Abstract
Interest in multiplicative vs. additive returns on bets has been revived by Peters, who proposes ergodicity and added noise are useful in understanding utility preferences. Peters requires a Monte Carlo simulation to demonstrate empirically a supposed paradox that arithmetic expectation is inappropriate for multiplicative gain distribution forecasting. Here I formalize the r operator notation, which significantly simplifies multiplicative problems, as an extension of the arithmetic group's Δ/d discrete and continuous operators into the multiplicative semigroup. I show how the annihilating (absorbing) element of the multiplicative semigroup at 0, not +/-∞, may be used to conveniently represent nonlinear sequence occurrences, such as running out of money, without the need for special computer rules outside the mathematics. I use this to solve Peters' expected-value paradox elegantly, without ergodicities nor noise. But Peters misses the real paradox, called “Just One More”: the outcome of an advantageous additive gamble is identical to the outcome of a similar disadvantageous multiplicative gamble, after one trial; hence, by induction, an agent will keep playing. I propose games “Hero or Heroin” and “American Roulette” to highlight this paradox. This may help in explaining addiction. The Supplement contains further visualizations and arguments against the need and applicability of ergodics for utility. The results contribute to the understanding of repeated multiplicative gambles with annihilating states, and their utility.
Multiplicative Gains, Non-ergodic Utility, and the Just One More Paradox (with Supplemental Information)
