Would I pay $24k to play a game where I had a 33⁄34 probability of winning an extra $3k? Let’s consult our good friend the Kelly Criterion.
We have a bet that pays 1/8:1 with a 33⁄34 probability of winning, so Kelly suggests staking ~73.5% of my bankroll on the bet. This means I’d have to have an extra ~$8.7k I’m willing to gamble with in order to choose 1b. If I’m risk-averse and prefer a fractional Kelly scheme, I’d need to start with ~$20k for a three-fourths Kelly bet and ~$41k for a one-half Kelly bet. Since I don’t have that kind of money lying around, I choose 1a.
In case 2, we come across the interesting question of how to analyze the costs and benefits of trading 2a for 2b. In other words, if I had a voucher to play 2a, when would I be willing to trade it for a voucher to play 2b? Unfortunately, I’m not experienced with such analyses. Qualitatively, it appears that if money is tight then one would prefer 2a for the greater chance of winning, while someone with a bigger bankroll would want the better returns on 2b. So, there’s some amount of wealth where you begin to prefer 2b over 2a. I don’t find it obvious that this should be the same as the boundary between 1a and 1b.
This is a problem because the 2s are equal to a one-third chance of playing the 1s. That is, 2A is equivalent to playing gamble 1A with 34% probability, and 2B is equivalent to playing 1B with 34% probability.
Equivalence is tricky business. If we look at the winnings distribution over several trials, the 1s look very different from the 2s and it’s not just a matter of scale. The distributions corresponding to the 2s are much more diffuse.
Surely, the certainty of having $24,000 should count for something. You can feel the difference, right? The solid reassurance?
A certain bet has zero volatility. Since much of the theory of gambling has to do with managing volatility, I’d say certainty counts for a lot.
Would I pay $24k to play a game where I had a 33⁄34 probability of winning an extra $3k? Let’s consult our good friend the Kelly Criterion.
We have a bet that pays 1/8:1 with a 33⁄34 probability of winning, so Kelly suggests staking ~73.5% of my bankroll on the bet. This means I’d have to have an extra ~$8.7k I’m willing to gamble with in order to choose 1b. If I’m risk-averse and prefer a fractional Kelly scheme, I’d need to start with ~$20k for a three-fourths Kelly bet and ~$41k for a one-half Kelly bet. Since I don’t have that kind of money lying around, I choose 1a.
In case 2, we come across the interesting question of how to analyze the costs and benefits of trading 2a for 2b. In other words, if I had a voucher to play 2a, when would I be willing to trade it for a voucher to play 2b? Unfortunately, I’m not experienced with such analyses. Qualitatively, it appears that if money is tight then one would prefer 2a for the greater chance of winning, while someone with a bigger bankroll would want the better returns on 2b. So, there’s some amount of wealth where you begin to prefer 2b over 2a. I don’t find it obvious that this should be the same as the boundary between 1a and 1b.
Equivalence is tricky business. If we look at the winnings distribution over several trials, the 1s look very different from the 2s and it’s not just a matter of scale. The distributions corresponding to the 2s are much more diffuse.
A certain bet has zero volatility. Since much of the theory of gambling has to do with managing volatility, I’d say certainty counts for a lot.