<p>It's a big topic. The relationship between group size and performance on a cognitive task is going to vary by several factors.</p>

<p>Here are a few thoughts:</p>

<ul>
<li>The form of interdependence adopted by the group on the task will matter. When everyone can just work independently (e.g., taking calls in a call centre), then it makes sense that output would increase almost linearly with increasing group size, in the absence of motivation or learning issues.</li>
<li>In the context of project teams, software teams, and so forth, I imagine the issues would be a lot more complex and that in some situations increasing group size might even reduce performance of the team. There a few articles cited <a href="http://www.teambuildingportal.com/articles/systems/teamperformance-teamsize">here</a> that discuss this.</li>
<li>A classic study in the context, albeit from a motor task, of the relationship between group-size and group performance is that by Ringelmann (see the <a href="http://en.wikipedia.org/wiki/Ringelmann_effect">wikipedia article</a>) that looked at rope pulling power as a function of group size. The basic finding was that average pulling power declined with group size. This has been interpreted as evidence of social loafing, but their are also coordination issues as well.</li>
<li>Some tasks are heavily influenced by the performance of the weakest or strongest member. For example, a musical group will often sound bad if only one member plays poorly. Or other tasks it will be strongest member, as you say if you have are playing at a trivia night, and you have one person with all the right answers on the table.</li>
</ul>

CC BY-SA 3.0
2012-01-23T11:21:02.143Z
The OP, however, is only interested in up-scaling the model. How do any of these factors influence the ease or level of difficulty in scaling up?
Kris
CC BY-SA 3.0
2012-01-23T11:31:21.823Z
If I understand your question then for the four dot points above the scaling would be: a. almost linear; b. increasing but possibly with an inverted u quadratic; c. monotonically increasing and monotonically decelerating; d. increasing or decreasing.
Jeromy Anglim
CC BY-SA 3.0
2012-01-23T11:33:31.930Z
@Kris but these are just some basic thoughts; I'm sure the question could be answered in many different ways; I'm keen to see what others have to say
Jeromy Anglim
CC BY-SA 3.0
2012-01-23T11:35:18.907Z
Thank you for the very useful info. I hope the OP finds this useful, too, in which case you could include it in the answer. I would wait know how each of the four are influenced.
Kris
CC BY-SA 3.0
2012-01-23T13:11:26.910Z
@JeromyAnglim: I am interested in collaboration on _cognitive_ tasks, so only the part on software development is answering my question. I am aware of Ringelmann effect (however, it's more about motivation/management than anything else). And yes, for different tasks the scaling is likely to be different.
Piotr Migdal
CC BY-SA 3.0
2012-01-23T06:51:15.757Z
2012-01-23T15:27:09.910Z
2012-01-23T15:27:09.910Z
Jeromy Anglim
<p>I recently read <a href="https://read.oecd-ilibrary.org/education/critical-masses-for-academic-research-groups-and-consequences-for-higher-education-research-policy-and-management_hemp-23-5kg0vswcm27g#page1" rel="nofollow noreferrer">a paper</a>, which showed a mathematical model for performance scaling of research groups in different scientific branches. I'm aware you were originally asking for smaller &quot;cognitive tasks&quot; and project-like group-processes in the comments, but output and quality of publications/patents is probably anyway a better and more objective measure on a larger scale. Also, the paper shows that optimal group sizes in different academic disciplines varies strongly, so comparing, e.g. single groups in math and biology solving tasks is probably not very meaningful (Jeromys 1st comment) for your original question and only valid for exactly that discipline/distinct cognitive task, which would make a general analysis/conclusions tricky/impossible.</p>
<p>Assumptions:</p>
<blockquote>
<p>The extent to which a group’s research quality is higher or lower than
average is a function of many parameters such as the innate calibre, education
and experience of its members, the quality of managerial support and of facilities
and infrastructure, the impact of external and international collaborations, the
prestige and confidence inspired by the institution and by past successes, etc. All
of these parameters contribute to the quality of an individual group but, in
general, research quality is size independent according to this argument. [...] However, it would be naive to retain this viewpoint alone because it
neglects the importance of interactions between group members.</p>
</blockquote>
<p>The model:</p>
<p>A group has <span class="math-container">$N$</span> members, therefore the total number of pairs of individuals along which channels of communication can open is</p>
<p><span class="math-container">$N_{T}=N(N-1)/2$</span> (quadratical scaling)</p>
<p>So group quality tends to rise continuosly with quantity. The main statement of the article however is that the rise of quality is independent of group quantity after a group reaches a upper critical mass (<span class="math-container">$NN_{C}/2$</span> linear scaling) and increaes only with much smaller slope further, basically because a single researcher can only communicate meaningfully with a limited number of other researchers (sub-groups build within big groups, specialization)</p>
<p>Comparison with available data:</p>
<p>I will not copy paste all the data fits here, look them up in the paper, well worth and interesting, many academic disciplines are compared in different countries and match the model and quadratic scaling quite well in different academic disciplines.</p>
<p><img src="https://i.sstatic.net/MWzUU.jpg" alt="enter image description here" /></p>
<p>So the scaling for groups is the same, as mainly determined by meaningful face-to-face communication, the different critical mass among different academic disciplines has probably to do with the necessary amount of researcher you need to run, for example, a high-technology research laboratory (many students who master the single instruments) vs. mathematics (formalized language, not the broad spectrum of problems like in natural sciences) vs. humanities (reading and discussing a lot of literature)</p>
<p>Implications:</p>
<p>To improve overall research performance in a country, increase groups below the size of the critical mass instead of further increasing of already abnormal big research groups, where a slope for performance scaling dependent on <span class="math-container">$N$</span> still exists, but is much smaller than below the critical mass</p>
<p>Side note: Apart from this critical research group size I'm aware of a new trend in bigger companies to forbid or restrict e-mailing for the majority of employees, because only a marginal amount of this e-mails is really important according to studies (15% I rememember mentioned by Volkswagen AG, but not the source currently)</p>
<p><strong>References</strong><br />
Kenna R, Berche B (2011). Critical masses for academic research groups and consequences for higher education research policy and management. Higher Education Management and Policy 23 (3), 9-29.</p>

CC BY-SA 3.0
2012-05-21T13:00:48.653Z
I am aware of that paper, and actually I like it very much (and had nice discussion with an author, and even [used it as an example on academia.SE](http://academia.stackexchange.com/a/1253/49)). However, scaling is quadratic/linear, depending on the group size. And discipline changes only critical group size, not the scaling properties (which may be more a subject of social interactions + some hidden factors correlating group size and a measure of performance).
Piotr Migdal
CC BY-SA 3.0
2012-05-21T14:12:58.687Z
But anyway, thanks and +1 :). (Though I clarified in the question, that I am the most interested in low-level task; a grade for the total research output, being a complex process by itself, is hardly low-level.)
Piotr Migdal
CC BY-SA 4.0
2012-05-21T12:06:04.353Z
2021-04-25T01:06:59.090Z
2021-04-25T01:06:59.090Z
Hauser
<p>Some intellectual, cognitive and perceptual tasks can be solved collaboratively. It is common knowledge that group performance is better than that of each single individual due to exchange of information (and without going into issues of motivation).</p>
<p>One widely known phenomenon is <em>Wisdom of Crowds</em> - when a group decision is much more accurate due to the averaging of individual biases. However, it is not the only possibility. Sometimes, a task can be divided into parts (and each participant works efficiently on his/ her own). Sometimes there are tasks too difficult to solve but easy to demonstrate that an answer is correct (and it suffices to have only one person in the group who answers correctly to make the group decision correct).</p>
<p>What I am interested in is, how the group performance scales with the group size?
(The measure of performance can be error rate, time needed to complete a certain task or the deviation from the right answer.)</p>
<p>The question is motivated by one of my recent works, <a href="https://arxiv.org/abs/1109.2044" rel="nofollow noreferrer">arXiv:1109.2044</a>.</p>
<p>EDIT (clarification):</p>
<p>I am the most interested in low-level cognitive tasks (e.g. perceptual or linguistic), especially two-choice decisions (but also multiple choice, counting of blobs, solving a puzzle...). However, other quantifiable measures for cognitive task (including, say, time to complete a certain programming project vs number of programmers) are of my interest as well.</p>
<p>When it comes to the relation to my paper, I am curious if I miss an important reference or if my theoretical results can be supported (or undermined) by any existing experimental results (or even related in other context).</p>

CC BY-SA 3.0
2012-05-21T13:31:59.067Z
@Piotr Migdal: What do you expect? For 2AFC tasks, your paper (which is nice, btw) summarizes the possible outcomes quite well, I guess. Depending on the knowledge about the group strategy, the model predictions should be quite precise. Hence, all you have to do now is asking the group which strategy they adopted. All empirical research that is left beyond your findings is the question which group strategy is adopted most often.
H.Muster
CC BY-SA 3.0
2012-05-21T14:04:09.260Z
@H.Muster My paper (thx, btw) deals with a specific 2AFC task and gives only theoretical predictions (but two of co-authors are now doing experimental part). I'm interested if there is known scaling of efficiency any other low-level tasks (e.g. solving a word puzzle, solving a multiple choice test, signal detection, counting the number of blobs...). I am curious if there are experimental results that either are compatible with my results, or undermine it.
Piotr Migdal
CC BY-SA 3.0
2012-05-21T14:14:11.667Z
I think that your models might be exhaustive for all 2AFC tasks, i.e., if you know the performance of each individual and the strategy of the group, the model will predict the group performance perfectly. Am I wrong?
H.Muster
CC BY-SA 3.0
2012-05-21T15:00:48.143Z
@H.Muster Not exactly. I put some assumptions that don't need to hold in the general context, but are motivated by some particular experimental outcomes. So even for 2AFC task one can came with a more general model (however, personally I would like rather to test/focus on a particular example, than generalize it even further).
Piotr Migdal
CC BY-SA 3.0
2012-01-23T05:47:30.967Z
I like this question (+1) as do a lot of others. However, I wonder if at present it is still a bit broad. There are many possible group tasks that we could discuss. I wonder whether the question could be improved by focusing on a particular class of tasks (e.g., are you thinking about software project teams; tasks involving coordination; etc.)?
Jeromy Anglim
CC BY-SA 3.0
2012-01-23T11:20:09.477Z
@JeromyAnglim I'd definitely not go for a more localized question.
Kris
CC BY-SA 3.0
2012-01-23T11:24:01.203Z
For an example of a broad question, see http://cogsci.stackexchange.com/questions/97/what-is-known-about-how-sleep-works-and-why-its-necessary
Kris
CC BY-SA 3.0
2012-01-23T11:50:11.310Z
@Kris I think that question on sleep is way too broad and multifaceted for the stack exchange format. The issue of scope seems like an interesting thing to discuss on meta; I posted a question http://meta.cogsci.stackexchange.com/questions/63/what-to-do-with-otherwise-good-questions-that-seem-a-bit-too-broad
Jeromy Anglim
CC BY-SA 3.0
2012-01-23T12:00:54.450Z
@JeromyAnglim One criterion for 'broad' could be 'sleep-inducing', hope you got the pun now. :)
Kris
CC BY-SA 3.0
2012-01-23T13:16:32.783Z
@JeromyAnglim: I am the most interested in low-level _cognitive_ tasks, esp. two-choice decisions (eg. perceptual or linguistic). However, other quantifiable measures for cognitive task (including, say, time to complete a certain programming project vs no of programmers) are of my interest as well.
Piotr Migdal
CC BY-SA 3.0
2014-04-08T17:59:41.800Z
this is an emerging area that depends a lot on the _framework_ by which the decision making is expressed/structured. see [collective intelligence](http://en.wikipedia.org/wiki/Collective_intelligence) a newly emerging field with deep study, and there is a lot of connection to online social networks etc
vzn
CC BY-SA 3.0
2015-05-19T17:53:30.963Z
The Polymath project (a massive collaboration of mathematicians) might be an interesting subject of study: http://m.youtube.com/watch?v=elWIDVI6b18
Lenar Hoyt
<decision-making><mathematical-psychology><problem-solving><collaboration><wisdom-of-crowds>
Performance of a group solving a cognitive task: How does it scale?
Piotr Migdal
# Performance of a group solving a cognitive task: How does it scale?

(Asked by: username_0 on 2012-01-18)

Some intellectual, cognitive and perceptual tasks can be solved collaboratively. It is common knowledge that group performance is better than that of each single individual due to exchange of information (and without going into issues of motivation).

One widely known phenomenon is *Wisdom of Crowds* - when a group decision is much more accurate due to the averaging of individual biases. However, it is not the only possibility. Sometimes, a task can be divided into parts (and each participant works efficiently on his/ her own). Sometimes there are tasks too difficult to solve but easy to demonstrate that an answer is correct (and it suffices to have only one person in the group who answers correctly to make the group decision correct).

What I am interested in is, how the group performance scales with the group size?
(The measure of performance can be error rate, time needed to complete a certain task or the deviation from the right answer.)

The question is motivated by one of my recent works, [arXiv:1109.2044](https://arxiv.org/abs/1109.2044).

EDIT (clarification):

I am the most interested in low-level cognitive tasks (e.g. perceptual or linguistic), especially two-choice decisions (but also multiple choice, counting of blobs, solving a puzzle...). However, other quantifiable measures for cognitive task (including, say, time to complete a certain programming project vs number of programmers) are of my interest as well.

When it comes to the relation to my paper, I am curious if I miss an important reference or if my theoretical results can be supported (or undermined) by any existing experimental results (or even related in other context).

**Tags:** decision-making, mathematical-psychology, problem-solving, collaboration, wisdom-of-crowds

> **Comments:**
> **username_2**: I like this question (+1) as do a lot of others. However, I wonder if at present it is still a bit broad. There are many possible group tasks that we could discuss. I wonder whether the question could be improved by focusing on a particular class of tasks (e.g., are you thinking about software project teams; tasks involving coordination; etc.)?  
> **username_3**: @JeromyAnglim I'd definitely not go for a more localized question.  
> **username_3**: For an example of a broad question, see http://cogsci.stackexchange.com/questions/97/what-is-known-about-how-sleep-works-and-why-its-necessary  
> **username_2**: @username_3 I think that question on sleep is way too broad and multifaceted for the stack exchange format. The issue of scope seems like an interesting thing to discuss on meta; I posted a question http://meta.cogsci.stackexchange.com/questions/63/what-to-do-with-otherwise-good-questions-that-seem-a-bit-too-broad  
> **username_3**: @JeromyAnglim One criterion for 'broad' could be 'sleep-inducing', hope you got the pun now. :)  
> **username_0**: @JeromyAnglim: I am the most interested in low-level _cognitive_ tasks, esp. two-choice decisions (eg. perceptual or linguistic). However, other quantifiable measures for cognitive task (including, say, time to complete a certain programming project vs no of programmers) are of my interest as well.  
> **username_1**: @username_0: What do you expect? For 2AFC tasks, your paper (which is nice, btw) summarizes the possible outcomes quite well, I guess. Depending on the knowledge about the group strategy, the model predictions should be quite precise. Hence, all you have to do now is asking the group which strategy they adopted. All empirical research that is left beyond your findings is the question which group strategy is adopted most often.  
> **username_0**: @username_1 My paper (thx, btw) deals with a specific 2AFC task and gives only theoretical predictions (but two of co-authors are now doing experimental part). I'm interested if there is known scaling of efficiency any other low-level tasks (e.g. solving a word puzzle, solving a multiple choice test, signal detection, counting the number of blobs...). I am curious if there are experimental results that either are compatible with my results, or undermine it.  
> **username_1**: I think that your models might be exhaustive for all 2AFC tasks, i.e., if you know the performance of each individual and the strategy of the group, the model will predict the group performance perfectly. Am I wrong?  
> **username_0**: @username_1 Not exactly. I put some assumptions that don't need to hold in the general context, but are motivated by some particular experimental outcomes. So even for 2AFC task one can came with a more general model (however, personally I would like rather to test/focus on a particular example, than generalize it even further).  
> **username_4**: this is an emerging area that depends a lot on the _framework_ by which the decision making is expressed/structured. see [collective intelligence](http://en.wikipedia.org/wiki/Collective_intelligence) a newly emerging field with deep study, and there is a lot of connection to online social networks etc  
> **username_5**: The Polymath project (a massive collaboration of mathematicians) might be an interesting subject of study: http://m.youtube.com/watch?v=elWIDVI6b18  

---

## 2 answers

### Answer by username_2 on 2012-01-23. Score: 12

It's a big topic. The relationship between group size and performance on a cognitive task is going to vary by several factors.

Here are a few thoughts:

* The form of interdependence adopted by the group on the task will matter. When everyone can just work independently (e.g., taking calls in a call centre), then it makes sense that output would increase almost linearly with increasing group size, in the absence of motivation or learning issues.
* In the context of project teams, software teams, and so forth, I imagine the issues would be a lot more complex and that in some situations increasing group size might even reduce performance of the team. There a few articles cited [here](http://www.teambuildingportal.com/articles/systems/teamperformance-teamsize) that discuss this.
* A classic study in the context, albeit from a motor task, of the relationship between group-size and group performance is that by Ringelmann (see the [wikipedia article](http://en.wikipedia.org/wiki/Ringelmann_effect)) that looked at rope pulling power as a function of group size. The basic finding was that average pulling power declined with group size. This has been interpreted as evidence of social loafing, but their are also coordination issues as well.
* Some tasks are heavily influenced by the performance of the weakest or strongest member. For example, a musical group will often sound bad if only one member plays poorly. Or other tasks it will be strongest member, as you say if you have are playing at a trivia night, and you have one person with all the right answers on the table.

> **Comments:**
> **username_3**: The OP, however, is only interested in up-scaling the model. How do any of these factors influence the ease or level of difficulty in scaling up?  
> **username_2**: If I understand your question then for the four dot points above the scaling would be: a. almost linear; b. increasing but possibly with an inverted u quadratic; c. monotonically increasing and monotonically decelerating; d. increasing or decreasing.  
> **username_2**: @username_3 but these are just some basic thoughts; I'm sure the question could be answered in many different ways; I'm keen to see what others have to say  
> **username_3**: Thank you for the very useful info. I hope the OP finds this useful, too, in which case you could include it in the answer. I would wait know how each of the four are influenced.  
> **username_0**: @JeromyAnglim: I am interested in collaboration on _cognitive_ tasks, so only the part on software development is answering my question. I am aware of Ringelmann effect (however, it's more about motivation/management than anything else). And yes, for different tasks the scaling is likely to be different.  

---

### Answer by username_6 on 2012-05-21. Score: 11

I recently read [a paper](https://read.oecd-ilibrary.org/education/critical-masses-for-academic-research-groups-and-consequences-for-higher-education-research-policy-and-management_hemp-23-5kg0vswcm27g#page1), which showed a mathematical model for performance scaling of research groups in different scientific branches. I'm aware you were originally asking for smaller "cognitive tasks" and project-like group-processes in the comments, but output and quality of publications/patents is probably anyway a better and more objective measure on a larger scale. Also, the paper shows that optimal group sizes in different academic disciplines varies strongly, so comparing, e.g. single groups in math and biology solving tasks is probably not very meaningful (Jeromys 1st comment) for your original question and only valid for exactly that discipline/distinct cognitive task, which would make a general analysis/conclusions tricky/impossible.

Assumptions:

> The extent to which a group’s research quality is higher or lower than
> average is a function of many parameters such as the innate calibre, education
> and experience of its members, the quality of managerial support and of facilities
> and infrastructure, the impact of external and international collaborations, the
> prestige and confidence inspired by the institution and by past successes, etc. All
> of these parameters contribute to the quality of an individual group but, in
> general, research quality is size independent according to this argument. [...] However, it would be naive to retain this viewpoint alone because it
> neglects the importance of interactions between group members.

The model:

A group has $N$ members, therefore the total number of pairs of individuals along which channels of communication can open is

$N\_{T}=N(N-1)/2$ (quadratical scaling)

So group quality tends to rise continuosly with quantity. The main statement of the article however is that the rise of quality is independent of group quantity after a group reaches a upper critical mass ($NN\_{C}/2$ linear scaling) and increaes only with much smaller slope further, basically because a single researcher can only communicate meaningfully with a limited number of other researchers (sub-groups build within big groups, specialization)

Comparison with available data:

I will not copy paste all the data fits here, look them up in the paper, well worth and interesting, many academic disciplines are compared in different countries and match the model and quadratic scaling quite well in different academic disciplines.

![enter image description here](https://i.sstatic.net/MWzUU.jpg)

So the scaling for groups is the same, as mainly determined by meaningful face-to-face communication, the different critical mass among different academic disciplines has probably to do with the necessary amount of researcher you need to run, for example, a high-technology research laboratory (many students who master the single instruments) vs. mathematics (formalized language, not the broad spectrum of problems like in natural sciences) vs. humanities (reading and discussing a lot of literature)

Implications:

To improve overall research performance in a country, increase groups below the size of the critical mass instead of further increasing of already abnormal big research groups, where a slope for performance scaling dependent on $N$ still exists, but is much smaller than below the critical mass

Side note: Apart from this critical research group size I'm aware of a new trend in bigger companies to forbid or restrict e-mailing for the majority of employees, because only a marginal amount of this e-mails is really important according to studies (15% I rememember mentioned by Volkswagen AG, but not the source currently)

**References**  
Kenna R, Berche B (2011). Critical masses for academic research groups and consequences for higher education research policy and management. Higher Education Management and Policy 23 (3), 9-29.

> **Comments:**
> **username_0**: I am aware of that paper, and actually I like it very much (and had nice discussion with an author, and even [used it as an example on academia.SE](http://academia.stackexchange.com/a/1253/49)). However, scaling is quadratic/linear, depending on the group size. And discipline changes only critical group size, not the scaling properties (which may be more a subject of social interactions + some hidden factors correlating group size and a measure of performance).  
> **username_0**: But anyway, thanks and +1 :). (Though I clarified in the question, that I am the most interested in low-level task; a grade for the total research output, being a complex process by itself, is hardly low-level.)  

---

