Now, if you take (iB-iA)/iA, which represents the percent increase in the expected value of iB over iA, you get the same number, as you stated.
(iB-iA)/iA = .0919 (rounded)
This number’s reciprocal represents the number of times greater the expected value of iA is than the marginal expected value of iB
iA/(iB-iA) = 10.88 (not rounded)
Now, take this number and divide it by the quantity p(iA wins)-p(iB wins). This represents how much you have to value the first $24000 you receive over the next $3000 to pick iA over iB. Keep in mind that 24⁄3 = 8, so if $1 = 1 utilon in all cases, you should pick iA only when this quotient is less than 8.
I have liabilities in excess of my assets of around $15000. That first $15000 is very important to me in a very quantized, thresholdy way, but it is not absolute. I can make the money some other way, but not needing to—having it available to me right now because of this game—represents more utility than a linear mapping of dollars to utility suggests, by a large factor.
The next threshold like this in my life that I can think of is “enough money to buy a house in Los Angeles without taking out a mortgage,” of which $3000 is a negligible portion.
I’d say that the utility I assign the first $24000 because of this lies between 370 and 1080 times the utility I assign the next $3000. This is why I take 1A and 2B given that this entire thing is performed only once. Once my debts are paid, all bets (on 1A) are off.
If we’re dealing with utilons rather than dollars, or I have repeated opportunity to play (which is necessary for you to “money pump” me) iB is the obvious choice in both cases.
So here’s why I prefer 1A and 2B after doing the math, and what that math is.
1A = 24000
1B = 26206 (rounded)
2A = 8160
2B = 8910
Now, if you take (iB-iA)/iA, which represents the percent increase in the expected value of iB over iA, you get the same number, as you stated.
(iB-iA)/iA = .0919 (rounded)
This number’s reciprocal represents the number of times greater the expected value of iA is than the marginal expected value of iB
iA/(iB-iA) = 10.88 (not rounded)
Now, take this number and divide it by the quantity p(iA wins)-p(iB wins). This represents how much you have to value the first $24000 you receive over the next $3000 to pick iA over iB. Keep in mind that 24⁄3 = 8, so if $1 = 1 utilon in all cases, you should pick iA only when this quotient is less than 8.
1A/(1B-1A)/[p(1A wins)-p(1B wins)] = 369.92
2A/(2B-2A)/[p(2A wins)-p(2B wins)] = 1088
I have liabilities in excess of my assets of around $15000. That first $15000 is very important to me in a very quantized, thresholdy way, but it is not absolute. I can make the money some other way, but not needing to—having it available to me right now because of this game—represents more utility than a linear mapping of dollars to utility suggests, by a large factor.
The next threshold like this in my life that I can think of is “enough money to buy a house in Los Angeles without taking out a mortgage,” of which $3000 is a negligible portion.
I’d say that the utility I assign the first $24000 because of this lies between 370 and 1080 times the utility I assign the next $3000. This is why I take 1A and 2B given that this entire thing is performed only once. Once my debts are paid, all bets (on 1A) are off.
If we’re dealing with utilons rather than dollars, or I have repeated opportunity to play (which is necessary for you to “money pump” me) iB is the obvious choice in both cases.