Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures

This paper derives explicit solutions for unconstrained and counter-monotonic inf-convolutions among agents with heterogeneous distortion risk measures, including risk-seeking preferences, showing that these optimal allocations can be represented by a generalization of distortion risk measures.

The Gist

Imagine a group of friends deciding how to split a giant, unpredictable pizza. Some friends are picky eaters (risk-averse) who hate surprises and want to make sure everyone gets a fair, predictable slice. Others are thrill-seekers (risk-seeking) who love the gamble; they'd rather have a 50% chance of getting the whole pizza and a 50% chance of getting nothing, rather than a guaranteed half-slice.

This paper is about figuring out the best way to split that "pizza" (which represents financial risk or loss) when the group is full of these different types of people, specifically focusing on the thrill-seekers.

Here is the breakdown of the paper's story, using simple analogies:

1. The Two Ways to Split the Pie

In the world of finance, there are two extreme ways to share risk:

• The "Team Huddle" (Comonotonicity): Everyone moves together. If the pizza gets bigger, everyone gets more. If it gets smaller, everyone gets less. This is what picky eaters (risk-averse people) prefer. They want to stand together so no one gets left behind.
• The "Winner Takes All" (Counter-monotonicity): This is the opposite. If the pizza is huge, one person gets it all, and the others get crumbs. If the pizza is tiny, one person gets the tiny bit, and the others get nothing. The paper calls this a "Jackpot" (if you are lucky) or a "Scapegoat" (if you are unlucky) arrangement. This is what thrill-seekers (risk-loving people) prefer. They want to bet on who gets the big win.

2. The Old Rules vs. The New Discovery

For a long time, economists studied groups where everyone was a "picky eater." They found that the best way to share risk was always the "Team Huddle." Everyone moves together, and the math is straightforward.

However, this paper asks: What happens if everyone in the group is a thrill-seeker? And what if they are different kinds of thrill-seekers? (e.g., one loves high stakes, another loves medium stakes).

The authors discovered that when thrill-seekers mix, the rules change completely:

• The "Team Huddle" is useless: If thrill-seekers try to move together, they miss out on the fun of the gamble.
• The "Winner Takes All" is the winner: The best way for thrill-seekers to share risk is to split the "lottery ticket." Instead of everyone getting a little bit of the risk, they split the chance of the big win.

3. The "Probability Share" Concept

The paper's biggest "Aha!" moment is this: When thrill-seekers share a fixed amount of money (like a guaranteed $100 pot), they don't split the money. Instead, they split the probability of winning it.

Imagine a pot of $100.

• Risk-Averse people would say: "Let's split it $33, $33, $34."
• Risk-Seeking people say: "Let's flip a coin! One person gets the whole $100, and the others get $0. But we will decide who gets the $100 based on our different risk appetites."

The paper provides a mathematical formula to figure out exactly who gets the biggest slice of the probability pie.

• If you are extremely risk-loving, you might get a 60% chance of winning the whole pot.
• If you are mildly risk-loving, you might get a 40% chance.

4. The Surprising Twist (The Two-Person Problem)

The paper found a weird quirk when there are only two thrill-seekers.
Usually, you'd think the person who loves risk more should get the bigger chance of winning. But sometimes, the math says the person who loves risk less should get the bigger chance!

Why?
Imagine two people betting on a coin flip. If the "rules of the game" (the math of their preferences) are slightly curved in a specific way, the person who is less crazy might actually end up taking the bigger risk to balance things out. It's like a seesaw where the heavier person has to sit closer to the middle to make it balance, even if they are heavier.

5. Why This Matters

This isn't just about pizza or coins. This applies to:

• Insurance: How companies share huge disasters.
• Investment: How hedge funds split profits and losses.
• Banking: How banks manage their risk portfolios.

The paper gives a "recipe book" for these scenarios. It tells us that if you have a group of gamblers (risk-seekers), you shouldn't try to smooth out their risks. Instead, you should let them gamble against each other in a structured way, where the "winner" takes the whole risk, and the math tells you exactly who should be the winner.

The Bottom Line

• Picky Eaters (Risk-Averse): Stick together, share the load equally, move in sync.
• Thrill-Seekers (Risk-Seeking): Split the "lottery ticket." Let one person take the whole risk while the others take none, but assign the odds based on how much they love the gamble.
• The Math: The authors wrote the code to calculate exactly how to split those odds, even when the thrill-seekers are all different from one another.

In short: When you are dealing with gamblers, don't try to make them safe. Give them a structured game where the "Jackpot" goes to one person, and use math to decide who that person should be.

Technical Summary

Here is a detailed technical summary of the paper "Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures" by Ghossoub, Ren, and Wang.

1. Problem Statement

The paper investigates Pareto-optimal risk sharing among a group of $n$ agents who face an aggregate risk $X$. The core challenge addressed is the characterization of optimal allocations when:

• Agents are Heterogeneous: Agents possess different risk preferences modeled by distinct distortion risk measures ($\rho_{h_i}$).
• Agents are Risk-Seeking: Unlike traditional literature which focuses on risk-averse agents (concave distortion functions), this paper explicitly studies agents who are risk-seeking (modeled by convex distortion functions).
• Dependence Structures: The study focuses on counter-monotonic allocations (the extreme opposite of comonotonicity), where agents' risks move in opposite directions. This is motivated by the "counter-monotonic improvement theorem," which suggests risk-seeking agents prefer such allocations.

The primary objective is to derive explicit solutions for the inf-convolution (the minimum aggregate risk value achievable) under three constraints:

1. Unconstrained: Allocations in the set $A_n(X)$.
2. Comonotonic: Allocations in $A_n^+(X)$.
3. Counter-monotonic: Allocations in $A_n^-(X)$.

2. Methodology and Framework

2.1 Mathematical Tools

• Distortion Riskmetrics: The authors utilize the class of distortion riskmetrics $\rho_h$, defined via a distortion function $h \in HBV$ (bounded variation).
  • Risk aversion corresponds to concave $h$.
  • Risk seeking corresponds to convex $h$.
  • The paper distinguishes between risk measures (where $h$ is increasing and normalized) and riskmetrics (more general, allowing non-monotonicity).
• Inf-Convolution: The central operator is the inf-convolution of risk measures:
  $$\square_{i=1}^n \rho_i(X) = \inf \left\{ \sum_{i=1}^n \rho_i(X_i) : \sum_{i=1}^n X_i = X \right\}$$
  The paper analyzes constrained versions: $\boxplus$ (comonotonic) and $\boxminus$ (counter-monotonic).
• Stochastic Representation: For $n \ge 3$, counter-monotonicity is characterized by a "jackpot" or "scapegoat" structure (Proposition 1), where the aggregate risk $X$ is partitioned into disjoint events $A_i$ such that $X_i = X \mathbb{1}_{A_i} + m_i$.

2.2 Key Theoretical Mechanisms

• Counter-Monotonic Improvement Theorem (Theorem 1): Citing Lauzier et al. (2024), the paper establishes that for risk-seeking agents, any allocation can be improved (in the convex order) by a counter-monotonic "jackpot" allocation. This justifies restricting the search space to counter-monotonic allocations for risk-seeking agents.
• Dual Functions: The analysis relies heavily on the dual distortion function $\tilde{h}(t) = h(1) - h(1-t)$ and the relationship $\rho_h(X) = -\rho_{\tilde{h}}(-X)$.
• Function Convolution: The authors define the inf-convolution and sup-convolution of the distortion functions themselves (mapping $[0,1] \to \mathbb{R}$) to derive the resulting risk metric.

3. Key Contributions and Results

3.1 Heterogeneous Risk-Averse Agents (Concave $h$)

The paper extends previous homogeneous results to heterogeneous settings:

• Ordering of Inf-Convolutions: In the homogeneous case, the order is fixed ($\boxminus \ge \boxplus = \square$). In the heterogeneous case, the relationship between the counter-monotonic ($\boxminus$) and comonotonic ($\boxplus$) inf-convolutions depends on the specific distortion functions.
• Theorem 2: Establishes conditions under which $\boxminus \ge \boxplus$. Specifically, if the combined distortion function $h = \bigvee h_i$ is dually subadditive, then $\boxminus \ge \boxplus = \rho_h \ge \square$.
• Corollary 1: If all agents are risk-averse and one agent is "least risk-averse" (dominating the others), the optimal allocation assigns the entire risk to that agent, making all three inf-convolutions equal.

3.2 Heterogeneous Risk-Seeking Agents (Convex $h$) - Main Contribution

This is the novel core of the paper.

• Proposition 2: If agents are risk-seeking (convex $h$) and the space is $L^\infty$ (unbounded), the inf-convolution is $-\infty$. Therefore, the analysis is restricted to $L^+$ (non-negative losses) or $L^-$ (non-negative gains).
• Theorem 3 (Explicit Solution): For risk-seeking agents with continuous convex distortion functions $h_i$:
  • The unconstrained and counter-monotonic inf-convolutions are equal: $\square_{i=1}^n \rho_{h_i}(X) = \boxminus_{i=1}^n \rho_{h_i}(X)$.
  • The result is a distortion riskmetric $\rho_g(X)$, where $g$ is derived from the inf-convolution of the distortion functions:
    • If $X \in L^+$: $g = \square_{i=1}^n h_i$ (inf-convolution of functions).
    • If $X \in L^-$: $\tilde{g} = \diamond_{i=1}^n \tilde{h}_i$ (sup-convolution of dual functions).
  • Crucial Insight: The resulting aggregate risk measure is not a distortion risk measure (it may not be monotone), but a distortion riskmetric.
• Inequality: For risk-seeking agents, the ordering is reversed compared to risk-averse agents:
  $$\boxplus_{i=1}^n \rho_{h_i}(X) \ge \square_{i=1}^n \rho_{h_i}(X) = \boxminus_{i=1}^n \rho_{h_i}(X)$$
  This confirms that risk-seeking agents strictly prefer counter-monotonic allocations over comonotonic ones.

3.3 Sharing a Constant Payoff (Section 5.3)

The paper analyzes the case where $X$ is a constant (no aggregate uncertainty).

• Probability Sharing: The optimal allocation becomes a "probability sharing" problem where agents share the probability of bearing the total loss.
• Proposition 3: For power distortions $h_i(x) = x^{\alpha_i}$ ($\alpha_i > 1$):
  • If $n \ge 3$, the probability of bearing the loss is monotonically increasing with the risk-seeking parameter $\alpha_i$ (more risk-seeking agents take more risk).
  • If $n = 2$, this monotonicity fails. The allocation depends on the curvature of the functions near the symmetric point, not just the magnitude of $\alpha$. A less risk-seeking agent may paradoxically bear a larger share of the loss.

4. Significance and Implications

1. Theoretical Advancement: The paper bridges the gap between risk-averse and risk-seeking literature. It proves that while risk-averse agents prefer comonotonicity (pooling), risk-seeking agents prefer counter-monotonicity (gambling/diversification of loss).
2. New Class of Aggregators: It demonstrates that the inf-convolution of convex distortion risk measures yields a distortion riskmetric, not a standard risk measure. This challenges the assumption that optimal risk sharing always results in a member of the same preference class.
3. Market Design: The results provide a theoretical foundation for "counter-monotonic markets" (markets where only counter-monotonic trades are allowed), which could be relevant for speculative markets or specific reinsurance structures involving risk-seeking participants.
4. Counter-Intuitive Behavior: The finding that in a two-agent risk-seeking market, the less risk-seeking agent might bear more risk (Proposition 3) highlights the complexity of heterogeneous preferences and the limitations of simple monotonicity heuristics in small groups.

Summary Conclusion

The paper successfully generalizes risk-sharing theory to heterogeneous, risk-seeking agents. By leveraging the counter-monotonic improvement theorem, it derives explicit formulas for optimal allocations, showing that for convex distortion functions, the optimal strategy is a counter-monotonic "jackpot" allocation. The resulting aggregate risk is characterized by the inf-convolution of the agents' distortion functions, fundamentally altering the nature of the risk metric from a measure to a metric.