Decision Theoretic Model of the Productivity Gap

Abstract

Using a decision theoretic model of scientists’ time allocation between potential research projects I explain the fact that on average women scientists publish less research papers than men scientists. If scientists are incentivised to publish as many papers as possible, then it is necessary and sufficient for a productivity gap to arise that women scientists anticipate harsher treatment of their manuscripts than men scientists anticipate for their manuscripts. I present evidence that women do expect harsher treatment and that scientists’ are incentivised to publish as many papers as possible, and discuss some epistemological consequences of this conjecture.

Appendix: Proofs

To explain the formal model underlying the above claims, it is necessary to introduce some terminology. An agent p is a pair of two functions $<G_{p}$, $C_{p}{>}$, G: [0,1] $\rightarrow \{0,1\}$ and C: [0,1] $\rightarrow [0, \infty ]$. $\hbox {G}_{p}$ tracks the minimal amount of effort p thinks they have to put into a project to get it published, and $\hbox {C}_{p}$ specifies how much credit they expect to receive from a project given how much effort they have put into it, conditional on it being published. Each agent is faced with the following choice scenario. They have a fixed budget of time to allocate as effort spent on projects, and may distribute this effort between k options $\{\mathcal {I}_{1}\ldots {\mathcal {I}}_{k}\}$. The set of options is called their associated idea set. Once chosen how to allocate their efforts a vector of length k is formed $<\hbox {x}_{1}\ldots \hbox {x}_{k}>$ where $\hbox {x}_{j}$ is the element of [0,1] allocated to ${\mathcal {I}}_{j}$, with the researcher’s time budget to allocate being 1. I call this vector the agent’s research profile (henceforth abbreviated to RP). The function #(RP) outputs the set of all $\hbox {x}\in \hbox {RP}$ s.t. $\hbox {G}_{p}(\hbox {x})=1$. This is the set of projects the researcher believes will result in published papers. I refer to the numbers which are elements of RP by the variables x, y, z, and the index of the ideas they are allocated to by the variables i, j, k. For any $\hbox {x}\in$RP it accrues the credit generated by the composite function $\hbox {G}_{p}(\hbox {x})\hbox {C}_{p}$(x); which is to say however much credit $\hbox {C}_{p}$ gives x providing G$_{p}$(x) is 1, and no credit otherwise. Let $\cup _{p}(\hbox {RP}) = \sum _{1}^{k}\hbox {G}_{p}(\hbox {x}_{k})\hbox {C}_{p}(\hbox {x}_{k}$). If $\cup _{p}(\hbox {RP}) >\cup _{p}$(RP′) then say that $\hbox {RP}>{\hbox {RP}}^{\prime}$. A parameterisation of the model consists of specifying the number of agents, the cardinality of their associated idea sets, and each agent’s G and C functions.

The three assumptions from Sect. 3 can now be stated formally.

Axiom 1

(Analytic Egalitarianism) In any parameterisation of the model, all agents are associated with the same cardinality idea sets.

Axiom 2

(Idea Homogeneity) All ideas have the same potential to generate credit. This can be broken into two parts

• a. Agents believe all ideas require the same amount of time allocated to them in order to be published. I.e. For all agents $\exists \hbox {x}\in$[0.1] s.t. $\forall \hbox {i }\in$ $[{\mathcal {I}}_{1}\ldots {\mathcal {I}}_{k}]$, $\forall \hbox {y }\in$ [0,1] $((\hbox {y}\ge \hbox {x} \rightarrow \hbox {G}(\hbox {y}_{i})=1)$ & $(\hbox {y}<\hbox {x} \rightarrow \hbox {G}(\hbox {y}_{i})=0))$

• b. For any two ideas with differing amount of effort allocated to them, the idea that has more time allocated to it generates more credit. $\forall \hbox {i}\forall \hbox {j }\in [{\mathcal {I}}_{1}\ldots {\mathcal {I}}_{k}]\,\forall \hbox {y }\in$ [0,1] $(\hbox {x}>\hbox {y} \rightarrow \hbox {C}(\hbox {x}_{i})>\hbox {C}(\hbox {y}_{j}$))

Axiom 3

(Credit Maximisation) Agents which to accrue as much credit to themselves as possible. I.e. Agents select an RP so as to maximise the value of $\cup$(RP).

Let $\hbox {RP}^{+}$ be the set of all top ranked elements of the agent’s choice set according to the preference ranking induced by Axiom 3. Let $\#^{max}$ be the set of highest cardinality sets generated by # when applied to all members of $\hbox {RP}^{+}$. That is to say, it is the set of all sets of papers published in agents’ most preferred research papers that have the most publications. Call the set elements of $\hbox {RP}^{+}$ that generate members of $\#^{max}\;\hbox {RP}^{max}$. Let $\#^{+}$ be the set of all sets generated by # when applied to all members of $\hbox {RP}^{+}$. I now characterise a productivity gap between agents m and w as occurring when one of the following sentence is made true by the parameterised model:

• Productivity Gap: $\exists \hbox {x} {\in}\#_{w}^{max}\,\forall \hbox {y} {\in} \#_{m}^{+}(|\hbox {x}|<|\hbox {y}|)$

In English, this says that a publication gap occurs when agent w’s top ranked research profiles with the most papers published contain less publications than any of the agent m’s top ranked research profiles.

Lemma 1

No agent would choose a research profile RP such that # $(RP) = \emptyset$, i.e. an agent will never choose to distribute their effort in a way that leaves them with no publications.

Proof

Suppose agent p chose a research profile RP which induced no publications, i.e. $\lnot \exists \hbox {x}\in \hbox {RP}$ s.t. $G_{p}(\hbox {x})=1$. Consider the research profile RP′ such that an element of p’s associated idea set, i, was allocated all their effort. Note that if RP is in p’s choice set then RP′ will be and that it follows from axiom 2a that $\hbox {G}_{p}(1_{i})=1$. Note that $\cup _{p}(\hbox {RP}) = 0$, whereas it follows from axiom 2b and the fact that $\hbox {C}_{p}$ is bounded above 0 by definition that $\hbox {C}_{p}(1_{i}) > 0$, and therefore that that $\cup _{p}$(RP′) $> 0$. Hence by axiom 3 p would never choose RP over RP′, and #(RP′) is not empty. $\square$

Lemma 2

No agent would choose a research profile RP that did not satisfy $\sum _{i=0}^{k}{x}_{k}\in {RP} = 1$, i.e. an agent would never leave some effort unallocated.

Proof

Note that the nature of the choice scenario ensures that $\sum _{i}^{k}\hbox {x}\in \hbox {RP }{\not {>}} 1$. Hence either $\sum _{i}^{k}\hbox {x}\in \hbox {RP }< 1$ or $\sum _{i}^{k}\hbox {x}\in \hbox {RP} = 1$. Suppose $\sum _{i}^{k}\hbox {x}\in \hbox {RP }< 1$. Let $\sum _{i}^{k}\hbox {x}\in \hbox {RP} =$ y and 1 − y = z. Note that by lemma 1 #(RP)$\not = \emptyset$. Now consider the alternate profile RP′ which is identical to RP except idea i has x + z effort allocated to it. Note that by axiom 2b C(x + z) > C(x) for any positive number z. By construction z is a positive number, hence $\cup (\hbox {RP}^{\prime}) >\cup (\hbox {RP})$. Hence by axiom 3 the agent would never choose RP over RP′. Hence $\sum _{i}^{k}\hbox {x}\in \hbox {RP} = 1$. $\square$

Lemma 3

No agent would choose a research profile RP such that $\exists {x}\in {RP}({x}>0 \& {G}({x})=0)$, i.e. an agent would never allocate effort to a project if they did not think that level of effort will result in a publication.

Proof

Note that since credit is allocated by the function $\hbox {G}_{p}(\hbox {x})\hbox {C}_{p}$(x) if $\hbox {G}_{p}(\hbox {x}_{i}) = 0$ then the agent gains no credit from idea i. Suppose $\exists \hbox {x}\in \hbox {RP}(\hbox {x}>0 \& \hbox {G}(\hbox {x}_{i})=0)$. By lemma 1 there exists an idea j in RP that has some amount of effort y$_{j}$ allocated to it such that $\hbox {G}(\hbox {y}_{j}) = 1$. Consider the alternate profile RP′ which is identical to RP except that j has y + x effort allocated to it. Note that by axiom 2b $\hbox {C}(y + x) > \hbox {C}(y)$ where x is a positive number. Hence if $x>0$ then $\cup$(RP′) $> \cup$(RP). Hence by axiom 3 an agent would never choose RP over RP′. $\square$

Lemma 4

If ${RP}>{RP'}$ and ${RP}^{\star }$ is a permutation of the elements of RP, then ${RP}^{\star }>{RP}^{\prime}$, i.e. if research profile A is preferred to research profile B, then research profile C that results from permuting the elements of A will also be preferred to B.

Proof

Note that the preference ordering over research profiles is formed by summing the credit generated by each element of the research profile. Note further that, since the credit function takes as input just a number representing the time allocated to an idea rather than that number indexed to a particular idea, the same amount of effort allocated to any two ideas will result in the same amount of credit allocated. Hence simply relabelling the ideas the effort is allocated to could never generate a change in the preference ordering. $\square$

Theorem 1

(Characterisation Theorem) $\exists x \in \#_{w}^{max}\; \forall y\; \in \#_{m}^{+}(|x|<|y|) \iff \; \exists \;RP\in RP_{w}^{max}\; \exists RP^{\star }\in RP_{m}^{+}\exists x_{i} \forall y\in RP\; [(y_{j}\not =x_{i} \rightarrow (y_{j}> 0 \rightarrow u_{j}^{\star } > 0))$ & $(x_{i}=0 \& z_{i}^{\star } > 0)]$, i.e. a productivity gap occurs if and only if one of the women’s most preferred research profiles which generates an element of # $^{max}_{w}$ has more ideas allocated 0 effort in it than one of the man’s most preferred research profiles.

A consequence of this characterisation theorem is that it suffices to tell whether a productivity gap will occur to simply count the number of 0s in an element of $\hbox {RP}_{w}^{max}$ and $\hbox {RP}_{m}^{+}$ respectively. The significance of this is that it shows that preference orderings in the model suffice to capture the occurrence of productivity gaps, and, as mentioned in Sect. 3, gives the model a greater generality than just representing my conjecture, since so long as one can calculate preference orderings over research profiles, there is a simple procedure for telling whether or not a productivity gap is predicted by a scientific time allocation model.

Proof From Left to Right: Informally the proof strategy will go as follows. An element of w’s most preferred research profiles which generates an element of #$^{max}_{w}$ that satisfies productivity gap (i.e. the antecedent) will be selected. It will then be shown that one can take an arbitrary element of m’s most preferred research profiles and, by means of permuting its elements, construct a research profile which demonstrably has at least one more non-0 element than the previously selected member of w’s most preferred research profiles. This, then, satisfies the consequent.

In formal detail, let W be a member of $\hbox {RP}_{w}^{+}$ that generates some element of $\#_{w}^{max}$, and let this be the witness for the existentially quantified statement in the antecedent. Take an arbitrary element element of $\hbox {RP}_{m}^{+}$ and call it M. Generate $\hbox {M}^{\star }$ as follows. For each $\hbox {i}^{W}\in$W, if $\hbox {i}^{W}$ is allocated some $\hbox {x}>0$ and i$^{M}$ is also allocated $\hbox {x}>0$ then $\hbox {i}^{M^{\star }}$ is allocated the same amount of effort as $\hbox {i}^{M}$. Whereas if $\hbox {i}^{W}$ is allocated some $\hbox {x}>0$ and $i^{M}$ is allocated 0 effort then find a $\hbox {j}^{M}$ such that $\hbox {j}^{M}$ is allocated some $\hbox {x}>0$, $\hbox {j}^{W}$ is allocated 0, and $\hbox {j}^{M}$ has not been used in a previous iteration of this process. Let $\hbox {i}^{M^{\star }} = j^{M}$, and $\hbox {j}^{M^{\star }} = 0$. Lemma 3 entails that if an element i of M or W has non 0 effort allocated then $\hbox {G}(\hbox {y}_{i}) = 1$; hence, since by the antecedent $|\#(\hbox {W})| < |\#(\hbox {M})|$, one will never run out of such j’s necessary for this constructive process. If $\hbox {i}^{W}$ is allocated 0 effort then $\hbox {i}^{M^{\star }} = \hbox {i}^{M}$. Note that by construction $\hbox {M}^{\star }$ is such that it is non-0 wherever W is non-0, and contains at least one element which is non-0 where W is 0. Now I need to show that $\hbox {M}^{\star }\in \hbox {RP}_{m}^{+}$, which is to say that $\hbox {M}^{\star }$ is amongst m’s top ranked research profiles. It follows from lemma 4 and the method of constructing $\hbox {M}^{\star }$ that $h^{\star }$ must be preferred to every RP that M was preferred to. Hence the relationship between $\hbox {M}^{\star }$ and W witnesses the consequent.

Proof From Right to Left: Call the $\hbox {RP}\in \hbox {RP}_{w}^{max}$ which witnesses the antecedent W, and the call the $\hbox {RP}\in \hbox {RP}_{m}^{+}$ which witnesses the antecedent M. Want to show that $|\#(\hbox {W})| < |\#(\hbox {M})|$. Note that by construction M has at least one more non-0 element than W. By lemma 3 if an element i of M or W has non 0 effort allocated then $G(y_{i}) = 1$. Hence #(M) has at least one more element than #(W).

Suppose the minimum amount of effort necessary to render a paper publishable according to the representative woman scientist’s G function is g. Let w be the largest integer such that $wg\le$1. I use m to represent the equivalent integer for the representative man scientist’s possible publications given their G function. Such integer’s are the representative scientists’ max. I refer to the cardinality of the idea sets the agents are working with by “n”.

Lemma 5

If an agent’s credit function is subadditive then for any $RP^{\star }\in RP^{+}$ the cardinality of $\#(RP^{\star }$) is whichever is lower out of n or the agents max, i.e. an agent with a subadditive credit function will publish as many papers as they can.

Proof

A subadditive credit function satisfies $\hbox {C}(x+y) < \hbox {C}(x) + \hbox {C}(y$). This can be interpreted as the agent expecting to be better rewarded for producing two minimally publishable units than producing one paper with twice as much effort put in. Let RP be a research profile such that a rational agent with G function equal to g has allocated k papers effort, which given lemma 3 is to say that there are k papers allocated at least g effort. By lemma 1 $k>$0. By lemma 2 the agent has distributed all their effort between these projects. I will show that RP is an element of $\hbox {RP}^{+}$, only if the cardinality of #(RP) is equal to n or the agents max. If $|\#(\hbox {RP})|=k=n$ then there does not exist a research profile with more papers published. Any candidate $\hbox {RP}^{*}$ that might be preferred to RP will therefore either have less than k papers allocated effort or will also have k papers allocated effort. I need only consider cases where the number of papers allocated effort in $\hbox {RP}^{*}$ is less than k. Consider a research profile RP$^{*}$ such that $|\#(\hbox {RP}^{*})| = |\#(\hbox {RP})|-$1. Given lemma 2, the agent would have to have redistributed effort from one element of RP among the $k-$1 non-0 elements of $\hbox {RP}^{*}$. Due to the nature of their credit function and given axiom 3, the agent would prefer to distribute the same amount of effort allocated to j papers among k papers, for any $j<k$, assuming that their G function permits them all to be published. By hypothesis the agent can allocate k papers at least g effort. Hence they prefer to publish k papers to $k-$1 papers. Hence the agent prefers RP to $\hbox {RP}^{*}$. The same reasoning would result in any paper with less publications than RP always being preferred to a paper with at least one more, hence for any $\hbox {RP}^{**}$ with less papers allocated effort than RP will always be dispreferred to RP by the transitivity of preference.

Suppose that $|\#(\hbox {RP})|=k<n$. Note that k cannot be greater than the agent’s max, since the agent cannot allocate at least g to more papers than their max since they only have 1 effort to distribute. Hence k must either be less than or equal to the agent’s max. If it is equal they cannot produce any more papers, and by the same reasoning as in the previous paragraph will prefer RP to any $\hbox {RP}^{*}$ with less papers allocated effort. If k is less then their max then by the definition of the max there exists an $\hbox {RP}^{!}$ such that $\hbox {RP}^{!}$ has more papers allocated g effort than RP. Once again the same reasoning as in the previous paragraph would show that the agent prefers $\hbox {RP}^{!}$ to RP. This did not depend on the value of k in particular, hence this generalises to any research profile that induces a slam dunk set with a cardinality less than the max or n. This covers all cases, and hence $\hbox {RP}\in \hbox {RP}^{+}$ only if $|\#(\hbox {RP})|$ is equal to the least of the agent’s max or n.

Lemma 6

$m>w$ if and only if $wG_{m}+G_{m}\le 1$, i.e. the representative man scientist’s max is greater than the representative woman scientist’s max if and only if the representative man scientist could allocate $G_{m}$ between w papers and still have at least $G_{m}$ effort left to allocate.

Proof

Recall that the definition of the max for agent J is defined as the largest integer, j, such that $j\hbox {G}_{j}\le 1$. From left to right, note that if $m>w$ then given the definition of maxes $w\hbox {G}_{m}<\hbox {mG}_{m}\le 1$. Given that both agents’ maxes must be integers, $m=w+k$ where $k\ge 1$. From these facts it follows that. $w\hbox {G}_{m}+\hbox {G}_{m}\le 1$. From right to left, suppose $m \le w$. Suppose $m=w$. Note that from this and the definition of maxes it follows that $w\hbox {G}_{m}+\hbox {G}_{m}=\hbox {mG}_{m}+\hbox {G}_{m}>$1. But this contradicts the initial assumption that $w\hbox {G}_{m}+\hbox {G}_{m}\le$1. Suppose $m<w$. From this and the definition of maxes it would follow that $1<m\hbox {G}_{m}+\hbox {G}_{m}<w\hbox {G}_{m}+\hbox {G}_{m}$. But $w\hbox {G}_{m}+\hbox {G}_{m}\le 1$. This exhausts the cases, hence $m>w$. $\square$

Theorem 2

Let the cardinality of both agents’ idea sets be n and suppose that both agents have subadditive credit functions. Then $\exists x \in \#_{w}^{max}\;\forall y \in \#_{m}^{+}(|x|<|y|) \iff \;wG_{m}+G_{m}\le 1$ and the representative woman scientist’s max is less than n, i.e. if both agents have subadditive credit functions then a productivity gap between the man and the woman representative scientists occurs when the man thinks they could produce to the woman’s max and then produce at least one more paper, and the woman does not think it possible for her to allocate her time in a way that will result in all of the ideas in her idea set being published.

Proof of Theorem 2

From right to left. By lemma 5 any element of $\hbox {RP}_{m}^{+}$ will have the least of either m or n papers assigned at least G$_{m}$ effort. Likewise $\hbox {RP}_{w}^{+}$’s elements will have the least of w or n elements assigned at least G$_{w}$ effort. By the antecedent we hence know that any element $\hbox {RP}_{w} \in$ $\hbox {RP}_{w}^{+}$ will be such that $|\#(\hbox {RP}_{w})| = w$. If $\hbox {RP}_{m}^{+}$ has n papers assigned at least $\hbox {G}_{m}$ effort then it will have a greater number of non-zero elements than $\hbox {RP}_{w}^{+}$. Suppose $\hbox {RP}_{m}\in \hbox {RP}_{m}^{+}$ has $m<n$ elements allocated non-zero effort. By the antecedent and lemma 6 we have $m>w$. Hence $\hbox {RP}_{m}^{+}$ has m papers assigned at least $g_{m}$ effort and hence has a greater number of non-zero elements than any element of $\hbox {RP}_{w}^{+}$. This covers all cases, and hence we know that any element of $\hbox {RP}_{m}^{+}$ has more non zero elements than any element of $\hbox {RP}_{w}^{+}$. Hence $\exists \hbox {x}{\in}\#_{w}^{max}$ $\forall \hbox {y}{\in}\#_{m}^{+}(|\hbox {x}|<|\hbox {y}|)$.

Going from left to right, suppose a productivity gap has occurred. By lemma 5 we know that if she could have produced n papers the representative woman scientist would have, but if she had done so then, given axiom 1, no strong productivity gap could have occurred by definition of a productivity gap. Hence the representative woman scientists’ max (w) is less than n. Want to show that $w\hbox {G}_{m}+\hbox {G}_{m}\le 1$. Suppose the representative man scientist had produced n papers. This would entail that largest integer m such that $\hbox {mG}_{m}\le$1 is greater than or equal to n. Whereas we already know that the representative woman scientist’s max is less than n. Hence $m>w$, and by lemma 6 $w\hbox {G}_{m}+\hbox {G}_{m}\le$1. Suppose the representative man scientist had produced $m<n$ papers. By the antecedent we know that there is a productivity gap between both agents, hence $\exists \hbox {x} \in \#_{w}^{max}\;\forall \hbox {y} \in \#_{m}^{+}(|\hbox {x}|<|y|)$. By lemma 5 all of the representative woman scientist’s most preferred research profiles will have w elements assigned non-zero effort, hence every element in $\#_{w}^{max}$ will have the same cardinality and hence every element of $\#_{w}^{max}$ will be such that it has a lower cardinality than every element of $\#_{m}^{max}$. By lemma 5 again the cardinality of any element of $\#_{w}^{max}$ is w and the cardinality of any element of SD$_{m}^{max}$ is m. Hence $m>w$, and by lemma 6 $w\hbox {G}_{m}+\hbox {G}_{m}\le$1. By lemma 5 this exhausts the possible cases, therefore $w\hbox {G}_{m}+\hbox {G}_{m}$.

Before concluding, as mentioned in Sect. 3 I consider an example of relaxing Idea Homogeneity, particularly axiom 2a, and modifying the manner in which C and G functions work. Two agents M and W idea sets are divided into high and low type ideas: ${\mathcal {I}}_{a} = {\mathcal {I}}_{h}\cup {\mathcal {I}}_{l}$, where ${\mathcal {I}}_{h}\cap {\mathcal {I}}_{l}=\emptyset$. The difference in effort/reward status of those subsets can be represented by typed C and G functions, with separate functions for elements of ${\mathcal {I}}_{h}$ and ${\mathcal {I}}_{l}$ respectively. Let both agents have identical C and G functions, as follows: $\hbox {C}_{h}(\hbox {x})=\hbox {x}^{2} + 2, \hbox {C}_{l}(\hbox {x})=\hbox {x}+0.2$, $\hbox {G}_{h}=0.5, \hbox {G}_{l}(\hbox {x})=\epsilon$. The respective G and C functions are applied according to whether the index of the element of the research profile is of a high or low type idea. Now suppose, finally, that both agents are associated with idea sets ${\mathcal {I}}_{a}$, $|{\mathcal {I}}_{a}|=4$. But agent M has $|{\mathcal {I}}_{Mh}|=1$, $|{\mathcal {I}}_{Ml}|=3$, whereas agent W has $|{\mathcal {I}}_{Wh}|=2$, $|{\mathcal {I}}_{Wl}|=2$. M maximises by investing .5 into the high type idea, earning them 2.25 credit, and distributing the rest of their time among low type ideas, earning them 1.1 credit. This earns M credit of 3.35 with 4 papers published. W maximises by investing all their effort into high type ideas, earning credit of 4.5 with 2 papers published. This modified model hence predicts a productivity gap between M and W in this scenario, even though axioms 1, 2b, and 3 are all satisfied, and the agents have identical C and G functions. This suggests that future research may fruitfully focus on relaxations of Idea Homogeneity.