Thoughts on Modal Distance

Duncan Pritchard’s modal luck/risk model in Epistemic Luck (2005) addresses a real problem, which can be illustrated as follows:

Say that Lottie has a lottery ticket that has a 1/50,000,000 chance of winning in an upcoming draw. And say that while walking home in relatively safe conditions, I have a 1/10,000,000 chance of being struck by lightning. I’m convinced that it would irrational for Lottie to decide to tear up her ticket before the draw, but it would be rational for me to think I won’t be struck by lightning and thus decide to walk home.

However, note that these are cases of prediction, such that what we are concerned with justifying here is not knowledge at all but confidence, i.e. justified likely prediction. Even if Lottie’s prediction that she won’t win turns out to be correct, the sense in which she “knew” the outcome is different from the sense of concurrent knowledge of an already true belief*. So, Pritchard’s model addresses the problem with relying solely on probability to justify confidence, not knowledge, as safe. I’m not sure if we can always rely solely on probability with the safety principle in justifying concurrent true belief— I haven’t seen any instances where this would be a problem, but there may be some.

Note that there could be a problem for Pritchard’s Modal Distance model, as a measure of physical difference between possible worlds, if there were exceptions to apparent closeness in particular instances, such as a particular instance where the energy required to make a given lottery draw when the needed ball is at the bottom of the pile is greater than the energy required in a particular instance for a loose piece of paper to jam the basement door in a well-maintained garbage chute. I thought that this problem could be solved by putting the Modal Distance together with probability, but the way I initially tried to do this was wrong**. The correct solution is to instead use the Average Modal Distance, as the sum of the Modal Distances of all the outcomes in close possible worlds over the number of outcomes: (MD₁ + MD₂ + … + MD_n)/n.

In any case, we can calculate the Modal Distance between two possible outcomes in an event as follows:

Where: P := conditions needing to obtain for any given outcome A to occur in an event E

Q := conditions needing to obtain for a target outcome B to occur in E

O := the total set of outcomes of E in close possible worlds

p := {x|(x ∈ P)  & (P ⊆ A)}

q := {y|(y ∈ Q)  & (Q ⊆ B)}

Modal Distance of A from B: MD[A, B] := |(p - q∩p)| i.e. the quantity of p minus the quantity of the intersection of q with p

In the lottery case, if the quantity of p, i.e. the relevant conditions needing to obtain for a non-winning draw A to occur, is 1.0, and the quantity of the intersection of q, i.e. the relevant conditions needing to obtain for the winning draw B to occur, with p is .95, then the Modal Distance of a non-winning draw from a winning draw is .05. In the lightning case, if the quantity of p, i.e. the relevant conditions needing to obtain for a safe walk home A to occur, is 1.0, and the quantity of the intersection of q, i.e. the relevant conditions needing to obtain for a deadly lightning strike B to occur, with p is .05, then the Modal Distance of a safe walk home from a deadly lightning strike is .95.

Thus, while the probability of being struck by lightning is actually greater than the probability of winning the lottery, the Modal Distance between winning and not winning the lottery is much closer than the Modal Distance between walking home safely and being struck by lightning. This latter disparity seems to explain why it would be irrational to tear up the ticket but rational to walk home.

But how does Modal Distance explain this? When should Modal Distance be appealed to? After all, we would still want to say that buying a lottery ticket with 1/50,000,000 odds is not a safe bet and thus is not a rational purchase, no matter how modally close any ticket is to winning.

I have a couple thoughts on how to answer these questions:

1. One solution that I don't particularly like is to just say that while modal distance can be measured quantitatively and does have an impact on our intuitions about rational decision making, e.g. in the lottery versus the lightning cases, these intuitions are actually emotion-based rather than strictly rational. If that were so, a purely rational subject (e.g. an AI), once fully apprised of all the external circumstances, would not have these intuitions and would base its decisions solely on simple probability over possible worlds.

Take Pritchard's bullet example: say the bullet that misses a soldier named Duncan by several meters is fired by a very accurate sniper who is sitting on a boat that just happens to be rocked by a wave at the moment he pulls the trigger; this boat had been steady all day and the wave at that moment is a fluke with a very low probability. Then, say the bullet that misses Duncan by only a few centimeters is fired by a sniper with a bent scope such that he routinely misses by a few centimeters, and thus his missing by a few centimeters has a very high probability. In this case, even if Duncan were to become aware of these facts, he might still want to say that the bullet that missed him by a few centimeters put him at greater risk than the one that missed him by several meters.

To explain this, Duncan might appeal to a concept of physical danger that we could only translate through something like the modal distance model, not through strict probability. But this physical danger concept might only be employed to capture a subjective emotional response to a situation as one experiences it in the moment, and not a strictly rational assessment. Perhaps we could avoid this conclusion by appealing to the idea of access to possible worlds within a given knowledge frame. Within the knowledge frame of Duncan's experience, he has access to all possible worlds resulting in the two bullets missing him, and it is rational for him to compare the Average Modal Distances of the shots over all those worlds, unrestricted by the facts of the sniper being rocked by a chance wave and the other sniper having a bent scope. This knowledge frame consideration might land us back in internalist territory, though, which I was trying to avoid because I prefer the strictly externalist Robust Anti-luck Epistemology account.  

2. Another solution is to take a closer look at the lottery case in particular and point out some considerations that might make this case exceptional. With a lottery draw where Lottie has already purchased a ticket, one can do a cost-benefit analysis: while there is a very small chance that she will have a huge payoff, now that she already has the ticket, it costs her nothing to keep it and check the result-- or at any rate, it actually costs her less energy to keep the ticket and check the result (in whatever fashion is most convenient to her) than it does to tear the ticket up and throw it away. So tearing up the ticket is irrational because it a loss to her. The value of the ticket before the draw may be much less than the $2 or whatever she paid for it, but because the payoff is so large, it's probably worth more than a penny-- it might be worth something between a nickel and dime. Most people don't throw away nickels and dimes. On the other hand, my being able to walk home might be of considerable benefit to me, much more, quantitatively, than the cost of risking the extremely low probability of being struck by lightning.

I like this solution better than previous one because it remains an externalist account, but I don't like that it deflates the interesting distinction between modal probability and modal (physical) potential through basically a game theory analysis***. 

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*How to interpret “knowledge” of future events is of course an ancient problem going back to Aristotle’s “sea battle” example, but more recent developments in modal logic look to have solved this problem.

**Initially I had thought that modal distance could just be put together with probability over possible worlds in a simple way, as a ratio, but I found that this yielded bad results, e.g. it would have said that a given outcome in a lottery ball draw with fifty possible outcomes was modally closer than a given outcome in a coin flip with two possible outcomes, even though the inertia displacement from outcome to outcome is roughly the same for both scenarios.

***Actually, I'm surprised Pritchard doesn't bring up game theory in his account of luck and risk. I wonder how he sees game theory fitting in with his account? I'm not sure that he really wants to rule out probability for modal distance altogether so much as say that the safety principle can't be reduced to a simple probability, i.e. 1/(# of possible outcomes). A game theoretical account could of course fully accommodate Bayesian induction.