# Time costs and the St. Petersburg paradox

In the St. Petersburg paradox, we end up with the problem that a rational agent should be willing to play the game for any wager, if we look at expected income or utility of expected income. The standard "solution" to this is to instead look at expected utility of income, where the marginal utility of income is decreasing.

Intuitively, decreasing marginal utility (for instance, where utility is logarithmic) of income makes sense, since one could argue that the utility of income comes from purchasing power, and standard indifference curves are convex.

But I've seen people (in a statistics department) point out that playing the game also has a time cost in addition to the monetary wager. This suggests potential other trade-offs that could affect decision making here, such as the fact that this time could be spent on more enjoyable pursuits, or in working and making money, even if the decreasing marginal utility is still the most convincing explanation for people's behavior to me.

How might be time-costs factored into the St. Petersburg problem, in terms of limiting the player's wager, and what literature might provide a good coverage of this?

Consider the version of the paradox from Wikipedia:

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, 16 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2k dollars, where k equals number of tosses. What would be a fair price to pay the casino for entering the game?

If there is a finite fixed per-period time cost, $c$, of playing the game then the paradox would not be resolved. The payoff from the lottery would then be $$-c+\sum_{t=1}^\infty \left[\frac{1}{2^t}\left(2^t-c\right)\right]=\sum_{t=1}^\infty \frac{1}{2^t}2^t-\sum_{t=0}^\infty \frac{c}{2^t}=-2c+1+1+1+\ldots=\infty$$

Thus, as in the original paradox, the value of the lottery is infinite.

An alternative might be to suppose that the marginal cost of time spent is not constant (e.g. it is increasing). However, it is hard to see how one would justify an increasing cost of time without invoking some idea of the dis-utility of time (but the whole purpose of this exercise is to get away from invoking a utility function).

Two solutions that are related (in the sense that they involve a time dimension) and that might be more appealing are:

(1) discounting. Even if your utility of wealth is linear, you might put a low weight on future payments because you have to forego some return during the period between now and when the payment occurs. If the future is discounted at rate $\delta<1$ then the lottery would have value $$\sum_{t=1}^\infty\delta^{t-1}\frac{1}{2^t}2^t=\frac{1}{1-\delta}.$$ Thus, the value of the lottery is finite for any $\delta<1$ and the lottery is more valuable to more patient people (i.e. people with larger $\delta$).

(2) finite lifespan. This is formally equivalent to discounting, but conceptually different. Now the idea is that the player is not going to live forever so there is a chance that they 'die' before having the opportunity to collect the lottery's payoff. In particular, suppose that the player dies at the end of each period with probability $\lambda$ and that the payoff is zero if s/he dies before the end of the lottery. The value of the lottery is therefore $$\sum_{t=1}^\infty\lambda^{t-1}\frac{1}{2^t}2^t=\frac{1}{1-\lambda}.$$ Thus, players who expect to live a long time have a higher willingness to pay for the lottery, but the value of the lottery is finite for anyone who does not expect to live forever.