# Negatively dependent optimal risk sharing

Jean-Gabriel Lauzier Department of Statistics and Actuarial Science,
University of Waterloo, Canada. ✉<EMAIL_ADDRESS>Liyuan Lin Department
of Statistics and Actuarial Science, University of Waterloo, Canada. ✉
<EMAIL_ADDRESS>Ruodu Wang Department of Statistics and Actuarial
Science, University of Waterloo, Canada. ✉<EMAIL_ADDRESS>
###### Abstract

We analyze the problem of optimally sharing risk using allocations that
exhibit counter-monotonicity, the most extreme form of negative dependence.
Counter-monotonic allocations take the form of either “winner-takes-all”
lotteries or “loser-loses-all” lotteries, and we respectively refer to these
(normalized) cases as jackpot or scapegoat allocations. Our main theorem, the
counter-monotonic improvement theorem, states that for a given set of random
variables that are either all bounded from below or all bounded from above,
one can always find a set of counter-monotonic random variables such that each
component is greater or equal than its counterpart in the convex order. We
show that Pareto optimal allocations, if they exist, must be jackpot
allocations when all agents are risk seeking. We essentially obtain the
opposite when all agents have discontinuous Bernoulli utility functions, as
scapegoat allocations maximize the probability of being above the
discontinuity threshold. We also consider the case of rank-dependent expected
utility (RDU) agents and find conditions which guarantee that RDU agents
prefer jackpot allocations. We provide an application for the mining of
cryptocurrencies and show that in contrast to risk-averse miners, RDU miners
with small computing power never join a mining pool. Finally, we characterize
the competitive equilibria with risk-seeking agents, providing a first and
second fundamental theorem of welfare economics where all equilibrium
allocations are jackpot allocations.

Keywords: Pareto optimality, Risk sharing, Counter-monotonicity, Risk seeking,
Rank-dependent expected utility, Cryptocurrency mining pools

##  1 Introduction

The problem of sharing risk and its mathematical underpinnings are pivotal in
understanding the economic behaviours of agents. When agents are risk-averse
expected utility maximizers, the risk sharing problem behaves similarly to the
general equilibrium of an exchange economy with aggregate risks (Arrow and
Debreu, 1954; Arrow, 1964; Radner, 1968). An important observation from this
literature is that, under strict risk aversion, Pareto-optimal allocations are
comonotonic, i.e., they are increasing functions of the total wealth. This can
be interpreted as agents being “on the same boat” when losses or gains occur.

As comonotonicity is an extreme form of positive dependence, one might wonder
if a converse statement exists for risk-seeking agents, i.e., the agents in an
optimal allocation being “in opposite boats” when losses or gains occur.
Unfortunately, it is well known that the most extreme form of negative
dependence is generally not tractable with three or more random variables, and
thus, the question is technically very challenging and not well understood.

This article addresses this gap by investigating negative dependence in risk
sharing, and in particular, the extreme form of negative dependence, which we
refer to as counter-monotonicity. To establish comonotonic optimal allocations
in the classic literature, a central mathematical tool is the _comonotonic
improvement theorem_ of Landsberger and Meilijson (1994), which states that
for any random vector, there exists a comonotonic random vector whose
components are _less risky_ than those of the given random vector, in the
sense of Rothschild and Stiglitz (1970). This establishes the important
intuition that risk-averse agents always prefer comonotonic allocations.

Parallel to this classic finding, our main result, Theorem 1, referred to as
the _counter-monotonic improvement theorem_ , states that for any random
vector bounded from below (or above), there exists a counter-monotonic random
vector whose components are _riskier_ than those of the given random vector.
The counter-monotonic improvement theorem uses the stochastic representation
of counter-monotonicity recently obtained by Lauzier et al. (2023a). In
Proposition 2, we provide a simplification of this stochastic representation
that makes transparent that any counter-monotonic allocation resembles extreme
forms of gambling as either “winner-takes-all” or “loser-loses-all” (drawing
straws) lotteries. We respectively define the normalized version of “winner-
takes-all” and “loser-loses-all” allocations as _jackpot_ and _scapegoat_
allocations.

To appreciate the optimality of the jackpot and the scapegoat allocations, we
need to depart from the standard utility theory of risk-averse agents. An
immediate consequence of our main theorem is that for the problem of sharing
risk among strictly risk-seeking agents, all Pareto-optimal allocations are
jackpot allocations. However, this set may be empty in some situation, and
this can be resolved by imposing constraints on the allocations or restricting
the effective domain of their Bernoulli utility function.

To understand the role of scapegoat allocations, we then analyze the problem
of sharing risk among agents that have the same discontinuous utility
function. We first consider the problem of sharing risk among Dirac utility
agents, defined as expected utility maximizers for which the Bernoulli utility
function is an indicator function. A key property of this problem is that
allocations which give a constant endowment to all agents but one are always
Pareto optimal. The choice of whose allocation varies can be random, as if all
agents were “drawing straws”. The optimal allocations must thus be (payoff
equivalent to) scapegoat allocations when the endowment is probabilistically
too small. In this case, Pareto-optimal allocations cannot be simultaneously
comonotonic and fair, where we define fairness as all agents having the same
expected utility.111Fair allocations maximize the Rawlsian social welfare
function in this particular setup. We show that a similar result holds for
agents with piecewise linear Bernoulli utility function with one jump,
demonstrating that this result does not rely on the satiation of the
underlying preference relation.

We proceed to consider agents modelled by rank-dependent expected utility
(RDU) of Quiggin (1993), with a particular focus on agents with inverted
S-shaped probability distortions as in the cumulative prospect theory of
Tversky and Kahneman (1992). RDU agents with inverted S-shaped distortion can
exhibit a combination of risk-averse and risk-seeking behaviours. Assuming
that all agents are modelled by the same RDU, we find conditions where these
agents prefer fair jackpot allocations to any other fair allocations. We show
that if the number of agents is large, then only jackpot allocations can be
both fair and Pareto optimal; as a consequence, comonotonic and fair
allocations cannot be Pareto optimal.

We conclude with a simplified game-theoretical model of cryptocurrency mining
where agents can choose to form a mining pool. Leshno and Strack (2020)
already observed that risk-averse agents have an incentive to form mining
pools because it allows them to reduce the variability of their payoff.
Clearly, the payoff of joining the pool is a mean-preserving contraction of
the payoff for mining alone when at least another agent joins the pool.
Joining the pool is thus a weakly dominant strategy for risk-averse agents,
with strict dominance if at least one other agent joins. However, RDU agents
can behave the opposite depending on the size of their computing power. RDU
agents with large computing power behave as risk-averse agents. But if their
computing power is small so that their probability of mining the coin is also
small, then mining alone can be a weakly dominant strategy, with strict
dominance whenever at least one agent joins the pool. These novel results
suggest that a richer model of pool formation is required to better understand
the interaction between crypto-miners.

It is natural to ask if and how the counter-monotone improvement theorem can
be used in other contexts than risk-sharing. We investigate competitive
equilibria with risk-seeking agents and obtain all results of typical interest
in general equilibrium, including the first and second fundamental theorem of
welfare economics. We also show that all jackpot allocations are Pareto
optimal, hereby extending the results of Section 4. The second welfare theorem
thus implies that all jackpot allocations are competitive equilibrium for some
initial endowments. The analysis of competitive equilibria is highly technical
and is thus relegated in Appendix A. We emphasize that we do not know if a
similar analysis can be performed for agents with other types of risk-seeking
decision criteria, as the construction of the equilibrium pricing measure is
quite delicate and tailored to the specific setting we study.

Next, we review the literature. Section 2 contains all preliminaries,
including a formal statement of the classic result of comonotonic improvement.
Section 3 states and proves our main result, the counter-monotonic improvement
theorem. This is also where we review the stochastic representation of
counter-monotonicity and define jackpot and scapegoat allocations. Sections 4,
5 and 6 consider respectively the risk sharing problem with risk-seeking
agents, agents with a discontinuous Bernoulli utility function and RDU agents.
Section 7 analyzes the choice of joining a crypto-currency mining pool, and
the conclusion discusses avenues for further research. Appendix A analyzes
competitive equilibria with risk-seeking agents, and Appendix B contains the
proofs.

###  1.1 Literature review

The technique of comonotonic improvement was initially introduced in
Landsberger and Meilijson (1994), and subsequently extended in Dana and
Meilijson (2003), Ludkovski and Rüschendorf (2008) and Carlier et al. (2012).
We refer to Rüschendorf (2013) for an up-to-date formal treatment and Section
2 for more details. For the formal treatment of counter-monotonicity, we refer
to Puccetti and Wang (2015) for a general overview and to Lauzier et al.
(2023a) for the stochastic representation of counter-monotonicity. In the
actuarial literature, counter-monotonicity in dimension greater than two is
also called mutual exclusivity; see Dhaene and Denuit (1999) and Cheung and Lo
(2014). See also our discussion in Section 3.

While the economics and finance literature does not always employ the terms
comonotonicity and comonotonic allocations, these concepts have long been a
subject of interest in these fields. Of direct relevance is the case of
economies with a constant aggregate endowment, where all allocations that are
constant across states are comonotonic. In this context, comonotonic
allocations are sometimes called “no-betting” or “risk-free allocations.” See,
for instance, in the literature on risk sharing under heterogeneous beliefs
and ambiguity: Billot et al. (2000), Rigotti et al. (2008), and Strzalecki and
Werner (2011). More recently, Beissner et al. (2023) analyzes no-betting
allocations on probability spaces with two RDU agents. Chateauneuf et al.
(2000) analyzes comonotonic allocations with aggregate risk when all agents
are ambiguity-averse Choquet expected utility maximizers.

In contrast to comonotonicity, the concept of counter-monotonicity received
much less attention in the economics and finance literature related to risk
sharing. A notable exception is quantile-based risk sharing. A key property of
counter-monotonic allocations and, more generally, of negatively dependent
allocations is their optimality in this setting. See Embrechts et al. (2018)
and Weber (2018) for quantile-based risk sharing problems on probability
spaces with Pareto-optimal counter-monotonic allocations and Embrechts et al.
(2020) for the case of heterogeneous beliefs. Lauzier et al. (2023b) contains
a risk sharing problem where the optimal allocations entail both positive and
negative dependence. Specifically, the authors show that when sharing risk
with agents that consider the inter-quantile difference as their measure of
variability, any Pareto-optimal allocation entails counter-monotonicity on the
tails of the distribution of the aggregate risk.

We refer to Quiggin (1993) for RDU agents, although we are mostly interested
in the inverted S-shaped probability distortion functions considered in the
cumulative prospect theory of Tversky and Kahneman (1992) (and Kahneman and
Tversky (1979)). We show that RDU agents using the Kahnmenan-Tversky inverted
S-shaped distortion function can behave as risk-seeking agents provided the
risk is to be shared among a large number of individuals.

Our crypto-currency example is inspired by the axiomatic characterization of
Leshno and Strack (2020), which obtains an impossibility result for risk-
averse miners. In a nutshell, mining pools can provide a reward scheme that is
a mean-preserving contraction of the payoff of individual mining. Risk-averse
agents always prefer to mine in a pool. On the opposite, we find conditions
for which RDU agents behave as risk-seeking agents and prefer to mine alone
despite having a concave Bernoulli utility function.

##  2 Preliminaries: risk sharing and comonotonic improvements

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$, and denote by
$\mathcal{X}$ a corresponding $L^{p}$ space, where almost surely equal objects
are treated as equal. While Sections 2 and 3 consider $\mathcal{X}=L^{1}$ for
generality, Sections 4, 5 and 6 consider the more standard setting
$\mathcal{X}=L^{\infty}$, the set of all bounded random variables. Let $n$ be
a positive integer and write $[n]:=\\{1,\dots,n\\}$. We are mainly interested
in the situation where $n\geq 3$ agents share a random outcome
$X\in\mathcal{X}$.

######  Definition 1.

An _allocation_ of $X\in\mathcal{X}$ is an element of the set

$\mathbb{A}_{n}(X):=\left\\{(X_{1},\dots,X_{n})\in\mathcal{X}^{n}:\sum_{i=1}^{n}X_{i}=X\right\\}.$

A foundational idea of risk sharing is that if all agents are strictly risk
averse and know the probability measure $\mathbb{P}$, then all Pareto-optimal
allocations are comonotonic (see e.g., Rüschendorf (2013)). At a formal level,
this is typically proved using the technique of comonotonic improvements, as
introduced in Landsberger and Meilijson (1994) for a finite state space. The
result was subsequently extended to $L^{\infty}$ (Dana and Meilijson, 2003)
and $L^{1}$ (Ludkovski and Rüschendorf, 2008). We provide some background to
understand the scope of the comonotonic improvement technique.

The two random variables $X,Y$ are said to be _comonotonic_ if

$\displaystyle(X(\omega)-X(\omega^{\prime}))(Y(\omega)-Y(\omega^{\prime}))\geq
0~{}~{}\mbox{for $(\mathbb{P}\times\mathbb{P})$-almost
every~{}}(\omega,\omega^{\prime})\in\Omega^{2},$

and the collection of random variables $X_{1},\dots,X_{n}$ is comonotonic if
all its component are pairwise comonotonic. Alternatively, the random
variables $X_{1},\dots,X_{n}$ are comonotonic if there exists a collection of
increasing functions $f_{i}:\mathbb{R}\rightarrow\mathbb{R}$, $i\in[n]$, and a
random variable $Z$ such that $X_{i}=f_{i}(Z)$ for all $i\in[n]$ (recall that
equalities are in the $\mathbb{P}$-almost sure sense). The latter definition
comes from the stochastic representation of comonotonicity given by
Denneberg’s Lemma (see Denneberg, 1994, Proposition 4.5), and if
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$, then one can set $Z=X$ in the
preceding definition.

A random variable $X$ is said to be smaller than a random variable $Y$ in the
convex order, denoted by $X\leq_{\rm{cx}}Y$, if
$\mathbb{E}[\phi(X)]\leq\mathbb{E}[\phi(Y)]$ for every convex function
$\phi:\mathbb{R}\to\mathbb{R}$ provided that both expectations exist (see
Rüschendorf (2013) and Shaked and Shanthikumar (2007)). The order
$X\leq_{\rm{cx}}Y$ means that $X$ is less risky than $Y$ in the sense of
Rothschild and Stiglitz (1970). Notice that if $X\leq_{\rm{cx}}Y$, then
$\mathbb{E}[X]=\mathbb{E}[Y]$, meaning that the convex order compares random
variable with the same mean. Similarly, $X$ is smaller than $Y$ in the
increasing convex order, denoted by $X\leq_{\rm icx}Y$, if
$\mathbb{E}[\phi(X)]\leq\mathbb{E}[\phi(Y)]$ for every increasing convex
function $\phi:\mathbb{R}\to\mathbb{R}$ provided the expectations exist. The
proof of the next proposition is in (Rüschendorf, 2013, Theorem 10.50).

######  Proposition 1 (Comonotonic improvements).

Let $X_{1},\dots,X_{n}\in L^{1}$ and $X=\sum_{i=1}^{n}X_{i}$. Then there
exists a $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that (i)
$(Y_{1},\dots,Y_{n})$ is comonotonic and (ii) for every $i\in[n]$ it is
$Y_{i}\leq_{\rm{cx}}X_{i}$.

Now, assume that $X$ represents a monetary payoff so that greater values are
preferred, and let $\rho_{i}:\mathcal{X}\rightarrow\mathbb{R}$ denote the
decision criterion used by agent $i\in[n]$. An allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is _Pareto-optimal_ in
$\mathbb{A}_{n}(X)$ if for any $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$
satisfying $\rho_{i}(Y_{i})\geq\rho_{i}(X_{i})$, all $i\in[n]$, we have
$\rho_{i}(Y_{i})=\rho_{i}(X_{i})$, all $i\in[n]$. Let
$(\lambda_{1},\dots,\lambda_{n})$ be a vector of positive numbers, usually
called a Negishi weight vector. We say that an allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is _sum-optimal_ in
$\mathbb{A}_{n}(X)$ with respect to
$\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ if $(X_{1},\dots,X_{n})$
maximizes $\sum_{i=1}^{n}\lambda_{i}\rho_{i}(X_{i})$ subject to the constraint
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$. We use the term sum optimality for
the case $\lambda_{i}=1$, all $i\in[n]$.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let all
agents have homogeneous beliefs, i.e., everyone agrees on the probability
measure $\mathbb{P}$. The concept of strict risk aversion translates to a
strict preference for random variables that are lower in the convex order.
That is, if $X_{i}<_{\textrm{cx}}Y_{i}$ (meaning $X_{i}\leq_{\rm cx}Y_{i}$ but
$Y_{i}\not\leq_{\rm cx}X_{i}$) then $\rho_{i}(X_{i})>\rho_{i}(Y_{i})$. We can
derive from Proposition 1 that the set of Pareto-optimal allocations contains
only comonotonic allocations when all agents are strictly risk averse.

##  3 Counter-monotonicity and counter-monotonic improvement

Comonotonicity is an extreme type of positive dependence. This article
contends with the opposite situation: negatively dependent optimal
allocations, which are much less studied in the literature. We first define
this dependence concept. In all statements, $X$ represents a random wealth, so
greater values are preferred; negative values of $X$ are allowed.

Throughout this section, we consider a fixed probability space
$(\Omega,\mathcal{F},\mathbb{P})$. Two random variables $X,Y$ are _counter-
monotonic_ if the two random variables $X,-Y$ are comonotonic. An allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is pairwise counter-monotonic if for
every $i\neq j$ the random variables $X_{i},X_{j}$ are counter-monotonic.
Pairwise counter-monotonicity is the generalization of counter-monotonicity
for the case $n\geq 3$, but the concept is not always well-defined for
dimensions $n\geq 3$. We use the simpler term counter-monotonicity throughout.

The next lemma, due to Dall’Aglio (1972), gives necessary conditions for a
random vector $(X_{1},\dots,X_{n})$ to be counter-monotonic.

######  Lemma 1 (Dall’Aglio (1972)).

If at least three of $X_{1},\dots,X_{n}$ are non-degenerate, counter-
monotonicity of $(X_{1},\dots,X_{n})$ means that one of the following two
cases holds true:

$\displaystyle\mathbb{P}(X_{i}>\operatorname*{ess\text{-}inf}X_{i},~{}X_{j}>\operatorname*{ess\text{-}inf}X_{j})=0\mbox{
for all $i\neq j$};$ (1)
$\displaystyle\mathbb{P}(X_{i}<\operatorname*{ess\text{-}sup}X_{i},~{}X_{j}<\operatorname*{ess\text{-}sup}X_{j})=0\mbox{
for all $i\neq j$}.$ (2)

A necessary condition for (1) is
$\sum_{i=1}^{n}\mathbb{P}(X_{i}>\operatorname*{ess\text{-}inf}X_{i})\leq 1,$
and a necessary condition for (2) is
$\sum_{i=1}^{n}\mathbb{P}(X_{i}<\operatorname*{ess\text{-}sup}X_{i})\leq 1.$

Let $\Pi_{n}$ be the set of all $n$-compositions of $\Omega$, that is,

$\Pi_{n}=\left\\{(A_{1},\dots,A_{n})\in\mathcal{F}^{n}:\bigcup_{i\in[n]}A_{i}=\Omega\mbox{~{}and~{}$A_{1},\dots,A_{n}$
are disjoint}\right\\}.$

In other words, a composition of $\Omega$ is a partition of $\Omega$ with
order. The next proposition simplifies the stochastic stochastic
representation of counter-monotonicity given in Theorem 1 of Lauzier et al.
(2023a).

######  Proposition 2.

For $X\in\mathcal{X}$, suppose that at least three of
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ are non-degenerate. Then,
$(X_{1},\dots,X_{n})$ is counter-monotonic if and only if there exist
constants $m_{1},\dots,m_{n}$ and $(A_{1},\dots,A_{n})\in\Pi_{n}$ such that

$\displaystyle X_{i}=(X-m)\mathds{1}_{A_{i}}+m_{i}~{}~{}~{}\mbox{for all
$i\in[n]$ with}~{}m=\sum_{i=1}^{n}m_{i}\leq\operatorname*{ess\text{-}inf}X,$
(3)

or

$\displaystyle X_{i}=(X-m)\mathds{1}_{A_{i}}+m_{i}~{}~{}~{}\mbox{for all
$i\in[n]$ with}~{}m=\sum_{i=1}^{n}m_{i}\geq\operatorname*{ess\text{-}sup}X.$
(4)

######  Remark 1.

As in Denneberg’s Lemma, the underlying probability measure $\mathbb{P}$ is
not used in the stochastic representation of counter-monotonicity of Lauzier
et al. (2023a), and so the allocations characterized in Proposition 2 are also
well defined on measurable spaces without specified probability, as long as
the null sets are specified.

The allocation $(X,0,\dots,0)$ is counter-monotonic by taking $A=\Omega$ and
$m=m_{1}=\operatorname*{ess\text{-}inf}X$, and it is trivial to verify that it
is also comonotonic. Notice now that the allocations defined in equation (3)
and (4) echo the allocations in Lemma 1. In (3), for every $\omega\in\Omega$,
at most one agent receives more than their essential infimum. Conversely, in
(4), at most one agent receives less than their essential supremum. This is
the “winner-takes-all” and “loser-loses-all” structure of counter-monotonic
allocations.

The most curious case of both (3) and (4) when $m_{1}=\dots=m_{n}=0$, given by

$\displaystyle X_{i}=X\mathds{1}_{A_{i}}~{}\mbox{for all $i\in[n]$, where
$(A_{1},\dots,A_{n})\in\Pi_{n}$},$ (5)

will draw our special attention. Note that $m=0$ implies that either $X\geq 0$
or $X\leq 0$ holds, resulting in two different cases.

######  Definition 2.

An allocation $(X_{1},\dots,X_{n})$ is a _jackpot allocation_ if (5) holds for
some $X\geq 0$, and it is a _scapegoat allocation_ if (5) holds for some
$X\leq 0$.

$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$A_{6}$$1$$0$$X(\omega)$ Figure 1: An
illustration of positively and negatively dependent allocations, where a
comonotonic allocation is $X_{i}=X/n$ for $i\in[n]$ (the area between two
dotted curves) and a jackpot allocation is $X_{i}=X\mathds{1}_{A_{i}}$ for
$i\in[n]$ with $\Omega=[0,1]$ (the area between two dashed lines).

A comparison of a jackpot allocation and a comonotonic allocation is
illustrated in Figure 1. Although sharing the formula (5), a jackpot
allocation and a scapegoat allocation have very different meanings. In a
jackpot allocation, the total wealth $X$ is nonnegative (e.g., a prize), and
for each realization of the world $\omega$, only one agent “wins”, i.e.,
receives all positive payoff, and all other agents receive nothing. In a
scapegoat allocation, the total wealth is nonpositive (e.g., a loss), and only
one agent “loses”, i.e., suffers the loss. Both types of allocations are often
observed in daily life. For instance, the simple lottery ticket (only one
winner) is a jackpot allocation, and the “designated driving” is a scapegoat
allocation.

The next result shows a special role of the jackpot and the scapegoat
allocations among all counter-monotonic allocations. Using Lemma 1, an
allocation $(X_{1},\dots,X_{n})$ is a jackpot allocation if and only if

$\displaystyle X_{i}\geq 0~{}\mbox{and}~{}\mathbb{P}(X_{i}\wedge
X_{j}>0)=0\mbox{ for all $i\neq j$},$ (6)

where $a\wedge b$ means $\min\\{a,b\\}$. Therefore, being a jackpot allocation
is a property of the joint distribution of $(X_{1},\dots,X_{n})$. The
probabilistic mixture of two random vectors with joint distributions $F$ and
$G$ is another random vector with joint distribution $\lambda F+(1-\lambda)G$
for some $\lambda\in[0,1]$. The next result yields that jackpot allocations
are closed under probabilistic mixtures. The same holds for scapegoat
allocations by symmetry.

######  Proposition 3.

A probabilistic mixture of two jackpot allocations is again a jackpot
allocation.

For two general counter-monotonic allocations other than jackpot and scapegoat
allocations, their mixture is not necessarily counter-monotononic, even if
they both belong to the same type (3) or (4).

Next is our main result.

######  Theorem 1.

Let $X_{1},\dots,X_{n}\in L^{1}$ be nonnegative and $X=\sum_{i=1}^{n}X_{i}$.
Assume that there exists a uniform random variable $U$ independent of $X$.
Then, there exists $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that (i)
$(Y_{1},\dots,Y_{n})$ is counter-monotonic; (ii) $Y_{i}\geq_{\rm cx}X_{i}$ for
$i\in[n]$; (iii) $Y_{1},\dots,Y_{n}$ are nonnegative. Moreover,
$(Y_{1},\dots,Y_{n})$ can be chosen as a jackpot allocation.

######  Remark 2.

The boundedness from below of $X_{1},\dots,X_{n}$ is necessary to obtain the
existence of jackpot allocations. A similar statement can be made for
scapegoat allocations, which then requires the boundedness of
$X_{1},\dots,X_{n}$ from above instead (e.g., $X_{i}\leq 0$ for all
$i\in[n]$). The proof follows from observing that in this case,
$-X_{1},\dots,-X_{n}$ satisfies the assumptions of Theorem 1 and is thus
omitted.

Theorem 1 gives a converse to the comonotonic improvements for bounded random
variables. We obtain that jackpot allocations will always be preferred by
risk-seeking agents.

Before moving on, we emphasize that the technical assumption that there exists
a uniform random variable $U$ independent of $X$ is not completely innocuous.
Intuitively, we can interpret it as assuming that any allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ can be “implemented” with
randomization devices like flipping coins or spinning roulette wheels. At a
technical level, this assumption guarantees that the inf-(sup-)convolution of
law-invariant functionals is law-invariant (see Liu et al, 2020).

##  4 Risk-seeking agents in expected utility theory

From now on, let us focus on $\mathcal{X}=L^{\infty}$. Counter-monotonic
allocations on probability spaces are not necessarily interesting when we
restrict our attention to the most popular preferences, which are modelled by
concave Bernoulli utility functions. To see why, let
$u_{i}:\mathbb{R}\rightarrow\mathbb{R}$ be a twice-differentiable Bernoulli
utility function, assume that every individual $i\in[n]$ shares the same risk
attitude and consider the expected utility criterion
$\rho_{i}(X_{i})=\mathbb{E}\left[u\left(X_{i}\right)\right]$. We can always
trivially find counter-monotonic Pareto-optimal allocations when all
individuals are risk neutral. The reader can convince themselves by simply
setting $u_{i}$ as the identity and observing that any allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is Pareto optimal.222We implicitly
consider a.s. bounded allocations. Thus, anything goes with risk-neutrality,
and there is little to say about counter-monotonicity in this context.

As mentioned, a foundational result in risk-sharing is the comonotonicity of
Pareto optimal allocations when individuals are strictly risk averse. This
result implies that there cannot be counter-monotonic Pareto-optimal
allocations (besides the trivial counter-monotonic allocations) when the
utility functions are strictly concave. It thus seems natural to consider
strictly risk-seeking individuals. However, the next informal argument shows
that Pareto-optimal allocations do not exist in the general case.

Suppose that $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is Pareto optimal and
consider two strictly risk-seeking individuals $i\neq j$. We can construct
another feasible allocation
$(X_{1}^{\prime},\dots,X_{n}^{\prime})\in\mathbb{A}_{n}(X)$ by finding a non-
trivial partition $A\cup B$ of $\Omega$ on which we create an arbitrary
transfer of wealth between $i$ and $j$. Say, if $\omega\in A$, then $i$ gives
one billion dollars to $j$ and vice-versa if $\omega\in B$. The strict
convexity of $u_{i}$ and $u_{j}$ implies, by Jensen’s inequality, that both
individuals are strictly better off, contradicting the Pareto optimality of
$(X_{1},\dots,X_{n})$.

We know of two ways to make the problem sensible. The first is to impose lower
bounds on the allocation, so $X_{i}\geq a$ for $a\in\mathbb{R}$ and $i\in[n]$.
The case $a=0$ is of particular interest because it can be interpreted as a
no-short selling/borrowing constraint. The second approach is to restrict the
set of allocations by restricting the effective domain of $u$; we emphasize
that this is a common strategy in the empirical literature, where concave
power utilities $u(x)=x^{\alpha}$, $0<\alpha\leq 1$, are used extensively. In
what follows, all utility functions are mappings from $\mathbb{R}$ to
$\mathbb{R}\cup\\{-\infty\\}$ and not constantly $-\infty$.

######  Theorem 2.

Let $X\geq 0$ in $L^{\infty}$ and such that there is a uniform independent of
it and let $u_{i}$ be increasing and strictly convex on $[0,\infty)$ and
taking value $-\infty$ for all $x<0$ for each $i$. Then all Pareto-optimal
allocations are jackpot allocations.

It is straightforward to see that we obtain a similar result for risk-seeking
agents if we restrict the set of feasible allocations to allocations
satisfying $X_{i}\geq 0$, all $i\in[n]$.

Theorem 2 obtains that all Pareto-optimal allocations must be counter-
monotonic allocations with $m_{i}=0$ for all $i\in[n]$, i.e., jackpot
allocation. It is natural to wonder whether the converse is true, that is,
whether all counter-monotonic allocations of $X$ satisfying $m_{i}=0$, all
$i\in[n]$, are Pareto optimal. Proposition 12 in Appendix A shows that the
answer is “Yes” in the general case. However, the construction is technical,
and the next proposition shows the simplified case $X=x>0$.

######  Proposition 4.

Let $x>0$ be given. Then

  1. (i)

If $u_{1},\dots,u_{n}$ are strictly increasing and concave functions, then all
comonotonic allocations of $x$ are Pareto optimal.

  2. (ii)

If $u_{1},\dots,u_{n}$ are increasing functions with $u_{i}(x)>u_{i}(0)$, all
$i\in[n]$ that are convex on $[0,\infty)$ and taking value $-\infty$ for all
$x<0$, then all jackpot allocations are Pareto optimal.

While all the allocations found above are Pareto optimal, not all are equal
from a welfare point of view. When all agents use the same decision criterion
and have the same Bernoulli utility, we say that an allocation is fair if all
agents achieve the same (Ex-ante) welfare.

######  Definition 3.

Let all agents have the same decision criterion $\rho$ and the same Bernoulli
utility function $u$. Then an allocation $(X_{1},\dots,X_{n})$ is fair if
$\rho_{i}(X_{i})=\rho_{i}(X_{j})$ for all $i\neq j$.

We emphasize that this notion of fairness is cardinal, and so we only define
it for the case where every agent is identical in order to avoid general
interpersonal comparisons of welfare.

######  Definition 4.

An allocation $(X_{1},\dots,X_{n})$ is distributionally fair if
$X_{i}\buildrel\mathrm{d}\over{=}X_{j}$ for all $i\neq j$.

For a counter-monotonic allocation to be distributionally fair, one must thus
require that the underlying $(A_{1},\dots,A_{n})\in\Pi_{n}$ be such that
$\mathbb{P}(A_{i})=\mathbb{P}(A_{j})=1/n$, all $i\neq j$. Clearly, a counter-
monotonic allocation is fair if it is distributionally fair, and all
distributionally fair allocations are fair when all agents share the same
decision criterion and Bernoulli utility. Equipped with this distinction, we
close this section with an example that highlights the role of counter-
monotonic allocations as a different way to take convex combinations.

######  Example 1.

Set $x=1$ and consider the problem

$\displaystyle\mbox{to
maximize~{}~{}~{}}\sum_{i=1}^{n}\mathbb{E}[u(X_{i})]~{}~{}~{}\mbox{subject
to}~{}(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(1)~{}\mbox{ and
}X_{1},\dots,X_{n}\geq 0.$

Set $a_{i}\in\mathbb{R}^{n}$ as $a_{1}=(1,0,\dots,0)$,
$a_{2}=(0,1,0,\dots,0)$, $\dots$ , $a_{n}=(0,\dots,1)$, and denote by $\delta$
the Dirac delta function. The collection
$\left(a_{i},\delta_{a_{i}}\right)_{i\in[n]}$ denotes all the allocations
giving the whole $x=1$ to one agent with certainty and corresponds to all the
extreme points of the utility possibility set. With risk-averse agents, one
must have comonotonic allocation, and we are taking “convex combinations along
the $a_{i}$s”. In this case, the only fair comonotonic allocation is the pair
$\left((1/n,\dots,1/n),\delta_{(1/n,\dots,1/n)}\right)$ that gives everyone
$x_{i}=1/n$ with certainty.

Counter-monotonic allocations are like “taking convex combination along the
$\delta_{a_{i}}$s”. In this case, all fair allocations must be like the
allocation $\left(a_{i},\delta_{a_{i}}/n\right)_{i\in[n]}$, i.e. they must
give $x_{i}=1$ to one agent with probability $1/n$. With strictly risk-seeking
agents, only this type of “convex combination” preserves Pareto optimality. In
the general case, the non-atomicity of the probability space combined with the
assumption that there exists a uniform independent of $X$ precisely guarantees
that we can find these “convex combinations along the probabilities”.

##  5 Discontinuous Bernoulli utilities

The previous section obtained the optimality of jackpot allocations when all
agents are risk-seeking (provided that Pareto-optimal allocations exist).
These allocations have a direct interpretation as a “winner-takes-all” lottery
where a (random) prize is (potentially non-randomly) given to only one winner.
We now turn our attention to scapegoat allocations and present situations
where they are optimal. In order to simplify the treatment, consider the
following:

######  Assumption 1.

The probability space $(\Omega,\mathcal{F},\mathbb{P})$ is atomless, the set
of random variables is $\mathcal{X}=L^{\infty}$, and $X\in\mathcal{X}$ is such
that there exists a uniform $U$ independent of $X$.

The sequel always assumes Assumption 1.

###  5.1 Pareto optimal allocations with Dirac utility

For every $i\in[n]$ set the decision criterion
$\rho_{i}:\mathcal{X}\rightarrow\mathbb{R}$ as
$\rho_{i}(X_{i})=\mathbb{E}\left[\alpha\mathds{1}_{\\{X_{i}\geq 1\\}}\right]$;
we will refer to these agents as Dirac agents. In Lauzier et al. (2023a), we
considered the special case of the risk-sharing problem with Dirac agents
where $X=1$ and $X_{i}\geq 0$. We interpreted the variable $X=1$ as an
indivisible good that was auctioned and the utility function as the net
utility of $n$ agents with the quasi-linear utilities $v(X,t)=\theta X-t$
having bid the same amount $\theta-t=\alpha$.

It is straightforward to see that the set

$\displaystyle\left\\{(\mathds{1}_{A_{1}},\dots,\mathds{1}_{A_{n}})\in\mathbb{A}_{n}(X):(A_{1},\dots,A_{n})\in\Pi_{n}\right\\}$

consists of Pareto-optimal jackpot allocations. We thus interpreted the
allocations satisfying $\mathbb{P}(A_{i})=\mathbb{P}(A_{j})$ for every $i\neq
j$ as the random tie-breaking rule. Those are distributionally fair, and thus,
they also are fair allocations because all agents have the same expected
utility. We observed that a fair lottery (which is counter-monotonic) is the
only fair way to distribute the indivisible good among people who value it
equally.

This section analyzes further the problem of sharing risk among Dirac agents.
We assume $\alpha=1$ for simplicity and without loss of generality. We first
establish that for every $X\in\mathcal{X}$ there exists a counter-monotonic
allocation which is Pareto optimal. Let us first set the Negishi weights to
one so that we search for the allocations $(X_{1},\dots,X_{n})$ that solve

$\displaystyle\mbox{to
maximize~{}~{}~{}}\sum_{i=1}^{n}\rho_{i}(X_{i})~{}~{}~{}\mbox{subject
to}~{}(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X).$

######  Proposition 5.

The allocation $X_{1}=X_{2}=\dots=X_{n-1}=1$ and $X_{n}=X-(n-1)$ is Pareto
optimal.

The trivial counter-monotonic allocation in Proposition 5 has the
characteristic that agent $n$ above potentially gives everything to its peers.
Notice now that

$\displaystyle\Upsilon(X):=\left\\{\left(\mathds{1}_{A_{1}}(X-(n-1))+\mathds{1}_{A_{1}^{c}},\dots,\mathds{1}_{A_{n}}(X-(n-1)\right)+\mathds{1}_{A_{n}^{c}})\in\mathbb{A}_{n}(X):(A_{1},\dots,A_{n})\in\Pi_{n}\right\\}$

consist exclusively of sum-optimal allocations. We can interpret this set as
the set of all allocations that “randomizes who gets or loses everything”,
much like drawing straws but with potentially unfair probabilities. When
$\mathbb{P}(X\leq n)=1$, the set $\Upsilon(X)$ boils down to

$\displaystyle\mathcal{T}(X):=\left\\{\left(\mathds{1}_{A_{1}}(X-n)+1,\dots,\mathds{1}_{A_{n}}(X-n)+1\right)\in\mathbb{A}_{n}(X):(A_{1},\dots,A_{n})\in\Pi_{n}\right\\},$

the set of scapegoats allocations of $X$ that are shifted so that it satisfies
$m_{i}=1$, all $i\in[n]$.

Let $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$, $\lambda_{i}>0$, be a
vector of Negishi weights and
$f:\mathcal{X}^{n}\times\mathbb{R}^{n}_{++}\rightarrow\mathbb{R}$ be

$f_{\bm{\lambda}}\left(\mathbf{X}\right)=\mathbb{E}\left[\sum_{i=1}^{n}\lambda_{i}\mathds{1}_{\\{X_{i}\geq
1\\}}\right]$

for $\mathbf{X}=(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$. By convention, we
consider $\lambda_{1},\dots,\lambda_{n}$ in decreasing order.

######  Proposition 6.

If $\mathbf{X}^{*}$ maximizes $f_{\bm{\lambda}}(\mathbf{X})$ then
$\mathbf{X}^{*}$ also maximizes $f_{\mathbf{1}}(\mathbf{X})$.

Proposition 6 informs us that with Dirac agents, it suffices to characterize
the set of sum-optimal allocations in order to understand the whole set of
allocations that are sum-optimal for some Negishi weights $\bm{\lambda}$.
This, of course, means that $\Upsilon(X)$ contains all allocations of
interest.

While all the allocations in $\Upsilon(X)$ maximize the sum of expected
utilities, not all are equal from a welfare point of view. Recall that a
Rawlsian social welfare function considers only the utility of the worst-off
agent and that maximizing a Rawlsian social welfare function involves focusing
on allocations where $\rho_{i}=\rho_{j}$, all $i\neq j$.333 In our context, a
social welfare function $\mathcal{W}$ is a mapping
$(\rho_{i})_{i\in[n]}=\bm{\rho}\mapsto\mathcal{W}(\bm{\rho})\in\mathbb{R}$
that ranks allocation, so higher value are better. The Rawlsian social welfare
function is $\mathcal{W}(\bm{\rho})=\min_{i\in[n]}\\{\rho_{i}$}. Thus if two
allocations $\mathbf{X},\mathbf{Y}$ are such that
$\sum_{i=1}^{n}\rho_{i}(X_{i})=\sum_{i=1}^{n}\rho_{i}(Y_{i})$, but $\bm{X}$ is
fair and $\bm{Y}$ is not, then a Rawlsian social welfare function ranks
$\bm{X}$ strictly higher than $\bm{Y}$. This motivates our focus on fair
allocations, where in our case we have that an allocation
$(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is fair if for every $i\neq j$,
$i,j\in[n]$, it is $\rho_{i}(X_{i})=\rho_{i}(X_{j})$.

Since we assumed that there exists a uniform $U$ independent of $X$, we can
always find a partition $(A_{1},\dots,A_{n})\in\Pi_{n}$ independent of $X$
such that $\mathbb{P}(A_{i})=1/n$ for all $i\in[n]$. Thus, the set
$\Upsilon(X)$ always contains distributionally fair allocations that give the
same expected utility to all agents, and $\Upsilon(X)$ always contains a fair
allocation.

######  Proposition 7.

If $\mathbb{P}(X<n)>0$ then there exists no allocation which is simultaneously
comonotonic, fair and sum-optimal.

To summarize, we obtained that the Pareto optimality of an allocation
$(X_{1},\dots,X_{n})$ requires that for every $\omega\in\Omega$, there is at
most one agent $i\in[n]$ for which $X_{i}(\omega)<1$. If $\mathbb{P}(X<n)=0$,
one can always find comonotonic allocations (satisfying $X_{i}\geq 1$, all
$i\in[n]$) that are Pareto optimal. In particular, fair ones exist. But when
$\mathbb{P}(X<n)>0$, it is no longer possible for a fair comonotonic
allocation to have at most one agent $i$ for which $X_{i}(\omega)<1$.

The idea of a scapegoat allocation and of drawing straws to randomize the
scapegoat is quite alien to standard welfare analysis, and one might be
interested in imposing a lower bound on the allocation. Once again, the
constraint $X_{i}\geq 0$, all $i\in[n]$, is particularly interesting because
it can be interpreted as a borrowing constraint. Clearly, imposing constraints
on the allocation can impact the aggregate welfare when $\mathbb{P}(X<n)>0$.
While we do not fully characterize the impacts of such constraints, we observe
that they sometimes imply that jackpot allocations are Pareto optimal, as in
the auction example above.

######  Corollary 1.

Let $X\geq 0$ and consider the constraints $X_{i}\geq 0$, all $i\in[n]$. If
$\mathbb{P}(X<2)=1$ then all Pareto-optimal allocations are payoff equivalent
to a jackpot allocation: if $(X_{1},\dots,X_{n})$ is constrained Pareto
optimal then there exists a feasible counter-monotonic allocation
$(Y_{1},\dots,Y_{n})$ such that $\rho_{i}(X_{i})=\rho(Y_{i})$ for all
$i\in[n]$.

Next, we extend our analysis to piecewise linear Bernoulli utility with one
jump.

###  5.2 Piecewise linear Bernoulli utility with one jump

Assume now that piecewise linear Bernoulli utility functions
$u_{i}(X_{i})=a_{i}X_{i}+b_{i}\mathds{1}_{\\{X_{i}\geq 1\\}}$, so that
$\rho_{i}(X_{i})=\mathbb{E}[u_{i}(X_{i})]=a_{i}\mathbb{E}[X_{i}]+b_{i}\mathbb{P}(X_{i}\geq
1)$. For simplicity, we assume that $a_{i}=a$ and $b_{i}=1$, for all
$i\in[n]$. These agents are a combination of risk-neutral agents and Dirac
agents, and it is easy to verify that $\rho_{i}$ is monotone.

Consider once again the set

$\displaystyle\Upsilon(X)=\left\\{\left(\mathds{1}_{A_{1}}(X-(n-1))+\mathds{1}_{A_{1}^{c}},\dots,\mathds{1}_{A_{n}}(X-(n-1)\right)+\mathds{1}_{A_{n}^{c}})\in\mathbb{A}_{n}(X):(A_{1},\dots,A_{n})\in\Pi_{n}\right\\}.$

As before, when $\mathbb{P}(X\leq n)=1$, the set $\Upsilon(X)$ boils down to

$\displaystyle\mathcal{T}(X)=\left\\{\left(\mathds{1}_{A_{1}}(X-n)+1,\dots,\mathds{1}_{A_{n}}(X-n)+1\right)\in\mathbb{A}_{n}(X):(A_{1},\dots,A_{n})\in\Pi_{n}\right\\},$

the set of Pareto-optimal scapegoat allocations satisfying $m_{i}=1$, all
$i\in[n]$. Once again by Assumption 1 we can find a partition
$(A_{1},\dots,A_{n})\in\Pi_{n}$ independent of $X$ such that
$\mathbb{P}(A_{i})=1/n$ for all $i\in[n]$, so $\Upsilon(X)$ contains a fair
allocation.

Suppose first that $\mathbb{P}(X<n)=0$. It is easy to verify that the
comonotonic allocation $X_{i}=X/n$, $i\in[n]$, is Pareto optimal and fair. In
other words, when $\mathbb{P}(X<n)=0$, the problem goes in a similar fashion
to the problem of sharing risk among Dirac agents. Our interest thus lies
again in the case where $\mathbb{P}(X<n)>0$. Let
$(X_{1},\dots,X_{n})\in\Upsilon(X)$ be fair, and for simplicity chose it so
that $(A_{1},\dots,A_{n})\in\Pi_{n}$ is independent of $X$. Computing the
expected utility for all $i\in[n]$ we have

$\displaystyle\mathbb{E}[u_{i}(X_{i})]$
$\displaystyle=a\mathbb{E}[X_{i}]+\mathbb{P}(X_{i}\geq 1)$
$\displaystyle=a\mathbb{E}[\mathds{1}_{A_{i}^{C}}+\mathds{1}_{A_{i}}(X-(n-1))]+\mathbb{P}(A_{i}^{C})+\mathbb{P}(A_{i})\mathbb{P}(X\geq
n)$
$\displaystyle=a\left(\mathbb{P}(A_{i}^{C})+\mathbb{P}(A_{i})\mathbb{E}[X]-\mathbb{P}(A_{i})(n-1)\right)+\mathbb{P}(A_{i}^{C})+\mathbb{P}(A_{i})\mathbb{P}(X\geq
n)$ $\displaystyle=\frac{a\mathbb{E}[X]+(n-1)+\mathbb{P}(X\geq n)}{n}$

since $\mathbb{P}(A_{i}^{C})=(n-1)/n=\mathbb{P}(A_{i})(n-1)$. Summing over
$i\in[n]$ we obtain

$\sum_{i=1}^{n}\mathbb{E}[u_{i}(X_{i})]=a\mathbb{E}[X]+(n-1)+\mathbb{P}(X\geq
n).$

######  Proposition 8.

Let $\mathbb{P}(X<n)>0$. Then a comonotonic and fair allocation
$(X_{1},\dots,X_{n})$ cannot be Pareto optimal.

To summarize, the key property of the jump in the Bernoulli utility function
is that it creates an incentive to concentrate losses on at most one agent.
This property is not driven by the satiation of preferences, as was implicitly
suggested by the risk-sharing problem with Dirac agents, but rather by the
sharp gains in utility at the discontinuity threshold. Thus, when there is a
positive probability of not having enough to share, the corresponding optimal
allocation cannot be simultaneously comonotonic and fair if it is to be Pareto
optimal, as the latter requires concentrating the losses.

##  6 Rank-dependent utility agent

We now analyze the problem of sharing risk among RDU agents. A function
$h:[0,1]\rightarrow[0,1]$ is called a probability distortion if it is non-
decreasing and satisfies $h(0)=0$ and $h(1)=1$. An agent is RDU if its
decision criterion is $\rho_{h}(X)=\int u(X)\,\mathrm{d}h\circ\mathbb{P}$,
where $u:\mathbb{R}\rightarrow\mathbb{R}$ is a Bernoulli utility function, $h$
is a probability distortion and where the integral is in the sense of Choquet.
When the Bernoulli utility function is linear, we obtain Yaari (1987)’s dual
utility with decision criterion $\rho_{h}(X)=\int
X\,\mathrm{d}h\circ\mathbb{P}$. Yaari agents are risk seeking (risk averse) if
the probability distortion function $h$ is concave (convex). We remind the
reader that when $0<\gamma<1$ the Kahneman-Tversky (KT) distortion function
$h_{\textrm{KT}}(t)=\frac{t^{\gamma}}{(t^{\gamma}+(1-t)^{\gamma})^{1/\gamma}}$
is inverted S-shaped, i.e. concave-convex.

######  Assumption 2.

The utility Bernoulli function $u:\mathbb{R}\rightarrow\mathbb{R}$ is
increasing and differentiable on $[0,\infty)$, weakly concave, satisfies
$u(0)=0$ and such that $u(x)=-\infty$ for all $x<0$. The probability
distortion function $h:[0,1]\rightarrow[0,1]$ is concave-convex.

Notice that the exponential Bernoulli utility $u(x)=x^{\alpha}$ for
$0<\alpha\leq 1$ satisfies Assumption 2. We emphasize that RDU agents
satisfying Assumption 2 with $u$ linear on $[0,\infty)$ are not Yaari agents.
This distinction will come back in our later discussion.

Denoting by $F_{X}^{-1}(1-t)$ the quantile function of $X$, the quantile
representation of $\rho_{h}(X)$ is

$\displaystyle\rho_{h}(X)=\int
u(X)\,\mathrm{d}h\circ\mathbb{P}=\int_{0}^{\infty}h\left(\mathbb{P}\left(u\left(X\right)>t\right)\right)\,\mathrm{d}t=\int_{0}^{1}u\left(F_{X}^{-1}(1-t)\right)\,\mathrm{d}h(t).$

We denote by $\overline{h}:[0,1]\rightarrow[0,1]$ the concave envelope of $h$,
which we define as the smallest concave function such that
$h(t)\leq\overline{h}(t)$ for all $t\in[0,1]$. We have that
$\overline{h}=h^{**}$ for $h^{**}$ the biconjugate of $h$; clearly
$\overline{h}$ is a probability distortion. Since $\overline{h}$ dominates $h$
pointwise for any $X\in\mathcal{X}$ we have

$\displaystyle\rho_{\overline{h}}(X)=\int_{0}^{1}u\left(F_{X}^{-1}(1-t)\right)\,\mathrm{d}\overline{h}(t)\geq\int_{0}^{1}u\left(F_{X}^{-1}(1-t)\right)\,\mathrm{d}h(t)=\rho_{h}(X)$

and $\rho_{\overline{h}}$ gives an upper-bound to the value attained by
$\rho_{h}$. By construction, if the Bernoulli utility is linear, then the
decision criterion $\rho_{\overline{h}}$ behaves as a risk-seeking agent,
similarly to Section 4. Recalling the KT distortion $h_{\textrm{KT}}$ is
always concave-convex when $0<\gamma<1$, with strict concavity on a subset, we
obtain that its concave envelope $\overline{h}_{\textrm{KT}}$ is strictly
concave on a subset.

Our goal now is to find conditions guaranteeing the Pareto-optimality of fair
jackpot allocations. To do so we introduce a strengthening of the notion of
distributional fairness:

######  Definition 5.

A random vector $(X_{1},\dots,X_{n})\in\mathcal{X}^{n}$ is exchangeable if
$\left(X_{1},\dots,X_{n}\right)\buildrel\mathrm{d}\over{=}\left(X_{\pi(1)},\dots,X_{\pi(n)}\right)$
for all permutation $\pi\in\mathfrak{S}_{n}$ where ${\mathfrak{S}_{n}}$ is the
set of all permutations on $[n]$.

Clearly if $\left(X_{1},\dots,X_{n}\right)\in\mathcal{X}^{n}$ is exchangeable
then $X_{i}\buildrel\mathrm{d}\over{=}X_{j}$ for all $i\neq j$, so
$\left(X_{1},\dots,X_{n}\right)$ is distributionally fair. Suppose that
$X_{i}\geq 0$ for all $i\in[n]$ and that there is a uniform distribution
independent of $\left(X_{1},\dots,X_{n}\right)$. Then
$\left(X_{1},\dots,X_{n}\right)$ has an exchangeable counter-monotonic
improvement $\left(Y_{1},\dots,Y_{n}\right)$ such that for all $i\in[n]$,
$Y_{i}\geq 0$ and $\mathbb{P}(Y_{i}>0)=1/n$. That is:

######  Corollary 2.

In the setting of Theorem 1, we further assume that $(X_{1},\dots,X_{n})$ is
exchangeable. Then, there exists an exchangeable jackpot allocation
$(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that $Y_{i}\geq_{\rm cx}X_{i}$
and $\mathbb{P}(Y_{i}>0)\leq 1/n$ for $i\in[n]$.

Having established that the counter-monotonic improvement of an exchangeable
allocation can be exchangeable, we now consider the risk-sharing problem with
agents that have a linear Bernoulli utility function on $[0,\infty)$.

######  Theorem 3.

Assume Assumptions 1 and 2. Let all agents be RDU agents with the same utility
function $u(x)=ax$ for some $a>0$ and probability distortion $h$ such that
$h=\bar{h}$ for $t\in[0,1/n]$. Let
$\left(X_{1},\dots,X_{n}\right)\in\mathbb{A}_{n}(X)$ be a non-negative
exchangeable allocation. Then there exists an exchangeable jackpot allocation
$\left(Y_{1},\dots,Y_{n}\right)\in\mathbb{A}_{n}(X)$ that Pareto improves upon
$\left(X_{1},\dots,X_{n}\right)$.

A direct implication of Theorem 3 is that if an agent with a linear Bernoulli
utility uses a distortion function $h$ that is strictly concave on the segment
$[0,1/n]$, then the same agent precisely behaves as the risk-seeking agent of
Section 4 when comparing the two allocations.

When is $n$ large and all agents have a linear Bernoulli utility, all fair
Pareto-optimal allocations are counter-comonotonic, and the exchangeable
jackpot allocation strictly Pareto dominates the exchangeable comonotonic
allocation. Fair Pareto-optimal allocations thus cannot be comonotonic.
However, we can have non-trivial unfair comonotonic allocations that are
Pareto optimal:

######  Example 2.

Assume all RDU agents have a linear Bernoulli utility function and the same
distortion function $h$. Let $X\geq 0$ and consider the following allocation:
$X_{1}=X_{2}=X/2$ and for all $i\neq 1,2$, $X_{i}=0$. This allocation is non-
trivially comonotonic, and one can build a jackpot allocation where
$Y_{1}=X\mathds{1}_{A_{1}}$ and $Y_{2}=X\mathds{1}_{A_{2}}$ with
$\mathbb{P}(A_{1})=\mathbb{P}(A_{2})=1/2$. Yet, the jackpot allocation need
not improve upon the original allocation. This happens when the condition
$h=\overline{h}$ for $t\in[0,1/2]$ is not satisfied. The economic intuition is
that the Pareto optimality of comonotonic allocations can happen when some
agents have a large enough share of the aggregate endowment so that gambling
with others does not create a high enough reward.

The symmetry of behaviour between RDU agents and the risk-seeking agents of
Section 4 goes beyond the cases considered above. Yaari agents are strictly
risk seeking when the distortion function $h$ is strictly concave, and we can
reproduce the argument of Section 4 to show that Pareto-optimal allocations do
not exist with strictly risk-seeking Yaari agents. Pareto-optimal allocations
also do not exist for Yaari agents with concave-convex distortions when the
number of agents $n$ is large. This is seen by concavifying the distortion
function $h$ and observing that if $\overline{h}$ is concave and $n$, our
Yaari agents behave as risk-seeking agents. Once again, the assumption that
$u(x)=-\infty$ for $x<0$ in Assumption 2 is instrumental in guaranteeing that
Pareto-optimal allocations exist; it might be replaced by constraining the set
of feasible allocations to non-negative allocations.

The observation that the exchangeable jackpot allocation sometimes leads to
strict Pareto improvements hints at the possibility that Theorem 3 might be
extended to some cases where the Bernoulli utility is strictly concave on
$[0,\infty)$. While the general case is still an open question, the next
section’s cryptocurrency example shows that counter-monotonic payoff can
indeed be preferred by RDU agents with concave Bernoulli utility when we
restrict their choice set.

##  7 Cryptocurrency mining: to pool or not to pool

Let us consider $n$ miners who need to decide whether they mine by themselves
or join a mining pool. For all $i\in[n]$, the actions set is
$\mathcal{A}_{i}=\mathcal{A}=\\{H,P\\}$, where $H$ denotes mining by
themselves (from “Home”), and $P$ denotes joining the pool. We consider two
types of miners. The first type of miner is behavioural, which we define as an
RDU agent with concave-convex probability distortion functions. Let
$y\in\mathbb{R}$ and let $\phi_{y}:[0,1]\rightarrow\mathbb{R}$ be
$\phi_{y}(x)=h(x)u(y/x)$. Behavioural miners satisfy the following assumption:

######  Assumption 3.

The utility Bernoulli function $u:\mathbb{R}\rightarrow\mathbb{R}$ is
increasing and differentiable on $[0,\infty)$, weakly concave, satisfies
$u(0)=0$ and such that $u(x)=-\infty$ for all $x<0$. The probability
distortion function $h:[0,1]\rightarrow[0,1]$ is concave-convex. Further,
there exists a unique a $p_{0}\in(0,1)$ such that for every $y\geq 0$,
$\phi_{y}(x)$ decreases on $(0,p_{0})$ and $\phi_{y}(p_{0})\geq\phi_{y}(z)$
for all $z>p_{0}$.

The second type of miner is the strictly risk-averse miner. We still denote by
$\rho(X)=\int u(X)\,\mathrm{d}h\circ\mathbb{P}$ their decision criterion to
unify notation, but do not specify the shape of $u$ or $h$ and simply assume
that $\rho$ is strictly risk-averse.444The criterion $\rho(X)=\int
u(X)\,\mathrm{d}h\circ\mathbb{P}$ can be risk-averse in the following three
cases: (1) $u$ concave and $h$ the identity; (2) $u$ linear and $h$ convex and
(3) $u$ concave can $h$ convex.

Let $k\in\mathbb{N}_{+}$ denote the number of behavioural miners so that there
are $n-k$ miners risk-averse miners. Each miner $i\in[n]$ has a computational
power $c_{i}$, and the probability of mining the next coin is proportional to
the total computing power as in Leshno and Strack (2020). Normalizing
$\sum_{i=1}^{n}c_{i}=1$ we can take an atomless probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and define the event

$\displaystyle A_{i}:=\\{\omega\in\Omega:\mbox{ agent $i$ mines the coin}\\}$

so that $\mathbb{P}(A_{i})=c_{i}$, all $i\in[n]$, and
$(A_{1},\dots,A_{n})\in\Pi_{n}$. Let $v>0$ denote the given value of the coin.
For all $i\in[n]$ we normalize the monetary payoff of mining from home as
$v\mathds{1}_{A_{i}}$ and set $u_{i}(0)=0$ so that the expected payoff of a
home miner $i$ is

$\rho_{i}(H)=h_{i}(c_{i})u_{i}(v)+(1-h(c_{i}))u_{i}(0)=h_{i}(c_{i})u_{i}(v).$

Let $\mathrm{Po}$ denote the set of agents that join the pool, i.e.
$\mathrm{Po}=\\{i\in[n]:a_{i}=P\\}$. We assume that the pool uses the
conditional mean risk-sharing rule so that the monetary payoff of agent
$i\in\mathrm{Po}$ conditional on the pool mining the coin is
$\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}$. Note that
$\bigcup_{j\in\mathrm{Po}}A_{j}$ is the event that the pool mines the coin,
and the unconditional monetary payoff of agent $i\in\mathrm{Po}$ is

$\displaystyle\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathds{1}_{\bigcup_{j\in\mathrm{Po}}A_{j}}.$

When $\mathrm{Po}$ is given the expected utility of miner $i\in\mathrm{Po}$ is
thus

$\displaystyle\rho_{i}(P)=h\left(\sum_{j\in\mathrm{Po}}c_{j}\right)u\left(\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\right).$

Since we focus on pure-strategy Nash equilibria, we slightly abuse notation
and denote by $a_{i}$ the strategy profile of agent $i\in[n]$ playing action
$a_{i}$ with probability one.555That is, we denote by $a_{i}$ the strategy
profile $\delta_{a_{i}}$, where $\delta$ is again the Dirac delta function.
Let $a_{-i}=(a_{j})_{j\neq i}\in\mathcal{A}^{n-1}$ and let
$\rho_{i}\left(a_{i},a_{-i}\right)$ be the expected utility of action $a_{i}$
given the action profile $a_{-i}$. We let
$\sigma_{i}:\mathcal{A}^{n-1}\rightrightarrows\mathcal{A}$ be the best-reply
correspondence of agent $i$ so that
$\sigma_{i}\left(a_{-i}\right)=\operatorname*{arg\,max}_{a_{i}\in\mathcal{A}}\rho_{i}(a_{i},a_{-i})$.
A pure-strategy Nash equilibrium of the crypto mining game is a profile of
actions $(a_{1}^{*},\dots,a_{n}^{*})\in\mathcal{A}^{n}$ such that for every
$i\in[n]$, $a_{i}^{*}\in\sigma_{i}\left(a_{-i}^{*}\right)$.

Notice now that

$\displaystyle\mathbb{E}\left[\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathds{1}_{\bigcup_{j\in\mathrm{Po}}A_{j}}\right]=\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathbb{P}\left(\bigcup_{j\in\mathrm{Po}}A_{j}\right)=vc_{i}=\mathbb{E}\left[v\mathds{1}_{A_{i}}\right]$

and $v\mathds{1}_{A_{i}}$ is a mean-preserving spread of
$\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathds{1}_{\bigcup_{j\in\mathrm{Po}}A_{j}}$
since both are Bernoulli random variables.

Clearly if $a_{-i}=(H,\dots,H)$ we have

$\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathds{1}_{\bigcup_{j\in\mathrm{Po}}A_{j}}=v\mathds{1}_{A_{i}}$

and $\rho_{i}\left(H,a_{-i}\right)=\rho\left(P,a_{-i}\right)$. The following
lemma is thus trivial:

######  Lemma 2.

The action profile $\mathbf{a}^{*}=(H,\dots,H)\in\mathcal{A}^{n}$ constitutes
a Nash equilibrium.

The lemma means that it can happen that the pool never forms in equilibrium.

######  Proposition 9.

Let agent $i$ be strictly risk averse. Then

$\displaystyle\sigma_{i}(a_{-i})=\begin{cases}\\{H,P\\}&\mbox{if $a_{j}=H$ for
all }j\neq i\\\ P&\mbox{otherwise}\end{cases}$

and $P$ is a weakly dominant strategy for agent $i$.

Of course, this implies that $H$ is a weakly dominated strategy for risk-
averse agents and the Nash equilibrium $(H,\dots,H)\in\mathcal{A}^{n}$ does
not survive the iterated elimination of weakly dominated strategies. Further,
if at least one agent chooses $P$, then all strictly risk-averse agents choose
$P$. The equilibria where all risk-averse agents choose $P$ are thus of
greater interest.

######  Proposition 10.

Let agent $i$ be a RDU agent with decision criterion $\rho_{h}(X)$ satisfying
Assumption 3. Then

  1. (i)

If $c_{i}\leq p_{0}$ and $\phi_{y}(x)$ strictly decreases on $(0,p_{0})$ we
have

$\displaystyle\sigma_{i}(a_{-i})=\begin{cases}\\{H,P\\}&\mbox{if $a_{j}=H$ for
all }j\neq i\\\ H&\mbox{otherwise}.\end{cases}$

  2. (ii)

If there is a $p^{*}\in[0,1]$ for which $h(t)/t$ is strictly increasing for
$t>p^{*}$ and if $c_{i}\geq p^{*}$, then

$\displaystyle\sigma_{i}(a_{-i})=\begin{cases}\\{H,P\\}&\mbox{if $a_{j}=H$ for
all }j\neq i\\\ P&\mbox{otherwise}.\end{cases}$

We immediately obtain the following corollary:

######  Corollary 3.

Let all behavioural agents $j\in[k]$ be such that $\phi_{y}(x)$ strictly
decreases on $(0,p_{0})$ and such that $c_{j}\in(0,p_{0})$. Then the action
profile $(a_{i}^{*})_{i\in[n]}$ is a pure-strategy Nash Equilibrium, where

$\displaystyle a_{i}^{*}=\begin{cases}H&\mbox{if }j\in k\\\
P&\mbox{otherwise}.\end{cases}$

Moreover, $(a_{i}^{*})_{i\in[n]}$ is the unique pure-strategy Nash Equilibrium
left after performing the iterated deletion of weakly dominated strategies.

The takeaway of Proposition 10 and its corollary is that behavioural agents
can have somewhat “bang-bang” strategies. On the one hand, they can behave as
risk-seeking agents and mine from home. This happens when they have
proportionally small computing power so that $c_{i}$ is in $(0,p_{0})$ a
subset of the concave part of $h$. This result both complements and contrasts
the results of the previous section, as RDU agents with strictly concave
Bernoulli utility functions can behave as risk-seeking agents when their
choice set is restricted.

On the other hand, behavioural agents can strictly prefer to join the pool
when their computing power is large in proportion to the total computer power
and their probability of winning is high. This happens for two reasons. First,
when $h(t)/t$ strictly increases for $t\in[p^{*},1]$, we have that
$c_{i}\in[p^{*},1]$ corresponds to a set of probability where the agent is
risk-averse. While the effect of risk aversion is clear, a second, less
obvious reason comes from our assumption that the pool uses the conditional
mean risk-sharing rule. This assumption implies that the high contribution of
player $i$ to the pool’s computing power translates into a large share of the
value of the coin. In other words, joining the pool lets player $i$ “hedge”
some of its risk. This property would not be obvious if the pool had a
different risk-sharing rule. For instance, imposing an upper bound on the
share that each individual miner can impact the equilibria.

The optimal strategy of behavioural agents is unclear when
$c_{i}\in(p_{0},p^{*})$. This is because the best-reply of agent $i$ can now
vary as a function of the other agents’ action. A complete equilibrium
analysis is out of the scope of this article, but we believe it would be
interesting to analyze the optimal pool formation as a function of both the
computing power of the agents, the risk-sharing rule and the ability to divide
its computing power among different pools.

##  8 Conclusion

Our main result, the counter-monotonic improvement theorem, lays the
foundation for analyzing risk sharing with counter-monotonic allocations, the
most extreme forms of negatively dependent allocation. This theorem allowed us
to shed light on Pareto-optimal allocations when the risk is to be shared
among risk-seeking agents, agents with a discontinuous Bernoulli utility
function and RDU agents with inverted S-shaped probability distortion
functions.

However, these characterizations of counter-monotonic Pareto-optimal
allocations beg for more questions than it answers. Can competitive equilibria
be counter-monotonic? If yes, under which conditions? What happens if we lift
the assumption of the underlying risk and probability space that is well-
understood by everyone?

The first two questions are natural extensions of our analysis of counter-
monotonic risk sharing. While we analyze the competitive equilibria with risk-
seeking agents in Appendix A, it is unclear to us if and how these questions
can be answered in the general case. The key issue is finding a price vector,
as this is usually done using fixed-point theorems relying on some continuity
property of the excess demand correspondence. Unfortunately, we do not
currently know how the excess demand correspondences behave in general, as the
Pareto optimality of counter-monotonic allocations requires “bang-bang”
behaviour of the underlying preferences of agents.

We have many reasons to believe that the counter-monotonicity of Pareto-
optimal allocations is possible when sharing risk under heterogeneous beliefs
or ambiguity. Our stochastic representation of counter-monotonicity holds on
general measurable spaces, and a superficial look at the no-betting
allocations literature suggests that imposing strong assumptions on the
beliefs held by agents might do the trick. In the case of ambiguity-averse
agents, Billot et al. (2000) suggests that the emptiness of the intersection
of the core of the agents’ capacity is likely to be a necessary condition for
the Pareto optimality of counter-monotonic allocations. This condition is
unlikely to be sufficient, as this assumption does not rule out the
comonotonicity of some agents’ allocation. We hope that further investigations
will shed light on this issue, as a characterization of extreme betting
behaviour with ambiguity-averse agents would be highly counter-intuitive.

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##  Appendix A Competitive equilibria with risk-seeking agents

Fix an atomless probability space $(\Omega,\mathcal{F},\mathbb{P})$. Let
$L^{\infty}_{+}$ be the set of nonnegative random variables in $L^{\infty}$
which are not constantly $0$.

######  Assumption 4.

All agents are expected utility agents with a common utility function $u$,
which is convex on $[0,\infty)$. The total wealth in the economy is $X\in
L^{\infty}_{+}$, and the vector of initial endowment, denoted by
$(\xi_{1},\dots,\xi_{n})\in\mathbb{A}_{n}(X)$, has nonnegative components.

We always make the above assumption.

###  A.1 Explicit construction of the equilibria

A pricing measure is a probability measure $Q$ with $Q(X>0)=1$. Consider the
individual optimization problem for agent $i\in[n]$:

$\displaystyle\mbox{maximize}~{}\mathbb{E}[u(X_{i})]~{}~{}~{}~{}\mbox{subject
to}~{}\mathbb{E}^{Q}[X_{i}]\leq\mathbb{E}^{Q}[\xi_{i}];~{}0\leq X_{i}\leq X.$
(7)

The tuple $(X_{1},\dots,X_{n},Q)$ is a _competitive equilibrium_ if (a)
individual optimality: $X_{i}$ solves (7) for each $i\in[n]$; and (b) market
clearance: $\sum_{i=1}^{n}X_{i}=X$. In this case, $(X_{1},\dots,X_{n})$ is an
_equilibrium allocation_ , and $Q$ is an _equilibrium pricing measure_.

######  Proposition 11.

Let $Q$ be given by

$\displaystyle\frac{\,\mathrm{d}Q}{\,\mathrm{d}\mathbb{P}}=\frac{u(X)}{X}\frac{1}{\mathbb{E}[u(X)/X]}\mbox{
~{}~{} with the convention $0/0=0$,}$ (8)

and let

$\displaystyle(X_{1},\dots,X_{n})=(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}})$
(9) $\displaystyle\mbox{~{}for some $(A_{1},\dots,A_{n})\in\Pi_{n}$ such that
$\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]=\mathbb{E}^{Q}[\xi_{i}]$ for $i\in[n]$}.$

Then $(X_{1},\dots,X_{n},Q)$ is a competitive equilibrium.

###### Proof.

Denote by $x_{i}=\mathbb{E}^{Q}[\xi_{i}]$ and $z=\mathbb{E}[u(X)/X]\geq 0$. It
follows that

$\mathbb{E}^{Q}[X_{i}]=\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]=x_{i},$

and hence the budget constraint is satisfied for each $i\in[n]$. Moreover,

$\mathbb{E}[u(X_{i})]=\mathbb{E}[u(X)\mathds{1}_{A_{i}}]=\mathbb{E}\left[X\frac{u(X)}{X}\mathds{1}_{A_{i}}\right]=z\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]=zx_{i}.$

For any $Y_{i}$ satisfying $0\leq Y_{i}\leq X$ and the budget constraint
$\mathbb{E}^{Q}\left[Y_{i}\right]\leq{x_{i}}$, using the fact that $x\mapsto
u(x)/x$ is increasing, we have

$\mathbb{E}[u(Y_{i})]=\mathbb{E}\left[X_{i}\frac{u(Y_{i})}{Y_{i}}\right]\leq\mathbb{E}\left[Y_{i}\frac{u(X)}{X}\right]=z{\mathbb{E}^{Q}\left[Y_{i}\right]}\leq
z{x_{i}}=\mathbb{E}[u(X_{i})].$

Therefore, $(X_{1},\dots,X_{n},Q)$ satisfies individual optimality. Market
clearance also holds, because $\sum_{i=1}^{n}\mathds{1}_{A_{i}}=1.$ ∎

######  Remark 3.

The proof of Proposition 11 only requires $x\mapsto u(x)/x$ to be an
increasing function. Under our assumption $u(0)=0$, this is a weaker condition
than convexity of $u$.

###  A.2 Uniqueness of the equilibrium

Let $\mathcal{L}$ be the set of random variables $Y$ in $L^{\infty}_{+}$ such
that there exists a standard uniform random variable independent of $Y$.

######  Theorem 4.

Suppose that $u$ is strictly convex on $[0,\infty)$ and $X\in\mathcal{L}$.

  1. (i)

All equilibrium allocations $(X_{1},\dots,X_{n})$ have the form (LABEL:eq:X).

  2. (ii)

If at least two of $\xi_{1},\dots,\xi_{n}$ are not $0$, then the equilibrium
pricing measures is uniquely given by (8).

###### Proof.

(i) Let $(X_{1},\dots,X_{n},Q)$ be a competitive equilibrium. By the counter-
monotonic improvement theorem, there exists an allocation
$(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that $Y_{i}\geq_{\rm cx}X_{i}$
and $Y_{i}\geq 0$ for each $i\in[n]$. If
$\mathbb{E}^{Q}[Y_{i}]<\mathbb{E}^{Q}[\xi_{i}]$ for some $i\in[n]$, then there
exists $b>0$ such that
$\mathbb{E}^{Q}[Y_{i}+b(X-Y_{i})]\leq\mathbb{E}^{Q}[\xi_{i}]$. Hence,
$Y_{i}+b(X-Y_{i})$ satisfies the budget constraint for agent $i$. Moreover, by
strict convexity of $u$ (implying strict increasing monotonicity), we have

$\mathbb{E}[u(Y_{i}+b(X-Y_{i}))]>\mathbb{E}[u(Y_{i})]\geq\mathbb{E}[u(X_{i})],$

and hence $X_{i}$ is not optimal for agent $i$, a contradiction. Therefore, we
conclude that $\mathbb{E}^{Q}[Y_{i}]\geq\mathbb{E}^{Q}[\xi_{i}]$ for all
$i\in[n]$. Since
$\sum_{i=1}^{n}\mathbb{E}^{Q}[Y_{i}]=\mathbb{E}^{Q}[X]=\sum_{i=1}^{n}\mathbb{E}^{Q}[X_{i}]$,
we further obtain $\mathbb{E}^{Q}[Y_{i}]=\mathbb{E}^{Q}[\xi_{i}]$ for all
$i\in[n]$. Hence, $(Y_{1},\dots,Y_{n})$ satisfies the budget constraint. For
each $i\in[n]$, individual optimality gives
$\mathbb{E}[u(X_{i})]\geq\mathbb{E}[u(Y_{i})]$ and convex order gives
$\mathbb{E}[u(X_{i})]\leq\mathbb{E}[u(Y_{i})]$, together leading to
$\mathbb{E}[u(X_{i})]=\mathbb{E}[u(Y_{i})]$, and thus
$Y_{i}\buildrel\mathrm{d}\over{=}X_{i}$. The rest of the proof follows from
the same argument as Theorem 2, justifying that $(X_{1},\dots,X_{n})$ is a
jackpot allocation, thus with the form
$(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}})$ in (LABEL:eq:X).

(ii) Suppose that $(X_{1},\dots,X_{n},Q)$ is a competitive equilibrium. Using
(i), we can write
$(X_{1},\dots,X_{n})=(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}})$ for some
$(A_{1},\dots,A_{n})\in\Pi_{n}$ in (LABEL:eq:X). Let $P$ be the conditional
probability measure of $\mathbb{P}$ on $\\{X>0\\}$, and let
$p=\mathbb{P}(X>0)$. Let $\eta=\,\mathrm{d}Q/\,\mathrm{d}P$ and define a
probability measure $R$ by

$\frac{\,\mathrm{d}R}{\,\mathrm{d}Q}=\frac{X}{c},~{}\mbox{where
$c=\mathbb{E}^{Q}[X]$},$

and Note that for any $A\in\mathcal{F}$, we have

$\mathbb{E}[u(X\mathds{1}_{A})]=\frac{1}{p}\mathbb{E}^{P}[u(X\mathds{1}_{A})]=\frac{1}{p}\mathbb{E}^{R}\left[\frac{\,\mathrm{d}P}{\,\mathrm{d}Q}\frac{\,\mathrm{d}Q}{\,\mathrm{d}R}u(X)\mathds{1}_{A}\right]=\frac{1}{p}\mathbb{E}^{R}\left[\frac{cu(X)}{\eta
X}\mathds{1}_{A}\right].$

Denote by $Z=cu(X)/(\eta X)$. Individual optimality of $(X_{1},\dots,X_{n})$
implies that for any $i\in[n]$ and any $A\in\mathcal{F}$ satisfying
$\mathbb{E}^{Q}[X\mathds{1}_{A}]\leq\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]$, we
have

$\mathbb{E}^{R}\left[Z\mathds{1}_{A}\right]=p\mathbb{E}[u(X\mathds{1}_{A})]\leq
p\mathbb{E}[u(X\mathds{1}_{A_{i}})]=\mathbb{E}^{R}\left[Z\mathds{1}_{A_{i}}\right].$

Note that
$\mathbb{E}^{Q}[X\mathds{1}_{A}]\leq\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]$ is
equivalent to $R(A)\leq R(\mathds{1}_{A_{i}})$. Take $A\in\mathcal{F}$ such
that $R(A)=R(\mathds{1}_{A_{i}})$ and $Z$ and $\mathds{1}_{A}$ are
comonotonic. Suppose that $Z$ is not a constant. The Fréchet-Hoeffding
inequality gives

$\mathrm{cov}(Z,\mathds{1}_{A_{i}})\geq\mathrm{cov}(Z,\mathds{1}_{A})\geq 0,$

and $\mathrm{cov}(Z,\mathds{1}_{A})>0$ if $R({A})\in(0,1)$. Since at least two
of $\xi_{1},\dots,\xi_{n}$ are not $0$, by (LABEL:eq:X), at least two of
${A_{1}},\dots,{A_{n}}$ have positive probability under $R$. Therefore,
$\mathrm{cov}(Z,\mathds{1}_{A_{i}})>0$ for at least one $i$. However,
$\sum_{i=1}^{n}\mathrm{cov}(Z,\mathds{1}_{A_{i}})=\mathrm{cov}(Z,1)=0$, a
contradiction. Hence, $Z$ is a constant. Therefore, $\eta$ is equal to a
constant times $u(X)/X$, showing that $Q$ has the form (8). ∎

In case only one of $\xi_{1},\dots,\xi_{n}$ is not $0$, say $\xi_{i}$, the
equilibrium allocation is $X_{i}=X$ and $X_{j}=0$ for
$j\in[n]\setminus\\{i\\}$, and the equilibrium pricing measure is arbitrary.

The equilibrium pricing density $\,\mathrm{d}Q/\,\mathrm{d}\mathbb{P}$ is
increasing in $X$, and which is more expensive for states with larger $X$.
This is in sharp contrast to the case of risk-averse expected utility agents,
where the pricing density is cheaper for states with larger $X$.

###  A.3 Existence of the equilibrium

The next lemma shows that the competitive equilibrium in Proposition 11 always
exists even without assuming the existence of a uniform random variable
independent of $X$.

######  Lemma 3.

For any probability $Q$, there exists $(A_{1},\dots,A_{n})\in\Pi_{n}$
satisfying (LABEL:eq:X).

###### Proof.

Let $U$ be a uniform transform of $X$ (i.e., $F^{-1}_{X}(U)=X$ a.s. and $U$ is
uniformly distributed on $[0,1]$), and let $A_{1}=\\{0\leq U\leq a_{1}\\}$
where $a_{1}$ satisfies

$\int X\mathds{1}_{\\{0\leq U\leq
a_{1}\\}}\,\mathrm{d}Q=\mathbb{E}^{Q}[\xi_{1}].$

Since $a\mapsto\int X\mathds{1}_{\\{0\leq U\leq a\\}}\,\mathrm{d}Q$ is
continuous on $[0,1]$ and takes value in $[0,\mathbb{E}^{Q}[X]]$, such $a_{1}$
exists. Next, let $A_{2}=\\{a_{1}\leq U\leq a_{2}\\}$ where $a_{2}$ satisfies

$\int X\mathds{1}_{\\{a_{1}\leq U\leq
a_{2}\\}}\,\mathrm{d}Q=\mathbb{E}^{Q}[\xi_{2}].$

Since $a\mapsto\int X\mathds{1}_{\\{a_{1}\leq U\leq a\\}}\,\mathrm{d}Q$ is
continuous on $[a_{1},1]$ and takes value in
$[0,\mathbb{E}^{Q}[X]-\mathbb{E}^{Q}[\xi_{1}]]$ (note that
$\mathbb{E}^{Q}[X]-\mathbb{E}^{Q}[\xi_{1}]\geq\mathbb{E}^{Q}[\xi_{2}]$), such
$a_{2}$ exists. Repeating this procedure yields the desirable
$(A_{1},\dots,A_{n})$. ∎

Let $Q$ be given in (8). Suppose $X\in\mathcal{L}$. In this case, the
composition in Lemma 3 is much simpler: we can take $A_{i}$ as an event
independent of $X$ with probability
$\mathbb{E}^{Q}[\xi_{i}]/\mathbb{E}^{Q}[X]$ for each $i\in[n]$. Then,
$(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}},Q)$ is a competitive
equilibrium. It has a simple interpretation: the probability of winning the
jackpot reward for agent $i$ is $\mathbb{E}^{Q}[\xi_{i}]$, which is
proportional to $\mathbb{E}[\xi_{i}u(X)/X]$. In particular, if $\xi_{i}=X/n$
for each $i\in[n]$, then we obtain a fair (exchangeable) jackpot allocation.

###  A.4 Welfare theorems

We now establish the first welfare theorem.

######  Proposition 12.

Every equilibrium allocation of $X\in\mathcal{L}$ is Pareto optimal.

###### Proof.

By Theorem 4, any equilibrium allocation $(X_{1},\dots,X_{n})$ has the form
$(X_{1},\dots,X_{n})=(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}})$ for some
$(A_{1},\dots,A_{n})\in\Pi_{n}$. Note that

$\sum_{i=1}^{n}\mathbb{E}[u(X_{i})]=\sum_{i=1}^{n}\mathbb{E}[u(X)\mathds{1}_{A_{i}}]=\mathbb{E}[u(X)].$

Convexity of $u$ implies $u(x+y)\geq u(x)+u(y)$ for all $x,y\geq 0$. For any
allocation $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$,

$\sum_{i=1}^{n}\mathbb{E}[u(Y_{i})]=\mathbb{E}\left[\sum_{i=1}^{n}u(Y_{i})\right]\leq\mathbb{E}\left[u\left(\sum_{i=1}^{n}Y_{i}\right)\right]=\mathbb{E}[u(X)].$

Therefore, $(X_{1},\dots,X_{n})$ is sum-optimal, and hence Pareto optimal. ∎

Proposition 12 also shows that all jackpot allocations are Pareto optimal in
this setting. Next we establish the second welfare theorem.

######  Proposition 13.

Suppose that $u$ is strictly convex on $[0,\infty)$. Every Pareto-optimal
allocation of $X\in\mathcal{L}$ is an equilibrium allocation for some initial
endowments.

###### Proof.

Suppose that $(X_{1},\dots,X_{n})$ is a Pareto-optimal allocation of $X$. By
Theorem 2, every Pareto-optimal allocation is a jackpot allocation; that is,
it admits a representation
$(X_{1},\dots,X_{n})=(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}})$ for some
$(A_{1},\dots,A_{n})\in\Pi_{n}$. Let $Q$ be given by (8). Further, let

$a_{i}=\frac{\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]}{\mathbb{E}^{Q}[X]}$

and $\xi_{i}=a_{i}X$ for each $i\in[n]$. It follows that

$\mathbb{E}^{Q}[X_{i}]=\mathbb{E}^{Q}[X\mathds{1}_{A_{i}}]=a_{i}\mathbb{E}^{Q}[X]=\mathbb{E}^{Q}[\xi_{i}].$

Therefore, (LABEL:eq:X) is satisfied. Using Proposition 11, we get that
$(X_{1},\dots,X_{n},Q)$ is a competitive equilibrium. ∎

To summarize all results, we obtain the following theorem.

######  Theorem 5.

Suppose that $u$ is strictly convex on $[0,\infty)$. For an allocation of
$X\in\mathcal{L}$, the following are equivalent.

  1. (i)

It is Pareto optimal;

  2. (ii)

it is an equilibrium allocation for some initial endowments;

  3. (iii)

it is a jackpot allocation.

##  Appendix B Omitted proofs

###  B.1 Proofs of Section 3

###### Proof of Proposition 2.

The “if” part follows from the fact that $\sum_{i=1}^{n}X_{i}=X$ and Lauzier
et al. (2023a, Theorem 1). We will show the “only if” part.

Assume $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is counter-monotonic. By
Lauzier et al. (2023a, Theorem 1), there exists
$(A_{1},\dots,A_{n})\in\Pi_{n}$ such that

$X_{i}=(X-m)\mathds{1}_{A_{i}}+m_{i}~{}~{}~{}\mbox{for all $i\in[n]$},$

where either $m_{i}=\operatorname*{ess\text{-}inf}X_{i}$ for $i\in[n]$ or
$m_{i}=\operatorname*{ess\text{-}sup}X_{i}$ for $i\in[n]$, and
$m=\sum_{i=1}^{n}m_{i}$. If $m_{i}=\operatorname*{ess\text{-}inf}X_{i}$ for
all $i\in[n]$, we have
$m=\sum_{i=1}^{n}\operatorname*{ess\text{-}inf}(X_{i})\leq\operatorname*{ess\text{-}inf}(\sum_{i=1}^{n}X_{i})\leq\operatorname*{ess\text{-}inf}X$.
If $m_{i}=\operatorname*{ess\text{-}sup}X_{i}$ for all $i\in[n]$, we have
$m=\sum_{i=1}^{n}\operatorname*{ess\text{-}sup}(X_{i})\geq\operatorname*{ess\text{-}sup}(\sum_{i=1}^{n}X_{i})\geq\operatorname*{ess\text{-}sup}X$.
∎

###### Proof of Proposition 3.

Let $(Z_{1},\dots,Z_{n})$ be a probabilistic mixture of $(X_{1},\dots,X_{n})$
and $(Y_{1},\dots,Y_{n})$, which are two jackpot allocations. It follows that
$Z_{i}\geq 0$ for all $i\in[n]$ and $\mathbb{P}(Z_{i}\wedge
Z_{j}>0)=\lambda\mathbb{P}(X_{i}\wedge
X_{j}>0)+(1-\lambda)\mathbb{P}(Y_{i}\wedge Y_{j}>0)=0$ for $i\neq j$.
Therefore, using (6) we know that $(Z_{1},\dots,Z_{n})$ is a jackpot
allocation. ∎

###### Proof of Theorem 1.

The case $X=0$ is trivial and will be excluded below. Let $U$ be a standard
uniform random variable independent of $X$. First we argue that we can assume
that $U$ is independent of $X_{1},\dots,X_{n}$. Otherwise, we can find two iid
standard uniform random variables $U$ and $V$ independent of $X$, and
$(\hat{X}_{1},\dots,\hat{X}_{n})$ measurable to $\sigma(X,V)$ such that
$(\hat{X}_{1},\dots,\hat{X}_{n},X)$ is identically distributed to
$(X_{1},\dots,X_{n},X)$. Clearly, $U$ is independent of
$(\hat{X}_{1},\dots,\hat{X}_{n},X)$, and all desired statements follow if we
could prove them for $(\hat{X}_{1},\dots,\hat{X}_{n})$.

Write

$Z_{i}=\frac{\sum_{j=1}^{i}X_{j}}{X}\mathds{1}_{\\{X>0\\}}~{}\mbox{for
$i\in[n]$ and $Z_{0}=0$.}$

Define the event $A_{i}=\\{Z_{i-1}\leq U<Z_{i}\\}$ for $i\in[n]$. Clearly,
$A_{1},\dots,A_{n}$ are disjoint and $\mathbb{P}(\bigcup_{i\in[n]}A_{i})=1$.
Let $Y_{i}=X\mathds{1}_{A_{i}}~{}\mbox{for $i\in[n]$}.$ It is clear that
$(Y_{1},\dots,Y_{n})$ is counter-monotonic. For $i\in[n]$,

$\displaystyle\mathbb{E}\left[Y_{i}\mid X_{1},\dots,X_{n}\right]$
$\displaystyle=\mathbb{E}\left[X\mathds{1}_{\\{Z_{i-1}\leq U<Z_{i}\\}}\mid
X_{1},\dots,X_{n}\right]$ $\displaystyle=\mathbb{E}\left[X(Z_{i}-Z_{i-1})\mid
X_{1},\dots,X_{n}\right]=X\frac{X_{i}}{X}\mathds{1}_{\\{X>0\\}}=X_{i},$

where in the last equality we used the fact that $X_{i}=0$ if $X=0$.
Therefore, $X_{i}$ is a conditional expectation of $Y_{i}$, yielding the
desired order $X_{i}\leq_{\rm cx}Y_{i}$ via Jensen’s inequality. ∎

###  B.2 Proofs of Section 4

###### Proof of Theorem 2.

Suppose $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is a Pareto-optimal
allocation. By Theorem 1, there is a jackpot allocation
$(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that for all $i\in[n]$ and
$Y_{i}\geq_{\rm cx}X_{i}$. As $u_{i}$ is strictly convex, we have
$\mathbb{E}[u(Y_{i})]=\mathbb{E}[u(X_{i})]$ by Pareto optimality of
$(X_{1},\dots,X_{n})$. By Shaked and Shanthikumar (2007, Theorem 3.A.43), we
obtain that $Y_{i}=_{\mathrm{st}}X_{i}$ in the usual stochastic order, and
thus $Y_{i}\buildrel\mathrm{d}\over{=}X_{i}$.

We now want to show that $(X_{1},\dots,X_{n})$ is counter-monotonic. Let
$\mathrm{var}$ and $\mathrm{cov}$ denote respectively the variance and
covariance. For any given $1\leq i<j\leq n$, we have that $Y_{i}$ and $Y_{j}$
are counter-monotonic, $X_{i}\buildrel\mathrm{d}\over{=}Y_{i}$ and
$X_{j}\buildrel\mathrm{d}\over{=}Y_{j}$. Therefore,
$\mathrm{cov}(X_{i},X_{j})\geq\mathrm{cov}(Y_{i},Y_{j})$. Furthermore, by the
fact that $\sum_{i=1}^{n}X_{i}=\sum_{i=1}^{n}Y_{i}=X$, we have

$\displaystyle\mathrm{var}(X)=\mathrm{var}\left(\sum_{i=1}^{n}Y_{i}\right)=\sum_{i=1}^{n}\sum_{j=1}^{n}\mathrm{cov}(Y_{i},Y_{j})\leq\sum_{i=1}^{n}\sum_{i=1}^{n}\mathrm{cov}(X_{i},X_{j})=\mathrm{var}\left(\sum_{i=1}^{n}X_{i}\right)=\mathrm{var}\left(X\right).$

Hence, $\mathrm{cov}(X_{i},X_{j})=\mathrm{cov}(Y_{i},Y_{j})$ for $i,j\in[n]$.

By the Hoeffding’s identity, we have for all $i\neq j$ and

$\displaystyle\iint\left(\mathbb{P}\left(X_{i}\leq t,X_{j}\leq
s\right)-\mathbb{P}\left(Y_{i}\leq t\right)\mathbb{P}\left(Y_{j}\leq
s\right)\right)\,\mathrm{d}t\,\mathrm{d}s$
$\displaystyle=\iint\left(\mathbb{P}\left(Y_{i}\leq t,Y_{j}\leq
s\right)-\mathbb{P}\left(X_{i}\leq t\right)\mathbb{P}\left(X_{j}\leq
s\right)\right)\,\mathrm{d}t\,\mathrm{d}s.$

Given that $(Y_{i},Y_{j})$ and $(X_{i},X_{j})$ have the same marginals and
$(Y_{i},Y_{j})$ is counter-monotonic, we have $\mathbb{P}\left(X_{i}\leq
t,X_{j}\leq s\right)\geq\mathbb{P}\left(Y_{i}\leq t,Y_{j}\leq s\right)$.
Therefore, $\mathbb{P}\left(X_{i}\leq t,X_{j}\leq
s\right)=\mathbb{P}\left(Y_{i}\leq t,Y_{j}\leq s\right)$ for almost every
$t,s\in\mathbb{R}$. Thus, for every $i\neq j$, it is
$\mathbb{P}\left(X_{i}>0,X_{j}>0\right)=\mathbb{P}\left(Y_{i}>0,Y_{j}>0\right)=0$
and $(X_{1},\dots,X_{n})$ is counter-monotonic, and further it is a jackpot
allocation by (6), as desired. ∎

###### Proof of Proposition 4.

(i) It is clear that any comonotonic allocation of $x$ is the set of
$(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ such that $\sum_{i=1}^{x}X_{i}=x$. We
first assume that the allocation $(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ is not
Pareto optimal; that is, there exists
$(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(x)$ such that
$\mathbb{E}[u_{i}(Y_{i})]\geq\mathbb{E}[u(x_{i})]=u(x_{i})$ for all $i\in[n]$,
with strict inequality for some $i\in[n]$. Let $y_{i}=\mathbb{E}[Y_{i}]$ for
$i\in[n]$. We have that $(y_{1},\dots,y_{n})\in\mathbb{A}_{n}(X)$ because

$\sum_{i=1}^{n}y_{i}=\sum_{i=1}^{n}\mathbb{E}[Y_{i}]=\mathbb{E}\left[\sum_{i=1}^{n}Y_{i}\right]=\mathbb{E}[x]=x.$

Since $u_{i}$ is concave we obtain that $u(y_{i})\geq\mathbb{E}[u(Y_{i})]\geq
u(x_{i})$ for all $i\in[n]$, with $u(y_{i})>u(x_{i})$ for some $i\in[n]$.
Furthermore, as $u_{i}$ is strictly increasing, we have $y_{i}\geq x_{i}$ for
all $i\in[n]$ and $y_{i}>x_{i}$ for some $i\in[n]$. Therefore,
$\sum_{i=1}^{n}y_{i}>\sum_{i=1}^{n}x_{i}=x$, a contradiction. Hence, the
allocation $(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ is Pareto optimal.

(ii) Let $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(x)$ be a counter-monotonic
allocation of $x$ satisfying $m_{i}=0$ for all $i\in[n]$. By Proposition 2 we
have

$(X_{1},\dots,X_{n})=\left(x\mathds{1}_{A_{1}},\dots,x\mathds{1}_{A_{n}}\right)\mbox{
for some }\left(A_{1},\dots,A_{n}\right)\in\Pi_{n}.$

Let $p_{i}=\mathbb{P}(X_{i})$ for $i\in[n]$. We have $\sum_{i=1}^{n}p_{i}=1$.

Assume that $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(x)$ is not Pareto optimal.
There is a $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(x)$ such that
$\mathbb{E}[u_{i}(Y_{i})]\geq\mathbb{E}[u_{i}(X_{i})]=p_{i}u_{i}(x)+(1-p_{i})u(0)$
for all $i\in[n]$, with strict inequalities for some $i\in[n]$. It is clear
that $Y_{i}\geq 0$ for all $i\in[n]$. By Theorem 1, we can always find a
nonnegative counter-monotonic allocation
$(Y^{\prime}_{1},\dots,Y^{\prime}_{n})\in\mathbb{A}_{n}(x)$ such that
$Y^{\prime}_{i}\geq_{\rm cx}Y_{i}$. As $\sum_{i=1}^{n}Y^{\prime}_{i}=x$, we
have

$Y^{\prime}_{i}=(x-m)\mathds{1}_{B_{i}}+m_{i}\mbox{ for some }m_{1},\dots
m_{n}\geq 0,m=\sum_{i=1}^{n}m\leq x,\mbox{ and
}\left(B_{1},\dots,B_{n}\right)\in\Pi_{n}.$

Furthermore, it is clear that the allocation
$\left(\hat{Y}_{1},\dots,\hat{Y}_{n}\right)=\left(x\mathds{1}_{B_{1}},\dots,x\mathds{1}_{B_{n}}\right)\in\mathbb{A}_{n}(x)$
satisfies
$\mathbb{E}\left[u_{i}\left(\hat{Y}_{i}\right)\right]\geq\mathbb{E}\left[u_{i}\left(Y^{\prime}_{i}\right)\right]\geq\mathbb{E}[u_{i}(X_{i})]$
for all $i\in[n]$ and
$\mathbb{E}\left[u_{i}\left(\hat{Y}_{i}\right)\right]>\mathbb{E}[u_{i}(X_{i})]$
for some $i\in[n]$. Let $q_{i}=\mathbb{P}(B_{i})$ for $i\in[n]$ so we have
$\sum_{i=1}^{n}q_{i}=1$. On the other hand, we also have

$q_{i}u_{i}(x)+(1-q_{i})u_{i}(0)=\mathbb{E}\left[u_{i}\left(\hat{Y}_{i}\right)\right]\geq\mathbb{E}[u(X_{i})]=p_{i}u_{i}(x)+(1-p_{i})u_{i}(0)$

for all $i\in[n]$ and strictly inequalities hold for some $i\in[n]$. That is,
we have $q_{i}\geq p_{i}$ for all $i\in[n]$ and $q_{i}>p_{i}$ for some
$i\in[n]$. As a result, $\sum_{i=1}^{n}q_{i}>\sum_{i=1}^{n}p_{i}=1$, a
contradiction. Hence, $(X_{1},\dots,X_{n})$ is Pareto optimal. ∎

###  B.3 Proofs of Section 5

###### Proof of Proposition 5.

The allocation $(X_{1},\dots,X_{n})$ is feasible because
$\sum_{i=1}^{n}X_{i}=n-1+(X-(n-1))=X$. Notice that since for all $i\in[n-1]$
it is $\mathbb{P}(X_{i}=1)=1$ and since $\mathbb{P}(X_{n}\geq
1)=\mathbb{P}(X\geq n)$ it holds that

$\sum_{i=1}^{n}\rho_{i}(X_{i})=\sum_{i=1}^{n}\mathbb{P}(X_{i}\geq
1)=n-1+\mathbb{P}(X\geq n).$

Consider an alternative allocation $(Y_{1},Y_{2},\dots,Y_{n})$ that satisfies
the feasibility condition $\sum_{i=1}^{n}Y_{i}=X$. It is

$\displaystyle\sum_{i=1}^{n}\rho_{i}(Y_{i})=\sum_{i=1}^{n}\mathbb{P}(Y_{i}\geq
1)=\mathbb{E}\left[\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq 1\\}}\right]\leq
n-1+\mathbb{P}(X\geq n)$

and $(Y_{1},\dots,Y_{n})$ cannot strictly improve upon $(X_{1},\dots,X_{n})$.
∎

###### Proof of Proposition 6.

Suppose by contraposition that $\mathbf{X}^{*}$ does not maximize
$f_{\mathbf{1}}(\mathbf{X})$. Recalling that
$\max_{\mathbf{X}}f_{\mathbf{1}}(\mathbf{X})=n-\mathbb{P}(X<n)$ we have that

$n-\mathbb{P}(X<n)>\mathbb{E}\left[\sum_{i=1}^{n}\mathds{1}_{\\{X_{i}^{*}\geq
1\\}}\right]=\sum_{i=1}^{n}p_{i}$

for $p_{i}=\mathbb{P}(X_{i}\geq 1)\in[0,1]$. Rearranging the previous
inequality yields $\sum_{i=1}^{n}(1-p_{i})>\mathbb{P}(X<n)$. Let
$\lambda_{1},\dots,\lambda_{n}$ be in decreasing order and notice that

$\sum_{i=1}^{n}\lambda_{i}(1-p_{i})\geq\sum_{i=1}^{n}\lambda_{n}(1-p_{i})>\lambda_{n}\mathbb{P}(X<n).$

Rearranging the previous inequality we have

$\max_{\mathbf{X}}f_{\bm{\lambda}}\left(\mathbf{X}\right)\geq\sum_{i=1}^{n}\lambda_{i}-\lambda_{n}\mathbb{P}(X<n)>\sum_{i=1}^{n}\lambda_{i}p_{i}=\mathbb{E}\left[\sum_{i=1}^{n}\lambda_{i}\mathds{1}_{\\{X_{i}^{*}\geq
1\\}}\right]=f_{\bm{\lambda}}(\mathbf{X}^{*}),$

and hence $\mathbf{X}^{*}$ does not maximize $f_{\bm{\lambda}}(\mathbf{X})$. ∎

###### Proof of Proposition 7.

Suppose by contradiction that $(X_{1},\dots,X_{n})$ is a comonotonic, fair and
optimal allocation.

Fairness implies that $\mathbb{P}(X_{1}\geq 1)=\dots=\mathbb{P}(X_{n}\geq 1)$.
We claim that comonotonicity of $X_{1},\dots,X_{n}$ implies that $\\{X_{i}\geq
1\\}=\\{X_{j}\geq 1\\}$ for any $i,j=1,\dots,n$; otherwise, there exist
$\omega\in\\{X_{i}\geq 1\\}\setminus\\{X_{j}\geq 1\\}$ and
$\omega^{\prime}\in\\{X_{j}\geq 1\\}\setminus\\{X_{i}\geq 1\\}$ such that
$(X_{i}(\omega)-X_{j}(\omega))(X_{i}(\omega^{\prime})-X_{j}(\omega^{\prime}))<0$.
Therefore, $\mathbb{P}(X_{i}\geq 1)=\mathbb{P}(X\geq n)$ for all
$i=1,\dots,n$. This implies

$\mathbb{E}\left[\sum_{i=1}^{n}\mathds{1}_{\\{X_{i}\geq
1\\}}\right]=n(1-\mathbb{P}(X<n))<n-\mathbb{P}(X<n)$

which shows that $(X_{1},\dots,X_{n})$ is not optimal. ∎

###### Proof of Corollary 1.

Assume that $(X_{1},\dots,X_{n})\in\mathbb{A}_{n}(X)$ is a constrained Pareto-
optimal allocation. Let $A_{i}=\\{X_{i}\geq 1\\}$. As $X_{i}\geq 0$ and
$\mathbb{P}(X<2)=1$, we have $\mathbb{P}(A_{i}\cap A_{j})=0$ for all
$i,j\in[n]$ such that $i\neq j$; otherwise, we will have $\mathbb{P}(X\geq
2)>0$. Let $B_{i}=\left(\bigcap_{j\neq i}^{n}A^{c}_{j}\right)\cap A_{i}$. It
is clear that $B_{1},\dots,B_{n}$ are disjoint. Furthermore,

$\mathbb{P}(A_{i})=\mathbb{P}(B_{i})+\mathbb{P}\left(A_{i}\cap\left(\bigcup_{j\neq
i}^{n}A_{j}\right)\right)=\mathbb{P}(B_{i})+\mathbb{P}\left(\bigcup_{j\neq
i}^{n}A_{i}\cap A_{j}\right)=\mathbb{P}(B_{i}).$

Let $X=\sum_{i=1}^{n}X_{i}$. Take the allocation

$Y_{i}=X\mathds{1}_{B_{i}}~{}\mbox{for}~{}i\in[n-1]~{}\mbox{and}~{}Y_{n}=X\mathds{1}_{\Omega\setminus\bigcup_{i=1}^{n-1}B_{i}}.$

It is clear that $(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$. Note that
$B_{i}\subseteq A_{i}\subseteq\\{X\geq 1\\}$ for $i\in[n]$. For $i\in[n-1]$,

$\mathbb{P}(Y_{i}\geq 1)=\mathbb{P}(X\mathds{1}_{B_{i}}\geq
1)=\mathbb{P}\left(\\{X\geq 1\\}\cap
B_{i}\right)=\mathbb{P}(B_{i})=\mathbb{P}(X_{i}\geq 1).$

For $i=n$, as $B_{n}\subseteq\Omega\setminus\bigcup_{i=1}^{n-1}B_{i}$, we have

$\mathbb{P}(Y_{n}\geq 1)=\mathbb{P}\left(\\{X\geq
1\\}\cap\left(\Omega\setminus\bigcup_{i=1}^{n-1}B_{i}\right)\right)\geq\mathbb{P}(B_{n})=\mathbb{P}(X_{n}\geq
1).$

Hence, $\rho_{i}(Y_{i})\geq\rho_{i}(X_{i})$ for all $i\in[n]$. As
$(X_{1},\dots,X_{n})$ is Pareto optimal, we have
$\rho_{i}(Y_{i})=\rho_{i}(X_{i})$ for all $i\in[n]$ ∎

###### Proof of Proposition 8.

Fairness implies that $a\mathbb{E}[X_{1}]+\mathbb{P}(X_{1}\geq
1)=\dots=a\mathbb{E}[X_{n}]+\mathbb{P}(X_{n}\geq 1)$. We claim that
comonotonicity of $X_{1},\dots,X_{n}$ implies that $\\{X_{i}\geq
1\\}=\\{X_{j}\geq 1\\}$ for any $i,j=1,\dots,n$; otherwise, there exist
$\omega\in\\{X_{i}\geq 1\\}\setminus\\{X_{j}\geq 1\\}$ and
$\omega^{\prime}\in\\{X_{j}\geq 1\\}\setminus\\{X_{i}\geq 1\\}$ such that
$(X_{i}(\omega)-X_{j}(\omega))(X_{i}(\omega^{\prime})-X_{j}(\omega^{\prime}))<0$.
Therefore, $\mathbb{P}(X_{i}\geq 1)=\mathbb{P}(X\geq n)$ for all
$i=1,\dots,n$. Summing we have

$\displaystyle\sum_{i=1}^{n}(a\mathbb{E}[X_{i}]+\mathbb{P}(X_{i}\geq
1))=a\mathbb{E}[X]+\sum_{i=1}^{n}\mathbb{P}(X_{i}\geq
1)=a\mathbb{E}[X]+n\mathbb{P}(X\geq n).$

Since $\mathbb{P}(X\geq n)<1$, the allocation
$(Y_{1},\dots,Y_{n})=(\mathds{1}_{A_{i}^{C}}+\mathds{1}_{A_{i}}(X-(n-1)))_{i\in[n]}$
Pareto dominates $(X_{1},\dots,X_{n})$, where $(A_{1},\dots,A_{n})\in\Pi_{n}$
is such that $\mathbb{P}(A_{i})=1/n$ for all $i\in[n]$. ∎

###  B.4 Proofs of Section 6

###### Proof of Corollary 2.

Following the proof of Theorem 1, we have

$\mathbb{P}(A_{i})=\mathbb{E}[\mathbb{E}[\mathds{1}_{Z_{i-1}\leq
U<Z_{i}}|X_{1},\dots,X_{n}]]=\mathbb{E}\left[\frac{X_{i}}{X}\mathds{1}_{\\{X>0\\}}\right].$

Since $(X_{1},\dots,X_{n})$ is exchangeable, we have
$\mathbb{P}(A_{i})=\mathbb{P}(A_{j})$ for all $i\neq j$. Hence,
$(Y_{1},\dots,Y_{n})=\left(X\mathds{1}_{A_{1}},\dots,X\mathds{1}_{A_{n}}\right)$
is an exchangeable jackpot allocation with $Y_{i}\geq_{\rm cx}X_{i}$ for all
$i\in[n]$. As $A_{1},\dots,A_{n}$ are disjoint and $(Y_{1},\dots,Y_{n})$ is
exchangeable, we have

$n\mathbb{P}(Y_{i}>0)=\sum_{j=1}^{n}\mathbb{P}(Y_{j}>0)\leq\mathbb{P}\left(\bigcup_{i=1}^{n}A_{i}\right)=1$

for all $i\in[n]$. Hence, we have (iv). ∎

###### Proof of Theorem 3.

By Corollary 2, there exists an exchangeable jackpot allocation
$(Y_{1},\dots,Y_{n})\in\mathbb{A}_{n}(X)$ such that $Y_{i}\geq_{\rm cx}X_{i}$
and $\mathbb{P}(Y_{i}>0)\leq 1/n$ for all $i\in[n]$.

As $\bar{h}\geq h$, we have $\rho_{h}(X)\leq\rho_{\bar{h}}(X)$. Since
$\mathbb{P}(Y_{i}>0)\leq 1/n$, we have
$\mathbb{P}(u(Y_{i})>0)=\mathbb{P}(Y_{i}>0)\leq 1/n$. Given that $h=\bar{h}$
for $t\in[0,1/n]$, we have

$\displaystyle\rho_{h}(Y_{i})$
$\displaystyle=\int_{0}^{\infty}h\left(\mathbb{P}\left(u(Y_{i})>t\right)\right)\,\mathrm{d}t=\int_{0}^{\infty}\overline{h}\left(\mathbb{P}\left(u(Y_{i})>t\right)\right)\,\mathrm{d}t=\rho_{\overline{h}}(Y_{i}).$

As $Y_{i}\geq_{\rm cx}X_{i}$, we have $u(Y_{i})=aY_{i}\geq_{\rm
cx}aX_{i}=u(X_{i})$. By the fact that $\bar{h}$ is concave, we get
$\rho_{\bar{h}}(Y_{i})\geq\rho_{\bar{h}}(X_{i})$. In conclusion, we have
$\rho_{h}(X_{i})\leq\rho_{\bar{h}}(X_{i})\leq\rho_{\bar{h}}(Y_{i})=\rho_{h}(Y_{i})$
for all $i\in[n]$. ∎

###  B.5 Proofs of Section 7

###### Proof of Proposition 9.

Suppose there is a $j\neq i$ such that $a_{j}=P$. Then
$v\mathds{1}_{A_{i}}>_{\mathrm{cx}\frac{vc_{i}}{\sum_{j\in\mathrm{Po}}c_{j}}\mathds{1}_{\bigcup_{j\in\mathrm{Po}}A_{j}}}$
and by strict risk-aversion we have
$\rho_{i}\left(P,a_{-i}\right)>\rho_{i}\left(H,a_{-i}\right)$, as desired. ∎

###### Proof of Proposition 10.

(i) If $c_{i}\leq p_{0}$ then by Assumption 3 we have

$\rho_{i}(H,a_{-i})=u_{i}(v)h_{i}(c_{i})=u_{i}\left(\frac{vc_{i}}{c_{i}}\right)h_{i}(c_{i})\geq
u_{i}\left(\frac{vc_{i}}{\sum_{j\in{\rm
Po}}c_{j}}\right)h_{i}\left(\sum_{j\in{\rm
Po}}c_{j}\right)=\rho_{i}(P,a_{-i}).$

If there is at least one coplayer $j\neq i$ such that $a_{j}=P$ then the
strict inequality obtains whenever $\phi_{y}$ strictly decreases on
$(0,p_{0})$.

(ii) By concavity of $u$ we have $au(x/a)\geq bu(x/b)$ for $a>b$. Suppose
there is a $p^{*}\in[0,1]$ for which $h(t)/t$ is strictly increasing for
$t>p^{*}$. If $c_{i}\geq p^{*}$ and there is at least one agent in the pool,
we obtain

$\displaystyle\rho_{i}(H,a_{-i})=c_{i}u_{i}\left(\frac{vc_{i}}{c_{i}}\right)\frac{h_{i}(c_{i})}{c_{i}}<\left(\sum_{j\in{\rm
Po}}c_{j}\right)u_{i}\left(\frac{vc_{i}}{\sum_{j\in{\rm
Po}}c_{j}}\right)\frac{h_{i}\left(\sum_{j\in{\rm
Po}}c_{j}\right)}{\sum_{j\in{\rm Po}}c_{j}}=\rho_{i}(P,a_{-i}).$

Hence, $\sigma_{i}(a_{-i})=P$. ∎