MoRA: Manipulability of Risk Aversion

• MoRA is defined as the ability to alter agents' effective risk aversion through modifications in the environment, policy, or information, impacting their payoff distributions.
• Theoretical analyses show that reduced risk aversion leads to broader and riskier payoff outcomes, with stochastic orders providing a framework for comparing these distributions.
• Applications in portfolio optimization and mechanism design demonstrate that the robustness of MoRA depends critically on market completeness and the structure of utility functions.

The manipulability of risk aversion (MoRA) refers to the capacity to influence, assess, or even fundamentally alter the effective risk preferences of agents—whether human, institutional, or artificial—through modifications to environment, mechanism, policy, or information. In economic theory and related domains, MoRA encompasses both the sensitivity of outcomes and policies to differences in risk aversion and the mechanisms or interventions that render observed risk aversion malleable or endogenous to context. This encyclopedic treatment synthesizes the core theoretical foundations, comparative statics, operationalizations, methodological results, and fragility considerations emerging from the analytical literature.

1. Utility Maximization, Risk Aversion, and Portfolio Distribution

The canonical foundation of risk aversion and its manipulability is the expected utility maximization framework. Investors are endowed with a strictly increasing, strictly concave utility function $U$ (commonly satisfying Inada conditions) and select dynamic trading strategies to maximize expected utility of terminal wealth:

$\max E[U(X_T)]$

subject to $X_t$ arising from self-financing admissible strategies.

Risk aversion is measured by the absolute risk aversion function $-U''(x)/U'(x)$. Higher risk aversion (greater curvature) leads to concentrating the distribution of $X_T$ around a safer central value, while lower risk aversion engenders a more dispersed, risky payoff. In complete continuous-time markets, duality theory characterizes the optimizer: if $Q$ is the unique local martingale measure,

$U'(\hat{X}_T) = y \frac{dQ}{dP}$

with $y$ a Lagrange multiplier.

Crucially, comparative statics analyzed by Dybvig and Wang and in continuous-time extensions establish that given two utility functions $U_{\text{RA}} \succeq U_{\text{RL}}$ (RA more risk averse than RL), then under suitable integrability and monotonicity conditions:

$$\hat{X}_T^{(\text{RA})} \leq_{MC} \hat{X}_T^{(\text{RL})}$$

i.e., the less risk-averse investor's payoff stochastically dominates that of the more risk-averse in the monotone convex order.

Implication: risk aversion is manifest in the entire distribution of outcomes, and "manipulating" risk preferences—e.g., reducing aversion—provably broadens and shifts the induced payoff distributions, with less risk averse agents accepting "extra noise" for higher risk premia.

2. Stochastic Orders, Fragility, and Completeness

The main analytical tool for formalizing MoRA is the family of stochastic orders: second-order stochastic dominance and the monotone convex order ($\leq_{MC}$). Given random variables $X, Y$,

$$X \leq_{MC} Y \iff E[(X - K)^+] \leq E[(Y - K)^+], \quad \forall K$$

This dominance is preserved in complete or weakly complete markets when risk aversion decreases, as distributional manipulations through risk preferences propagate into stronger allocation of mass toward higher payoffs and fatter right tails for the less risk averse.

However, this ordering is "fragile" in incomplete markets or under perturbations. The paper constructs counterexamples where even events of arbitrarily small probability disrupt the monotone convex order—e.g., a rare branch can force the more risk-averse investor to obtain wealth exceeding that attainable by any strategy of the less risk-averse agent. This demonstrates that the economic intuition underlying MoRA holds only under restrictive (market completeness and certain utility structure) conditions.

Summary table:

Domain	Stochastic Order Holds	Fragility
Complete markets	Yes	Robust
Incomplete markets	No (counterexamples)	Violated by small-probability events

3. Power Utilities, Lévy Markets, and Robust Ordering

For investors with power utility functions,

$U(x) = \frac{x^{1-p}}{1-p}, \quad p > 0, p \neq 1$

where $p$ is the constant relative risk aversion (CRRA) parameter, the manipulability results are extended. In models where asset returns are modeled as stochastic exponentials of Lévy processes or, more generally, processes with (conditionally) independent increments, the following convex order is shown for the centered terminal wealth:

$$\left(\hat{X}_T^{(\text{RA})} - E[\hat{X}_T^{(\text{RA})}]\right) \leq_C \left(\hat{X}_T^{(\text{RL})} - E[\hat{X}_T^{(\text{RL})}]\right)$$

Here, the magnitude of the risky allocation (fraction $\hat{\pi}$ of wealth in risky asset) is monotonically increasing as risk aversion decreases ($$|\hat{\pi}_{\text{RA}}| \leq |\hat{\pi}_{\text{RL}}|$$), confirming the general theory but with explicit, model-tractable expressions for the ordering.

Through discrete-time Euler approximation, the preservation of the stochastic order under $L^1$ convergence, and conditional independence, the results are further generalized to time-inhomogeneous increments and models with stochastic risk factors. In these settings, MoRA is robust and the orderings are preserved even as the noise structure varies, provided the increments maintain the required independence.

4. Mechanisms and Fragility under Market and Model Perturbations

MoRA is inherently sensitive to the structure of the economic environment:

• Market completeness: The key stochastic dominance result requires market completeness or sufficient weak completeness. Any move toward incompleteness (e.g., discrete state expansions) can destroy the orderings that constitute MoRA.
• Utility restrictions: Results are strongest for CRRA, CARA, or other tractable utility families; irregular preferences may break monotonicity of risk aversion mappings.
• Admissibility and constraints: Additional constraints, such as consumption or leverage restrictions, may interact nontrivially with MoRA, sometimes amplifying fragility.

The ability to "manipulate away" risk aversion and guarantee dominance relations applies only under the intersection of these technical conditions. In practical applications, robust mechanistic mapping from subjective preferences to outcome distributions is therefore highly sensitive to modeling and environmental details.

5. Applications, Real-World Manipulability, and Policy Implications

The mathematical results establish that MoRA—operationalized as the lawlike dominance of terminal payoffs disfavoring higher risk aversion—underpins practical approaches in portfolio optimization and mechanism design:

• In financial practice, agents or institutions adjusting risk attitude (for instance, by explicit policy or through advisory algorithms) can predictably shift the distribution of returns, offering a trade-off between expected value and payout risk profile.
• Mechanism designers can, to the extent the model conditions are met, safely ignore "risk aversion heterogeneity" in allocation rules, focusing instead on mean-optimizing mechanisms and relying on payment "insurance" transformations when needed (see the literature on mechanism design with risk-averse agents (Dughmi et al., 2012)).
• Regulatory intervention or product design aimed at influencing investors' (or consumers') effective risk aversion can systematically achieve predictable shifts in outcome distributions, subject to the caveats about fragility in incomplete/infected markets.

In summary, the manipulability of risk aversion is mathematically rigorous in complete, properly structured settings but can be highly sensitive or outright fail under mild departures from these conditions. Its operational deployment requires careful attention to environmental assumptions, and real-world implementations should account for robustness limits exposed by the theory.