This is an interactive version of Figure 3 and Figure 4 of the article Taming Fitness: Organism-Environment Interdependencies Preclude Long-Term Fitness Forecasting (Doulcier, Takacs and Bourrat, 2021).

Number of rounds (n _max): Number of repeats (k _max):

Let :

• X_n be the score of a given player at round n.
• Y_n be the average score for several rounds, Y_n = X_n/n
• Z_k be the average score for several independent repeats of the game, Z_k = Σ_k Y_n^(k)

The die is cast before each coin toss: No irreversible changes

The average-over-all-possible coins score for the heads player is 0 (0 for the red coin and 0 for the blue coin. E = (0 + 0)/2 = 0).

Independant rounds variant

The properly weighted average-over-all-possible coins score for the heads player is 0 (0 × 0 for the red coin and 0 × 0 for the blue coin. E = (0×0 + 0×0) = 0).

Both the average score per round Y_n and the average score per outcome Z_k converge toward the properly weighted average-over-all-possible coins score.

Markovian rounds variant

The properly weighted average-over-all-possible coins score for the heads player is 0 (0 × 0 for the red coin and 0 × 0 for the blue coin. E = (0×0 + 0×0) = 0). [The weights are computed using the eigenvalues of the transition matrix.]

Both the average score per round Y_n, and the average score per outcome Y_k converge toward the same value for the Tails player. (The convergence may only become visible for large sample size, but can be analytically proven using Markov-chains theory).

The die is cast once at the beginning: Irreversible changes

Even though both red and blue coins are "possible" the average score per round Y_n never converges toward the average-over-all-possible-coins score. This is because once the die has been cast, the colour of the coin never changes. Note that the average score per outcome Z_k still converges toward the properly weighted average-over-all-possible-coins score of for the heads player.

Rare events change the environement and rare events change the side of a player: Mutations

Note that both the average score per round Y_n, and the average score per outcome Z_k converge toward the same value for both players. (The convergence may only become visible for large sample size, but can be analytically proven using Markov-chains theory).

References

Source: main.js Software used: d3js.

The article: