Utility and fitness
Wants versus needs
2025-06-05 — 2025-06-05
Locally around a given phenotype or decision context, fitness functions in evolutionary biology and utility functions in economics can both be approximated by affine (linear plus constant) models, enabling shared analytical techniques. In evolutionary quantitative genetics, the selection gradient is defined as the slope of a linear regression of relative fitness on phenotype, arising naturally from log-linear representations of fitness such that
making
with constant trade-off weights
1 Mathematical Background on Fitness Functions
1.1 Definitions and Selection Gradients
In evolutionary biology, an individual’s fitness
where
so that the selection gradient
averaged over the phenotypic distribution of the population (Morrissey and Goudie 2022).. Equivalently, one can fit a linear regression of relative fitness on traits:
where
1.2 Quadratic and Multivariate Extensions
To account for curvature in the fitness landscape, one can introduce a quadratic term:
where
1.3 Local Linear Approximation of Fitness
Even if the true fitness function is highly nonlinear, around a focal phenotype
Because
which yields a local linear fitness approximation of the form
Hence, selection gradients always define a local linearization of the fitness landscape (Morrissey and Goudie 2022).
2 Mathematical Background on Utility Functions
2.1 Von Neumann–Morgenstern Utility and Local Approximations
In economic theory, a decision-maker’s preferences over lotteries
If outcomes are vector-valued
where each
showing that marginal utilities
2.2 Multi-Attribute Utility Theory
When more than two attributes are present, additional independence conditions (e.g., mutual utility independence, quasi-additivity) allow the utility function to remain an additive polynomial, or to adopt a multiplicative form when necessary. Under mutual utility independence (MUI) for
where the marginal and interaction coefficients
is sufficient or at least a reasonable local approximation when interactions are negligible (Sarin 2013).
3 Formal Connection Between Fitness and Utility
3.1 Log-Linear Fitness as Utility
Since both fitness and utility can be seen as objectives to be maximized, earlier scholars suggested that in evolution, the “utility” of an organism is indeed its fitness, and that natural selection performs a form of “utility maximization” (Orr 2007). Formally, consider a log-linear fitness function:
Interpreting
with
3.2 Selection Gradient = Marginal Utility
In a population with phenotypic distribution
Thus, for small deviations around the population mean
meaning that each component
3.3 Phenotypic Variance-Covariance and Maximization
Lande’s identity in quantitative genetics expresses the response to selection as
where
4 Consequences When the Analogy Holds
4.1 Predictions of Evolutionary Stable Strategies (ESS)
When a population’s fitness function is approximable by a concave (downward-curved) log-linear or quadratic form, the selection gradients define a local optimum at
4.2 Risk Preferences and Bet Hedging
In uncertain environments, expected fitness itself may be insufficient; natural selection acts on geometric mean fitness, leading to bet-hedging strategies that trade off mean and variance in reproductive success. This is analogous to a risk-averse economic agent who maximizes
preferring a certain outcome to a gamble with the same mean. In evolutionary terms, if fitness is multiplicative across generations, the effective “utility” is
which is inherently concave, yielding risk-averse (“pessimistic”) strategies insofar as they maximize long-term growth Orr (2007). Thus, risk preferences in economics can map to bet-hedging in biology, both emerging from concavity in the objective function.
5 Limitations and Divergences
5.1 Niche Construction and Dynamic Fitness Landscapes
Niche construction occurs when organisms modify their environments, thereby altering the fitness landscape dynamically. In such cases, fitness
5.2 Adaptive Preferences and Endogenous Utility
In economic settings, agents’ preferences (utility functions) are often assumed exogenous and fixed. By contrast, organisms’ “preferences” for certain phenotypic states may coevolve with genetic or cultural transmission, leading to adaptive preferences that change in response to ecological and social contexts. Consequently, there is no fixed utility function
5.3 Frequency-Dependent Selection and Non-Additivity
When fitness depends on the frequency of phenotypes in the population—frequency-dependent selection—the fitness function is
5.4 Multi-Level Selection
In multi-level selection, group fitness and individual fitness may diverge: group fitness
6 Mathematical Formalization and Example
6.1 First-Order Taylor Approximation
Let
where
Since
In practice, one fits a linear regression of
6.2 Corresponding Utility Approximation
Consider a utility function
where
7 Implications for Cross-Disciplinary Models
7.1 Evolutionary Game Theory and Utility
In evolutionary game theory, fitness functions often depend on payoff matrices akin to utility matrices. If players adopt strategies
a linear function of the population state. Here, the selection gradient is constant and directly analogous to the expected utility of a strategy under a belief distribution
7.2 Machine Learning and Fitness-Driven Optimization
In genetic algorithms and evolutionary computation, a fitness function
where
8 Concluding Remarks
- The identification of selection gradients with marginal utilities holds when fitness can be locally approximated by a log-linear function and environmental factors remain stable.
- Under these conditions, evolutionary change can be analyzed using tools from convex optimization and comparative statics in economics.
- However, the dynamic nature of fitness landscapes (niche construction), evolving preferences (adaptive preferences), and non-additive or frequency-dependent interactions break the correspondence.
- Multi-level selection introduces divergences between individual and group “utilities,” requiring more complex analogies from social choice theory.
In summary, viewing utility functions as local linear approximations to fitness provides a powerful cross-disciplinary framework, but its validity is bounded by assumptions of additivity, environmental constancy, and fixed preferences.