There are other cool things about this system as well. So you are worried about stat dumping? I'm not. A PC could theoretically dump a stat, but due to the opposed roll system, the bonus disparity effect is lessened and there are diminising returns to min/maxing.

The statistical hit you take from lowering one stat, to further increase one that is already high quickly outweighs the benefit. Say you have a STR 16 and a WIS 10. Mathematically speaking you are worse off overall by going to WIS 9 and STR 17. The value gained by rolling d20 +7 instead of d20 +6 is less than the drawback of going from d20 to d20 -1 for WIS checks and saves.

Dumping a stat makes you far far more vulnerable to a spell that targets one of your dump stats. There is less incentive to try to get that 18 STR, if its going to leave you with a CHA of 8. A -2 penalty to all your charm saves? You'll get to roll that massive attack bonus all right. Against your allies because the DM's monster dominated you.

First off, that was a great post and I think you might be onto something. I do have a quibble with what you wrote above, though. I know you read my

earlier post about opposed checks, but for everyone else I want to clarify this aspect:

In an opposed check, the further apart the attacker's bonus a and the defender's bonus b are, the less difference a small change in either of those bonuses makes. (For example, if a-d is 10, increasing the attack by 1 or decreasing the defense by 1 changes the probability of success by about 2.5%) When a and b are nearly the same, however, a +1 or -1 change does change the probability of success by close to the 5% with which we're familiar. (I've assumed that defenders win ties, and for simplicity ignored anything special that might happen on a natural 1 or 20).

What this means is that small changes to ability scores or other bonuses have the largest effect in closely matched contests, such as when a character is attacking a level-appropriate monster's best defense, and the least effect when there is already a large disparity.

Now, it is true that increasing a large ability score does have diminishing returns when attacking a poor defense, compounded by the fact that in general it is best to attack a target's weakest defense whenever possible. However, the exact same math applies to changing one's dump stats, meaning a small decrease to an already low stat won't greatly affect the probability of success against it.

Improving a dump stat by 1 when fighting an opponent with a low attack (a-d is close to 0) will improve the chance of saving by about 5%. Most creatures tend not to attack with such low scores, so this will be a rare occurrence. More common is fighting an opponent with a high attack (a-d close to 10), where increasing the dump stat by 1 will improve it only by about 2.5%.

Where does this leave the min-maxer? Suppose one is trying to decide whether to increase their best stat or a dump stat by 1. If they use their best stat every round for attacks, then every round they can expect to be at least 2.5% more successful if they increase it by 1. Although they try to attack a target's lowest score, sometimes that isn't possible so it is likely to actually be somewhere in between 2.5% and 5% overall. On those occasions where the high stat is used for defense, it probably gives closer to the 5% benefit because the attacker is probably attacking with a high stat. If they increase their dump stat, then they can expect to be at least 2.5% more likely to save when defending with the dump stat. Since an opponent's attacks are almost always made using their good ability scores and bonuses (so a-d is still large), they should probably expect this to stay close to the 2.5% mark. Furthermore, if they expect to use their good stat more frequently than their dump stat, the improvement of the good stat is expected to be more impactful in the long term.

Therefore your statement that "The value gained by rolling d20 +7 instead of d20 +6 is less than the drawback of going from d20 to d20 -1" is misleading. In any given situation the one that is better will depend on the magnitude of a-d, and even if we fix the initial a-d in the opposed check and assume each increase would be used equally often, improving the dump stat is only slightly better. For example, if a-d is +5, then an attack has a 70% chance of success. Increasing a by 1 (such as from improving the attack stat) gives a-d=+6, which has a 73.75% chance of success, a +3.75% change. Improving the defense by 1 gives a-d=4 and the attack has a 66% chance of succeeding, or a -4% change. In other words, the change due to improving one's defense is only .25% better than the change due to improving one's attack in these analogous situations. (This .25% advantage for improving the defense holds for any +/- 1 change in a-d when a-d is initially positive. If a-d <= 0, then improving the attack has the .25% advantage.) In short, this sort of one-to-one trade is basically a wash in terms of pure probability if all a-d situations were equally likely, but by the arguments in the last paragraph we expect situations that favor the higher stat to appear more frequently.

Having given it this additional thought, I suppose I need to modify the conclusion of my earlier post. Opposed rolls introduce some interesting tensions:

1) The greater returns for small changes in mid-range ability scores compared to those at either extreme will tend to incentivize players to keep the middle ground.

2) Despite this, decreasing an infrequently used low stat to increase a frequently used high stat on a 1-for-1 basis will be a favorable trade.

3) In any case, large disparities in scores have less impact than they would in 3.5/4e.

These suggest to me that opposed checks may tolerate min/maxing (and other wide disparities) more so than actively discourage it. For games with rolled stats that is a clear win. For point-buy games #2 suggests that some escalating cost for higher scores will still be needed.

Oof, for a quibble that took a bit more effort than I intended.