Equitable exchange for multiplicative gambles

Peters (02019) argues for reframing economic theory considering ergodicity. He contends that the traditional economic models of expected wealth from an investment are unrealistic and fail to capture what happens over time unless the wealth dynamic is additive rather than multiplicative. His motivating example is a gamble on the outcome of a fair coin in which your wealth increases by 50% if the coin lands heads and decreases by 40% if tails. Expected change in wealth for Peters's paradigmatic gamble (PPG) is positive, but most players accepting it have a decrease in expected wealth. Peters and Gell-Mann (02016) contrast such multiplicative iterated gambles, where the amount won or lost each time is proportional to the amount bet, with additive gambles in which, “the expectation value would reflect how the individual fares over time”. While others have noted the economics literature that the analysis in Peters (02019) ignores, this brief paper looks at a simpler fundamental problem with the Peters analysis. Specifically, I examine an assumption central to the analysis presented by Peters and, using only basic mathematics, show it to be false. By considering other parameter choices I show that iterated multiplicative gambles can have expectations that reflect how individuals (not just ensembles) fare over time, and I set out a sufficient criterion for when that is so.