Bot rollouts, however, show no detectable difference in the average length of a game if the Jacoby Rule is introduced, so something doesn't seem quite right here.
Tim, I don't think we can conclude that because Jacoby or not makes little difference in game length with bot-perfect play that it also makes little difference with human players of various strengths. Bots miss few double/passes and not many double/takes. People do, and compound those errors by continuing to play on when they shouldn't, thinking that they're still not good enough, or too good. Sometimes they even know that they should now double out but hope to salvage a second point, having missed (they now realize) a huge double.
I was thinking also about your comments about the rule being socially, not technically oriented. I largely agree with your observations, but not entirely.
I was thinking, first, suppose that a Jacoby chouette player plays on in an obvious drop situation. Why should his better opponent object? He shouldn't (probably -- there is the time vs money argument), but in this game, in this position, his equity is something greater than minus one until he has to pass. Why he should mind that? But then I thought, such small differences can't amount to much. The advantage Jacoby gives to the better player over the late doubler is elsewhere. With or without the rule, he gains equity on every turn that a proper double/take or double/pass is not given. But with the rule, the gains can be huge:
Blue on roll, cube decision?
| White | 162 | ||
|---|---|---|---|
                      | |||
| Blue | 140 | ||
| Position ID: 4PPgAUgzbuABMA Match ID: cAkAAAAAAAAE | |||
Above, a clear double/take. If Blue doesn't double, White's better off by 0.080. But look ahead two turns after an excellent sequence for Blue:
Blue on roll, cube decision?
| White | 164 | ||
|---|---|---|---|
   | |||
| Blue | 120 | ||
| Position ID: 4PPgAWDbOIMBMA Match ID: cAkAAAAAAAAE | |||
Blue's already missed out on a chunk of equity by not doubling, with this sequence landing in Too Good territory. But without Jacoby, despite the cube error, the game's still worth about 1.350 points to Blue. With Jacoby, it's only worth one point. That's a big difference.
Do you have a reference to the bot rollouts you cite? I don't doubt that most doubles, takes and passes are doubles, takes and passes with or without Jacoby, but I'd like to see numbers on how much difference it makes when they're not.