Agent-Level Utility Functions

• Agent-level utility functions are mathematical formulations that assign values to outcomes, actions, or trajectories, capturing risk and time preferences.
• The inverse utility theory approach deduces underlying utility functions from observed behavioral patterns using conditions like Black’s PDE in stochastic markets.
• Differences between deterministic and stochastic settings highlight that while many utility functions can match observed actions, consistency conditions in uncertain environments narrow the admissible solutions.

Agent-level utility functions formalize and quantify individual agent preferences over outcomes, states, actions, or trajectories, providing the foundational objective by which agents evaluate possible decisions. Across economics, finance, AI, and multi-agent systems, these functions mediate between fundamental assumptions about rationality and observable agent behavior, while their precise form encodes critical properties such as risk aversion and time consistency. Recent research has expanded the scope of agent-level utility function analysis through inverse inference from behavior, mechanism design in strategic environments, incorporation of risk, and adaptation to multi-objective, time-inconsistent, and uncertain environments.

1. Inverse Utility Theory: Inferring Utility from Observed Behavior

The classical paradigm in utility theory assumes that agents maximize a known utility function, leading to tractable predictions for consumption and investment (Cox et al., 2011). The inverse utility approach reverses this logic: starting from observable behavior—specifically, a consumption function $c(t, w)$ and, in stochastic settings, an investment strategy $\pi(t, w)$—one asks whether there exists a utility function $u$ for which these choices are optimal. In the deterministic setting, the Lagrangian first-order condition is

$u_c(t, c(t, w(t, x))) = \lambda(x),$

where $\lambda(x)$ is a Lagrange multiplier and $w(t, x)$ is defined by the wealth trajectory $w_t = -c(t, w)$.

In stochastic (Black-Scholes) markets, an analogous duality yields

$u_c(t, c(t, W_t)) = \lambda Z_t,$

with $Z_t$ the state-price density. The critical insight is that, while there may be infinitely many utility functions producing a given deterministic consumption plan, the stochastic setting imposes a strong consistency condition (a specific PDE—see Section 3), reducing this degeneracy.

This inverse perspective enables:

• Direct behavioral inference: Preference parameters (risk aversion, etc.) are deduced from actions rather than imposed.
• Consistency testing: If observed strategies fail necessary PDE conditions, they cannot be optimal for any classical utility.
• Sharp risk attitude recovery: Characteristics such as decreasing absolute risk aversion (DARA) can be traced precisely to derivatives of observed decisions.

2. Deterministic vs. Stochastic Settings

The utility inference methodology bifurcates depending on the presence of uncertainty:

• Deterministic case: Given infinite-horizon consumption, utility functions $u$ satisfying the first- and second-order conditions can always be constructed. Indeterminacy arises: infinitely many $u$ can support observed $c(t, w)$ unless further constraints are imposed.
• Stochastic case (Black-Scholes): The agent’s wealth follows

$$dW_t = r W_t dt - c(t, W_t) dt + \pi(t, W_t)\sigma (dB_t + \theta dt),$$

and utility maximization involves the expected discounted utility of consumption. Here, the existence of an optimally-generating $u$ requires $(c, \pi)$ to solve Black’s PDE (see next section). The stochastic structure tightly couples risk-taking to observed strategies, removing much but not all of the indeterminacy seen in deterministic worlds.

3. Consistency Conditions and Black’s PDE

A central result in stochastic settings is the identification of a necessary and sufficient condition for the observability of an agent-level utility function: Black’s PDE. For given $(c(t, w), \pi(t, w))$, this takes the form

$$\int_1^w \frac{c_t(t, \xi)}{\pi^2(t, \xi)} d\xi + \frac{\sigma^2}{2} \pi_w(t, w) + \frac{c(t, w)}{\pi(t, w)} - \frac{r w}{\pi(t, w)} = \phi(t),$$

where $\phi(t)$ is a function of time only, $\pi_w$ denotes the partial derivative of $\pi$ in $w$, and $c_t$ the time derivative. This PDE must be satisfied for $(c, \pi)$ to correspond to the optimum of any classical utility function.

Given $u_c(t, c(t, w)) = \int_w^\infty D(\xi)d\xi$ (with $D(\cdot)$ a weighting function), Black’s PDE ensures that both optimality and market dynamics are consistent with utility maximization. Violation implies observed behavior cannot be rationalized under any expected utility principle.

4. Deducing Risk Attitudes from Choices

The inverse approach reveals how risk aversion—a fundamental agent property—can be deduced from consumption/investment strategies. The absolute risk aversion is defined by

$\rho(t, c) = -\frac{u_{cc}(t, c)}{u_c(t, c)}.$

Utilizing the relationship between $u_c$ and the observed $c$, differentiability conditions can be framed entirely in terms of $c$ and $\pi$. For example, the agent is DARA (decreasing absolute risk aversion) if and only if

$$\frac{\pi_w(t, w)}{\pi(t, w)} \ge -\frac{c_{ww}(t, w)}{c_w(t, w)},$$

evaluated along optimal wealth paths.

Thus, local convexity or other features in consumption/investment rules can be interpreted as empirical evidence for DARA, CARA, or IARA, with implications for both theoretical modeling and product design in finance.

5. Indeterminacy and Model Implications

A significant finding is the persistence of indeterminacy:

• Deterministic case: Even perfectly observed $c(t, w)$ does not determine $u$ uniquely; infinitely many compatible utility functions exist (offset by different weighting schemes $D(x)$).
• Stochastic case: Black’s PDE necessitates consistency between $c$ and $\pi$, winnowing the class of admissible $u$, but without further side conditions, uniqueness is not achieved, only regularity within the admissible class.

Practical implications include the ability to:

• Test observed consumption/investment data for compatibility with utility maximization.
• Estimate market parameters (e.g., volatility $\sigma$) from strategic behavior if the agent is assumed to be rational under expected utility.
• Deduce population-level risk attitude distributions, facilitating improved behavioral finance models.

The framework also readily generalizes—potential extensions include incomplete markets, environments with model/belief uncertainty, or multiperiod/infinite-horizon settings.

6. Applications and Future Directions

Applications emerging from the inverse utility approach include:

• Behavioral finance: Data-driven estimation of agent utility functions and attitudes without the need for laboratory experiments.
• Financial engineering: Tailoring products (insurance, derivatives) to empirically-observed risk attitudes.
• Testing rationality: Empirical validation of the expected utility hypothesis via PDE-based tests on observed data.

Potential future directions highlighted in the paper involve

• Extending the inverse approach beyond the Black-Scholes framework to settings with incompleteness or model ambiguity.
• Joint inference of both utility and belief formation (model uncertainty), crucial for robust economic modeling.
• Investigation of utility function recovery in multi-agent or aggregate environments where agent heterogeneity is prominent.

This theoretical platform thus enables rigorous, data-oriented modeling of agent-level utility functions, providing a bridge between observable consumption/investment decisions and the latent structure of risk preferences, even as it exposes the practical limits of identifiability and the importance of dynamic consistency.