Projects – an inevitable part of college life, good news for some, nightmares for others. The most dreadful part of a project is not its workload, but the disagreements and conflicts and the night before deadline it brings along. Why do people still find doing projects a good way to learn?

Okay, time to get serious. After the project has been submitted, it is time for peer evaluation (as if the professor thinks not enough blood has been spilled). We want to find out how much each team member has contributed, and who were the free-riders.

However, dividing the pie may not be as trivial as it sounds. Often when two people work together, their total contribution is greater than the sum of the contributions when each of them work on their own. This is called “**synergy**”.

The amount of “synergy” depends on how effective people work together. Some groups may have large synergies, while some may have none. For example, A may be a perfect partner for B so they two create a lot, but A may not work well with C so there is no synergy between them. In that case, if A,B,C work together, who should be credited for the synergy?

Note that in the discussions below, we assume the synergy to be non-negative. That is, when two people form a group, in the worst case there will be no cooperation and the total outcome is the sum of the individual outputs.

**Suppose A,B,C works together in a project worth of 100 points. How many points should each of them get?**

### Criteria for dividing contribution

- Efficiency (i.e.
**Adds up to 100%**) Points received by A,B,C should add up to 100. It would be absurd if each of them gets only 10 points, or 50 points. - Symmetry (i.e.
**Can I be replaced?**) If the output of any group won’t change by substituting B with C, then B and C should get the same number of points, since they are doing the same job. - Null player (i.e.
**Free-rider**) If the output of any group won’t change no matter whether A is in the group or not, A should get 0 points since he has effectively done nothing. - Linearity (i.e.
**Just add them up**) Suppose the project is divided into part 1 (60 points) and part 2 (40 points), then the number of points for A, is the sum of the points he gets in part 1 and part 2 respectively.

These criteria look quite intuitive and logical, but they also seem to be quite loose. This is the time when the magic of game theory steps in, as it has been proved that the four rules above already determine **a unique way** of dividing total contribution – the **Shapley value**.

### Shapley value

The idea behind Shapley value is simple: Form the whole group by recruiting people one-by-one, and see how much more output is **generated by each recruit**.

Let’s look at the example of the project done by A,B,C. If only A works on it, he will only get 10 points. Similarly, points received by B and C if working solo are 20 and 30 respectively.

Then we consider the cases where two people work together. Since A and B make a great pair, they can get 60 points by working together, much better than 10+20. B and C don’t really get along, so they get 50 points if working together. Finally, A and C together can get 65 points.

Consider the sequence A-B-C, which means that A works on the project first, then B joins to work together, and finally C also joins. When A starts, output increases from 0 to 10. Then, when B joins, output increases from 10 to 60. Next, when C joins, output increases from 60 to 100. Hence, for this sequence A-B-C, the contribution of (A,B,C) is (10,50,40).

Of course, there are other sequences. For example, for B-C-A, output increases from 0 to 20 when B starts, then increases to 50 when C joins, and finally to 100 when A also joins, hence contribution of (A,B,C) is (50,20,30)

There are totally 3!=6 sequences. You can verify the contribution for the remaining 4 sequences:

C-A-B: (A,B,C)=(35,35,30); A-C-B: (A,B,C)=(10,35,55); C-B-A: (A,B,C)=(50,20,30); B-A-C: (A,B,C)=(40,20,40)

Now, take average of the contributions in these six sequences:

A=(10+50+35+10+50+40)/6=195/6=**32.5**

B=(50+20+35+35+20+20)/6=180/6=**30**

C=(40+30+30+55+30+40)/6=225/6=**37.5**

These are called the Shapley values of the three people.

For A, even though he can only get 10 points if working alone, he is very cooperative and able to work well with both B and C, so his Shapley value is high. We can say that his ability score is 10, and his cooperation score is 22.5. Meanwhile, the cooperation scores of B and C are 10 and 7.5 respectively.

The idea of Shapley value can be applied to other areas, such as cost allocation. For example, 4 factories have decided to build a power plant together to supply the electricity needed, how much of the construction cost should each factory pay? Just treat cost as negative points can calculate the Shapley value of each factory. That’s a fair way to solve the problem.

Shapley value not only considers the ability of each member, but also takes into account the cooperation among the members. Thus, it is a fair evaluation system. **If you think those four criteria are applicable, then Shapley’s allocation is the only way you should follow.**