The Gambler's First Fallacy First-Order Ergodicity Misjudgment: The Ergodicity Gap, a Three-Rung Ladder, and the Tail Index as a Thermometer of Ergodicity Breaking

Abstract

Abstract A person who anchors his subjective expectation perfectly on the ensemble average, and whose every estimate is exactly right, still systematically overstates his own long-run growth rate in a multiplicative process. The core of the phenomenon is neither greed nor estimation bias but operator misapplication: averaging in the multiplicative group and then reading the result as if it had been averaged in the additive group. The phenomenon is accordingly given a formal name, First-Order Ergodicity Misjudgment, with the Gambler’s First Fallacy retained as the vernacular name. Four results. The ergodicity-gap identity: the Jensen gap equals exactly the Kullback–Leibler divergence between the real measure and the growth-tilted measure, involving no approximation and remaining well defined in fat tails where the log-variance does not exist. The Three-Rung Ladder Theorem: the time, ensemble and narrative growth rates are strictly ordered, the two gaps being the forward and reverse divergences and their sum the Jeffreys divergence. An exhaustiveness proposition: on the empirical array of agents by instants, any systematic discrepancy between a subjective expectation and the time average can be truncated along only two axes. The Ergodicity–Power-Law Correspondence Theorem: under Kesten random-affine dynamics the cross-sectional tail index obeys an exact sign correspondence with the ensemble growth rate. This version makes one correction, one notational unification, one completion, and — in a second pass — four repairs internal to the draft. All are listed here and argued where they occur. Correction. The previous version stated that distinguishability is orthogonal to the tail index. It is not: distinguishability has the closed form I_d = m·κ_★, of which the tail index is one factor. The guardrail still holds — the tail index is not a thermometer of vitality — but its reason is rewritten: head turnover is orthogonal to the tail index, distinguishability is not. Notational unification. Distinguishability is written I_d throughout and never D, since D denotes diffusion and hopping coefficients in the companion papers. Completion: the cognitive penalty. One and the same volatility-to-drift ratio levies a threefold penalty — growth, structural and cognitive — and substituting the optimal delay bound gives the corollary that in the ergodicity-breaking window one is both dying and least able to tell. The four repairs. One. The limit of the tail index as the bet fraction approaches one is not one. On an unfavourable coin paying 1±f with win probability p_w, the losing branch vanishes and the moment equation degenerates to p_w·2^κ = 1, so the index tends to log₂(1/p_w) — 1.0589 for the parameter set of Table 6, not 1.0000. The conclusion that over-betting can never cross the Zipf point stands and is in fact strengthened, since the index is bounded away from one by a computable margin (§7.3). Two. Power laws in general are not rare; only the sub-unit regime is. The window in which the index lies below one has width exactly the ergodicity gap, which grows with volatility, so calling it narrow has the direction backwards. The phrasing that a power law requires a steady-state system to refuse to be a steady state is retracted here as it has been in the classification companion (§7.6). Three. A steady state does not by itself give a large tail index. The index is approximately one plus twice the ratio of the absolute ensemble rate to the log variance, so what makes it large is that ratio and not the smallness of either quantity: an ensemble rate of −0.0001 with a log variance of 0.01 gives an index of 1.02, not a large one (§7.6). Four. The exchange of the two probabilities under the h-transform is exact only far from the barrier. At an up-probability of 0.4 the conditioned up-probabilities are 0.6842, 0.6192 and 0.6016 at positions three, six and twelve, converging to 0.6; the conditioned walk is spatially inhomogeneous and its up-probability equals the original down-probability only in the limit. The draft stated the exchange as exact and placed the caveat where it applies to a general multiplicative process (§6.2). Five, new in this pass. Section 3.1 said that half the variance fails where it matters but gave no size for the failure. The new §3.5 supplies it: the gap is the sum of all cumulants of log growth from the second onward, half the variance is its first term, and on the Pareto family the gap has the closed form ln[α/(α−1)] − 1/α, finite for every tail index above one while the variance of the growth factor itself diverges at two. The half-variance figure understates the gap by about two-thirds of the reciprocal of the tail index — 23 percent at an index of three — and substituting the wrong variance overstates it 5.2-fold there and returns infinity at an index of two. Nothing else in the paper changes; no prediction is added or altered