1 The modal account of luck

The notion of luck has broad relevance in contemporary philosophy, in particular in moral philosophyFootnote 1, legal philosophyFootnote 2, and epistemology.Footnote 3 Two leading accounts of luck are what, following Lackey (2008), we can call the lack of control account of luck Footnote 4 and the modal account of luck. The modal account, which characterizes luck in terms counterfactual robustness, has over the past decadeFootnote 5 gained considerable traction, and this is especially the case in recent mainstream epistemology.Footnote 6

Duncan Pritchard (2005, p. 128) articulates the modal account of luck as followsFootnote 7:

If an event is lucky, then it is an event that occurs in the actual world but which does not occur in a wide class of the nearest possible worlds where the relevant conditions for that event are the same as in the actual world.

The modal account has quite a bit of surface plausibility. Suppose, for example, you win a (fair) lottery. In virtue of what is this event lucky? The modal account tells us it’s lucky because although the event of your winning the lottery obtains in the actual world, it doesn’t obtain in a wide class of near-by possible worlds where the relevant initial conditions for this event are the same as in the actual world–viz., worlds where you continue to buy a lottery ticket, the lottery remains free and fair and with long odds, etcetera.Footnote 8 As Pritchard (2007, 278) puts it: “Indeed, in most near-by possible worlds that meet this description, I am right now tearing up my lottery ticket in disgust.”Footnote 9

Recent work in epistemology suggests that the modal account is in better shape than the lack-of-control account, but this point won’t concern us here.Footnote 10 Rather, we think the modal account—at least as it is canonically formulated—is open to an intractable sort of counterexample, one which has to this point been overlooked in the literature.

2 A counterexample

Imagine that some event E occurs: (i) in the actual world, and (ii) in a small class of the nearest possible worlds where the relevant conditions for E are the same as in the actual world. Also consider some other event E*, which occurs (iii) in the actual world, and (iv) in the very same small class of previously mentioned nearest possible worlds, and (v) in every possible world located just a tiny bit further away. According to the canonical formulation of the modal account of luck, both E and E* will be classed as equally lucky events. This is, however, a problematic conclusion.

In order to see why, note that E* is surely less lucky than E because E* occurs in many more worlds than E does. So suppose we construct the example such that the number of nearest worlds in which E occurs is just at the limit at which E would no longer count as a lucky event. That is, if E had occurred in, say, one or a few more of the nearest worlds, then E would not have been lucky. However, because E was in fact lucky, proponents of the modal account must concede that E* is (by the same rationale) lucky too. But this is absurd. By construction, E* occurs in the same worlds as E and in every possible world located just a tiny bit further way, and those somewhat more remote worlds should surely count for something.

In order to bring this point into sharp relief, imagine that you are in Edinburgh but wish to meet up with your sister in London tonight. You decide to take the East Coast Express from Edinburgh to London. To your surprise, the train actually arrives on time at 7.59 p.m. This event, call it E, is a lucky event because the East Coast Express usually arrives in London at least ten minutes after schedule. However, E is fairly close to being a non-lucky event. If the on-time performance had just been a little bit better, E would have occurred in “too many” nearby possible world and would thus not have counted as lucky.

Now consider event E*. This is the event in which you arrive in London no later than 7.59 p.m. There is a very reliable express coach running from Peterborough to London that is scheduled to reach its destination before 7.59 p.m. In a large number of the possible worlds in which you sit on the East Coast Express you notice that the train is running late as it stops in Peterborough. In those worlds you therefore transfer to the express coach and arrive on time in London no later than 7.59.

Is event E* a lucky event? Our intuition is that it is not. E* occurs in the same worlds as E and in every possible world located just a tiny bit further way. Those somewhat more distant worlds make the scales tip over. E* is not a lucky event.

We can also illustrate our point in a lottery example. Let E be the event that you win £1000 in your yacht club’s annual lottery. Suppose that there are 100 winning tickets and that you buy one of them. If the total number of tickets in the lottery is of the right size, then E is a lucky event. Let us stipulate that if there had been just one or two more winning tickets in the lottery, then the proportion of winning tickets would have been so large that E would no longer have been a lucky event. By construction, E is right at the border between a lucky and non-lucky event.

Now imagine that E* is the event that you win £1000 in your golf club’s annual lottery. There are 100 winning tickets and if the total number of winning tickets is the same as above, meaning that if there had been just one or two more winning tickets then E* would no longer have been a lucky event. However, unlike in the previous lottery, you will also win the £1000 prize in all of the somewhat more remote worlds in which some member of the club wins next year’s British Open.Footnote 11 Because the members of the golf club are amateurs, this is a more remote possibility than the event in which you happen to buy one of the 100 winning tickets. However, because the event in which you win £1000 was already on the verge of being non-lucky, this additional possibility of winning £1000 makes the scales tip over. Winning the £1000 is no longer a lucky event. Unfortunately, and for the reasons explained above, the canonical formulation of the modal account lacks the resources to account for this intuition.

The counterexamples outlined here are equally applicable if one accepts, as Pritchard (2015) Sosa (2015) and Carter (2016) have, that luck comes in degrees. In that case we construct the examples such that E occurs in the actual world and is lucky to degree x (“very lucky”, “somewhat lucky”, etc.) although E is just at the limit at which an event is no longer lucky to degree x. While E and E* both occur in the very same small class of previously mentioned nearest possible worlds, the difference between E and E* is that the latter event also occurs in every possible world located just a tiny bit further away. Our intuition is that those somewhat more remote worlds count. Therefore, E* is lucky to a lower degree than E. But the standard formulation of the modal account lacks the resources to acknowledge this. On the standard modal account, all that matters is what happens in the nearest worlds.

If one believes that luck is a vague concept, all one has to do to get the counter example to work is to make sure that E is a clear example of a lucky event and E* is a clear example of a non-lucky event. The fact that there might be a borderline area in between the two events makes no difference, as long as the difference between E and E* is large enough.

3 Lackey’s counterexample

Jennifer Lackey (2008) has proposed a different counterexample to the modal account of luck. It goes as follows:

BURIED TREASURE: Sophie, knowing that she had very little time left to live, wanted to bury a chest filled with all of her earthly treasures on the island she inhabited. As she walked around trying to determine the best site for proper burial, her central criteria were, first, that a suitable location must be on the northwest corner of the island—where she had spent many of her fondest moments in life—and, second, that it had to be a spot where rose bushes could flourish—since these were her favorite flowers. As it happens, there was only one particular patch of land on the northwest corner of the island where the soil was rich enough for roses to thrive. Sophie, being excellent at detecting such soil, immediately located this patch of land and buried her treasure, along with seeds for future roses to bloom, in the one and only spot that fulfilled her two criteria. One month later, Vincent, a distant neighbor of Sophie’s, was driving in the northwest corner of the island—which was also his most beloved place to visit—and was looking for a place to plant a rose bush in memory of his mother who had died ten years earlier—since these were her favorite flowers. Being excellent at detecting the proper soil for rose bushes to thrive, he immediately located the same patch of land that Sophie had found one month earlier. As he began digging a hole for the bush, he was astonished to discover a buried treasure in the ground (Lackey 2008, p. 261).

Lackey’s point is that Vincent’s discovery of the buried treasure is an example of a lucky event, even though it does not satisfy the conditions for a lucky event as specified by the modal account.Footnote 12 The problem is that the event that Lackey considers to be lucky occurs in a wide class of the nearest possible worlds in which the relevant conditions are the same as in the actual world. In all relevant nearby worlds, Vincent digs in the same spot as Sophie has buried her treasure.

The difference between Lackey’s and our examples is illustrated in Figs. 1a, b and 2. A solid line means that the event occurs, and a dashed line that it doesn’t. The actual world is at the center of the circle, and the more remote a possible world is the further from the center it is. Our intuition is that the fact that the event occurs in a large number of more remote worlds should make the scales tip over, but according to the standard formulation of the modal account of luck it is irrelevant what happens in these worlds.

Fig. 1
figure 1

a By construction, this event (E) is lucky but very close to not being lucky. b According to the modal account this event (E*) is also lucky because it occurs in equally many of the nearest possible worlds. However, our intuition is that it is not lucky: it occurs in so many more remote worlds that the scales tip over

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Fig. 2
figure 2

In Lackey’s example the event is lucky despite the fact that it occurs in all the nearest possible worlds

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4 Saving the modal account: modal weighted likelihood

What can we learn from these counterexamples? Unlike Lackey, we do not believe that every modal account of luck is doomed to fail. On the contrary, it seems that the following revised version of the modal account can overcome both counterexamples: If an event is lucky, then it is an event that occurs in the actual world but whose modal weighted likelihood is above some appropriate threshold.

The term “modal weighted likelihood” refers to a measure that considers which worlds E occurs in and the distance of those worlds from the actual world, such that (i) the weight assigned to the occurrence of E in a world decreases as the distance from the actual world increases, and (ii) an event is less lucky the more worlds it occurs in, everything else being equal.

When making the notion of modal expected likelihood sharp, we have to keep in mind that we are dealing with an infinite number of possible worlds. This creates some technical difficulties. If, for instance, both E and not-E occur in infinitely many worlds this should not lead us to believe that E and not-E are equally lucky. This is because the density of E-worlds might be much lower than the density of not-E-worlds. The mathematical example mentioned in footnote 9 suffices for explaining this point: We know that the countably infinite set of positive integers is no larger than the countably infinite set of perfect squares. Despite this, the perfect squares become increasingly scarcer as we move upwards from 1 towards infinity. There is, therefore, a sense in which one is luckier if one by random happens to pick a perfect square than a non-perfect square. Mathematicians use the notion of density for articulating the observation that the perfect squares are scarcer than the positive integers. For our present purposes an intuitive understanding of density will be sufficient.Footnote 13

Let d(E, x) denote the density of E at distance x from the actual world. For each distance x from the actual world, d(E, x) assigns a value [0,1] that represents the density of E-worlds at distance x from the actual world, such that d(E, x) = 0 if and only if E occurs in no world at distance x, and d(E, x) = 1 if and only if E occurs in all worlds at distance x.

Let w(x) denote the weight assigned to events that occur at distance x from the actual world. It is plausible to assume that w(x) approaches 0 as x approaches \(\infty \) and that w(x) approaches 1 as x approaches 0. The expected modal likelihood ml of E can then be defined as follows:

$$\begin{aligned} ml(\hbox {E})=\mathop {\int }\nolimits _{x=0}^\infty w(x)\cdot ({1-d({E,x})}) \end{aligned}$$

It can be easily verified that the expected modal likelihood measure has a number of attractive properties:

Property 1 If E occurs in the actual world but in no other possible world, then E is maximally lucky: \(ml(\hbox {E})=\mathop {\smallint }\nolimits _{x=0}^\infty w(x)\).

Property 2 If E occurs in the actual world and in all other possible worlds, then E is minimally lucky (that is, not lucky at all): \(ml(\hbox {E})=0\).Footnote 14

Property 3 If \(\hbox {E}_{1}\) occurs in the actual world and with density d in the possible worlds at distance \(\hbox {x}_{1}\), and \(\hbox {E}_{2}\) occurs in the actual world and with the same density d in the possible worlds at distance \(\hbox {x}_{2}\), and the worlds at distance \(\hbox {x}_{2 }\) are further away than the worlds at distance \(\hbox {x}_{1}\), then \(\hbox {E}_{1}\) is more lucky than \(\hbox {E}_{2}: ml(\hbox {E}_1)=k\cdot w({\hbox {x}_1 })>ml({\hbox {E}_2 })=k\cdot w({\hbox {x}_2})\).

Property 4 If \(\hbox {E}_{1}\) occurs in the actual world and in some possible worlds at distance \(\hbox {x}_{1}\), and \(\hbox {E}_{2}\) occurs in the actual world and in some possible worlds at distances \(\hbox {x}_{2}\) and \(\hbox {x}_{3,}\) such that \(\hbox {x}_{3 }>\hbox { x}_{1} >\hbox { x}_{3,}\) then \(\hbox {E}_{1}\) and \(\hbox {E}_{2}\) will be equally lucky events for some \(d(\hbox {E}_{1},\hbox { x}_{1}), d(\hbox {E}_{2},\hbox { x}_{2})\), and \(d(\hbox {E}_{2},\hbox { x}_{3})\) provided that d is a continuous function.

The modal expected likelihood account is non-binary in that it allows for degrees of luck.Footnote 15 By selecting a suitable threshold \(\hbox {T}\) this account can be turned into a binary notion of luck by stipulating that E is lucky if and only if \(ml(\hbox {E})>\hbox {T}\).

The modal weighted likelihood account of luck gives the right answer in our lottery example. Because E* occurs in many more worlds, the density of this event is high. Therefore, our account is able to explain why E is lucky and E* is not. Furthermore, the new account also gives the right answer in Lackey’s example. Although Sophie and Vincent were almost certain to dig in the same spot, Vincent’s modal weighted likelihood of finding Sophie’s treasure was low.Footnote 16 By acknowledging that events in all possible worlds count, advocates of the weighted likelihood account can explain why Vincent’s discovery of the buried treasure was a lucky event.

5 Conclusion

There are many advantages to thinking about luck in terms of counterfactual robustness rather than control, and these advantages recommend a modal account of luck. We hope to have shown, however, that standard formulations of the modal account of luck face a hitherto unexplored problem, one which is highlighted by the counterexamples we raise in Sect. 2. The problem, in short, can be stated simply: it is a mistake for traditional modal accounts of luck to focus, when assessing a given event for luckiness, on events distributed over just the nearest possible worlds. However, rather than to abandon the modal account, as Lackey has recommended, we’ve opted for a more attractive way to defend the modal account, the modal weighted likelihood model, which avoids the objection we raise to the traditional construal of the account. Moreover, we’ve shown how the modal weighted likelihood model also has a further advantage: the account can straightforwardly handle another kind of counterexample which Lackey regards as devastating to the traditional formulation.