A Chinese version is here: 署名博弈.

Background and Motivation

In collaborations such as doing research work and writing a paper together, often, the relative contribution from each team member determines the order of authors of the paper. We call this the authorship/credit allocation problem. Ideally, one should have a system to record contributions from all members at each step and also have a metric to evaluate their contributions. However, in reality, there is no such explicit system, and it often relies very much on each member's memory of the contribution records.

The previous research already found that people intend to exaggerate/overestimate his/her own contribution when the group effort leads to something good (and likely to underestimate when it leads to something bad)^[1]. In particular, ^[2] examined this phenomenon on authorship allocation and found that overall the total estimated contribution, which is the sum of the relative contribution (in percentage) of each member estimated by each author him/herself, is larger than [math]\displaystyle{ 100\% }[/math].

This is rather a serious issue. First of all, it will be good for scientometric studies to have a truthful author list, which more or less describes the true contributions of the authors. Moreover, more importantly, if one believes that his/her contribution is not properly recognized in the author list, he/she might stop further even more fruitful/needed collaboration, thus hinder the development of Sciences.

So, can we have a tool to help all collaborators to get more satisfied (by each member or at least by majority) authorship allocation, without needing the above tool of recording and evaluating contributions?

BTW, the tool of recording and evaluating contributions is, of course, also a project that should be implemented by our scientometric people.

Another motivation is from the studies of game theory, the ultimatum game (UG) in particular. In the ultimatum game, two players need to determine how to allocate a given amount of money. The first player suggests the plan, says how much will be given to the second player, while the second player decides either to accept or to reject the offer. If accepted, the allocation will be done as suggested. If rejected, neither player gets any money.

Usually, in UG, people found that the first player usually offers [math]\displaystyle{ 30-50\% }[/math] to the second player while both players understand that if both of them are rational, the second player will accept even a minimal offer. In a sense, the summation of both players' expected relative payoff is less than [math]\displaystyle{ 100\% }[/math]: let us rephrase the game in this way, each player sets a lower bound of his/her own payoff [math]\displaystyle{ p_{1}, p_{2} }[/math], as long as the summation is less than [math]\displaystyle{ 100\% }[/math], the allocation will succeed. In this sense, we say that it has been shown in previous studies that in UG, real human players often have [math]\displaystyle{ p_{1}+p_{2}\lt 100\% }[/math]. This is, in a sense, very different from the authorship allocation issue, where [math]\displaystyle{ p_{1}+p_{2}+\cdots \gt 100\% }[/math]. The difference between the two situations is that in UG, there is no previous effort from each player, while in the authorship allocation issue, each member needs to work together for the publication.

So is it possible that the previous effort is the key that determines either [math]\displaystyle{ p_{1}+p_{2}+\cdots \gt 100\% }[/math] or [math]\displaystyle{ p_{1}+p_{2}+\cdots \lt 100\% }[/math]?

In this project, we want to look into the above two issues: whether or not we can design an authorship/credit allocation game to help collaborators to reach higher satisfaction of the allocation, and whether or not the existence of previous effort makes the difference between [math]\displaystyle{ p_{1}+p_{2}+\cdots \gt 100\% }[/math] and [math]\displaystyle{ p_{1}+p_{2}+\cdots \lt 100\% }[/math].

The Authorship/Credit Allocation Game

Our authorship/credit allocation game (AAG) is very simple: at each round, all authors [math]\displaystyle{ i }[/math], which will be referred as players from now on, report his/her estimated relative contribution [math]\displaystyle{ p_{i} }[/math] (if necessary, we can limit the amount of options, say [math]\displaystyle{ 0\%, 5\%, 10\%, \cdots }[/math]) and then we if [math]\displaystyle{ \sum_{i}p_{i}\lt 100\% }[/math] the author list will be created accordingly and otherwise, there will be another round, till an author list is created.

AAG can be done before or after the publication of the paper. Let us call them pre-publication AAG (PAAG) and after-publication AAG (AAAG).

We can also run artificial AAG with monetary payoffs (AAGM). Say with a given amount of money [math]\displaystyle{ M }[/math], each player suggest certain amount of money for his/herself [math]\displaystyle{ p_{i} }[/math], if [math]\displaystyle{ \sum_{i}p_{i}\lt 100\% }[/math] M will be allocated accordingly, and otherwise, every player get no money at all.

Experimental Design

1. Field experiment: recruit authors of the same publication to perform PAAG or AAAG, and their satisfaction of the author list before and after the game will be measured, via a questionnaire or via EEG and other brain imaging techniques^[3][4].
2. Lab experiment: run a collaborative task first in the Lab and then run AAGM instead of the authorship, again their satisfaction with the allocation will be measured.
   1. Two-step setting: payoffs to players are cleared at the end of each sub-games.
   2. Compound game setting: payoffs to players are cleared at the end of both game, where total payoffs of both games are combined and then allocated according to the relative contributions resulted from AAGM.
   3. AAGM only: run AAGM only without the collaborative task. This is a setup similar to UG.

One example of such a collaborative task can be the public goods (PG) game. In the public goods game, each player is given initially the same amount of money ([math]\displaystyle{ d }[/math]) and then each player decide how much ([math]\displaystyle{ v_{i}\lt d }[/math]) out of this initial amount will be invested in the public goods. The invested money will be rewarded with a constant rate [math]\displaystyle{ R }[/math] and in the end, each player will get an even part of the reward so that at the end [math]\displaystyle{ E^{i}=d-v_{i}+\frac{R}{N}\left(\sum_{i=1}^{N}v_{i}\right) }[/math].

In our randomized public goods (RPG) game, we allow the rate of rewording [math]\displaystyle{ R }[/math] to be a random variable, such as [math]\displaystyle{ R\in \left[rN-\Delta, rN+\Delta\right] }[/math], to mimic the uncertainty of successful publication and to allow the players to have certain room for their estimation. Otherwise, in the case of two players, if there is no random variable, each player knows exactly the relative contribution of each player. Even with more than two players, their own relative contribution is known at least to themselves. Furthermore, if needed, we can set [math]\displaystyle{ \Delta=0 }[/math] to get rid of this random variable.

Note that even when [math]\displaystyle{ \Delta=0 }[/math] one still might overestimate his/her own contribution: in the game with three players or more, one knows exactly their own contribution but not that of each of other players. This might motivate the players to an overestimation.

In the two-step setting, [math]\displaystyle{ E^{i}=d-v_{i}+\frac{R}{N}\left(\sum_{i=1}^{N}v_{i}\right) }[/math] is payed to the player [math]\displaystyle{ i }[/math] at the end of RPG and then at the end of AAGM, [math]\displaystyle{ M }[/math] is determined by [math]\displaystyle{ p_{i} }[/math]. In the compond game, at the end of AAGM, [math]\displaystyle{ R\left(\sum_{i=1}^{N}v_{i}\right)+M }[/math] is allocated to each player according to [math]\displaystyle{ p_{i} }[/math].

We now have four different PGs, which will be used as the collaborative task in the step-one game in Lab Experiment. There are two-player deterministic PG (2DPG), two-player random PG (2RPG), three-player deterministic PG (3DPG), three-player random PG (3RPG). Of course, one can go beyond that, say with more players. In 2DPG, both players know precisely the contribution of each player. In 2RPG, there are rooms for each player to downplay or overestimate his/her own contribution. In 3DPG, each player knows precisely his/her own contribution, and thus the sum of the other two players but not of the individual players. This also gives some room for each player to exaggerate his/her own contribution. In 3RPG, there are even more rooms for each player to overestimate his/her own contribution.

Research questions (hypothesis) corresponding to each treatment and their comparisons

For the field experiments, we have two treatments: PAAG and AAAG. Satisfaction levels are measured before and after the game via satisfaction questionnaires (SQ). For each treatment, our hypothesis is the same: AAG can improve each author or most authors' satisfaction level with the recognition of their relative contribution.

For the lab experiments, we have five treatments:

1. T1: AAG
2. T2: 2DPG+(1 round of)AAG+SQ+(as many rounds as needed till successful allocation)AAG+SQ
3. T3: 3DPG+(1 round of)AAG+SQ+(as many rounds as needed till successful allocation)AAG+SQ
4. T4: 2RPG+(1 round of)AAG+SQ+(as many rounds as needed till successful allocation)AAG+SQ
5. T5: 3RPG+(1 round of)AAG+SQ+(as many rounds as needed till successful allocation)AAG+SQ

T1 is a symmetric version of UG. T2 is a fully deterministic setting with full information. T3 is a fully deterministic setting with partial information. T4 is a random setting. T5 is a random setting with even less information.

Our hypothesis when comparing results from T1 and others is: It is the existence of previous effort that makes [math]\displaystyle{ \sum_{i}p_{i}\gt 1 }[/math], otherwise [math]\displaystyle{ \sum_{i}p_{i}\leq 1 }[/math].

Our hypothesis when comparing results of SQs (before and after AAG) from each of T2-T5 on the level of satisfaction is: AAG can improve each participant's satisfaction level with the recognition of their relative contribution.

We then compare results from T2-T5 on the following quantity: [math]\displaystyle{ S^{n}=\sum_{i}p^{n}_{i}-1 }[/math], where [math]\displaystyle{ n=2,3,4,5 }[/math] corresponding to T2-T5. We can then check whether or not [math]\displaystyle{ S^{2} \gt 0 }[/math], [math]\displaystyle{ S^{2} \lt S^{3} \lt S^{5} }[/math], [math]\displaystyle{ S^{2} \lt S^{4} \lt S^{5} }[/math], and [math]\displaystyle{ S^{3} \lt S^{4} }[/math].

1. [math]\displaystyle{ S^{2} \gt 0 }[/math] means even in fully deterministic setting with full information, people exaggerate their own contributions.
2. [math]\displaystyle{ S^{2} \lt S^{3} \lt S^{5} }[/math] means with less and less deterministic setting with less and less information, people exaggerate their own contributions more and more.
3. [math]\displaystyle{ S^{3} \lt S^{4} }[/math] or [math]\displaystyle{ S^{3} \lt S^{4} }[/math] is a competition between the two kind of randomness, the expected uncertainty of other players on "my" contribution (T3) and the uncertainty from [math]\displaystyle{ \Delta }[/math] (T4).

One can also adjust the value of [math]\displaystyle{ \Delta }[/math] to see how the two uncertainties trade off.

References