It’s the more-than-one-round calculation that I’m currently trying to wrap my brain around, rather than the sum of a series of halves adding to one. If there’s a 1⁄3 chance of each round continuing, then that also adds up, with 1⁄9 of the second round’s value, and 1⁄27 of the third’s, and so on—it doesn’t add up to one, but it does add up to more than 1⁄3. Ditto if there’s a 3⁄4 chance of a next round, or a 99% chance.
In the p=1/3 case, there is a 2⁄3 chance of lasting exactly 1 round, 2⁄9 of lasting exactly 2 rounds, 2⁄27 three rounds. This does add up to 1. It will always add up to 1.
We seem to be talking past each other. Yes, the total odds add up to 100%; but the sum of how important each individual round is, differs.
Let’s say that the factor is 2⁄3. Then the first round contributes a total of 2⁄3 its nominal score to the expected value; the second round contributes (2/3)^2=4/9 of its score; and already that adds up to more than 1 - meaning that the effects of future rounds are more likely to outweigh the benefits of a defection-based strategy.
Okay, I understand the issue now, I think. So, summing up the effect of all the future rounds in exactly the way you are describing is something you would do to determine if grim trigger is an equilibrium strategy. (If you defect now, you get punished in ALL future rounds) However, in tit for tat, your punishment for defecting only lasts for 1 round, so you don’t have to add all that up.
It’s the more-than-one-round calculation that I’m currently trying to wrap my brain around, rather than the sum of a series of halves adding to one. If there’s a 1⁄3 chance of each round continuing, then that also adds up, with 1⁄9 of the second round’s value, and 1⁄27 of the third’s, and so on—it doesn’t add up to one, but it does add up to more than 1⁄3. Ditto if there’s a 3⁄4 chance of a next round, or a 99% chance.
In the p=1/3 case, there is a 2⁄3 chance of lasting exactly 1 round, 2⁄9 of lasting exactly 2 rounds, 2⁄27 three rounds. This does add up to 1. It will always add up to 1.
We seem to be talking past each other. Yes, the total odds add up to 100%; but the sum of how important each individual round is, differs.
Let’s say that the factor is 2⁄3. Then the first round contributes a total of 2⁄3 its nominal score to the expected value; the second round contributes (2/3)^2=4/9 of its score; and already that adds up to more than 1 - meaning that the effects of future rounds are more likely to outweigh the benefits of a defection-based strategy.
Okay, I understand the issue now, I think. So, summing up the effect of all the future rounds in exactly the way you are describing is something you would do to determine if grim trigger is an equilibrium strategy. (If you defect now, you get punished in ALL future rounds) However, in tit for tat, your punishment for defecting only lasts for 1 round, so you don’t have to add all that up.