Okay, I’ll play along. Lets see where this takes us. The math here is not going to be strict and I’m going to use infinities to mean “sufficiently large”, but it will hopefully help us make some sense of this proposition.
P(W) = Probability of the mechanism working
P(T) = Probability that the mugger is being truthful
P(M) = Probability that the mugger is “maximizing”
h = Amount of harm threatened
a = Amount being asked for in the mugging
We want to know if PT(h) PW h > a for sufficiently large h, without really specifying a.
As the amount of harm threatened gets larger, the probability that the mugger is maximizing approaches unity.
Since you’re claiming that the probability that the mugger is maximizing is dependent on the amount of harm threatened, we can rewrite P(M) as a function of h, so lets call it PM(h) such that lim h->∞ PM(h) = 1
As the probability that the mugger is engaged in maximizing approaches unity, the likelihood that the mugger’s claim is true approaches zero.
We can compose the functions to get our relationship between PM and PT: lim h->∞ PT(PM(h)) = 0 which simplifies to: lim h->∞ PT(h) = 0
If we express the original question using this notation, we want to know if PT(h) PW h > a for large enough values of h. If we take our limits, we get: lim h->∞ PT(h) PW h > a which evaluates to 0 PW ∞ > a
This doesn’t really work, since we have 0 * ∞, but remember that our infinity means “sufficiently large” and our zero therefore has to mean “very very low probability”.
...the evidence that the mugger is maximizing can lower the probability below that of the same harm when no mugger has claimed it. [...] the claim can become less believable than if it hadn’t been expressed.
This part here has to mean that for sufficiently large h, PT(h) PW h < PW h, which is not hard to believe since it’s just adding another probability, but it also doesn’t solve the original problem of telling us that we can be justified in not paying the mugger. In order to do that, we’d need some assurance that for sufficiently large h and some a, PT(h) PW h < a. To get that assurance, PT(h) h would have to have some upper bound, and the theory you presented doesn’t give us that.
It’s a fun theory to play with and I would encourage you to try to flesh it out more and see if you can find a good mathematical relationship to model it.
Okay, I’ll play along. Lets see where this takes us. The math here is not going to be strict and I’m going to use infinities to mean “sufficiently large”, but it will hopefully help us make some sense of this proposition.
P(W) = Probability of the mechanism working P(T) = Probability that the mugger is being truthful P(M) = Probability that the mugger is “maximizing” h = Amount of harm threatened a = Amount being asked for in the mugging
We want to know if PT(h) PW h > a for sufficiently large h, without really specifying a.
Since you’re claiming that the probability that the mugger is maximizing is dependent on the amount of harm threatened, we can rewrite P(M) as a function of h, so lets call it PM(h) such that lim h->∞ PM(h) = 1
We can compose the functions to get our relationship between PM and PT: lim h->∞ PT(PM(h)) = 0 which simplifies to: lim h->∞ PT(h) = 0
If we express the original question using this notation, we want to know if PT(h) PW h > a for large enough values of h. If we take our limits, we get: lim h->∞ PT(h) PW h > a which evaluates to 0 PW ∞ > a
This doesn’t really work, since we have 0 * ∞, but remember that our infinity means “sufficiently large” and our zero therefore has to mean “very very low probability”.
This part here has to mean that for sufficiently large h, PT(h) PW h < PW h, which is not hard to believe since it’s just adding another probability, but it also doesn’t solve the original problem of telling us that we can be justified in not paying the mugger. In order to do that, we’d need some assurance that for sufficiently large h and some a, PT(h) PW h < a. To get that assurance, PT(h) h would have to have some upper bound, and the theory you presented doesn’t give us that.
It’s a fun theory to play with and I would encourage you to try to flesh it out more and see if you can find a good mathematical relationship to model it.