The Tragedy of the Confidence Commons Optimal Belief-Update Rate, Adaptive Overconfidence, and the Tail Index as Commons

Abstract

Abstract Twenty-five centuries of moral exhortation have not visibly made people better calibrated. This paper offers an explanation: the person being exhorted has not made a mistake. In adversarial environments the individually optimal belief bias Δ★ may be strictly positive while the collectively optimal bias is zero. That is the structure of a commons tragedy. The work of this paper is to write each of its components as a computable object. (i) "How fast should one update" has a unique optimum. If the truth performs a random walk with per-step variance v and observations carry noise variance s², the steady-state error of exponentially weighted updating is unimodal in the update rate α, with its minimum at the steady-state Kalman gain; for small α it is L(α) = α·s²/2 + v/(2α), minimised at α★ = √(v/s²). α → 1 is being led by noise (the self-abasing type); α → 0 is a frozen belief (the Ah-Q type). These are not two ailments but the two ends of one U-shaped curve, and at the optimum the marginal costs of the two errors are equal — the first-order condition of "the golden mean". v = 0 gives α★ = 0: in a genuinely stationary world, not updating is optimal; "you must update" is a theorem whose premise is that the world is non-stationary. (ii) Defining rigidity as the reciprocal of the update rate, k ≡ 1/α, gives optimal rigidity k★ = s/√v and an irreducible loss s·√v. The faster the environment drifts, the softer the optimal dogma; and modernity, by raising both v and s, raises the minimum loss itself — the crisis of meaning is not "the rigidity is mis-set" but "whatever the rigidity, the floor is rising". (iii) The U has one critical point and no barrier, in every coordinate. This holds for every positive v and s and for every monotone identification of rigidity with the update rate. In logarithmic deviation the U is exactly symmetric; the asymmetry that survives every coordinate is that, over the admissible range of convex-combination updating, excessive speed costs at most the observation noise s² — which can still be about k★ times the minimum loss — while excessive rigidity in a drifting world has no ceiling. (iv) Individually optimal overconfidence is converted, through the single arrow "belief determines position size", into multiplicative volatility for the population. Under Kesten dynamics the tail index of a mixture is the minimum over its components: κ★_pop = min_i κ★( f_i ) Not the average. The minimum. Which agent attains it depends on the sign of the edge. On an unfavourable game it is the most leveraged, and the index stays above one. On a favourable game, the only kind on which super-heavy tails with no finite mean can arise, κ★ = 1 − 2/c rises with the relative position c beyond the ruin point c = 2, so the tail is set not by the boldest agent but by the one closest to the ruin point, and pulling a position back toward that point lowers the index. The minimum governs only beyond a crossover scale: at finite thresholds the weights of the components still matter, so restraint that is not refilled by others lowers exceedance probabilities even when it leaves the index unchanged.