Why a fair-coin game has infinite expected payout but finite willingness to pay, how a 64-unit cap gives a mean of 7, and why log utility does not supply a universal ticket price.
St Petersburg paradox
The St Petersburg paradox is the tension between an ideal coin-toss game’s infinite expected monetary payout and a person’s finite willingness to pay to enter it. For a quant, it exposes a valuation trap: a mathematically correct expectation need not be a useful purchase price when the payoff is unbounded, the payer has limited resources, or the buyer cares about wealth beyond its mean.
A game that ends, with a mean that does not
Toss a fair coin independently until its first head. If that head arrives on toss N, pay X=2N units: 2 for a first-toss head, 4 for a second-toss head, 8 for a third-toss head, and so on. There is no fee in the payout definition, no discounting and no maximum prize. All monetary amounts below use the same payout unit.
Reaching toss n requires n−1 tails and then a head. Doubling the prize exactly offsets halving its probability:
E[X]=n=1∑∞2n2−n=n=1∑∞1=∞.
Every extra first-head position adds another unit to the expectation. Yet the game ends almost surely: the probability of no head in the first m tosses is 2−m, which tends to zero. Each realised prize is finite with probability one. A finite outcome almost surely does not imply a finite mean.
This is not a contradiction inside probability theory. The paradox appears when “maximise expected money” is treated as sufficient reason to pay any finite entry fee. Subtracting a fixed finite fee leaves the ideal game’s expected net payout infinite, but says nothing about whether that fee is affordable or attractive to this buyer.
Daniel Bernoulli’s historical treatment uses a version starting at one ducat, half the payouts used here. Its divergence is the same. His distinction between money and the utility of wealth motivates the decision question; it does not change the probability calculation.
A 64-unit cap is not a six-toss deadline
Suppose the organiser instead promises Y=min(2N,64). Continue until the first head, but never pay more than 64. The cap makes the distribution finite even though the stopping time remains unbounded.
For N=1,…,5, each weighted contribution is still 1. If N≥6, all those outcomes pay 64. Their combined probability is the chance that the first five tosses are tails, 1/32:
E[Y]=n=1∑52n2−n+64P(N≥6)=5+64(321)=7.
Fair-coin game: each uncapped first-head position contributes 1 payout unit to expectation. Capping the payout at 64 gives five contributions of 1 plus a tail contribution of 2, for a mean of 7. Paying zero after six tails instead gives a mean of 6.
Original exact calculation for the stipulated game, not a simulation. A payout cap preserves the tail payment; a deadline with zero payment removes it.
Now change the contract: stop after six tosses, pay 2N if a head has appeared, and pay zero if all six tosses are tails. This different payoff is
Z={2N,0,N≤6,N>6.E[Z]=n=1∑62n2−n=6.
The one-unit difference comes entirely from six tails: probability 1/64, payment 64 under the cap, payment zero under the deadline. No probability should be silently discarded or renormalised. The experimental design described by Cox and co-authors includes finite games with this zero-after-the-deadline convention.
More generally, a payout cap at 2m gives expected payout m+1, while a zero-payment deadline after m tosses gives m. The contract’s tail rule is part of the model, not an incidental implementation choice.
Why “log utility gives a price of 4” skips a step
For the uncapped game, taking the natural logarithm of the numerical payout gives a finite expectation:
E[lnX]=ln2n=1∑∞n2−n=2ln2≈1.386294.
Here X is expressed as a number of payout units, so the logarithm is dimensionless. Exponentiating gives 4. That is the certainty equivalent of the payout considered in isolation under this log transformation. It is not a universal maximum ticket price.
Expected utility and risk aversion normally concern final wealth. Let initial wealth be W>0, the entry fee be c, and utility be u(w)=ln(w/wref) for a fixed positive reference unit. With no other wealth changes, an indifference fee satisfies
E[u(W−c+X)]=u(W).
Compare only affordable fees under the assumed borrowing rules, and require positive final wealth in every relevant outcome. The fee is inside the utility calculation. Changing W, preferences, the cap or the organiser’s ability to honour the prize changes the decision. Finite expected log payout alone has not specified any of those things.
Concave utility provides a coherent preference model for this game; it is not a complete empirical explanation of every response to St Petersburg lotteries. Cox and co-authors report similar reluctance in waiting-time experiments even where separate choices indicated risk seeking. Their finite experimental games do not directly test an enforceable infinite-prize promise.
For modelling, keep three questions distinct: What exactly is paid? Does its monetary expectation exist as a finite number? How does the buyer value the resulting final wealth? A finite simulation run produces finite prizes and a finite sample average; that does not establish a finite population mean for the ideal game. A real cap should enter the payoff definition before valuation.
For the probability-weighting foundations, use Expectation, Variance and Covariance. Its public page previews the lesson; the full selected primer lesson requires a free signed-in account.
