Consider a simple Agent-Based Model (ABM). On a grid, a group of agents each carry an action (strategy) and move randomly; at each step, they pair up with a neighbour, play a round of a game, receive a payoff, then update their action. The update rule is straightforward β look at the neighbours' scores, and in the next step switch to the strategy with the higher score.
There is only one key variable: which score do you compare?
Model 1: compare this step's payoff $P_1$. Whoever earns more this round is the one I copy.
Model 2: compare the cumulative payoff $P_2$ from all historical games. Whoever has earned the most in total so far is the one I copy.
At the micro level, the difference is only "look at the current step" versus "look at history." But mathematical derivation tells us that these two models correspond to two entirely different dynamic systems β Model 1 is a first-order ordinary differential equation (Replicator Dynamics), where the payoff difference determines the "velocity" of evolution; Model 2 is a second-order integro-differential equation (Inertial Dynamics), where the payoff difference determines the "acceleration." Velocity versus acceleration, first order versus second order, memoryless versus inertial. The mathematical distinction is clear.
But I want to do something else: translate these two models into the language of biology.
The First Step of the Translation
This ABM is itself a simulation of natural selection, so the translation is almost a literal one:
- Action = phenotype
- Each step = one generation
- Imitating the better strategy = better genes spreading between generations
So what are $P_1$ and $P_2$? They both resemble fitness. Model 1 selects by current fitness, Model 2 by cumulative fitness. Intuitively, natural selection in Model 1 is responsive, while Model 2 β because of its "inertia" β is sluggish, and natural selection is weakened.
That rough line of reasoning is broadly right, but three things need to be corrected.
$P_1$ and $P_2$ Are Both Fitness, but Not the Same Kind
$P_1$ is more like instantaneous fitness or current realised fitness β how well an organism performs in the current environment in this generation. $P_2$ is more like cumulative fitness or lifetime reproductive success β the total performance of an individual so far.
The difference is not just "look at one step" versus "look at many steps." $P_2$, as a simple sum, mixes in factors like "lived longer, accumulated more comparisons." An individual with a high $P_2$ may not have done so because their strategy is genuinely better, but simply because they have played more rounds. In other words, what $P_2$ compares is not pure biological fitness, but fitness mixed with "accumulated duration."
If a cleaner biological interpretation is wanted, the common approach is to change $P_2$ to the average payoff per interaction, or to normalise it by age or number of interactions. That way, you compare "performance per game" rather than "total performance."
But in our ABM, a particular design choice makes this question more subtle.
Immortal Agents
In our model, all agents are "permanently" alive β death does not exist. Each step is just the time step abstracted as a generation, so there is no individual lifespan difference of the kind classical biology has, where older individuals accumulate more reproductive opportunities simply by living longer.
Does the "lived longer" bias problem then disappear?
No. It just takes a different form.
$P_2$ still carries forward the early payoffs and creates path dependence. Even without an "age bias," there is still a "historical trajectory bias": if a strategy is dominant early on, even if the environment changes later, its cumulative score still carries that early advantage.
The problem is not "lived longer." It is that the weight of past information is too heavy.
"Inertia" Does Not Mean "Natural Selection Is Weakened"
This is the most important correction.
The intuitive argument: $P_2$ has a bigger influence $\to$ the system has inertia $\to$ natural selection is weakened. The direction is right, but the conclusion needs to be made more precise.
What $P_2$ introduces at its core is not inertia per se, but memory / path dependence / time-averaging. It makes strategy updating depend less on the current single-shot performance and more on the cumulative outcome over time. Biologically, this is closer to "long-term average performance" or "history influences current fitness," rather than inertia in the classical physics sense.
To say "natural selection is weakened" is inaccurate. The more accurate statement is: it shifts natural selection from "instantaneous selection" to "historical selection."
The distinction between the two is not "strong" versus "weak," but "looking at what":
- Instantaneous selection: if you perform well this generation, the next generation gains an advantage.
- Historical selection: if your total historical performance is strong, the next generation gains an advantage.
In a stable environment, historical selection can even make the system look more "stable" β it dampens short-term fluctuations, effectively smoothing the signal over time. In a rapidly changing environment, it is the one that appears to "fall behind" β outdated cumulative information drags on the new changes.
"Slower to react" is a statement relative to environmental change β it is an adaptive lag, not an absolute property.
Two Modes of Natural Selection
After these corrections, the translation can be completed:
Model 1: selection acts on current phenotype performance; adaptation is responsive and local. Natural selection is based on current fitness, is responsive, and has no memory.
Model 2: selection acts on historical accumulated performance; adaptation has memory and path dependence. Natural selection is based on cumulative fitness, and exhibits memory and path dependence.
Under the "each step = one generation" interpretation, $P_1$ is more like "this generation's reproductive success," and $P_2$ is more like "total reproductive success accumulated across generations" or "long-term fitness trajectory." Model 2 is more appropriate for explaining delayed payoff, experience accumulation, or long-term strategy evaluation. The difference between the two modes is not the strength of selection, but the basis on which selection operates β the present or the past.
Model 1 approximates selection based on instantaneous fitness, leading to a fast and locally responsive evolutionary update.
Model 2 approximates selection based on cumulative fitness, introducing memory and delayed response to environmental change.
Closing Remarks
We do not actually know how natural selection operates on "cooperation" in games β is it instantaneous selection, or historical selection? That is an empirical question, not one that mathematics can answer.
But if we assume Model 1 and Model 2 correspond to these two modes respectively, then it is fair to say that natural selection in Model 2 "reacts more slowly" β as long as "slower" is understood precisely: not weaker selection pressure, but reduced sensitivity to environmental change, with an adaptive lag built in.
Model 1: selection is more responsive and memoryless.
Model 2: selection is more history-dependent and has a lag.
What is natural selection "looking at," in the end? Perhaps the question itself is miscast β natural selection is not "looking" at anything. It is just a statistical outcome.
The one really "looking," I suspect, is the motive behind every free will's decision.