X+2x\).
Treatment T2 endowed the respondents with a 50–50 lottery of \(3X=90\) ETB or 0, which was the maximum risky investment level in T1. The respondents were then offered to sell all or part of the lottery and would then get a payment of one-third of the lottery winning value they would sell. If they sold y out of 3X, they would get y/3 as payment (y was in multiples of 15, so that the safe amounts received were in multiples of 5). Losers of the game would get y/3 and winners would get \(3X-y+y/3=3X-2/3y\).
Details of the experimental protocols (English version) for the treatments are provided in Appendix 2. These were translated to the local language, Tigrinya, which was the language used in the field. The enumerators were trained with both versions and we ensured that the translations were accurate and that the enumerators understood the questions correctly and used the same wording in the local language for all the questions and explanations.
EUT vs. PT
To an applied development economist, it may not be obvious that the one-shot risky investment game invokes loss aversion. S/he may therefore interpret the experimental results through the lens of Expected Utility Theory (EUT). It is especially not common to assume that monetary endowments induce endowment effects due to loss aversion. EUT has for long dominated economic thinking related to risky choice among applied economists. Within the EUT framework under narrow bracketing,Footnote 2 risk preferences are captured by the utility curvature over the risky and safe amounts in the one-shot risky investment game [treatment T1 (safe base)]
$$\begin{aligned} \max EUT(x)=0.5u(30-x)+0.5u(30+2x). \end{aligned}$$
(1)
Risk aversion, captured by the concavity of the utility function, is necessary to get interior solutions for x with \(0\le x^{*}\le 30\). The optimal level of \(x_{i}=x_{i}^{*}\) for each subject i is identified with the standard experiment. When imposing a specific functional form on the utility function such as a Constant Relative Risk Aversion (CRRA) function, the relative risk aversion parameter (r) and its distribution in a sample population may be derived from the observed investment distribution based on the one-shot standard game, see Fig. 1.
For treatment T2, the EUT maximization problem can be stated as
$$\begin{aligned} \max EUT(y)=0.5u(90-(2/3)y)+0.5u(y/3). \end{aligned}$$
(2)
With behavior according to EUT, subject’s allocation decisions should not vary across treatments in our experiment as behavior according to EU implies no endowment effects (reference point bias) due to loss aversion or probability weighting. This may be verified as the relationship between x in Eq. (1) and y in Eq. (2) is \(x=30-y/3\) or \(y=90-3x\). By plugging in for x in Eq. (2), we see that it becomes identical to Eq. (1). The optimal investment level will be the same across the two treatments for a subject i behaving according to EUT:
$$\begin{aligned} EUT: x_{i}^{*}(T1)=x_{i}^{*}(T2). \end{aligned}$$
(3)
Given a specific functional form of the utility function such as CRRA, this, therefore, leads to the same individual risk aversion (utility curvature) parameter derived for each subject based on her/his optimal \(x^{*}\) allocation that would be identical across the two treatments. Using the one-shot game to measure risk aversion would then lead to no bias in the estimation of risk aversion. Given a CRRA-utility function, no integration of prospect money with background wealth, no endowment effect, and objective probability judgment, the relationship between CRRA-r and optimal investment level is illustrated in Fig.A1 in Appendix 1.Footnote 3
However, if real behavior deviates from EUT because of reference point effects, loss aversion, and/or probability weighting, the conversion between x and y and Eq. (3) will not hold and using T1 based on EUT to estimate the utility curvature as the measure of risk aversion would lead to biased estimates.
Alternatively, the decisions in the game may be modeled based on the Prospect Theory (PT) to assess whether this theoretical framework is better as a basis to explain behavior across T1–T2. One may even need inspiration from Third-Generation Prospect Theory (PT3) to think of there being endowment effects associated with monetary prospects in the one-shot risky investment game (Schmidt et al. 2008).Footnote 4 This model assumes that the reference point is the endowment at the decision point in treatments T1 and T2 and it is only deviations from the reference point endowment that matter. PT typically assumes diminishing sensitivity around the reference point, implying a convex value function in the loss domain and a concave value function in the gains domain. Loss aversion is represented as a kink in the value function at the reference point. Assuming PT for T1 (safe base), the reference point is the sure amount of 30. The decision-maker then maximizes the following expression (denoting loss aversion as \(\lambda \)):
$$\begin{aligned} \max PT(T1)=w^{+}(0.5)v(2x)- w^{-}(0.5)\lambda v(|x|). \end{aligned}$$
(4)
For T2 (full risk), it is the subjective value of the risky lottery yielding 90 with 0.5 probability which is the reference point. We denote this (endogenous) reference point R. Under PT, the decision-maker seeks to maximize
$$\begin{aligned} \max PT(T2)=w^{+}(0.5)v(90-2/3y-R)-w^{-}(0.5)\lambda v(|(y/3 -R|). \end{aligned}$$
(5)
This model holds as long as \(90-2/3y-R\ge 0\). Respondents will choose optimal \(y^{*}\), such that they avoid violation of this inequality. Given two respondents i, j with reference points \(R_i>R_j\) who are identical in all other respects than their reference points, will choose optimal levels of \(y^{*}\), such that \(y^{*}_i<y^{*}_j\). If T2 gives a higher reference point than T1 (\(R>30\)), combined with loss aversion, the optimal investment level will be higher in T2 than in T1. The game does not allow us to identify respondents’ reference points in T2 or how these reference points are associated with w(0.5) and \(\lambda \). However, a significantly higher investment level in T2 than in T1 allows us to reject EUT and is a clear indication of significant endowment effects in the game. Interior solutions in the game are also an indication of non-linear value functions.
The standard one-shot game has been shown to give significant gender differences with women investing significantly less than men in most earlier studies (Charness and Gneezy 2012). It is still a mystery why this game tends to give stronger gender differences than other games used to investigate gender differences in risk preferences (Filippin and Crosetto 2016). We ask whether this could be associated with an endowment effect bias that may be stronger for women. We test whether the gender difference is stronger in T1 than in T2, or whether it is eliminated or goes even in the opposite direction in T2. If the gender difference remains strong and in the same direction in all three treatments, we interpret this as an indication that women are less risk tolerant than men but not more loss averse.