I've asked this question on many places already & this forums comes highly recommended; perhsps the solution is provided here.
I've been working in a spreadsheet, inputing the formulas which give a result not desired or intended. I do wonder if the math side of things is correct.
That being said, here's the relevant data, & the formulas applied are further below.
LEGEND
A1 ....... balance (bal), or oustanding debt carried over
A4 ....... base unit (bu)
B1-B4 .... multiplier (mul), whatever B value is, minus 1
C1-C4 .... correct result (CR).. for the formulas once done
D1-D4 .... value aimed at .. CR for D1,D2 ≈1 .. D3,D4 ≈0
E1-E4 .... D non-aproximated
| 0 | A | B | C | D | E |
|---|---|---|---|---|---|
| 1 | -2.15 | 9 | 1.78 | 1 | 1 |
| 2 | 6 | 2.67 | 1 | 1 | |
| 3 | 4 | 3.44 | -0.01 | -0.009999999999999787 | |
| 4 | 1 | 2.34 | 5.88 | -0.0108 | -0.010800000000000143 |
The issue I have with set of formulas in C1-C4 is that all those must take into the account any change in the cells B1-B4 (hard-typed, input manually on the go), or said differently any change in B1-B4 shall affect the results in all cells C1-C4.
The intended result in D1,D2 are 1, & D3,D4 ±0 .. or as close by as possible.
Practically speaking: based on the B-values, we are to determine the best equilibrium in wagers C1-C4, so that C1,C2 gives us +1 unit or very nearby, & C3,C4 basically break even.
When I change any value in set B, all the wagers in C1-C4 should adjust into a newly-established equilibrium, constituting that intended result in D1-D4.
To finalize, regarding the equation .. the values in columns B (odds) & D (result) are fixed, the only variable is the wager (C1-C4).
Formulas
My logic in creating the formulas on the most fundamental level is
C1,C2 wager multiplied by odds, minus sum of the balance & all other wagers .. resulting in +1 (multiplication overcoming the sum of wagers, & oustanding debt + 1 unit extra)
C3,C4 wager multiplied by odds, minus all other wagers (multiplication voiding the cost of other wagers = break even)
Is my logic correct?
Spreadsheet results show otherwise, implying there's something I do not account for, yet ..
To clarify, the results are falling a bit short .. so that when I increase the base unit (where bu=1 means that the C1,C2 should be 1, bu=5 → C1,C2=5) to 1.22, I finally get D1-D4 to +1,0 respectively, but not in the completely optimized way equilibrium-related.
Exact formulas
C1 = (bu-bal+wag2+wag3+wag4) / (odds-1) = (A4-A1+C2+C3+C4) / (B1-1) = ( A4-A1 + (A4/(B2-1)) + (A4/(B3-1)) + (A4/(B4-1)) / (B1-1)
C3 = (wag1+wag2+wag4) / (odds3 -1) (C1+C2+C4) / (B3-1) = (A4/(B1-1)) + (A4/(B2-1)) + (A4/(B4-1)) / (B3-1)
The formulas in column D are: for
D1 = (C1 x B1) - (C2+C3+C4) + A1.
D2 = (C2 x B2) - (C1-C3-C4) + A1.
D3 = (C3 x B3) - (C1+C2+C4).
D4 = (C4 x B4) - (C1+C2+C3).
For C3 & C4 you'd just adjust the formulas above repectively with correct shift of numerals.
What is that I am not accounting for yet?
What is that I am not accounting for yet?
Or, how to achieve the perfect equilibrium in wagers C1-C4 optimized to get D1,D2= +1 & D3,D4= ±0, with minimal offset distributed across all four (bold figure under C in the table at the top)?