Joint desirability foundations of social choice and opinion pooling

Abstract

We develop joint foundations for the fields of social choice and opinion pooling using coherent sets of desirable gambles, a general uncertainty model that allows to encompass both complete and incomplete preferences. This leads on the one hand to a new perspective of traditional results of social choice (in particular Arrow’s theorem as well as sufficient conditions for the existence of an oligarchy and democracy) and on the other hand to using the same framework to analyse opinion pooling. In particular, we argue that weak Pareto (unanimity) should be given the status of a rationality requirement and use this to discuss the aggregation of experts’ opinions based on probability and (state-independent) utility, showing some inherent limitation of this framework, with implications for statistics. The connection between our results and earlier work in the literature is also discussed.

Appendices

Appendix A: Representation of a social rule Γ as a bundle of functions

A social rule Γ can be represented as a bundle of functions $\{{\Gamma }_{f}: f \in {\mathscr{L}}\}$, where Γ_f identifies those profiles for which the gamble f belongs to the aggregated set

$$ \begin{array}{@{}rcl@{}} &&{\kern-1.5pc}{\Gamma}_{f}: \mathbb{D}^{n}\rightarrow \{0,1\} \\ &&[\mathcal{D}_{i}] \hookrightarrow \begin{cases} 1 &\text{ if } f\in {\Gamma}([\mathcal{D}_{i}]) \\ 0 &\text{ otherwise}. \end{cases} \end{array} $$

The fact that the social set of desirable gambles is coherent imposes some minimal requirements on these functions:

• G1. ${\Gamma }_{{\mathscr{L}}^{+}}=1$.

• G2. Γ₀ = 0.

• G3. $f,g\in {\mathscr{L}}\Rightarrow {\Gamma }_{f+g}\geq {\Gamma }_{f} \cdot {\Gamma }_{g}$.

• G4. $f\in {\mathscr{L}},\lambda >0\Rightarrow {\Gamma }_{f}={\Gamma }_{\lambda f}$.

We can reformulate additional properties that a social rule may satisfy using this equivalent representation. In particular:

• ◦ Completeness is equivalent to:

  $$ f\neq 0\Rightarrow {\Gamma}_{f} + {\Gamma}_{-f}=1, $$

  and strict completeness can be expressed as

  $$ \epsilon>0\Rightarrow{\Gamma}_{f}+{\Gamma}_{-f+\epsilon}\geq 1. $$

• ◦ Independence of irrelevant alternatives. Let us define the function

  $$ \begin{array}{@{}rcl@{}} \pi_{f}:&\mathbb{D}^{n}\rightarrow \{0,1\}^{n} \\ &[\mathcal{D}_{i}] \hookrightarrow \vec{z}, \end{array} $$

  where

  $$\vec{z}_{i}:=\begin{cases} 1 &\text{ if } f \in\mathcal{D}_{i} \\ 0 &\text{ otherwise}. \end{cases}$$

  In other words, $\pi _{f}([\mathcal {D}_{i}])=(\mathbb {I}_{\mathcal {D}_{1}}(f),\dots ,\mathbb {I}_{\mathcal {D}_{n}}(f))$. Then independence of irrelevant alternatives can be equivalently expressed as

  $$ (\forall f\in\mathcal{L})\ \pi_{f}([\mathcal{D}_{i}])=\pi_{f}([\mathcal{D}^{\prime}_{i}]) \Rightarrow {\Gamma}_{f}([\mathcal{D}_{i}])={\Gamma}_{f}([\mathcal{D}^{\prime}_{i}]). $$

• ◦ Weak Pareto is equivalent to

  $$ (\forall f\in\mathcal{L})\ \pi_{f}^{-1}(\{\vec{1}\})\subseteq {\Gamma}_{f}^{-1}(\{1\}). $$

• ◦ Given a set of individuals $\mathcal {G}$, it is almost decisive for f if and only if

  $$ \pi_{f}^{-1}(\vec{1}_{\mathcal{G}},\vec{0}_{\mathcal{H}\setminus\mathcal{G}})\subseteq {\Gamma}_{f}^{-1}(\{1\}), $$

  where $\vec {1}_{\mathcal {G}},\vec {0}_{{\mathscr{H}}\setminus \mathcal {G}}$ is the vector $\vec {z}$ given by

  $$ z_{i}:=\begin{cases} 1 &\text{ if } i\in\mathcal{G} \\ 0 &\text{ if } i\notin\mathcal{G}, \end{cases} $$

  whereas $\mathcal {G}$ is decisive for a gamble f if and only if

  $$ \pi_{f}^{-1}(\vec{1}_{\mathcal{G}})\subseteq {\Gamma}_{f}^{-1}(\{1\}). $$

Appendix B: Results in terms of lower previsions and sets of finitely additive probabilities

In this paper, we have assumed that the beliefs of each voter are represented by means of a coherent set of desirable gambles $\mathcal {D}_{i}$, and that the outcome of the social rule Γ on a profile $[\mathcal {D}_{i}]$ is also a coherent set of desirable gambles. The main motivation behind this choice lies in the flexibility and generality of sets of desirable gambles, that can be linked to preference relations and also encompass as particular cases coherent lower previsions or sets of probability measures.

For completeness, in this appendix we translate the results we have established in the first part of the paper to social rules defined on coherent lower previsions ${\underline {P}}$ or on closed convex sets of probability measures ${\mathscr{M}}$. We shall make use of the one-to-one correspondence between these models and coherent sets of strictly desirable gambles: in the case of a coherent lower prevision $\underline {P}$, its associated set is given by (3); given a closed convex set of probability measures ${\mathscr{M}}$, we consider the set of strictly desirable gambles associated with its lower prevision $\underline {P}:=\min \limits {\mathscr{M}}$.

As a consequence, we shall say for instance that a social rule Γ on coherent lower previsions satisfies weak Pareto if and only if the social rule ${\Gamma }^{\prime }$ that we can determine on the associated sets of strictly desirable gambles by

$$ {\Gamma}^{\prime}([\mathcal{D}_{i}])=\{f: {\Gamma}([{\underline{P}}_{i}])(f)>0\}\cup\mathcal{L}^{+}, \text{ where } {\underline{P}}_{i} \text{ is induced by} \mathcal{D}_{i} through (1) $$

(9)

satisfies weak Pareto. Similar considerations hold for the other axioms, and for social rules defined on credal sets.

Our next result summarises how the different axioms and government systems are represented in this context:

Theorem 6

Let Γ be a social rule defined on a profile $[{\underline {P}}_{i}]$ (resp., $[{\mathscr{M}}_{i}]$) given by coherent lower previsions (resp., credal sets). Then:

• 1. Γ satisfies weak Pareto $\Leftrightarrow (\forall [{\underline {P}}_{i}] \in \mathcal {A}) {\underline {P}} \geq \min \limits _{i} {\underline {P}}_{i},$ $\Leftrightarrow (\forall [{\mathscr{M}}_{i}] \in \mathcal {A}) {\mathscr{M}} \subseteq \overline {\text {ch}(\cup _{i} {\mathscr{M}}_{i})}$.

• 2. Γ satisfies strict completeness $\Leftrightarrow (\forall [{\underline {P}}_{i}] \in \mathcal {A}) {\underline {P}}$ linear $\Leftrightarrow (\forall [{\mathscr{M}}_{i}] \in \mathcal {A}) |{\mathscr{M}}|=1$.

• 3. A set $\mathcal {G} \subseteq {\mathscr{H}}$ is decisive $\Leftrightarrow (\forall [{\underline {P}}_{i}] \in \mathcal {A}) {\underline {P}}\geq \min \limits _{i\in \mathcal {G}}{\underline {P}}_{i}$ $\Leftrightarrow (\forall [{\mathscr{M}}_{i}] \in \mathcal {A}) {\mathscr{M}} \subseteq \overline {\text {ch}(\cup _{i \in \mathcal {G}} {\mathscr{M}}_{i})}$.

• 4. A set $\mathcal {G} \subseteq {\mathscr{H}}$ is an oligarchy $\Leftrightarrow (\forall [{\underline {P}}_{i}] \in \mathcal {A}) {\underline {P}}\geq \min \limits _{i\in \mathcal {G}}{\underline {P}}_{i}$ and ${\overline {P}}\geq \max \limits _{i\in \mathcal {G}}P_{i}$.

• 5. An individual $j \in {\mathscr{H}}$ is a dictator $\Leftrightarrow (\forall [{\underline {P}}_{i}] \in \mathcal {A}) {\underline {P}}\geq {\underline {P}}_{j}$ $\Leftrightarrow {\mathscr{M}} \subseteq {\mathscr{M}}_{j}$.

• 6. Γ satisfies anonymity $\Leftrightarrow (\forall [{\underline {P}}_{i}],[{\underline {P}}_{\sigma (i)}] \in \mathcal {A}) {\underline {P}}={\underline {P}}_{\sigma } \Leftrightarrow (\forall [{\mathscr{M}}_{i}], [{\mathscr{M}}_{\sigma (i)}] \in \mathcal {A}) {\mathscr{M}}= {\mathscr{M}}_{\sigma }$, where σ is a permutation of ${\mathscr{H}}$ and P_σ (resp. $[{\mathscr{M}}_{\sigma }])$ is the coherent lower prevision (resp. credal set) given by Γ([P_σ(i)]) (resp. ${\Gamma }([{\mathscr{M}}_{\sigma (i)}])$).

• 7. Γ satisfies independence of irrelevant alternatives $\Leftrightarrow (\forall [{\underline {P}}_{i}],[{\underline {P}}^{\prime }_{i}] \in \mathcal {A}) (\forall f \notin {\mathscr{L}}^{+})$:

  $$ (\forall i \in \mathcal{H}) {\underline{P}}_{i}(f){\underline{P}}^{\prime}_{i}(f)>0 \Rightarrow {\underline{P}}(f){\underline{P}}^{\prime}(f)>0, $$

  where ${\underline {P}}^{\prime }$ denotes the coherent lower prevision that is obtained by means of ${\Gamma }([{\underline {P}}^{\prime }_{i}])$.

More generally speaking, if we consider a social rule ${\Gamma }^{\prime }$ defined on coherent sets of desirable gambles, we wonder if similar equivalences hold with respect to the social rule Γ on coherent lower previsions or credal sets that is associated with it. However, this will not be the case in general. To see just one example, consider the social rule ${\Gamma }^{\prime }$ given by ${\Gamma }^{\prime }([\mathcal {D}_{i}])=\mathcal {D}_{j\triangleright }$, for some fixed $j\in {\mathscr{H}}$. Then ${\Gamma }^{\prime }$ will not satisfy weak Pareto in general (for instance with $\mathcal {D}_{i}=\mathcal {D}_{j}$ for every i, and the latter being non strictly desirable), but its associated Γ will be given by ${\Gamma }([\underline {P}_{i}])=\underline {P}_{j}\geq \min \limits _{i} \underline {P}_{i}$.

As a corollary, which follows immediately from Theorem 6 and (2), we can characterise the above conditions in the particular case where the profile is precise:

Corollary 3

Let Γ be a strict complete social rule defined on a profile [P_i] given by linear previsions. Then:

• 1. A set $\mathcal {G}\subseteq {\mathscr{H}}$ is decisive $\Leftrightarrow (\forall [P_{i}] \in \mathcal {A}) {\Gamma }([P_{i}]) \in \text {ch}(\{P_{i}\mid i\in \mathcal {G}\})$.

• 2. A set $\mathcal {G}\subseteq {\mathscr{H}}$ is an oligarchy $\Leftrightarrow (\forall [P_{i}] \in \mathcal {A}), P_{j}=P_{j^{\prime }} \ \forall j,j^{\prime }\in \mathcal {G} \text { and } {\Gamma }([P_{i}])=P_{j}$.

• 3. An individual $j \in {\mathscr{H}}$ is a dictator $\Leftrightarrow (\forall [P_{i}] \in \mathcal {A}) {\Gamma }([P_{i}])=P_{j}$.

We see then that the notion of oligarchy does not make much sense in the precise case, because it forces all the individuals that form the oligarchy to have exactly the same opinion.

Appendix C: Proofs

In the following proofs, it will be convenient at times to reformulate the condition of avoiding partial loss in the following manner:

Lemma 5

Given a set $\mathcal {K}$ of desirable gambles, $\text {ch}(\mathcal {K}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset \Leftrightarrow 0 \notin \text {posi}(\mathcal {K} \cup {\mathscr{L}}^{+}).$ As a consequence, if any of the two conditions above hold, it follows that $\mathcal {K}$ has a coherent (maximal) superset.

Proof

It is equivalent to prove that

$$\text{ch}(\mathcal{K}) \cap (\mathcal{L}^{-} \cup\{0\})\neq \emptyset\Leftrightarrow 0 \in \text{posi}(\mathcal{K} \cup \mathcal{L}^{+}).$$

To prove that this is the case, note that

$$ \begin{array}{@{}rcl@{}} &&0\in\text{posi}(\mathcal{K} \cup \mathcal{L}^{+})\\ &&\Leftrightarrow (\exists f\in\mathcal{L}^{+}\cup \{0\}, f_{j}\in\mathcal{K}, \lambda_{j}>0, j=1,\dots,r, r \ge 1) \ 0=f+\sum\limits_{j=1}^{r} \lambda_{j} f_{j} \\ &&\Leftrightarrow (\exists f\in\mathcal{L}^{+}\cup \{0\}, f_{j}\in\mathcal{K}, \lambda_{j}>0, j=1,\dots,r, r \ge 1) \ -f=\sum\limits_{j=1}^{r} \lambda_{j} f_{j} \\ &&{\kern-.9pc}\Leftrightarrow {}(\exists f\in\mathcal{L}^{+}\cup \{0\}, f_{j}\in\mathcal{K}, \lambda_{j}>0, j{}={}1,\dots,r, r \ge 1) -\frac{f}{{\sum}_{j=1}^{r} \lambda_{j}} =\sum\limits_{j=1}^{r} \frac{\lambda_{j}}{{\sum}_{j=1}^{r} \lambda_{j}} f_{j}, \end{array} $$

and since the left-hand side belongs to ${\mathscr{L}}^{-}\cup \{0\}$ and the right-hand side belongs to $\text {ch}(\mathcal {K})$, we deduce that this is equivalent to $\text {ch}(\mathcal {K}) \cap ({\mathscr{L}}^{-} \cup \{0\})\neq \emptyset$.

For the second statement, if any of these conditions hold we deduce that the set $\text {posi}(\mathcal {K}\cup {\mathscr{L}}^{+})$ is a coherent set of desirable gambles that trivially includes $\mathcal {K}$. Since any coherent set of desirable gambles has a coherent maximal superset [12, Corollary 4], it follows in particular that $\mathcal {K}$ has a coherent maximal superset. □

In order to prove Lemmas 1 and 2, we also need first to establish the following auxiliary result:

Lemma 6

Assume that $|\mathcal {Z}|\geq 3$. Then for every gamble f such that $f \notin ({\mathscr{L}}^{+} \cup {\mathscr{L}}^{-} \cup \{0\})$ there always exist two gambles {h, t} such that − f = f + h + t and

• ◦ $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, $\text {ch}(\{-f,h,f+h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ and $\text {ch}(\{-f,h,-(f+h)\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$;

• ◦ $\text {ch}(\{f+h, t\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, $\text {ch}(\{-(f+h),t, -f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ and $\text {ch}(\{-(f+h),t,f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$.

Proof

Since f is such that $f \notin ({\mathscr{L}}^{+} \cup {\mathscr{L}}^{-} \cup \{0\})$ then there exist at least two points $z_{1},z_{2} \in \mathcal {Z}$ with z₁ ≠ z₂ such that f(z₁) = f₁ > 0 and f(z₂) = f₂ < 0. We consider a number of possibilities:

• ∘ If there is some z₃ ≠ z₁, z₂ such that f(z₃) ≥ 0, then we define, for a fixed 𝜖 > 0:

  h(z) := { f₁ > 0 if z = z₁ −f₂/2 > 0 if z = z₂ − 2f(z) − 𝜖 otherwise and t(z) := { − 3f₁ < 0 if z = z₁ − 3f₂/2 > 0 if z = z₂ 𝜖 otherwise.

  Then by construction − f = f + h + t. To see that they fulfil the conditions in the lemma, note that:

  • ◦ $\text {ch}(\{f,h\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$ (use z₁);

  • ◦ $\text {ch}(\{-f,h,f+h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$: if 0 ≥ λ_−f(−f) + λ_hh + λ_f+h(f + h), then it should be on the one hand λ_−f ≥ λ_h + 2λ_f+h because of z₁, and on the other 2λ_−f + λ_h ≤ λ_f+h because of z₂; these two conditions are incompatible.

  • ◦ $\text {ch}(\{-f,h,-(f+h)\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₂);

  and also

  • ◦ $\text {ch}(\{f+h,t\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$: if λ_f+h(f + h) + λ_tt ≤ 0, it should be on the one hand λ_f+h ≤ 0.6 because of z₁ and on the other λ_f+h ≥ 0.75 because of z₂;

  • ◦ $\text {ch}(\{-(f+h),t, -f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₂);

  • ◦ $\text {ch}(\{-(f+h),t,f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₃ and z₁).

• ∘ If f(z₃) < 0 for every z₃ ≠ z₁, z₂, then we define:

  h(z) := { −f₁ < 0 if z = z₁ −f₂/2 > 0 if z = z₂ − 2f(z) otherwise and t(z) := { −f₁ < 0 if z = z₁ − 3f₂/2 > 0 if z = z₂ 0 otherwise.

  Then by construction − f = f + h + t. To see that they fulfil the conditions in the lemma, note that:

  • ◦ $\text {ch}(\{f,h\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$: if λ_f(f) + λ_hh ≤ 0, it should be on the one hand λ_f ≤ 0.5 because of z₁ and on the other $\lambda _{f}\geq \frac {2}{3}$ because of z₃;

  • ◦ $\text {ch}(\{-f,h,f+h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₃);

  • ◦ $\text {ch}(\{-f,h,-(f+h)\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₂);

  and also

  • ◦ $\text {ch}(\{f+h,t\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$ (use z₂ and z₃);

  • ◦ $\text {ch}(\{-(f+h),t, -f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₂);

  • ◦ $\text {ch}(\{-(f+h),t,f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$: if 0 ≥−λ_−(f+h)(f + h) + λ_tt + λ_ff, then it should be λ_t ≥ λ_f so that this combination is non-positive on z₁, but then this implies that it is positive on z₂.

□

Proof of Lemma 1

Note that we can assume without loss of generality that $\mathcal {G}$ is a proper subset of ${\mathscr{H}}$, since otherwise the thesis follows immediately from the property of weak Pareto. In addition, we can then assume that the gamble f does not belong to ${\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$.

We must prove that $(\forall [\mathcal {D}_{i}])\ \cap _{i \in \mathcal {G}} \mathcal {D}_{i} \subseteq {\Gamma }([\mathcal {D}_{i}])$. Consider thus a profile $[\mathcal {D}_{i}]$ (eventually in $\hat {\mathbb {D}}^{n}$), and let $g\in \cap _{i\in \mathcal {G}}\mathcal {D}_{i}$. We can assume without loss of generality that $g\notin {\mathscr{L}}^{+}$, since in that case it trivially belongs to ${\Gamma }([\mathcal {D}_{i}])$. There are a number of possibilities:

• 1. Assume first of all that g = f + h, where:

  $$ \begin{array}{@{}rcl@{}} & \text{ch}(\{f,h\}) \cap (\mathcal{L}^{-} \cup\{0\})= \emptyset, \end{array} $$

  (C.1)
  $$ \begin{array}{@{}rcl@{}} & \text{ch}(\{-f,h,g\}) \cap (\mathcal{L}^{-} \cup\{0\})= \emptyset, \end{array} $$

  (C.2)
  $$ \begin{array}{@{}rcl@{}} & \text{ch}(\{-f,h,-g\}) \cap (\mathcal{L}^{-} \cup\{0\})= \emptyset. \end{array} $$

  (C.3)

  Hence, it holds that:

  • ◦ By Lemma 5, if $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$, there exists a coherent set $\mathcal {D}^{1}$ (possibly maximal) of desirable gambles that includes both f, h, and as a consequence also g by additivity.

  • ◦ Since $\text {ch}(\{-f,h,-g\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$, then the coherent set $\mathcal {D}^{2}:=\text {posi}(\{-f,h,-g\}\cup {\mathscr{L}}^{+})$ (or a maximal set that contains it) of desirable gambles includes h and not f and g.

  • ◦ Finally, let $\mathcal {D}^{3}:=\text {posi}(\{-f,h,g\}\cup {\mathscr{L}}^{+})$ (or a maximal set that contains it). It includes h, g but not f.

  Let us now define the following profile $[\mathcal {D}^{\prime }_{i}]$:

  • ◦ For every $i\in \mathcal {G}$, let $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{1}$ (possibly the maximal one). Then $f,h\in \mathcal {D}^{\prime }_{i}$, whence also $g\in \mathcal {D}^{\prime }_{i}$ by additivity.

  • ◦ Given $i\in {\mathscr{H}}\setminus \mathcal {G}$, if $g\notin \mathcal {D}_{i}$ then we let $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{2}$ (possibly the maximal one). Then $h\in \mathcal {D}^{\prime }_{i}$ and $f,g\notin \mathcal {D}^{\prime }_{i}$.

  • ◦ Given $i\in {\mathscr{H}}\setminus \mathcal {G}$, if $g\in \mathcal {D}_{i}$ then we let $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{3}$ (possibly the maximal one). Then $h,g\in \mathcal {D}^{\prime }_{i}$ and $f\notin \mathcal {D}^{\prime }_{i}$.

  It follows then that $h\in \cap _{i}\mathcal {D}^{\prime }_{i}$, whence $h\in {\Gamma }([\mathcal {D}^{\prime }_{i}])$, using the weak Pareto property; moreover, $f\in \cap _{i\in \mathcal {G}}\mathcal {D}^{\prime }_{i}$ and $f\notin \cup _{i\in {\mathscr{H}}\setminus \mathcal {G}}\mathcal {D}^{\prime }_{i}$. Since $\mathcal {G}$ is almost decisive for f, we deduce that $f\in {\Gamma }([\mathcal {D}^{\prime }_{i}])$. Since the latter is a coherent set of desirable gambles, we deduce by additivity that $g=f+h\in {\Gamma }([\mathcal {D}^{\prime }_{i}])$. And since $\pi _{g}([\mathcal {D}_{i}])=\pi _{g}([\mathcal {D}^{\prime }_{i}])$ by construction, we deduce from the independence of irrelevant alternatives that also $g\in {\Gamma }([\mathcal {D}_{i}])$.

  Now note that the reasoning above can be applied in particular for those $g\gneq f$ (i.e., when $h\in {\mathscr{L}}^{+}$). To see that this is the case, note that when $h\in {\mathscr{L}}^{+}$:

  • (C.1) $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ is equivalent to {f} avoiding partial loss, which holds because $f\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$ by assumption.

  • (C.2) $\text {ch}(\{-f,h,g\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ becomes equivalent to $\text {ch}(\{-f,g\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$. But if this intersection was non-empty we would deduce that for some λ_−f + λ_g = 1 it holds that 0 ≥ λ_−f(−f) + λ_gg ≥ (λ_g − λ_−f)f, which contradicts the assumption that $f\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$.

  • (C.3) $\text {ch}(\{-f,h,-g\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ becomes equivalent to {−g} avoiding partial loss, which holds because it cannot be $g\in {\mathscr{L}}^{-}\cup \{0\}$ since we are assuming that $g\in \cap _{i\in \mathcal {G}} \mathcal {D}_{i}$.

  This allows us to deduce that

  $$ \mathcal{G} \text{ is decisive for any } g\gneq f. $$

  (C.4)

  Next we establish that:

  $$ \mathcal{G} \text{ almost decisive for } g \Rightarrow \mathcal{G} \text{ decisive for } -g. $$

  (C.5)

  To prove this, note that we can assume without loss of generality that $g\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$. Applying Lemma 6, we can find two gambles h, t such that − g = g + h + t. Moreover, by construction in the proof of that lemma we have that $\text {ch}(\{g,h\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$, $\text {ch}(\{-g,h, (g+h)\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$ and $\text {ch}(\{-g,h, -(g+h)\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$. Applying point 1, we deduce that $\mathcal {G}$ is decisive for g + h, and therefore also almost decisive. Moreover, we also have from the proof of the lemma that $\text {ch}(\{g+h,t\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$, $\text {ch}(\{-(g+h),t, -g\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$ and $\text {ch}(\{-(g+h),t, g\})\cap ({\mathscr{L}}^{-}\cup \{0\})=\emptyset$. Applying again point 1, we deduce that $\mathcal {G}$ is decisive for g + h + t = −g.

  Thus, (C.5) holds. Applying twice this condition we deduce in particular that $\mathcal {G}$ is decisive for − f, and also for f.

• 2. Assume now that g = f + h, with $h\in {\mathscr{L}}^{-}$. Then $g\lneq f$, or, equivalently $-f\lneq -g$. Since $\mathcal {G}$ is decisive at − f, it is in particular almost decisive at this gamble, whence by (C.4) $\mathcal {G}$ is also almost decisive at $-g\gneq -f$. Applying now (C.5), we deduce that $\mathcal {G}$ is also decisive at g.

• 3. Finally, consider a gamble $h\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$, and let g := f + h. Then we can rewrite g as g = f − h⁻ + h⁺, where h⁺ and h⁻ are respectively the positive and the negative part function of h, i.e. $h^{+}:=\max \limits \{0, h\}$ and $h^{-}:=-\min \limits \{0, h\}$. It then follows from point 2 that $\mathcal {G}$ is decisive on g = f − h⁻ because $-h^{-}\in {\mathscr{L}}^{-}$, and as a consequence it is also almost decisive on this gamble. If we now apply (C.4) we deduce that $\mathcal {G}$ is decisive on $g=(f-h^{-} )+ h^{+}\gneq f-h^{-}$.

□

Proof of Lemma 2

Let us prove that there is a proper subset $\mathcal {G}^{\prime }$ of $\mathcal {G}$ that is also decisive. For this, we consider a partition of $\mathcal {G}$ into non-empty and disjoint subsets $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$, and we proceed to establish that one of these two sets is also decisive.

Consider $z_{1},z_{2}\in \mathcal {Z}$ with z₁ ≠ z₂, and let us define the gambles f₁, f₂, f₃ by:

$$f_{1}(z):=\begin{cases} -1 & \text{if }z=z_{1} \\ 1 & \text{if }z=z_{2} \\ 0 & \text{otherwise,} \end{cases} \quad f_{2}(z):=\begin{cases} 0 & \text{if }z=z_{1} \\ -2 & \text{if }z=z_{2} \\ 1 & \text{otherwise,} \end{cases} \quad f_{3}:= f_{1}+f_{2}.$$

Let z₃ denote an element different from z₁, z₂, existing because $|\mathcal {Z}|\geq 3$. These gambles satisfy the following conditions:

• ◦ $\text {ch}(\{f_{1},f_{2},f_{3}\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ (use z₃ and z₂);

• ◦ $\text {ch}(\{-f_{1},f_{2}, -f_{3}\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ (use z₁ and z₃);

• ◦ $\text {ch}(\{f_{1},-f_{3}\}) \cap ({\mathscr{L}}^{-} \cup \{0\})= \emptyset$ (use z₂).

Then, thanks to Lemma 5 we can consider the following profile $[\hat {\mathcal {D}}_{i}] \in \hat {\mathbb {D}}^{n}$:

• ◦ for all $i\in \mathcal {G}_{1}, \hat {\mathcal {D}}_{i}:= M_{1} \supseteq \text {posi}(\{f_{1},f_{2},f_{3}\}\cup {\mathscr{L}}^{+})$, with $M_{1} \in \hat {\mathbb {D}}$;

• ◦ for all $i\in \mathcal {G}_{2}, \hat {\mathcal {D}}_{i}:= M_{2} \supseteq \text {posi}(\{-f_{1},f_{2},-f_{3}\}\cup {\mathscr{L}}^{+})$, with $M_{2} \in \hat {\mathbb {D}}$;

• ◦ for all $i\notin \mathcal {G}, \hat {\mathcal {D}}_{i}:= M_{3} \supseteq \text {posi}(\{f_{1},-f_{3}\}\cup {\mathscr{L}}^{+})$, with $M_{3} \in \hat {\mathbb {D}}$.

Note then that $-f_{1} \notin \hat {\mathcal {D}_{i}}$ for any $i\notin \mathcal {G}_{2}$, and that $f_{3}\notin \hat {\mathcal {D}_{i}}$ for any $i\notin \mathcal {G}_{1}$, because these sets of gambles are maximal. Since $f_{2}\in \cap _{i\in \mathcal {G}} \hat {\mathcal {D}}_{i}$, it follows from decisiveness of $\mathcal {G}$ that $f_{2}\in {\Gamma }([\hat {\mathcal {D}}_{i}])$. Since this set is maximal, then it either includes f₁ or − f₁.

• 1. If $f_{1} \in {\Gamma }([\hat {\mathcal {D}}_{i}])$, then $f_{3} \in {\Gamma }([\hat {\mathcal {D}}_{i}])$ by additivity. Let us prove that in that case $\mathcal {G}_{1}$ is almost decisive with respect to f₃.Consider any profile $[\hat {\mathcal {D}}'_{i}] \in \hat {\mathbb {D}}^{n}$, such that $f_{3}\in \cap _{i\in \mathcal {G}_{1}}\hat {\mathcal {D}}'_{i}$ and $f_{3}\notin \cup _{i\notin \mathcal {G}_{1}}\hat {\mathcal {D}}'_{i}$. Then $\pi _{f_{3}}([\hat {\mathcal {D}}'_{i}])=\pi _{f_{3}}([\hat {\mathcal {D}}_{i}])$, and applying the independence of irrelevant alternatives we deduce that $\mathcal {G}_{1}$ is almost decisive on f₃. Applying Lemma 1, we deduce that it is also decisive.

• 2. If $-f_{1}\in {\Gamma }([\mathcal {D}_{i}])$, then let us show that $\mathcal {G}_{2}$ is almost decisive for − f₁.Consider any profile $[\hat {\mathcal {D}}'_{i}] \in \hat {\mathbb {D}}^{n}$, such that $-f_{1}\in \cap _{i\in \mathcal {G}_{2}}\hat {\mathcal {D}}'_{i}$ and $-f_{1}\notin \cup _{i\notin \mathcal {G}_{2}}\hat {\mathcal {D}}'_{i}$. It follows that $\pi _{-f_{1}}([\hat {\mathcal {D}}'_{i}])=\pi _{-f_{1}}([\hat {\mathcal {D}}_{i}])$, and applying the independence of irrelevant alternatives we deduce that $-f_{1}\in {\Gamma }([\hat {\mathcal {D}}'_{i}])$. Thus, $\mathcal {G}_{2}$ is almost decisive on − f₁, and applying Lemma 1 we deduce that it is also decisive.

□

Proof of Theorem 1

The case of n = 1 is trivial, so let us assume n ≥ 2. By weak Pareto condition we know that ${\mathscr{H}}$ is decisive. By repeatedly applying Lemma 2, we can eventually arrive to a decisive individual, who must, thus, be a dictator. □

Proof of Proposition 1

Let j be the dictator. Since we have unlimited maximal domain, the social rule must be given by ${\Gamma }([\mathcal {D}_{i}])=\mathcal {D}_{j}$ for any profile $[\mathcal {D}_{i}]$, and as a consequence it satisfies independence of irrelevant alternatives. □

Proof of Lemma 3

Suppose $\mathcal {G}$ and $\mathcal {G}^{\prime }$ are both oligarchies, and assume that $i^{*}\in \mathcal {G}^{\prime }\setminus \mathcal {G}$. Let $f\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$, and let us consider a profile (if needed made up only by maximal coherent sets of desirable gambles) $[\mathcal {D}_{i}]$ such that $f \in \mathcal {D}_{i}$ for all $i \in \mathcal {G}$ and $-f \in \mathcal {D}_{i}$ for all $i \in \mathcal {G}^{\prime } \setminus \mathcal {G}$: it suffices to make

$$ \mathcal{D}_{i}:=\begin{cases} \text{posi}(\{f\}\cup\mathcal{L}^{+}) &\text{ or a maximal superset, if } i\in\mathcal{G} \\ \text{posi}(\{-f\}\cup\mathcal{L}^{+}) &\text{ or a maximal superset, if } i\notin\mathcal{G}. \end{cases} $$

Then since $\mathcal {G}^{\prime }$ is an oligarchy we should have $f \notin {\Gamma }([\mathcal {D}_{i}])$, but $\mathcal {G}$ decisive implies $f \in {\Gamma }([\mathcal {D}_{i}])$, a contradiction. □

Proof of Theorem 2

Since by Lemma 3 there can be at most one oligarchy, we only need to establish its existence.

By the weak Pareto condition, the set ${\mathscr{H}}$ of all individuals is decisive. Let $\mathcal {G}$ be a decisive set of minimal size, meaning that there does not exist $\mathcal {G}^{\prime } \subset \mathcal {G}$ that is also decisive. We shall demonstrate that $\mathcal {G}$ is an oligarchy. If $\mathcal {G}$ contains a single member, then $\mathcal {G}$ is trivially an oligarchy because any dictatorship is. Let us then consider the case in which $|\mathcal {G}| \ge 2$. Note that we only need to prove that O2 holds, because O1 follows immediately because $\mathcal {G}$ is decisive.

We consider first of all the case where Γ has unlimited domain.

Consider a profile $[\mathcal {D}_{i}]$, and let $f\in \cup _{i\in \mathcal {G}}\mathcal {D}_{i}$. We must prove that $-f\notin {\Gamma }([\mathcal {D}_{i}])$. We may assume without loss of generality that $f\notin {\mathscr{L}}^{+}\cup {\mathscr{L}}^{-}\cup \{0\}$; the result otherwise is trivial. We can partition the group $\mathcal {G}$ into the following sets:

• ◦ $A:=\{ i \in \mathcal {G} | f \in \mathcal {D}_{i} \}$,

• ◦ $B:=\{ i \in \mathcal {G} | -f \in \mathcal {D}_{i} \}$,

• ◦ $C:=\{ i \in \mathcal {G} | f \notin \mathcal {D}_{i}, -f \notin \mathcal {D}_{i} \}$,

where by assumption A is non-empty, but B, C may be.

Since $f \notin ({\mathscr{L}}^{+} \cup {\mathscr{L}}^{-} \cup \{0\})$, there exist two points $z_{1}, z_{2} \in \mathcal {Z}$ with z₁ ≠ z₂ such that f(z₁) = f₁ > 0 and f(z₂) = f₂ < 0. Let us prove that there always exist two gambles {h, g} such that f = h + g and

• ◦ $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, and $g,-g \notin \mathcal {D}^{1}:=\text {posi}(\{f,h\}\cup {\mathscr{L}}^{+})$;

• ◦ $\text {ch}(\{-f, h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, and $-g \in \mathcal {D}^{2} :=\text {posi}(\{-f,h\}\cup {\mathscr{L}}^{+})$;

• ◦ $\text {ch}(\{-g, h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, and $f,-f \notin \mathcal {D}^{3}:=\text {posi}(\{-g,h\}\cup {\mathscr{L}}^{+})$;

• ◦ $\text {ch}(\{-f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$, whence − g and as a consequence $h \notin \mathcal {D}^{4}:=\text {posi}(\{-f\}\cup {\mathscr{L}}^{+})$.

We consider a number of possibilities:

• 1. If there is some z₃ ≠ z₁, z₂ such that f(z₃) > 0, then we define:

  $$ h(z):=\begin{cases} 3f_{1}/2> 0 & \text{if } z=z_{1} \\ f_{2}/2<0 & \text{if } z=z_{2} \\ f(z)-\epsilon & \text{otherwise} \end{cases} \quad\text{and}\quad g(z):=\begin{cases} -f_{1}/2<0 & \text{if } z=z_{1} \\ f_{2}/2<0 & \text{if } z=z_{2} \\ \epsilon & \text{otherwise}, \end{cases} $$

  where 𝜖 > 0 is small enough for the conditions to be satisfied.

  Then by construction f = h + g. To see that they fulfil the conditions, note that:

  • ◦ $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₁);

    • $g \notin \mathcal {D}^{1}$ (use z₁);

    • $-g \notin \mathcal {D}^{1}$ (use z₃ with 0 < 𝜖 < f(z₃));

  • ◦ $\text {ch}(\{-f, h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$: If λ_−f(−f) + λ_hh ≤ 0 it should be on the one hand λ_−f ≥ 0.6 because of z₁ and on the other $\lambda _{-f} \le \frac {1}{3}$ because of z₂. Then $-g \in \mathcal {D}^{2}$ by additivity;

  • ◦ $\text {ch}(\{-g, h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₁);

    • $f \notin \mathcal {D}^{3}$: If f ≥ λ_−g(−g) + λ_hh for some non-negative λ_−g, λ_h (with at least one positive), we have that

      $$ f_{2} \geq - \lambda_{-g} f_{2}/2 + \lambda_{h} f_{2}/2, $$

      whence

      $$ -1 \geq \lambda_{-g}/2 - \lambda_{h}/2 $$

      dividing by the positive number − f₂. This means that

      $$ \lambda_{h} \geq 2+ \lambda_{-g} \geq 2. $$

      Now, this means that

      $$ \lambda_{h} h(z_{1}) + \lambda_{-g} (-g(z_{1}))= \lambda_{h} 3f_{1}/2 + \lambda_{-g} f_{1}/2 \geq 3 f_{1}> f_{1}, $$

      a contradiction;

    • $-f \notin \mathcal {D}^{3}$ (use z₁);

  • ◦ $\text {ch}(\{-f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ by definition of f;

    • $-g \notin \mathcal {D}^{4}$: If − g ≥ λ(−f) it should be on the one hand λ ≤ 0.5 because of z₂ and on the other λ ≥ 𝜖/f(z₃) > 0.5 if we choose 0 < f(z₃)/2 < 𝜖 < f(z₃), because of z₃.

• 2. If there is some z₃ ≠ z₁, z₂ such that f(z₃) = 0, then we define:

  $$ h(z):=\begin{cases} 3f_{1}/2> 0 & \text{if } z=z_{1} \\ f_{2}/2<0 & \text{if } z=z_{2} \\ f(z)-\epsilon & \text{otherwise} \end{cases} \quad\text{and}\quad g(z):=\begin{cases} -f_{1}/2<0 & \text{if } z=z_{1} \\ f_{2}/2<0 & \text{if } z=z_{2} \\ \epsilon & \text{otherwise}, \end{cases} $$

  where 𝜖 > 0 is small enough for the conditions to be satisfied.

  Notice that h and g are the same of the previous case, so to show that they fulfil the conditions we indicate only the cases that involve the value of f(z₃).

  • ◦ It only changes $-g \notin \mathcal {D}^{1}$: If − g ≥ λ_ff + λ_hh for some non-negative λ_f, λ_h (with at least one positive), we have that

    $$ - \epsilon \ge \lambda_{f}(f(z_{3})) + \lambda_{h} (f(z_{3})- \epsilon)=-\epsilon \lambda_{h}, $$

    from which follows λ_h ≥ 1, which does not work for z₁ because − g(z₁) < h(z₁) and h(z₁),f₁ are both positive;

  • ◦ same as before;

  • ◦ same as before;

  • ◦ it only changes $-g \notin \mathcal {D}^{4}$ (use z₃).

• 3 Finally, if f(z₃) < 0 for every z₃ ≠ z₁, z₂, then we define:

  $$ h(z):=\begin{cases} f_{1}> 0 & \text{if } z=z_{1} \\ 3f_{2}/2<0 & \text{if } z=z_{2} \\ 0 & \text{otherwise} \end{cases} \quad \text{and}\quad g(z):=\begin{cases} 0 & \text{if } z=z_{1} \\ -f_{2}/2>0 & \text{if } z=z_{2} \\ f(z) & \text{otherwise}. \end{cases} $$

  Then by construction f = h + g. To see that they fulfil the conditions, note that:

  • ◦ $\text {ch}(\{f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₁);

    • $g \notin \mathcal {D}^{1}$ (use z₁);

    • $-g \notin \mathcal {D}^{1}$ (use z₁) ;

  • ◦ $\text {ch}(\{-f,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₃ and z₁) and $-g \in \mathcal {D}_{2}$ by additivity;

  • ◦ $\text {ch}(\{-g,h\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ (use z₃ and z₁);

    • $f \notin \mathcal {D}^{3}$ (use z₃);

    • $-f \notin \mathcal {D}^{3}$ (use z₁) ;

  • ◦ $\text {ch}(\{-f\}) \cap ({\mathscr{L}}^{-} \cup \{0\}) = \emptyset$ by definition of f;

    • $-g \notin \mathcal {D}^{4}$: (use z₂).

Let us consider now a profile $[\mathcal {D}^{\prime }_{i}]$ where $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{1}$ if i ∈ A, $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{2}$ if i ∈ B, $\mathcal {D}^{\prime }_{i}=\mathcal {D}^{3}$ if i ∈ C and $\mathcal {D}^{\prime }_{i}$ is either ${\mathscr{L}}^{+}$ or $\mathcal {D}^{4}$ if $i\notin \mathcal {G}$ (as needed below so as to include − f if necessary). We shall use these sets to establish that $-f\notin {\Gamma }([\mathcal {D}_{i}])$. We have a number of possible scenarios:

• ◦ If A, B, C are all non-empty and $-f\in {\Gamma }([\mathcal {D}_{i}])$, then we can consider a profile $[\mathcal {D}^{\prime }_{i}]$ as above so that $\pi _{-f}([\mathcal {D}_{i}])=\pi _{-f}([\mathcal {D}^{\prime }_{i}])$. Applying independence of irrelevant alternatives, it follows that $-f\in {\Gamma }([\mathcal {D}^{\prime }_{i}])$. Since $\mathcal {G}$ is decisive and $h\in \cap _{i\in \mathcal {G}}\mathcal {D}^{\prime }_{i}$, we also have that $h\in {\Gamma }([\mathcal {D}^{\prime }_{i}])$, we deduce that − g = −f + h belongs to ${\Gamma }([\mathcal {D}^{\prime }_{i}])$. By construction, $\pi _{-g}([\mathcal {D}^{\prime }_{i}])=(\vec {1}_{B\cup C})$. Applying independence of irrelevant alternatives, for any other profile $[\mathcal {D}^{\prime \prime }_{i}]$ such that $-g\in \cap _{i\in B\cup C}\mathcal {D}^{\prime \prime }_{i}$ and $-g\notin \cup _{i\notin B\cup C}\mathcal {D}^{\prime \prime }_{i}$, it holds that $-g\in {\Gamma }([\mathcal {D}^{\prime \prime }_{i}])$. But this means that B ∪ C is almost decisive with respect to g, and applying Lemma 1, it is decisive. This contradicts that $\mathcal {G}$ is a decisive set of minimal size.

• ◦ If A≠∅ ≠ B and C = ∅, we reason as in the previous case, and end up concluding that B is a decisive set.

• ◦ If A≠∅ ≠ C and B = ∅, we reason as in the first case, and end up concluding that C is a decisive set.

• ◦ Finally, if A ≠ ∅ and B = C = ∅, it holds that $f\in \cap _{i\in \mathcal {G}}\mathcal {D}_{i}$, whence, since $\mathcal {G}$ is decisive, $f\in {\Gamma }([\mathcal {D}_{i}])$. Since the latter is a coherent set, this means that $-f\notin {\Gamma }([\mathcal {D}_{i}])$.

We consider next the case of unlimited maximal domain. Recall that since $\mathcal {G}$ is decisive, condition O1 is satisfied. Let us consider next condition O2. Consider a profile $[\hat {\mathcal {D}_{i}}] \in \hat {\mathbb {D}}^{n}$. Assume that $f\in \cup _{i\in \mathcal {G}}\hat {\mathcal {D}_{i}}$. Since all the sets in the profile are maximal, we can partition again the group $\mathcal {G}$ into the following sets:

• ◦ $A:=\{ i \in \mathcal {G} | f \in \hat {\mathcal {D}_{i}} \}$,

• ◦ $B:=\{ i \in \mathcal {G} | -f \in \hat {\mathcal {D}_{i}} \}$,

where by assumption A is non-empty, but B may be. We need to show that $-f\notin {\Gamma }([\hat {\mathcal {D}}_{i}])$. There are two possibilities:

• ◦ if both A, B are non-empty and $-f\in {\Gamma }([\hat {\mathcal {D}_{i}}])$, applying independence of irrelevant alternatives, for any other profile $[\hat {\mathcal {D}^{\prime }_{i}}]$ such that $- f \in \cap _{i\in B}\hat {\mathcal {D}}'_{i}$ and $-f \notin \cup _{i\in A}\hat {\mathcal {D}}'_{i}$, it holds that $-f\in {\Gamma }([\hat {\mathcal {D}}'_{i}])$. But this means that B is almost decisive with respect to − f, and applying Lemma 1, it is decisive. This contradicts that $\mathcal {G}$ is a decisive set of minimal size;

• ◦ if instead A≠∅ and B = ∅, it holds that $f\in \cap _{i\in \mathcal {G}}\hat {\mathcal {D}_{i}}$, whence, since $\mathcal {G}$ is decisive, $f\in {\Gamma }([\hat {\mathcal {D}_{i}}])$. Since the latter is a coherent set, this means that $-f\notin {\Gamma }(\hat {[\mathcal {D}_{i}]})$.

□

Proof of Proposition 2

Assume ex-absurdo that there exists a profile $[\mathcal {D}_{i}]$ such that ${\Gamma }([\mathcal {D}_{i}]) \supsetneq \cap _{i \in \mathcal {G}} \mathcal {D}_{i}$, and let us consider $f\in {\Gamma }([\mathcal {D}_{i}]) \setminus \cap _{i \in \mathcal {G}} \mathcal {D}_{i}$, so that $\emptyset \neq A=\{i\in \mathcal {G}: f \in \mathcal {D}_{i}\}\neq \mathcal {G}$. For every $j\in \mathcal {G}\setminus A$, let $\mathcal {D}^{\prime }_{j}$ be a maximal set of desirable gambles that includes − f, and let $\mathcal {D}^{\prime }_{i}:=\mathcal {D}_{i}$ for every $i\in A \cup \mathcal {G}^{c}$. Then since $-f\in \cup _{i \in \mathcal {G}} \mathcal {D}^{\prime }_{i}$, it follows from O2 that $f\notin {\Gamma }([\mathcal {D}^{\prime }_{i}])$. But on the other hand f belongs to the same sets in the profiles $[\mathcal {D}_{i}]$ and $[\mathcal {D}^{\prime }_{i}]$, so $f\in {\Gamma }([\mathcal {D}_{i}]), f\notin {\Gamma }([\mathcal {D}^{\prime }_{i}])$ is a contradiction with independence of irrelevant alternatives. □

Proof of Theorem 3

From Theorem 2 there exists a unique oligarchy $\mathcal {G}$. Then we must show that if the social rule satisfies also anonymity, $\mathcal {G}= {\mathscr{H}}$. Assume ex-absurdo that $\mathcal {G}\neq {\mathscr{H}}$, and take $i^{*}\notin \mathcal {G}$.

By the hypothesis of unlimited (maximal) domain we can consider a profile (possibly composed only by maximal coherent sets of desirable gambles) $[\mathcal {D}_{i}]$ such that $f \in \mathcal {D}_{i^{*}}$ and $-f \in \mathcal {D}_{j} \ \forall j\neq i^{*}$. Since $\mathcal {G}$ is an oligarchy, it follows that $-f \in {\Gamma }([\mathcal {D}_{i}])$.

Consider now a permutation σ of ${\mathscr{H}}$ such that $\sigma (i^{*})\neq i^{*}, \sigma (i^{*}) \in \mathcal {G}$, and let $[\mathcal {D}^{\prime }_{i}]:=[\mathcal {D}_{\sigma (i)}]$ denote the associated profile. Then there exists some $j\in \mathcal {G}$ such that $f\in \mathcal {D}^{\prime }_{j}$, whence, by definition of oligarchy, $-f \notin {\Gamma }([\mathcal {D}^{\prime }_{i}])$. But this means that ${\Gamma }([\mathcal {D}_{\sigma (i)}])\neq {\Gamma }([\mathcal {D}_{i}])$, meaning that anonymity is violated. □

Proof of Proposition 3

Thanks to Proposition 2 we know that for every profile (possibly composed only by maximal coherent sets of desirable gambles) $[\mathcal {D}_{i}]$, ${\Gamma }([\mathcal {D}_{i}])= \cap _{i} \mathcal {D}_{i}$. Hence it satisfies anonymity. □

Proof of Lemma 4

That $\mathcal {E}$ is coherent follows from [35, Prop. 29].

From (6), any $f \in \mathcal {E}$ can be written as

$$ f= \sum\limits_{i \in \mathcal{H}} \lambda_{i} \mathbb{I}_{i} \otimes f_{i} + f_{0}, $$

with $f_{0} \in {\mathscr{L}}^{+}({\mathscr{H}} \times \mathcal {Z}) \cup \{0\}, f_{i} \in \mathcal {D}_{i}\cup \{0\}, \lambda _{i}>0, f \neq 0$. Then $f \in \text {Marg}_{\mathcal {Z}}(\mathcal {E})$ iff $f \in \mathcal {E}$ and

$$ (\forall i,j \in \mathcal{H})(\forall z \in \mathcal{Z})\ f(i,z):= \lambda_{i} \mathbb{I}_{i} \otimes f_{i} (i,z) + f_{0}(i,z) = \lambda_{j} \mathbb{I}_{j} \otimes f_{j} (j,z)+ f_{0}(j,z)=\!: f(j, z). $$

This is equivalent to

$$ (\forall i,j \in \mathcal{H})(\forall z \in \mathcal{Z})\ \lambda_{i} f_{i}(z)+ f_{0}(i,z) = \lambda_{j} f_{j}(z) + f_{0}(j,z). $$

You can observe that, fixing i and j, the left term of this equation is a gamble in $\mathcal {D}_{i}$ and the right term is a gamble in $\mathcal {D}_{j}$, considering also that f≠ 0. Thus, $f \in \text {Marg}_{\mathcal {Z}}(\mathcal {E})$ if and only if it depends only on $z \in \mathcal {Z}$ and, as a function of only $z \in \mathcal {Z}$, it belongs to $\cap _{i \in {\mathscr{H}}} \mathcal {D}_{i}$. Hence we have the thesis. □

Proof of Theorem 4

Let us address the points of the statement in turn.

• 1. The equality of the sets in (6) and Theorem 4 is well known, see for instance [35, Prop. 29].

• 2. The converse implication is trivial: if such an $\mathcal {E}^{\prime }$ exists, then ${\Gamma }([\mathcal {D}_{i}])=\text {Marg}_{\mathcal {Z}}(\mathcal {E}^{\prime })\supseteq \text {Marg}_{\mathcal {Z}}(\mathcal {E})$.

  For the direct implication, consider $\mathcal {E}^{\prime }$ as defined in (7). It includes $\mathcal {E}$ by definition. To prove that it is coherent, note that $\mathcal {E}^{\prime }=\text {posi}({\Gamma }([\mathcal {D}_{i}])\otimes {\mathscr{H}} \cup \cup _{i} \mathcal {D}|i)$ (this is due again to the transformation referenced in point 1). It includes ${\mathscr{L}}^{+}({\mathscr{H}}\times \mathcal {Z})$ because for every $f\in {\mathscr{L}}^{+}({\mathscr{H}}\times \mathcal {Z})$, the restriction of $\mathbb {I}_{i}\otimes f$ on ${\mathscr{L}}(\mathcal {Z})$ belongs to $\mathcal {D}_{i}\cup \{0\}$ for all $i\in {\mathscr{H}}$. Thus, coherence holds if and only if $0\notin \text {posi}({\Gamma }([\mathcal {D}_{i}])\otimes {\mathscr{H}} \cup \cup _{i} \mathcal {D}|i)$.

  To prove that this is the case, let us reason by contradiction. Note that the zero gamble can in principle be produced only by adding $f_{0}\otimes {\mathscr{H}}$ with ${\sum }_{i\in {\mathscr{H}}}\mathbb {I}_{i}\otimes f_{i}$, for some $f_{0} \in {\Gamma }([\mathcal {D}_{i}])\cup \{0\},(\forall i\in {\mathscr{H}}) f_{i}\in \mathcal {D}_{i}\cup \{0\}$, not all of them zero. In order for the sum to yield zero, f₀ must be different from zero. And since $f_{0}\otimes {\mathscr{H}}$ is $\mathcal {Z}$-measurable, in order to yield zero it must hold that $f_{1}=\dots =f_{n}=-f_{0}$. This implies that $-f_{0}\in \cap _{i\in {\mathscr{H}}}\mathcal {D}_{i}$. We are showing the direct implication, so it follows by hypothesis ${\Gamma }([\mathcal {D}_{i}])\supseteq \text {Marg}_{\mathcal {Z}}(\mathcal {E})$ and Lemma 4, that $-f_{0}\in {\Gamma }([\mathcal {D}_{i}])$. But then ${\Gamma }([\mathcal {D}_{i}])$ contains both f₀ and − f₀. This contradicts the coherence of ${\Gamma }([\mathcal {D}_{i}])$. Therefore the sum of two gambles in $\mathcal {E}^{\prime }$ must be different from zero.

  To conclude, we show that ${\Gamma }([\mathcal {D}_{i}])=\text {Marg}_{\mathcal {Z}}(\mathcal {E}^{\prime })$. Since the direct inclusion is trivial, we focus on the converse inclusion. Remember that $\text {Marg}_{\mathcal {Z}}(\mathcal {E}^{\prime })=\mathcal {E}^{\prime }\cap {\mathscr{L}}_{\mathcal {Z}}({\mathscr{H}}\times \mathcal {Z})$. Let us consider a $\mathcal {Z}$-measurable gamble $f\in \mathcal {E}^{\prime }$: $f=f_{0}\otimes {\mathscr{H}}+{\sum }_{i\in {\mathscr{H}}}\mathbb {I}_{i}\otimes f_{i}$. The case $f_{1}=\dots =f_{n}=0$ is trivial, so let us assume that there is $i\in {\mathscr{H}}$ such that f_i≠ 0. Since f is $\mathcal {Z}$-measurable, this means that

  $$ (\forall i,j \in \mathcal{H})(\forall z \in \mathcal{Z})\ f(i,z)= f_{0}(z)+ f_{i}(z) = f_{0}(z) + f_{j}(z) = f(j, z). $$

  This implies f_j = f_i for all j≠i. As a consequence, $f_{i}\in \text {Marg}_{\mathcal {Z}}(\mathcal {E})$ and by hypothesis then $f_{i}\in {\Gamma }([\mathcal {D}_{i}])$. It follows that $f_{0}+f_{1}+\dots +f_{n}\in {\Gamma }([\mathcal {D}_{i}])$, whence $f\in {\Gamma }([\mathcal {D}_{i}])$.

• 3. Assume by contradiction that there is such a set $\mathcal {E}^{\prime \prime }$: $\mathcal {E}^{\prime }\supsetneq \mathcal {E}^{\prime \prime }\supseteq \mathcal {E}$. Since $\mathcal {E}^{\prime }$ strictly contains $\mathcal {E}^{\prime \prime }$, there must be some $f_{0}\in {\Gamma }([\mathcal {D}_{i}])$ such that $f_{0}\otimes {\mathscr{H}}\notin \mathcal {E}^{\prime \prime }$. Note that $f_{0}\notin \cap _{i\in {\mathscr{H}}}\mathcal {D}_{i}$, otherwise it would belong to $\mathcal {E}^{\prime \prime }$ given that $\mathcal {E}^{\prime \prime }\supseteq \mathcal {E}$. Whence $f_{0}\in {\Gamma }([\mathcal {D}_{i}])\setminus \cap _{i\in {\mathscr{H}}}\mathcal {D}_{i}$ and by marginalising $\mathcal {E}^{\prime \prime }$ we obtain a set that does not contain f₀; therefore ${\Gamma }([\mathcal {D}_{i}])\neq \text {Marg}_{\mathcal {Z}}(\mathcal {E}^{\prime \prime })$. This is a contradiction.

• 4. Remember that $\mathcal {E}^{\prime }|i=\{f\in \mathcal {E}^{\prime }:f=\mathbb {I}_{i}f\}$. Since $f_{0}\otimes {\mathscr{H}}$, in the definition of $\mathcal {E}^{\prime }$, is constant on the elements of ${\mathscr{H}}$, any gamble $f=\mathbb {I}_{i}f$ must be such that f₀ = 0. The thesis then follows immediately.

• 5. Remember again that $\text {Marg}_{{\mathscr{H}}}(\mathcal {E}^{\prime })=\mathcal {E}^{\prime }\cap {\mathscr{L}}_{{\mathscr{H}}}({\mathscr{H}}\times \mathcal {Z})$. Let us consider a ${\mathscr{H}}$-measurable gamble $f\in \mathcal {E}^{\prime }$: $f=f_{0}\otimes {\mathscr{H}}+{\sum }_{i\in {\mathscr{H}}}\mathbb {I}_{i}\otimes f_{i}$ for some $f_{0} \in {\Gamma }([\mathcal {D}_{i}])\cup \{0\},(\forall i\in {\mathscr{H}}) f_{i}\in \mathcal {D}_{i}\cup \{0\}$, not all of them zero. ${\mathscr{H}}$-measurability requires that

  $$ (\forall i \in \mathcal{H})(\forall z,z' \in \mathcal{Z})\ f(i,z)= f_{0}(z)+ f_{i}(z) = f_{0}(z') + f_{i}(z') = f(i, z'). $$

  This is valid if and only if f_i + f₀ = k_i constant $\forall i\in {\mathscr{H}}$. Hence we have that $f \in \text {Marg}_{{\mathscr{H}}}(\mathcal {E}^{\prime })$ is a gamble $f\in \mathcal {E}^{\prime }$: $f=f_{0}\otimes {\mathscr{H}}+{\sum }_{i\in {\mathscr{H}}}\mathbb {I}_{i}\otimes f_{i}$ such that f_i(z) = −f₀(z) + k_i for all $z \in \mathcal {Z}$, where k_i is a constant for all $i \in {\mathscr{H}}$.

  Now we can distinguish two cases:

  • (a) If f₀(z) = k for all $z \in \mathcal {Z}$, then it must be k ≥ 0, considering that $f_{0}\in {\Gamma }([\mathcal {D}_{i}])\cup \{0\}$. It then follows that f_i(z) = k_i − k for all $z \in \mathcal {Z}$, for all $i \in {\mathscr{H}}$, and it must be k_i − k ≥ 0, since otherwise $f_{i} \in {\mathscr{L}}^{-}(\mathcal {Z})$ and this would contradict $f_{i} \in \mathcal {D}_{i}\cup \{0\}$. We deduce that $(\forall i \in {\mathscr{H}})(\forall z \in \mathcal {Z})\ f(i,z)=k+k_{i}-k \ge 0$ and so $f \in {\mathscr{L}}^{+}({\mathscr{H}})$. Note that f≠ 0 otherwise $f \notin \mathcal {E}^{\prime }$.

  • (b) Assume next that f₀ is not constant (and in particular non-zero). This means that it must be f_i = k_i − f₀≠ 0 for every $i\in {\mathscr{H}}$, and in particular $f_{i}\in \mathcal {D}_{i}$ for every i.

    For all $i \in {\mathscr{H}}$ we can define

    $$ {\overline{P}}_{i}(f_{0}):= \inf\{ \lambda \in \mathbb{R} : \lambda - f_{0} \in \mathcal{D}_{i} \}, $$

    the upper prevision of f₀ obtained from $\mathcal {D}_{i}$. Thus, $k_{i} \ge {\overline {P}}_{i}(f_{0})$, for all $i \in {\mathscr{H}}$, and the equality can only hold if for some $i \in {\mathscr{H}}$, ${\overline {P}}_{i}(f_{0})$ is a mininum, i.e. if ${\overline {P}}_{i}(f_{0})-f_{0} \in \mathcal {D}_{i}$. This means that, given that f(i, z) = k_i for every $i\in {\mathscr{H}},z\in \mathcal {Z}$,

    $$ f(i,z) \ge {\overline{P}}_{i}(f_{0}), $$

    and that the inequality is strict unless ${\overline {P}}_{i}(f_{0})-f_{0}\in \mathcal {D}_{i}$. Now, given the fourth statement, we know that for every $i\in {\mathscr{H}},g\in {\mathscr{L}}(\mathcal {Z})$,

    $$ {\overline{P}}_{i}(g)= {\overline{P}}(g|i), $$

    so, defining the conditional upper prevision ${\overline {P}}(g| {\mathscr{H}}):= {\sum }_{i} \mathbb {I}_{i}{\overline {P}}(g|i)$, we have that $f \ge {\overline {P}}(f_{0}| {\mathscr{H}})$, and that the inequality is strict unless ${\overline {P}}(f_{0}|i)-f_{0} \in \mathcal {D}_{i}$.

    This establishes the form of $\text {Marg}_{{\mathscr{H}}}(\mathcal {E}^{\prime })$.

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Proof of Corollary 1

Weak Pareto means that if a gamble f is strictly desirable for any P_i, $i=1,\dots ,n$, then it should also be strictly desirable for P, or, equivalently, that $P\geq \min \limits _{i} P_{i}$ (Theorem 6(1)). But this means that P belongs to the credal set associated with the coherent lower prevision $\underline {P}:=\min \limits _{i} P_{i}$, and as a consequence that it is a convex combination of $P_{1},\dots ,P_{n}$. □

Proof of Theorem 5

If π is not degenerate, then we can find j₁ ≠ j₂ in ${\mathscr{H}}$ such that π(j₁),π(j₂) > 0. Consider a profile $[\hat {\mathcal {D}_{i}}] \in \mathcal {A}$ such that $\hat {\mathcal {D}_{i}}$ assigns all the mass to (s₁, x₁) if i = j₁ and all the mass to (s₂, x₂) if i≠j₁, where s₁ ≠ s₂ and j₁ ≠ j₂. Then we obtain

$$ P(s_{1},x_{1})=\pi(j_{1})=P(s_{1})=P(x_{1}), $$

meaning that the linear prevision induced by ${\Gamma }([\hat {\mathcal {D}_{i}}])$ is not state independent. This is a contradiction. □

Proof of Corollary 2

• 1. It follows from Theorem 2 and definition of weak dictatorship.

• 2. This is a consequence of Theorem 1 and definition of strong dictatorship.

• 3. It follows from Theorem 3 and definition of weak dictatorship.

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Proof of Theorem 6

We shall establish the first equivalence; the second equivalence follows taking into account that

$$ {\underline{P}} \geq {\underline{Q}} \iff \mathcal{M}({\underline{P}})\subseteq \mathcal{M}({\underline{Q}}), $$

that the credal set associated with $\min \limits _{i} {\underline {P}}_{i}$ is $\overline {\text {ch}(\cup _{i} {\mathscr{M}}({\underline {P}}_{i}))}$ and that the one associated with $\max \limits _{i} {\underline {P}}_{i}$ is $\cap _{i} {\mathscr{M}}({\underline {P}}_{i})$, where the credal set associated with a coherent lower prevision is given by (2).

We shall denote by $\mathcal {D}_{i}$ the coherent set of strictly desirable gambles associated with ${\underline {P}}_{i}$ by means of (3), and by $\mathcal {D}$ the coherent set of strictly desirable gambles associated with $\underline {P}={\Gamma }([P_{i}])$.

• 1. Let us establish the direct implication. Assume that Γ satisfies weak Pareto, and as a consequence that $\cap _{i \in {\mathscr{H}}} \mathcal {D}_{i}\subseteq {\Gamma }^{\prime }([\mathcal {D}_{i}])$, where ${\Gamma }^{\prime }$ is associated with Γ by means of (1). It follows that, for any $f \in {\mathscr{L}}$,

  $$ {\underline{P}}(f):= \sup \{ \mu \in \mathbb{R}: f- \mu \in {\Gamma}^{\prime}[\mathcal{D}_{i}] \} \ge \sup \{ \mu \in \mathbb{R}: f- \mu \in \cap_{i \in \mathcal{H}} \mathcal{D}_{i} \}. $$

  Let us prove that the right-hand side is greater than or equal to $\min \limits _{i} {\underline {P}}_{i}(f)$. Assume that $\min \limits _{i} {\underline {P}}_{i}(f)={\underline {P}}_{j}(f)$. Then it follows from (1) that for any 𝜖 > 0, $f-\underline {P}_{j}(f)+\epsilon$ belongs to $\mathcal {D}_{j}$, and for any $j^{\prime }\neq j$ it holds that $f-\underline {P}_{j}(f)+\epsilon \geq f-\underline {P}_{j^{\prime }}(f)+\epsilon \in \mathcal {D}_{j^{\prime }}$. As a consequence, $f-\underline {P}_{j}(f)+\epsilon \in \cap _{i} \mathcal {D}_{i}$, whence

  $$ \sup \{ \mu \in \mathbb{R}: f- \mu \in \cap_{i \in \mathcal{H}} \mathcal{D}_{i} \}\geq{\underline{P}}_{j}(f)-\epsilon. $$

  Since this holds for any 𝜖 > 0, it follows that

  $$ \sup \{ \mu \in \mathbb{R}: f- \mu \in \cap_{i \in \mathcal{H}} \mathcal{D}_{i} \}\geq{\underline{P}}_{j}(f)=\min_{i} {\underline{P}}_{i}(f). $$

  We conclude that ${\underline {P}}(f)\geq \min \limits _{i} {\underline {P}}_{i}(f)$ for every gamble f.

  To see the converse, given a gamble $f\in (\cap _{i} \mathcal {D}_{i})\setminus {\mathscr{L}}^{+}$, it holds that ${\underline {P}}_{i}(f)>0$ for every i, whence $0<\min \limits {\underline {P}}_{i}(f)\leq {\underline {P}}(f)$, and as a consequence f belongs to the set of strictly desirable gambles associated with $\underline {P}$. Since trivially $(\cap _{i} \mathcal {D}_{i})\cap {\mathscr{L}}^{+}\subseteq {\mathscr{L}}^{+}\subseteq \mathcal {D}$, we deduce that ${\Gamma }^{\prime }$ satisfies weak Pareto and therefore so does Γ.

• 2. This follows from (1) and (3) and the definition of strict completeness and maximal strict desirability.

• 3. This follows applying the first statement to $\mathcal {G}$ instead of ${\mathscr{H}}$.

• 4. The first part is a consequence of the third statement. Let us now prove that

  $$ \left[(\exists i \in \mathcal{G})(f \in \mathcal{D}_{i})) \Rightarrow -f \notin {\Gamma}^{\prime}([\mathcal{D}_{i}])\right] \Leftrightarrow {\overline{P}}\geq \max_{i\in\mathcal{G}}{\underline{P}}_{i}. $$

  (C.6)

  To see the direct implication, consider a gamble f, and let ${\underline {P}}_{j}(f)=\max \limits _{i\in \mathcal {G}} {\underline {P}}_{i}(f)$. If ${\underline {P}}_{j}(f)>0$, then $f\in \mathcal {D}_{j}$, whence $-f\notin {\Gamma }^{\prime }([\mathcal {D}_{i}])$ and, from (1) and the conjugacy relation $\overline {P}(\cdot )=-\underline {P}(-\cdot )$, it follows that $\overline {P}(f)\geq 0$. Moreover, it must be $\underline {P}(f)>0$: otherwise by considering $f^{\prime }:= f-\frac {\underline {P}_{j}(f)}{2}$, we would obtain $\underline {P}_{j}(f^{\prime })>0$ and $\overline {P}(f^{\prime })<0$. Therefore, we conclude that $\max \limits _{i\in \mathcal {G}} \underline {P}_{i}(f)>0 \Rightarrow \overline {P}(f)>0$. But since both $\max \limits _{i\in \mathcal {G}} \underline {P}_{i}$ and $\overline {P}$ satisfy constant additivity, this means that $\overline {P}\geq \max \limits _{i\in \mathcal {G}}\underline {P}_{i}$.

  Conversely, if there is a gamble f such that $f\in \mathcal {D}_{j}$ for some $j\in \mathcal {G}$ while $-f\in {\Gamma }^{\prime }([\mathcal {D}_{i}])$, it necessarily must be $f\notin {\mathscr{L}}^{+}$. It then follows that $\max \limits _{i\in \mathcal {G}}\underline {P}_{i}(f)\geq \underline {P}_{j}(f)>0$, while $0<\underline {P}(-f)=-\overline {P}(f)$, meaning that $\overline {P}(f)<0$. This is a contradiction.

  As a consequence, (C.6) holds. This concludes the proof of this statement.

• 5. This is a particular case of the third statement with $|\mathcal {G}|=1$.

• 6. Γ satisfies anonymity if and only if ${\Gamma }^{\prime }([\mathcal {D}_{i}]) = {\Gamma }^{\prime }([\mathcal {D}_{\sigma (i)}])$. It follows that, for any $f \in {\mathscr{L}}$,

  $$ {\underline{P}}(f):= \sup \{ \mu {}\in{} \mathbb{R}{}:{} f- \mu \in {\Gamma}^{\prime}([\mathcal{D}_{i}]) \} {}={} \sup \{ \mu{} \in{} \mathbb{R}{}:{} f- \mu \in {\Gamma}^{\prime}([\mathcal{D}_{\sigma(i)}]) \} =\!: {\underline{P}}_{\sigma}(f). $$

  To see the converse, let us suppose without loss of generality that there is a gamble $f \in {\Gamma }^{\prime }([\mathcal {D}_{i}]) \cap ({\Gamma }^{\prime }([\mathcal {D}_{\sigma (i)}])^{c}$ for some permutation σ of ${\mathscr{H}}$. Then it follows that $\underline {P}(f)>0$ and $\underline {P}_{\sigma }(f)\le 0$ and this contradicts $\underline {P}(f)=\underline {P}_{\sigma }(f)$.

• 7. Let us consider $f \notin {\mathscr{L}}^{+}$ and two profiles $[{\underline {P}}_{i}], [{\underline {P}}^{\prime }_{i}]$ such that $(\forall i \in {\mathscr{H}}) {\underline {P}}_{i}(f){\underline {P}}^{\prime }_{i}(f)>0$. If Γ satisfies independence of irrelevant alternatives, then $f \in {\Gamma }^{\prime }([\mathcal {D}_{i}]) \Leftrightarrow {\Gamma }^{\prime }([\mathcal {D}^{\prime }_{i}])$. As a consequence, we have $\underline {P}(f)>0 \Leftrightarrow \underline {P}^{\prime }(f)>0$, whence $\underline {P}(f)\underline {P}^{\prime }(f)>0$.

  Conversely, if ${\underline {P}}(f){\underline {P}}^{\prime }(f)>0$ then $f \in {\Gamma }^{\prime }([\mathcal {D}_{i}]) \Leftrightarrow {\Gamma }^{\prime }([\mathcal {D}^{\prime }_{i}])$ by (3), whence Γ satisfies independence of irrelevant alternatives.

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