Consistent updating of social welfare functions

Abstract

It is one of the central doctrines for ‘rational’ decision making that an agent should be forward-looking and not be bound by bygones. We argue that this is not an appealing principle for collective decision making, and that bygones have necessary and substantive roles. We consider an explicitly dynamic process of social welfare orderings, and propose a dynamic constraint which is acceptable even after rejecting the principle that one should be forward-looking. It is a conjunction of two assertions: (i) the process must be dynamically consistent, which means an ex ante welfare judgment must be respected by the ex post ones and there should be no contradiction between them; (ii) the structure of a welfare judgment should be recurrent under consistent updating, in the sense that if a postulate is satisfied by an ex ante welfare judgment, then it is also satisfied by any of the ex post ones that follow this ex ante judgment. Based on this standpoint, we present a set of axioms for social welfare orderings which are recurrent under consistent updating, and characterize a set of social welfare functions which are closed under updating. With such a class of social welfare functions, we characterize the roles that can be played in the updating stage by the past and things known not to have occurred.

Appendices

Appendix 1: The class of static social welfare functions

1.1 Basic axioms and characterizations

This section presents a set of axioms, which in the main text are translated into a dynamic setting and shown to be recurrent under consistent updating with the passing of time and the resolution of uncertainty. It presents a class of social welfare functions characterized by that.

Let I be the set of individuals. For technical reason, we assume $|I|\ge 3$. We assume that individual utilities to are given as cardinal and interpersonally comparable objects. Let $\mathbb {R}^{I}$ be the domain of such individual utilities. We consider a social welfare ordering $\succsim$ defined over $\mathbb {R}^{I}$.

We consider the following axioms.

• Order $\succsim$ is complete and transitive.

• Continuity $\succsim$ is a closed subset of $\mathbb {R}^{I} \times \mathbb {R}^{I}$.

• Pareto for all $U,V\in \mathbb {R}^{I}$, $U_{i}\ge V_{i}$ for every $i\in I$ implies $U\succsim V$, and the conclusion is strict if $U_{i}> V_{i}$ for some $i\in I$ in addition.

• Inequality aversion for all $U,V\in \mathbb {R}^{I}$ and $c \in [0,1]$, $U\sim V$ implies $c U+(1-c)V \succsim U$.

• Separability for all $J\subset I$, for all $U_{J},V_{J}\in \mathbb {R}^{J}$ and $U_{-J},V_{-J}\in \mathbb {R}^{I{\setminus } J}$, $(U_{J},U_{-J})\succsim (V_{J},U_{-J})$ holds if and only if $(U_{J},V_{-J})\succsim (V_{J},V_{-J})$.

• Shift covariance for all $U,V\in \mathbb {R}^{I}$ and $c\in \mathbb {R}$, $U\succsim V$ implies $U+c \mathbf {1}\succsim V+c\mathbf {1}$.

This last, the shift covariance axiom, is concerned with interpersonal comparison of utilities, and says that adding ‘equal utilities’ to everybody does not change the social welfare ranking. This means that the attitude toward inequality is independent of the absolute level of utilities. While homogeneity, another prominent independence property, is not recurrent under consistent updating when translated to dynamic settings, shift covariance can be shown to be recurrent under consistent updating. Note, however, that it relies on the assumption that individual utility functions fall in the class of additively separable ones.

Let $\Delta ^{I} = \left\{ a \in \mathbb {R}^{I}_{+}:\sum _{i\in I} a_{i} =1\right\}$ and $int \Delta ^{I}$ denote the relative interior of $\Delta ^{I}$.

Proposition 11

A social welfare ordering $\succsim$ satisfies order, continuity, Pareto, inequality aversion, separability and shift covariance if and only if either of the following two cases holds:

• (i) there exist a vector $a \in int \Delta ^{I}$ and a number $\lambda >0$ such that $\succsim$ can be represented in the form

  $$\begin{aligned} \Phi (U)=-\sum _{i\in I}a_{i}e^{-\lambda U_{i}}. \end{aligned}$$

  We call this class of orderings and representations the exponential class.

• (ii) there exists a vector $a \in int \Delta ^{I}$ such that $\succsim$ can be represented in the form

  $$\begin{aligned} \Phi (U)=\sum _{i\in I}a_{i}U_{i}. \end{aligned}$$

  We call this class of orderings and representations the additive class.

Moreover, in case (i) a and $\lambda$ are unique and in case (ii) a is unique.

Proof

This is an asymmetric extension of the argument in Roberts (1980), Theorem 6 and Moulin (1989), Theorem 2.6.

The necessity of the axioms is routine. We prove sufficiency.

From order, continuity and separability, $\succsim$ allows the additive representation (see Debreu 1960)

$$\begin{aligned} \Phi (U)=\sum _{i\in I} \phi _{i}(U_{i}), \end{aligned}$$

which is unique up to a positive affine transformation. From Pareto, each $\phi _{i}$ is strictly increasing.

From shift covariance, both $\sum _{i\in I} \phi _{i}(U_{i})$ and $\sum _{i\in I} \phi _{i}(U_{i}+c)$ are additive representations of the same ranking for all $c\in \mathbb {R}$, hence they are cardinal equivalent: there exist real-valued functions $\psi$ and $\zeta _{i}$ with $\psi$ being positive such that

$$\begin{aligned} \phi _{i}(U_{i}+c)=\psi (c) \phi _{i}(U_{i})+\zeta _{i}(c) \end{aligned}$$

for all i.

This is the generalized Pexider equation, which has a strongly increasing and weakly concave solution either in the form

$$\begin{aligned} \phi _{i}(U_{i})=-A_{i}e^{-\lambda _{i}U_{i}} \end{aligned}$$

with $A_{i}>$ and $\lambda _{i}>0$, or

$$\begin{aligned} \phi _{i}(U_{i})=A_{i} U_{i} \end{aligned}$$

with $A_{i}>0$.

Because $\psi$ is the same for all individuals, we must have

$$\begin{aligned} \phi _{i}(U_{i})=-A_{i}e^{-\lambda U_{i}}\quad \text {for some }\lambda \text { independent of }i \end{aligned}$$

for all $i\in I$ or

$$\begin{aligned} \phi _{i}(U_{i})=A_{i} U_{i} \end{aligned}$$

for all $i\in I$.

By normalizing $a_{i}=A_{i}/\sum _{j\in I}A_{j}$ for each $i\in I$, we obtain the representation.

Uniqueness for the exponential case, suppose both $-\sum _{i\in I}a_{i}e^{-\lambda U_{i}}$ and $-\sum _{i\in I}a_{i}^{\prime }e^{-\lambda ^{\prime } U_{i}}$ represent the same ranking. Since they are additive representations of the same ranking, we have cardinal equivalence: there exist constants C, D with $C>0$ such that

$$\begin{aligned} -a_{i}^{\prime }e^{-\lambda ^{\prime } U_{i}}=-Ca_{i} e^{-\lambda U_{i}}+D \end{aligned}$$

for all i.

Suppose $D\ne 0$. Then as $U_{i}\rightarrow \infty$, we have $-a_{i}^{\prime }e^{-\lambda ^{\prime } U_{i}} \rightarrow 0$. But $-Ca_{i} e^{-\lambda U_{i}}+D \rightarrow D \ne 0$, a contradiction. Hence $D=0$.

By letting $U_{i}=0$, we have $a_{i}^{\prime }=Ca_{i}$. Since $\sum _{i\in I}a_{i}=1$, we obtain $C=1$, which implies $a_{i}^{\prime }=a_{i}$. Then it is immediate to see $\lambda ^{\prime }=\lambda$.

For the additive case, suppose both $\sum _{i\in I}a_{i} U_{i}$ and $\sum _{i\in I}a_{i}^{\prime } U_{i}$ represent the same ranking. Since they are additive representations of the same ranking, we have cardinal equivalence: there exist constants C, D with $C>0$ such that

$$\begin{aligned} -a_{i}^{\prime } U_{i}=-Ca_{i} U_{i}+D \end{aligned}$$

for all i.

By letting $U_{i}=0$, we obtain $D=0$. By letting $U_{i}=1$, we have $C=1$, which implies $a_{i}^{\prime }=a_{i}$. $\square$

We will also consider a weaker version of Shift Covariance. This is because ‘equality’ does not necessarily mean ‘equality of utilities’, depending on the situation. This is particularly the case when the ‘scaling’ of utility is different for different individuals. As we have seen in the main text, when individuals’ subjective weights on the future differ, it may be natural to say that their scalings of future utilities are treated as different too, and when individuals’ beliefs on events differ it may be natural to say that their scalings of utilities contingent on events are treated as different too. The axiom below says that up to some scalings, adding ‘equal conditions’ to everybody does not change the social welfare ranking.

• General shift covariance there exists $W\in \mathbb {R}_{++}^{I}$ such that for all $U,V\in \mathbb {R}^{I}$ and $c\in \mathbb {R}$, $U\succsim V$ implies $U+c W\succsim V+cW$.

Proposition 12

A social welfare ordering $\succsim$ satisfies order, continuity, Pareto, inequality aversion, separability and general shift covariance if and only if either of the following two cases holds:

• (i) there exists a vector $a \in int \Delta ^{I}$ and a vector $\lambda \in \mathbb {R}^{I}_{++}$ such that $\succsim$ can be represented in the form

  $$\begin{aligned} \Phi (U)=-\sum _{i\in I}a_{i}e^{-\lambda _{i} U_{i}}. \end{aligned}$$

  We call this class of orderings and representations the generalized exponential class.

• (ii) there exists a vector $a \in int \Delta ^{I}$ such that $\succsim$ can be represented in the form

  $$\begin{aligned} \Phi (U)=\sum _{i\in I}a_{i}U_{i}. \end{aligned}$$

  We again call this class of orderings and representations the additive class.

Moreover, in case (i) a and $\lambda$ are unique and in case (ii) a is unique.

Proof

The necessity of the axioms is routine. We prove sufficiency.

Define $\succsim ^{\star }$ by

$$\begin{aligned} U \succsim ^{\star } V \Longleftrightarrow (W_{i}U_{i}/\overline{W})_{i\in I} \succsim (W_{i}V_{i}/\overline{W})_{i\in I}, \end{aligned}$$

where $\overline{W}=\sum _{i\in I}W_{i}$.

Then $\succsim ^{\star }$ satisfies order, continuity, Pareto, shift covariance, inequality aversion and separability, and it follows from Theorem 11 that one of the following two cases holds:

• (i) there exists $\lambda ^{\star }>0$ and a vector $a^{\star } \in \mathbb {R}_{++}^{I}$ such that $\succsim ^{\star }$ can be represented in the form

  $$\begin{aligned} \Phi ^{\star }(U)=-\sum _{i\in I}a_{i}^{\star }e^{-\lambda ^{\star } U_{i}} \end{aligned}$$

• (ii) there exists a vector $a^{\star } \in \mathbb {R}_{++}^{I}$ such that $\succsim$ can be represented in the form

  $$\begin{aligned} \Phi ^{\star }(U)=\sum _{i\in I}a_{i}^{\star }U_{i} \end{aligned}$$

  In case (i), we have

  $$\begin{aligned} U\succsim V\Longleftrightarrow & {} (\overline{W}U_{i}/W_{i})_{i\in I} \succsim ^{\star } (\overline{W}V_{i}/W_{i})_{i\in I} \\\Longleftrightarrow & {} -\sum _{i\in I}a_{i}^{\star }e^{-\lambda ^{\star } \overline{W}U_{i}/W_{i}} \ge -\sum _{i\in I}a_{i}^{\star }e^{-\lambda ^{\star } \overline{W}V_{i}/W_{i}}. \end{aligned}$$

  Therefore, by letting $\lambda _{i}=\lambda ^{\star }\overline{W}/W_{i}$ for each $i\in I$ and $a=a^{\star }$, we obtain the representation. In case (ii), we have

  $$\begin{aligned} U\succsim V\Longleftrightarrow & {} (\overline{W}U_{i}/W_{i})_{i\in I} \succsim ^{\star } (\overline{W}V_{i}/W_{i})_{i\in I} \\\Longleftrightarrow & {} \sum _{i\in I}a_{i}^{\star } \overline{W}U_{i}/W_{i} \ge -\sum _{i\in I}a_{i}^{\star } \overline{W}V_{i}/W_{i} \end{aligned}$$

  Therefore, by letting $a_{i}=\frac{a_{i}^{\star }\overline{W}/W_{i}}{\sum _{j\in I}a_{j}^{\star }\overline{W}/W_{j}}$ for each $i\in I$ we obtain the representation.

Uniqueness for the exponential case, suppose both $-\sum _{i\in I}a_{i}e^{-\lambda _{i} U_{i}}$ and $-\sum _{i\in I}a_{i}^{\prime }e^{-\lambda _{i}^{\prime } U_{i}}$ represent the same ranking. Since they are additive representations of the same ranking, we have cardinal equivalence: there exist constants $C>0$ and $(D_{i})_{i\in I}$ such that

$$\begin{aligned} -a_{i}^{\prime }e^{-\lambda _{i}^{\prime } U_{i}}=-Ca_{i} e^{-\lambda _{i} U_{i}}+D_{i} \end{aligned}$$

for all i.

Suppose $D_{i}\ne 0$. Then as $U_{i}\rightarrow \infty$, we have $-a_{i}^{\prime }e^{-\lambda _{i}^{\prime } U_{i}} \rightarrow 0$. But $-Ca_{i} e^{-\lambda _{i} U_{i}}+D_{i} \rightarrow D_{i} \ne 0$, a contradiction. Hence $D_{i}=0$.

By letting $U_{i}=0$, we have $a_{i}^{\prime }=Ca_{i}$. Since $\sum _{i\in I}a_{i}=1$, we obtain $C=1$, which implies $a_{i}^{\prime }=a_{i}$. Then it is immediate to see $\lambda _{i}^{\prime }=\lambda _{i}$, which is true for all i.

Uniqueness for the additive case is immediate. $\square$

1.2 Comparative inequality aversion

Here we discuss the normative content of the parameters in the social welfare function characterized above. For the exponential class and additive class, one can make a straightforward interpretation of the parameters, which is an analogue of the standard argument about risk aversion: a explains the welfare weights on the individuals and $\lambda$ explains the degree of inequality aversion.

We extend this interpretation to the generalized exponential class, in which the notion of ‘equality’ may depend on the different scalings of utilities of different individuals, and hence the degree of inequality aversion may be depend on the individual.

We define comparative inequality aversion as follows.

Definition 5

$\succsim$ is more inequality averse than $\succsim ^{\prime }$ if there exists a vector $W\in \mathbb {R}^{I}_{++}$ such that for all $c\in \mathbb {R}$ and $U\in \mathbb {R}^{I}$, $U\succsim cW$ implies $U\succsim ^{\prime } cW$.

Here the ray spanned by W reflects what is regarded as ‘equal’ by the given social welfare judgment. This includes the standard definition of inequality aversion as the special case in which W is proportional to $\mathbf {1}$.

Proposition 13

Suppose $\succsim$ and $\succsim ^{\prime }$ fall in the generalized exponential class, where $(a,\lambda )$ describes $\succsim$ and $(a^{\prime },\lambda ^{\prime })$ describes $\succsim ^{\prime }$. Then $\succsim$ is more inequality averse than $\succsim ^{\prime }$ if and only if $a=a^{\prime }$ and $\lambda =\mu \lambda ^{\prime }$ for some $\mu \ge 1$.

Proof

The ‘if’ part is routine. We prove the ‘only if’ part.

Consider the ‘marginal rate of substitution’ between individual utilities associated with $\succsim$, which is given by

$$\begin{aligned} MRS(U)=\left( \frac{\lambda _{i}a_{i}e^{-\lambda _{i}U_{i}}}{\lambda _{1}a_{1}e^{-\lambda _{1}U_{1}}}\right) _{i \in I {\setminus } \{1\}}. \end{aligned}$$

Note that

$$\begin{aligned} MRS(cW)=\left( \frac{\lambda _{i}a_{i}}{\lambda _{1}a_{1}}\right) _{i \in I {\setminus } \{1\}} \end{aligned}$$

for all c, where $W=(1/\lambda _{i})_{i\in I}$. Do the same argument for $MRS^{\prime }$ and $W^{\prime }=(1/\lambda ^{\prime }_{i})_{i\in I}$ associated with $\succsim ^{\prime }$.

For $\succsim$ and $\succsim ^{\prime }$ to be comparable, the indifference curves passing through the origin given by $\succsim$ and $\succsim ^{\prime }$ must be tangent to each other at the origin, for otherwise they cross and the comparison fails along any ray. Thus, we conclude that $MRS(\mathbf {0})=MRS^{\prime }(\mathbf {0})$. In other words,

$$\begin{aligned} \frac{\lambda _{i} a_{i}}{\lambda _{1} a_{1}}=\frac{\lambda ^{\prime }_{i} a^{\prime }_{i}}{\lambda ^{\prime }_{1} a^{\prime }_{1}}, \end{aligned}$$

for all $i\in I {\setminus } \{1\}$. It follows that $MRS(cW)=MRS^{\prime }(c^{\prime }W^{\prime })$ for all $c,c^{\prime }>0$.

Then it must be that W and $W^{\prime }$ span the same ray passing through the origin. Suppose not. Then the indifference curves given by $\succsim$ are parallel along the ray spanned by W yielding the same vector of MRS, and those given by $\succsim ^{\prime }$ are parallel along the ray spanned by $W^{\prime }$ yielding the same vector of $MRS^{\prime }$, while keeping $MRS=MRS^{\prime }$. Hence, by convexity, they must cross somewhere between the two rays and cannot be tangent to each other anywhere.

Therefore, $\lambda =\mu \lambda ^{\prime }$ for some $\mu >0$. Since MRS and $MRS^{\prime }$ must be the same along the ray, we have $a=a^{\prime }$. By comparing the second-order derivatives, we obtain $\mu \ge 1$. $\square$

Now it is immediate to see the following claim. Note that in the exponential class, $\lambda$ reduces to a scaler.

Corollary 7

Suppose $\succsim$ and $\succsim ^{\prime }$ fall in the exponential class, where $(a,\lambda )$ describes $\succsim$ and $(a^{\prime },\lambda ^{\prime })$ describes $\succsim ^{\prime }$. Then $\succsim$ is more inequality averse than $\succsim ^{\prime }$ if and only if $a=a^{\prime }$ and $\lambda \ge \lambda ^{\prime }$.

Appendix 2: Proofs for Sect. 2

1.1 Proof of Proposition 1

Order obvious.

DU-Pareto let $\sum _{\tau =t+1}^{\infty } \beta ^{\tau -t} u_{i\tau } \ge \sum _{\tau =t+1}^{\infty } \beta ^{\tau -t} v_{i\tau }$ for all $i\in I$. Then $u_{it}+\sum _{\tau =t}^{\infty } \beta ^{\tau -t} u_{i\tau } \ge u_{it}+\sum _{\tau =t}^{\infty } \beta ^{\tau -t} v_{i\tau }$ for all $i\in I$. Since DU-Pareto was assumed to hold for $\succsim _{\mathbf {u}^{t-1}}$, we have $(u_{t},\mathbf {u}_{t+1}) \succsim _{\mathbf {u}^{t-1}} (u_{t},\mathbf {v}_{t+1})$. By dynamic consistency, $\mathbf {u}_{t+1} \succsim _{(\mathbf {u}^{t-1},u_{t})} \mathbf {v}_{t+1}$. The strict case is proved similarly.

DU-continuity let $\{U_{t+1}^{\nu }\}$ be a sequence in $\mathbb {R}^{I}$ converging to $U_{t+1}$, and let $\{V_{t+1}^{\nu }\}$ be a sequence in $\mathbb {R}^{I}$ converging to $V_{t+1}$. Suppose $U_{t+1}^{\nu }\succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}^{\nu }$ for all $\nu$. By dynamic consistency, we have $(u_{it}+\beta U_{i,t+1}^{\nu })_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta V_{i,t+1}^{\nu })_{i\in I}$ for all $\nu$. Since DU-continuity was assumed to hold for $\succsim _{\mathbf {u}^{t-1}}$, we have $(u_{it}+\beta U_{i,t+1})_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta V_{i,t+1})_{i\in I}$. By dynamic consistency, $U_{t+1} \succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}$.

DU-inequality aversion let $U_{t+1}\sim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}$. By dynamic consistency, we have $(u_{it}+\beta U_{i,t+1})_{i\in I} \sim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta V_{i,t+1})_{i\in I}$. Since DU-inequality aversion was assumed to hold for $\succsim _{\mathbf {u}^{t-1}}$, we have $(u_{it}+\beta (c U_{i,t+1}+(1-c)V_{i,t+1}))_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta U_{i,t+1})_{i\in I}$. By dynamic consistency, $cU_{t+1}+(1-c)V_{t+1} \succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} U_{t+1}$.

DU-separability let $(U_{J,t+1},U_{-J,t+1})\succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} (V_{J,t+1},U_{-J,t+1})$. By dynamic consistency, this holds if and only if $(u_{Jt}+\beta U_{Jt}, u_{-Jt}+\beta U_{-Jt}) \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{Jt}+\beta V_{Jt}, u_{-Jt}+\beta U_{-Jt})$. Since DU-separability was assumed to hold for $\succsim ^{*}_{\mathbf {u}^{t-1}}$, this holds if and only if $(u_{Jt}+\beta U_{Jt}, u_{-Jt}+\beta V_{-Jt}) \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{Jt}+\beta V_{Jt}, u_{-Jt}+\beta V_{-Jt})$. By dynamic consistency, this holds if and only if $(U_{J,t+1},V_{-J,t+1})\succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} (V_{J,t+1},V_{-J,t+1})$.

DU-shift covariance let $U_{t+1}\succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}$. By dynamic consistency this holds if and only if $(u_{it}+\beta U_{it})_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta V_{it})_{i\in I}$. Since DU-shift covariance was assumed to hold for $\succsim _{\mathbf {u}^{t-1}}$, this holds if and only if $(u_{it}+\beta (U_{it}+c) )_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta (V_{it}+c))_{i\in I}$. By dynamic consistency, this holds if and only if $U_{t+1}+c \mathbf {1} \succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}+c \mathbf {1}$.

1.2 Proof of Proposition 2

It follows from the fact that $\succsim _{\mathbf {u}^{t-1}}^{*}$ satisfies all the conditions in Proposition 11.

1.3 Proof of Theorem 1

Note that for the exponential class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{0}|\emptyset )= & {} -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda (\emptyset )\sum _{\tau =0}^{\infty }\beta ^{\tau }u_{i\tau } \right) \\= & {} -\sum _{i\in I}a_{i}(\emptyset )\exp \left( -\lambda (\emptyset )\sum _{\tau =0}^{t-1}\beta ^{\tau }u_{i\tau }\right) \exp \left( -\lambda (\emptyset )\beta ^{t} \sum _{\tau =t}^{\infty }\beta ^{\tau -t}u_{i\tau }\right) , \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{t}$ as $\Phi (\mathbf {u}_{t}|\mathbf {u}^{t-1})= -\sum _{i\in I}a_{i}(\mathbf {u}^{t-1}) \exp \left( -\lambda (\mathbf {u}^{t-1})\sum _{\tau =t}^{\infty }\beta ^{\tau -t}u_{i\tau }\right)$ does. Note that it is obviously impossible to switch from the exponential class to the additive class through updating, and vice versa.

For the additive class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{0}|\emptyset )= & {} \sum _{i\in I}a_{i}(\emptyset ) \sum _{\tau =0}^{\infty }\beta ^{\tau }u_{i\tau } \\= & {} \sum _{i\in I}a_{i}(\emptyset ) \sum _{\tau =0}^{t-1}\beta ^{\tau }u_{i\tau } +\beta ^{t}\sum _{i\in I}a_{i}(\emptyset ) \sum _{\tau =t}^{\infty }\beta ^{\tau -t}u_{i\tau }, \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{t}$ as $\Phi (\mathbf {u}_{t}|\mathbf {u}^{t-1})= \sum _{i\in I}a_{i}(\mathbf {u}^{t-1}) \sum _{\tau =t}^{\infty }\beta ^{\tau -t}u_{i\tau }$ does.

1.4 Proof of Proposition 4

We only prove that DU-general shift covariance is recurrent, since the rest is similar to the case of homogeneous discounting.

Let $U_{t+1}\succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}$. By dynamic consistency this holds if and only if $(u_{it}+\beta _{i} U_{it})_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta _{i} V_{it})_{i\in I}$. Since DU-general shift covariance was assumed to hold for $\succsim _{\mathbf {u}^{t-1}}$, for some $W_{\mathbf {u}^{t-1}}$ this holds if and only if $(u_{it}+\beta _{i} (U_{it}+cW_{i,\mathbf {u}^{t-1}}/\beta _{i})_{i\in I} \succsim ^{*}_{\mathbf {u}^{t-1}} (u_{it}+\beta _{i} (V_{it}+cW_{i,\mathbf {u}^{t-1}}/\beta _{i}))_{i\in I}$. By dynamic consistency, this holds if and only if $U_{t+1}+c W_{\mathbf {u}^{t-1},u_{t}} \succsim ^{*}_{\mathbf {u}^{t-1},u_{t}} V_{t+1}+c W_{\mathbf {u}^{t-1},u_{t}}$, where $W_{\mathbf {u}^{t-1},u_{t}} =( W_{i,\mathbf {u}^{t-1},u_{t}}/\beta _{i} )_{i\in I}$.

1.5 Proof of Proposition 5

It follows from the fact that $\succsim _{\mathbf {u}^{t-1}}^{*}$ satisfies all the conditions in Proposition 12.

1.6 Proof of Theorem 2

For the generalized exponential class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{0}|\emptyset )= & {} -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda _{i}(\emptyset )\sum _{\tau =0}^{\infty }\beta _{i}^{\tau }u_{i\tau }\right) \\= & {} -\sum _{i\in I}a_{i}(\emptyset )\exp \left( -\lambda _{i}(\emptyset )\sum _{\tau =0}^{t-1}\beta _{i}^{\tau }u_{i\tau }\right) \exp \left( -\lambda _{i}(\emptyset )\beta _{i}^{t} \sum _{\tau =t}^{\infty }\beta _{i}^{\tau -t}u_{i\tau }\right) , \end{aligned}$$

which under dynamic consistency yields the same ranking over $\mathbf {u}_{t}$ as $\Phi (\mathbf {u}_{t}|\mathbf {u}^{t-1})= -\sum _{i\in I}a_{i}(\mathbf {u}^{t-1}) \exp \left( -\lambda _{i}(\mathbf {u}^{t-1})\sum _{\tau =t}^{\infty }\beta _{i}^{\tau -t}u_{i\tau }\right)$ does.

For the additive class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{0}|\emptyset )= & {} \sum _{i\in I}a_{i}(\emptyset ) \sum _{\tau =0}^{\infty }\beta _{i}^{\tau }u_{i\tau } \\= & {} \sum _{i\in I}a_{i}(\emptyset ) \sum _{\tau =0}^{t-1}\beta _{i}^{\tau }u_{i\tau } +\sum _{i\in I}a_{i}(\emptyset )\beta ^{t}_{i} \sum _{\tau =t}^{\infty }\beta _{i}^{\tau -t}u_{i\tau }, \end{aligned}$$

which under dynamic consistency yields the same ranking over $\mathbf {u}_{t}$ as $\Phi (\mathbf {u}_{t}|\mathbf {u}^{t-1})= \sum _{i\in I}a_{i}(\mathbf {u}^{t-1}) \sum _{\tau =t}^{\infty }\beta _{i}^{\tau -t}u_{i\tau }$ does.

Appendix 3: Proofs for Sect. 3

1.1 Proof of Proposition 6

Order obvious.

EU-Pareto let $\int _{E_{t+1}}u_{i}(s)p(ds|E_{t+1}) \ge \int _{E_{t+1}}v_{i}(s)p(ds|E_{t+1})$ for all $i\in I$ for all $i\in I$. Then we have

$$\begin{aligned}&\int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +\int _{E_{t+1}}u_{i}(s)p(ds|E_{t}) \\&\quad \ge \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + \int _{E_{t+1}}v_{i}(s)p(ds|E_{t}) \end{aligned}$$

for all $i\in I$. Since EU-Pareto is assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, we have $(\mathbf {u}_{E_{t+1}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} ) \succsim _{\mathbf {u}_{-E_{t}}} (\mathbf {v}_{E_{t+1}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} )$. By dynamic consistency, $\mathbf {u}_{E_{t+1}} \succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} } \mathbf {v}_{E_{t+1}}$. The strict case is proved similarly.

EU-continuity let $\{U_{E_{t+1}}^{\nu }\}$ be a sequence in $\mathbb {R}^{I}$ converging to $U_{E_{t+1}}$, and let $\{V_{E_{t+1}}^{\nu }\}$ be a sequence in $\mathbb {R}^{I}$ converging to $V_{E_{t+1}}$. Suppose $U_{E_{t+1}}^{\nu } \succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}^{\nu }$ for all $\nu$. By dynamic consistency, we have

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{i,E_{t+1}}^{\nu } \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})V_{i,E_{t+1}}^{\nu }\right) _{i\in I} \end{aligned}$$

for all $\nu$. Since EU-continuity was assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, we have

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{i,E_{t+1}} \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})V_{i,E_{t+1}} \right) _{i\in I}. \end{aligned}$$

By dynamic consistency, $U_{E_{t+1}}\succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}$.

EU-inequality aversion let $U_{E_{t+1}}\sim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}$. By dynamic consistency, we have

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{i,E_{t+1}} \right) _{i\in I} \\&\quad \sim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})V_{i,E_{t+1}} \right) _{i\in I}. \end{aligned}$$

Since EU-inequality aversion was assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, we have

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) (cU_{i,E_{t+1}}+(1-c)V_{i,E_{t+1}}) \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})V_{i,E_{t+1}} \right) _{i\in I}. \end{aligned}$$

By dynamic consistency, $cU_{E_{t+1}}+(1-c)V_{E_{t+1}} \succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} U_{E_{t+1}}$.

EU-separability let $(U_{J,E_{t+1}},U_{-J,E_{t+1}})\succsim ^{*}_{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} } (V_{J,E_{t+1}},U_{-J,E_{t+1}})$. By dynamic consistency, this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{J,E_{t+1}}, \right. \\&\quad \quad \left. \int _{E_{t}{\setminus } E_{t+1}}u_{-J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{-J,E_{t+1}}\right) \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) V_{J,E_{t+1}}, \right. \\&\quad \quad \left. \int _{E_{t}{\setminus } E_{t+1}}u_{-J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{-J,E_{t+1}}\right) . \end{aligned}$$

Since separability was assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{J,E_{t+1}}, \right. \\&\quad \quad \left. \int _{E_{t}{\setminus } E_{t+1}}u_{-J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) V_{-J,E_{t+1}}\right) \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) V_{J,E_{t+1}}, \right. \\&\quad \quad \left. \int _{E_{t}{\setminus } E_{t+1}}u_{-J}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) V_{-J,E_{t+1}}\right) \end{aligned}$$

By dynamic consistency, this holds if and only if $(U_{J,E_{t+1}},V_{-J,E_{t+1}})\succsim ^{*}_{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} } (V_{J,E_{t+1}},V_{-J,E_{t+1}})$ .

EU-shift covariance let $U_{E_{t+1}}\succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}$. By dynamic consistency, this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) U_{i,E_{t+1}} \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})V_{i,E_{t+1}} \right) _{i\in I}. \end{aligned}$$

Since EU-shift covariance was assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) +p(E_{t+1}|E_{t}) (U_{i,E_{t+1}}+c) \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p(ds|E_{t}) + p(E_{t+1}|E_{t})(V_{i,E_{t+1}}+c) \right) _{i\in I}. \end{aligned}$$

By dynamic consistency, this holds if and only if $U_{E_{t+1}}+c\mathbf {1} \succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}+c\mathbf {1}$.

1.2 Proof of Proposition 7

It follows from the fact that $\succsim _{\mathbf {u}_{-E_{t}}}^{*}$ satisfies all the conditions in Proposition 11.

1.3 Proof of Theorem 3

Note that for the exponential class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{\Omega }|\emptyset )= & {} -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda (\emptyset )\int _{\Omega }u_{i}(s)p(ds)\right) \\= & {} -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda (\emptyset ) \int _{\Omega {\setminus } E_{t}}u_{i}(s)p(ds)\right) \\&\times \exp \left( -\lambda (\emptyset ) p(E_{t} )\int _{E_{t}}u_{i}(s)p(ds|E_{t})\right) , \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{E_{t}}$ as $\Phi (\mathbf {u}_{E_{t}}|\mathbf {u}_{-E_{t}})= -\sum _{i\in I}a_{i}(\mathbf {u}_{-E_{t}}) \exp \big (-\lambda (\mathbf {u}_{-E_{t}})\int _{E_{t}}u_{i}(s)p(ds|E_{t})\big )$ does.

For the additive class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{\Omega }|\emptyset )= & {} \sum _{i\in I}a_{i}(\emptyset ) \int _{\Omega }u_{i}(s)p(ds)\\= & {} \sum _{i\in I}a_{i}(\emptyset ) \int _{\Omega {\setminus } E_{t}}u_{i}(s)p(ds) + p(E_{t})\sum _{i\in I}a_{i}(\emptyset ) \int _{E_{t}}u_{i}(s)p(ds|E_{t}) , \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{E_{t}}$ as $\Phi (\mathbf {u}_{E_{t}}|\mathbf {u}_{-E_{t}})= \sum _{i\in I}a_{i}(\mathbf {u}_{-E_{t}}) \int _{E_{t}}u_{i}(s)p(ds|E_{t})$ does.

1.4 Proof of Proposition 9

We only prove that EU-general shift covariance is recurrent, since the rest is similar to the case of homogeneous beliefs.

Let $U_{E_{t+1}}\succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}$. By dynamic consistency this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p_{i}(ds|E_{t}) +p_{i}(E_{t+1}|E_{t}) U_{i,E_{t+1}} \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p_{i}(ds|E_{t}) + p_{i}(E_{t+1}|E_{t})V_{i,E_{t+1}} \right) _{i\in I}. \end{aligned}$$

Since EU-general shift covariance was assumed to hold for $\succsim _{\mathbf {u}_{-E_{t}}}$, for some $W_{\mathbf {u}_{-E_{t}}}$ this holds if and only if

$$\begin{aligned}&\left( \int _{E_{t}{\setminus } E_{t+1}}u_{i}(s)p_{i}(ds|E_{t}) +p_{i}(E_{t+1}|E_{t}) \left( U_{i,E_{t+1}}+c\frac{W_{\mathbf {u}_{-E_{t}}}}{p_{i}(E_{t+1}|E_{t})}\right) \right) _{i\in I} \\&\quad \succsim _{\mathbf {u}_{-E_{t}}}^{*} \left( \int _{E_{t}{\setminus } E_{t+1}}\!u_{i}(s)p_{i}(ds|E_{t}) \!+\! p_{i}(E_{t+1}|E_{t})\left( V_{i,E_{t+1}}+c\frac{W_{\mathbf {u}_{-E_{t}}}}{p_{i}(E_{t+1}|E_{t})}\right) \right) _{i\in I}. \end{aligned}$$

By dynamic consistency, this holds if and only if $U_{E_{t+1}}+cW_{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} } \succsim _{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }^{*} V_{E_{t+1}}+ cW_{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }$, where $W_{\mathbf {u}_{-E_{t}}, \mathbf {u}_{E_{t}{\setminus } E_{t+1}} }=\left( W_{i,\mathbf {u}_{-E_{t}}}/p_{i}(E_{t+1}|E_{t}) \right) _{i\in I}$.

1.5 Proof of Proposition 10

It follows from the fact that $\succsim _{\mathbf {u}_{-E_{t}}}^{*}$ satisfies all the conditions in Proposition 12.

1.6 Proof of Theorem 4

For the generalized exponential class, we have

$$\begin{aligned}&\Phi (\mathbf {u}_{\Omega }|\emptyset ) = -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda _{i}(\emptyset )\int _{\Omega }u_{i}(s)p_{i}(ds)\right) \\&\quad = -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda _{i}(\emptyset )\left( \int _{\Omega {\setminus } E_{t}}u_{i}(s)p_{i}(ds)+ p_{i}(E_{t})\int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t})\right) \right) \\&\quad = -\sum _{i\in I}a_{i}(\emptyset ) \exp \left( -\lambda _{i}(\emptyset ) \int _{\Omega {\setminus } E_{t}}u_{i}(s)p_{i}(ds)\right) \\&\quad \quad \times \exp \left( -\lambda _{i}(\emptyset ) p_{i}(E_{t} )\int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t})\right) , \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{E_{t}}$ as $\Phi (\mathbf {u}_{E_{t}}|\mathbf {u}_{-E_{t}})= -\sum _{i\in I}a_{i}(\mathbf {u}_{-E_{t}}) \exp \big (-\lambda _{i}(\mathbf {u}_{-E_{t}})\int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t})\big )$ does.

For the additive class, we have

$$\begin{aligned} \Phi (\mathbf {u}_{\Omega }|\emptyset )= & {} \sum _{i\in I}a_{i}(\emptyset ) \int _{\Omega }u_{i}(s)p_{i}(ds)\\= & {} \sum _{i\in I}a_{i}(\emptyset )\left( \int _{\Omega {\setminus } E_{t}}u_{i}(s)p_{i}(ds)+ p_{i}(E_{t})\int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t})\right) \\= & {} \sum _{i\in I}a_{i}(\emptyset ) \int _{\Omega {\setminus } E_{t}}u_{i}(s)p_{i}(ds) + \sum _{i\in I}a_{i}(\emptyset ) p_{i}(E_{t}) \int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t}) , \end{aligned}$$

which under dynamic consistency must yield the same ranking over $\mathbf {u}_{E_{t}}$ as $\Phi (\mathbf {u}_{E_{t}}|\mathbf {u}_{-E_{t}})= \sum _{i\in I}a_{i}(\mathbf {u}_{-E_{t}}) \int _{E_{t}}u_{i}(s)p_{i}(ds|E_{t})$ does.