2.1. Choice scenarios and rules

The model we use is obtained by combining ingredients from the study of welfare distributions (Sen Reference Sen1970) and decision-making under ignorance (Resnik Reference Resnik1987; Peterson Reference Peterson2017). A key ingredient is the class of choice scenarios, which serve as the input for individual and social choice.Footnote ⁵

A choice scenario is a compact representation of a situation where some agent – an individual member of society, a social planner or a group of persons – has to make a certain choice between several competing (mutually exclusive) alternatives. Here, $N$ represents the set of individuals that make up society and $A$ consists of the alternatives (options, choices, policies, courses of action) that may be chosen from. The set $S$ represents the ignorance of the decision-maker, i.e. $S$ is the set of states the decision-maker considers possible. Members of $A\times S$ are called the (possible) outcomes of the scenario. For each outcome $(a,s)$ , the distribution function $d$ determines the welfare (level) $n\in\mathbb{R}$ of each individual $i\in N$ at $s$ , given $a$ . Note that the representation in terms of real numbers implies that welfare levels (of individuals, given states and alternatives) are interpersonally comparable and totally ordered. Depending on the application, one may also interpret them as cardinal welfare levels. In particular, some of the choice rules introduced below for illustrative purposes rely on notions of utility maximization, thus requiring such a stronger interpretation to make sense at all. However, none of our characterization results hinges on this. We work with finite models to keep our examples simple, but our characterization result also does not depend on this assumption.

Figure 1 represents a simple choice scenario with two individuals $1$ and $2$ , three alternatives, and two states. Here, the couples $(n,m)$ represent the distribution function, where $n = d(1,a,s)$ and $m = d(2,a,s)$ . For example, at outcome $(b,s_2)$ individual $1$ ’s welfare is $1$ whereas individual $2$ ’s welfare is $3$ , so that individual $2$ is considered better off than individual $1$ at that outcome. Throughout this article, we use this choice scenario (and variations of it) as our running example.

Figure 1. Choice scenario ${\mathfrak{C}}_1$ , with two individuals, two states, and three alternatives.

We consider two types of functions that are defined on the basis of choice scenarios: social choice rules and individualistic choice rules. Given any choice scenario ${\mathfrak{C}}= \langle N,A,S,d \rangle$ , a social choice rule $\bf {S}$ determines a set of socially admissible alternatives ${\bf S}({\mathfrak{C}}) \subseteq A$ , whereas an individualistic choice rule $\bf{R}$ determines a set of individually admissible alternatives ${\bf{R}}({\mathfrak{C}},i) \subseteq A$ for each individual $i \in N$ .Footnote ⁶ In addition, social and individualistic choice rules are supposed to satisfy some weak axioms that will be spelled out below. We will continue to use choice rules to refer to any rule that selects a set of admissible alternatives from the available ones, whether or not relative to some individual.

It should be noted that, while the specific individual and social choice rules that we will use to illustrate our results all maximize a certain social welfare ordering, our results themselves do not presuppose such an ordering at all. Put differently, the results concern choice rules in a very general sense, i.e. rules that select some (possibly empty) subset of the available options. All we require is that this selection is invariant under some minimal permutations of the models — as will become clear below.

The choice rules that we consider in this paper all satisfy Milnor’s Symmetry condition (Milnor Reference Milnor, Thrall, Coombs and Davis1954), which means that they are invariant under any re-labelling of states or alternatives. In order to spell out this condition below, we define two equivalence relations on choice scenarios. First, two scenarios are S-label variants whenever they can be obtained from each other by a mere relabelling of the states. An example in case is given in Figure 2.

Figure 2. Scenario ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are S-label variants, with $\sigma \left( {{s_1}} \right) = s{{\rm{'}}_2}$ and $\sigma \left( {{s_2}} \right) = s{{\rm{'}}_1}$ .

Analogously, A-label variants are obtained by re-labelling the alternatives, holding all other parts of the scenario fixed. This is illustrated in Figure 3, where $\alpha(a) = c'$ , $\alpha(b) = a'$ , and $\alpha(c) = b'$ . Formally:

Figure 3. Scenario ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are A-label variants, with $\alpha \left( a \right) = c{\rm{'}}$ , $\alpha \left( b \right) = a{\rm{'}}$ , and $\alpha \left( c \right) = b{\rm{'}}$ .

Finally, we will also presuppose that choice rules are invariant under the replication of states, as illustrated in Figure 4 and made precise by Definition 4.

Figure 4. Scenario ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_1^\Delta$ , for $\Delta = \{\delta_1,\delta_2\}$ .

• • $S^\Delta = S \times \Delta$ and

• • for all $i \in N$ , $a \in A$ , $(s,\delta) \in S^\Delta:\; d^\Delta(i,a,(s,\delta)) = d(i,a,s)$ .

2.2. Choice rules

In what follows, we provide examples of individualistic (section 2.2.1) and social (section 2.2.3) choice rules and give exact definitions of the respective classes of such rules (sections 2.2.2 and 2.2.4). It should be emphasized that, while the concrete choice rules that we define below are well-known and have been defended in a range of contexts, we do not endorse or intend to defend them here; we merely use them for the didactic purpose of illustrating our formal framework and its implications.

2.2.1. Examples of individualistic choice rules

Let us briefly recall two well-known individual choice rules to set the stage for later discussions. First, the maximin rule tells us to choose any alternative that maximizes the value of the worst possible outcome.Footnote ⁷ More precisely, where ${\sf min}(X)$ denotes the ≤-minimal element of a set $X$ of real numbers, we have:

Definition 5 (Maximin admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario, $i \in N$ , and $a \in A\!:$ $a \in {\bf{R}}^{\rm m}({\mathfrak{C}},i)$ iff for all $b\in A: {\sf min}\{ d(i,a,s)\! \mid\! s \in S \}\geq {\sf min}\{ d(i,b,s)\! \mid\! s \in S \}$ .

For example, in Figure 1, alternatives $a$ and $b$ are maximin admissible for individual $1$ whereas for individual $2$ only alternative $b$ is admissible.Footnote ⁸

Second, the expected utility rule tells us to choose any alternative that maximizes expected utility. However, recall that we do not assume that individuals have expectations about the relative likelihood of states. In order to perform expected utility calculations, one may however rely on the principle of insufficient reason or principle of indifference (Keynes Reference Keynes1921), which states that in the absence of relevant evidence, individuals should assume that every state is equally likely.Footnote ⁹

Notation 1. Let ${\mathfrak{C}}= \langle N,A,S,d \rangle$ be a choice scenario. Where $a \in A$ and $i \in N$ , we write ${\sf eu}_i(a)$ to denote the expected utility of $a$ for $i$ , i.e.

$${\sf eu}_i(a) = \sum_{s\in S} {{d(i,a,s)}\over{|S|}} .$$

Definition 6 (Expected utility admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario, $i \in N$ , and $a \in A$ : $a \in {\bf{R}}^{\sf eu}({\mathfrak{C}},i)$ iff for all $b\in A: {\sf eu}_i(a) \geq {\sf eu}_i(b)$ .

In our running example (Figure 1), we have ${\sf eu}_1(a) = 1.5$ , ${\sf eu}_1(b) = 1$ , and ${\sf eu}_1(c) = 3$ . Hence, the ranking for individual $1$ induced by expected utility is $c \succ a \succ b$ . For individual $2$ , we have $b \succ a \succ c$ since ${\sf eu}_2(a) = 1.5$ , ${\sf eu}_2(b) = 2.5$ , and ${\sf eu}_2(c) = 1$ . In conclusion, only alternative $c$ is expected utility admissible for individual $1$ , whereas only alternative $b$ is expected utility admissible for individual $2$ .

2.2.2. An exact characterization of individualistic choice rules

In general, we define individualistic choice rules as individual choice rules that satisfy certain axioms. Recall that by an individual choice rule we mean a choice rule that takes as input any choice scenario and individual, and gives as output a set of alternatives in the scenario that are admissible for that individual in that scenario. An individualistic choice rule is an individual choice rule that satisfies Individualism, Symmetry and State Replication Indifference. We spell out these three axioms in turn. First, Individualism (I) requires that for each individual, its set of admissible alternatives does not depend on the payoffs of other individuals in the same scenario. This implies that whatever is admissible for one individual does not change when we change the payoffs of other individuals. In order to define this axiom precisely, we introduce another equivalence relation on choice scenarios:

Definition 7 (i-equivalence). Let ${\mathfrak{C}}= \langle N,A,S,d \rangle$ and ${\mathfrak{C}'} = \langle N,A,S,d' \rangle$ be choice scenarios and let $i \in N$ . Scenarios $\mathfrak{C}$ and ${\mathfrak{C}'}$ are $i$ -equivalent iff for all $a \in A$ and all $s \in S$ : $d'(i,a,s) = d(i,a,s)$ .

In Figure 5, scenarios ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are $1$ -equivalent because individual $1$ receives exactly the same payoffs in ${\mathfrak{C}}_1$ as in ${\mathfrak{C}}_2$ . By contrast, individual $2$ receives different payoffs in both scenarios. Individualism requires that if $a$ is admissible for individual $1$ in ${\mathfrak{C}}_1$ , then $a$ should be admissible for individual $1$ in ${\mathfrak{C}}_2$ as well.

Figure 5. Scenario ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are $1$ -equivalent but not $2$ -equivalent.

Individualism If ${\mathfrak{C}}$ and ${\mathfrak{C}'}$ are $i$ -equivalent then ${\bf{R}}({\mathfrak{C}'},i) = {\bf{R}}({\mathfrak{C}},i)$ .

As a second axiom, we presuppose the aforementioned Symmetry condition from Milnor (Reference Milnor, Thrall, Coombs and Davis1954). This condition can be further subdivided into two parts: Column Symmetry requires that a choice rule is not sensitive to the way states are labelled; Row Symmetry requires that the label of alternatives is immaterial. Relying on the notions of invariance defined in section 2.1, we can state these conditions as follows:

Column Symmetry If $\mathfrak{C}$ and ${\mathfrak{C}'}$ are S-label variants then for all $i\in N:{\bf{R}}({\mathfrak{C}},i) = {\bf{R}}({\mathfrak{C}'},i)$ .

Row Symmetry Let ${\mathfrak{C}}= \langle N,A,S,d\rangle$ and ${\mathfrak{C}'} =\langle N,A',S,d'\rangle$ be A-label variants, with $\alpha: A\rightarrow A'$ the underlying bijection. Then for all $i \in N$ and $a \in A$ : $a\in {\bf{R}}({\mathfrak{C}},i)$ iff $\alpha(a) \in {\bf{R}}({\mathfrak{C}'},i)$ .

In what follows, Symmetry is simply the conjunction of both properties, Column Symmetry and Row Symmetry.

Third and last, we require State Replication Indifference. This axiom is similar to Column Symmetry, but replacing S-label variants with state replication:

State Replication Indifference For all choice scenarios ${\mathfrak{C}}= \langle N,A,S,d\rangle$ , all $i \in N$ , and all non-empty and finite $\Delta\!: {\bf{R}}({\mathfrak{C}},i) = {\bf{R}}({\mathfrak{C}}^\Delta,i)$ .

State Replication Indifference is weaker than the property known as Column Duplication (Milnor Reference Milnor, Thrall, Coombs and Davis1954; Binmore Reference Binmore2008).Footnote ¹⁰ The latter allows for adding single copies of just one state, whereas State Replication only allows one to copy all states a given number of times.

Let us discuss each of these three axioms in turn, in order to explain their motivation and intended interpretation. First, Individualism is integral to the concept of an original position argument. Rawls writes: ‘The essential idea is that we want to account for the social values, for the intrinsic good of institutional, community, and associative activities, by a conception of justice that in its theoretical basis is individualistic’ (Rawls Reference Rawls1999 [1971]: 233). Hence, the attractiveness of original position arguments relies on the fact that some social choice rules can be reduced to the rational decision-making of self-interested persons (i.e. they are individualistic) under hypothetical circumstances considered fair.Footnote ¹¹

Second, Symmetry is both common and plausible for models that represent the ignorance of the decision-maker in unidimensional terms, i.e. as a single set of atomic states. Given such a representation, if the label of a state or alternative really matters for what counts as a rational choice, then it seems that something went wrong when stipulating the welfare levels in the first place. For instance, this may mean that depending on the labels, one should downgrade or upgrade the outcome (for certain individuals), associating a different payoff with the alternative at that state. Put differently, the whole point of representing a concrete situation by means of a decision scenario is that one must ensure that rational choice is really a function of the payoffs at each state, not of the labels or ordering of the states and alternatives themselves.Footnote ¹² Moreover, as shown in the Appendix, without Column Symmetry, the notion of an original position argument as we characterize it is trivialized (Theorem 4).

Some readers may worry that our very starting point – i.e. that a decision-maker’s ignorance can be represented by a unique set of atomic states (that cannot be further analysed) – is problematic in the context of original position arguments. After all, when placed in the original position, a decision-maker may well be ignorant about both the state of the world and their own identity. So even if in other contexts this is unproblematic, it would be a bad idea to collapse these two types of ignorance into one set of states. We have little to say in response to this critique, other than that it requires spelling out in detail what would go wrong if we do collapse all ignorance into a single dimension or set of atomic states. Ultimately this is what our paper does: our results indicate that at least on some accounts of fair social choice rules, this model is overly simplistic. Our paper thus establishes a conditional claim: if one buys into the model defined here, then original position arguments can be characterized by the axioms that we discuss below. In section 5 we reconsider this point and discuss an alternative, richer account of ignorance.

Third and last, State Replication Indifference, though less well-known, seems just as natural as Symmetry in the context of choice under ignorance. Intuitively there should be no difference between scenarios such as ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_1^\Delta$ in Figure 4. In particular, even if some bad, good or mediocre payoff will appear more often in the state replication variant of a given scenario than in the initial scenario, the ratio between such states and other states remains constant.Footnote ¹³

In sum, we obtain the following:

Definition 8 (Individualistic choice rule). ${\bf{R}}$ is an individualistic choice rule iff ${\bf{R}}$ is an individual choice rule that satisfies Individualism, Symmetry and State Replication Indifference.

Clearly, both maximin and the expected utility rule satisfy Individualism since both determine a set of admissible alternatives for each individual $i \in N$ and only take into account the payoffs of that individual. Likewise, both rules satisfy Symmetry and State Replication Indifference.Footnote ¹⁴ Thus, they are both individualistic choice rules in the above sense.

2.2.3. Examples of social choice rules

In this section, we introduce four distinct social choice rules. These social choice rules will be used as examples throughout the article.

Difference Principle The difference principle states that we should ‘arrange social and economic inequalities in such a way that they are to the benefit of the least advantaged’ (Rawls Reference Rawls1999 [1971]: 20). Conceived as a social choice rule, the difference principle tells us to choose any alternative that maximizes the prospects of the least well-off. However, once we are dealing with a context of choice under ignorance about the state of the world, it is ambiguous what exactly ‘least well-off’ means, and hence one may specify this principle in conceptually distinct ways (see De Coninck and Van De Putte (Reference De Coninck and Van De Putte2023) for an overview of these approaches).

Let us first focus on what is called the ‘basic approach’ in De Coninck and Van De Putte (Reference De Coninck and Van De Putte2023). On the basic approach, we maximize welfare, ignoring the distinction between different states and different individuals.

Definition 9 (Difference admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario and $a \in A\!:$ $a \in {\bf{S}}^{\sf d}(\mathfrak{C})$ iff for all ${b \in A:{\sf min}\{d(i,a,s)\! \mid\! i \in N \, \, s \in S \}\geq {\sf min}\{d(i,b,s)\! \mid\! i \in N\, \, s \in S \}}$ .

For example, in the scenario in Figure 1, we have ${\sf min}\{d(i,a,s)\! \mid i \in N \, \, s \in S \} = 1$ , ${\sf min}\{d(i,b,s)\! \mid\! i \in N\, \, s \in S \} = 1$ , and ${\sf min}\{d(i,c,s)\! \mid\! i \in N\, \,s \in S \} = 0$ . Hence, both $a$ and $b$ are difference admissible, but $c$ is not.

Average Expected Utility The second social choice rule that we discuss is the average expected utility rule. It tells us to choose any alternative that maximizes the average expected utility of all individuals.

Notation 2. Let ${\mathfrak{C}}= \langle N,A,S,d \rangle$ be a choice scenario. Where $a \in A$ , we write ${\sf aeu}(a)$ to denote the average expected utility of $a$ , i.e.

$${\sf aeu}(a) = {{\sum_{i\in N} {\sf eu}_i(a)}\over{|N|}} .$$

Definition 10 (Average expected utility admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario and $a \in A\!:$ $a \in {\bf{S}}^{\sf aeu}(\mathfrak{C})$ iff for all $b\in A: {\sf aeu}(a) \geq {\sf aeu}(b)$ .

Applying the average expected utility rule to our running example (cf. Figure 1), we have ${\rm{aeu}}\left( a \right) = 1.5$ , ${\rm{aeu}}\left( b \right) = 1.75$ , and ${\rm{aeu}}\left( c \right) = 2$ , and hence the ranking induced by average expected utility is $c \succ b \succ a$ . Notice that the expected utility for individual 1 under $c$ is very high ( ${\rm{e}}{{\rm{u}}_1}\left( c \right) = 3$ ), whereas for individual 2 it is relatively low ( ${\rm{e}}{{\rm{u}}_2}\left( c \right) = 1$ ). Still, the rule picks $c$ because the average social utility is skewed upwards by the great prospects of individual 1, even though it is the worst alternative for individual $2$ . For this reason, the average expected utility rule is sometimes criticized on the grounds that it allows for the ‘sacrificing’ of those who are less well-off if doing so would be offset by a sufficient benefit to others.Footnote ¹⁵ In order to remedy this, one should give greater weight to the expectations of those who are less well-off. More generally, one may consider ways of combining the difference principle and expected utility. In what follows, we consider two such combinations originally introduced by Mongin and Pivato (Reference Mongin and Pivato2021).Footnote ¹⁶

Difference Expected Utility According to this social choice rule we maximize the social value of alternatives, where the social value of an alternative is the expected utility of the individual who has the worst prospects under that alternative.

Definition 11 (Difference expected utility admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario and $a \in A$ : $a \in {\bf S}^{\sf deu}(\mathfrak{C})$ iff for all $b\in A: {\sf min}\{{\sf eu}_i(a) \mid i \in N \} \geq {\sf min}\{{\sf eu}_i(b) \mid i \in N \}$ .

In our running example (cf. Figure 1), we have ${\sf min}\{{\sf eu}_i(a)\! \mid\! i \in N \} = 1.5$ , ${\sf min}\{{\sf eu}_i(b)\! \mid\! i \in N \} = 1$ , and ${\sf min}\{{\sf eu}_i(c)\! \mid\! i \in N \} = 1$ . Hence, the difference expected utility ranking is $a \succ b\sim c$ .

Maximin Expected Utility On this social choice rule, to determine the social value of an alternative, we obtain for each outcome the value of the worst-off person at that outcome and then apply the expected utility rule to these values.

Notation 3. Let ${\mathfrak{C}}= \langle N,A,S,d \rangle$ be a choice scenario. Where $a \in A$ , we write ${\sf meu}(a)$ to denote the maximin expected utility of alternative $a$ , i.e.

$${\sf meu}(a) = \sum_{s \in S} {{{\sf min}\{d(i,a,s)\! \mid\! i \in N \}}\over{|S|}}.$$

Definition 12 (Maximin expected utility admissibility). Where ${\mathfrak{C}}= \langle N,A,S,d\rangle$ is a choice scenario and $a \in A$ , $a \in {\bf S}^{\sf mau}(\mathfrak{C})$ iff for all $b\in A: {\sf meu}(a) \geq {\sf meu}(b)$ .

Looking at our running example once more, we have ${\sf meu}(a) = 1.5$ , ${\sf meu}(b) = 1$ , and ${\sf meu}(c) = 0$ . Hence, the induced ranking is $a \succ b \succ c$ . An overview of the various rankings induced by the social choice rules introduced so far is given by Table 1.

Table 1. The choice rules applied to the running example (cf. Figure 1)

3.1. A definition of original position arguments

We present the general format for evaluating original position arguments introduced in De Coninck and Van De Putte (Reference De Coninck and Van De Putte2023). The starting point is the view that the original position does not correspond to a particular scenario: instead, for each particular choice scenario, we can construct a corresponding choice scenario which has the characteristics of an original position. The latter is called the original position transformation of the initial choice scenario.

• • $S^* = S \times \Pi$ and

• • for all $i\in N$ , $a\in A$ , and $(s,\pi)\in S^*$ : $d^*(i,a,(s,\pi)) = d(\pi(i),a,s)$ .

In other words, given some choice scenario ${\mathfrak{C}}$ , we obtain its OP-transformation ${\mathfrak{C}}^*$ by combining the ignorance in the original model with ignorance about the individual’s identities and the way these identities affect the level of welfare one receives. We illustrate this by means of our running example. Figure 6 displays (on the left-hand side) the choice scenario ${\mathfrak{C}}_1$ , and (on the right-hand side) its OP-transformation. Here, $\pi_{=}$ is the identity relation, and $\pi_{\ne}$ swaps the two individuals, i.e. $\pi_{=}\!(1) = 1$ , $\pi_{=}\!(2) = 2$ , $\pi_{\ne}\!(1) = 2$ , and $\pi_{\ne}\!(2) = 1$ . If we apply the maximin rule to the OP-transformation, we find that both $a$ and $b$ are admissible for individual $1$ , while $c$ is not.

Figure 6. A choice scenario ( ${\mathfrak{C}}_1$ ) and its OP-transformation ( ${\mathfrak{C}}_1^*$ ).

With this in place, we can give an exact definition of what it means for a social choice rule to be supported by an original position argument.

The social choice rule $\bf{S}$ can be supported by an original position argument iff there is an individualistic choice rule ${\bf{R}}$ such that $\bf{S}$ can be original position derived from ${\bf{R}}$ .

Note that, while social choice rules can be OP-derived from or supported by an (arbitrary) individual choice rules, in order to obtain an original position argument, we require the individual choice rule in question to satisfy the axioms of Symmetry, State Replication Indifference and Individualism. This allows us to consider what would happen to the notion of an original position argument if we lift some of these axioms (cf. our Appendix).

3.2. Two examples of original position arguments

Recall that both Rawls and Harsanyi claimed that their favoured social choice rules can be supported by an original position argument. We will show that we can verify counterparts of these claims, in the context of choice under ignorance. In particular, we will show that the difference principle can be OP-derived from the maximin rule (Proposition 1) and that the principle of average expected utility can be OP-derived from the expected utility rule (Proposition 2). We also provide an example of a social choice rule that cannot be supported by an original position argument: the difference expected utility rule (Proposition 3).

Before we get to the examples, it will be helpful to introduce some extra notation. In particular, we work with multisets, i.e. sets that can contain multiple instances of the same member. To distinguish a multiset from a regular set, we use rectangular brackets $[$ , $]$ instead of $\{,\}$ .

We start by observing that there is a specific relation between the payoffs of all individuals in a choice scenario and the payoffs of a fixed individual in its original position transformation:

Fact 1. For all choice scenarios ${\mathfrak{C}}= \langle N,A,S,d \rangle$ , $i \in N$ and $a \in A$ :

$$[d(j,a,s)\! \mid\!\, j \in N \, \, s \in S ] = [d(i,a,s)\! \mid\! s \in S^* ] .$$

Fact 1 says that, given an alternative, the multiset of possible payoffs any individual can receive in a choice scenario is identical to the multiset of possible payoffs a fixed individual can receive in its OP-transformation. This holds because the OP-transformation of any scenario is constructed precisely so that each individual in the original position considers the possibility of receiving the payoffs of each individual in the pre-transformed scenario.

Compare scenarios ${\mathfrak{C}}_{1}$ and ${\mathfrak{C}}_{2}$ . In scenario ${\mathfrak{C}}_{1}$ , both alternative $a$ and alternative $b$ are admissible according to ${\bf S}^{\sf deu}$ , since the expected utility of the worst-off person under both is $0$ . By contrast, in ${\mathfrak{C}}_{ 2}$ only alternative $a$ is admissible since the expected utility of both individuals is $0.5$ , whereas the expected utility for each under alternative $b$ is $0$ . Hence, ${\bf S}^{\sf deu}({\mathfrak{C}}_{ 1}) \ne {\bf S}^{\rm deu}({\mathfrak{C}}_{2})$ .

The proof now proceeds by reductio. Suppose that there is some individualistic choice rule ${\bf{R}}$ such that the social choice rule ${\bf S}^{\sf deu}$ can be OP-derived from it. Hence, $x \in {\bf S}^{\sf deu}({\mathfrak{C}}_{1})$ iff $x \in {\bf{R}}({\mathfrak{C}}_{ 1}^*,i)$ for all $i\in N$ . Similary for scenario ${\mathfrak{C}}_{ 2}$ we have $x \in {\bf S}^{\sf deu}({\mathfrak{C}}_{2})$ iff $x \in {\bf{R}}({\mathfrak{C}}_{\ 2}^*,i)$ for all $i \in N$ . By Column Symmetry and Individualism, for all $i \in N$ , $x \in {\bf{R}}({\mathfrak{C}}_{ 1}^*,i)$ iff $x \in {\bf{R}}({\mathfrak{C}}_{2}^*,i)$ . Following this chain of equivalences allows us to conclude that $x \in {\bf S}^{\sf deu}({\mathfrak{C}}_{ 1})$ iff $x \in {\bf S}^{\sf deu}({\mathfrak{C}}_{2})$ , which contradicts our earlier observation that ${\bf S}^{\sf deu}({\mathfrak{C}}_{ 1}) \ne {\bf S}^{\sf deu}({\mathfrak{C}}_{ 2})$ .□

So far, we have illustrated how one can show that a social choice rule can be supported by an original position argument (Propositions 1 and 2) as well as how one can argue that it is impossible to do so (Proposition 3).Footnote ²¹ In doing so, we had to rely on the specific properties of the social choice rules in question. However, one may also ask whether there are general properties that make original position arguments tick. This would allow us to reduce the question of whether a given social choice rule can be supported by an original position argument to the question of whether it satisfies these properties.

3.3. Indifference axioms

In this section, we introduce and discuss two axioms that we show to be characteristic of original position arguments. First, Indifference to Intra-State Distribution of Payoffs to Persons (IISD) states that social admissibility should not depend on how the payoffs within states are distributed across individuals. To put it plainly, given any particular outcome, it should not matter whether Bob gets a cake and Alice an apple or the other way around. Let us make this more precise.

Figure 7 below depicts two choice scenarios that are $\Pi$ -variants of each other. Note that in scenario ${\mathfrak{C}}_2$ the payoffs of individuals 1 and 2 are switched at $s_1$ compared with $s_1$ in ${\mathfrak{C}}_1$ , i.e. $\pi_{s_1}(i) = j, \pi_{s_1}(j) =i$ and $\pi_{s_2}(i) = i,\pi_{s_2}(j) = j$ . In general, for different states $s$ and $s'$ one may have different permutations: $\pi_s \ne \pi_{s'}$ .

Figure 7. Two $\Pi$ variants.

One way to interpret the IISD axiom is to view it as securing what might be called ex post anonymity; i.e. once a particular state is fixed, the labels of individuals should not matter. An example of a social choice rule that satisfies IISD is the maximin expected utility rule. The maximin expected utility rule is only sensitive to the utility values within states, and any permutation of the individual’s payoffs at those states does not affect the utility values at those state. By contrast, the difference expected utility rule does not satisfy IISD. For example, in Figure 7, the difference expected utility rule considers alternative $a$ admissible in scenario ${\mathfrak{C}}_1$ , whereas both alternative $a$ and alternative $b$ are considered admissible in scenario ${\mathfrak{C}}_2$ .

In light of the preceding, IISD is strictly stronger than Anonymity, i.e. the axiom that tells us that the labels of individuals do not matter (Sen Reference Sen1970). Since we need not rely on Anonymity for our results to go through, we will omit a detailed discussion of this principle.

Our second axiom, Indifference to Intra-Person Distribution of Payoffs at States (IIPD), says that social admissibility should not depend on how each individual’s payoffs are distributed across states. For example, it should not make a difference whether Bob gets a cake in state $s$ and an apple in state $s'$ but only that Bob either gets an apple or a cake.

Figure 8 depicts two $\Sigma$ -variants. In scenario ${\mathfrak{C}}_2$ , the payoffs of individual 1 are switched between ${s_1}$ and ${s_2}$ compared with scenario ${\mathfrak{C}}_1$ , whereas the payoffs of individual $2$ are untouched, i.e. $\sigma_{1}(s_1) = s_2, \sigma_{1}(s_2) = s_1$ and $\sigma_{2}(s_1) = s_1,\sigma_{2}(s_2) = s_2$ .

Figure 8. Two $\Sigma$ -variants.

The difference expected utility rule satsifies IIPD, whereas the maximin expected utility rule does not. The difference expected utility rule satisfies IIPD because any permutation of an individuals’ payoffs across states does not change the expected utility of that individual. Of course, this only holds because we are assuming that each state is equally likely and hence switching payoffs between equally likely states does not make a difference. To see that the maximin expected utility rule does not satisfy IIPD, consider Figure 8. In scenario ${\mathfrak{C}}_1$ , only alternative $a$ is maximin expected utility admissible, whereas in scenario ${\mathfrak{C}}_2$ all alternatives are admissible.

A little reflection on the definition of IIPD reveals that it implies Column Symmetry for social choice rules. To see why, suppose ${\mathfrak{C}}$ and ${\mathfrak{C}'}$ are S-label variants (cf. Definition 2). Hence, there is some $\sigma: S \to S$ such that for all $i \in N$ , $a \in A$ and $s \in S:$ $d'(i,a,\sigma(s)) = d(i,a,s)$ . Given $\sigma$ , we let $\sigma_i = \sigma$ for each $i \in N$ . This gives us exactly what is needed to satisfy the definition of $\Sigma$ -variants (Definition 17). By IIPD, ${\bf S}({\mathfrak{C}}) = {\bf S}({\mathfrak{C}'})$ . So we have:

Taken together, IISD and IIPD say that it does not matter which individual gets what and under what circumstances. More precisely, any permutation that swaps the payoffs for a given individual $i$ at a state $s$ with the payoffs of a (possibly different) individual $j$ at a (possibly different) state $s'$ – and that does so uniformly for each of the choices $a,b,\ldots$ – should leave the set of admissible choices unaltered. While these are rather strong conditions, both the basic difference rule and the average expected utility rule satisfy them.Footnote ²² In contrast, as explained above, Difference Expected Utility does not satisfy IISD, while Maximin Expected Utility does not satisfy IIPD. Table 2 gives an overview of which social choice rules satisfy which of the axioms discussed so far.

Table 2. Overview of the axioms governing social choice rules

4.1. Left to right

In what follows we show that if a social choice rule can be supported by an original position argument, it satisfies IISD (Theorem 1) and IIPD (Theorem 2).

The following lemma establishes that if two scenarios are $\Pi$ -variants, then for any individual, the same alternatives will be admissible in their OP-transformations. In what follows, if $f$ and $g$ are functions, we use $g \circ f$ to denote the composition of $f$ and $g$ , i.e. $g \circ f(x) = g(f(x))$ .

Proof. Let ${\mathfrak{C}}_1 = \langle N,A,S,d_1 \rangle$ and ${\mathfrak{C}}_2 = \langle N,A,S,d_2 \rangle$ be choice scenarios and let ${\bf{R}}$ be an individualistic choice rule. Suppose that ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are $\Pi$ -variants. We show that ${\mathfrak{C}}^*_1$ and ${\mathfrak{C}}^*_2$ are S-label variants. By the supposition, for all $s \in S$ there is a bijection $\pi_s: N \to N$ such that for all $a \in A$ and $i \in N: d_2(\pi_s(i),a,s) = d_1(i,a,s)$ (Definition 16). We define $\sigma^*: S^* \to S^*$ as follows. Given some arbitrary state $(s,\pi) \in S^*$ , let $\sigma^*(s,\pi) = (s,\pi_s \circ \pi)$ . Let $i \in N$ , $a \in A$ , $(s,\pi) \in S^*$ and $m \in \mathbb{R}$ be arbitrary. We have:

$$\hskip 32pt{\rm{iff\quad\quad\quad Definition\;}}14{\rm{\;}}\left( {{\rm{OP}} - {\rm{transformation}}} \right)$$

$$\hskip-118pt{{d_1}\left( {\pi \left( i \right),a,s} \right) = m}$$

$${\rm{{\hskip40pt iff}\quad\quad\quad\quad\quad\quad \hskip12ptDefinition\;}}16{\rm{\;}}\left( {{\rm{\Pi }} - {\rm{variants}}} \right)$$

$$\hskip-132pt{{d_2}\left( {{\pi _s} \circ \pi \left( i \right),a,s} \right) = m}$$

$$\hskip 32pt{\rm{iff\quad\quad\quad Definition\;}}14{\rm{\;}}\left( {{\rm{OP}} - {\rm{transformation}}} \right)$$

$$\hskip-135pt{d_2^{\rm{*}}\left( {i,a,\left( {s,{\pi _s} \circ \pi } \right)} \right) = m.}$$

It follows that $\sigma^*$ is as required so that ${\mathfrak{C}}^*_1$ and ${\mathfrak{C}}^*_2$ are S-label variants. Since ${\bf{R}}$ satisfies Column Symmetry, it follows that for all $i \in N: {\bf{R}}({\mathfrak{C}}_1^*,i) = {\bf{R}}({\mathfrak{C}}_2^*,i)$ .

With Lemma 1 in place, proving the following result is now straightforward.

Proof. Suppose $\bf{S}$ is a social choice rule that can be supported by an original position argument. Hence, there is some individualistic choice rule ${\bf{R}}$ such that for all scenarios ${{\mathfrak{C}}}= \langle N,A,S,d \rangle$ and all individuals $i \in N: {\bf{R}}({\mathfrak{C^*}},i) = {\bf S}({\mathfrak{C}})$ . Take two arbitrary $\Pi$ -variants ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ . Let $i \in N$ and $a \in A$ be arbitrary. We have

$$\hskip-160pt{a \in {\bf S}({\mathfrak{C}}_1)}$$

$$\hskip40pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Definition\;}}15{\rm{\;}}\left( {{\rm{OP}} - {\rm{argument}}} \right)}$$

$$\hskip-152pt{a \in {\bf{R}}({\mathfrak{C}}_1^*,i)}$$

$${\hskip-46pt \rm iff}\hskip92pt {\rm Lemma\;}1$$

$$\hskip-152pt{a \in {\bf{R}}({\mathfrak{C}}_2^*,i)}$$

$${\hskip42pt\rm iff}\hskip89pt{\rm Definition\ } 15{\;}\left( {\rm OP} - {\rm argument} \right)$$

$$\hskip-158pt{a \in {\bf S}({\mathfrak{C}}_2).}$$

Hence, $\bf{S}$ satisfies IISD.

Our next step is to show that if $\bf{S}$ can be supported by an original position argument, it has to satisfy the IIPD axiom. We prove this in a way analogous to our proof of Theorem 1. More precisely, we show that if two scenarios are $\Sigma$ -variants, then if we apply any individualistic choice rule to their OP-transformations, we end up with the same set of admissible alternatives (Corollary 1). However, proving this claim requires a bit more preparatory work. Lemma 2 shows that the property of being $\Sigma$ -variants is preserved under OP-transformations. Lemma 3 establishes that if two scenarios are $\Sigma$ -variants, then the application of any individualistic choice rule on those scenarios themselves will yield the exact same recommendations. Corollary 1 follows immediately from these two properties.

Proof. Suppose that ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ $\Sigma$ -variants. Hence, for all individuals $i \in N$ there is some bijection $\sigma_i: S_1 \to S_2$ such that for all alternatives $a \in A:$ $d_1(i,a,s) = d_2(i,a,\sigma_i(s))$ . For every $i \in N$ and state $(s,\pi) \in S^*$ , let

$$ \sigma^{*}_i(s,\pi) = (\sigma_{\pi(i)}(s),\pi) $$

Let $i \in N$ , $a \in A$ , $(s,\pi) \in S^*$ , and $m \in \mathbb{R}$ be arbitrary. We have:

$$\hskip-95pt{d_1^{\rm{*}}\left( {i,a,\left( {s,\pi } \right)} \right) = m}$$

$${\hskip105pt{\rm iff}}{\hskip26pt{\rm Definition\;}}{14\;}\left( {{\rm OP}} - {\rm{transformation}} \right)$$

$$\hskip-90pt{{d_1}\left( {\pi \left( i \right),a,s} \right) = m}$$

$$\hskip108pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Definition\;}}17{\rm{\;}}\left( {{\rm{\Sigma }} - {\rm{variants}}} \right)}$$

$${\hskip-115pt{d_2}\left( \pi \left( i \right),a,\sigma _{\pi \left( i \right)}\left( s \right) \right) = m}$$

$$\hskip105pt{\rm iff}\;\;\;\;\;\;\;\;\;\;{\rm Definition\;}14{\rm{\;}}\left( {\rm{OP}} - {\rm{transformation}} \right)$$

$$\hskip-115pt{d_2^{\rm{*}}(i,a,\left( {{\sigma _{\pi \left( i \right)}}\left( s \right),\pi } \right) = m.}$$

Hence, for every $i \in N$ , $\sigma^*_i$ is as required so that ${\mathfrak{C}}^*_1$ and ${\mathfrak{C}}^*_2$ are $\Sigma$ -variants.□

Proof. Let ${\mathfrak{C}}_1 = \langle N,A,S,d_1 \rangle$ and ${\mathfrak{C}}_2 = \langle N,A,S,d_2 \rangle$ be choice scenarios. Fix an arbitrary individual $i \in N$ and let ${\bf{R}}$ be an individualistic choice rule. Suppose that ${\mathfrak{C}}_1$ and ${\mathfrak{C}}_2$ are $\Sigma$ -variants. Hence, there is a bijection $\sigma_i: S \to S$ such that for all alternatives $a \in A: d_1(i,a,s) = d_2(i,a,\sigma_i(s))$ . Let ${\mathfrak{C}}^\circ_i = \langle N,A,S,d^\circ_i \rangle$ , where $d^\circ_i$ is defined as:

$$d^\circ_i(j,a,s) =_{df} \cases{ d_1(j,a,s)& if j = i \cr d_2(j,a,\sigma_i(s))& otherwise}$$

We show that ( $\star$ ) ${\mathfrak{C}}^\circ_i = \langle N, A,S,d^\circ_i \rangle$ and ${\mathfrak{C}}_2$ are S-label variants. Let $\sigma = \sigma_i$ . Let $j \in N$ , $a \in A$ , $s\in S$ , and $m \in \mathbb{R}$ be arbitrary. There are two cases. The case for $j \ne i$ follows immediately from the definition of $d^\circ_i$ and $\sigma$ . If $j = i$ , we have

$$\hskip-65pt{d_i^ \circ \left( {i,a,s} \right) = m}$$

$$\hskip80pt{ {\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Definition\;of\;}}{d}_{{i}}^ \circ}$$

$$\hskip-65pt{{d_1}\left( {i,a,s} \right) = m}$$

$$\hskip80pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Definition\;of\;}}\sigma}$$

$$\hskip-69pt{ d_2\left( i,a,\sigma \left( s \right) \right) = m.}$$

Let $i \in N$ , and $a \in A$ be arbitrary. We have:

$${\hskip50pt \rm iff}\;\;\;\;\;\;\;\;{\rm Individualism\ and\ definition\ of\ } d_{{i}}^ \circ$$

$${\hskip45pt \rm iff}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm Column\ Symmetry\ and\ }\left( \star \right)$$

$$\hskip-155pt {a \in \bf{R}}({\mathfrak{C}}_2,i).$$

4.2. Right to left

In this section, we show that if a social choice rule $\bf{S}$ satisfies IISD and IIPD, then $\bf{S}$ can be supported by an original position argument (Theorem 3). To prove this, we show that given any social choice rule $\bf{S}$ satisfying IISD and IIPD, we can define an individualistic choice rule ${\bf{R}}^{\bf S}_{=}$ such that $\bf{S}$ can be original position derived from ${\bf{R}}^{\bf S}_{=}$ . In order to give a precise definition of ${\bf{R}}^{\bf S}_{=}$ , we first introduce some additional notation.

In words, the scenario ${\mathfrak{C}}_{=i}$ is the scenario where at every outcome every individual receives the same payoff as individual $i$ receives at that outcome in scenario ${\mathfrak{C}}$ . To see how this works, an illustration is helpful (cf. Figure 9).

Figure 9. A choice scenario ( $\mathfrak{C}$ ) and the scenario ( ${\mathfrak{C}}_{=1}$ ), where $1$ is the ‘first’ individual.

In the next step, we define the choice rule ${\bf{R}}^{\bf S}_{=}$ using the definition of ${\mathfrak{C}}_{=i}$ as follows.

The first lemma that we establish shows that ${\bf{R}}^{\bf S}_{=}$ is an individualistic choice rule if $\bf{S}$ satisfies IIPD.

1. (i) Column Symmetry

2. (ii) Row Symmetry

3. (iii) State Replication Indifference

4. (iv) Individualism

Hence, for all $j \in N$ we have $d'_{=i}(j,a,\sigma(s)) = d_{=i}(j,a,s)$ . This implies that ${\mathfrak{C}}_{=i}$ and ${\mathfrak{C}'}_{=i}$ are S-label variants. Since $\bf{S}$ is a social choice rule, it satisfies Column Symmetry, and so we have ${\bf S}({\mathfrak{C}}_{=i}) = {\bf S}({\mathfrak{C}'}_{=i})$ . By Definition 19, ${\bf{R}}^{\bf{S}}_{=}({\mathfrak{C}},i) = {\bf{R}}^{\bf{S}}_{=}({\mathfrak{C}'},i)$ . Thus, ${\bf{R}}$ satisfies Column Symmetry.

Hence, for all $j \in N$ we have $d'_{=i}(j,\alpha(a),s) = d_{=i}(j,a,s)$ . This implies that ${\mathfrak{C}}_{=i}$ and ${\mathfrak{C}'}_{=i}$ are A-label variants. Since $\bf{S}$ is a social choice rule, it satisfies Row Symmetry, and so we have that for all $a \in A$ , $a\in {\bf S}({\mathfrak{C}}_{=i})$ iff $\alpha(a)\in {\bf S}({\mathfrak{C}'}_{=i})$ . By Definition 19, for every $i\in N$ and $a \in A$ , $a\in {\bf{R}}^{\bf S}_{=}({\mathfrak{C}},i)$ iff $\alpha(a)\in {\bf{R}}^{\bf S}_{=}({\mathfrak{C}'},i)$ . Thus, ${\bf{R}}$ satisfies Row Symmetry.

Before we come to the main result of this section, we prove two lemmas. First, if a social choice rule satisfies IISD, then the set of admissible alternatives is preserved under OP-transformations (Lemma 5). Second, for each individual $i$ , the scenario ${\mathfrak{C}}_{=i}^*$ is a $\Sigma$ -variant of ${\mathfrak{C}}^*$ (Lemma 6).

Proof. Let ${\mathfrak{C}}= \langle N,A,S,d \rangle$ be a choice scenario and let $\bf{S}$ be a social choice rule that satisfies IISD. Where $\Pi$ is the set of all bijections $\pi: N \rightarrow N$ , ${\mathfrak{C}}^\Pi = \langle N,A,S^\Pi,d^\Pi \rangle$ is a State Replication variant of ${\mathfrak{C}}$ (Definition 4). We now show that ${\mathfrak{C}}^\Pi$ and ${\mathfrak{C}}^*$ are $\Pi$ -variants. For every $i \in N$ and state $(s,\pi) \in S^\Pi$ , let

$${\delta _{\left( {s,\pi } \right)}}\left( i \right) = {\pi ^{ - 1}}\left( i \right)$$

Let $i \in N$ , $a \in A$ , $(s,\pi) \in S^\Pi$ , and $m \in \mathbb{R}$ be arbitrary. We have:

$$\hskip-140pt{{d^{\rm{\Pi }}}\left( {i,a,\left( {s,\pi } \right)} \right) = m}$$

$$\hskip80pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;State\;Replication\;Indifference}}}$$

$$\hskip-116pt{d\left( {i,a,s} \right) = m}$$

$$\hskip85pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;OP}} - {\rm{transformation\;}}\left( {{\rm{Definition\;}}14} \right)}$$

$${\hskip87pt \rm{iff}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\delta _{{\left( {{\rm{s}},\pi } \right)}}\left( {\rm{i}} \right) = {\pi ^{ - 1}}\left( {\rm{i}} \right)$$

Relying on our intermediate steps, we have the following, for every $a \in A$ :

$$\hskip-120pt{a \in {\bf S}({\mathfrak{C}})}$$

$${\hskip55pt {\rm iff}}\;\;\;\;\;\;\;\;\;\;\;{\rm State\ Replication\ Indifference}$$

$$\hskip52pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;IISD}}}$$

Proof. Fix a choice scenario ${\mathfrak{C}}= \langle N,A,S,d \rangle$ and individual $i \in N$ . Because of the definition of $\Sigma$ -variants (Definition 17) we must show that for all $j \in N$ , there is a function $\sigma^j: S^* \to S^*$ such that for all $a \in A$ and $(s,\pi) \in S^*$ :

$$d_{ = i}^{\rm{*}}(j,a,\left( {s,\pi } \right) = {d^{\rm{*}}}\left( {j,a,{\sigma ^j}\left( {s,\pi } \right)} \right)$$

For every $j \in N$ , we define $\sigma^j$ as follows. Given some arbitrary state $(s,\pi) \in S^*$ , let $\sigma^j(s,\pi) = (s,\pi')$ , where $\pi'$ is such that:

1. (1) $\pi(i) = \pi'(j)$ ,

2. (2) $\pi(j) = \pi'(i)$ , and

3. (3) $\pi(k) = \pi'(k)$ for all $k \in N \setminus \{1,j\}$ .

Note that ${\sigma ^j}$ is well-defined. First, there is always exactly one $\pi'$ that satisfies conditions (1)–(3). Second, $\sigma^j$ is a bijection; i.e. if $\sigma^j(s,\pi) = (s,\pi')$ , then $\sigma^j(s,\pi') = (s,\pi)$ . As our next step, we show that we have the following property:

$$\left( \dagger \right){\rm{\;for\ all\ j}} \in {\rm{N}},{\rm{a}} \in {\rm{A\ and\ }}\left( {{\rm{s}},\pi } \right) \in {{\rm{S}}^{\rm{*}}}:{d^{\rm{*}}}\left( {j,a,{\sigma ^j}\left( {s,\pi } \right)} \right) = {d^{\rm{*}}}\left( {i,a,\left( {s,\pi } \right)} \right).$$

Let $j \in N$ , $a \in A$ , $(s,\pi) \in S^*$ be arbitrary and let $m \in \mathbb{R}$ . Given $(s,\pi)$ , let $\pi {\rm{'}}$ be such that each of (1)–(3) is satisfied. We have:

$$\hskip-115pt{{d^{\rm{*}}}\left( {j,a,{\sigma ^j}\left( {s,\pi } \right)} \right) = m}$$

$$\hskip75pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;by\;the\;definition\;of\;}}{\sigma ^{\rm{j}}}}$$

$$\hskip-110pt{{d^{\rm{*}}}\left( {j,a,\left( {s,\pi {\rm{'}}} \right)} \right) = m}$$

$$\hskip73pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;by\;condition\;}}\left( 1 \right)}$$

$$\hskip-103pt{{d^{\rm{*}}}\left( {i,a,\left( {s,\pi } \right)} \right) = m.}$$

By the definition of $d^*_{=i}$ (Definition 18): for all $j \in N$ , $a \in A$ and $(s,\pi) \in S^*$ : $d^*_{=i}(j,a,(s,\pi)) = d^*(i,a,(s,\pi))$ . Hence, invoking our property $(\dagger)$ , we conclude that for all $j \in N$ , $a \in A$ and $(s,\pi) \in S^*$ : $d^*_{=i}(j,a,(s,\pi)) = d^*(j,a,\sigma^j(s,\pi))$ .

Proof. Let $\bf{S}$ be a social choice rule satisfying IISD and IIPD. We establish that for all choice scenarios ${\mathfrak{C}}= \langle N,A,S,d \rangle$ and all $i \in N$ : ${\bf S}({\mathfrak{C}}) = {\bf{R}}^{\bf S}_{=}({\mathfrak{C}}^*,i)$ , where ${\bf{R}}^{\bf S}_{=}$ is given by Definition 19. Note that by Lemma 4, ${{\bf R}}^{\bf S}_{=}$ is an individualistic choice rule. Let $i \in N$ and $a \in A$ be arbitrary. We have

$$\hskip-122pt{a \in {\bf S}({\mathfrak{C}})}$$

$$\hskip25pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Lemma\;}}5}$$

$$\hskip25pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;Lemma\;}}6{\rm{\;and\;IIPD}}}$$

$$\hskip20pt{{\rm{iff\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Definition\;}}19}$$