Commander Upgrades and cost benefit analysis

Maybe this will interest people, maybe this will glaze some eyes over. I've mentioned in some comments previously that the commander upgrades (CO +3 and TFC +5) are not worth their costs. I haven't figured out how to make any dice give me cute plots of dice statistics, but for a pool of 2 dice, I can show it in Excel.

To start with here is best of 2d6, no bonus for skill.

Best of 2d6

And here is 2d6 with a Init skill of 5.

Init 5+

This is your default commander, you can't take anything less out of the box.

Init 4+ and 3+ follow:

Init 4+

Init 3+

Or to summarize, a normal CGL as Force Leader has an average roll of 4.58333 and 3/36 chance of getting a 7, while for 3 points you go to an average roll of 4.72222 and 5/36 chance of getting a 7, all the way up to 4.91667 and 7/36 for points. That is an increase on the average of only .33334.

If each player has the same Init skill, they each have a 40.8% chance to win initiative, with the 18.4% balance being ties. After rerolling enough times, both players have a 50% chance of winning.

The skew become is player A has a 3+ and player B has a 5+, player A has a 48.1% chance to win initiative, while player B has only a 34.7% chance, chance of a tie is 17.2%. Repeating to eliminate ties gets you to 58% to 42% for a 5 point upgrade. (53.5 vs 46.5 for 4+ vs 5+ for the curious)

Assuming you are only looking at the 4 rounds of the game, there isn't enough deviation to say you should win 3 out of 4. You're more likely to win any given roll, but not by enough to really weight the outcome. In fact, with a 3+ commander, assuming they don't die, the normalized chance of winning initiative all 4 rounds is only ~11%.

All that said, I hope that in 4e, upgraded commanders are either cheaper or better. Maybe a +1 on the result rather than a skill increase.