[Ganchrow]

This is the more general case where we relax the constraint that all single-bet Kelly stakes need to be equal. For the sake of sanity, the weightings are defined recursively.

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Given n uncorrelated binary bets, the "Kelly-optimal allocation" is the set of weightings for each of the 2ⁿ-1 n-or-fewer-team parlays (where a single bet is considered a 1-team parlay) that can be created from the n-single bets, which maximizes the expected logarithm of the bankroll.

Let o_i = decimal odds on the i^th bet,
Let p_i = win probability of the i^th bet,
Let k_i = i^th single-bet Kelly stake = MAX[(p_i*o_i-1)/(o_i-1), 0],

Define κ(n,m,{B}) as the sum of the Kelly optimal weights for all m-team parlays made up of all bets included the set {B}, then

                       n 
κ(n,m,{B}) = ∏ ki  -  ∑ κ(n,i,{B})
           i Є {B}   i=m+1

Example:

given:
k1 = 1%
k2 = 2%
k3 = 3%
k4 = 4%
k5 = 5%

κ(5,5,{1,2,3,4,5})
  = (weighting of the 5-team parlay as % of bankroll)
  = k1*k2*k3*k4*k5 
  = 1%*2%*3%*4%*5%
  = 0.0000012%

κ(5,4,{1,2,3,4})
  = (weighting of the 4-team parlay consisting of bets {1,2,3,4} as % of bankroll)
  = k1*k2*k3*k4 - κ(5,5,{1,2,3,4})
  = 1%*2%*3%*4% - κ(5,5,{1,2,3,4,5})
  = 0.0000228%

κ(5,4,{1,2,3,5})
  = (weighting of the 4-team parlay consisting of bets {1,2,3,5} as % of bankroll)
  = k1*k2*k3*k5 - κ(5,5,{1,2,3,5})
  = 1%*2%*3%*5% - κ(5,5,{1,2,3,4,5})
  = 0.0000288%

κ(5,3,{1,2,3})
  = (weighting of the 3-team parlay consisting of bets {1,2,3} as % of bankroll)
  = k1*k2*k3 - κ(5,4,{1,2,3}) - κ(5,5,{1,2,3})
  = 1%*2%*3% - κ(5,4,{1,2,3,4}) - κ(5,4,{1,2,3,5}) - κ(5,5,{1,2,3,4,5})
  = 0.00054720%

etc.

And now a non-recursive statement of the same. Please forgive the abuse of notation.

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Given n uncorrelated binary bets, define the "Kelly-optimal allocation" is the set of weightings for each of the 2n-1 n-or-fewer-team parlays (where a single bet is considered a 1-team parlay) that can be created from the n-single bets, which maximizes the expected logarithm of the bankroll.

Let o_i = decimal odds on the i^th bet,
Let p_i = win probability of the i^th bet,
Let {k} = the set of all n single-bet Kelly stakes,
where k_i = i^th single-bet Kelly stake = MAX[(p_i*o_i-1)/(o_i-1), 0],

Define {P(k)} = the power set of {k}

Define {S({B},i)} = the set of all sets, {s} Є {P(k)} such that |{s}| = i, {S} ⊇ k_{B},
where k_{B} is the set of the single-bet Kelly weights associated with the elements of {B}

Define κ(n,m,{B}) as the Kelly optimal weight for the m-team parlay made up of all bets included in the set {B} (where |{B}| = m).

             n
κ(n,m,{B}) = ∑( (-1)i-m * ∏ kj)
             i = m              j Є {S({B},i)}