Fractional odds at less than 1:1

[Courtenay]

What exactly is the rule for fractional odds if the odds are less than 1:1 and one is using fractional odds and the 2d10 table? MWiF is using a different interpretation of the rules than I do, but I am not at all sure that MWiF is wrong; I just reread the rules, and they are ambiguous. The fractional odds rule reads:

Option 41: (Fractional odds) Round to a whole number in favor of the defender, then work out how far to the next odds ratio you are. Round this in favor of the defender to the next 10%. Roll a die just before rolling the combat die (you could roll it with the combat die if you want), to see if you find the result on the lower odds or the higher odds. If you roll the percentage or less, you resolve it on the next higher odds, otherwise on the lower odds.

The 2d10 table rules state:

Section 11.16.6. 2D10 Odds Modifier
When playing with Fractional Odds (see Section 11.6.5), for odds of 1:1 and higher, the odds modifiers are considered linear (e.g., 3.65:1 gives you 7.3 die roll modifiers, while 3.64:1 gives you 7.2).

I have always interpreted this to mean that if one has odds of less than 1:1, one does not use fractional odds. However, that is not what the rule says -- it says what to do if the odds are >= 1:1, not what to do if the odds are less. MWiF is giving some fractional odds to attacks at less than 1:1. For all I know, this is correct. Equally, it might be incorrect. Does anyone know for sure? It is not in the WiFFE FAQ.

(For that matter, what is the rule if using the 1d10 table?)

[composer99]

ORIGINAL: Courtenay

What exactly is the rule for fractional odds if the odds are less than 1:1 and one is using fractional odds and the 2d10 table? MWiF is using a different interpretation of the rules than I do, but I am not at all sure that MWiF is wrong; I just reread the rules, and they are ambiguous. The fractional odds rule reads:

Option 41: (Fractional odds) Round to a whole number in favor of the defender, then work out how far to the next odds ratio you are. Round this in favor of the defender to the next 10%. Roll a die just before rolling the combat die (you could roll it with the combat die if you want), to see if you find the result on the lower odds or the higher odds. If you roll the percentage or less, you resolve it on the next higher odds, otherwise on the lower odds.

The 2d10 table rules state:

Section 11.16.6. 2D10 Odds Modifier
When playing with Fractional Odds (see Section 11.6.5), for odds of 1:1 and higher, the odds modifiers are considered linear (e.g., 3.65:1 gives you 7.3 die roll modifiers, while 3.64:1 gives you 7.2).

I have always interpreted this to mean that if one has odds of less than 1:1, one does not use fractional odds. However, that is not what the rule says -- it says what to do if the odds are >= 1:1, not what to do if the odds are less. MWiF is giving some fractional odds to attacks at less than 1:1. For all I know, this is correct. Equally, it might be incorrect. Does anyone know for sure? It is not in the WiFFE FAQ.

(For that matter, what is the rule if using the 1d10 table?)

I can't see why you wouldn't implement fractional odds when attacking at odds ratios of less than 1:1. For the 1d10 table, the rule can be implemented exactly as written (with the benefit that the computer works out 'how far to the next odds ratio you are').

The difference, at least for the 2d10 table, is that odds ratios of 1:1 or better translate precisely into die roll modifiers, where each increase in the odds ratio is an additional +2 DRM (and each fractional increase in the odds ratio a fractional increase in the DRM).

But at lower odds ratios, the decimal version of the ratio is always a number between 0 and 1, while the die roll modifiers range from negative to +1 (a 1:1 ratio gives a +2 die roll modifier), so it's not a linear relationship anymore. So instead of translating the odds ratios directly into die roll modifiers, like you can do for >= 1:1 odds, you have to work it out the way the rule does for the 1d10 table.

[paulderynck]

You work out how far off you are from the next DRM up, and the fractional portion to resolve is pro rata of that amount versus the difference between the lower and higher DRMs.

[Courtenay]

Thank you. Another rule that I had been playing wrong for years. I wish that the rules had an example of how to do fractional odds for combats at less than one to one, rather than the one for odds at greater than one to one, which I could have figured out on my own.

As is all too often the case, my signature applies.

Again, thanks.

[ertiyu28]

Hi

<1: 1
1: divide the attack by defense
2: Multiply by -2
3: Add +4
4: Round to the nearest tenth (for the defense)

Ex1:

Def=20
Att=10
20/10=2
2*-2=-4
-4+4=0

Ex2:
Def=20
Att=15
20/15=1.3333333
1.333333*-2=-2.666
-2.666+4=1.33
Round=1 & Fractional= 1 to 3 for bonus 2

Ex3:
Def=20
Att=3
20/3=6.666
6.666*-2=-13.333
13.333+4=-9.333
bonus=-9 & Fractional= 1 to 7 for bonus -8

Sorry for my english I'm french.

[Joseignacio]

Thanks for that, pretty much interesting.

[Orm]

ORIGINAL: ertiyu28

Hi

<1: 1
1: divide the attack by defense
2: Multiply by -2
3: Add +4
4: Round to the nearest tenth (for the defense)

Ex1:

Def=20
Att=10
20/10=2
2*-2=-4
-4+4=0

Ex2:
Def=20
Att=15
20/15=1.3333333
1.333333*-2=-2.666
-2.666+4=1.33
Round=1 & Fractional= 1 to 3 for bonus 2

Ex3:
Def=20
Att=3
20/3=6.666
6.666*-2=-13.333
13.333+4=-9.333
bonus=-9 & Fractional= 1 to 7 for bonus -8

Sorry for my english I'm french.

Thank you. [:)]

[Courtenay]

ORIGINAL: ertiyu28

Hi

<1: 1
1: divide the attack by defense
2: Multiply by -2
3: Add +4
4: Round to the nearest tenth (for the defense)

Ex1:

Def=20
Att=10
20/10=2
2*-2=-4
-4+4=0

Ex2:
Def=20
Att=15
20/15=1.3333333
1.333333*-2=-2.666
-2.666+4=1.33
Round=1 & Fractional= 1 to 3 for bonus 2

Ex3:
Def=20
Att=3
20/3=6.666
6.666*-2=-13.333
13.333+4=-9.333
bonus=-9 & Fractional= 1 to 7 for bonus -8

Sorry for my english I'm french.

You say "divide the attack by the defense", but in your examples you divide the defense by the attack. I assume the examples are right and the algorithm should be corrected?

[ertiyu28]

OOOUUUPPPPSSS

"Divide the Defense by the attack" is ready

Sorry

[paulderynck]

ORIGINAL: ertiyu28

Hi

<1: 1
1: divide the attack by defense
2: Multiply by -2
3: Add +4
4: Round to the nearest tenth (for the defense)

Ex1:

Def=20
Att=10
20/10=2
2*-2=-4
-4+4=0

Ex2:
Def=20
Att=15
20/15=1.3333333
1.333333*-2=-2.666
-2.666+4=1.33
Round=1 & Fractional= 1 to 3 for bonus 2

Ex3:
Def=20
Att=3
20/3=6.666
6.666*-2=-13.333
13.333+4=-9.333
bonus=-9 & Fractional= 1 to 7 for bonus -8

Sorry for my english I'm french.

Examples 2 and 3 are not quite right, so I don't think this is the best procedure for odds less than 1:1.

Let's look at Example 2:
15 attacking 20. You need 13.33 to get a 2:3, which from the table on the 2D10 chart gives a DRM of 1. The next odds up to get a DRM of 2 is 1:1. To get a 1:1 against 20 defense factors, you need 20 attack factors, and you have 15 - 13.33 = 1.67 "extra factors". Divide the extra by the number needed to get there and you have the fractional portion. So the fractional is 1.67 divided by (20 minus 13.33) = 1.67/6.67 = .254.

Over the board, the fractional roll is 1D10 and you always lose the fraction beyond the tenths, so you would need to roll a 1 or 2 to get a DRM of +2, otherwise the attack is a DRM of +1. In MWiF, if memory serves, the fractional is computed based on thousandths, so in MWiF a random number between 0 and 999 is generated and if it is 254 or less, you get a DRM of +2, otherwise it is a DRM of +1.

Example 3:
3 attacking 20. From the 2D10 chart, all odds equal to or lower than 1:6 give a DRM of -10. To get 1:7 against 20, you need 2.86. You have 0.14 "extra factors". Even if you made the fractional roll [which would be 0.14/(3.33-2.86) = 0.14/.473 = .295], you'd still have at best a 1:6, so the DRM in this case is -10 and there's no point rolling for the fractional.

Another example illustrating the general procedure of working out how far off you are from the next DRM up, and the fractional portion to resolve is pro-rata of that amount versus the difference between the lower and higher DRMs:
First, if you look at the 2D10 chart you get a +1 DRM for every "half" an odds. Here's a portion of what the full chart would look like when using fractionals:

Odds DRM
1:4 == -4
2:7 == -3
1:3 == -2
2:5 == -1
1:2 == 0
2:3 == +1
1:1 == +2
3:2 == +3
2:1 == +4
5:2 == +5
3:1 == +6

13 attacking 34. Q. What odds are attained before the fractional is considered? A. 1:3 giving a DRM of -2

Q. How many factors are "extra"? A. 34/3 = 11.33 are needed of the 13, so 13 - 11.33 means 1.67 are "extra".

Q. How many factors are needed to get an odds ratio of 2:5 and thus the next higher DRM of -1? A. 34/2.5 = 13.6

Q. What is the pro-rata of the distance to that odds level? A. 1.67/(13.6-11.33) = 1.67/2.27 = .735

Thus the fractional roll to go up one DRM to a -1 attack is 7 or less with a 1D10 in WiFFE and .735 in MWiF.

[BrianJH]

Are you still confused how these calculations work?

Gee, I sure was. So I thought about different ways on how to simplify things a little. I simply tried converting everything to decimals. Lets see how this looks...


I've modified and extended the table paulderynck posted.




Example 2
15 attacking 20 15/20 = 0.75

Refering to the table above
2:3 = 0.667
<------------------ 0.75 15/20 Lies somewhere in between here
1:1 = 1

Excess factors above 2:3 is 0.75 - 0.667 = 0.083
Total difference between odds is 1 - 0.667 = 0.333

Fractional odds as a percentage is 0.083/ 0.333 = 0.249 = 24.9%



Example 3
3 attacking 20 3/20 = 0.15

1:7 = 0.143
<------------------ 0.15 3/20 Lies somewhere in between here
1:6 = 0.167

Given that the upper and lower odds both attract a -10 modfier its not worth calculating the fractional odds. The modifier would be -10 either way. But lets calculate it anyway just for fun!

Excess factors above 1:7 is 0.15 - 0.143 = 0.007
Total difference between odds is 0.167 - 0.143 = 0.024

Fractional odds as a percentage is 0.007/0.024 = 0.292 = 29.2%



Example 4.

13 attacking 34 13/34 = 0.3824
1:3 = 0.333
<------------------ 0.3824 13/34 Lies somewhere in between here
2:5 = 0.4

Excess factors above 1:3 0.3824 - 0.333 = 0.0494
Total difference between odds is 0.4 - 0.333 = 0.067

Fractional odds as a percentage is 0.0494/0.067 = 0.737 = 73.7%



Example 5 (Odds > 1:1 Ex.1 - 11.16.5 RAC Page 79)

12 attacking 7 12/7 = 1.714

Again, from the table above
3:2 = 1.5
<------------------ 1.714 12/7 Lies somewhere in between here
2:1 = 2

Excess factors above 3:2 1.714 - 1.5 = 0.214
Total difference between odds is 2.0 - 1.5 = 0.5

Fractional odds as a percentage is 0.214/0.5 = 0.428 = 42.8% (RAC has 42.9%)


There is a slight descepency with the figures 'paulderynck' generated (and with the RAC), I simply put this down to rounding errors. The calculations are consistent to 'paulderynck' (and the RAC) to within 2 decimal places.

Brian.

[paulderynck]

Yes we agree very closely. Consider also that you don't round, you just lose whatever amount is past the last digit of significance, tenths over the board, thousandths in MWiF.

But to reduce the complications in FTF play I use a Visual Basic program I wrote to calculate the odds..

[Ginella1946]

Above 1/1

ATT / DEF * 2
Nearest tenth

ATT = 20
DEF = 8

20/8 = 2.5
2.5 * 2 = 5 DRM

ATT = 22
DEF = 8

22/8 = 2.75
2.75 * 2 = +5.5 DRM

The fraction is 1 to 5 to 6 DRM

The formula for lower 1/1 was validated by Harry Rowland in 2008

See Strategikon forum
onglet world in flame page 9

[Courtenay]

What is the correct algorithm for odds less than 1:1? Only one algorithm has been given, and people are saying that that one is incorrect.

Thank you.

[paulderynck]

See posts 10 and 11. Two approaches that yield almost the same results. Close enough depending on how many decimal places are significant.

I'll try to express my approach algebraically:

Fraction = (F-f)/(D - f)

Where: 'F' = attack factors in the attack, 'f' = attack factors needed to get the odds that would be used if no fractionals were to be resolved, (in other words, the odds rounded to the defender's benefit), 'D' is the number of factors needed to get the odds that would yield the corresponding DRM, PLUS one.

'f' and 'D' are computed as fractions to the number of decimal places considered significant, only 'F' is an integer.

The tricky part is that in 2D10 the DRM goes up one per half an odds increase, as shown by the tables in both posts 10 and 11.

I did not look at the Strategicon forum referenced above, but the process I gave is mathematically correct.