The Revenge of Ecological Rationality: Strategy-Selection by Meta-Induction Within Changing Environments

Abstract

According to the paradigm of adaptive rationality, successful inference and prediction methods tend to be local and frugal. As a complement to work within this paradigm, we investigate the problem of selecting an optimal combination of prediction methods from a given toolbox of such local methods, in the context of changing environments. These selection methods are called meta-inductive (MI) strategies, if they are based on the success-records of the toolbox-methods. No absolutely optimal MI strategy exists—a fact that we call the “revenge of ecological rationality”. Nevertheless one can show that a certain MI strategy exists, called “AW”, which is universally long-run optimal, with provably small short-run losses, in comparison to any set of prediction methods that it can use as input. We call this property universal access-optimality. Local and short-run improvements over AW are possible, but only at the cost of forfeiting universal access-optimality. The last part of the paper includes an empirical study of MI strategies in application to an 8-year-long data set from the Monash University Footy Tipping Competition.

Appendix

Proof of Theorem 2

For 2.1(a): Since xMI’s imitates for each time n > 1 some deceiver P_i, xMI’s score for all times > 1 is 0, and so limsuc(xMI) = 0. For 3.1(b): On average, xMI imitates each player equally often, with a limiting frequency of 1/m. So the frequency of times for which each player earns a maximal score of 1 is (m − 1)/m. For (2.2): Let “p(fav-P_i)” be the limiting frequency of times for which player P_i was xMI’s favorite, and limsuc(P_i|fav) be player P_i’s limit success conditional on these times. Then by probability theory, limsuc(xMI) = Σ_1≤i≤mp(fav-P_i)·limsuc(P_i|fav), which implies that limsuc(xMI) ≤ max({limsuc(P_i|fav):1 ≤ i ≤ m}). By the negative correlation assumption, limsuc(P_i|fav)) < limsuc(P_i) holds for all i ∈ {1, …, m}; so limsuc(xMI) < max({limsuc(P_i):1 ≤ i ≤ m}) =_def maxlimsuc. Hence xMI is not long-run optimal.□

Proof of Theorem 3

For 3.1(a): Before the convergence time s, TTB may be, in the worst case, permanently deceived by the non-MI-players (or cues), by negatively correlated success-oscillations. So TTB’s worst-case success until time s is zero, whence his worst-case loss at times n ≥ s is s/n. After time point s, TTB’s earns for each kth-best player (or cue) ${\text{P}}_{{{\text{s}}_{\text{k}} }}$ the sum-of-scores earned by ${\text{P}}_{{{\text{s}}_{\text{k}} }}$, for all time points at which ${\text{P}}_{{{\text{s}}_{\text{k}} }}$ but no player better than ${\text{P}}_{{{\text{s}}_{\text{k}} }}$ delivered a prediction. The sum-expression in 3.1(a) is identical with the sum of these scores divided by time n. For 3.1(b): Additionally we assume that after time s each player’s validity is better than the success of a random guess, which is 1/2 in a binary prediction game. This implies that suc_n(TTB) > maxsuc_n − (s/n), where maxsuc_n ≥ suc_n(ITB) since ITB approximates maxsuc_n from below (by Theorem 1). (3.2) follows from (3.1) in the explained way, since lim_n→∞(s/n) = 0. □