Abstract
We experimentally investigate preference for randomization in social settings, in which the dictator chooses probabilistically between two allocations for herself and an anonymous recipient. We observe substantial proportions of subjects choosing to randomize under various circumstances. The observed patterns have rich implications for various assumptions in social preference models and shed light on recent studies on ex-ante and ex-post social preferences.
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Notes
Machina (1989) considers an example in the context of the social planner’s problem, in which a mother is to allocate an indivisible good between two children whom she likes equally well. While the mother is indifferent between allocating the good to either child, she strictly prefers randomizing her choice as in a coin flip. Ex-post preference respecting first-order stochastic dominance implies that any probabilistic allocation has the same valuation as either of the two degenerate allocations. Thus, preference for randomization cannot be explained by ex-post preference respecting dominance.
In this example, the incompatibility between ex-post preference and preference for randomization remains valid for a wider set of utility specifications with the risk aggregator function respecting first-order stochastic dominance. This class of utility functions includes the rank-dependent utility in Quiggin (1982) and those adopting the betweenness approach (Chew 1983; Dekel 1986). Meanwhile, a utility function that is quasiconcave in probabilities, e.g., quadratic utility (Chew et al. 1991), could be compatible with preference for randomization. In fact, the original form of quadratic utility in Chew et al. (1991) still permits first-order stochastic dominance. Ex-post preference with a quadratic utility relaxing dominance is compatible the preference ranking in the example. See Epstein and Segal (1992) for an application of quadratic utility in a social setting, and Machina (1985) for an analysis of stochastic choice in the context of decision making under risk.
Trautmann and Wakker (2010) investigate the implications of process fairness and outcome fairness on dynamic consistency.
Dana et al. (2007) consider cases when subjects are able to leave the relationship between their actions and resulting outcomes uncertain, which gives subjects the moral wiggle room to behave self-interestedly. In the deterministic setup, traditional dictator/ultimatum games examine deterministic social preference along the hypotenuse (see Camerer 2003 for a review) as the subjects choose how to distribute a pie of fixed size. Andreoni et al. (2003) extend the analysis to inside the triangle with a convex ultimatum game, in which the responder can choose to shrink the pie rather than the take-it-or-leave-it option in the standard ultimatum game.
This view echoes the intuition in the convex ultimatum game in Andreoni et al. (2003), in which the responders can choose to shrink the pie instead of accepting or rejecting the offer.
It is possible to have \(U\left( {x_1 ,x_2 } \right) =U\left( {y_1 ,y_2 } \right) \) in some menus in our setup, and ex-post preference remains silent in these cases.
Fehr and Schmidt (1999) admit the form \(x_1 -\alpha \max \left\{ {x_1 -x_2 ,0} \right\} -\beta \hbox {max}\left\{ {x_2 -x_1 ,0} \right\} \). Bolton and Ockenfels (2000) take the form \(x_1 -\alpha \max \left\{ {\left( {x_1 -x_2 } \right) /\left( {x_1 +x_2 } \right) ,0} \right\} -\beta \hbox {max}\left\{ {\left( {x_2 -x_1 } \right) /\left( {x_1 +x_2 } \right) ,0} \right\} \). Charness and Rabin (2002) admit the form \(\left( {1-\gamma } \right) x_1 +\gamma \left( {\delta \min \left\{ {x_1 ,x_2 } \right\} +{\,}\left( {1-\delta } \right) \left( {x_1 +x_2 } \right) } \right) \). Andreoni and Miller (2002) take the form \(\left( {\delta x_1^\alpha +\left( {1-\delta } \right) x_2^\alpha } \right) ^{1/\alpha }\). Cox et al. (2007) admit the form \(\left( {x_1^\alpha +\theta x_2^\alpha } \right) /\alpha \). In all of these models, varying p in \(\left( {px_1 ,px_2 } \right) \) does not change the relative rank between \(px_1 \) and \(px_2 \), and thus would not change the parameters in these models when evaluating \(\left( {x_1 ,x_2 } \right) \) and \(\left( {px_1 ,px_2 } \right) \). Given the linearity in the first three functional forms and the CES form in the latter two, all of these models exhibit proportional monotonicity.
Levati et al. (2014) test the single-peakedness property of social preference, which is implied by concavity.
Given that \({\Theta }\) admits the expected utility with a homogeneous utility function u, the ex-ante allocation for menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) under any p, which is \(\left( {u^{-1}\left( {pu\left( {x_1 } \right) } \right) ,u^{-1}\left( {pu\left( {x_2 } \right) } \right) } \right) \), lies on the line extended by \(\left( {x_1 ,x_2 } \right) \) and \(\left( {0,0} \right) \). Krawczyk and Le Lec (2014) suggest that several observations on ex-ante preferences in Brock et al. (2013) can be rationalized by risk aversion.
Similar results hold for the two menus on the bottom line of the triangle.
Rohde (2010) axiomatizes deterministic Fehr–Schmidt specification. See Saito (2013) for discussions on the connection between Rohde (2010) and Saito (2013). Neilson (2006) considers the applicability of the standard separability axiom for both risk and other-regarding preferences and the resulting representation coincides with Fehr–Schmidt specification for other-regarding preferences, and prospect theory for risk preferences. We show in “Appendix 2” that the incompatibility between interior choice and Saito (2013) is due to the axioms of quasi-comonotonic independence and dominance.
These two incidences of choosing \(p=0\) are similar to those giving more than 50% in the standard dictator game (e.g., Camerer 2003). Our results are robust to the exclusion of these two subjects. If we consider the five subjects choosing interior p for \(\left( {{\begin{array}{ll} 10,&{}\quad 10 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) as noises, our results are also robust to the exclusion of these five subjects.
Alternatively, we use the Chi-square test, which yields similar statistics.
Similar to the intuition in Prediction 2B, it can be shown that a more restrictive prediction of strict concavity is that the proportion of subjects choosing \(p\le 0.5\) in menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) equals the proportion of subjects choosing \(p=0\) in menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0.5x_1,&{}\quad 0.5x_2 \\ \end{array} }} \right) \). In Experiment I, the proportion of subjects choosing \(p=0\) in \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 10 \\ \end{array} }} \right) \) is 22.3%, which is significantly less than 42.7%, the corresponding proportion of subjects choosing \(\,p\le 0.5\) in \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 10 \\ \end{array} }} \right) \) (proportion test, \(p < 0.001\)). Similarly, the proportion of subjects choosing \(p=0\) in \(\left( {{\begin{array}{ll} 4,&{}\quad 16 \\ 2,&{}\quad 8 \\ \end{array} }} \right) \) is 2.3%, which is significantly less than 10.2%, the corresponding proportion of subjects choosing \(p\le 0.5\) in \(\left( {{\begin{array}{ll} 4,&{}\quad 16 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) (proportion test, \(p < 0.013\)). For Experiment II, the proportion of subjects choosing \(p=0\) is 18.4% in menu \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 10 \\ \end{array} }} \right) \), which is significantly less than 48.9%, the proportion of subjects choosing \(p\le 0.5\) in menu \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) (proportion test, \(p < 0.002\)).
We also check whether the chosen probabilities result in violations of transitivity. For instance, \(p=1\) in \(\left( {{\begin{array}{ll} 20,&{}\quad 0 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \), \(p=0\,\) in \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \), and \(p=0\,\)in \(\,\left( {{\begin{array}{ll} 20,&{}\quad 0 \\ 0,&{}\quad 20 \\ \end{array} }} \right) \) violates transitivity. No subject violates transitivity.
If the generated ex-ante allocation does not lie in between \(\left( {x_1 ,x_2 } \right) \) and \(\left( {0.5x_1 ,0.5x_2 } \right) \), then the chosen probability must be 0 in menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0.5x_1,&{}\quad 0.5x_2 \\ \end{array} }} \right) \) to be consistent with strict convexity. Moreover, with 11 discrete probabilities in each menu, sometimes the chosen probabilities cannot generate exactly the same ex-ante allocation. We allow for a difference of 0.05 in the probabilities when counting for ex-ante type.
We check the behavioral patterns consistent with pure selfishness, efficiency, and inequality aversion motives. The selfish type chooses 1 in all the menus except for menus \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) and \(\left( {{\begin{array}{ll} 0,&{}\quad 20 \\ 0,&{}\quad 10 \\ \end{array} }} \right) \); 86 subjects (54.8%) belong to this group. The efficient type chooses 1 in all of the menus to maximize efficiency except for menus \(\left( {{\begin{array}{ll} 20,&{}\quad 0 \\ 0,&{}\quad 20 \\ \end{array} }} \right) \) and \(\left( {{\begin{array}{ll} 16,&{}\quad 4 \\ 4,&{}\quad 16 \\ \end{array} }} \right) \); 67 subjects (42.7%) belong to this group. In contrast, no subject belongs to the inequality aversion type, who is supposed to minimize the inequality. The prediction of the selfish type is independent of ex-ante and ex-post concerns, i.e., maximizing either the ex-ante or ex-post selfishness utility results in the same predictions. Similarly, ex-ante and ex-post efficiency concerns predict the same choice patterns. The implication for inequality aversion is slightly different, and ex-post and ex-ante inequality-averse subjects may behave differently in menus \(\left( {{\begin{array}{ll} 20,&{}\quad 0 \\ 0,&{}\quad 20 \\ \end{array} }} \right) \) and \(\left( {{\begin{array}{ll} 16,&{}\quad 4 \\ 4,&{}\quad 16 \\ \end{array} }} \right) \). For instance, a quadratic ex-ante inequality-averse agent chooses 0.5, whereas a quadratic ex-post inequality-averse subject is indifferent among all of the probabilities.
Efficiency and inequality aversion cannot account for the overall behavior because efficiency constantly predicts the choice of \(p=1\) in all menus except for \(\left( {{\begin{array}{ll} 20,&{}\quad 0 \\ 0,&{}\quad 20 \\ \end{array} }} \right) \) and \(\left( {{\begin{array}{ll} 16,&{}\quad 4 \\ 4,&{}\quad 16 \\ \end{array} }} \right) \), whereas inequality is minimized along the \(45^{^{\circ }}\) line in the allocation triangle. The presence of only efficiency and inequality aversion motives would imply that we should observe more corner choices for menus lying on the \(45^{^{\circ }}\) line, which is incompatible with the overall data.
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We are grateful to an anonymous referee, the editor, and the conference participants at the XVI Foundations of Utility and Risk Conference at Erasmus University 2014, and the 5th Annual Xiamen University International Workshop on Experimental Economics 2014, and seminar participants at Erasmus University for their helpful comments. Miao wishes to thank the China National Science Foundation (Grant 71503158) for financial support, and Zhong wishes to acknowledge financial support from the Ministry of Education—Singapore (R-122-000-237-112).
Appendices
Appendix 1: Appended tables
Appendix 2: Analysis of axioms in Saito (2013)
We demonstrate that the two axioms, namely Quasi-comonotonic Independence and Dominance, imply the monotonicity of the combination preference in p for \(\left( {\left( {x_1 ,x_2 } \right) ,p;\left( {0,0} \right) ,\left( {1-p} \right) } \right) \). The two axioms are as follows:
Quasi-comonotonic Independence: For all \(p\in \left( {0,1} \right] \), and \(\left( {x_1 ,x_2 } \right) \), \(\left( {y_1 ,y_2 } \right) \), \(\left( {z_1 ,z_2 } \right) \) that are pairwise comonotonic, \(\left( {x_1 ,x_2 } \right) \succeq \left( {y_1 ,y_2 } \right) \) if and only if \(p\left( {x_1 ,x_2 } \right) +\left( {1-p} \right) \left( {z_1 ,z_2 } \right) \succeq p\left( {y_1 ,y_2 } \right) +\left( {1-p} \right) \left( {z_1 ,z_2 } \right) \).
Dominance: Given two contingent allocations \(\left( {\left( {x_1 ,x_2 } \right) ,p;\left( {z_1 ,z_2 } \right) ,\left( {1-p} \right) } \right) \) and \(\left( {\left( {y_1 ,y_2 } \right) ,q;\left( {w_1 ,w_2 } \right) ,\left( {1-q} \right) } \right) \), \({\mathrm{E}}_{\left( {px_1 +\left( {1-p} \right) z_1 ,px_2 +\left( {1-p} \right) z_2 } \right) }\ge {\mathrm{E}}_{\left( {qy_1 +\left( {1-q} \right) w_1 ,qy_2 +\left( {1-q} \right) w_2 } \right) } \) and \(p{\mathrm{E}}_{\left( {x_1 ,x_2 } \right) } +\left( {1-p} \right) {\mathrm{E}}_{\left( {z_1 ,z_2 } \right) } \ge q{\mathrm{E}}_{\left( {y_1 ,y_2 } \right) } +\left( {1-q} \right) {\mathrm{E}}_{\left( {w_1 ,w_2 } \right) }\) imply \(\left( {\left( {x_1 ,x_2 } \right) ,p;\left( {z_1 ,z_2 } \right) ,\left( {1-p} \right) } \right) \succeq \left( {\left( {x_1 ,x_2 } \right) ,q;\left( {w_1 ,w_2 } \right) ,\left( {1-q} \right) } \right) \), where \({\mathrm{E}}\) denotes the equality equivalent for an allocation, i.e., \(\left( {{\mathrm{E}}_{\left( {x_1 ,x_2 } \right) } ,{\mathrm{E}}_{\left( {x_1 ,x_2 } \right) } } \right) \sim \left( {x_1 ,x_2 } \right) \).
Intuitively, the Quasi-comonotonic Independence axiom states that the preference over deterministic allocations satisfies the usual independence axiom separately in the upper and lower parts of the triangle, which directly implies that the preference over deterministic allocations is monotonic along straight lines passing through 0 because all of the allocations along the line belong to either the upper or lower part of the triangle. The Quasi-comonotonic Independence axiom taken together with the Dominance axiom implies the monotonicity of combinational preference. Formally, suppose \(\left( {x_1 ,x_2 } \right) \succ \left( {0,0} \right) \). Then the Quasi-comonotonic Independence implies \(p\left( {x_1 ,x_2 } \right) +\left( {1-p} \right) \left( {0,0} \right) \succeq q\left( {x_1 ,x_2 } \right) +\left( {1-q} \right) \left( {0,0} \right) \) for \(p\ge q\). Therefore, we have \({\mathrm{E}}_{\left( {px_1 ,px_2 } \right) } \ge {\mathrm{E}}_{\left( {qx_1 ,qx_2 } \right) } \), which in turn implies \(\left( {\left( {x_1 ,x_2 } \right) ,p;\left( {0,0} \right) ,\left( {1-p} \right) } \right) \succeq \left( {\left( {x_1 ,x_2 } \right) ,q;\left( {0,0} \right) ,\left( {1-q} \right) } \right) \) with Dominance. In sum, the two axioms imply corner choices in all the menus \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \). The argument in situation \(\left( {x_1 ,x_2 } \right) \prec \left( {0,0} \right) \) is similar.
Appendix 3: Analysis of the proposed behavioral model
Assume the utility function for deterministic allocations takes the following form:
and the combinational utility is a weighted average \(\lambda {\Phi }_{\text {ex-ante}} +\left( {1-\lambda } \right) {\Phi }_{\text {ex-post}} \).
For menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) with \(x_1 <x_2 \), the utility for an interior chioce p is
and we obtain the following FOC characterizing the optimal choice of p:
The optimal solution is \(\frac{x_1 +\delta \left( {x_1 +x_2 } \right) -\left( {1-\lambda } \right) \beta \left( {x_1 -x_2 } \right) ^{2}}{2\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}=\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{2\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{1-\lambda }{2\lambda }\).
For menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0.5x_1,&{}\quad 0.5x_2 \\ \end{array} }} \right) \) with \(x_1 <x_2 \), the utility for an interior choice p is as follows:
and we obtain the following FOC:
The optimal solution is given by: \(\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{3+\lambda }{4\lambda }\). The optimal chosen probabilities may lie in the interior or the corner, depending on parameter values. It is possible to have \(\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{3+\lambda }{4\lambda }>\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{2\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{1-\lambda }{2\lambda }\) given \(\,\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{2\beta \left( {x_1 -x_2 } \right) ^{2}}>\frac{1+3\lambda }{4}\). If we have \(\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{3+\lambda }{4\lambda }>0>\frac{x_1 +\delta \left( {x_1 +x_2 } \right) }{2\lambda \beta \left( {x_1 -x_2 } \right) ^{2}}-\frac{1-\lambda }{2\lambda }\), a subject chooses \(p=0\) in menu \(\left( {{\begin{array}{ll} x_1,&{}\quad x_2 \\ 0,&{}\quad 0 \\ \end{array} }} \right) \) and an interior probability in menu \(\left( \begin{array}{ll} x_1,&{}\quad x_2 \\ 0.5x_1,&{}\quad 0.5x_2 \\ \end{array}\right) \). Moreover, we have optimal chosen probabilities decreasing from the upper to the lower part of the triangle if \(\alpha <\beta \). Thus, this behavioral model could be compatible with the observed behavior.
Appendix 4: Experimental instructions and decision sheets
1.1 Experimental Instructions
In this decision making experiment, the task involves a pair of participants, yourself and another participant in this room. You will make decisions regarding possible payment for both of you as shown the decision table example below. Note that the numbers here are for illustrative purpose only.
Allocation 1 | Allocation 2 | ||
|---|---|---|---|
You | Other | You | Other |
7 | 8 | 10 | 0 |
p: | \({\varvec{1-p:}}\) | ||
Example 1
In this decision, there are two allocations: Allocation 1, you get $7 and the other participant gets $8; Allocation 2, you get $10, and the other participant gets $0. You are asked to choose a probability o to implement Allocation 1 and \({\varvec{1-p}}\) to implement Allocation 2. You can choose any probability including 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.
Allocation 1 | Allocation 2 | ||
|---|---|---|---|
You | Other | You | Other |
3 | 5 | 5 | 3 |
\({\varvec{p:}}\) | \({\varvec{1-p:}}\) | ||
Example 2
In this decision, there are two allocations: Allocation 1, you get $3 and the other participant gets $5; Allocation 2, you get $5, and the other participant gets $3. You are asked to choose a probability p to implement Allocation 1 and \({\varvec{1-p}}\) to implement Allocation 2. You can choose any probability including 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.
You will make a number of choices similar to the examples. There is neither correct nor wrong answer to the tasks, and you choose your preferred probability for each of the decision tables. At the end of the experiment, we will randomly choose one participant to implement one of his or her choices, and match him/her to the other participant in this room.
The probability will be implemented by drawing one card from a set of 10 cards numbered from 1 to 10. If you choose p to be 0, Allocation 1 will be not implemented and Allocation 2 will be implemented regardless of the card you draw. If you choose p to be 0.1, Allocation 1 will be implemented if you draw number 1, otherwise Allocation 2 will be implemented. If you choose p to be 0.2, Allocation 1 will be implemented if you draw number 1 or 2, otherwise Allocation 2 will be implemented. If you choose p to be 0.3, Allocation 1 will be implemented if you draw number 1, 2, or 3, otherwise Allocation 2 will be implemented. And so on.
Exercise
While calculating payoffs seems easy, it is important that everyone understands. So, below we ask you to calculate the payoffs of both players for some specific examples. After you finish, we will go over the correct answers together.
Exercise 1.
Allocation 1 | Allocation 2 | ||
|---|---|---|---|
You | Other | You | Other |
7 | 8 | 10 | 0 |
p: 0.4 | \({\varvec{1-p:}}\)0.6 | ||
Suppose the table above is chosen for implementation, and you choose 0.4 for Allocation 1 and 0.6 for Allocation 2.
At the end of the experiment, a card is randomly drawn from a set of 10 cards numbered from 1 to 10.
If the card drawn is 3, your payment will be ____, and the payment of the other participant will be ____.
If the card drawn is 9, your payment will be ____, and the payment of the other participant will be ____.
Exercise 2.
Allocation 1 | Allocation 2 | ||
|---|---|---|---|
You | Other | You | Other |
3 | 5 | 5 | 3 |
p: 1 | \({\varvec{1-p:}}\)0 | ||
Suppose the table above is chosen for implementation, and you choose 1 for Allocation 1 and 0 for Allocation 2.
At the end of the experiment, a card is randomly drawn from a set of 10 cards numbered from 1 to 10.
If the card drawn is 3, your payment will be ____, and the payment of the other participant will be ____.
If the card drawn is 9, your payment will be ____, and the payment of the other participant will be ____.
This is the end of the instruction. Should you have any question, please raise your hand.
Sample Decision Sheet
Allocation 1 | Allocation 2 | ||
|---|---|---|---|
You | Other | You | Other |
0 | 20 | 0 | 0 |
p: | \({\varvec{1-p:}}\) | ||
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Miao, B., Zhong, S. Probabilistic social preference: how Machina’s Mom randomizes her choice. Econ Theory 65, 1–24 (2018). https://doi.org/10.1007/s00199-016-1015-y
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DOI: https://doi.org/10.1007/s00199-016-1015-y