Suppose you are a a timeless decision theory agent playing this modified Prisoner’s Dilemma with an actor that will always pick “defect” no matter what your strategy is. Clearly, your best move is to cooperate, gaining you 1 util instead of no utility, and giving your opponent his maximum 3 utils instead of the no utility he would get if you defected. Now suppose you are playing against another timeless decision theory agent. Clearly, the best strategy is to be that actor which defects no matter what.
Here is what I believe to be the standard explanation.
Unfortunately, you don’t have the option of playing the same strategy as a “perfect defector” since you are currently a hypothetical TDT agent. You can of course play the strategy of being a hypothetical TDT agent that turned itself into a perfect defector. However, from the point of view of your TDT opponent this is a different strategy. In particular, a TDT will cooperate when confronted with a “true” perfect defector but defect§ when faced with an ex-TDT that turned itself into one. Therefore, even though the perfect defector would gain 3 utils, there is no strategy you as a TDT can follow that will mimic the perfect defector so you might as well act like a true TDT and agree to cooperate.
This does, however, raise interesting questions about why you aren’t winning.
BTW, the standard name for this prisoner’s dilemma variant is chicken.
§ Edit: Actually after thinking about it I realized that what a TDT would do is cooperate with probability 2/3-ε and defect with probability 1/3+ε. This gives him a higher utility, 2/3-ε instead of 0, and still leaves you with a utility of 2-3ε, which is still enough to make you wish you had played a strait TDT strategy and cooperated.
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.
Here is what I believe to be the standard explanation.
Unfortunately, you don’t have the option of playing the same strategy as a “perfect defector” since you are currently a hypothetical TDT agent. You can of course play the strategy of being a hypothetical TDT agent that turned itself into a perfect defector. However, from the point of view of your TDT opponent this is a different strategy. In particular, a TDT will cooperate when confronted with a “true” perfect defector but defect§ when faced with an ex-TDT that turned itself into one. Therefore, even though the perfect defector would gain 3 utils, there is no strategy you as a TDT can follow that will mimic the perfect defector so you might as well act like a true TDT and agree to cooperate.
This does, however, raise interesting questions about why you aren’t winning.
BTW, the standard name for this prisoner’s dilemma variant is chicken.
§ Edit: Actually after thinking about it I realized that what a TDT would do is cooperate with probability 2/3-ε and defect with probability 1/3+ε. This gives him a higher utility, 2/3-ε instead of 0, and still leaves you with a utility of 2-3ε, which is still enough to make you wish you had played a strait TDT strategy and cooperated.
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.