Foundations of utilitarianism under risk and variable population

Abstract

Utilitarianism is the most prominent social welfare function in economics. We present three new axiomatic characterizations of utilitarian (that is, additively-separable) social welfare functions in a setting where there is risk over both population size and individuals’ welfares. We first show that, given uncontroversial basic axioms, Blackorby et al.’s (J Popul Econ 11:1–20, 1998) Expected Critical-Level Generalized Utilitarianism is equivalent to a new axiom holding that it is better to allocate higher utility-conditional-on-existence to possible people who have a higher probability of existence. The other two characterizations extend and clarify classic axiomatizations of utilitarianism from settings with either social risk or variable-population, considered alone.

A Appendix: Proofs of the results

1.1 A.1 Proof of Theorem 1

Step 1: Proof that ECLGU satisfies all the properties. It can easily be checked that an ECLGU social welfare ordering satisfies the Basic Principles and NoLD.

Step 2: Proof that $\succsim$ satisfies Property 1. We first show that, if social welfare ordering $\succsim$ on P satisfies the Basic Principles and NoLD then it satisfies the following property:

Property 1:

For all p, $p^{\prime }\in P$, if there exist three alternatives u, v, w in U such that $N(u)\cap N(v)=\emptyset$, $N(w)=\{k\}$ with $k\notin N(v)$, and $p(uv)=1/3$, $p(w)=2/3$, $p^{\prime }(u)=1/3$, $p^{\prime }(vw)=1/3$ and $p^{\prime }(w)=1/3$, then $p\sim p^{\prime }$.

To prove that the property must be satisfied, consider p and $p^{\prime }\in P$ like those described in Property 1 and assume by contradiction that $p\succ p^{\prime }$. By Social expected-utility (and the definition of p and $p^{\prime }$), it must be the case that

$$\begin{aligned} \tfrac{1}{3}V(uv)+\tfrac{2}{3}V(w)>\tfrac{1}{3}V(u)+\tfrac{1}{3}V(vw)+\tfrac{1}{3}V(w), \end{aligned}$$

where V is the function in the statement of Social expected utility.

Let $w^{\prime }$ be an alternative such that $N(w^{\prime })=\{k^{\prime }\}$, with $k^{\prime }\notin (N(v)\cup \{k\})$ and $w_{k^{\prime }}^{\prime }>w_k$. By Strong Pareto and Anonymity, $V(w^{\prime })>V(w)$. But we can find $0<\varepsilon <1/3$ small enough so that:

$$\begin{aligned} \tfrac{1}{3}V(uv)+\tfrac{2}{3}V(w)-\tfrac{1}{3}V(uw)-\tfrac{1}{3}V(v)-\tfrac{1}{3}V(w)>\varepsilon \big (V(w^{\prime })-V(w)\big ). \end{aligned}$$

(A.1)

Let ${\hat{p}}$ be the lottery such that ${\hat{p}}(u)=1/3$, ${\hat{p}}(vw)=1/3$, ${\hat{p}}(w)=1/3-\varepsilon$ and ${\hat{p}}(w^{\prime })=\varepsilon$. By Eq. (A.1),

$$\begin{aligned} \tfrac{1}{3}V(uv)+\tfrac{2}{3}V(w)>\tfrac{1}{3}V(u)+\tfrac{1}{3}V(vw)+\left( \tfrac{1}{3}-\varepsilon \right) V(w)+\varepsilon V(w^{\prime }). \end{aligned}$$

By Social expected-utility, this implies that $p\succ {\hat{p}}$. But, by NoLD we should have $p\prec {\hat{p}}$. Indeed, in ${\hat{p}}$ we have increased the probability of existence of $k^{\prime }$ and decreased by the same amount that of k, where $k^{\prime }$ has higher utility than k, while maintaining the probability of existence of other people.

The contradiction shows that we cannot have $p\succ p^{\prime }$. We can similarly prove that we cannot have $p\prec p^{\prime }$ (now taking $k^{\prime }$ with lower utility than k).

Step 3: Proof that $\succsim$ satisfies Ind-a. Using Property 1, we can show that $\succsim$ satisfies Ind-a.

To see why this is the case, consider any u, $u^{\prime }$, and v, like described in the axiom Ind-a. Let w be any alternative such that $N(w)=k$, with $k\notin N(v)$. Let us denote p, $p^{\prime }$, q, $q^{\prime }\in P$ such that:

• $p(uv)=1/3$, $p(w)=2/3$, $p^{\prime }(u)=1/3$, $p^{\prime }(vw)=1/3$ and $p^{\prime }(w)=1/3$;

• $q(u^{\prime }v)=1/3$, $q(w)=2/3$, $q^{\prime }(u^{\prime })=1/3$, $q^{\prime }(vw)=1/3$ and $q^{\prime }(w)=1/3$.

By Property 1, we know that $p\sim p^{\prime }$ and $q\sim q^{\prime }$ so that $p\succsim q$ if and only if $p^{\prime }\succsim q^{\prime }$. By Social expected-utility, this means that:

$$\begin{aligned}&\tfrac{1}{3}V(uv)+\tfrac{2}{3}V(w)\ge \tfrac{1}{3}V(u^{\prime }v)+\tfrac{2}{3}V(w)\Longleftrightarrow \tfrac{1}{3}V(u)+\tfrac{1}{3}V(vw)+\tfrac{1}{3}V(w)\\&\quad \ge \tfrac{1}{3}V(u^{\prime })+\tfrac{1}{3}V(vw)+\tfrac{1}{3}V(w). \end{aligned}$$

The equivalence simplifies to $V(uv)\ge V(u^{\prime }v)\Longleftrightarrow V(u)\ge V(u^{\prime })$. By Social expected-utility, this implies that $uv\succsim u^{\prime }v$ if and only if $u\succsim u^{\prime }$.

Step 4: Conclusion. $\succsim$ is a social welfare ordering that satisfies the Basic Principles and Ind-a. By Proposition 2, we thus know that there exist two continuous and increasing functions $\varphi :\mathbb {R}\rightarrow \mathbb {R}$ and $g:\mathbb {R}\rightarrow \mathbb {R}$ such that $\varphi (0)=g(0)=0$, and for all p, $p^{\prime }\in P$:

$$\begin{aligned} p\succsim p^{\prime }& \Longleftrightarrow \sum _{u\in supp(p)}p(u)\times \varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) \nonumber \\& \quad \ge \sum _{u\in supp(p^{\prime })}p^{\prime }(u)\times \varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) . \end{aligned}$$

(A.2)

Consider the situation in Property 1, where there are three alternatives u, v, w in U such that $N(u)\cap N(v)=\emptyset$, $N(w)=\{k\}$ with $k\notin N(v)$, and $p(uv)=1/3$, $p(w)=2/3$, $p^{\prime }(u)=1/3$, $p^{\prime }(vw)=1/3$ and $p^{\prime }(w)=1/3$. Assume that $w_k=c$, with c the critical-level parameter in Eq. (A.2).

Given that $\succsim$ satisfies Property 1, we know that $p\sim p^{\prime }$. Using the representation in Eq. (A.2), this implies:

$$\begin{aligned}&\tfrac{1}{3}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+\sum _{j\in N(v)}\big [g(v_j)-g(c)\big ]\right) \\&\quad =\tfrac{1}{3}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) +\tfrac{1}{3}\varphi \left( \sum _{j\in N(v)}\big [g(v_j)-g(c)\big ]\right) \end{aligned}$$

Denoting a the real number $a=\sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]$ and b the real number $b=\sum _{j\in N(v)}\big [g(v_j)-g(c)\big ]$, we thus get the equality

$$\begin{aligned} \varphi (a+b)=\varphi (a)+\varphi (b). \end{aligned}$$

We can actually get this equality for any pair of real numbers $(a,b)\in \mathbb {R}^2$. Indeed, any real number can be reached as a sum of transformed utilities minus the transformed critical level.²³ We thus get the Cauchy functional equation $\varphi (a+b)=\varphi (a)+\varphi (b)$ for all $(a,b)\in \mathbb {R}^2$. Given that $\varphi$ is continuous, we know that there must exist a real number $\alpha$ such that $\varphi (a)=\alpha a$ for all $a\in \mathbb {R}$ ( Aczél 1966, Chap. 2). Given that $\varphi$ is increasing, we actually know that $\alpha >0$.

So we can conclude that there exist a continuous and increasing function $g:\mathbb {R}\rightarrow \mathbb {R}$ such that $g(0)=0$, and a real number $c\in \mathbb {R}$, such that, for all p, $p^{\prime }\in P$:

$$\begin{aligned}&p\succsim p^{\prime }\Longleftrightarrow \sum _{u\in supp(p)}p(u)\times \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) \\&\quad \ge \sum _{u\in supp(p^{\prime })}p^{\prime }(u)\times \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) . \end{aligned}$$

The social welfare ordering $\succsim$ is an ECLGU social welfare ordering.

1.2 A.2 Proof of Proposition 3

Given that $\succsim$ on F satisfies the Basic Principles, by Proposition 1, there exist a function $W:\mathbb {N}\times \mathbb {R}\rightarrow \mathbb {R}$ increasing and continuous in its second argument, and for each $n\in \mathbb {N}$ a continuous, increasing, symmetric and normalized function $\Xi _n$ such that, for all p, $p^{\prime }\in P$:

$$\begin{aligned} p\succsim p^{\prime }\Longleftrightarrow \sum _{u\in supp(p)}p(u)\times W\Big (n(u),\Xi _{n(u)}(u)\Big )\ge \sum _{u\in supp(p^{\prime })}p^{\prime }(u)\times W\Big (n(u),\Xi _{n(u)}(u)\Big ).\end{aligned}$$

(A.3)

Consider any $e\in \mathbb {R}$ and any $m\in \mathbb {N}$. Let u, v, w in U be three alternatives such that $n(u)=n(v)=1$, $n(w)=m$, $N(u)\cap N(v)=N(u)\cap N(w)=N(v)\cap N(w)=\emptyset$, and $u_i=v_j=w_k=e$ for all $i\in N(u)$, $j\in N(v)$ and $k\in N(w)$. By Social risk neutrality in population size for perfect equality, if p and $p^{\prime }\in P$ are such that $p(uvw)=1/2$, $p(w)=1/2$, $p^{\prime }(uw)=1/2$ and $p^{\prime }(vw)=1/2$, then $p\sim p^{\prime }$. By Eq. (A.3), this means that:

$$\begin{aligned} \tfrac{1}{2}W(m+2,e)+\tfrac{1}{2}W(m,e)=\tfrac{1}{2}W(m+1,e)+\tfrac{1}{2}W(m+1,e). \end{aligned}$$

Let us denote $\theta _e:\mathbb {N}\rightarrow \mathbb {R}$ the function such that $\theta _e(n)=W(n,e)$. The equality implies that for all $m\in \mathbb {N}$,

$$\begin{aligned} \theta _e(m+2)=2\theta _e(m+1)-\theta _e(m). \end{aligned}$$

Let us prove that $\theta _e(n)=\Phi _e+n\Psi _e$, for all $n\in \mathbb {N}$, where $\Phi _e=2\theta _e(1)-\theta _e(2)$ and $\Psi _e=\theta _e(2)-\theta _e(1)$. For $n=1$ and $n=2$, this is obviously true. Assume that for some $m\in \mathbb {N}$, $\theta _e(m)=\Phi _e+m\Psi _e$ and $\theta _e(m+1)=\Phi _e+(m+1)\Psi _e$: let us show that it is also the case for $m+2$. By the equality above, we obtain

$$\begin{aligned} \theta _e(m+2) & = 2\theta _e(m+1)-\theta _e(m)=2\big (\Phi _e+(m+1)\Psi _e\big )-\big (\Phi _e+m\Psi _e\big )\\ & = \Phi _e+\big [2(m+1)-m\big ]\Psi _e=\Phi _e+(m+2)\Psi _e.\end{aligned}$$

Hence, for any $e\in \mathbb {R}$ and any $n\in \mathbb {N}$, we have that:

$$\begin{aligned} W(n,e)=\big [2W(e,1)-W(e,2)\big ]+n\big [W(e,2)-W(e,1)\big ]. \end{aligned}$$

Denoting $\Phi :\mathbb {R}\rightarrow \mathbb {R}$ the continuous function such that $\Phi (e)=2W(e,1)-W(e,2)$ for all $e\in \mathbb {R}$, and $\Psi :\mathbb {R}\rightarrow \mathbb {R}$ the continuous function such that $\Psi (e)=W(e,2)-W(e,1)$ for all $e\in \mathbb {R}$, we obtain the result.

1.3 A.3 Proof of Theorem 2

Step 1: Proof that ECLGU satisfies all the properties. It can easily be checked that an ECLGU social welfare ordering satisfies the Basic Principles, Ind-l, and Social risk neutrality in population size for perfect equality.

Step 2: function V can only be specific transformations of an additively separable social welfare function. Given that $\succsim$ on P satisfies the Basic Principles and Ind-l (and therefore Ind-a), we know by Proposition 2 that there exist two continuous and increasing functions $\varphi :\mathbb {R}\rightarrow \mathbb {R}$ and $g:\mathbb {R}\rightarrow \mathbb {R}$ such that $\varphi (0)=g(0)=0$, and for all p, $p^{\prime }\in P$:

$$\begin{aligned} p\succsim p^{\prime } &\Longleftrightarrow \sum _{u\in supp(p)}p(u)\times \varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) \nonumber \\&\quad \ge \sum _{u\in supp(p^{\prime })}p^{\prime }(u)\times \varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) .\end{aligned}$$

(A.4)

Consider p, $p^{\prime }$, q, $q^{\prime }\in P$, such that there exist distinct alternatives u, v, $v^{\prime }$, w, and $w^{\prime }$ in U for which:

• $N(u)\cap N(v)=N(u)\cap N(v^{\prime })=N(u)\cap N(w)=N(u)\cap N(w^{\prime })=\emptyset$;

• $p(uv)=p(uv^{\prime })=q(uw)=q(uw^{\prime })=p^{\prime }(v)=p^{\prime }(v^{\prime })=q^{\prime }(w)=q^{\prime }(w^{\prime })=1/2$.

By Ind-l, and using Eq. (A.4), we obtain that:

$$\begin{aligned}&\tfrac{1}{2}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+\sum _{i\in N(v)}\big [g(v_i)-g(c)\big ]\right) +\tfrac{1}{2}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+\sum _{i\in N(v^{\prime })}\big [g(v^{\prime }_i)-g(c)\big ]\right) \\&\quad \ge \tfrac{1}{2}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+\sum _{i\in N(w)}\big [g(w_i)-g(c)\big ]\right) +\tfrac{1}{2}\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+\sum _{i\in N(w^{\prime })}\big [g(w^{\prime }_i)-g(c)\big ]\right) \\&\quad \Longleftrightarrow \tfrac{1}{2}\varphi \left( \sum _{i\in N(v)}\big [g(v_i)-g(c)\big ]\right) +\tfrac{1}{2}\varphi \left( \sum _{i\in N(v^{\prime })}\big [g(v^{\prime }_i)-g(c)\big ]\right) \\&\quad \ge \tfrac{1}{2}\varphi \left( \sum _{i\in N(w)}\big [g(w_i)-g(c)\big ]\right) +\tfrac{1}{2}\varphi \left( \sum _{i\in N(w^{\prime })}\big [g(w^{\prime }_i)-g(c)\big ]\right) \end{aligned}$$

Denote $a=\sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]$, $x=\sum _{i\in N(v)}\big [g(v_i)-g(c)\big ]$, $x^{\prime }=\sum _{i\in N(v^{\prime })}\big [g(v^{\prime }_i)-g(c)\big ]$, $y=\sum _{i\in N(w)}\big [g(w_i)-g(c)\big ]$ and $y^{\prime }=\sum _{i\in N(w^{\prime })}\big [g(w^{\prime }_i)-g(c)\big ]$. We get that, for any real numbers a, x, $x^{\prime }$, x and $y^{\prime }$:

$$\begin{aligned} \varphi (a+x)+\varphi (a+x^{\prime })\ge \varphi (a+y)+\varphi (a+y^{\prime })\Longleftrightarrow \varphi (x)+\varphi (x^{\prime })\ge \varphi (y)+\varphi (y^{\prime }). \end{aligned}$$

Let $I=\varphi (\mathbb {R})$, which is an open interval in $\mathbb {R}$ because $\varphi$ is continuous and increasing. First fix a and denote $\psi _a:I\rightarrow \mathbb {R}$ the continuous function such that $\psi _a(x)=\varphi \big (a+\varphi ^{-1}(x)\big )$. By letting $z=\varphi (x)$, $z^{\prime }=\varphi (x^{\prime })$, $t=\varphi (y)$ and $t^{\prime }=\varphi (y^{\prime })$, the above equivalence can be written:

$$\begin{aligned} \psi _a(z)+\psi _a(z^{\prime })\ge \psi _a(t)+\psi _a(t^{\prime })\Longleftrightarrow z+z^{\prime }\ge t+t^{\prime }, \end{aligned}$$

and it holds for all z, $z^{\prime }$, t and $t^{\prime }$ in I. So, there must exist an increasing function $\Psi _a:\mathbb {R}\rightarrow \mathbb {R}$ such that, for all z, $z^{\prime }\in I$: $\psi _a(z)+\psi _a(z^{\prime })=\Psi _a(z+z^{\prime })$. This is a Pexider functional equation, and it is known that in that case $\Psi _a$ and $\psi _a$ must be affine (Aczél 1966). Hence there exist $\alpha _a\in \mathbb {R}_{++}$ and $\beta _a\in \mathbb {R}$ such that $\psi _a(z)=\alpha _az+\beta _a$.

Define the functions $\alpha :\mathbb {R}\rightarrow \mathbb {R}_{++}$ and $\beta :\mathbb {R}\rightarrow \mathbb {R}$ by $\alpha (a)=\alpha _a$ and $\beta (a)=\beta _a$ for all $a\in \mathbb {R}$. By definition of function $\psi _a$, we obtain that, for all $x\in \mathbb {R}$, $\psi _a(\varphi (x))=\varphi \big (a+\varphi ^{-1}\circ \varphi (x)\big )=\varphi (a+x)$. But by our result above, it is also the case that $\psi _a(\varphi (x))=\alpha (a)\varphi (x)+\beta (a)$. We thus end up with the functional equation: $\varphi (a+x)=\alpha (a)\varphi (x)+\beta (a)$ for all $(a,x)\in \mathbb {R}^2$. By Corollary 1 (pp. 150–151) in Aczél (1966), this equation implies that either $\varphi$ is affine or that it is a positive affine transformation of the function $x\rightarrow \alpha e^{\alpha x}$ for some $\alpha \ne 0$.

Step 3: Conclusion. By Step 2, we know that

$$\begin{aligned} V(u)=\varphi \left( \sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]\right) , \end{aligned}$$

where V is the function in the statement of Social expected utility, and $\varphi$ is affine or that it is a positive affine transformation of the function $x\rightarrow \alpha e^{\alpha x}$ for some $\alpha \ne 0$.

Also given that $\succsim$ satisfies RiskNeu, we know by Proposition 3 that there exist a positive number a, a number b, two continuous functions $\Psi :\mathbb {R}\rightarrow \mathbb {R}$ and $\Phi :\mathbb {R}\rightarrow \mathbb {R}$, and for each $n\in \mathbb {N}$ a continuous, increasing, symmetric and normalized function $\Xi _n$ such that, for u such that $n(u)>0$:

$$\begin{aligned} V(u)=a \Big [\Phi \left( \Xi _{n(u)}(u)\right) +n(u)\times \Psi \left( \Xi _{n(u)}(u)\right) \Big ]+b. \end{aligned}$$

By contradiction, suppose that $\varphi$ is a positive affine transformation of the function $x\rightarrow \alpha e^{\alpha x}$ for some $\alpha \ne 0$. Consider any $n\in \mathbb {N}$ and $x\in \mathbb {R}$, and let u be such that $n(u)=n$ and $u_i=x$ for all $i\in N(u)$. We must have:

$$\begin{aligned} \alpha e^{\alpha n(g(x)-g(c))}=a \Big [\Phi \left( x\right) +n\times \Psi \left( x\right) \Big ]+b. \end{aligned}$$

If $x\ne c$, then the left-hand side is an exponential function in n, whereas the right-hand side is an affine function in n. But this cannot be true. Hence only the affine form is possible for $\varphi$.

1.4 A.4 Proof of Proposition 4

Consider any $N\in {\mathcal {N}}$ and let $n=\vert N\vert$. For any $i\in N$ let $\succsim _i$ be the ordering on $P_N$ such that, for any p, $p^{\prime }\in P_N$, $p\succsim _i p^{\prime }$ if and only if $p_i\succsim p^{\prime }_i$. By Social expected-utility, defining $W_i(p)=\sum _{u\in U_i}p_i(u)V(u)$, we have that $p\succsim _i p^{\prime }$ if and only if $W_i(p)\ge W_i(p^{\prime })$. Similarly, defining $W_0(p)=\sum _{u\in U_N}p(u)V(u)$, we have that $p\succsim p^{\prime }$ if and only if $W(p)\ge W(p^{\prime })$.

Consider $F=(W_0,(W_i)_{i\in N}):P_N\rightarrow \mathbb {R}^{n+1}$. By definition $F(P_N)$ is convex because $W_i(\kappa p+(1-\kappa )q)=\kappa W_i(p)+(1-\kappa ) W_i(q)$ for each $i\in N$ or $i=0$. IDom implies that, if $W_i(p)>W_i(q)$ for all $i\in N$, then $W_0(p)> W_0(q)$. By Proposition 2 in De Meyer and Mongin (1995), there must exist non-negative numbers $\lambda _i$ and a number $\gamma$ such that, for each $p\in P_N$:

$$\begin{aligned} W_0(p)=\sum _{i\in N}\lambda _i W_i(p)+\gamma . \end{aligned}$$

Focusing on lotteries yielding sure outcomes, we get that, for all u, $v\in U_N$,

$$\begin{aligned} u\succsim v\Longleftrightarrow \sum _{i\in N}\lambda _i V(u_i)\ge \sum _{i\in N}\lambda _i V(v_i). \end{aligned}$$

By Anonymity and Strong Pareto, it must be the case that all the $\lambda _i$ must be the same positive number $\lambda$. Denoting g the function such that $g(u_i)=V(u_i)$ for each $u_i\in \mathbb {R}$, we obtain that for all p, $q\in P_N$,

$$\begin{aligned} p\succsim q\Longleftrightarrow \sum _{u\in U_N}p(u)\left[ \sum _{i\in N}g(u_i)\right] \ge \sum _{u\in U_N}p(u)\left[ \sum _{i\in N}g(u_i)\right] . \end{aligned}$$

Observe that, by definition and Anonymity, the function g does not depend neither on i nor on N.

By Social expected-utility, and given that a VNM utility function is defined up to an increasing affine transformation, it must be the case that, for each $N\in {\mathcal {N}}$, for each $u\in U_N$:

$$\begin{aligned} V(u)=F(N)\left[ \sum _{i\in N}g(u_i)\right] +G(N), \end{aligned}$$

for some positive F(N) and some number G(N). By Anonymity, F(N) and G(N) only depend on population size.

1.5 A.5 Proof of Theorem 3

Step 1: Proof that ECLGU satisfies all the properties. It can easily be checked that an ECLGU social welfare ordering satisfies the Basic Principles, IDom, IndPUnc and CCL-rf.

Step 2: Proof that $\succsim$ has a constant critical level. We show that if $\succsim$ satisfies the Basic Principles and IndPUnc, then it satisfies CCL-rf.

By Minimal existence of a critical level, there exist $u\in U$, $c\in \mathbb {R}$, and $i\notin N(u)$, such that if $v\in U$ is defined by $N(v)=N(u)\cup \{i\}$, $v_i=c$ and $v_j=u_j$ for all $j\in N(u)$, then $v\sim u$. Let $\varepsilon$ be some positive number. Define ${\tilde{v}}\in U$ by $N({\tilde{v}})=N(v)$, ${\tilde{v}}_i=c+\varepsilon$ and ${\tilde{v}}_j=u_j$ for all $j\in N(u)$. By Strong Pareto, we must have ${\tilde{v}}\succ v \sim u$, which by Social expected-utility implies that $V({\tilde{v}})>V(u)$.

Consider any $u^{\prime }\in U$ and any $k\notin N(u^{\prime })$, and define ${\tilde{v}}^{\prime }\in U$ such that $N({\tilde{v}}^{\prime })=N(u^{\prime })\cup \{k\}$, ${\tilde{v}}^{\prime }_k=c+\varepsilon$ and ${\tilde{v}}^{\prime }_l={\tilde{u}}_l$ for all $l\in N(u^{\prime })$. Let p, $p^{\prime }\in P$ such that $p(u^{\prime })=p^{\prime }({\tilde{v}}^{\prime })=q$ and $p(u)=p^{\prime }({\tilde{v}})=1-q$, with $q\in (0,1)$. IndPUnc requires that, if $p^{\prime }\succ p$ for some q, this should be true whatever q is. But $p^{\prime }\succ p$ means that:

$$\begin{aligned} q V({\tilde{v}}^{\prime })+ (1-q) V({\tilde{v}})>q V(u^{\prime })+ (1-q) V(u) \end{aligned}$$

which can be written

$$\begin{aligned} V({\tilde{v}})-V(u)>\tfrac{q}{1-q}\Big [V(u^{\prime })-V({\tilde{v}}^{\prime })\Big ]. \end{aligned}$$

Given that $V({\tilde{v}})>V(u)$, this must be true, whatever the values $V({\tilde{v}}^{\prime })$ and $V(u^{\prime })$, for small enough value of q.

Thus, whatever q is, we have:

$$\begin{aligned} q V({\tilde{v}}^{\prime })+ (1-q) V({\tilde{v}})>q V(u^{\prime })+ (1-q) V(u) \end{aligned}$$

which can be written

$$\begin{aligned} V({\tilde{v}}^{\prime })-V(u^{\prime })>\tfrac{1-q}{q}\Big [V({\tilde{v}})-V(u)\Big ]. \end{aligned}$$

Given that $\tfrac{1-q}{q}$ can be as low as we want, we need to have $V({\tilde{v}}^{\prime })-V(u^{\prime })\ge 0$, and therefore ${\tilde{v}}^{\prime }\succsim u^{\prime }$.

So for every $\varepsilon >0$, ${\tilde{v}}^{\prime }\succsim u^{\prime }$, where ${\tilde{v}}^{\prime }\in U$ is such that $N({\tilde{v}}^{\prime })=N(u^{\prime })\cup \{k\}$, ${\tilde{v}}^{\prime }_k=c+\varepsilon$ and ${\tilde{v}}^{\prime }_l={\tilde{u}}_l$ for all $l\in N(u^{\prime })$. By continuity, if $v^{\prime }\in U$ such that $N(v^{\prime })=N(u^{\prime })\cup \{k\}$, $v^{\prime }_k=c$ and $v^{\prime }_l=u^{\prime }_l$ for all $l\in N({\tilde{u}})$, then $v^{\prime }\succsim u^{\prime }$.

We can prove that $u^{\prime }\succsim v^{\prime }$ in the same way by using a negative $\varepsilon <0$. Hence, $u^{\prime }\sim v^{\prime }$.

In conclusion, there exists $c\in \mathbb {R}$, such that for all $u^{\prime }\in U$ and for all $k\notin N(u^{\prime })$, if $v^{\prime }\in U$ is such that $N(v^{\prime })=N(u^{\prime })\cup \{k\}$, $v^{\prime }_k=c$ and $v^{\prime }_l=u^{\prime }_l$ for all $l\in N(u)$, then $u^{\prime }\sim v^{\prime }$. The social ordering $\succsim$ satisfies CCL-rf.

Step 3: Conclusion. Given that $\succsim$ satisfies the Basic Principles and IDom, we know by Proposition 4 that it is an ENWGU social ordering. Then there exist a continuous and increasing function $g:\mathbb {R}\rightarrow \mathbb {R}$, and two functions $F:\mathbb {N}\rightarrow \mathbb {R}$ and $G:\mathbb {N}\rightarrow \mathbb {R}$ such that for any $u\in U$ the VNM social utility function is

$$\begin{aligned} V(u)=F\big (n(u)\big )\sum _{i\in N(u)}g(u_i)+G\big (n(u)\big ). \end{aligned}$$

By Step 3, we also know that $\succsim$ satisfies CCL-rf. Hence, there exists a $c\in \mathbb {R}$ such that, for any $n\in \mathbb {N}$ and any $x\in \mathbb {R}$, if $u, v\in U$ are such that $n(u)=n$, $N(v)=N(u)\cup \{i\}$ (with $i\notin N(u)$), $u_j=v_j=x$ for all $j\in N(u)$ and $v_i=c$, then $V(u)=V(v)$, so that:

$$\begin{aligned} F(n)n g(x)+G(n)=F(n+1)\big [n g(x)+g(c)\big ]+G(n+1). \end{aligned}$$

Given that g is increasing in x and that this equality must be true for all $x\in \mathbb {R}$, we must have $F(n+1)=F(n)$. Hence, we also obtain by the equality that $G(n+1)-G(n)=-g(c)$. Hence, by iterating on n (given that these results are true for all $n\in \mathbb {N}$), we have $F(n)=F(0)=a$ and $G(n)=G(0)-ng(c)=b-ng(c)$, where $a>0$ and b is a real number. Hence,

$$\begin{aligned} V(u)=a\sum _{i\in N(u)}\big [g(u_i)-g(c)\big ]+b. \end{aligned}$$