The Significance, Persistence,Contingency Framework
William MacAskill *, Teruji Thomas* , and Aron Vallinder†

1 Introduction
The world, considered from beginning to end, combines many different
features, or states of affairs, that contribute to its value. 1 The value ofeach feature can be factored into its significance—its average value perunit time—and its persistence—how long it lasts. Sometimes, though,we want to ask a further question: how much of the feature’s value
can be attributed to a particular agent’s decision at a particular point
in time (or to some other originating event)? In other words, to what
extent is the feature’s value contingent on the agent’s choice? For this,
we must also look at the counterfactual: how would things have turned
out otherwise?
In this note we give a way to formalise the ideas of significance, persistence, and contingency. We call this the SPC framework. It must beemphasised that the main goal is to help estimate the instrumental value* Global Priorities Institute, University of Oxford† Forethought Foundation for Global Priorities ResearchThis is version 1.1, dated August 2022. We thank Max Daniel for his comments.
1 For these purposes, a ‘feature’ can be any source of value (or disvalue). For example, one could consider a particular headache, the future existence of human civilisation, or the prevalence of QWERTY keyboards. A feature may contribute by having
intrinsic value, or by entailing or promoting other things with intrinsic value, such as
human welfare.

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(typically interpreted as the expected value) of an event, compared to therelevant counterfactual. There may be different ways to do this, suitable for different situations. It seems clear to us that thinking in termsof significance, persistence, and contingency can be a useful heuristic
when thinking about the importance of historical events and the consequences of our choices for the long-run future; see MacAskill’s What
We Owe the Future for many applications. This note shows how thatheuristic can be related to a more formal theory of evaluation. Doingso helps to clarify when the heuristic can be useful; gives some discipline
to its application; and opens the door to its use in more formal analyses.
Still, different variations on the SPC framework might be more useful or perspicacious in some situations; in others, the SPC framework
may only be useful for evaluating some aspects of an event; and sometimes one may want to use a different framework altogether. So while
we hope that this note will be useful to decision-makers, we (of course)
make no claim to have the final word.
In section 2, we explain the framework as it appears in What We Owe
the Future. We’ll sketch an alternative approach in section 3. Section 4
discusses how the SPC framework can contribute to the overall evaluation of our options. It explains the ‘ITN framework’, which evaluates
problems in terms of their importance, tractability, and neglectedness, andindicates how the two frameworks can be combined.

2 The Basic Framework
We want to describe the value contributed by some feature F of theworld as a result of some originating event E . We will formalise significance, persistence, and contingency in terms of the total value V
contributed by F and its duration T , given the occurence of E . Let∆V and ∆T be the changes that E makes to V and T , relative to thesalient alternative or status quo. Then we can formally define
Sig = ∆V /∆T
Per = T
Con = ∆T /T.

(1)

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The product of these three factors is the change in value:∆V = Sig × Per × Con .

(2)

Thus, estimating Sig, Per, and Con leads to an estimate for ∆V . By
considering different features of the world, we can then reach an overall
evaluation of the event E . (For more on this overall evaluation, seesection 4.) Even quite rough estimates may be informative, insofar as
we expect some events to differ in value by many orders of magnitude.
While these definitions and the resulting product decompositionof ∆V make logical sense quite generally, they make most intuitive
sense in a scenario where the primary effect of E on F is to increase
its duration, i.e., to add extra time during which F is present, withoutchanging the value that F contributes during the time it would havebeen present anyway. Con is high (equal to 1) when the duration of F
fully depends on E (F would not have been present but for E ), andlow (equal to 0) when the event E makes no difference to the duration.
More generally, Con tracks the effect of E on the duration of F , and,in the stated scenario, Sig is the average value per unit time contributed
during the extra time due to E .
For example, suppose that the value potentially contributed by Fat each time increases exponentially, as in the following table:
TIME PERIOD
1
CONTRIBuTION 1

2 3
2 4

4
8

Let us suppose that, in the status quo, F would be present in periods 2
and 3, and the effect of E is to make F present in periods 1 and 4 aswell. Then Per = 4, since, given E , F lasts for four periods; Con = 1/2(half the duration of F is due to E ); and Sig = 9/2, the average value
contributed per period during the extra two periods attributable to E .
The product decomposition (2) gives ∆V = 9. We could, of course,
have found this answer directly by adding up the contributions of F inperiods 1 and 4.
As this example indicates, Sig differs from both the average value
per period in the status quo (i.e., 6/2) and the average value per period
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given E (i.e., 15/4), though these might both serve as rough approximations. Sig depends on what happens in the status quo as well as what
happens given E . It might be conceptually cleaner to have definitions
according to which ‘significance’ and ‘persistence’ both depend only on
what happens given E , with ‘contingency’ alone taking account of thecounterfactual. We will consider such alternative definitions in section
3.
2.1 Marginal Analysis
The originating event E can often be thought of as a marginal change
in some parameter X (for example, the level of funding for a particular
project). The definitions above then become:Sig =dV /dXdT /dX

Per = TCon =dT /dX.
T(3)

The reason we do not simplify Sig to dV /dT is to emphasise that theindependent variable is X , which we are assuming has an effect on both
V and T as dependent variables.
Note that Con measures the relative change in T , that is, the change
in T as a proportion of T itself. In the economics literature this would
sometimes be called the semi-elasticity of persistence. A convenient notation for the semi-elasticity is %dT /dX ; thus we can equivalently write
Con =

%dT

dX

.

(4)

At any rate, as the marginal case of equation (2), we have the productdecompositiondV= Sig × Per × Con .
dX(5)

2.2 Uncertainty
The definitions just given can be used in different ways when there is
uncertainty about the outcome of E , or about what happens in the4counterfactual. Here, we will assume that the fundamental rule forchoice under uncertainty is maximising expected value. So we would
like to calculate the change in expected value, ∆EV , contributed by
the possible feature F .
The change in expected value is the same as the expected change invalue, E∆V . Thus, one possibility is to use the definitions of significance, persistence, and contingency to calculate ∆V in each state, andthen to calculate its expectation.
An alternative, which we think will often be more convenient, is
to extend the definitions of Sig, Per, and Con to take uncertainty into
account in each factor:
Sig =

∆EV
∆ET

Per = ET

Con =

∆ET
.
ET

(6)

These ‘ex ante’ interpretations of significance, persistence, and contingency agree with the previous definitions (1) in the absence of uncertainty. We obtain a product decomposition∆EV = Sig ×Per × Con(7)in the same way as before. The marginal version is exactly similar:
Sig =dEV /dX
dET /dX

Per = ET

Con =

%dET

dX

(8)

with the result thatdEV
= Sig ×Per × Con.
dX

(9)

It is worth emphasising that Sig, as defined by (6), is not the same as
the expected significance, i.e., E(∆V /∆T ), nor is Con the same as the
expected contingency E(∆T /T ). Nor is ∆EV equal to the product of
expected significance, expected persistence, and expected contingency:
the expectation of a product is not usually equal to the product of expectations. Similar comments apply to (8) and (9).
Extending our previous discussion, this picture, and especially thedefinition of Con, makes most intuitive sense when the key margin for5influence is the expected duration, ET . This can involve some combination of increasing the probability that feature F is present and increasing the duration of F , conditional on its presence. In the next section
we present an alternative framework that treats these, along with significance, as separate margins for influence.

3 An Alternative Framework
While the framework as described in section 2 captures our intuitions
about significance, persistence, and contingency in many situations, weare generally agnostic about what formal framework may be most usefulfor actual decision-makers, so it may be useful to sketch an alternative
which has some conceptual advantages.
First, we will interpret significance simply as the average value contributed by F per unit time, given E :Sig′ = V /T.

(10)

Persistence will be the same as before:
Per′ = T.

(11)

Thus, note, V = Sig′ × Per′ . Previously, contingency was interpreted as
tracking how the duration of F depended on the originating event E ;now, it will track the overall dependence of V on E :Con′ =∆VV

or in the marginal case

%dV

dX

.

(12)

Again, %dV is short for V1 dV . We obtain the usual product decomposition (focusing now on the marginal case):dV= Sig′ × Per′ × Con′ .
dX

(13)

The definitions just given agree with those in section 2 in the special casewhere F contributes a fixed amount of value per unit time, unaffectedby E .
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A conceptual advantage of this framework is that significance, likepersistence, is understood in a way that is independent of the counterfactual. On the other hand, this makes contingency harder to estimate.
However, in the marginal analysis we will have 2
′

Con =

%dV

dX

=

%d Sig′dX+%d Per′

dX

.

(14)

So we can interpret contingency in terms of the semi-elasticities of significance and of persistence. There may be situations where these semielasticities are relatively easy to understand.
What about uncertainty? There are again several ways to proceed.
We could use Sig′ , Per′ , and Con′ to calculate dV /dX , and then calculate expectations. Or we can extend the definition of each factor to
include uncertainty, as we did in section 2.2, taking Sig′ = EV /ET ,Per′ = ET , and Con′ = %dEV /dX , which can again be expanded as%d Sig′ /dX + %d Per′ /dX .
However, in some cases it may be perspicuous to treat the probability of F ’s presence as a separate variable, Prob. For example, smallchanges in the probability that human civilisation continues beyond
this millennium are often thought to be a significant source of value.
To do this, let EF V be the expected value contributed by F conditional on the presence of F . (It equals EV / Prob.) Similarly, let EF T be
the expected duration of F conditional on the presence of F . We thenhave
E V
EV = F × EF T × Prob .
(15)
EF T
Now define
EF V EV=
EF T ET
Per′ = EF T
Sig′ =

Con′ =
′

%dEV

dX

=

(16)

%d SigdX′%d Per′+dX

2 Here we use the fact that %dV /dX

Sig × Per′ gives log V = log Sig′ + log Per′ .

7+%d ProbdX.

= d log V /dX ; the decomposition V =We obtain the familiar decompositiondEV= Sig′ × Per′ × Con′ .
dX

(17)

In short, in this alternative framework we treat significance, persistence, and probability as different factors making up the value contributed by F , and define contingency to be the sum of their semielasticities.

4 The ITN Framework and Prioritisation
We can embed the SPC framework into the Importance, Tractability,
Neglectedness (ITN) framework for prioritising among global problems, which was first proposed by Holden Karnofsky. 3
The ITN framework is a framework for assessing problems withrespect to the marginal benefit of investing work (or other resources) intrying to solve them. The assessment proceeds in terms of three factors.
The only one immediately relevant to our discussion is the first:
Importance: how much value there is in solving the problem, or making progress on it. Another word used for thesame idea is scale.
To formalise this idea, let us make use of a parameter X that measuresprogress on solving the problem. This could directly reflect the percentage of the problem that has been solved, or it could be some other variable that is a proxy for progress. The importance of marginal progresson the problem (as opposed to the importance of solving it completely)is then naturally identified withImp =dEVtotdX(18)

where Vtot is the overall value of the world. Here we have gone directly
for a definition of importance that takes into account uncertainty about
3 See Karnofsky (2014), ‘Narrowing down U.S. policy areas’, [link]. A variety ofdifferent terminology for the three factors can be found, some of which we’ll mention
below.

8the value of making progress on the problem, e.g., because of uncertainty about downstream effects. Notice that we do, in principle, wantto take into account all downstream effects, hence the use of Vtot .
How is this related to the SPC framework? We earlier consideredthe value V contributed by some feature F of the world, and the change
in V attributable to some originating event, such as an increase in thevariable X . Since we may want to consider different features of theworld, let us now call that value VF . The role of the SPC framework,in this context, is to help estimate dEVF /dX (i.e., Sig × Per × Con) for
different features F , contributing to an overall estimate of Imp. In thesimplest case, we can identify a list of features F1 , F2 , . . . such that
Vtot = VF1 + VF2 + · · · .

(19)Then Imp is also a sum,
Imp =

dEVF1
dX

+

dEVF2
dX

+ ··· .

(20)

Of course, it may often be difficult to identify a list of easily evaluable
features that contribute additively to the value of the world, as in equation (19), although, on the other hand, even rough approximations canbe informative.
4.1 Tractability, Neglectedness, and Leverage
We conclude with some further comments about the ITN framework(which are, however, not necessary for understanding the SPC framework).
The second factor used in the ITN framework is
Tractability: how easy it is to solve the problem, or to makeprogress. Another word used for the same idea is solvability.
At a first pass, the more important a problem, and the more tractable itis, the better to work on it; the ideal would be to accomplish something
very good with very little effort (one could then spend more effort on9

other things). Let W be the amount of work done on the problem (orthe amount of other resources invested in it). The marginal value ofworking on the problem is thendEVtot dEVtot dX
=
×
.
dW
dX
dW

(21)

The first factor on the right-hand side is Imp. It would be natural
to identify ‘tractability’ with the second factor, dX /dW , the marginal
progress made by extra work. However, it is traditional (and sometimes
useful) to separate out of the second factor another consideration:
Neglectedness: how much work is already being done. A
word used for the opposite idea is crowdedness.
The idea here is that many problems show diminishing returns; that is,
if a lot of work has been done, then the easiest aspects of the problem
will already have been solved, making it harder to make progress. Owen
Cotton-Barratt proposed identifying ‘neglectedness’ with 1/W , leading
to the following decomposition:
dX
dX
1
=
× .
dW %dW W(22)

He then identifies ‘tractability’ with the first factor on the right-handside, dX /%dW . 4
These are perfectly sensible definitions which are useful in many
contexts. However, this definition of tractability is arguably most intuitive when there are diminishing returns, and most of all when there
4 See Cotton-Barratt (2016),‘Prospecting for Gold’, [link], and also Wiblin(2019), ‘A framework for comparing global problems in terms of expected impact’,[link]. Note that our variable X corresponds formally to Cotton-Barratt’s log S . The
issue here is whether we are interested in absolute or relative increases in our measure of progress. For example, if X is the ‘percentage of the problem solved’ then it
seems most natural to think in terms of absolute changes in X , as we do in this report.
However, it may sometimes be more convenient to focus on some concrete statistic
such as (say) ‘number of bednets distributed’. Letting S be this number, it may thenbe convenient to think in terms of relative changes in S , and thus to define importance as dEVtot /%dS (equivalently, dEVtot /d log S ) and tractabilility as the elasticity%dS /%dW (equivalently d log S /%dW ). Thanks to Cotton-Barratt for discussion.

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are logarithmic returns. This is connected to the point that a high valueof 1/W only intuitively speaks in favour of additional work if there arediminishing returns. In the simplest case of logarithmic returns, whereX = a log W , we find dX /%dW = a. Although, in one sense, we mightwant to say that the problem becomes ‘less tractable’ as W increases(there are diminishing returns), the constant scale factor a is related tothe overall difficulty of the problem—the total work that would be required to solve it. In contrast, consider a case in which there are linear
returns: X = W . Then dX /%dW = W . It could be somewhat counterintuitive to think of tractability increasing with W in this case, and thecurrent value of W does not give a sense of the overall tractability of the
problem, either.
An alternative approach is to identify ‘tractability’ with the overall
difficulty of the problem. Let W0 be the total amount of work thatwould be required to solve the problem (or to achieve some relevantbenchmark), and let X0 be the value of X that would count as a full
solution. (The simplest way to think about this is that X is ‘the percent
solved’, so that X0 = 100) . We can then writeX
dX /dW
dX
= 0 ×
.
dW W0
X0 /W0(23)

The first factor on the right-hand side, X0 /W0 , measures the overall
easiness of solving the problem, while the second, (dX /dW )/(X0 /W0 ),measures how much easier it is to make progress at the current margin.
In other words, it captures directly the extent to which there are lowerthan-average hanging fruit. To avoid confusion, let us call this second
factor leverage. Leverage is a form of ‘neglectedness’ (the low-hanging
fruit haven’t been picked), but high leverage does not imply that little
work is being done. When there are diminishing returns, leverage decreases with W (for fixed values of X0 and W0 ), but when (for example)
returns are linear, leverage is constant: the marginal worker has as much
impact as the average one.

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