- [[Fisher's theorem]] # Idea We can use replicator dynamics models to study [[learning]] and [[evolution]]. Distributions exist and they change over time depending on their proportions and payoff ([[fitness]]). These models tell us the relationship between copying what other agents are doing and doing whatever has the highest payoff. It can explain the [[collective action problem]] (see [[Bradsher 2002 high and mighty]]), such as why people choose to drive SUVs instead of compacts. ## Basic setup - set of types: $\{1,2,3,4,5,6,7, . . \mathrm{N}\}$ - payoff for (or fitness of) each type: $\pi(i)$ - proportion of each type: $P(i)$ A [[rational]] agent aims to maximize payoff. A rule-based agent might choose to copy the most common strategy. $weight = \pi(i) \times P(i)$ ## Replicator equation ${P}_{t+1}(i)=\frac{{P}_{t}(i) \pi(i)}{\sum_{j=1}^{N} {P}_{t}(j) \pi(j)}$ - ${P}_{t+1}(i)$: probability of playing strategy $i$ in time $t+1$ - ${P}_{t}(i) \pi(i)$: weight of strategy $i$ at time $t$ - $\sum_{j=1}^{N} {P}_{t}(j) \pi(j)$: sum of all different weights for all possible actions # References - https://www.coursera.org/learn/model-thinking/lecture/kTMre/replicator-dynamics - https://www.coursera.org/learn/model-thinking/lecture/6Pey7/the-replicator-equation - [[Bradsher 2002 high and mighty]]