Timeline for I know that you know...
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 23, 2012 at 20:13 | comment | added | Michael Greinecker | Yey, but there is no reason why this probability should be nonzero without further assumptions. | |
| Nov 23, 2012 at 19:37 | comment | added | Emil Jeřábek | So? As far as I can see, the expected utility of choosing the predecessor is strictly larger than the expected utility of choosing another ordinal $\beta$, as long as the probability that the opponent’s choice is larger than $\beta$ is nonzero. | |
| Nov 23, 2012 at 19:13 | comment | added | Michael Greinecker | A best response is a expected utility maximizing choice given the beliefs, which I have restricted here to point beliefs (in oder not to worry about measurability). | |
| Nov 23, 2012 at 18:39 | comment | added | Emil Jeřábek | But I am not assuming any common knowledge, I am in fact not assuming anything whatsoever about the opponent. You defined rationality as “not using a choice you know is not a best response”, and I know choosing an ordinal smaller than the predecessor is not the best response, because choosing the predecessor is a better response. I have no idea whether this conforms to one formal definition or another, but if not, then IMHO the example merely illustrates that the definition is contrived rather than the phenomena mentioned by the OP. | |
| Nov 23, 2012 at 17:55 | comment | added | Michael Greinecker | @Emil: No using weakly dominated strategies is stronger requirement than rationality. And having common knowledge of not playing weakly dominated strategies might be impossible. There is a well known paper by L. Samuelson, in which he show that there are games in which common knowledge of not-playing weakly dominated strategies is inconsistent (alturl.com/d2s5m). The notion I used coincides with point-rationalizability in the sense of Bernheim (alturl.com/fc9oe). | |
| Nov 23, 2012 at 16:43 | comment | added | Emil Jeřábek | Playing an ordinal $\beta$ smaller than the predecessor of $\alpha$ is never rational, because there is the possibility that the opponent may play something larger than $\beta$, which would make me lose, whereas if I play the predecessor, it is either a win for me or a draw. I know this a priori, without considering what the opponent knows. | |
| Nov 23, 2012 at 16:15 | comment | added | Michael Greinecker | @Emil It is correct that playing the predecessor is always compatible with common knowledge of rationality up to some ordinal $\beta<\alpha$. But there are many more choices that are compatible with common rationality up to a lower level. If you want to role them out, you have to go all the way. The notion of rationality being used is not using a choice you know is not a best response. | |
| Nov 23, 2012 at 16:03 | comment | added | Emil Jeřábek | I don’t get the example. Whatever the other player knows, thinks, or plays, I’m always better off playing the predecessor of $\alpha$ than any other ordinal, hence rationality dictates I do just that, without any iteration. | |
| Nov 23, 2012 at 15:51 | history | answered | Michael Greinecker | CC BY-SA 3.0 |