It does not matter what probability of defecting if you expect the other agent to defect you precommit to, just so long as it is greater than 1⁄3. This is because if you do precommit to defecting with probability > 1⁄3 in that situation, the probability of that situation occurring is exactly 0.
Of course, that assumes mutual perfect information about each others’ strategy. If beliefs about each others’ strategy is merely very well correlated with reality, it may be better to commit to always defecting anyway, because if your strategy is to defect with probability slightly greater than 1⁄3, and the other agent expects a high probability that that is your strategy, but also some probability that you will chicken out and cooperate with with probability 1, he might decide that defecting is worthwhile. If he does, that indicates that your probability of defecting was too low. Of course, having a higher chance of defecting conditional on him defecting does hurt you if he does, so the best strategy will not necessarily be to always defect; it depends on the kind of uncertainty in the information. But the point is, defecting with probability 1/3+ε is not necessarily always best.
Fair enough, and thanks for supplying the name.
It does not matter what probability of defecting if you expect the other agent to defect you precommit to, just so long as it is greater than 1⁄3. This is because if you do precommit to defecting with probability > 1⁄3 in that situation, the probability of that situation occurring is exactly 0. Of course, that assumes mutual perfect information about each others’ strategy. If beliefs about each others’ strategy is merely very well correlated with reality, it may be better to commit to always defecting anyway, because if your strategy is to defect with probability slightly greater than 1⁄3, and the other agent expects a high probability that that is your strategy, but also some probability that you will chicken out and cooperate with with probability 1, he might decide that defecting is worthwhile. If he does, that indicates that your probability of defecting was too low. Of course, having a higher chance of defecting conditional on him defecting does hurt you if he does, so the best strategy will not necessarily be to always defect; it depends on the kind of uncertainty in the information. But the point is, defecting with probability 1/3+ε is not necessarily always best.