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3 outcome answered question

Hi all,

I've been having trouble finding the Kelly Criterion bet size for an each-way bet.

The above link shows the solution to a problem with 3 distinct and mutually exclusive outcomes. In an each-way bet there are three distinct outcomes (Lose all bets, horse places but doesn't win, horse places and does win).

An each way bet is a matched bet of one bet on the outright win and one on the placing of the horse (1st or 2nd or 3rd for example). These bets are of an equal amount.

I have calculated the Kelly criterion decimal bet for the win only and place bets independently and these are: 0.06 and 0.23 respectively but since to place this each way bet, I must choose an equal stake for each bet, hence my problem.

One option is a simple average which brings us to 0.145

The problem to use the Kelly criterion method as linked is that I can not know the odds for the second possible outcome (places but doesn't win) as the odds for place (win or not) or outright win can only be known. Does anyone have any insights into this?

Thanks for reading and any forthcoming help!

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The question to which you link to, answers your question too, as you can treat these as 3 different outcomes. For example, if you back a horse E/W with x-x @5/1 odds with 1/5 E/W for places 1-2-3, then, in David Speyer's answer, you need to substitute for no place $b_1=-2$, for place 2-3 $b_2=1-1=0$, and for win $b_3=1+5=6$ (where the numbers are nice because I've chosen the parameters so).

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