Here's a "what would you have done?" question - I just encountered a situation in which I was last to act on the final round of a tournament which payed 70% of the prize pool to 1st place, and 30% to 2nd. I was faced with the choice of standing on the BJ I had been dealt, and taking a guaranteed 2nd place (with no chance of 1st), or risking the DD against a dealer 10 to go for 1st (and risk getting nothing). At the time I figured the DD was probably the better play, but could not bring myself to do it. I think the fact that I've just come off a mini losing streak had a big influence on my decision-making here. The DD outcome probabilites are win: 50.9%, push: 6.9%, lose: 42.2%; so the EV comparison is .... 30% guaranteed if I stand, versus (50.9% * 70%) + (6.9% * 30%) = 37.7% if I DD. The EV of the DD is better, but there is that nagging 42.2% chance that I'll come away with nothing. I imagine the absolute value of the prize money involved will be the biggest factor that might influence people to take the risk-averse route. In this case it wasn't much at all - 1st: $112, 2nd: $48. (Compared to the entry fee of $20+2) - and I really was being a bit cowardly. However, add a few noughts and even the most ardent chaser of the maximal expected value might think twice. So how much do you think a bird in the hand is really worth?
a good utility function for personal investments (for someone who is not in poverty) is the Expectation of the logarithm of the your total worth. If your net worth is large enough, that simplifies (roughly) to the (linear) Expectation of gains. So here, the correct strategy would be to DD. If the potential gain is large compared to your total worth, you have to calculate Expectation of ln(worth+gain) and you can see that makes you more risk averse. There's a point where the correct strategy is to stay.
Thanks for that Now that you mention it, I recall that I have come across this before. There is a section in Griffin's 'The Theory of Bllackjack' which comes up with formulae to determine whether or not to insure a BJ, based on the amount of your bankroll you have invested in the hand, not just the simple 1/3 test of the proportion of tens remaining. I confess the factors influencing me were more psychological than coolly analytical. Twenty seconds to decide, and the two interested parties, who each stood to lose or gain by my decision, whispering contradictory advice in my ear. I don't really have the mathematics to understand the underlying justification for the principle of maximising the average logarithm of your capital. From my naive perspective, a big factor in this particular decision was that it involved a jump from a locked-in return to a coin-flip. Could it not be argued that there is value in being extra-cautious in such circumstances, in order to 'smooth out' the randomness? If I were faced with this same decision a thousand times in succession, it must be quite probable that my actual return from doubling each time would be less than the guaranteed return from standing. (Of course it's more probable that the return would be higher.)
