Communism and the Incentive to Share in Science


Communism and the Incentive to Share in
Science∗

Remco Heesen†

August 24, 2015

Abstract
The communist norm requires that scientists widely share the re-

sults of their work. Where did this norm come from, and how does it
persist? Michael Strevens provides a partial answer to these questions
by showing that scientists should be willing to sign a social contract
that mandates sharing. However, he also argues that it is not in an
individual credit-maximizing scientist’s interest to follow this norm.
I argue against Strevens that individual scientists can rationally con-
form to the communist norm, even in the absence of a social contract
or other ways of socially enforcing the norm, by proving results to
this effect in a game-theoretic model. This shows that the incentives
provided to scientists through the priority rule are sufficient to explain
both the origins and the persistence of the communist norm, adding
to previous results emphasizing the benefits of the incentive structure
created by the priority rule.

∗Thanks to Kevin Zollman, Michael Strevens, Stephan Hartmann, Teddy Seidenfeld,
Thomas Boyer, Lee Elkin, Liam Bright, and audiences at the Bristol-Groningen Conference
in Formal Epistemology and the Logic Colloquium in Helsinki for valuable comments and
discussion. This work was partially supported by the National Science Foundation under
grant SES 1254291.
†Department of Philosophy, Baker Hall 161, Carnegie Mellon University, Pittsburgh,

PA 15213-3890, USA. Email: rheesen@cmu.edu

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mailto:rheesen@cmu.edu


1 Introduction
The social value of scientific work is highest when it is widely shared. Work
that is shared can be built upon by other scientists, and utilized in the wider
society. Work that is not shared can only be built upon or utilized by the
original discoverer, and would have to be duplicated by others before they
can use it, leading to inefficient double work.1

To put the point more strongly, it can be argued that work that is not
widely shared is not really scientific work. Insofar as science is essentially
a social enterprise, representing the cumulative stock of human knowledge,
work that other scientists do not know about and cannot build upon is not
science (cf. the distinction between Science and Technology in Dasgupta and
David 1994). The sharing of scientific work is thus a necessary condition
not merely for the success of science, but in an important sense for its very
existence.

The sociologist Robert Merton first noticed that there exists an insti-
tutional norm in science that mandates widely sharing results. He called
this the communist norm, according to which “[t]he substantive findings of
science. . . are assigned to the community. . . The scientist’s claim to ‘his’ intel-
lectual ‘property’ is limited to that of recognition and esteem” (Merton 1942,
p. 121). Subsequent empirical work by Louis et al. (2002) and Macfarlane
and Cheng (2008) confirms that over ninety percent of scientists recognize
this norm of sharing. Moreover, most scientists (if not as many as ninety
percent) consistently conform to the communist norm.

The existence of this norm raises two questions. Where did it come from?
And how does it persist? In light of what I said above, these are important
questions. A good understanding of what makes the communist norm persist
tells us which aspects of the institutional (incentive) structure of science can

1Of course scientific work is often duplicated by others even when it is shared (so-called
replications). But this is not inefficient in the same way, as after the replication is shared
the work is known by all to be more certainly established than if only one or the other
instance was shared.

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be changed without affecting the communist norm. Understanding its origins
might allow us to reinstate the communist norm if it disappeared for whatever
reason. Insofar as we value the existence and success of science, these are
things we should want to know.

There must be some sense in which it is in scientists’ interests to uphold
the communist norm and conform to it, or else it would disappear.2 One
such sense is given by Strevens (forthcoming). He gives what he calls a
“Hobbesian vindication” of the communist norm by showing that scientists
should be willing to sign a contract that enforced sharing. The claim is that,
from a credit-maximizing perspective, it is not beneficial for an individual
scientist to share her work (which would help other scientists more than her),
but every scientist is better off if everyone shares than if no one shares.

As Strevens is well aware, this only partially answers the question of the
persistence of the communist norm, and says little about its origins. In con-
trast, I argue that sharing is rational from a credit-maximizing perspective
for an individual scientist. If my argument is successful, it provides a much
more detailed account of both the origins and the persistence of the commu-
nist norm. It also adds to a tradition of work in philosophy and economics
that has emphasized the power of the priority rule to incentivize scientists
to organize themselves in ways that further the aims of science (e.g., Kitcher
1990, Dasgupta and David 1994, Strevens 2003).

Because the existence of a norm can itself change what is in scientists’
interests to do, the sense in which sharing is or is not rational or beneficial
to scientists needs to be clarified. For this purpose, I rely on the terminology
for social norms developed by Bicchieri (2006). I explain this terminology
in section 2 and use it to state Strevens’ position more precisely than I did
above.

Section 3 sets out my own position by explaining how the idea that sci-
2It is perhaps debatable whether there would be a norm worth speaking of if scientists

recognized an obligation to share but never acted on it, but since that is not the case
nothing in this paper turns on that definitional question.

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entists can publish and claim credit for intermediate results can be used to
establish the rationality of sharing. Section 4 makes this more precise by
describing a game-theoretic model of scientists working on a research project
needing to decide whether to share their intermediate results.3

I then show that rational credit-maximizing scientists should indeed be
expected to share in two versions of the model (sections 5 and 6). In section 7
I use these results to give an explanation of the persistence of the communist
norm, and I consider some objections. I extend my explanation to include
the origins of the norm in section 8, which involves considering boundedly
rational scientists and some historical evidence. A brief conclusion wraps up
the paper.

2 Social Norms and Communism
The question that this paper focuses on is whether it is in a scientist’s interest
to behave in accordance with the communist norm. Here, the crucial turning
point is what is meant by a scientist’s “interest”. The specific question I want
to raise is whether it would be in scientists’ interest to share their work even
in the absence of a norm telling them to do so. To clarify the distinction I
have in mind, I use some terminology defined by Bicchieri (2006). She defines
a social norm as follows:

Let R be a behavioral rule for situations of type S, where S can
be represented as a mixed-motive game. We say that R is a social
norm in a population P if there exists a sufficiently large subset
Pcf ⊆ P such that, for each individual i ∈ Pcf:

Contingency: i knows that a rule R exists and applies to situa-
tions of type S;

3The idea of using game theory to get a better understanding of norms in science goes
back at least as far as Bicchieri (1988).

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Conditional preference: i prefers to conform to R in situations of
type S on the condition that:

(a) Empirical expectations: i believes that a sufficiently large sub-
set of P conforms to R in situations of type S;

and either

(b) Normative expectations: i believes that a sufficiently large
subset of P expects i to conform to R in situations of type S;

or

(b′) Normative expectations with sanctions: i believes that a suf-
ficiently large subset of P expects i to conform to R in situations
of type S, prefers i to conform, and may sanction behavior. (Bic-
chieri 2006, p. 11)

The crucial feature of this definition is the requirement of normative ex-
pectations. This says that an individual’s preference to conform to the norm
is conditional on others’ expectations (possibly enforced by sanctions). For
example, norms surrounding the sharing of food are plausibly social norms:
in the absence of others expecting them to share, many people might prefer
not to share even if they knew that a norm existed and most people followed
it. In contrast, if an individual knows that in a particular country there
exists a norm to drive on the right side of the road which is followed by most
people, she would probably prefer to conform to that norm even if others
had no expectations about her behavior.

In other words, a social norm actively works to change people’s prefer-
ences: the norm makes it such that people expect each other to conform to
it, and this expectation from others is itself necessary to make individuals
prefer to conform. Other kinds of norms, such as descriptive norms and con-
ventions, do not have this feature. They merely work with already existing
preferences to help coordinate behavior.

The language of game theory is useful to sharpen these ideas. Recall that
a (Nash) equilibrium is a situation in which each individual involved in the

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situation behaves in such a way that no individual has an incentive to deviate
unilaterally. That is, keeping everyone else’s behavior fixed, it is not in an
individual’s interest to change her behavior.

Consider a situation of type S and a putative norm R. If knowledge of R
and empirical expectations (that others will conform to R) are sufficient to
make an individual prefer to conform to R, then by definition R is an equi-
librium of the underlying game that is being played in situations of type S.
But if normative expectations are required, that is, if individuals only prefer
to conform to R if others expect them to conform (and, possibly, are willing
to back this up with sanctions), then R is not an equilibrium of the “original”
game: it is only made into an equilibrium by the existence of the norm itself.
So the existence of a social norm—unlike other kinds of norms—transforms
the underlying game by changing people’s preferences, thus creating a new
equilibrium (Bicchieri 2006, pp. 25–27).

Is the communist norm a social norm in this sense, i.e., are normative
expectations a necessary ingredient to make it in scientists’ interest to share
their work? In order to answer this question, one needs to know what scien-
tists’ interests are. In particular, an account of their motivations is needed
that is independent of the communist norm, so that the question can be
asked whether a self-interested scientist would share her work in the absence
of a normative expectation that she do so.

A scientist’s achievements create for her a stock of credit. This credit
is the means by which she advances her career, which determines both her
income and her status in the profession. Insofar as a scientist is someone
who is interested in building a career in science, it is then in her interest to
maximize credit. This claim has been defended by philosophers and sociol-
ogists as diverse as Hull (1988, chapter 8), Kitcher (1990), Strevens (2003),
Merton (1957, 1969), and Latour and Woolgar (1986, chapter 5).

This is not to deny that the scientist may have other interests, either
as a scientist (e.g., to advance human knowledge) or apart from being a
scientist (e.g., to have time for other pursuits). But these are idiosyncratic.

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I aim to show that sharing is beneficial to scientists as a consequence of an
interest that all scientists share. Credit maximization is, in my view, the
only candidate here.

What kind of achievements does the scientist get credit for? The answer
is simple: scientific discoveries. The institutions of science put a premium on
originality. Credit is awarded to the first scientist to publish some particular
result, and the amount of credit awarded is roughly proportional to the sig-
nificance of the result. This feature of science is known as the priority rule,
and the extent to which it shapes scientists’ behavior is well-documented
(Merton 1957, 1969, Kitcher 1990, Strevens 2003).

By rewarding only the first scientist, the priority rule encourages scientists
to work and publish quickly (Dasgupta and David 1994). In this way, it
seems that the priority rule creates an incentive for scientists to share their
work. However, “the same considerations give you a powerful incentive not
to share your results before you have extracted every last publication from
them” (Strevens forthcoming, p. 2). If results were shared before publication,
this would improve other scientists’ chances of scooping important discoveries
for which those results are relevant. So, Strevens argues, there is a split in
the motivations provided by the priority rule:

The priority rule motivates a scientist to keep all data, all technol-
ogy of experimentation, all incipient hypothesizing secret before
discovery, and then to publish, that is to share widely, anything
and everything of social value as soon as possible after discovery
(should a discovery actually be made). The interests of soci-
ety and the scientist are therefore in complete alignment after
discovery, but before discovery, they appear to be diametrically
opposed. (Strevens forthcoming, pp. 2–3)

Of course, any sharing that happens after a discovery has been made
does not help science in coming to that discovery faster. Thus, at the crucial
stage at which science can be sped up by sharing, the priority rule provides
no incentive to do so, according to Strevens.

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Strevens then goes on to show that a social contract, in which all scientists
agree to widely share their work (even before discovery), would be beneficial
to all scientists. In doing so, he shows that the problem of sharing has
the structure of a Prisoner’s Dilemma: every scientist would be better off if
every scientist shared, but each individual scientist has an incentive not to
share. The communist norm is thus a social norm on Strevens’ view: without
normative expectations to transform the game (into something that looks
more like a Stag Hunt), widely sharing scientific work is not an equilibrium.

Strevens is not the only one to make this claim. For example, Resnik
(2006, p. 135) observes that “the desire to protect priority, credit, and in-
tellectual property” can motivate scientists to keep scientific results secret.
Similarly, “[the priority rule] sets up an immediate tension between coop-
erative compliance with the norm of full disclosure (to assist oneself and
colleagues in the communal search for knowledge), and the individualistic
competitive urge to win priority races” (Dasgupta and David 1994, p. 500).4

Claims like these are also made by Arzberger et al. (2004, p. 146), Borgman
(2012, p. 1072), and Soranno et al. (2015, p. 70), among others.

3 Communism and Intermediate Results
In this paper I argue that, given the priority rule, it is in a scientist’s own
interest to share her work widely, at least whenever she expects other scien-
tists to do the same. In other words, sharing widely is an equilibrium of the
relevant game even in the absence of normative expectations. The problem
of sharing is thus not like a Prisoner’s Dilemma: the role of the communist
norm is not to change scientists’ preferences to make sharing attractive (at

4Dasgupta and David (1994, p. 502) go on to semi-formally characterize a situation very
similar to the model of Boyer (2014) and this paper, but they draw the opposite conclusion:
they agree with Strevens that conforming to the communist norm is structurally similar
to cooperating in a Prisoner’s Dilemma.

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least not primarily).5 It merely describes a rule of behavior that it is in
scientists’ own best interests to follow.

An important part of my argument is the insight that major discoveries
can often be split into multiple smaller discoveries that were made along the
way. Newton’s famous comment “If I have seen further it is by standing on
the shoulders of giants” illustrates this accumulative nature of science. Boyer
(2014, p. 18 and p. 21) gives some more detailed examples: the construction
of the first laser can be split into a theoretical development and the actual
construction based on that theory, and the experimental test of the EPR
thought experiment by Aspect et al. (1982) was preceded by a number of
papers defining and refining the experiment.

It is noteworthy that in these cases each of the smaller discoveries was
published as soon as it was done, rather than after the major discovery was
completed. It is not obvious that it is always in an individual scientist’s best
interest to behave as these scientists did. On the one hand, credit can be
claimed for the smaller discovery. On the other hand, the advantage that
the smaller discovery gives on the way toward the major discovery is lost
by publishing (and hence widely sharing) it. In fact, Schawlow and Townes
seem to have lost the race to build the first working laser at least partially
because their publication of the theoretical idea spurred on other teams.

Boyer (2014) provides a model to analyze this tradeoff. He shows that
in some idealized circumstances the benefits of sharing these intermediate
results outweigh the costs, with costs and benefits both measured in credit
assigned via the priority rule. Although Boyer does not use these terms, his
result could be used to argue that normative expectations are not necessary
for the communist norm to arise: the priority rule encourages wide sharing
of scientific work even before the potential of future discoveries based on
this work has been exhausted, i.e., “before you have extracted every last

5I do not deny that normative expectations calling on scientists to share their work
exist, as they in fact appear to do (Louis et al. 2002, Macfarlane and Cheng 2008). The
point is rather that these are not required to explain the origins or persistence of the norm.
I return to this point in section 7.

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publication” (Strevens forthcoming, p. 2).
Unfortunately, things are not that simple. A number of objections can

be made. The remainder of this section describes two such objections, which
motivate the construction and analysis of a formal model in sections 4–6. In
section 7 I flesh out the explanation of the communist norm suggested by
this model, and I consider some further objections.

One may worry that Boyer’s result is not general enough to support claims
about the origins or persistence of the communist norm. By his own admis-
sion, he only shows that “there exist simple and plausible research situations
for which the [credit] incentive to publish intermediate steps is sufficient”
(Boyer 2014, p. 29). I aim to show that in fact all or most research situa-
tions are such that there is a credit incentive to publish intermediate results,
which requires a more general model. The results I obtain may be viewed
as generalizations of Boyer’s—relaxing the assumptions that there are only
two scientists, that the scientists are equally productive, and that scientists
share either all or no intermediate results—although speaking strictly math-
ematically they are not (because Boyer uses discrete time steps and I use
continuous time).

The second worry questions the relevance of equilibria. The worry may
be either that showing that the communist norm is an equilibrium is not
sufficient to show that one should expect real scientists to share, especially
when there are also other equilibria (this is known as the equilibrium selection
problem). Or alternatively one may disagree with Bicchieri that any observed
behavioral rule has to be the equilibrium of some underlying game. I alleviate
both of these worries by showing that the communist norm is not merely an
equilibrium, but an equilibrium that one should expect to be realized by both
fully rational and boundedly rational scientists. Thus, regardless of what
one thinks of the general relevance of equilibria, the particular equilibrium
considered here has behavioral implications.

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4 A General Game-Theoretic Model of Inter-
mediate Results

The game-theoretic model I develop in this section is intended to investigate
scientists’ incentives when they are working on a project that can be divided
into a number of intermediate stages. An intermediate stage is a part of the
project which, when completed successfully, yields a publishable intermediate
result in the sense of Boyer (2014, section 2). I assume that stages can only
be completed in one order.6 The number of intermediate stages of the project
is denoted k.

Competition plays a central role in the model. I assume that scientists
are aware that other scientists are working on the same project (or at least
believe this to be the case). Merton (1961) argued for the ubiquity of multiple
discoveries in science, which suggests that scientists should almost always
expect other scientists to be working on the same project. I thus assume
that n ≥ 2, where n is the number of scientists (or research groups) working
on the project. Note that by “scientist” I mean not just one working in the
natural sciences, but also the social sciences, the humanities, or any other
creative field where the priority rule applies.

Whenever a scientist completes an intermediate stage, she has to make a
choice: she can either publish the result, or keep it to herself.7 Publishing
benefits the scientist, because she thereby claims credit for completing that
intermediate stage as well as any preceding stages that remain unpublished,
in accordance with the priority rule. I assume that all stages are equally
valuable, so the amount of credit obtained is equal to the number of stages

6This linearity assumption may seem restrictive and unrealistic. However, any alterna-
tive assumption would only make sharing more attractive by improving the chance that a
scientist can claim credit for an intermediate result without hurting her chances of future
credit (because, e.g., other scientists are following a different path within the research
project and thus are not helped by the publication of the intermediate result).

7By assumption, the result is publishable, i.e., if she decides to publish it, it will be
accepted by a journal.

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published. Publishing also benefits the scientific community: other scientists
no longer need to work independently on the stages that have been published.
Publishing thus “expedites the flow of knowledge”. I use E to denote this
strategy.

The way that the scientific community benefits from publications is a
potential downside to the individual scientist: if she keeps her results secret
instead, she can start working on the next stage before anyone else can.
This improves her chance of being the first to successfully complete the next
stage, thus allowing her to claim credit for more stages later (at the risk
that someone else claims credit for the one she did not publish). Holding
onto a discovery until a more expedient time might thus be beneficial to the
scientist. Call this strategy H.

When a scientist completes the last stage there is no incentive (within the
model) to keep her from publishing. So when a scientist completes stage k she
always publishes, claiming credit for all unpublished stages and concluding
this instance of the model.

Note that I assume that scientists care only about credit8, and that the
only way to get credit is by publishing. Scientists are thus not assumed to
have any inherent preference for or against sharing their work. In particular,
expectations (normative or otherwise) from other scientists are not built into
the individual scientist’s preferences.

An interesting feature of the priority rule is its uncompromising nature.
According to the priority rule, there are no second prizes, even if the time
interval between the two discoveries is very small. This feature was noted
by Merton (1957, p. 658), who quotes the French scientist François Arago as
saying: “‘about the same time’ proves nothing; questions as to priority may
depend on weeks, on days, on hours, on minutes.”9

8More precisely: I investigate the incentives provided to scientists through credit, in-
dependent of any other interests or incentives they might have.

9Merton (1957, pp. 658–659) goes on to argue that this is a pathological extreme:
when the interval between two discoveries is so small, “priority has lost all functional
significance.” I agree with Strevens (2003, section IV.1) that this is not obviously correct.

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To incorporate this feature in the model, it needs to be able to distinguish
arbitrarily small time intervals. This suggests a continuous-time model: a
model using discrete time units might place two discoveries in the same time
unit even though in reality one of them happened (slightly) earlier than the
other. This would fail to adequately model the uncompromising nature of
the priority rule.

This means that a continuous-time probability distribution is needed to
model the waiting time: the time it takes a given scientist to complete an
intermediate stage. For this purpose I use the exponential distribution, the
only candidate that has significant empirical support behind it (Huber 2001,
more on this below). In particular, I assume that the time scientist i takes
to complete any intermediate stage follows an exponential distribution with
parameter λi. The parameter can be interpreted as the average number
of stages completed by the scientist per unit time. The parameter may be
different for different scientists, indicating the possibility that some scientists
work faster than others, or are part of a larger or more efficient research
group.

The assumption that waiting times are exponential is equivalent to the
assumption that scientists’ productivity is a Poisson process with a param-
eter that is constant over time. Empirical work has shown that scientists’
productivity fits a Poisson distribution quite well, and the percentage of au-
thors who experience significant trends or surges over time is small. Huber
(1998a,b) has established this for the rate at which patents are produced by
inventors, Huber and Wagner-Döbler (2001a) for publications in mathemati-
cal logic, Huber and Wagner-Döbler (2001b) for publications in 19th century
physics, and Huber (2001) for publications in modern physics, biology, and

A version of the priority rule which gives shared credit when the time interval between
discoveries is below a certain threshold would create a different incentive structure for
scientists, and it is an open question whether that incentive structure would be better or
worse. In any case, here I simply take the uncompromising version of the priority rule as
given.

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psychology.10

The assumption that waiting times are exponential means that the prob-
ability that it will take scientist i more than t time units to complete a
given stage is exp{−tλi}.11 This distribution has some formal features that
I will make use of (Norris 1998, section 2.3). First, it is “memoryless”. This
means that after a certain amount of time has passed and the waiting time
has not ended yet, the distribution of the remaining waiting time is equal
to the original distribution of the waiting time. Second, the minimum of n
independent exponential random variables with parameters λi (i = 1, . . . ,n)
is itself exponentially distributed with parameter λ = λ1 + . . . + λn. Thus,
the waiting time until at least one of the scientists completes a stage of the
project is exponentially distributed with parameter λ. Third, the probability
that scientist i is the first one to finish the stage she is working on is λi/λ.

The memorylessness property may seem odd, as it suggests that the scien-
10The fact that scientists’ total career productivity follows a Poisson distribution (if

accepted) does not imply exponential waiting times. One could generate Poisson distribu-
tions in other ways. But the evidence regarding trends and surges, as well as the fact that
the evidence includes scientific careers cut short, suggests the stronger claim that at any
given time in a scientist’s career the Poisson distribution is a good model for her produc-
tivity up to that point. On this interpretation it is a simple mathematical consequence
that the waiting times are exponential.

11Compare this with Boyer’s assumption that there is a fixed probability λ that a given
scientist will solve a given stage in a given time unit. As noted above, by using discrete time
units this model provides no way of applying the priority rule when two scientists finish
the same stage in the same time unit. This problem can be addressed by using smaller
time units. Suppose that what was previously one time unit is now x time units, and in
each unit the scientist completes the stage with probability λ/x. Unfortunately, the same
problem may still arise, and this will be true for any (finite) magnification factor x. The
problem is solved by taking the limit as x goes to infinity. For any finite x, the probability
that the scientist has not completed the stage at time t (measured in the original time
units before magnification) is (1 −λ/x)tx. In the limit the probability that the scientist
has not completed the stage at time t is limx→∞(1 −λ/x)tx = exp{−tλ}. So, in addition
to being independently empirically justified, exponential waiting times naturally arise as
the limiting case of Boyer’s model where the priority rule can be applied unambiguously.

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tist herself never knows whether she is making any progress on the problem.
Moreover, if she starts working on a given stage much later than another
scientist she has the same chance of completing it first as she would have had
if both had started at the same time (conditional on the fact that the other
scientist does not complete the stage before she starts).

While these features of the exponential distribution do not seem to mesh
well with the subjective experience of working on a research project, I want
to insist that Huber’s empirical evidence should be given more weight than
subjective experience. The following consideration may help to smooth the
apparent conflict.

In the model, scientists only make decisions after they have just completed
a stage. So for the model it only matters that when a scientist completes a
stage, she views the time she needs to complete future stages and the time
other scientists need to complete stages as exponentially distributed. I do
not need to insist that the scientist views the time needed to complete stages
as exponentially distributed while she is in the middle of one.

How does my model compare to the one given by Strevens (forthcoming)?
Perhaps the key difference is that contrary to Strevens I have described a
zero-sum game. In my model it is implicitly assumed that the scientists will
always eventually complete the entire research project.12 Since each stage is
worth one unit of credit, and the first scientist to complete stage k always
claims credit for it and any unclaimed stages, this means that at the end of
the game the scientists have always divided k units of credit between them.
So any change in strategy that leads to one scientist improving her (expected)
credit must always lead to a decrease for at least one other scientist.

In contrast, a key component of Strevens’ model is the chance each scien-
tist has of successfully completing the research project “in isolation”, which
leaves room for the scenario in which the research project is never completed
by anyone. By sharing their progress, Strevens assumes, the scientists im-
prove each other’s chances of completing the research project. In fact this

12More precisely: the scientists complete all k stages in finite time with probability one.

15



appears to be the main driving force behind his result that scientists should
be willing to sign a social contract that enforces sharing: in his model sharing
“creates” expected credit (by improving the overall chance that any credit
is awarded at all), and as long as this “extra” credit is divided in such a
way that everyone benefits at least a little (in expectation), it is clear that
everyone will be better off if everyone shares.

By allowing for a chance that no scientist completes the research project,
Strevens’ model is arguably more realistic than mine. But I claim that this
is a strength rather than a weakness of my model. Working with a zero-sum
game reflects a strictly more pessimistic assumption about the benefits of
sharing than working with a model like Strevens’. The result that sharing
is incentive-compatible which I state and prove below is thus a somewhat
surprising result: it is stronger than the result Strevens proved, while his
model makes a more optimistic assumption about the benefits of sharing.
Insofar as I show that the priority rule is sufficient for a communist norm to
arise (without a need for normative expectations) in my model, this result
should hold a fortiori in a more realistic (not zero-sum) model.

There are other ways to change the model that would make it no longer
zero-sum. For example, Boyer and Imbert (forthcoming, section 4) argue
that the relevant notion to consider is credit per unit time (rather than “total
credit” which I use). This incorporates the idea that if the research project
finishes faster the competing scientists will be free to work on other (po-
tentially credit-worthy) projects sooner. Then sharing benefits all scientists
to some extent by decreasing the expected completion time of the research
project; Boyer and Imbert call this a “speedup effect”. So considering credit
per unit time instead of total credit also invalidates the zero-sum property.
Since, as above, it does so in a way that makes sharing more attractive, the
result I get in my model holds a fortiori when credit per unit time is used.

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5 A Backwards Induction Analysis
The previous section described a game-theoretic model of scientists working
on a project that requires some number of intermediate stages to be com-
pleted. The game consists of a sequence of (probabilistic) events, in which
the scientists can intervene at specific points through their choice of strat-
egy by publishing their work (E) or keeping it secret (H). Each scientist
attempts to maximize her credit.

In the simplest version of the game there are two scientists (n = 2) and
the research project has two stages (k = 2). The extensive form of the game
is given in figure 1.

N

1

N

(2, 0)

λ1/λ

(1, 1)

λ2/λ

E

N

(2, 0)

λ1/λ

2

N

(1, 1)

λ1/λ

(0, 2)

λ2/λ

E

N

(2, 0)

λ1/λ

(0, 2)

λ2/λ

H

λ2/λ

H

λ1/λ

2

N

(1, 1)

λ1/λ

(0, 2)

λ2/λ

E

N

1

N

(2, 0)

λ1/λ

(1, 1)

λ2/λ

E

N

(2, 0)

λ1/λ

(0, 2)

λ2/λ

H

λ1/λ

(0, 2)

λ2/λ

H

λ2/λ

Figure 1: Extensive form of the game with n = 2 and k = 2

At the root node (marked “N”) Nature decides which of the two scientists
is the first one to complete the first stage of the project. As indicated, Nature
picks scientist 1 with probability λ1/λ and scientist 2 with probability λ2/λ.

17



Suppose Nature picks scientist 1. This leads to a decision node marked
“1”, indicating that scientist 1 is the one to make a decision at this node.
If scientist 1 publishes (strategy E), she collects one unit of credit. Both
scientists now know the solution to stage 1 of the project, so they start
working on stage 2.

Nature again decides with the indicated probabilities which of the two
scientists completes the second stage first. In either case the game ends. If
Nature picks scientist 1, she gets credit for completing both stages of the
project and scientist 2 gets nothing (as indicated by the payoff pair (2, 0)
in the figure). If Nature picks scientist 2, she gets credit for completing the
second stage, and since scientist 1 had already claimed credit for the first
stage, both scientists end up with one unit of credit.

What if scientist 1 chooses not to publish her solution to the first stage of
the project (strategy H at the node marked “1”)? Then scientist 1 does not
collect a unit of credit, and scientist 2 does not learn the solution to stage 1.
So now scientist 1 starts working on stage 2, while scientist 2 continues to
work on stage 1.

Once again Nature decides which of the two scientists finishes the stage
she is working on first (due to the memorylessness of the exponential distribu-
tion, scientist 2 is not more likely to finish fast despite having already spent
some time working on stage 1). If Nature picks scientist 1, she completes the
project. The game ends and scientist 1 gets both units of credit.

If Nature picks scientist 2, she now has a decision to make (at the node
marked “2”). She can claim one unit of credit by playing strategy E, or defer
by playing H. In either case, both scientists can now work on stage 2.

Nature makes its final decision by picking a scientist who completes the
second stage first. That scientist gets both units of credit (and the other gets
nothing) if scientist 2 chose strategy H, whereas if scientist 2 chose E she
gets one unit of credit for sure and the scientist picked by Nature gets the
other unit.

The right-hand side of the figure (associated with Nature picking scien-

18



tist 2 at the root node) works similarly.
If the first scientist to complete stage 1 plays H, and the other scientist

completes stage 1 before the first scientist finishes the project, it is rational
for the other scientist to play E: this makes it certain that she will get
one unit of credit, without reducing either her probability of completing the
second stage or her payoff if she does so, and without giving the first scientist
any information she does not already have.

This is a so-called “backwards induction” argument: if a certain node is
reached, then it is rational for the scientist who has to make a decision at that
node to choose x; therefore, other scientists may assume that if that node is
reached, x will be played. Applying this argument to the terminal decision
nodes in figure 1 leads to a truncated game tree, as shown in figure 2.

N

1

N

(2, 0)

λ1/λ

(1, 1)

λ2/λ

E

N

(2, 0)

λ1/λ

2

(λ1/λ, 1 + λ2/λ)

E

λ2/λ

H

λ1/λ

2

N

(1, 1)

λ1/λ

(0, 2)

λ2/λ

E

N

1

(1 + λ1/λ,λ2/λ)

E

λ1/λ

(0, 2)

λ2/λ

H

λ2/λ

Figure 2: Truncated game tree for the game with n = 2 and k = 2

Here it is assumed that the second scientist to complete stage 1 always
plays strategy E. The (expected) payoff of that strategy is one plus the
probability of being the first to complete stage 2 for the scientist who just
completed stage 1, and just the probability of being the first to complete

19



stage 2 for the other scientist.
Now consider the decision scientist 1 has to make if she completes stage 1

first. If she plays strategy E, her payoff is 2 with probability λ1/λ and 1 with
probability λ2/λ, so her expected payoff is 2 ·λ1/λ + 1 ·λ2/λ.

If she plays strategy H instead, her payoff is 2 with probability λ1/λ and
λ1/λ with probability λ2/λ. So in this case her expected payoff is 2 ·λ1/λ +
λ2/λ ·λ1/λ.

Since λ1/λ < 1 and λ2/λ > 0, the expected payoff of E is strictly greater
than the expected payoff of H. So scientist 1 should play E if she is the first
to complete stage 1 (and a similar argument applies to scientist 2).

Thus, the backwards induction solution of this game is for both scientists
to play E at both of their decision nodes. Like any backwards induction
solution, this is an equilibrium. The expected payoff if this equilibrium is
played is 2λ1/λ to scientist 1 and 2λ2/λ to scientist 2.

Nothing in the above analysis depended on the values of λ1 and λ2. More-
over, it can be shown that a similar analysis goes through when the number
of scientists or the number of stages is changed, as stated in the following
theorem (see appendix A for a proof).

Theorem 1. In the (unique) backwards induction solution to this game with
n ≥ 2 scientists and k ≥ 1 stages, every scientist plays E at every decision
node. Moreover, there are no equilibria that are behaviorally distinct13 from
the backwards induction solution.

Backwards induction thus yields a unique prediction for this game. But
under what circumstances should scientists be expected to behave according
to the backwards induction solution? A sufficient condition is that all scien-
tists are rational (maximizing expected credit) and that this fact is common
knowledge among the scientists (Aumann 1995).

13That is, while there may be other equilibria, these differ only in that some scientists
make different decisions at decision nodes that will not actually be reached in the game
(given the strategies of the other scientists).

20



Common knowledge of rationality is a very strong assumption. It requires
that scientists expect each other to behave rationally, even when they have
already been seen to behave irrationally (making it a common belief rather
than common knowledge). In the next section, I relax this assumption as
well as an unrealistic assumption about the information that is available to
the scientists in this game.

6 A Game of Imperfect Information
The analysis in section 5 uses backwards induction, in which one works
through the game tree from the end of the game back to the beginning.
This type of analysis relies on the scientists having very specific knowledge
about the state of the game.

For example, in the case with two scientists and two stages I argued
that it is rational for a scientist to play strategy E if she completes the first
stage after the other scientist has already done so. This argument relies
on the assumption that she can distinguish between the situation in which
the other scientist has already completed the first stage but decided not to
publish this information and the situation in which the other scientist does
not have a solution to the first stage yet. Without this assumption the
backwards induction analysis never gets off the ground.

Is it realistic to assume that scientists know the results their peers have
obtained even when they have not published them? I think this differs from
field to field. In small fields where everyone knows what everyone else is
working on word gets around when one of the labs has solved a particular
problem, even when they manage to keep the details to themselves. Or, with
pre-registration of clinical trials becoming more and more common, scientists
might know that some other scientist knows, say, whether a particular drug
is effective, without knowing whether the answer is yes or no.

But in other fields this kind of information might not be available. In
this section I analyze a version of the model in which scientists do not know

21



if other scientists have any unpublished results. I retain the assumption that
once a result is published all scientists know about it. This yields a game of
imperfect information.14

N

1

N

(2, 0)

λ1/λ

(1, 1)

λ2/λ

E

N

(2, 0)

λ1/λ

2

N

(1, 1)

λ1/λ

(0, 2)

λ2/λ

E

N

(2, 0)

λ1/λ

(0, 2)

λ2/λ

H

λ2/λ

H

λ1/λ

2

N

(1, 1)

λ1/λ

(0, 2)

λ2/λ

E

N

1

N

(2, 0)

λ1/λ

(1, 1)

λ2/λ

E

N

(2, 0)

λ1/λ

(0, 2)

λ2/λ

H

λ1/λ

(0, 2)

λ2/λ

H

λ2/λ

Figure 3: Extensive form of the game of imperfect information with n = 2
and k = 2

Figure 3 shows the extensive form of the game of imperfect information
in its simplest form (n = 2 and k = 2). The only difference compared to
figure 1 is the appearance of the dashed lines between decision nodes. These
indicate so-called information sets: sets of decision nodes that the scientist
who has to make a decision cannot distinguish between (i.e., she must play
the same strategy at each node in the set).

14This is a technical term for a game in which players cannot distinguish certain decision
nodes. Not to be confused with a game of incomplete information, where the players may
not know each other’s preferences or possible strategies.

22



What is rational for the scientists to do in this version of the game? On
the one hand the information sets have made the problem harder, because
backwards induction can no longer be used. But on the other hand they have
also made the problem easier by reducing the number of possible strategies.
Previously, each scientist had four possible strategies: they could play either
E or H independently at either of their decision nodes. As they can no longer
distinguish between their decision nodes, conditional strategies are no longer
allowed, so each scientist only has two possible strategies: E and H.

E H

E
(
2λ1
λ
, 2λ2

λ

) (
2λ1
λ

+ λ
2
1
λ2

λ2
λ
, 2λ2

λ
− λ

2
1
λ2

λ2
λ

)
H

(
2λ1
λ
− λ1

λ

λ22
λ2
, 2λ2

λ
+ λ1

λ

λ22
λ2

) (
2λ

2
1
λ2

+ 4λ
2
1
λ2

λ2
λ
, 2λ

2
2
λ2

+ 4λ1
λ

λ22
λ2

)

Table 1: Expected credit for each scientist as a function of scientist 1’s
strategy (row) and scientist 2’s strategy (column)

Table 1 gives the expected credit for each scientist as a function of the
scientists’ choice of strategy and the values of λ1 and λ2. This game only has
one equilibrium (including mixed-strategy equilibria) regardless of the values
of λ1 and λ2: both scientists play strategy E. This can be seen by noting
that strategy E is the unique best response if the other scientist plays E as
well, and for at least one of the two scientists (possibly both, depending on
the values of λ1 and λ2)15 strategy E is also the unique best response if the
other scientist plays H.

Moreover, this is a strict equilibrium. An equilibrium is strict if, keeping
the other scientists’ strategies fixed, deviating from the equilibrium strictly

15Strategy E is the unique best response to strategy H for scientist 1 if 2λ1
λ

+ λ
2
1
λ2

λ2
λ
>

2λ
2
1
λ2

+ 4λ
2
1
λ2

λ2
λ
, which happens if and only if 2λ2 > λ1. Similarly, strategy E is the unique

best response to strategy H for scientist 2 if 2λ1 > λ2. At least one of these conditions
always holds, and both of them hold whenever 12λ2 < λ1 < 2λ2, i.e., when the values of
λ1 and λ2 are “close”.

23



decreases a scientist’s expected credit. This is a stronger requirement than
that for an equilibrium because that definition allows for cases in which a
deviating scientist is equally well off.

In the general version of the game (with n and k possibly greater than 2)
each scientist has to formulate a strategy (E or H) for each information set.
At an information set, the scientist knows which stage was the last one to
be completed and shared by some scientist, and how many stages she has
since completed herself. However, she does not know how many stages have
been completed but not shared by other scientists. As a result, the number
of possible strategies is smaller than in the game of perfect information of
section 5 (but greater than two if k > 2). It turns out that the general
version of the game also has only one equilibrium.

Theorem 2. The game with imperfect information with n ≥ 2 scientists and
k ≥ 1 stages has a unique equilibrium in which all scientists play strategy E
at every information set. Moreover, this is a strict equilibrium.

A proof of this fairly strong theorem in favor of the sharing of intermediate
results is given in appendix A.

What does this result say about what it is rational for a scientist to
do? It says that if not every scientist immediately shares any stage that she
completes, there is at least one scientist who is irrational in the sense that
she would have had a higher expected credit if she had played a different
strategy. So the only way these scientists can all be rational is if they all
share every stage. In other words, if all scientists are rational expected credit
maximizers they will all share.

7 Explaining the Persistence of the Commu-
nist Norm

Above I have shown in a game-theoretic model that it is rational for credit-
maximizing scientists subject to the priority rule to share their intermediate

24



results. I take this result to give an explanation for the persistence of the
communist norm.

The explanation runs as follows. Suppose the communist norm is in place,
i.e., scientists are sharing their intermediate results. If a given scientist devi-
ates by not sharing an intermediate result, she thereby lowers her expected
credit (this is just what it means for sharing to be a strict equilibrium). Hence
the scientist has a credit incentive to return to conforming to the norm. So
credit incentives can correct small deviations from the norm (and, since there
are no other equilibria, arguably also larger deviations).

Note that I do not claim that real scientists are rational credit-maximizers.
This is not necessary for my explanation. I have shown that rational credit-
maximizing scientists would conform to the norm. All that follows for real
scientists is that they have a credit incentive to conform to the norm. This
fact, combined with the fact that real scientists are at least somewhat sensi-
tive to credit incentives (more on this in section 8), constitutes my explana-
tion of the persistence of the norm.

Here I want to point out a number of peculiar features of my explanation
and consider some objections based on those features.

My explanation relies on only three basic principles: scientists’ sensitivity
to credit incentives, the credit-worthiness of intermediate results, and the
priority rule as the mechanism for assigning credit. These ingredients are
sufficient to explain the persistence of the norm. In particular, there is no
need for a social contract, normative expectations, or altruism.

This leads to a potential objection. On my construal, the communist
norm is not strictly a social norm in Bicchieri’s sense, as normative expec-
tations have no role in the explanation. But the sociological evidence cited
above seems to refute this: scientists do view the communist norm as a social
norm, they (normatively) expect other scientists to conform to it, and they
feel the weight of this expectation when making their own decisions (Louis
et al. 2002, Macfarlane and Cheng 2008). This appears to be at odds with
my model: since the game is zero-sum, other scientists benefit when a given

25



scientist deviates from the norm, so from a credit-maximizing perspective
they should actually be encouraging each other to deviate.16

To answer this objection I note that the model considers only those sci-
entists who are directly competing on a given research project. While those
scientists may stand to gain if their competitors fail to share their interme-
diate results, the wider scientific community stands to lose, as it will take
longer to complete the research project. It is this wider community, I argue,
that is the source of any normative expectations regarding sharing behavior.
The normative expectations can then also be explained from self-interest, as
the completion of the research project may benefit other scientist’ research.17

This yields an empirical prediction that might be used to help decide be-
tween Strevens’ explanation and mine. On Strevens’ explanation a deviation
from the communist norm is a breach of a social contract which most di-
rectly impacts the immediate competitors of the scientist within the research
project, who may legitimately regard it as unfair. On my explanation a devi-
ation actually benefits the immediate competitors; the most direct negative
impact is on those scientists who work on nearby projects. An examination
of which scientists (direct competitors or those working on nearby projects)
tend to most vocally object to deviations from the communist norm may
thus shed light on the question which of these explanations is closer to the
truth.

Because my explanation depends on the claim that it is rational for credit-
maximizing scientists to share their intermediate results, which is supported
by a game-theoretic model, the explanation depends on the generality of that
model. While my model is more general than Boyer’s in allowing an arbitrary
number of competing scientists, arbitrary differences in productivity among
the scientists, and in considering a large strategy space, still some assump-

16I thank Michael Strevens and Liam Bright for pressing me on this point.
17Alternatively or additionally, normative expectations may arise simply because ev-

eryone in the community is in fact behaving in a certain way. Bicchieri points out that
“[s]ome conventions may not involve externalities, at least initially, but they may become
so well entrenched that people start attaching value to them” (Bicchieri 2006, p. 40).

26



tions had to be made. Which of these assumptions are truly restrictive?
A number of assumptions are, perhaps surprisingly, not restrictive. The

reason is that realistic ways of relaxing these assumptions would actually
make sharing more attractive rather than less, and thus would not affect the
results obtained in theorems 1 and 2. These are: (1) the assumption that the
scientists always finish the project, (2) the assumption that scientists maxi-
mize (total) credit rather than credit per unit time, (3) the assumption that
the project can only be completed by finishing these particular intermediate
stages, and (4) the assumption that the scientists know in advance which
intermediate stages need to completed.

This leaves two crucial assumptions: exponential waiting times and equal
credit for different stages. It is possible that using a different distribution for
the waiting times would lead to a model in which the equivalent of theorems 1
and 2 does not hold. But I claim that any such deviation would actually make
the model less realistic, citing once again the empirical evidence obtained by
Huber (1998a,b, 2001) and Huber and Wagner-Döbler (2001a,b).

I have given no real defense of the assumption that each intermediate
stage has the same value (in terms of credit). In fact it seems quite realistic
that the scientist to finish the last stage (“puts it all together”) might get
more credit. And this is exactly the circumstance in which my results may
fail: if later stages are worth more credit than earlier ones, there may be an
incentive not to share.18

Here I have little to add to Boyer (2014, section 4.3.1). From a descrip-
tive perspective, this might be the kind of cases where scientists do not share
their intermediate results, and with good reason. From a normative perspec-
tive, this could be viewed as an argument against giving extra credit to the

18Boyer (2014, theorem 3) suggests (for the case where n = 2 and k = 2) that if the
second stage is worth up to twice as much credit as the first, there may still be an incentive
to share. This would indicate some fairly significant robustness of the result. However, my
own investigations suggest that this is an artifact of Boyer’s assumption that the scientists
have equal productivity: the larger the differences in productivity among the scientists
the less robust the incentive to share.

27



scientist who finishes the last stage (because the more equal the division of
credit, the more incentive scientists have to share).

Another feature of my explanation is that it explains sharing behavior
only for “intermediate results”, i.e., results that are significant enough to be
publishable in their own right. Strevens points out that on this view, “nothing
will be shared until something relevant is ready for publication, and worse, it
is only what characteristically goes into the journals that gets broadcast, so
details of experimental or computational methods and raw data will remain
hidden” (Strevens forthcoming, p. 5). This constitutes an objection to my
explanation, as according to Strevens the communist norm requires that any
and all results should be shared, regardless of their credit-worthiness.

To this worry I reply that it is not clear that the communist norm makes
such strong requirements. When the material under consideration is too little
or too detailed to be considered publishable, scientists’ actual compliance
with a putative norm of sharing drops off steeply (Louis et al. 2002, Tenopir
et al. 2011).19 If Strevens’ aim is to explain a norm of sharing for these cases,
he may be trying to explain something that does not exist.

Strevens may reply to this that regardless of the content of the norm
currently in place, it would be good to have a maximally inclusive communist
norm. After all, scientists would benefit most from each other’s work (thus
speeding up the overall progress of science) if they shared results even before
they had achieved publishable size and without hiding crucial details. By
using the framework of a social contract to point out the benefits of more
widespread sharing, Strevens could argue, it might be possible to help the
scientific community get to such an improved norm.

That would be both a fair point and a laudable goal. However, the re-
19If it is assumed that material that cannot be published in a journal is worth zero credit

when shared, then my model would of course predict that nothing would be shared. This
prediction is not borne out empirically: while there is much less sharing of this kind of
material, there is still some sharing. Perhaps this behavior is simply unexplainable from a
pure credit-maximizing perspective. However, the assumption that this material is worth
zero credit may not need to be granted. See Piwowar (2013) and the discussion below.

28



sults from my model can do the same. They suggest a clear way to make it
incentive-compatible for scientists to share work below publishable size: al-
low smaller publications. And sharing crucial details can similarly be made
incentive-compatible just by giving credit for it (Tenopir et al. 2011, Gor-
ing et al. 2014). If getting scientists to share these minor results or crucial
details is a goal that scientists and policy makers consider important, the
model gives clear directions on how to get there (but it may not be possible
or desirable to do this, cf. Boyer 2014, section 4.4). Modern information
technology readily suggests ways in which this can be done without over-
burdening existing scientific journals. Developments in this area are already
underway (Piwowar 2013). In this sense, the results from this paper are more
actionable than Strevens’.

8 Explaining the Origins of the Communist
Norm

Above I argued that the results from the game-theoretic model explain the
persistence of the communist norm. It could be argued that they also explain
the origins of the norm: the uniqueness clauses in theorems 1 and 2 guarantee
that behavior in accordance with the communist norm is the only pattern
that rational credit-maximizing scientists could settle on.

But such an argument would make stringent demands on the scientists’
rationality which real scientists are unlikely to satisfy. This section investi-
gates the question whether less than perfectly rational scientists would also
learn to share their intermediate results, thus giving a more robust account
of the origins of the communist norm.

To answer this question I consider a boundedly rational learning rule that
makes only minimal assumptions on the cognitive abilities of the scientists.
In particular, it requires only that the scientists know which strategies are
available to them and that they can compare the credit earned on the previous
round to that earned on the current round (where a “round” is one instance

29



of the game of imperfect information; to evaluate this bounded rationality
rule one needs to assume the game is played repeatedly).

The rule I consider is probe and adjust. A scientist using probe and adjust
follows the following simple procedure: on each round, play the same strategy
as the round before with probability 1 − ε, or “probe” a new strategy with
probability ε (with 0 < ε < 1; ε is usually “small”). In case of a probe, she
picks a new strategy uniformly at random from all possible strategies. After
playing this strategy for one round, the probe is evaluated: if the payoff for
the round in which she probed is higher than the payoff in the previous round,
keep the probed strategy (at least until the next probe); if the payoff is lower,
return to the old strategy; if payoffs are equal, return to the old strategy with
probability q and retain the probe with probability 1 − q (0 < q < 1).

Note that this is not quite the same as asking whether the probed strategy
is a better reply to the other scientists’ strategy than the old strategy, as
other scientists may have changed their strategy as well. In particular, if
all scientists are using probe and adjust, simultaneous probes and probes
on subsequent rounds prevent this rule from necessarily always picking the
better reply.

Consider a population of n ≥ 2 scientists using probe and adjust to de-
termine their strategy in repeated plays of the game of imperfect information
with the number of stages k ≥ 1 fixed. Assume all scientists use the same
values of ε and q (this assumption can be relaxed, see Huttegger et al. 2014,
pp. 837–838). Then the following result can be proven (see appendix A).

Theorem 3. For any probability p < 1, if the probe probability ε > 0 is small
enough there exists a T such that on an arbitrary round t with t > T, all
scientists play strategy E at every information set with probability at least p.

If, on a given round, all scientists play strategy E at every information
set, they may be said to have learned to share their intermediate results. The
theorem says that the probability of this happening can be made arbitrarily
high by choosing a small enough probe probability and a long enough waiting
time. Moreover, the theorem says that once the scientists learn to share their

30



intermediate results they continue to do so on most subsequent rounds. So
even on this very cognitively simple learning rule both the origins and the
persistence of the communist norm can be explained on the basis of credit
incentives.

Having already shown the same to be the case for highly rational scientists
in sections 5 and 6, I suggest that similar results should be expected for in-
termediate levels of rationality.20 Conforming to the communist norm is then
shown to be incentive-compatible for credit-maximizing scientists regardless
of their level of rationality.

I have suggested that credit incentives are responsible for the origins of
the communist norm. How historically plausible is this? It is not entirely
clear how one should evaluate this question. But a necessary condition for
my explanation to be the correct one is that credit for scientific work, and in
particular credit awarded in accordance with the priority rule, predates the
communist norm. The remainder of this section argues that this condition
is satisfied.

As Merton (1957) points out, scientists’ concern for priority goes back at
least as far as Galileo. In 1610, he used an anagram to report seeing Saturn
as a “triple star” (the first sighting of the rings of Saturn). The device of the
anagram served “the double purpose of establishing priority of conception
and of yet not putting rivals on to one’s original ideas, until they had been
further worked out” (Merton 1957, p. 654). If Galileo was concerned about
establishing priority for his ideas, it seems that the priority rule must already
have been in effect in 1610. Priority disputes also go back at least as far, as
Galileo wrote multiple polemics to defend his priority on various discoveries
(Galilei 1607, 1623).

The communist norm, on the other hand, was not established as a norm
of science until the 1660s, in the controversy between Boyle and Hobbes over

20Because the equilibrium in the game of imperfect information is both strict and unique,
various other learning rules and evolutionary dynamics can easily be shown to converge
to it. Examples include fictitious play, the best-response dynamics, and the replicator
dynamics.

31



the air-pump (Shapin and Schaffer 1985). Part of what was at stake in this
controversy were the norms for establishing a “matter of fact”, i.e., a scientific
fact. Boyle (who ended up “winning” the controversy) argued that

An experience, even of a rigidly controlled experimental perfor-
mance, that one man alone witnessed was not adequate to make
a matter of fact. If that experience could be extended to many,
and in principle to all men, then the result could be constituted
as a matter of fact. In this way, the matter of fact is to be seen as
both an epistemological and a social category. The foundational
item of experimental knowledge, and of what counted as properly
grounded knowledge generally, was an artifact of communication
and whatever social forms were deemed necessary to sustain and
enhance communication. (Shapin and Schaffer 1985, p. 25)

Scientific facts are attributed to the community rather than the indi-
vidual, echoing Merton’s definition of the communist norm, and this leads
directly to a call for enhanced communication, i.e., sharing. If I am right that
here the communist norm is being first established, the necessary condition
that the priority rule predates the communist norm is satisfied.

9 Conclusion
In the introduction I argued that the sharing of scientific results (mandated
by the communist norm) is important to the success of science and indeed
to the existence of science as we know it. Theorems 1, 2, and 3 show that
the priority rule gives scientists an incentive to share any and all intermedi-
ate results. These results can be used to explain both the origins and the
persistence of the communist norm, answering the questions I raised in the
introduction.

If my explanation is accepted, the crucial features of the social struc-
ture of science that maintain the communist norm are seen to be the fact

32



that scientists respond to credit incentives, the priority rule, and the credit-
worthiness of intermediate results. Tinkering with these features thus risks
undercutting one of the most central aspects of science as a social enterprise.

By emphasizing credit incentives moderated by the priority rule, this
paper falls in the tradition of Kitcher (1990), Dasgupta and David (1994),
and Strevens (2003). Like those papers, I have picked one aspect of the social
structure of science, and shown how the priority rule has the power to shape
that aspect to science’s benefit.

I take my results to show that no special explanation (using, e.g., nor-
mative expectations and/or a social contract) is required for the communist
norm, contra Strevens (forthcoming). However, this only applies to whatever
is publishable (or otherwise credit-worthy) in a given scientific community.
Sharing scientific work that is too insignificant to be published is not incen-
tivized in the same way. But insofar as this is a problem it suggests its own
solution: give credit in accordance with the priority rule for whatever one
would like to see shared, and scientists will indeed start sharing it.

A A Unique Nash Equilibrium
Let n ≥ 2 be the number of scientists and k ≥ 1 the number of stages. Let
G
p
n,k denote the game with perfect information described in section 5 and let

Gmn,k denote the game with imperfect information described in section 6.
As is commonly done in game theory, I use ui(si,s−i) to denote the payoff

(expected units of credit at the end of the game) to scientist i if she plays
strategy si and s−i gives the strategies of all scientists other than scientist i
(call this an “incomplete strategy profile”).

One strategy is of particular interest. Let sEi denote the strategy for
scientist i in which she plays E (that is, shares and claims credit for her
most recently completed stage) at every decision node in Gpn,k or at every
information set in Gmn,k (so technically, sEi denotes two strategies, one for each
game, but they share a lot of features which I use below). Let sE−i denote

33



the incomplete strategy profile (in either game) where every scientist i′ other
than scientist i plays strategy sEi′ . Let SE denote the strategy profile (in
either game) in which every scientist i plays strategy sEi .

Lemma 4. In both Gpn,k and Gmn,k, for any scientist i, the payoff when every
scientist always shares any stages she completes immediately is

ui
(
sEi ,s

E
−i

)
= k

λi
λ
.

Proof. Scientist i is the first to complete stage 1 with probability λi/λ. If
she does she immediately claims one unit of credit. If any other scientist
completes stage 1 before scientist i, that scientist immediately claims one
unit of credit. Thus scientist i’s expected credit from the first stage is λi/λ.
Then all scientists simultaneously start working on the next stage. So by the
same reasoning, scientist i’s expected credit from any given stage is λi/λ.
The result follows.

Lemma 5. In both Gpn,k and Gmn,k, if scientist i plays strategy sEi but not every
other scientist always shares any stages she completes, the payoff to scientist i
is strictly higher than the payoff given in lemma 4. More precisely: let s−i
denote any incomplete strategy profile such that at least one scientist i′ plays
some strategy other than sEi′ (this can be either a different pure strategy, or
any mixed strategy which plays strategy sEi′ with probability less than one).
In the case of Gpn,k, add the further assumption that this involves a deviation
on the equilibrium path, i.e., there is at least one scientist i′ who plays a
strategy si′ (or a mixed strategy in which si′ is played with positive probability)
such that if every other scientist i′′ plays strategy sEi′′ then there is a positive
probability of reaching a decision node at which strategy si′ plays strategy H.
Then

ui
(
sEi ,s−i

)
> k

λi
λ
.

Proof. Note that in the case described by lemma 4, i.e., when the strategy
profile SE is being played, the outcome of a single instance of the game can

34



be described by a sequence (i1, i2, . . . , ik), where the first member denotes
the first scientist who completes a stage, the second member the second
scientist to complete a stage (not necessarily a different scientist than the
first), and so on. Because every scientist i plays strategy sEi , each member of
the sequence also denotes the claiming of one unit of credit by that scientist.
The probability of such a sequence describing the outcome of the game is

λi1
λ
·
λi2
λ
· · ·

λik
λ
.

Now suppose that there is at least one scientist i′ playing a strategy different
from sEi′ . Let si′ 6= sEi′ be some strategy that scientist i′ plays with some
positive probability p (where p = 1 if scientist i′ plays a pure strategy),
and assume that si′ involves a deviation on the equilibrium path in the case
of Gpn,k.

A sequence like (i1, i2, . . . , ik) can still be used to describe the first k
scientists to complete a stage, but because not everyone always claims credit,
this may not completely describe the outcome of the game: if a scientist
completed a stage but did not claim credit for it either immediately or later,
it is possible that not all k units of credit have been claimed after k scientists
have completed a stage.

However, regardless of whether credit is being claimed, the probability of
the sequence remains unchanged due to the memorylessness property of the
exponential distribution. Moreover, because scientist i plays strategy sEi , she
is still claiming a unit of credit whenever she occurs in the sequence. Thus,
all possible sequences (i1, i2, . . . , ik) still occur with the same probability,
and scientist i claims the same amount of credit in them. So scientist i now
expects to accrue kλi/λ units of credit during the time it takes for k scientists
to complete a stage.

But, by assumption, there is at least one sequence (i1, i2, . . . , ik) in which
i′ occurs and (with probability p) plays strategy H at the corresponding
decision node or information set, and i′ does not occur in the remainder of
that sequence. As a result, at the end of that sequence at most k−1 units of

35



credit have been claimed. In the remainder of that game, there is a positive
probability (at least λi/λ, the probability that she is the very next one to
complete a stage) that scientist i gains more credit, credit that she would not
have obtained if scientist i′ had played strategy sEi′ . Since p > 0 and λi′′ > 0
for all i′′, it follows that

ui
(
sEi ,s−i

)
≥ k

λi
λ

+
λi1
λ
·
λi2
λ
· · ·

λik
λ
·p ·

λi
λ
> k

λi
λ
.

Theorem 6. Let S be any strategy profile for Gmn,k other than SE, or let S
be any strategy profile for Gpn,k that involves deviations on the equilibrium
path relative to SE. Then there exists at least one scientist i playing strategy
si 6= sEi such that she would be strictly better off playing strategy sEi :

ui
(
sEi ,s−i

)
> ui (si,s−i) .

Proof. Note that the game is zero-sum: regardless of strategies, there are k
units of credit to be divided, and so if one scientist increases her payoff, that
of another must decrease. This fact, combined with lemmas 4 and 5, yields
the theorem. Distinguish three cases:

1. There is only one scientist i playing a (pure or mixed) strategy si 6= sEi .
Then every scientist i′ other than scientist i is playing strategy sEi′ and
so by by lemma 5 is getting a payoff greater than kλi′/λ. Because the
game is zero-sum, it follows that ui(si,s−i) < kλi/λ. By lemma 4,
ui(sEi ,s−i) = kλi/λ, and the result follows.

2. There is at least one scientist i′ playing strategy sEi′ and at least two
scientists playing some other strategy. Then any scientist i′ who is
playing strategy sEi′ is getting a payoff greater than kλi′/λ by lemma 5.
Because the game is zero-sum, at least one of the remaining scientists,
say scientist i, must be getting a payoff less than kλi/λ. But if scientist i
changed her strategy to sEi , by lemma 5 she would get a payoff greater
than kλi/λ. So ui(sEi ,s−i) > kλi/λ > ui(si,s−i).

36



3. Every scientist i′ is playing some strategy si′ 6= sEi′ . Because the game
is zero-sum, it is impossible for every scientist i′ to be getting a greater
payoff than kλi′/λ. So there is at least one scientist, say scientist i,
such that ui(si,s−i) ≤ kλi/λ. By lemma 5, ui(sEi ,s−i) > kλi/λ, and
the result follows.

Theorem 6 plays an important role in the proofs of the results in the main
text.

Proof of theorem 1. Consider the game Gpn,k. In any profile (of pure or mixed
strategies) at least one scientist has an incentive to change her strategy,
unless every scientist i plays strategy sEi or a strategy that deviates from
sEi only off the equilibrium path. Thus no profile is a Nash equilibrium
unless every scientist i plays strategy sEi or a strategy that deviates from sEi
only off the equilibrium path. But since the backwards induction solution
is a Nash equilibrium, it follows that in the backwards induction solution
(which is guaranteed to exist for any finite game of perfect information)
every scientist i must play strategy sEi or a strategy that deviates from sEi
only off the equilibrium path. So in the backwards induction solution every
scientist immediately shares and claims credit for any stage she completes.

A direct proof using backwards induction is also possible. This proof
yields the slightly stronger result that in the backwards induction solution
every scientist plays strategy E at every decision node (including those off
the equilibrium path) and is available from the author upon request.

Proof of theorem 2. Let S be any profile (of pure or mixed strategies) for the
game Gmn,k. If S 6= SE, then at least one scientist has an incentive to change
her strategy, and so S is not a Nash equilibrium. Thus there is at most one
Nash equilibrium: SE.

That SE is indeed a Nash equilibrium, and in fact a strict Nash equi-
librium, also follows from theorem 6 by considering the special case where
s−i = sE−i. This shows that a scientist i who deviates unilaterally makes
herself strictly worse off.

37



To prove theorem 3, some terminology and a result from Huttegger et al.
(2014) are needed. Define a weakly better reply path to be a sequence of
profiles (S1, . . . ,S`) such that for any j < `, profile Sj differs from profile Sj+1

only in one scientist’s strategy, say scientist i (so sj−i = s
j+1
−i ), and ui(Sj+1) ≥

ui(Sj), i.e., scientist i changes to a strategy that is a (weakly) better reply
to the other scientists’ strategies. Define a weakly better reply game to be
a game in which for every profile S there exists a weakly better reply path
from S to a strict Nash equilibrium.

Let G be a weakly better reply game with n scientists. Assume the
scientists play G repeatedly, adjusting their strategy using probe and adjust
and using the same values of ε and q. Let St be the profile of strategies
played on round t.

Theorem 7 (Huttegger et al. (2014)). For any probability p < 1, if the probe
rate ε > 0 is sufficiently small, then the profile St is a strict Nash equilibrium
of G for all sufficiently large t with probability at least p.

Theorem 3 is a corollary of theorems 2, 6 and 7.

Proof of theorem 3. By theorem 2, the strategy profile in which every scien-
tist plays strategy E at every information set is the only strict Nash equilib-
rium of the game. If Gmn,k is a weakly better reply game, the desired result
follows from theorem 7.

That the game is a weakly better reply game follows straightforwardly
from theorem 6. At any strategy profile, for at least one scientist i whose
strategy differs from sEi switching to strategy sEi is a better reply for her.
This switch leads to a profile which is either the strict Nash equilibrium or
in which the same is true for some other scientist. The result is a path of
length at most n from any profile to the strict Nash equilibrium, in which
at each step along the path one scientist i switches her strategy to sEi , and
improves her payoff by doing so.

38



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	Introduction
	Social Norms and Communism
	Communism and Intermediate Results
	A General Game-Theoretic Model of Intermediate Results
	A Backwards Induction Analysis
	A Game of Imperfect Information
	Explaining the Persistence of the Communist Norm
	Explaining the Origins of the Communist Norm
	Conclusion
	A Unique Nash Equilibrium
	References