<p>The correct answer is Option (b): <span class="math-container">$xA$</span>=4.5 and <span class="math-container">$xB$</span>=12.5.</p>
<p>Calculations:</p>
<ol>
<li><p><span class="math-container">$V(A)$</span>= 2, <span class="math-container">$V(B)$</span>= 10, <span class="math-container">$V(A,B)$</span>= 4+16-3=17</p>
</li>
<li><p>Calculation of Marginal Contributions (MC) by <span class="math-container">$A$</span> and <span class="math-container">$B$</span>:</p>
</li>
</ol>
<p><img src="https://i.sstatic.net/gwu5SwRI.png" alt="Table: Marginal Contributions, <span class="math-container">$A$</span> and <span class="math-container">$B$</span>.1" /></p>
<ol start="3">
<li><p>Shapley Values = Expected Marginal Contributions:</p>
<p><span class="math-container">$xA$</span>: {(1/2)*2+ (1/2)*7} = {(1/2)*9} = 4.5;</p>
<p><span class="math-container">$xB$</span>: {(1/2)*15+ (1/2)*10} = {(1/2)*25} = 12.5</p>
</li>
</ol>
<p>Note:</p>
<p>Shapley Value:
&quot;Average Marginal Contribution” of agent <span class="math-container">$i$</span>, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition.</p>
<p>(Within any such sequence of additions, look at agent <span class="math-container">$i$</span>′s MC at the time he is added)</p>
<p>References:</p>
<ol>
<li><a href="https://faculty.econ.ucdavis.edu/faculty/bonanno/teaching/122/Shapley.pdf" rel="nofollow noreferrer">https://faculty.econ.ucdavis.edu/faculty/bonanno/teaching/122/Shapley.pdf</a></li>
<li><a href="https://www.cs.ubc.ca/%7Ekevinlb/teaching/cs532l%20-%202007-8/lectures/lect23.pdf" rel="nofollow noreferrer">https://www.cs.ubc.ca/~kevinlb/teaching/cs532l%20-%202007-8/lectures/lect23.pdf</a></li>
</ol>

CC BY-SA 4.0
2024-12-03T19:29:47.347Z
Thank you for your answer!
opre
CC BY-SA 4.0
2024-12-14T06:48:43.427Z
@pragyashankar, homework ready solutions are not welcome on the site. You are encouraged to give a general approach to the problem (i.e. some symbolic theory behind the matter as a tip to OP, and as a explanation to other readers). Anyway, welcome on the SE.
garej
CC BY-SA 4.0
2024-12-14T18:23:23.737Z
Just to clarify, this wasn't a homework question, I wrote it myself as I was unclear how individual investments should be considered in Shapley values. The answer and calculations by (@)pragya shankar, though could be helpful for others, were already implicitly in my question and did not really help me much in fully clarifying how any individual investments should enter the Shapley values in general. I'd be very happy if there might be any further comments on this.  Thank you!
opre
CC BY-SA 4.0
2024-12-16T02:22:34.020Z
I thought you are looking for the basic concept regarding Shapley Value. You could check the following sources:                     
  Selcuk Ozyurt 
 https://youtu.be/HWRHKs3O_Us?si=VorKd3EsBCOYlOR6                      Game Theory Online https://youtu.be/qcLZMYPdpH4?si=EL9wfC_E_-4Pioic
pragya shankar
CC BY-SA 4.0
2024-12-16T05:44:17.147Z
Would request you to specify your doubt clearly.  Looked like calculation of Shapley Value (based on options given in the qn).
pragya shankar
CC BY-SA 4.0
2024-12-02T08:24:00.670Z
2024-12-21T06:00:56.493Z
2024-12-21T06:00:56.493Z
pragya shankar
<p>In my view, since by definition, profits are calculated after deducting all types of costs from the revenue, there is no need to subtract the cost of investment from the joint profits (again). The game presented above can be represented as a coalitional game with the following valuation function:
<span class="math-container">$$v(\emptyset)=0, v(\{A\})=2, v(\{B\})=10, v(\{A,B\})=4+16=20$$</span>
Therefore, Shapley Value of A is <span class="math-container">$\frac{1}{2}(2+10)=6$</span> and of B is <span class="math-container">$\frac{1}{2}(10+18)=14$</span>.</p>
<p>Besides, please note that the amount invested cannot be taken as the cost. Investment augments the capital stock. Cost of investment is the interest paid on the amount borrowed for the purpose of investment. There is also a possibility of depreciation which can also be treated as cost for the accounting purposes. Therefore, it is incorrect to subtract the investment amounts from the individual or joint profits to arrive at various valuations.</p>

CC BY-SA 4.0
2024-12-21T09:08:36.093Z
2024-12-21T09:40:51.077Z
2024-12-21T09:40:51.077Z
Amit
<p>Suppose firm <em>A</em> has a profit of 2 and firm <em>B</em> has a profit of 10.</p>
<p>The firms can make a joint investment of 3 to increase their profits. Suppose they share the investment by 1/3 and 2/3, respectively, i.e., firm <em>A</em> will invest 1, and firm <em>B</em> will invest 2.</p>
<p>If both firms make their respective investments, their profits will increase. The new profit of firm <em>A</em> will be 4 and the new profit of firm <em>B</em> will be 16.  Otherwise, both profits remain as earlier.</p>
<p>What is the correct Shapley Value here?</p>
<ul>
<li><p>Is it <em>xA</em>=5 and <em>xB</em>=12 due to <em>v(A)</em>=2-1=1, <em>v(B)</em>=10-2=8, and <em>v(A,B)</em>=17 (taking into account individual investment regardless of joint investment), or</p>
</li>
<li><p>Is it <em>xA</em>=4.5 and <em>xB</em>=12.5 due to <em>v(A)=2</em>, <em>v(B)=10</em>, and <em>v(A,B)=17</em> (considering that they will not individually invest but will only jointly invest)?</p>
</li>
</ul>
<p>Thank you for your help in clarifying!</p>

CC BY-SA 4.0
2024-12-14T06:46:23.170Z
is it a homework?
garej
CC BY-SA 4.0
2024-12-14T18:18:40.747Z
@garej, no, this wasn't a homework question, I wrote it myself as I was unclear how individual investments should be considered in Shapley values. The answer by (@)pragya shankar, though could be helpful for others, was already part of my question and did not really help me much in fully clarifying how any individual investments should enter the Shapley values in general. That's why I did not mark it as an accepted answer yet in case there might be further responses.  If you have any further comments or resources, I'd be very happy if you can also let me know. Thank you!
opre
<cooperative-game-theory>
How to take investments into account in Shapley value calculation?
opre
# How to take investments into account in Shapley value calculation?

(Asked by: username_0 on 2024-12-01)

Suppose firm *A* has a profit of 2 and firm *B* has a profit of 10.

The firms can make a joint investment of 3 to increase their profits. Suppose they share the investment by 1/3 and 2/3, respectively, i.e., firm *A* will invest 1, and firm *B* will invest 2.

If both firms make their respective investments, their profits will increase. The new profit of firm *A* will be 4 and the new profit of firm *B* will be 16. Otherwise, both profits remain as earlier.

What is the correct Shapley Value here?

* Is it *xA*=5 and *xB*=12 due to *v(A)*=2-1=1, *v(B)*=10-2=8, and *v(A,B)*=17 (taking into account individual investment regardless of joint investment), or
* Is it *xA*=4.5 and *xB*=12.5 due to *v(A)=2*, *v(B)=10*, and *v(A,B)=17* (considering that they will not individually invest but will only jointly invest)?

Thank you for your help in clarifying!

**Tags:** cooperative-game-theory

> **Comments:**
> **username_1**: is it a homework?  
> **username_0**: @username_1, no, this wasn't a homework question, I wrote it myself as I was unclear how individual investments should be considered in Shapley values. The answer by (@)username_2 username_2, though could be helpful for others, was already part of my question and did not really help me much in fully clarifying how any individual investments should enter the Shapley values in general. That's why I did not mark it as an accepted answer yet in case there might be further responses.  If you have any further comments or resources, I'd be very happy if you can also let me know. Thank you!  

---

## 2 answers

### Answer by username_2 on 2024-12-02. Score: 1

The correct answer is Option (b): $xA$=4.5 and $xB$=12.5.

Calculations:

1. $V(A)$= 2, $V(B)$= 10, $V(A,B)$= 4+16-3=17
2. Calculation of Marginal Contributions (MC) by $A$ and $B$:

![Table: Marginal Contributions, <span class=](https://i.sstatic.net/gwu5SwRI.png)$A$ and $B$.1" />

3. Shapley Values = Expected Marginal Contributions:

   $xA$: {(1/2)\*2+ (1/2)\*7} = {(1/2)\*9} = 4.5;

   $xB$: {(1/2)\*15+ (1/2)\*10} = {(1/2)\*25} = 12.5

Note:

Shapley Value:
"Average Marginal Contribution” of agent $i$, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition.

(Within any such sequence of additions, look at agent $i$′s MC at the time he is added)

References:

1. <https://faculty.econ.ucdavis.edu/faculty/bonanno/teaching/122/Shapley.pdf>
2. [https://www.cs.ubc.ca/~kevinlb/teaching/cs532l%20-%202007-8/lectures/lect23.pdf](https://www.cs.ubc.ca/%7Ekevinlb/teaching/cs532l%20-%202007-8/lectures/lect23.pdf)

> **Comments:**
> **username_0**: Thank you for your answer!  
> **username_1**: @username_2, homework ready solutions are not welcome on the site. You are encouraged to give a general approach to the problem (i.e. some symbolic theory behind the matter as a tip to OP, and as a explanation to other readers). Anyway, welcome on the SE.  
> **username_0**: Just to clarify, this wasn't a homework question, I wrote it myself as I was unclear how individual investments should be considered in Shapley values. The answer and calculations by (@)username_2 username_2, though could be helpful for others, were already implicitly in my question and did not really help me much in fully clarifying how any individual investments should enter the Shapley values in general. I'd be very happy if there might be any further comments on this.  Thank you!  
> **username_2**: I thought you are looking for the basic concept regarding Shapley Value. You could check the following sources:                        Selcuk Ozyurt   https://youtu.be/HWRHKs3O_Us?si=VorKd3EsBCOYlOR6                      Game Theory Online https://youtu.be/qcLZMYPdpH4?si=EL9wfC_E_-4Pioic  
> **username_2**: Would request you to specify your doubt clearly.  Looked like calculation of Shapley Value (based on options given in the qn).  

---

### Answer by username_3 on 2024-12-21. Score: 0

In my view, since by definition, profits are calculated after deducting all types of costs from the revenue, there is no need to subtract the cost of investment from the joint profits (again). The game presented above can be represented as a coalitional game with the following valuation function:
$$v(\emptyset)=0, v(\{A\})=2, v(\{B\})=10, v(\{A,B\})=4+16=20$$
Therefore, Shapley Value of A is $\frac{1}{2}(2+10)=6$ and of B is $\frac{1}{2}(10+18)=14$.

Besides, please note that the amount invested cannot be taken as the cost. Investment augments the capital stock. Cost of investment is the interest paid on the amount borrowed for the purpose of investment. There is also a possibility of depreciation which can also be treated as cost for the accounting purposes. Therefore, it is incorrect to subtract the investment amounts from the individual or joint profits to arrive at various valuations.


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