I wish you'd come up with a way to quantify aversion to fluctuation.
Certain-equivalent dollars is fairly simple, and consistent. You just need
a number to quantify your aversion to risk. Traditionally, that number is
called "bankroll," but it doesn't necessarily mean ready cash.
To use CEq, replace each uncertain outcome by its Certain Equivalent. Yuri
used a formula with logarithms that was correct, but too slow to work out in
my head. I prefer an approximation that's accurate enough as long as the
numbers are each smaller than my "bankroll."
For a bankroll of B, each payoff P gets replaced by its certain equivalent,
C, calculated as
2B
C = P * -----
2B+P
So, for a half-million dollar bankroll, 2B is a million dollars (Tom is
playing $5 NSUD, presumably with only about a 0.6% cash back, and must have
a bankroll that large for the game to be worth playing in the first place).
A 20,000 royal has a certain equivalent of about 19,608; less $108 of tips
leaves about 19,500, the number that I use as what the jackpot is worth for
every decision which can result in winning it. In particular, I would draw
for that royal only when the expected cost of drawing for it is less than
19,500 per extra royal hit.
The certain equivalent of a number that is small relative to the bankroll is
nearly unchanged by this substitution. A number which is one percent of the
bankroll is reduced by about half a percent; a number which is ten percent
of the bankroll is reduced about 5 percent.
The $250,000 capped dealt royal progressive at the Venetian had a certain
equivalent against that same half-million bankroll of only $200,000, which
amortized to 30.7 cents per pull, while the underlying game returned 97.87%
without dealt royals and after allowing for a $40 tip for each drawn royal;
at $15 of coin-in per hand, that comes to $14.68 returned per initial hand
dealt. Add in the $.045 for (Gold) cashback, and the game had a certain
equivalent of 3.2 cents per pull: about $20 per hour. For players with
smaller risk tolerance / bankroll, the risk-adjusted expectation of the game
was negative.
Incidentally, B, bankroll, in the above is the full-Kelly bankroll, and can
also be used in the contexts where Kelly calculations are performed.
···
--
Randy Hudson