More generally, optimal game/stake selection does depend on what future investment opportunities will be available to you. The premise of the question is a little misleading, because, in fact, whether or not to take ANY risky opportunity today (even in choosing a game to play regularly and "reach the long run" with) depends on how many times you intend to play that game. The intuition, however, is clearest when thinking about rare opportunities with abnormally high risk.
For example, is it possible that it's "not worth" playing the WSOP Main Event because it is so high-variance and only occurs once a year (so you "can't reach the long run"), but that the exact same tournament would be worth playing if it occurred more often, such as once a month? It can be tough to reason through this with intuition alone, but it turns out to be pretty easy to show as a consequence of either rational risk aversion and/or progressive tax rates.
An example of a unique opportunity
Let's take the usual example of a "typical" poker player with our usual after-tax utility function, $80k net worth prior to this year, and $40k income on the year (assumed to be all from poker, otherwise tax effects will shift).
Suppose this player were to come across the unique, one-time opportunity on December 31st to risk $10k on a weighted coin flip with a 51% probability of winning $10k and a 49% probability of losing $10k. A +EV opportunity, but a significant percentage of the small-stakes grinder's bankroll. What to do?
If he passes on the opportunity, his total after-tax utility payoff for the year is the utility of his $80k in existing wealth plus what remains of his $40k in income after paying taxes. If he takes the opportunity, his expected after-tax utility is 0.51 * util($80k + tax($50k)), plus 0.49 * util($80 + tax($30k)).
Which of these two outcomes is higher and whether or not the player should take this risky opportunity will depend on both his risk preferences and his tax bracket. We can look at his optimal decision for each different possible value of his year-to-date poker winnings:
The rows denote his current year-to-date poker winnings and the columns denote how many opportunities he will have available to take this coin flip (in this example, only 1).
A red box denotes that he should pass on the opportunity because taking it reduces his expected after-tax utility. A blue box denotes that he should take the opportunity.
Here, since his year-to-date poker winnings are $40k, he should pass on the opportunity. He would need to be up $100k on the year for it to become a profitable opportunity after adjusting for risk aversion and taxes. We see that it again becomes unprofitable in a higher region, when he has $170k or $180k, which we will discuss later.
If the opportunity is not unique
We modify the above situation by making it so that the player will have the option (but not requirement) to take this same coin flip opportunity up to 30 times. For example, maybe it's December 1st, and he knows that he will have the option of taking this coin flip opportunity once per day for the rest of the year.
This chart is simply an expansion of the first one, with additional columns added. The leftmost column agrees with the first chart, and it treats the case where the player will have only one option to take the opportunity. From that, we can calculate the next column over, where he will have 2 separate opportunities to take the coin flip.
If he passes on the opportunity when there are 2 days left, his year-to-date winnings will stay the same, and his expected utility will be the same as his expected utility at that bankroll level of $40k for when there is 1 day left (as he will again have the option of taking or passing on the opportunity). If he takes the opportunity when there are 2 days left, his expected utility will be equal to 0.51 times the expected utility of having year-to-date winnings of $50k with 1 opportunity left, plus 0.49 times the expected utility of having year-to-date winnings of $30k with 1 opportunity left. In this manner, we can "work backwards" iteratively from the known case of 1 day left to find his optimal decisions for all previous days.
The black boxes denote regions which are impossible to reach. Since the player's initial year-to-date winnings are $40k with 30 days left, he'll never be able to reach, say, a $100k bankroll with 29 days left. He'll also never reach year-to-date winnings of -$80k, because that would mean he would have taken the opportunity when his year-to-date winnings were $-70k, but the player would never be able to profitably risk the last $10k of his net worth on any sort of uncertain outcome.
Before we discuss the results, let's first look at what's causing the different shapes between the red and blue regions. Is the particular form of his decision strategy being shaped by risk aversion, tax effects, or both?
Without risk aversion or taxes
If we set our player's risk aversion to zero and remove the effect of taxes, then, as we might expect, he always takes the opportunity:
Without risk aversion or tax effects, all that matters is whether or not the opportunity has a positive expected value. It always does, so he always takes it.
With risk aversion, without taxes
If we return the player's risk aversion level to ρ=0.8 but still ignore taxes, then he starts to pass on the opportunity when his bankroll is too small to afford the risk:
Here, we see that, with only 1 chance at the opportunity, the player will need year-to-date winnings of $130k or more in order for the coin flip to be profitable after adjusting for risk.
If he had only $120k, then he should pass on the coin flip if it were a unique opportunity. However, with more than 1 option to take the coin flip, he should take it when his year-to-date winnings are $120k! This is because, while the immediate expected utility of taking the coin flip might be slightly negative, the player gets additional benefits to expected utility in the future when he happens to win the coin flip and can then take a profitable opportunity the next day. This turns out to be enough to make his expected utility of taking the opportunity positive relative to passing on it.
We see some other effects at the $110k and $100k levels; if there will be enough options to take the same opportunity in the future, that can be enough to turn an unprofitable opportunity into a profitable one.
With taxes, without risk aversion
If we go back to ignoring risk aversion but add in the effects of federal (and the negligible effects of NJ state) income tax, we see some different shapes in the chart:
Without risk aversion, the only factor that would keep the player from taking this +EV coin flip would be adverse tax consequences. Here, with 1 opportunity left (the leftmost column), we see a few regions where the player will pass on the coin flip. These turn out to be the regions where his tax bracket would change based on the outcome of the opportunity.
For example, the player passes on the coin flip when his year-to-date winnings are $0, because he gets no tax deduction if he loses and hits -$10k, but will have to pay some income tax if he wins and hits +$10k, which turns out to be enough to turn the +EV opportunity into a -EV opportunity after taxes.
The effect is similar at $10k, $30k, $40k, and $170k; a win or loss at any of these points will cause a significant jump in the individual's marginal federal tax rate. This is a great illustration of how progressive taxes create additional risk aversion when a risky opportunity has the possibility of changing one's tax bracket (cough cough).
Just as in the previous case, these tax bracket risks are mitigated when there will be additional options to take the opportunity. With enough time left before the end of the year, the player will take the opportunity regardless of his level of wealth. Essentially, the probability of ending up on the threshold of a different tax bracket gets lower the further out from December 31st you go, and the fundamental +EV nature of the opportunity outweighs this risk if there's enough time left.
With both risk aversion and taxes
Combining both risk aversion and taxes, we get back to the first full chart, where both "shapes" of red regions can be seen together:
A few observations:
- While a pass (red) can turn into a take (blue) when additional options are added (when moving to the right on the chart), the opposite can never happen. Since the additional opportunities are not mandatory, the player could simply blindly pass on some number of them and then be able to realize the full expected utility of the point to the left on the chart. So additional time until utility realization (year-end in our model) can only result in an increase in willingness to take on risk.
- Tax effects are only eliminated entirely when the player's annual income (poker or otherwise) is high enough that the risky opportunity could never move him out of the highest tax bracket. Otherwise, unless the risky opportunity has a large enough pure expected value, risks will never be taken near the boundaries of tax brackets.
- The fact that the opportunities are optional does matter. For example, while the player takes the opportunity at a year-to-date winnings level of $90k when there will be 9 total options to take the opportunity available to him, he would not accept the opportunity if he had to be locked in to taking the coin flip all 9 times. If he had to commit to taking all 9 flips, it turns out that he would need $110k to take that opportunity. The optionality lets him quit in the middle if he ends up losing too much. Strategic options always have nonnegative value, and that is as true in this decision theory problem as it is in game theory.
Keep in mind that the implications about unique situations here are about situations with a uniquely high mean/variance tradeoff compared to normal play. All that matters is the shape of the probability distribution, not the particular nature of the opportunity.
For example, if you play $100 heads-up sit-and-goes for a living where you win 55% of the time and lose 45% of the time (ignore rake), and you happened to come across a one-off investment opportunity (perhaps a prop bet) where you could have a 55% chance of winning $100 and a 45% chance of losing $100, that's exactly the same as your usual heads-up sit-and-go, and you should take it. It doesn't matter that this particular opportunity will only happen once; the laws of probability only care about the distribution of the payoffs, and this will "reach the long run" along with your usual results. It doesn't matter that you "can't reach the long run" with a unique opportunity if you'll be able to reach it with other opportunities with similar (or riskier) payoff distributions.
While coding this algorithm in more complicated cases becomes more difficult, there are some powerful prospects for expansion of this model:
- The discrete nature of this model, while it would be only an approximation for a continuous-time problem in traditional finance, is actually a perfect fit for any poker situation, where the number of sessions or risky opportunities will always be discrete.
- The same approach can be used to compare any number of different possible games. For example, the player could choose between 4 options: playing his regular medium-stakes game, a lower-stakes game (higher mean relative to variance), or a higher-stakes game (lower mean relative to variance), or not playing at all. Expanding the number of opportunities to a larger number, such as 365 opportunities (playing one session each day for a year), would give a very powerful dynamic stake selection model. Rather than just working off of bankroll "rules of thumb", this model would optimize adaptively based on the running total of year-to-date winnings and provide strong risk- and tax-adjusted mathematical guidelines for decisions such as when to move up, when to "take a shot" in a particularly soft game, when to move down after significant losses, etc.
- Similarly, this model can answer the question of at what pot size it becomes beneficial to pay $1 to run it twice. The discrete occurrences of pots over a certain threshold size can be plugged into this model as the two choices (one being to not run it twice, the other being to run it twice) and compared.
- The model could also be expanded to capture randomness in other external investments, such as the player's stock portfolio.