On fraud

Abstract

Preferably scientific investigations would promote true rather than false beliefs. The phenomenon of fraud represents a standing challenge to this veritistic ideal. When scientists publish fraudulent results they knowingly enter falsehoods into the information stream of science. Recognition of this challenge has prompted calls for scientists to more consciously adopt the veritistic ideal in their own work. In this paper I argue against such promotion of the veritistic ideal. It turns out that a sincere desire on the part of scientists to see the truth propagated may well promote more fraud rather than less.

Appendix: Proofs

A publication market model consists of a question, consisting of a set of possible answers {$a_{1} \ldots a_{n}$}, some scholars {$v_{1} \ldots v_{m}$}, and Nature. Nature first selects an element of the question set, and then sends a signal to each element of the scholar set. Each signal is paired with one (and only one) answer, such that if scholar $v_{j}$ receives signal $s_{i}$ this raises the $v_{j}$ credence in answer $a_{i}$ without effecting $v_{j}$’s judgements of relative credence between $a_{j}$ and $a_{k}$ (where $a_{j}\not =a_{i}$ and $a_{k}\not =a_{i}$). Call the element of the question set selected by Nature the truth, and the signal scholar $v_{j}$ receives the evidence proposition received by scholar $v_{j}$. I call a specification of the question, the scholar set, and the decisions made by Nature a configuration of the model.

Each scholar chooses an answer to offer to their community. The election scholars take part in can be represented by a function from the profile of answers scholars offer to a winning answer. For ease of representation’s sake I assume there is a unique winner, nothing will turn upon this assumption. Each scholar taking part in the election knows how the election function operates. Finally, the election function satisfies: if $a_{1}$ is victorious from profile $\mathcal {P_{A}}$ and $\mathcal {P_{B}}$ differs from $\mathcal {P_{A}}$ only in the fact that all votes besides $a_{1}$ have equal or less votes in $\mathcal {P_{B}}$ as in $\mathcal {P_{A}}$ and $a_{1}$ has more votes in $\mathcal {P_{B}}$ than in $\mathcal {P_{A}}$, then $a_{1}$ is victorious in $\mathcal {P_{B}}$.

Each scholar has a utility type, as per Sect. 2. I deal with three types in this paper. First, there is the pure credit seeker, who gains payout of 1 if the evidence proposition they announce is the winner of the election, and payout of zero otherwise. Second, there is the pure truth seeker, who gains payout of 1 if the winner of the election is the proposition Nature selected, and payout of 0 otherwise. Third, there is the mixed credit/truth seeker, who gains payout of 1 if they announce the proposition that wins the election and the winner of the election is the answer Nature selected, and 0 otherwise.

Each scholar has a credence function. This is a probability distribution over all elements of a relativised state space, defined as follows. A relativised state space for scholar $v_{j}$ is a set of pairs $\langle p, a \rangle$ where p is the voting profile of every scholar except for $v_{j}$, and a is the answer chosen by Nature. So, for instance, if there are two scholars $v_{X}$ and $v_{Y}$ and two answers $a_{1}$ and $a_{2}$ the relativised state space for $v_{X}$ would be: $\langle a^{Y}_{1} \rangle _{1}$, $\langle a^{Y}_{1} \rangle _{2}$, $\langle a^{Y}_{2} \rangle _{1}$, $\langle a^{Y}_{2} \rangle _{2}$. This would represent, respectively, the case where $v_{Y}$ announces $a_{1}$ and Nature has favoured 1, where where $v_{Y}$ announces $a_{1}$ and Nature has favoured 2, where $v_{Y}$ announces $a_{2}$ and Nature has favoured 1, and where where $v_{Y}$ announces $a_{2}$ and Nature has favoured 2.

Define a response procedure set as follows. Each element of $v_{i}$’s relativised state space is an announcement profile specifying, first, which answer Nature has favoured and, second, how everybody but the scholar under consideration has voted. The response procedure set for a given element of the state space is constructed as follows. Let c be an element of $v_{i}$’s relativised state space. Let $W_{i}$ be a function from $v_{i}$’s relativised state space to the power-set of answers. $W_{i}(c)$ outputs the set of all answers that could win in c, depending on how $v_{i}$ themselves announces. The output $W_{i}(c)$ is the response procedure set for c.

With the response procedure defined I can divide the relativised state space for a scholar into four types:

$\alpha$: An element of $v_{i}$’s relativised state space is an $\alpha$ element if and only if there is more than one element of its response procedure set.

$\beta$: An element of $v_{i}$’s relativised state space is a $\beta$ element if and only if its response procedure set is a singleton.

$\gamma$: An element of $v_{i}$’s relativised state space is a $\gamma$ element if and only if the answer initially chosen by Nature in that element of the state space is in their response procedure set.

$\delta$: An element of $v_{i}$’s relativised state space is an $\delta$ element if and only if the answer initially chosen by Nature in that element of the state space is not in their response procedure set.

Note that $\alpha /\beta$ and $\gamma /\delta$ are partitions of the state space. Finally, note that the following is true of response procedure sets:

Lemma 1

(Lemma 1) If $a_{1}$ is in S’s response procedure set for scholar $v_{1}$ then if $v_{1}$ votes for $a_{1}$ in S $a_{1}$ shall be victorious in the election over S.

Suppose S is a $\beta$ type set. Then no matter what $v_{1}$ votes for $a_{1}$ shall emerge victorious. Hence if $v_{1}$ votes for $a_{1}$ it shall be. Suppose S is an $\alpha$ type set. Since $a_{1}$ is in S’s response procedure set there must be some vote $v_{1}$ could offer such that $a_{1}$ would win. Consider any such vote that isn’t $a_{1}$, and call S filled in with that vote $S*$. Compare $S*$ to $S^{\star }$, which is S filled in with $v_{1}$’s vote for $a_{1}$. Note that in $S*$ $a_{1}$ is victorious, and $S^{\star }$ is identical $S*$ with except that $a_{1}$ has one more vote. Hence if $a_{1}$ is victorious in $S*$ then it must also be in $S^{\star }$, and hence the lemma is proven.

I assume scholars are expected utility maximisers; scholars select an announcement to make which, given their beliefs about how likely they are to be in different elements of their relativised state space and their utility type, they expect to generate the highest return. I say that a scholar is incentivised to fraud if it would not be expected utility maximising to announce the evidence proposition that Nature sent them.

As mentioned in Sect. 2, I begin with a short demonstration that my representation of the mixed credit/truth seeker is superior to what might seem like a natural alternative. According to the alternative method of tempering the credit motive by way of truth, the scholar has a nuanced non-binary preference structure: receiving credit > no credit but truth victorious > other outcomes. That is to say, this scholar receives payout of 1 if the answer they vote for is victorious, $1>r>0$ if an answer they did not vote for but which Nature selects in this element of the state space is victorious, and 0 otherwise. Call this agent the nuanced credit seeker.

Lemma 2

A nuanced credit seeker would vote for answer a if and only if a pure credit seeker in the same position would vote for a

Two scholars are in the same position if they have the same relativised state space, the same credence over that state space, and received the same signal from Nature. Suppose nuanced credit seeker $v^{n}$ was in the same position as pure credit seeker $v^{p}$. Consider the expected utility of $v^{p}$ announcing $a_{i}$. It is equal to $\sum c_{k}$ for all elements k of $v^{p}$’s relativised state space wherein $c_{k}$ is a positive number and voting $a_{i}$ attains payout of 1 in that element of the state space. Call the announcement that would maximise $v^{p}$’s expected utility $a_{W}$. Consider the expected utility of $v^{n}$ announcing $a_{i}$. It is equal to $\sum c_{k} + \sum r c_{j}$. This is the sum of all elements k of $v^{n}$’s relativised state space wherein $c_{k}$ is a positive number and voting $a_{i}$ attains payout of 1 in that element of the state space, added to the sum of all elements j of $v^{n}$’s relativised state space wherein $c_{j}$ is a positive number and $v^{n}$ voting $a_{i}$ results in the community has voted for whatever the nuanced agents middle option is. Suppose $a_{W}$ did not maximise $v^{n}$’s expected utility, but some other answer $a_{F}$ did. Note that since $v^{p}$ and $v^{n}$ face the same situation $\sum c^{W}_{k} \ge \sum c^{F}_{k}$—otherwise $v^{p}$ would also prefer $a_{F}$. Hence it must be that $\sum r c^{F}_{j} > (\sum c^{W}_{k} - \sum c^{F}_{k}) + \sum r c^{W}_{j}$. Consider when $r c_{F}$ will be earned. These are cases where the nuanced agents middle option but not top option is attained. That is to say, the community has selected the answer Nature selected, but $v^{n}$ has not voted for it. If in such a case the community does not vote for $a_{W}$ then $r c^{W}_{j} = r c^{F}_{j}$ would also have been earned by $v^{n}$ in this scenario, since what Nature selects does not depend on what $v^{n}$ voted for. Hence such cases cannot contribute to the left hand term being greater than the right in this inequality. However, if the community does vote for $a_{W}$ then such cases contribute to $\sum c^{W}_{k}$ and since $v^{p}$ and $v^{n}$ have the same credences over states this case actually contributes more to the right hand side than the left hand side of the inequality. Hence whether or not the community votes for $a_{W}$ the inequality cannot be satisfied. Hence $a_{W}$ must also be $v^{n}$’s expected utility maximising option.

Theorem 1

(Pessimistic theorem) First, mixed credit/truth seekers can be incentivised to lie in scenarios where the pure credit seeker is not. Second, if a pure credit seeker is incentivised to lie in a scenario where a mixed credit/truth seeker is not, then in this scenario the scholar does not believe their vote will affect what proposition the scientific community comes to accept.

Suppose without loss of generality that $v_{i}$ received signal $a_{1}$. I consider the possible states $v_{i}$ could believe themselves to be in, and how their behaviour would differ depending on whether they were a pure credit seeker or a mixed credit and truth seeking type.

$\alpha , \gamma$ elements of the state space. These are element where the scholar, $v_{i}$ is decisive and their vote can bring about victory for the option they believe Nature to have favoured in this element of their relativised state space. Note that if $a_{1}$ is also the signal $v_{i}$ believes Nature to favour in this state space, then $v_{i}$ cannot be incentivised to dishonesty in this state space. If, however, $a_{1}$ is not the signal they believe Nature to favour, then the incentives of pure credit seekers and mixed credit/truth seekers can diverge. In particular, the following is true: if the signal $v_{i}$ received is in the response procedure set for an $\alpha -\gamma$ element of the state space where the signal favoured by Nature is not identical with the signal $v_{i}$ received, a credit seeker cannot be incentivised to dishonesty while a mixed credit/truth seeker can. By lemma 1 a credit seeker could vote for the answer Nature signaled to them, and would expect to receive payout of 1 in such a scenario. Whereas, again by lemma 1, the mixed credit/truth seeker will only receive payout 1 if they vote for the option Nature favoured, which by hypothesis is not the answer Nature signaled to them. Hence in $\alpha -\gamma$ elements of the state space a pure credit seeker can never be incentivised to fraud while a mixed credit/truth seeker can.

$\beta , \gamma$ elements of the state space. In such elements both pure credit seekers and mixed credit/truth seekers are incentivised to fraud just under the same conditions, namely just in case $a_{1}$ is not the sole element of the response procedure set.

$\delta$ elements of the state space. Note that in these elements a mixed credit/truth seeking scholar is in a state of despair: no matter what they vote for they believe they will get payout of 0. Hence they cannot be incentivised to fraud in any $\delta$ type state space. Whereas a pure credit seeker can be, depending on whether the signal Nature sent them is in their response procedure set.

To summarise: the mixed credit/truth seeker will not be incentivised to commit fraud, whereas the pure credit seeker might be, in scenarios where they believe they cannot bring the scientific community to accept the answer they believe to be true. However, in situations where, first, they believe the community is going to accept the truth however they vote, and, second, they do not think the evidence they received from Nature is representative of what is true, the pure credit seeker and mixed credit/truth seeker will be incentivised to lie at just the same times. What is more, if the two diverge and the scholar thinks they can bring the community to accept the truth or the answer their evidence supports depending on what they announce when they publish, then the mixed/credit truth seeker can actually be incentivised to dishonesty where the pure credit seeker would not be.

Now consider the behaviour of pure truth seekers.

Theorem 2

(Du Bois’ conjecture) A pure truth seeker is incentivised to lie on strictly fewer occasions than the mixed credit/truth seeker.

Note that all of the above argument in theorem (1) would be identical for the pure credit seeker, with one exception. In $\beta ,\gamma$ elements of the state space the pure credit seeker takes themselves to be a guaranteed a payout of (1) no matter how they vote. Hence in such scenarios they cannot be incentivised to fraud, where the mixed credit/truth seeker would be. As such, the class of scenarios which, if believed to be most likely, would incentivise a pure truth seeker to dishonesty is a strictly proper subset of the class of scenarios which, if believed to be most likely, would incentivise a mixed credit/truth seeker to dishonesty.

The first behavioural posit mentioned in Sect. 2 is now explored.

Axiom 1

(Cost of fraud) Suppose scholar $v_{i}$’s received signal $s_{h}$ from Nature. Then for every element of the state space the utility of making any announcement $a_{k \not =h}$ is small $\epsilon _{1}>0$ less than it would otherwise be in that element of the state space given $v_{i}$’s utility type.

This behavioural posit can be seen as representing the idea that there is $\epsilon _{1}$ cost to fraud. This could be brought about if scholars attributed some small probability that they will be caught and punished for fraud. Note that this generates the following behavioural changes:

Theorem 3

(Active honesty) If scholars obey Cost of Fraud, then the class of $\alpha , \gamma$ elements of the state space, where the signal the scholar received is in the response set, wherein a pure credit seeking scholar is incentivised to be honest is a super set of the class of $\alpha$ elements of the state space where a truth seeking scholar is incentivised to be honest.

Say a scholar is incentivised to be honest wherein announcing the signal they received from Nature is their unique expected utility maximising option in a configuration. Consider an $\alpha , \gamma$ element of agent $v_{i}$’s state space where the signal $v_{i}$ received from Nature is in their response set. If it is, then a pure credit seeking $v_{i}$ who obeys Cost of Fraud will, by Lemma 1, always be incentivised to honesty. This is because whatever $v_{i}$ votes for will win, so they are guaranteed to be on the winning side, and Cost of Fraud gives $v_{i}$ a preference for being on the winning side with their honest announcement. However, a pure or mixed truth seeking $v_{i}$ may still fail to be incentivised to honesty. In particular, suppose in the $\alpha , \gamma$ element in question the signal they received from Nature is $a_{1}$ but they believe Nature to have favoured a separate answer, $a_{2}$, where $a_{2}$ is itself an element of the response procedure set. In this scenario the pure credit seeker would be strictly incentivised to honesty while the mixed or pure truth seeker would be strictly incentivised to dishonesty!

The second behavioural posit mentioned Sect. 2 is now explored. Consider the following behavioural posit.

Axiom 2

(Self confidence) Suppose scholar $v_{j}$ received evidence proposition $a_{k}$ from Nature. Let $c_{k}$ be an arbitrary element of the state space wherein Nature favoured $a_{k}$ and $c_{\lnot k}$ be an arbitrary element of the state space wherein Nature favoured some answer other than $a_{k}$. Scholar $v_{j}$ assigns small $\epsilon _{2} \ge 0$ credence to any such $c_{\lnot k}$.

Informally—scholars believe the results of their own research, and in particular if their research suggests Nature favours $a_{k}$ then no matter what results they think their colleagues are going to report they still believe Nature favours $a_{k}$, assigning any other alternative such a small probability as to be swamped out in expected utility calculations.

Theorem 4

(No insecurity) If scholars satisfy self-confidence and are pure or mixed credit/truth seekers then they would never be incentivised to commit fraud.

As before, truth seeking scholars could never be incentivised to commit fraud in a $\delta$ type case, whereas a pure credit seeker still could. By definition and granting Self Confidence if the agent was in a $\gamma$ type case they would believe the signal they received from Nature was in the response set. Hence whether they believed an $\alpha$ or $\beta$ type element of their relativised state space was most likely, the truth-motivated scholar would not have incentive to lie if they were Self Confident.