## Authors’ Contributions Conjecture

Many years ago I conducted an experiment. I asked several sets of friends who had written joint math papers what they thought their individual contributions were. I asked them separately, of course. As the result of this experiment I formulated the conjecture:

The total of what joint authors estimate their contributions to be is always more than 100%.

Here is an actual example of answers I received from the two authors of a joint paper.

Author 1: My contribution is 80%. I suggested a breakthrough idea that made this paper possible. He just typed everything.

Author 2: My contribution is 80%. I did all the work. She just suggested a good idea.

You can see how the answers are synchronized. It is clear that both are telling the truth. People just tend to over-value their own input.

In other cases each author thinks that she or he generated the main idea. It doesn’t mean that one of them is lying. Very often they are absolutely sincere. Take this example of Alice and Bob, who are working on a paper together. Alice suggests that they might have better progress on their theorem if they consider graphs with symmetries first. Bob is engrossed in his thoughts and doesn’t register Alice’s suggestion. Next day, he comes up with an idea to add a group action on graphs. He sincerely believes that this was his own idea. It would be hard to know whether this had been provoked by Alice’s suggestion, or had come to Bob independently. Alice assumes that they are working on her idea.

When you acknowledge other people’s contribution, keep in mind that their perception might be different from yours. If you do not want to hurt other people’s feelings, you might consider inflating your gratitude.

The conjecture doesn’t apply to single-author papers. First of all, mathematicians never claim their contribution is 110% as non-mathematicians do. In many cases, especially when there are acknowledgements in the paper, it would be illogical to claim 100% contribution. Most mathematicians are logical, so if they are gracious enough to acknowledge the help of others, they are unlikely to claim 100%.

I would be curious to continue the experiment and either prove or disprove my conjecture. I’d appreciate your help. If you want to be part of this experiment, you can provide the following numbers in your comments: your average contribution to your own papers; and also your weighted average contribution to your joint papers.

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1. #### Gabe:

It would be interesting to see how much this phenomenon holds true in other disciplines (and how different the “total contributions” are from discipline to discipline). Also, I wonder if the total reported contribution on 3-author papers is markedly higher than on 2-author papers.

2. #### Jonathan:

I worked in a department (not math) where the pros (not me) had created some things of value. There were about 6-7 pros. For each thing of value, two or three of the pros had worked, but the contributions were clearly not equal. There was also an immediate administrator who did much support and a little direct contribution to the things of value, and a big administrator who did some dealing with the big company that bought or commissioned the work of value, but who himself did not contribute.

They had to decide how to divide royalties on each thing of value. They held a special sort of auction.

For Thing of Value 1 they each, all the pros and the two administrators, wrote on a piece of paper the percentage that each should receive. Some numbers were easy. If pro #3 had done no work on this Thing, all the papers correctly showed his share as 0. But there were, say, two pros, plus the two administrators. And on the 8 papers the numbers did not agree. They posted all of the proposed divisions on the board (without the names of the proposers) and studied them. No discussion. And then they filled in a second ballot. This time the numbers came closer. Again, posted, studied, but no discussion. They continued for several rounds, until the numbers were a perfect match. And then they moved to the Second Thing of Value.

And it worked. I am still amazed that their numbers converged, but they did.

3. #### Maria Roginskaya:

I think you discribe a bad collaboration. I had once a collaboration where the sum of percieved contribution was more than 100% and it was a disaster – at the end we managed to set up a paper (based mutual agreement on the fact that we both are mature adults), but agreed avoid working together. Since that I’m careful to choose my collaborators (I have 5 with whom I work whenever we got a joint interest, and a few more whom we had been invited to one or another project). I write about 2 papers a year (all with collaborators) and in all of those (as far as I know) the sum of percieved contribution is less than 100%. It probably due to the fact that we all have enough experience to recognize when what the other do with ease (and even find pleasant) would take oneself ages to do (for example writting/proof-reading or using standard techniques from the other person’s area). And when it comes to the ideas it is for the most a result of a dialog, which every morning starts with “I have thought about your yesterday suggestion, and it seems to me it may work if we do it a bit differently…” So, for the most it is impossible to say whos idea it was – sometimes my collaborators forget what we had discussed a week ago, and think that I’m inventing something new, when I just repeat their ideas. Quite often one person find a method for a special case and the other is able to recognize it as a method (where the first was just fiddling with an example) an so on.

I think the difference depends more from the person than from the curcumstances, and a good collaboration is like a good marriage – requires wisdom and appreciation from all the participants (I can observe, when I’ve learned them better, that with all the differences the only common feature of all my collaborators is that they are all lucky in their marriages).

And, finally, we almost never discuss the contribution. If the discussion arrises because sometimes one of us feel that own contribution was too little, the other is simply reminding that it is not our last article and revokes the fourth Hardy-Littlewood rule (see for example https://www.math.ufl.edu/misc/hlrules.html ).

4. #### Maria Roginskaya:

I can give a life example (happened in one of my projects) opposite to the one above of Alice and Bob:

Author 1: We need to prove that the estimate on the function is continuous with respect to the parameters. As we know it is differentiable, one can replace the pointwise estimate by the estimate on the integral of the derivative which is usually easier. Unfortunately, I have no idea how to get continuity with respect to the parameters for the derivatives either… I can recursevly express derivative via the function, but then we are back to proving the continuity in the parameter for the function, and it just goes in circles :-(.

Author 2: Yes, it seems difficult. I don’t think I can help you – I haven’t dealt with differential equations for ages, if ever. May be we can ask somebody …

Author 1: My God, you are right! It IS a differental equation. Then the continuity with respect to the parameters follows from the most standart theorems – I have taught it in Calculus III just a year ago.

After looking it up in a text book the lemma is proved. Author 2 considers his contribution close to zero. Author 1 thinks that Author 2 provided the main idea of the proof (as the original problem comes from a field which is far from Differential Equations).

5. #### Tanya Khovanova:

Maria,

Thank you for sharing your experience. I am glad that you provided a counterexample to my conjecture – I was hoping that counterexamples exist.

Another thought. I wonder if the perception of contribution is related to gender.

6. #### Maria Roginskaya:

After some consideration, I should say that perception of contribution (and as a matter of fact actual contribution) depends from how well collaborators share their thoughts about the question. As both our examples show, if a collaborator doesn’t listen to the partner – ideas will have to be discovered twice and this overlap leads to increased contibution (percieved and actual) which is needed to complete the project. While it may be some correlation with gender based on traditional gender roles (girls are supposed to listen, boy are supposed let female talking fly above thier heads ðŸ™‚ ), I have enough of male collaborators to contradict any firm relation. I guess the matter is the original respect to each other mathematical abilities, i.e. you should genuinly believe that whatever your collaborator saggests is worth to listen (not as absolute truth, but at the same level as your own working hypotheses).

And it is important that this feeling is mutual, as if one of the collaborators neglects the other, the other will faced with the dilemma: “The person, whos opinion I respect thinks I’m not clever enough”. Whether the solution would be to resent the collaborator or to get unsure about her/himself, one will stop to share ideas and it destroys the purpose of collaboration.

7. #### Christ Schlacta:

my contribution to my own papers is usually around 10%, and I’ve never done a join paper, but I’ll be generous and estimate 20% for myself on all join projects as an average of all joint projects.