No-betting Pareto under ambiguity

Abstract

It has been argued that Pareto-improving trade is not as compelling under uncertainty as it is under certainty. The former may involve agents with different beliefs, who might wish to execute trades that are no more than betting. In response, the concept of no-betting Pareto dominance was introduced, requiring that putative Pareto improvements must be rationalizable by some common probabilities, even though the participants’ beliefs may differ. In this paper, we argue that this definition might be too narrow for use when agents are not Bayesian. Agents who face ambiguity might wish to trade in ways that can be justified by common ambiguity, though not necessarily by common probabilities. We accordingly extend the notion of no-betting Pareto dominance to characterize trades than are “no-betting Pareto” ranked according to the maxmin expected utility model.

Appendix: Proofs

1.1 Proof of Theorem 1

We will show that the following eight statements, about two allocations, $x=\left( x_{i}^{k}\right) _{i\le n,k\le m}$ and $y=\left( y_{i}^{k}\right) _{i\le n,k\le m}$, are equivalent:

Statement 1: There exists a convex, closed set of probabilities on S, $C\subseteq \Delta ^{n-1}$ such that, for each $k\le m,$

$$\begin{aligned} \min _{p\in C}p\cdot x^{k}\le \min _{p\in C}p\cdot y^{k}. \end{aligned}$$

Statement 2: There are 2m probability vectors, $\left( p^{k}\right) _{k\le m},\left( q^{k}\right) _{k\le m}\in \Delta ^{n-1}$ such that, for each $k\le m$,

$$\begin{aligned} p^{k}\cdot x^{k}\le q^{k}\cdot y^{k} \end{aligned}$$

(2)

and, for all $k,l\le m$

$$\begin{aligned} p^{k}\cdot x^{k}\le & {} p^{l}\cdot x^{k},q^{l}\cdot x^{k} \nonumber \\ q^{k}\cdot y^{k}\le & {} p^{l}\cdot y^{k},q^{l}\cdot y^{k}. \end{aligned}$$

(3)

Statement 3: The following linear programming problem is feasible:

$$\begin{aligned} (P)\qquad Max_{\left( p_{i}^{k}\right) _{i\le n,k\le m},\left( q_{i}^{k}\right) _{i\le n,k\le m}}\sum _{k=1}^{m}\sum _{i=1}^{n}0\cdot p_{i}^{k}+\sum _{k=1}^{m}\sum _{i=1}^{n}0\cdot q_{i}^{k} \end{aligned}$$

subject to

$$\begin{aligned} \sum _{i=1}^{n}x_{i}^{k}\cdot p_{i}^{k}-\sum _{i=1}^{n}y_{i}^{k}\cdot q_{i}^{k}\le & {} 0\qquad \forall k\le m\qquad \text {(P1)} \\ \sum _{i=1}^{n}x_{i}^{k}\cdot p_{i}^{k}-\sum _{i=1}^{n}x_{i}^{k}\cdot p_{i}^{l}\le & {} 0\qquad \forall k,l\le m,\quad k\ne l\qquad \text {(P2)} \\ \sum _{i=1}^{n}x_{i}^{k}\cdot p_{i}^{k}-\sum _{i=1}^{n}x_{i}^{k}\cdot q_{i}^{l}\le & {} 0\qquad \forall k,l\le m\qquad \text {(P3)} \\ \sum _{i=1}^{n}y_{i}^{k}\cdot q_{i}^{k}-\sum _{i=1}^{n}y_{i}^{k}\cdot p_{i}^{l}\le & {} 0\qquad \forall k,l\le m\qquad \text {(P4)} \\ \sum _{i=1}^{n}y_{i}^{k}\cdot q_{i}^{k}-\sum _{i=1}^{n}y_{i}^{k}\cdot q_{i}^{l}\le & {} 0\qquad \forall k,l\le m,\quad k\ne l\qquad \text {(P5)} \\ \sum _{i=1}^{n}p_{i}^{k}= & {} 1\qquad \forall k\le m\qquad \text {(P6)} \\ \sum _{i=1}^{n}q_{i}^{k}= & {} 1\qquad \forall k\le m\qquad \text {(P7)} \\ p_{i}^{k},q_{i}^{k}\ge & {} 0\qquad \forall i\le n,k\le m. \end{aligned}$$

Statement 4: The following linear programming problem is bounded:

$$\begin{aligned}&(D)\qquad Min_{\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right)_{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}} \\&\qquad \qquad \qquad \sum _{k=1}^{m}0\cdot \lambda ^{k}+\sum _{k=1}^{m}\sum\limits_{\begin{subarray}{c} l=1 \\ l\ne k\end{subarray}}^{m} 0\cdot \alpha ^{kl}+\sum _{k=1}^{m}\sum _{l=1}^{m}0\cdot \beta ^{kl}+ \\&\qquad \qquad \qquad \sum _{k=1}^{m}\sum _{l=1}^{m}0\cdot \gamma ^{kl}+\sum _{k=1}^{m}\sum\limits_{\begin{subarray}{c} l=1 \\ l\ne k\end{subarray}}^{m} 0\cdot \delta ^{kl}+\sum _{k=1}^{m}\mu ^{k}+\sum _{k=1}^{m}\nu ^{k} \\ \end{aligned}$$

subject to

$$\begin{aligned} &x_{i}^{k}\lambda ^{k}+x_{i}^{k}\left[{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\alpha ^{kl}+\sum _{l=1}^{m}\beta ^{kl}\right] -{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}x_{i}^{l}\alpha ^{lk}-\sum _{l=1}^{m}y_{i}^{l}\gamma ^{lk}+\mu ^{k}\ge 0\qquad \forall i\le n,k\le m\qquad \text {(D1)} \\ &-y_{i}^{k}\lambda ^{k}+y_{i}^{k}\left[ \sum _{l=1}^{m}\gamma ^{kl}+{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\delta ^{kl}\right] - \sum _{l=1}^{m}x_{i}^{l}\beta ^{lk}-{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}y_{i}^{l}\delta ^{lk}+\nu ^{k}\ge 0\qquad \forall i\le n,k\le m\qquad \text {(D2)} \\ &\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}\ge 0. \end{aligned}$$

Statement 5: For every $\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}} ,\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}\ge 0$ and every real numbers $\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}$,

IF

$$\begin{aligned} x_{i}^{k}\lambda ^{k}+x_{i}^{k}\left[ {\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\alpha ^{kl}+\sum _{l=1}^{m}\beta ^{kl}\right] -{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}x_{i}^{l}\alpha ^{lk}-\sum _{l=1}^{m}y_{i}^{l}\gamma ^{lk}+\mu ^{k}\ge & {} 0\qquad \forall i\le n,k\le m \\ -y_{i}^{k}\lambda ^{k}+y_{i}^{k}\left[ \sum _{l=1}^{m}\gamma ^{kl}+{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\delta ^{kl}\right] - \sum _{l=1}^{m}x_{i}^{l}\beta ^{lk}-{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}y_{i}^{l}\delta ^{lk}+\nu ^{k}\ge & {} 0\qquad \forall i\le n,k\le m \end{aligned}$$

THEN

$$\begin{aligned} \sum _{k=1}^{m}\mu ^{k}+\sum _{k=1}^{m}\nu ^{k}\ge 0. \end{aligned}$$

Statement 6: For every $\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}} ,\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}\ge 0$, we have

$$\begin{aligned} \sum _{k=1}^{m}\mu ^{k}+\sum _{k=1}^{m}\nu ^{k}\ge 0 \end{aligned}$$

where

$$\begin{aligned} \mu ^{k}\equiv & {} \max _{1\le i\le n}\left[ -x_{i}^{k}\lambda ^{k}-x_{i}^{k}\left({\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\alpha ^{kl}+\sum _{l=1}^{m}\beta ^{kl}\right) +{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}x_{i}^{l}\alpha ^{lk}+\sum _{l=1}^{m}y_{i}^{l}\gamma ^{lk}\right] \nonumber \\ \nu ^{k}\equiv & {} \max _{1\le i\le n}\left[ y_{i}^{k}\lambda ^{k}-y_{i}^{k}\left( \sum _{l=1}^{m}\gamma ^{kl}+{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}\delta ^{kl}\right) +\sum _{l=1}^{m}x_{i}^{l}\beta ^{lk}+{\mathop{\mathop{\mathop{\sum}\limits_{l=1 }}\limits_{ l\ne k}}\limits^{m}}y_{i}^{l}\delta ^{lk}\right] \end{aligned}$$

(4)

for every $i\le n,k\le m$.

Statement 7: For every $\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}} ,\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}\ge 0$, for every $k\le m$ there exist $i_{k},j_{k}\le n$ such that

$$\begin{aligned} \sum _{k=1}^{m}\left[ \begin{array}{c} -x_{i_{k}}^{k}\lambda ^{k}-x_{i_{k}}^{k}\left( \sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\alpha ^{kl}+\sum _{l=1}^{m}\beta ^{kl}\right) +\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}x_{i_{k}}^{l}\alpha ^{lk}+\sum _{l=1}^{m}y_{i_{k}}^{l}\gamma ^{lk}+ \\ y_{j_{k}}^{k}\lambda ^{k}-y_{j_{k}}^{k}\left( \sum _{l=1}^{m}\gamma ^{kl}+\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\delta ^{kl}\right) +\sum _{l=1}^{m}x_{j_{k}}^{l}\beta ^{lk}+\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}} ^{m}y_{j_{k}}^{l}\delta ^{lk} \end{array} \right] \ge 0. \end{aligned}$$

Statement 8: For every $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{k,l\le m},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{k,l\le m}\ge 0$, for every $k\le m$ there exist $i_{k},j_{k}\le n$ such that

$$\begin{aligned} \sum _{k=1}^{m} \begin{array}{c} \lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \end{array} \ge \sum _{k=1}^{m}\left[ \begin{array}{c} \sum _{l=1}^{m}\alpha ^{kl}\left( x_{i_{k}}^{k}-x_{i_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\beta ^{kl}\left( x_{i_{k}}^{k}-x_{j_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\gamma ^{kl}\left( y_{j_{k}}^{k}-y_{i_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\delta ^{kl}\left( y_{j_{k}}^{k}-y_{j_{l}}^{k}\right) \end{array} \right] \,. \end{aligned}$$

Proof that Statements (1) and (2) are equivalent:

Assume first that Statement (1) holds for some set C. For each k, choose $p^{k},q^{k}\in C$ such that

$$\begin{aligned} \min _{p\in C}p\cdot x^{k}=\, & {} p^{k}\cdot x^{k} \\ \min _{p\in C}p\cdot y^{k}=\, & {} q^{k}\cdot y^{k}. \end{aligned}$$

Since $p^{k}\in \arg \min _{p\in C}p\cdot x^{k}$, and $q^{k}\in \arg \min _{p\in C}p\cdot y^{k}$, (3) follows. And Statement (1) guarantees that the minimal payoff for x is bounded above by the minimal payoff for y, that is, that (2) holds as well.

Conversely, assume that Statement (2) holds. Define

$$\begin{aligned} C=conv\left[ \left\{ p^{k}\right\} _{k\le m}\cup \left\{ q^{k}\right\} _{k\le m}\right] \end{aligned}$$

and observe that it satisfies the requirement of Statement (1).

Proof that Statements (2) and (3) are equivalent:

The feasible set of (P) in Statement (3) is a restatement of Statement (2), with (i) constraints (P1) corresponding to (2), (ii) constraints (P2) and (P3) guaranteeing the first set of inequalities in (3), having to do with $p^{k}\in \arg \min _{p\in C}p\cdot x^{k}$ , and (iii) $q^{k}\in \arg \min _{p\in C}p\cdot y^{k}$ being equivalent to constraints (P4) and (P5). Finally, constraints (P6) and (P7), combined with the nonnegativity constraints, imply that $p^{k}$ and $q^{k}$ are indeed probability vectors for each k.

Proof that Statements (3) and (4) are equivalent:

(D) is the dual problem of (P), with $\left( \lambda ^{k}\right) _{k\le m}$ corresponding to the inequalities (P1), and $\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}$ corresponding to the sets of inequalities (P2),(P3),(P4), and (P5), respectively, and $\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}$ corresponding to the sets of equalities (P6) and (P7), respectively. (P) is a Max problem, and so (D) is a Min problem. The sets of inequalities (P1)-(P5) have the natural direction, and thus the corresponding variables $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}$ are constrained to be nonnegative, while the equalities (P6) and (P7) leave $\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}$ unconstrained. The constraints of (D) denoted (D1) correspond to $\left( p_{i}^{k}\right)$, while those denoted (D2) correspond to $\left( q_{i}^{k}\right)$. As both $\left( p_{i}^{k}\right)$ and $\left( q_{i}^{k}\right)$ are sets of nonnegative variables in (P), the direction of the inequalities in (D) is the natural one, namely $\ge$.

Note that the objective function in (P) was set of be identically zero, and thus all right sides in (D) are zero and the problem is homogenous. (Note that its feasible set includes the origin and it is a convex cone.) As the right side in (P1)-(P5) (in (P)) are zero, the variables $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}$ have zero coefficients in the objective function of (D), so that the latter is

$$\begin{aligned} Min_{\left( \lambda ^{k}\right) _{k\le m},\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},\left( \beta ^{kl}\right) _{k,l\le m},\left( \gamma ^{kl}\right) _{k,l\le m},\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}}\sum _{k=1}^{m}\mu ^{k}+\sum _{k=1}^{m}\nu ^{k}. \end{aligned}$$

Proof that Statements (4) and (5) are equivalent:

As Problem (D) is homogeneous, it is bounded if and only if it is bounded below by 0. This means that for every set of variables in the feasible set (i.e., that satisfy the constraints (D1) and (D2) and the relevant nonnegativity constraints), the objective function is nonnegative.

Proof that Statements (5) and (6) are equivalent:

For every $k\le m$, $\mu ^{k}$ is bounded below by n constraints in (D1), while $\nu ^{k}$ is bounded below by n constraints in (D2). Apart from these constraints, $\left( \mu ^{k}\right) _{k\le m},\left( \nu ^{k}\right) _{k\le m}$ are unconstrained, and therefore the inequality $\sum _{k=1}^{m}\mu ^{k}+\sum _{k=1}^{m}\nu ^{k}\ge 0$ would hold if and only if it holds for their minimal values. These minimal values are the maxima of the lower bounds, as given in (4).

Proof that Statements (6) and (7) are equivalent:

If Statement (6) holds, one can choose $i_{k}$ and $j_{k}$ to be in the argmax of the expressions defining $\mu ^{k}$ and $\nu ^{k}$, respectively, to obtain the conclusion of Statement (7). Conversely, if Statement (7) holds and such $i_{k}$ and $j_{k}$ exist, the summation of the maxima is obviously nonnegative as well.

Proof that Statements (7) and (8) are equivalent:

Statement (7) can also be written as: For every $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{\begin{subarray}{c} k,l\le m \\ k\ne l \end{subarray}}\ge 0$, for every $k\le m$ there exist $i_{k},j_{k}\le n$ such that

$$\begin{aligned} \sum _{k=1}^{m}\left[ \begin{array}{c} \lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \\ -\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\alpha ^{kl}x_{i_{k}}^{k}+\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\alpha ^{lk}x_{i_{k}}^{l} \\ -\sum _{l=1}^{m}\beta ^{kl}x_{i_{k}}^{k}+\sum _{l=1}^{m}\beta ^{lk}x_{j_{k}}^{l} \\ -\sum _{l=1}^{m}\gamma ^{kl}y_{j_{k}}^{k}+\sum _{l=1}^{m}\gamma ^{lk}y_{i_{k}}^{l} \\ -\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\delta ^{kl}y_{j_{k}}^{k}+\sum _{\begin{subarray}{c} l=1 \\ l\ne k \end{subarray}}^{m}\delta ^{lk}y_{j_{k}}^{l} \end{array} \right] \ge 0 \end{aligned}$$

or – allowing for arbitrary $\left( \alpha ^{kk},\delta ^{kk}\right) _{k}$ –

$$\begin{aligned} \sum _{k=1}^{m} \begin{array}{c} \lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \end{array} \ge \sum _{k=1}^{m}\left[ \begin{array}{c} \sum _{l=1}^{m}\alpha ^{kl}x_{i_{k}}^{k}-\sum _{l=1}^{m}\alpha ^{lk}x_{i_{k}}^{l} \\ +\sum _{l=1}^{m}\beta ^{kl}x_{i_{k}}^{k}-\sum _{l=1}^{m}\beta ^{lk}x_{j_{k}}^{l} \\ +\sum _{l=1}^{m}\gamma ^{kl}y_{j_{k}}^{k}-\sum _{l=1}^{m}\gamma ^{lk}y_{i_{k}}^{l} \\ +\sum _{l=1}^{m}\delta ^{kl}y_{j_{k}}^{k}-\sum _{l=1}^{m}\delta ^{lk}y_{j_{k}}^{l} \end{array} \right] \,\,. \end{aligned}$$

By rearranging the terms (so that $\alpha ^{lk}$ appears in the l’th row), we obtain the equivalent inequality

$$\begin{aligned} \sum _{k=1}^{m} \begin{array}{c} \lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \end{array} \ge \sum _{k=1}^{m}\left[ \begin{array}{c} \sum _{l=1}^{m}\alpha ^{kl}x_{i_{k}}^{k}-\sum _{l=1}^{m}\alpha ^{kl}x_{i_{l}}^{k} \\ +\sum _{l=1}^{m}\beta ^{kl}x_{i_{k}}^{k}-\sum _{l=1}^{m}\beta ^{kl}x_{j_{l}}^{k} \\ +\sum _{l=1}^{m}\gamma ^{kl}y_{j_{k}}^{k}-\sum _{l=1}^{m}\gamma ^{kl}y_{i_{l}}^{k} \\ +\sum _{l=1}^{m}\delta ^{kl}y_{j_{k}}^{k}-\sum _{l=1}^{m}\delta ^{kl}y_{j_{l}}^{k} \end{array} \right] \end{aligned}$$

or

$$\begin{aligned} \sum _{k=1}^{m} \begin{array}{c} \lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \end{array} \ge \sum _{k=1}^{m}\left[ \begin{array}{c} \sum _{l=1}^{m}\alpha ^{kl}\left( x_{i_{k}}^{k}-x_{i_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\beta ^{kl}\left( x_{i_{k}}^{k}-x_{j_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\gamma ^{kl}\left( y_{j_{k}}^{k}-y_{i_{l}}^{k}\right) \\ +\sum _{l=1}^{m}\delta ^{kl}\left( y_{j_{k}}^{k}-y_{j_{l}}^{k}\right) \end{array} \right] \,\,. \end{aligned}$$

1.2 Proof of Remark 1

Assume indeed that there is but a single agent, $m=1$. We first observe that Condition (B) boils down to

• Condition B(1): For all $\lambda ,\beta ,\gamma \ge 0$, there exists states i and j with

  $$\begin{aligned} \lambda (y_{j}-x_{i})\ge \beta (x_{i}-x_{j})+\gamma (y_{j}-y_{i}). \end{aligned}$$

To see this, consider the right side of Condition (B). As there is only one individual, $k=l=1$ and it follows that $i_{k}$ must equal $i_{l}$ and, similarly, $j_{k}=j_{l}$. Hence the expressions involving $\alpha$ and $\delta$ vanish. Condition (B) then becomes Condition B(1).

We now wish to show that (when $m=1$), Condition B(1) holds iff there exists a state $\ell$ with $y_{\ell }\ge x_{\ell }$. First assume that such an $\ell$ exists. To see that Condition B(1) holds, let there be given $\lambda ,\beta ,\gamma \ge 0$ and choose $i=j=\ell$ so that the inequality above becomes $\lambda (y_{\ell }-x_{\ell })\ge 0$ which indeed holds.

Conversely, if Condition B(1) holds, apply it to $\lambda =\gamma =1$ and $\beta =0$ to obtain i and j such that $y_{j}-x_{i}\ge y_{j}-y_{i}$ or $y_{i}\ge x_{i}$ (so that the condition holds for $\ell =i$).

1.3 Proof of Remark 3

Assume that Condition (GSS’) holds. Given $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{k,l\le m},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{k,l\le m}\ge 0$, use Condition (GSS’) for $\left( \lambda ^{k}\right) _{k\le m}\ge 0$ to find a state $\ell$ such that $\sum _{k=1}^{m}\lambda ^{k}(y_{\ell }^{k}-x_{\ell }^{k})\ge 0$. Setting, for each k, $i_{k},j_{k}=\ell$, the inequality (1) boils down to $\sum _{k=1}^{m}\lambda ^{k}(y_{\ell }^{k}-x_{\ell }^{k})\ge 0$, so that Condition (B) holds.

1.4 Proof of Remark 4

We already established that Condition (GSS’) implies Condition (B), regardless of whether $\left( x^{k}\right) _{k}$ are constant (across states for each agent). To complete the proof we need to show that, if $\left( x^{k}\right) _{k}$ are indeed constant (that is, $x_{i}^{k}=x_{j}^{k}$ $\forall i,j,k$) the converse also holds. Let x be such an allocation, and let there be given $\left( \lambda ^{k}\right) _{k\le m}\ge 0$ as in Condition (GSS’). Set all of $\left( \alpha ^{kl}\right) _{k,l\le m},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{k,l\le m}$, apart from $\left( \delta ^{k1}\right) _{k\le m}$, to zero, and $\delta ^{k1}=\lambda ^{k}$ for $k\le m$. Apply Condition (B) to $\left( \lambda ^{k}\right) _{k\le m},$ $\left( \alpha ^{kl}\right) _{k,l\le m},$ $\left( \beta ^{kl}\right) _{k,l\le m},$ $\left( \gamma ^{kl}\right) _{k,l\le m},$ $\left( \delta ^{kl}\right) _{k,l\le m}$ (with the original $\left( \lambda ^{k}\right) _{k\le m}$ given in the antecedent of Condition (GSS’)). Inequality (1) then implies

$$\begin{aligned} \sum _{k=1}^{m}\lambda ^{k}\left( y_{j_{k}}^{k}-x_{i_{k}}^{k}\right) \ge \sum _{k=1}^{m}\lambda ^{k}\left( y_{j_{k}}^{k}-y_{j_{1}}^{k}\right) \end{aligned}$$

or $\sum _{k=1}^{m}\lambda ^{k}\left( y_{j_{1}}^{k}-x_{i_{k}}^{k}\right) \ge 0$. However, as $x^{k}$ is constant across states, we have, in particular, $x_{i_{k}}^{k}=x_{j_{1}}^{k}$ and we can set $\ell =j_{1}$ to obtain the desired result.