Utility and fitness

1.1 Definitions and Selection Gradients

In evolutionary biology, an individual’s fitness \(W\) is a function that quantifies expected reproductive success as a function of its phenotype \(\mathbf{z}\). The simplest local model assumes a log-linear form:

\[ W(\mathbf{z}) \;=\; \exp\!\bigl(\alpha + \mathbf{b}^\top \mathbf{z}\bigr), \]

where \(\alpha\in\mathbb{R}\) is an intercept and \(\mathbf{b}\in\mathbb{R}^n\) is a vector of coefficients on phenotypic traits \(\mathbf{z}\) ((Morrissey and Goudie 2022), Morrissey and Goudie (2022)). Taking the natural logarithm yields

\[ \ln W(\mathbf{z}) \;=\; \alpha + \mathbf{b}^\top \mathbf{z}, \]

so that the selection gradient \(\boldsymbol\beta\) is given by the vector of partial derivatives

\[ \beta_i \;=\; \frac{\partial \ln W(\mathbf{z})}{\partial z_i} \;=\; b_i, \]

averaged over the phenotypic distribution of the population (Morrissey and Goudie 2022).. Equivalently, one can fit a linear regression of relative fitness on traits:

\[ \omega(\mathbf{z}) \;=\; \alpha' + \boldsymbol\beta^\top \mathbf{z} \;+\;\text{residual}, \]

where \(\omega(\mathbf{z})\) is relative fitness and \(\beta_i\) quantifies directional selection on \(z_i\) (Coop 2025 ch 8)).

1.2 Quadratic and Multivariate Extensions

To account for curvature in the fitness landscape, one can introduce a quadratic term:

\[ \omega(\mathbf{z}) \;=\; \alpha'' + \boldsymbol\beta^\top \mathbf{z} \;+\; \tfrac{1}{2}\,\mathbf{z}^\top \mathbf{G}\,\mathbf{z}, \]

where \(\mathbf{G}\) is a symmetric matrix of second derivatives representing stabilizing or disruptive selection The vector \(\boldsymbol\beta\) remains the first selection gradient (directional selection), while \(\mathbf{G}\) yields quadratic and correlational gradients. In generalized linear models (GLMs) with log link functions, the selection gradients can be derived by integrating over trait distributions, linking regression coefficients \(\mathbf{b}\) directly to \(\boldsymbol\beta\) (Morrissey and Goudie 2022).

1.3 Local Linear Approximation of Fitness

Even if the true fitness function is highly nonlinear, around a focal phenotype \(\mathbf{z}_0\) the first-order Taylor expansion is

\[ W(\mathbf{z}) \approx W(\mathbf{z}_0) \;+\; \nabla W(\mathbf{z}_0)^\top (\mathbf{z} - \mathbf{z}_0). \]

Because \(\nabla \ln W(\mathbf{z}_0) = \nabla W(\mathbf{z}_0)/W(\mathbf{z}_0) = \boldsymbol\beta\), this is equivalent to

\[ \ln W(\mathbf{z}) \approx \ln W(\mathbf{z}_0) \;+\; \boldsymbol\beta^\top (\mathbf{z} - \mathbf{z}_0), \]

which yields a local linear fitness approximation of the form

\[ \omega(\mathbf{z}) \approx \text{const} \;+\; \boldsymbol\beta^\top \mathbf{z}. \]

Hence, selection gradients always define a local linearization of the fitness landscape (Morrissey and Goudie 2022).

2.1 Von Neumann–Morgenstern Utility and Local Approximations

In economic theory, a decision-maker’s preferences over lotteries \(\ell\) (probability distributions over outcomes) are represented by a utility function \(u: \mathcal{X}\to\mathbb{R}\) satisfying the von Neumann–Morgenstern axioms. Under risk, the expected utility of a lottery that yields outcome \(x\) with probability \(p\) is

\[ \mathbb{E}[u(x)] \;=\; \sum_{x} p(x)\,u(x). \]

If outcomes are vector-valued \(\mathbf{x} \in \mathbb{R}^n\), and under additive independence of attributes (i.e., marginal preferences are independent), one obtains an additive utility representation:

\[ u(\mathbf{x}) \;=\; \sum_{i=1}^n k_i\,u_i(x_i), \]

where each \(u_i\) is a single-attribute utility function on attribute \(i\), and \(k_i\) are nonnegative trade-off weights summing to one (Sarin 2013). In the simplest case, if each \(u_i\) itself is differentiable and one examines \(\mathbf{x}\) near a reference point \(\mathbf{x}_0\), then a first-order Taylor expansion yields

\[ u(\mathbf{x}) \approx u(\mathbf{x}_0) \;+\; \sum_{i=1}^n \left.\frac{\partial u}{\partial x_i}\right|_{\mathbf{x}_0} (x_i - x_{0,i}), \]

showing that marginal utilities \(\partial u/\partial x_i\) act as local “weights” on small attribute changes (Sarin 2013).

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 \(n\) attributes, one gets a multilinear utility form:

\[ u(\mathbf{x}) \;=\; \sum_{i} k_i\,u_i(x_i) \;+\; \sum_{i<j} k_{ij}\,u_i(x_i)\,u_j(x_j) \;+\; \cdots \;+\; k_{1\cdots n}\,\prod_{i}u_i(x_i), \]

where the marginal and interaction coefficients \(k_i, k_{ij}, \dots\) satisfy normalization constraints (Sarin 2013). Nevertheless, in many economic models (e.g., expected utility theory under additive outcomes), the additive form

\[ u(\mathbf{x}) = \sum_{i} k_i\,x_i \]

is sufficient or at least a reasonable local approximation when interactions are negligible (Sarin 2013).

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:

\[ \ln W(\mathbf{z}) = \alpha + \sum_{i=1}^n b_i\,z_i. \]

Interpreting \(\ln W(\mathbf{z})\) as a utility function \(u(\mathbf{z})\), one obtains

\[ u(\mathbf{z}) = \alpha + \sum_{i} b_i\,z_i, \]

with \(b_i\) serving as marginal “utilities” of trait \(z_i\). This identification holds locally whenever fitness is well-approximated by a log-linear function (Morrissey and Goudie 2022).

3.2 Selection Gradient = Marginal Utility

In a population with phenotypic distribution \(p(\mathbf{z})\), the average directional selection gradient is

\[ \boldsymbol\beta = \mathbb{E}_p\bigl[\nabla_{\mathbf{z}} \ln W(\mathbf{z})\bigr] = \mathbb{E}_p[\mathbf{b}] = \mathbf{b}. \]

Thus, for small deviations around the population mean \(\bar{\mathbf{z}}\), the linearized fitness landscape is

\[ \ln W(\mathbf{z}) \approx \ln W(\bar{\mathbf{z}}) + \mathbf{b}^\top (\mathbf{z} - \bar{\mathbf{z}}), \]

meaning that each component \(b_i\) is both the selection gradient on \(z_i\) and the marginal utility weight on attribute \(z_i\) if one views \(\ln W\) as a utility function (Morrissey and Goudie 2022).

3.3 Phenotypic Variance-Covariance and Maximization

Lande’s identity in quantitative genetics expresses the response to selection as

\[ \Delta \bar{\mathbf{z}} = \mathbf{G}\,\boldsymbol\beta, \]

where \(\mathbf{G}\) is the additive genetic variance–covariance matrix of traits ((Coop 2025 ch 8), Morrissey and Goudie (2022)). One can interpret this as stating that evolution moves the mean phenotype “uphill” in \(\ln W\), analogous to a constrained gradient ascent on a utility surface, where the genetic covariances shape the feasible direction of movement ((Coop 2025 ch 8), Morrissey and Goudie (2022)). In this sense, multi-trait evolution under selection is formally akin to an economic agent maximizing expected utility under budget constraints and correlation structures among decision variables.

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 \(\boldsymbol\beta = \mathbf{0}\). A phenotype \(\mathbf{z}^*\) satisfying \(\nabla_{\mathbf{z}}\ln W(\mathbf{z}^*) = 0\) is an evolutionarily stable strategy (ESS). This mirrors the condition for a utility maximizer in economics: set \(\nabla u(\mathbf{x}^*) = \mathbf{0}\). Under concavity of \(\ln W\), \(\mathbf{z}^*\) is a global maximum of fitness, making it dynamically stable against small perturbations (Morrissey and Goudie 2022). Similarly, in consumer theory, a concave utility function yields a unique global maximum for an unconstrained agent.

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

\[ U[\text{lottery}] = \mathbb{E}[u(x)] \quad\text{with concave }u, \]

preferring a certain outcome to a gamble with the same mean. In evolutionary terms, if fitness is multiplicative across generations, the effective “utility” is

\[ u = \ln(\text{fitness}), \]

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.

5.1 Niche Construction and Dynamic Fitness Landscapes

Niche construction occurs when organisms modify their environments, thereby altering the fitness landscape dynamically. In such cases, fitness \(W(\mathbf{z}, E(\mathbf{z}))\) depends on both phenotype and an environment \(E\) shaped by past phenotypic distributions. Because \(E\) evolves with \(\mathbf{z}\), \(\ln W\) cannot be treated as a static utility function: the selection gradient at time \(t\), \(\nabla_{\mathbf{z}}\ln W(\mathbf{z}, E_t)\), shifts with environmental feedback, invalidating a fixed linear approximation around a single reference point (Schulz 2014; Coop 2025 ch 8).

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 \(u(\mathbf{z})\): instead, preferences adapt to past payoffs, creating path dependence and shifting marginal weights \(\partial u/\partial x_i\) over time (Schulz 2014; Okasha 2009).

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 \(W(\mathbf{z}; p_t(\cdot))\). Here, \(\nabla_{\mathbf{z}}\ln W\) changes as the distribution \(p_t\) evolves, so a linear approximation around \(\bar{\mathbf{z}}\) may be valid only at a specific time. Moreover, nonlinear interactions (e.g., coevolutionary arms races) introduce higher-order terms that cannot be captured by first-order (linear) utility analogues (Morrissey and Goudie 2022).

5.4 Multi-Level Selection

In multi-level selection, group fitness and individual fitness may diverge: group fitness \(W_g\) might not equal the sum or average of individual fitnesses. Social choice theory analogues show that even if individual utility is additive, group utility need not be (Okasha 2009; Orr 2007). Hence, at the group level, a single additive utility (fitness) function may not exist without further independence assumptions.

6.1 First-Order Taylor Approximation

Let \(W:\mathbb{R}^n \to \mathbb{R}^+\) be the true fitness function, assumed twice differentiable. Around a focal phenotype \(\mathbf{z}_0\), the first-order expansion is

\[ W(\mathbf{z}) = W(\mathbf{z}_0) + \nabla W(\mathbf{z}_0)^\top(\mathbf{z} - \mathbf{z}_0) + \tfrac{1}{2}(\mathbf{z} - \mathbf{z}_0)^\top \mathbf{H}(\mathbf{z}_0)(\mathbf{z} - \mathbf{z}_0) + \cdots, \]

where \(\mathbf{H}\) is the Hessian matrix of second derivatives (Morrissey and Goudie 2022). If we restrict to the linear term only, then

\[ W_{\text{lin}}(\mathbf{z}) = W(\mathbf{z}_0) + \nabla W(\mathbf{z}_0)^\top(\mathbf{z} - \mathbf{z}_0). \]

Since \(\nabla \ln W(\mathbf{z}_0) = \nabla W(\mathbf{z}_0)/W(\mathbf{z}_0)\), we can write

\[ \ln W(\mathbf{z}) \approx \ln W(\mathbf{z}_0) + \boldsymbol\beta^\top (\mathbf{z} - \mathbf{z}_0), \quad\text{with }\boldsymbol\beta = \nabla_{\mathbf{z}}\ln W(\mathbf{z}_0). \]

In practice, one fits a linear regression of \(\ln W\) on \(\mathbf{z}\) to estimate \(\boldsymbol\beta\) (Morrissey and Goudie 2022).

6.2 Corresponding Utility Approximation

Consider a utility function \(u(\mathbf{x})\) capturing preferences over a choice vector \(\mathbf{x}\in\mathbb{R}^n\). Near a reference \(\mathbf{x}_0\), a first-order Taylor expansion gives

\[ u(\mathbf{x}) \approx u(\mathbf{x}_0) + \mathbf{u}'(\mathbf{x}_0)^\top (\mathbf{x}-\mathbf{x}_0), \]

where \(\mathbf{u}'(\mathbf{x}_0) = \nabla_{\mathbf{x}}u(\mathbf{x}_0)\). If \(u\) is additive across attributes (i.e., \(u(\mathbf{x}) = \sum_i k_i\,x_i\)), then \(\nabla u(\mathbf{x}) = (k_1,\dots,k_n)\) is constant and the linear approximation is exact. Whenever \(u\) is nonlinear but smooth, the linear term still captures local marginal utility, analogous to \(\boldsymbol\beta\) in the fitness case (Sarin 2013; Morrissey and Goudie 2022).

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 \(s\) with frequency \(p\), the fitness of strategy \(i\) is

\[ W_i(p) = \sum_j A_{ij}\,p_j, \]

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 \(p\) (Morrissey and Goudie 2022).

7.2 Machine Learning and Fitness-Driven Optimization

In genetic algorithms and evolutionary computation, a fitness function \(f(\mathbf{x})\) assigns a score to candidate solutions \(\mathbf{x}\). Often these functions are approximated linearly (fitness approximation) to reduce computational cost, by fitting a regression model

\[ \hat{f}(\mathbf{x}) = \boldsymbol\theta^\top \mathbf{x}, \]

where \(\boldsymbol\theta\) is updated periodically. This approach mirrors the use of linear utility approximations in reinforcement learning (value function approximation).