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Time resolution
A resolution of the St. Petersburg paradox is possible by considering the time-average performance of the lottery. Although expressible in mathematical terms identical to the resolution from expected logarithmic utility, the time resolution is obtained using a conceptually different approach. This avoids the need for utility functions, the choice of which is to a large extent arbitrary, and the expectation values thereof, in which the interest of an investor in the lottery has little a priori justification.

Peters pointed out that computing the naive expected payout is mathematically equivalent to considering multiple outcomes of the same lottery in parallel universes. This is irrelevant to the individual considering whether to buy a ticket since he exists in only one universe and is unable to exchange resources with the others. It is therefore unclear why expected wealth should be a quantity whose maximization should lead to a sound decision theory. Indeed, the St. Petersburg paradox is only a paradox if one accepts the premise that rational actors seek to maximize their expected wealth. The classical resolution is to apply a utility function to the wealth, which reflects the notion that the "usefulness" of an amount of money depends on how much of it one already has, and then to maximise the expectation of this. The choice of utility function is often framed in terms of the individual's risk preferences and may vary between individuals: it therefore provides a somewhat arbitrary framework for the treatment of the problem.

An alternative premise, which is less arbitrary and makes fewer assumptions, it that the performance over time of an investment better characterises an investor's prospects and, therefore, better informs his investment decision. In this case, the passage of time is incorporated by identifying as the quantity of interest the average rate of exponential growth of the player's wealth in a single round of the lottery,


 * $$\bar{g}(w,c) = \sum_{k=1}^\infty p_k \ln \left(\frac{w-c+D_k}{w}\right) \mathrm{per\ round},$$

where $$D_k$$ is the $$k$$th (positive finite) payout and $$p_k$$ is the (non-zero) probability of receiving it. In the standard St. Petersburg lottery, $$D_k=2^{k-1}$$ and $$p_k=2^{-k}$$.

Although this is an expectation value of a growth rate, and may therefore be thought of in one sense as an average over parallel universes, it is in fact equivalent to the time average growth rate that would be obtained if repeated lotteries were played over time. While $$\bar{g}$$ is identical to the rate of change of the expected logarithmic utility, it has been obtained without making any assumptions about the player's risk preferences or behaviour, other than that he is interested in the rate of growth of his wealth.

Under this paradigm, an individual with wealth $$w$$ should buy a ticket at a price $$c$$ provided


 * $$\bar{g}(w,c)>0.$$

It should be noted that this strategy counsels against paying any amount of money for a ticket that admits the possibility of bankruptcy, i.e.


 * $$w-c+D_k = 0,$$

for any $$k$$, since this generates a negatively divergent logarithm in the sum for $$\bar{g}$$ which can be shown to dominate all other terms in the sum and guarantee that $$\bar{g}<0$$. If we assume the smallest payout is $$D_1$$, then the individual will always be advised to decline the ticket at any price greater than


 * $$c_\mathrm{max} = w+D_1,$$

regardless of the payout structure of the lottery. The ticket price for which the expected growth rate falls to zero will be less than $$c_\mathrm{max}$$ but may be greater than $$w$$, indicating that borrowing money to purchase a ticket for more than one's wealth can be a sound decision. This would be the case, for example, where the smallest payout exceeds the player's current wealth, as it does in Menger's game.

It should also be noted in the above treatment that, contrary to Menger's analysis, no higher-paying lottery can generate a paradox which the time resolution - or, equivalently, Bernoulli's or Laplace's logarithmic resolutions - fail to resolve, since there is always a price at which the lottery should not be entered, even though for especially favourable lotteries this may be greater than one's worth.