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I remember watching "The Dungeons & Dragons Experience" and one of the people featured in the documentary mentioned how him and his friends had developed a way to blend and randomize attributes of their PCs and their NPC wives to create their next character via their retired character's offspring(time marker 20:13).
What homebrew system best does this?
Some years ago I made just such a system; I've made it work for D&D, RuneQuest 2/3, Pendragon & Ars Magica, so I suppose it is reasonably generic.
This is a "pure genetic" system, and does not allow for the effects of environment.
The first step is to retrieve - or recreate - the original dice rolls involved in creating the parent characters' characteristics. For example (assuming 3D6 characteristics):
Strength Intelligence ...
Mum: 6 + 3 + 4 = 13 1 + 6 + 5 = 12 ...
Dad: 5 + 6 + 5 = 16 6 + 2 + 6 = 14 ...
The trick here is - if you didn't record your exact rolls at character generation - to roll dice for each characteristic until the combination of \$n-1\$ rolls in \$n\text{D}x\$ is such that the last die must be between 1 and \$x\$ for the characteristic total. In addition, where possible, write down the die rolls in the order they were rolled or in random order, not in any sorted order. Make sure that these die rolls are written down for future reference.
Imagine, then, that each die rolled in a characteristic score is a gene. We can simplify the whole complexity of Mendelian genetics by stating that a child inherits a gene from one or other of his/her parents. So, for each "gene" (i.e. die score), flip a coin (or use some other form of 50/50 random result generation), and using whatever rules you want as long as you stay consistent, select the die score from either the mother or the father in the same column, e.g.:
Strength Intelligence ...
Kid 1: 6 + 6 + 5 = 17 1 + 2 + 5 = 8 ...
Kid 2: 5 + 3 + 4 = 12 6 + 6 + 6 = 18 ...
Kid 3: 6 + 3 + 5 = 14 6 + 2 + 5 = 13 ...
...
This provides fairly variable results, yet can be shown to be derived entirely from the parents.
An alternative is to sort the die rolls (in a system where that is possible - this wouldn't work for Ars Magica characteristic pairs where one D10 is positive and the other is negative) from low to high; this would reduce variability somewhat, e.g.:
Strength Intelligence ...
Mum: 3 + 4 + 6 = 13 2 + 3 + 6 = 11 ...
Dad: 5 + 5 + 6 = 16 1 + 4 + 5 = 10 ...
Kid 1: 3 + 4 + 6 = 13 1 + 3 + 5 = 9 ...
Kid 2: 5 + 4 + 6 = 15 2 + 4 + 5 = 11 ...
Kid 3: 5 + 5 + 6 = 16 2 + 4 + 6 = 12 ...
...
To allow for things such as racial bonuses/penalties, my solution was to average them; this worked easily when the average was a whole number, but when the average is a fraction, my solution was to roll a die or flip a coin to decide whether to round up or down.
As I stated earlier, this discounts the effects of environment. Scientific studies have shown that in separated-identical-twin studies, if one is given as much stimulation as possible in a particular area, and the other is neglected as much as is ethical in the same area, the stimulated child will be significantly better in that area than the neglected child. The question is then how to simulate this...
In order to add extra variability in a believable and accountable manner, we can allow for mutation. Depending upon how frequently the GM wants mutation to occur, once the child's characteristic dice have been determined, roll a D(whatever) for each child die, and if the D(whatever) comes the minimum value, subtract 1 from that child die (down to the minimum possible), if it comes up as the maximum value, add one (up to that die's maximum possible), otherwise do nothing.
It is also possible to mutate a racial bonus if desired. For the most part, treat it as another die, except that it can also be negative, and the GM may want to impose limits upon its magnitude. For 3D6-based games, ±0–3 seems to work for most cases, and ±4–6 for highly exceptional cases.
If you wanted very frequent mutation, use a D6, otherwise a D20 or even a D100 might be more realistic.
I've seen used a system where you simply adjust the child's attribute in the direction of the parents' attributes.
The method I used was, presuming same race, adjust one point in the direction of the closest parent's roll; if equidistant, do not adjust.
Example: Cugel Junior's Strength. Mom was Str 11, Dad Str 15.
Roll of:
3-10 up one toward mom's 11
11 No adjustment - mom's is closest
12 down 1 toward mom
13 no adjustment - equidistant
14 one up towards dad's 15
15 no adjustment - dad's is same
16-18 one down towards dad.
This had a major flaw - no 3/18 results.
Borrowing from later games, I would switch to roll to see which parent is dominant in each attribute. If that parent is higher, move the roll 1 point towards the parent's roll.
Given that most games use 3d6 for attribute generation, it's possible to reduce that number of dice and replace the reduced dice with a value derived from a parent's attribute.
If replace 1 die with a value equal to 1/3 the parent's attribute, you typically preserve the population's range. It would be wise to randomly select which parent for each attribute. One could, alternatively, replace two dice, each die replaced with 1/3 of the score from one parent, and thus have only 1 die random.
Note that many people use 4d6 drop lowest; in such a case, replacing one of those 4 dice with a value of 1/3 the parent's score preserves the potential for 18 scores; again, randomize which parent.
Another system, used by friends, was to use the average of the parents' attribute modifiers directly as modifiers to attribute. (Worked OK in Moldvay/Mentzer, but not in AD&D, where the mods are not a simple single table. Not so good for 3.X, due to the higher modifiers.)
Some searching turned up a thread on EN World devoted to this very question. It features a bunch of extra discussion about alignment generation and personalities based on zodiac signs, but the ability score method is as follows:
Now, we can easily see that this quickly creates insanely high-score children:
In an OSR game where there is no automatic ability score advancement, these statistics are probably okay, though I would personally leave out the bonus dice from parents' levels. In a more modern system where ability scores improve fairly rapidly throughout the course of play, this would be a problem. It's also important to factor out racial bonuses to scores, otherwise you'll end up including them twice (for the parents and later for the child), which isn't really that fair.
Let's imaginify that we have two PCs who have a "standard heroic" ability spread of 16-14-14-12-10-10, which kinda-sorta approximates results from four dice.
Now, if you keep in the level adjustment bonus dice, then things start to increase pretty fast. After a few generations of adventurer-on-adventurer eugenics, you could pretty easily engineer a family that consistently produced children with scores in the range 14-18 for everything. Unless you want to build such concepts into your setting, I'd stay away from the level adjustments.
