Download Deducing Social Risk Preferences from Personal Risk Preferences and Social Preferences and more Slides Statistics in PDF only on Docsity!
Linking Social and Personal Preferences:
Theory and Experiment∗
William R. Zame, Bertil Tungodden, Erik Ø. Sørensen,
Shachar Kariv, and Alexander W. Cappelen†
September 25, 2019
Abstract
The goal of this paper is to link attitude toward risk over personal consumption with attitude toward risk over social consumptions. Because many everyday choices involve risk, these attitudes enter virtually every realm of individual decision-making. We provide necessary and sufficient conditions for deducing preferences over risky social choices (which have consequences both for the Decision Maker and for others) from risky personal choices (which have consequences only for the Decision Maker) and riskless social choices, and we offer an experimental test of the theory. The experiments generate a rich dataset that enables completely non-parametric revealed preference tests of the theory at the level of the individual subject. Many subjects behave as predicted by the theory but a substantial fraction do not. JEL Classification Numbers: C91, D63, D81. Keywords: social preferences, risk preferences, revealed preference, experiment. ∗We are grateful to Daniel Silverman, Bob Powell, Benjamin Polak, Daniel Markovitz, Edi Karni, Dou- glas Gale, Raymond Fisman and Chris Chambers for helpful discussions and encouragement and for sug- gestions from a number of seminar audiences. Financial support was provided by the National Science Foundation, the Research Council of Norway, and the Peder Sather Center for Advanced Study. Any opin- ions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of any funding agency. Experiments reported in this paper were con- ducted by the Choice Lab at the Centre for Experimental Research on Fairness, Inequality and Rationality (FAIR) at NHH Norwegian School of Economics. † Authors in reverse alphabetical order. Zame: University of California, Los Angeles (zame@econ.ucla.edu); Tungodden: Norwegian School of Economics (bertil.tungodden@nhh.no); Sørensen: Norwegian School of Economics (erik.sorensen@nhh.no); Kariv: University of California, Berkeley (kariv@berkeley.edu); Cappelen: Norwegian School of Economics (alexander.cappelen@nhh.no).
1 Introduction
Many individuals must make choices that have consequences only for themselves – choices in the personal domain – and choices that have consequences both for themselves and for others – choices in the social domain. Many of these choices involve risk, so a full understanding of choice behavior in these domains requires an understanding of both the individual’s preferences over consequences and the individual’s attitude toward risk. It is natural to ask: Is there a connection between an individual’s attitude toward risk in the personal domain and that individual’s attitude toward risk in the social domain? This paper offers formalizations of this question, theoretical answers to this question, and an experimental test of the theory.
Our motivation for asking (and answering) this question arises not only from intellectual curiosity but also from pragmatism because we often choose – or at least influence – which individuals will be in a position to make choices that have consequences both for themselves and for us: Chairs, Deans, Mayors, Governors, Congresspersons, Senators – even Presidents. And we certainly care not only whether the President prefers peace to war, but what actions the President would be prepared to take to alter the risks of peace or war. Blockading Soviet ships bound for Cuba (as John Kennedy did) risks war, and putting forward both tax reform and civil rights legislation simultaneously (as Kennedy also did) risks accomplishing neither.^1
But can we draw any inferences at all about the President’s risky choices in the social domain from the fact that the President chooses to conduct an illicit affair or smoke in secret or invest aggressively or exaggerate his accomplishments (to mention just a few personal choices that have made headlines in recent memory)? What these personal choices have in common is that they involve (personal) risk – to the President’s marriage, health, finances, reputation. Drawing inferences about (present and future) risky social choices from knowledge of a (past and present) risky personal choices would seem useful, and it would seem possible as well – provided that there is a linkage between attitude toward risk in the personal domain and attitude toward risk in the social domain.
In this paper, we give a precise formalization of these ideas, establish a theoretical linkage between preferences and risk attitudes in the social domain and in the personal domain, and provide an experimental test of the theory. One important qualification needs (^1) And in fact, Kennedy accomplished neither tax reform nor civil rights legislation; both were pushed through by Lyndon Johnson after Kennedy’s assassination.
extension to the preference relation over the full domain of lotteries L(Ω) on social states? If we make the (relatively weak) assumption that the DM’s preferences obey the usual axioms of individual choice under uncertainty – Completeness, Transitivity, Continuity, Reduction of Compound Lotteries and the Sure Thing Principle – together with a natural axiom that we call State Monotonicity, then a necessary and sufficient condition that it be possible to deduce the entire preference relation from the sub-preference relation 0 is that the DM finds every social state to be indifferent to some personal state.^3
The theoretical result seems clean and satisfying but it is another question entirely whether is also descriptive of reality. To address this latter question, we designed and executed an experiment in which subjects were confronted with choices in three domains:
- Personal Risk The objects of choice are risky personal choices (equiprobable binary lotteries whose consequences are monetary outcomes for self alone).
- Social The objects of choice are riskless social choices (deterministic divisions of money between self and one other ).
- Social Risk The objects of choice are risky social choices (equiprobable binary lotteries whose consequences are divisions of money between self and other ).
In the experimental setting, each decision problem is presented as a choice from a two-dimensional budget line using a graphical interface developed by Choi, Fisman, Gale, and Kariv (2007b). The Personal Risk domain is identical to the (symmetric) risk experiment of Choi, Fisman, Gale, and Kariv (2007a). The Social domain is identical to the (linear) two-person dictator experiment of Fisman et al (2007).^4 The Social Risk (^3) As we discuss below, State Monotonicity is a much weaker assumption than Independence because it compares only lotteries whose outcomes are primitives – social states – rather than lotteries whose outcomes are themselves lotteries. Almost all decision-theoretic models that have been proposed as alternatives to Expected Utility obey State Monotonicity. (^4) Ahn, Choi, Gale, and Kariv (2014) extended the work in Choi et al. (2007a) on risk to settings with ambiguity. Choi, Kariv, M¨uller, and Silverman (2014) investigated the correlation between individual behavior and demographic and economic characteristics in the CentERpanel (a representative sample of Dutch households). These datasets have also been analyzed by others, including Halevy, Persitz, and Zrill (2018) and Pollison, Quah, and Renou (2018). Fisman, Jakiela, Kariv, and Markovits (2015b), Fisman, Jakiela, and Kariv (2015a), Fisman, Jakiela, and Kariv (2017) and Li et al (2017) build on the work in Fisman, Kariv, and Markovits (2007) to study distributional preferences. Since all experimental designs share the same graphical interface, we are building on expertise we have acquired in previous work.
domain is new and intended to represent the choice problem over lotteries over pairs of consumption for self and for other.^5
The approach has a number of advantages over earlier approaches. First, the choice of a bundle subject to a budget constraint provides more information about preferences than a typical binary choice. Second, because the interface is extremely user-friendly, it is possible to present each subject with many choices in the course of a single experimental session, yielding much larger data set. This makes it possible to analyze behavior at the level of the individual subject, without the need to pool data or assume that subjects are homogenous. And third, it is easy to make direct comparisons of choices across the three domains using an attractive non-parametric econometric approach that builds on classical revealed preference analysis.
For some of our subjects, the experimental data does not provide testable implications of our theory unless we make additional assumptions about the form, parametric or otherwise, of the underlying utility function. However, for two classes of subjects, the predictions of our theory are testable:
- For subjects who are selfish – those who, in the Social domain give nothing to other
- the theory predicts that choice behavior in the Personal Risk domain should coincide with choice behavior in the Social Risk domain.
- For subjects who are egalitarian – those who, in the Social domain, treat other symmetrically to self – the theory predicts that choice behavior in the Social domain should coincide with choice behavior in the Social Risk domain. (That is, risk attitude should be irrelevant in the Social Risk domain.)^6
The theory of revealed preference allows us to provide strong and completely non- (^5) It is of course possible that presenting choice problems graphically biases choice choice behavior in some particular way – and that is a useful topic for experiment – but there is no evidence that this is the case. For instance: behavior in the Social domain elicited graphically Fisman et al. (2007) is quite consistent with behavior elicited by other means (Camerer, 2003), and behavior in the Personal Risk domain elicited graphically Choi et al. (2007b) is quite consistent with behavior elicited by other means (Holt and Laury, 2002). 6 The objects of choice in the Social domain are payout pairs (x, y) where x is the payout to other and y is the payout to self. In the experimental setting, we offer choices from linear budget sets px + qy ≤ w. We identify behavior in the Social domain to be selfish if the choice subject to the budget constraint px + qy ≤ w is always of the form (0, y) (payout to other is 0). We identify behavior in the Social domain to be egalitarian if (a, b) is chosen subject to the budget constraint px + qy ≤ w if and only if (b, a) is chosen subject to the mirror-image budget constraint qx + py ≤ w.
with respect to what they observe.^7
We assume Ω is finite; this avoids subtle issues about the topology of Ω and the con- tinuity properties of preferences. We also assume that P ⊂ Ω contains at least two states that the DM does not find indifferent (there are states A, B ∈ P with A B); this avoids degeneracy. For any subset Θ ⊂ Ω, we write L(Θ) for the set of finite lotteries over states in Θ. We frequently write
i piωi^ for the lottery whose outcome is the state^ ωi^ with prob- ability pi. We refer to lotteries in L(P ) as personal lotteries and to lotteries in L(Ω) as social lotteries.
We assume that the DM has a preference relation on L(Ω) that satisfies the familiar requirements: Completeness, Transitivity, Continuity, Reduction of Compound Lotteries and the Sure Thing Principle. ∑ These imply that we may – and do – identify the lottery
i piω^ with the certain state^ ω. Throughout, we also assume that^ ^ obeys the following requirement, which we call State Monotonicity.
State Monotonicity If ωi, ω′ i ∈ Ω for i = 1,... k, ωi ω′ i for each i and p = (p 1 ,... , pk) is a probability vector, then
∑^ k
i=
piωi
∑^ k
i=
piω i′
State Monotonicity is equivalent to a condition that (Grant, Kajii, and Polak, 1992) call Degenerate Independence.^8 For ease of comparison, recall that the familiar (von Neumann and Morgenstern, 1944) Independence Axiom is
Independence If Wi, W (^) i′ ∈ L(Ω) for i = 1,... k, Wi W (^) i′ for each i and p = (^7) In earlier versions of this paper we were more specific about the nature of Ω, P. We assumed that Ω has a product structure: Ω ⊂ X × Z; where, given a state ω = (x, z) the component x represents the personal component of ω and z represents the social component of ω. We assumed a given reference social component z 0 ∈ Z and identified P with the product X × {z 0 }. However, because the particular structure played no role in the actual analysis, we prefer to use the more abstract formulation given here. (^8) As we do here, Grant et al. (1992) asks to what extent all preference comparisons can be deduced from a subset of preference comparisons. However, because our intent is different from Grant et al. (1992) we face quite different issues so the differences are greater than the similarities. The discussion in the Conclusion elaborates on that.
(p 1 ,... , pk) is a probability vector, then ∑^ k
i=
piWi
∑^ k
i=
piW (^) i′
Notice that the difference between these two Axioms is precisely that the Independence Axiom posits comparisons between lotteries over lotteries, while State Monotonicity only posits comparisons between lotteries over states.^9 As we discuss below, the difference is enormous. We note that almost all decision-theoretic models that have been proposed as alternatives to Expected Utility of which we are aware obey State Monotonicity, including Weighted Expected Utility (Dekel, 1986; Chew, 1989), Rank Dependent Utility (Quiggin, 1982, 1993), and (much of) Prospect Theory (Tversky and Kahneman, 1992).
To see what State Monotonicity does and does not imply, suppose that Ω consists of three mutually non-indifferent states Ω = {A, X, B}; without loss, assume that A X B and picture the familiar Marschak-Machina triangle in which each point represents a lottery (pA, pX , pB ) over the states A, X, B (pA = 0 on the horizontal edge, pX = 0 on the hypotenuse, and pB = 0 on the vertical edge). Continuity requires that X be indifferent to some lottery over A, B; say X ∼ 12 A + 12 B.^10 Assuming the other axioms, Independence implies that the preference relation admits an Expected Utility representation, so that the indifference curves in the triangle are parallel straight lines. Hence knowledge that X ∼ 12 A + 12 B completely determines on the entire triangle. In particular, 12 A + 12 X ∼ 3 4 A^ +^
1 4 B,^
1 2 X^ +^
1 2 B^ ∼^
1 4 A^ +^
3 4 B^ and so forth – see Figure 1A. In Weighted Expected Utility, Betweenness, which is a weaker axiom than Indepen- dence, implies that all indifference curves are again straight lines but they need not be parallel; in particular, it may be that 12 A + 12 X ∼ 18 A + 78 B and 12 X + 12 B ∼ 101 A + 109 B
- see Figure 1B. In this example, the indifference curves “fan out,” becoming steeper (corresponding to higher risk aversion) when moving northeast in the triangle. In Rank Dependent Utility and Prospect Theory, indifference curves can change slopes and also “fan out” and “fan in,” especially near the triangle boundaries – see Figure 1C.
To see what State Monotonicity implies, consider a lottery aA+xX +bB (0 ≤ a, x, b ≤ 1 (^9) We have formulated State Monotonicity in terms of weak preference, rather than indifference, be- cause the two are not generally equivalent. We have formulated the Independence Axiom in terms of weak preference, rather than indifference, only to highlight the difference between State Monotonicity and Independence. (^10) Some caution must be exercised here. In the absence of the Independence Axiom, which we do not assume, the Continuity Axiom is stronger than the Archimedean Axiom.
Figure 1. Independence, Betweenness, Rank-dependence, and State Monotonicity in the Marschak-Machina triangle
𝐴
X 𝐵
. 5 A +. 5 𝐵 . 75 𝐴 +. 25 𝐵 . 5 𝑋 +. 5 𝐴
X 𝐵
. 5 𝐴 +. 5 𝐵 . 875 𝐴 +. 125 𝐵 . 5 𝐴 +. 5 𝑋
𝐴
. 25 𝐴 +. 75 𝐵 . 5 𝑋 +. 5 𝐵 . 1 𝐴 +. 9 𝐵 . 5 𝑋 +. 5 𝐵
Panel A Panel B
Panel C Panel D
X 𝐵
. 5 𝐴 +. 5 𝐵 . 875 𝐴 +. 125 𝐵 . 5 𝐴 +. 5 𝑋
𝐴
. 1 𝐴 +. 9 𝐵 . 5 𝑋 +. 5 𝐵
X
. 5 𝐴 +. 5 𝐵 . 875 𝐴 +. 125 𝐵 . 5 𝐴 +. 5 𝑋
𝐴
. 1 𝐴 +. 9 𝐵 . 5 𝑋 +. 5 𝐵 𝐵
Figure 1: Independence, Betweenness, Rank-dependence, and State Monotonicity in the Marschak-Machina triangle
1 , consider once again the setting in which Ω consists of the three states A, X, B with A X B and picture the Marschak-Machina triangle in which each point represents a lottery (pA, pX , pB ) over the states A, X, B. Suppose that A, B are personal states and X is a social state (and hence is not equivalent to any personal state).
If we observe 0 we observe the ordering A 0 X 0 B and the ordering of lotteries between A, B – but no others – shaded gray in Figure 2A. State Monotonicity assures us that from these observations we can infer the ordering of lotteries between A, X and lotteries between X, B – see Figure 2B. Continuity assures us that X is indifferent to some lottery aA + (1 − a)B – but we do not observe which lottery. If we observe 1 then we do observe which lottery – but that is all – see Figure 2C. However, if we observe 1 and we assume that obeys Independence – and hence has an Expected Utility representation – then observing which lottery completely determines – see Figure 2D.
This illustrates how deducing the DM’s entire preference relation from a sub- preference relation depends on the amount that can be observed/infrared about the DM’s preferences as well as on the degree of rationality we ascribe to the DM, and in particular on whether the observer believes/assumes that the DM’s preferences obey the axioms of Expected Utility or believes/assumes only that the DM’s preferences obey some weaker criteria. We next show that if we assume only State Monotonicity, then a necessary and sufficient condition that it be possible to deduce the entire preference relation from 0 is that the DM finds every social state to be indifferent to some personal state.
3 Deducing Preferences
The question we have in mind can now be formulated in the following way: If we ob- serve the sub-preference relation 0 can we deduce the entire preference relation ? In different words: is the unique entire preference relation that extends the sub-preference relation 0 and obeys the same axioms (Completeness, Transitivity, Continuity, Reduction of Compound Lotteries, the Sure Thing Principle and State Monotonicity)? We next pro- vide necessary and sufficient conditions that this be the case. When these necessary and sufficient conditions are not satisfied, there will be many lotteries in L(Ω) over which the preference ordering of the DM cannot be deduced from 0.
Theorem 1 Assume that the DM’s preference relation satisfies Completeness, Transi- tivity, Continuity, Reduction of Compound Lotteries, the Sure Thing Principle and State
Monotonicity. In order that can be deduced from 0 it is necessary and sufficient that the DM finds every social state ω ∈ Ω/P to be indifferent to some personal state ω˜ ∈ P.
Proof. To see that this condition is sufficient, assume that every social state ω admits a personal state equivalent ˜ω. State Monotonicity implies that
piωi ∼
pi ω˜i for every lottery
piωi ∈ L(Ω). Hence given two lotteries
piωi,
qj ωj it follows from transitivity that (^) ∑ piωi,
qj ωj ⇔
pi ω˜i
qj ω˜j ⇔
pi ω˜i 0
qj ω˜j.
That is, can be deduced from 0.
To see that this condition is necessary, we suppose that there is some social state X that the DM does not find indifferent to any personal state and construct a preference relation that agrees with on L(P ) and on Ω but not on all of L(Ω). We will need to take some care in order to be sure that the preference relation we construct obeys Continuity and State Monotonicity.
Because is continuous (by assumption) and L(Ω) can be identified with a finite- dimensional simplex, which is a separable metric space, we can use Debreu’s representation theorem (Debreu 1954) to find a utility function u : L(Ω) → R that represents , that is
∀Γ, Γ′^ ∈ L(Ω) : Γ Γ′^ ⇔ u(Γ) ≥ u(Γ′).
Without loss, assume that the range of u contained in the interval [0, 1]. We construct a new utility function U : L(Ω) → R that agrees with u on L(P ) and induces the same ordering as u on Ω but does not induce the same ordering as u on L(Ω).
To make the remainder of the proof more transparent, consider again three mutually non-indifferent states Ω = {A, X, B}, two of which are personal states P = {A, B}, and assume without loss that A B. As our earlier discussion of State Monotonicity suggests, in this setting it is easy construct a utility function U on L(Ω) with the desired properties
- but it is a little less easy to do so in a way that generalizes to the general setting with finitely many states. It is convenient to distinguish three cases: (i) A X B, (ii) X A B, and (iii) A B X:
- Case (i) A X B: Continuity guarantees that there is some γ ∈ (0, 1) such that X ∼ γA + (1 − γ)B; equivalently, u(X) = u(γA + (1 − γ)B). We construct a continuous utility function U on L(Ω) that agrees with u on L(P ) and for which
U (A) > U (X) > U (B) – so that U induces the same ordering as u on Ω – but U (X) 6 = u(X) = u(γA + (1 − γ)B) – so that the preference relation U induced by U does not agree with on L(Ω).
To understand the idea behind the construction, consider once again the Marschak- Machina triangle – see Figure 1D. In order that U satisfy State Monotonicity, U must be strictly increasing (from bottom to top) along vertical lines and strictly decreasing (from left to right) along horizontal lines. To construct U we first define two auxiliary functions f, g:
f (aA + xX + bB) = u(aA + xA + bB) g(aA + xX + bB) = u(aA + xB + bB)
for every lottery aA + xX + bB ∈ L(Ω). Because u is continuous, both f and g are continuous. Moreover, f is constant on vertical lines and strictly decreasing on horizontal lines, while g is strictly increasing on vertical lines and constant on horizontal lines.
Hence for every λ ∈ (0, 1) the convex combination λf + (1 − λ)g is strictly increasing on vertical lines and strictly decreasing on horizontal lines. Choose λ so that
λf (X) + (1 − λ)g(X) = λu(A) + (1 − λ)u(B) 6 = u(X),
define the utility function U = λf + (1 − λ)g, and let U be the preference relation induced by U. It is evident that U satisfies Completeness, Transitivity, Reduction of Compound Lotteries and the Sure Thing Principle. Because u, f, g are continuous, so is U ; hence U satisfies Continuity. By construction, U is strictly increasing along vertical lines and strictly decreasing along horizontal lines, so U satisfies State Monotonicity. Finally, note that U agrees with u on L(P ) and
U (A) = u(A) > U (X) = λu(A) + (1 − λ)u(B) > u(B) > U (B)
so that U is an extension of . Finally, because we have chosen λ so that U (X) 6 = u(X) = u(γA + (1 − γ)B) it follows that U 6 =, so the argument is complete.
- Case (ii) X A B: This is easier than Case (i). Continuity guarantees that there is some ν ∈ (0, 1) for which A ∼ νX + (1 − ν)B. Choose λ > 0 for which u(νA + (1 − ν)B) + λν 6 = u(A) and set
U (xX + aA + bB) = u(xA + aA + bB) + λx
that X ∼ γA + (1 − γ)B. As before, define auxiliary functions f, g : L(Ω) → R by
f (Γ) = u(ΓA + x(Γ)A + ΓB) g(Γ) = u(ΓA + x(Γ)B + ΓB)
where x(Γ) =
ωi∈X pi. Choose^ λ^ so that^ λf^ (X) + (1^ −^ λ)g(X)^6 =^ u(γA^ + (1^ −^ γ)B and define U = λf + (1 − λ)g. It is easily checked that the preference relation U induced by U satisfies all the desired axioms; because U (X) 6 = U (γA + (1 − γ)B) we conclude that U 6 =.
- Case (ii) A = ∅ and B 6 = ∅: Because there are at least two inequivalent personal states, we can choose B′^ ∈ B with B B′. Continuity guarantees there is some ν ∈ (0, 1) for which B ∼ νX+(1−ν)B′. Choose λ > 0 so that u(νB+(1−ν)B′)−λν 6 = u(B) and set U (Γ) = u(x(Γ)B + ΓB) − λx(Γ) It is easily checked that the preference relation U induced by U satisfies all the desired axioms; because U (νX + (1 − ν)B′) = u(νB + (1 − ν)B′)− 6 = u(B) we conclude that U 6 =.
- Case (iii) A 6 = ∅ and B = ∅: Because there are at least two inequivalent personal states, we can choose A′^ ∈ A for which A′^ A. Continuity guarantees there is some η ∈ (0, 1) for which A ∼ ηA′^ + (1 − η)X. Choose λ > 0 for which u(ηA′^ + (1 − η)A) + λη 6 = u(A) and set U (Γ) = u(ΓA + x(Γ)A) + λx(Γ) It is easily checked that the preference relation U induced by U satisfies all the desired axioms; because U (ηA′^ + (1 − η)X) = u(ηA′^ + (1 − η)A) + λη 6 = u(A) we conclude that U 6 =.
In each case we have constructed a preference relation that extends 0 , satisfies all of our axioms and differs from , so the proof is complete.
4 Testable Implications
This section provides a bridge between the general theory described above and our exper- iment, designed to test the implications of the theory. We first define a special setting that serves as the domains/environments in the experimental design and describe their
theoretical properties. Then we develop a number of theoretical results in this setting that are testable on the basis of observed choices.
In order to connect the theoretical predictions of the Theorem with our experimental data, it is convenient to isolate the argument for sufficiency and extend it to a setting in which we consider only a particular set of lotteries. To this end, fix a non-empty set Π of probability vectors. For each non-empty subset Θ ⊂ Ω, let LΠ(Θ) be the set of lotteries of the form p 1 θ 1 +... + pkθk, where (p 1 ,... , pk) ∈ Π and θ 1 ,... , θk ∈ Θ. In view of the Sure Thing Principle, we may identify the lottery p 1 θ +... + pkθ with θ itself, so Θ ⊂ LΠ(Θ).
If Π is the set of all probability vectors then LΠ(Ω) = LΠ(Ω). In that context, recall that observing 0 is just the same as observing the restrictions of to Ω and to L(P ). Hence the following proposition generalizes the sufficient condition in Theorem 1.
Proposition 2 Let Π is a non-empty set of probability vectors and let be a preference relation on LΠ(Ω) that satisfies Completeness, Transitivity, Continuity, the Sure Thing Principle and State Monotonicity. In order that can be deduced from its restrictions Ω to Ω and LΠ(P ) to LΠ(P ), it is sufficient that the DM finds social state ω ∈ Ω/P to be indifferent to some personal state ω˜ ∈ P.
Proof. By assumption, every social state ω ∼Ω ω˜ for some personal state ˜ω ∈ P. (We do not require that ˜ω be unique.) State Monotonicity implies that if (p 1 ,... , pk) ∈ Π and ω 1 ,... , ωk ∈ Ω then (^) ∑ piωi ∼
pi ω˜i
Hence given two lotteries
piωi,
qj ωj ∈ LΠ(Ω) it follows from State Monotonicity and Transitivity that ∑ piωi
qj ωj ⇔
pi ω˜i
qj ω˜j ⇔
pi ω˜i LΠ(P )
qj ω˜j.
That is, can be deduced from Ω and L(P ), as asserted.
In the experiment there is a subject self (the DM) and an (unknown) other so the set of social states Ω consists of monetary payout pairs (a, b), where b ≥ 0 is the payout for self and a ≥ 0 is the payout for other. Because the set of lotteries we can present to (human) subjects is limited, we restrict the set L(Ω) of lotteries on the set Ω of social states to binary lotteries with equal probabilities 12 (a, b) + 12 (c, d). In the framework of the Proposition 2, we are restricting the set of lotteries to Π = {( 12 , 12 )}. To simplify notation,
that choice behavior in the Personal Risk domain coincides with choice behavior in the Social Risk domain; For impartial DM, we show that choice behavior in the Social Choice domain coincides with choice behavior in the Social Risk domain (so an impartial DM is immune to social risk).
In our experiments, we present subjects with a sequence of standard consumer decision problems: selection of a bundle of commodities from a standard budget set:
B = {(x, y) ∈ Ω : pxx + pyy = m}
where px, py, m > 0. A choice of the allocation (x, y) from the budget line represents an allocation between accounts x, y (corresponding to the usual horizontal and vertical axes). The actual payoffs of a particular choice in a particular domain/environment are determined by the allocation to the x and y accounts, according to the particular domain
- Personal Risk, Social Choice, Social Risk. With these preliminaries in hand, we can state the theoretical predictions for selfish preferences and impartial preferences in the experimental setting (but these results hold for arbitrary choice sets):
Proposition 3 If preferences SC in the Social Choice domain are selfish then prefer- ences PR in the Personal Risk domain coincide with preferences SR in the Social Risk domain. In particular: if preferences SC in the Social Choice domain are selfish then choice behavior in the Personal Risk domain and choice behavior in the Social Risk domain coincide – so if B is a budget set then [x, y] ∈ arg max(B) in the Personal Risk domain if and only if 〈x, y〉 ∈ arg max(B) in the Social Risk domain.
Proof. 〈x, y〉 ∈ arg max(B) in the Personal Risk domain if and only if 〈x, y〉 PR 〈x,ˆ yˆ〉 for every 〈ˆx, ˆy〉 ∈ B. Unwinding the notation, this means that
〈x, y〉 =^1 2
(0, x) +^1 2
(0, y) PR^1 2
(0, xˆ) +^1 2
(0, yˆ) = 〈ˆx, yˆ〉
for every 〈x,ˆ yˆ〉 ∈ B. If preferences SC are selfish (x, y) ∼SC (0, y) for all (x, y) this will hold if and only if
[x, y] =
2 (y, x) +
2 (x, y)^ SR
2 (ˆy,^ ˆx) +
2 (ˆx,^ ˆy) = [ˆx,^ yˆ]
for every [ˆx, ˆy] ∈ B. We conclude that preferences PR in the Personal Risk domain coincide with preferences SR in the Social Risk domain. Since preferences in the two domains coincide, choice behavior coincides as well.
Proposition 4 If preferences SC in the Social Choice domain are impartial then SC coincide with preferences SR in the Social Risk domain. In particular: if preferences SC in the Social Choice domain are impartial then choice behavior in the Social Choice domain and choice behavior in the Social Risk domain coincide – so if B is a budget set then (x, y) ∈ arg max(B) in the Social Choice domain if and only if [x, y] ∈ arg max(B) in the Social Risk domain.
Proof. Assume preferences SC in the Social Choice domain are impartial. Consider two choices [x, y] and [ˆx, yˆ] in the Social Risk domain and suppose [ˆx, yˆ] SR [x, y]. When we express this explicitly in terms of lotteries, this means
1 2 (ˆx,^ yˆ) +^
1 2 (ˆy,^ ˆx)^ SR^
1 2 (x, y) +^
1 2 (y, x)
State Monotonicity implies that either (ˆx, yˆ) SC (x, y) or (ˆy, xˆ) SC (y, x); Impartiality implies that if either of these is true then they both of these are true. Hence we conclude that if [ˆx, yˆ] SR [x, y] in the Social Risk domain then (ˆx, yˆ) SC (x, y) in the Social Choice domain. Conversely if (ˆx, yˆ) SC (x, y) in the Social Choice domain then
1 2 (ˆx,^ yˆ) +^
1 2 (ˆy,^ ˆx)^ SR^
1 2 (x, y) +^
1 2 (y, x)
That is: [ˆx, yˆ] SR [x, y] in the Social Risk domain. Putting these together we conclude that preferences SC in the Social Choice domain coincide with preferences SR in the Social Risk domain. Since preferences in the two domains coincide, choice behavior coincides as well.
5 Experimental Design and Sample
In this section, we present the experimental design and the sample participating in the experiment.
5.1 Experimental Design
At the beginning of the session, the participants received general instructions and were told that the experiment would consist of four parts. The instructions were read aloud by an experimenter.
