Suppose you are offered the following game: We will toss an honest two-sided coin until the first head shows up. You will win 2^n dollars where n is the number of tosses that gets you to that first head. So, if the sequence is H (the very first toss is a head), you win $2; if it is TH (i.e., you toss first a Tail, then a Head) you win $4. TTH is worth $8, etc. What would you pay to play the game? The expected winnings are the sum of an unlimited number of terms, each showing the probability of a particular sequence times the payoff for that sequence:

(0.5 * 2) + (0.25 * 4) + (0.125 * 8 ) + . . . (an unlimited number of terms)

which equals

1 + 1 + 1 + . . . (also an unlimited number of terms)

or, in short, the expected winnings are infinite. No one, of course, would be willing to pay an infinite amount to play this game.

Is any of this news? Well, no. This is the famous St. Petersburg Paradox, much discussed and resolved the better part of three hundred years ago. Here is another description of the game, maybe better written:

The expected utility hypothesis stems from Daniel Bernoulli's (1738) solution to the famous St. Petersburg Paradox posed in 1713 by his cousin Nicholas Bernoulli (it is common to note that Gabriel Cramer, another Swiss mathematician, also provided effectively the same solution ten years before Bernoulli). The Paradox challenges the old idea that people value random ventures according to its expected return. The Paradox posed the following situation: a fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2n ducats. How much should one pay to play this game? The paradox, of course, is that the expected return is infinite. . .Yet while the expected payoff is infinite, one would not suppose, at least intuitively, that real-world people would be willing to pay an infinite amount of money to play this!

Daniel Bernoulli's solution involved two ideas that have since revolutionized economics: firstly, that people's utility from wealth, u(w), is not linearly related to wealth (w) but rather increases at a decreasing rate - the famous idea of diminishing marginal utility, (Chips: here it gives some expressions using the notation of the calculus with the intuitive meaning that each additional dollar contributes something to the owner's utility, but that the increase is less for each subsequent dollar) ; (ii) that a person's valuation of a risky venture is not the expected return of that venture, but rather the expected utility from that venture.

I make it a point to mention the age of this problem and its solution because I have seen some implausible claims of originality in FIRE posts. I make no such claim. My material comes from classes I took decades ago.

I have posted my arguments based on utility theory and life expectancy in another thread here at NFB's FIRE Board. I have not posted these ideas at Hocus' board even though he suggested that the ideas I raised would be appropriately discussed over there. If he thinks so, I have not made myself sufficiently clear. What I am suggesting is subversive to the entire notion of safe withdrawal rates as usually defined. I want instead to derive mathematically a schedule of withdrawals over many years that maximizes the expected total utility of the withdrawals. The expectation depends on the owner's life expectancy as we show by making explicit use of mortality tables. The utility depends on the planners' current level of wealth and, to be plausible, must grow at a declining rate. (The logarithmic utility function satisfies these requirements. That's the one I have used in my experiments.)

Withdrawal schedules set up in the "conventional way" aim at avoiding bankruptcy for a specified number of years while taking annual withdrawals with fixed purchasing power. This is not likely to maximize the total utility of the withdrawals, depending on what utility function the planner uses. Further, this approach pays too little attention to the planner's not knowing how long he will live. Hocus can and, I suppose, should see these ideas as disruptive to his SWR research and therefore quite appropriately kept off his board. I have posted occasionally on that board, but only when I think my comments are not at all controversial or disruptive. He has a right to pursue his goals in peace; I just don't happen to share them.

I am glad to hear that posts at TMF have considered that the risk in an investment is less significant for someone with ample FIRE assets than it is for someone with marginal FIRE assets. However, the basic idea (the utility of the next ducat diminishes with the number of ducats already owned) is approaching three centuries in age. I would not be surprised to find that there have been mentions of Bernoulli and St. Petersburg over there at TMF too. I would like to see some posts that mention (and use) standard mortality tables as somehow significant in FIRE planning. As ataloss put it, a 50-year old man with a million dollars is somehow richer than a 100-year old man with a million dollars.

(Editted after the fact to fix an error spotted by Wise and Lucky and Perceptive: The line in the quotation that reads " if the first head appears on the nth toss, then the payoff is 2n ducats." should read " if the first head appears on the nth toss, then the payoff is 2^n ducats." I won't pretend that I copied it correctly in the first place.)

He who has lived obscurely and quietly has lived well. [Latin: Bene qui latuit, bene vixit.]

Chips