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Originally Posted by Art Vandeleigh
I don't get what you're trying to say.
Let's say Mr. X has a bankroll of $4,000
He bet 25% or $1,000 at +100 odds on NFC +6
This turned out to be a bad bet. To safely hedge at AFC -6 he must lay -120.
So if Mr. X bets $1090 on AFC -6, he would get back $908
If AFC win by more than 7, would lose $1000-$908= $92
If AFC wins by 6, both bets push
If NFC wins SU or loses by 5 or fewer, would lose $1090-$1000=$90
So MR. X is going to lose about 90/4000=2.25% of bankroll.
What are you trying to optimize here? I'm so not with the program.
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In this example we're maximizing the player's
expected utility. Putting it another way, we're attempting to find the optimal trade-off between risk (which is bad) and return (which is good).
If the player were only interested in maximizing return (in other words, if he were risk neutral) the optimal decision would be not to hedge at all. This is because both hedge candidates have negative expected value.
If the player were only interested in minimizing risk (in other words, if he were infinitely risk averse), he would behave largely as you've described. (An infinitely risk averse player would want the same financial outcome regardless of the game's outcome. He would do this by betting less on AFC -6 -120 than in your example, and more on AFC -6½ -109. So assuming that the player bet $1,000 on NFC +6 +100, the proper hedge for the infinitely risk averse player would be $1,000 on AFC -6, and $86.92 on AFC -6½. This way, regardless of the game's outcome the financial result would be a loss of $86.92.)
What I demonstrated was how a player with logarithmic preferences (which are the preferences that the Kelly criterion requires of a player for it to be that player's optimal staking strategy) would go about optimally hedging the initial bet, given the two hedge candidate choices.
The generalized form of the expected utility function (which is given in the initial post) is:
E(U) = ( K/K-1) * ∑i{p(outcomei) * [1 + (outcomei profit or loss)]1-1/K} for K ≠ 1
E(U) = ∑i{p(outcomei) * ln[1 + (outcomei profit or loss)]} for K = 1
where constant K is the Kelly multiplier.