The Sampling Distortion

In population games, agents are supposed to observe the distribution of opponents’ strategies and best-respond. In practice, they observe a finite sample. The standard approach treats this as noise — the agent sees an imprecise version of the true distribution and responds noisily. Sampling logit equilibrium reveals that the noise is not just imprecision but distortion: it systematically changes the payoffs the agent is responding to.

When agents evaluate actions using finite samples of opponents’ plays and respond according to a logit choice rule, the equilibrium approximates a logit equilibrium in a virtual game — one where the payoffs are the true payoffs plus distortion terms arising from the sampling noise. The distortion is endogenous: it depends on the distribution of play, which depends on the distorted payoffs, which depend on the distribution.

The distortion terms can select among equilibria. In games with multiple equilibria, the limited-information environment doesn’t just add noise to the selection — it shifts the payoff landscape so that different equilibria become stable. Agents coordinate on outcomes that wouldn’t survive under perfect information, not because they’re irrational but because the information structure makes those outcomes genuinely optimal given what each agent can observe.

The through-claim: finite sampling is not a degraded version of full information — it’s a different game. The payoffs under sampling are not the true payoffs plus noise; they are structurally different payoffs that arise from the interaction between the sampling process and the strategic environment. The distortion is not an error to be corrected but a feature of the informational ecology. This means studying games under the assumption of full information tells you about a game that no agent is actually playing.