Open Archive 6
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Posted by Brh on 5 June 1999, at 4:40 a.m., in response to Contradiction between...., posted by Clarke Cant on 2 June 1999, at 6:52 a.m.
you will. Yet the Yamashita (I am willing to use that label for clarity even though I was first with such formulas in
1982) formula leads to taking your normal expectation, expressed as ev, and being able to track bankroll growth as ev..
I hope back in 82 you provided the existence and continuity proof which Yamashita failed to provide. Also Yamashita did not generalise the idea to fixed bettors and provide the connection between optimal betting spreads, real dollar betting schemes and risk of ruin. Karel Janacek independently, at the same time as myself, produced the proof that a unit spread count be constructed, but I showed the additional connection to fixed schemes and ROR.
The full optimal betting theory is a non-linear functional which satisfies certain continuity and consistency conditions. Yamashita showed how to construct a betting scheme in situ given a real Kelly bankroll. What was missing was that the procedure could be split into two stages, the first being the construction of the optimal unit spread, which is independent of Kelly or ROR, and the second stage, which depending on whether the bettor was Kelly or fixed, is the construction of the optimal dollar betting scheme from the optimal spread. The optimal spread simply maximises DI, or minimises N0, without reference to any bankroll considerations. A player using a Kelly betting scheme will continue to use the same optimal spread, regardless of the raising or lowering of the bankroll, only the unit dollar bet $B changes, producing a new scheme for each bankroll size.
To sum up, optimal spread theory has nothing to do with Kelly or ROR, or any monetary unit whatsoever, it is simply the unit spread which maximizes the unit ev to sd ratio, or minimizes the long run index.
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