<p>Perhaps there is an elegant solution, but the problem is simple enough that one can reason graphically. In the case where
<span class="math-container">$$
0 \leq p_A,p_B \leq 1
$$</span>
and
<span class="math-container">$$
p_{AB} \leq p_A + p_B
$$</span>
we have something like this:</p>
<p><a href="https://i.sstatic.net/uzuCZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uzuCZ.png" alt="enter image description here" /></a></p>
<p>In the <span class="math-container">$(v_A,v_B)$</span> coordinate system the green area shows where the buyer achieves the highest surplus by buying <span class="math-container">$A$</span> only, so where
<span class="math-container">$$
v_A + v_B - p_{AB} \leq v_A - p_A  \text{ and } 0 \leq v_A - p_A.
$$</span>
Blue area shows where buyer buys <span class="math-container">$B$</span> only, and in the red area they buy both products. Since the probability distributions are independent and uniform, the probabilibities are proportional to these areas, thus you can get them as not too complicated formulas of <span class="math-container">$p_A,p_B,p_{AB}$</span>. Then you can sketch the expected profit
<span class="math-container">$$
P(A \text{ only}) p_A + P(B \text{ only}) p_B + P(A \text{ and } B) p_{AB}, 
$$</span>
and maximize this w.r.t. the prices. The problem is that these geometric formulas are only valid when the above constraints hold, otherwise you have to rethink them, and there could potentially be many cases; though my guess is that the real solution lies within the above constraints.</p>

CC BY-SA 4.0
2023-02-26T19:26:18.067Z
Your answer is actually as far as I got. I find $P(A$ only $)=1-p_A, P(B$ only $)=1-p_B$, and $p(A$ and $B)=1-\frac{\left(P_{A B}\right)^2}{2}, \mathrm{I}$ believe. However, I am not sure if this expected profit is correct since there is some overlap with these probabilities. For example, you can't buy A and B and then buy A individually.
Student
CC BY-SA 4.0
2023-02-26T19:43:07.593Z
@AaronZheng $P(A \text{ only}) \neq 1 - p_A$. That is equal to the total area to the right of the green line. The green rectangle - defined by the intersections of the green and red lines - is smaller than this.
Giskard
CC BY-SA 4.0
2023-02-26T19:43:50.103Z
"*there is some overlap with these probabilities*" The three events described above are mutually exclusive, notice the words "only" in the descriptions. The inequalities
$$
v_A + v_B - p_{AB} \leq v_A - p_A  \text{ and } 0 \leq v_A - p_A
$$
iny my answer are about the choices of "$A$ only" and "$A$ and $B$" being mutually exclusive.
Giskard
CC BY-SA 4.0
2023-02-26T21:44:12.983Z
Letting $p(A \text { and } B)=p\left(v_A+v_B-p_{A B} \geq v_A-p_A \wedge v_A+v_B-p_{A B} \geq v_B-p_B \wedge v_A+v_B \geq p_{A B}\right)$, and a few lines of code, we obtain that $p_A,p_B=0.66$ and $p_{AB}=0.862$! This answer is given in Borger's book on page 28.
Student
CC BY-SA 4.0
2023-02-27T06:49:26.517Z
@AaronZheng If you have a reference for the problem (Book X, problem Y), would you please include that at the top of your question? This will help other users find it in search.
Giskard
CC BY-SA 4.0
2023-02-26T18:24:02.923Z
2023-02-27T06:51:37.050Z
2023-02-27T06:51:37.050Z
Giskard
<p>Suppose a seller has two distinct indivisible goods, good <span class="math-container">$A$</span> and good <span class="math-container">$B$</span>, for sale. For simplicity, we assume that the seller values the goods at zero and is risk-neutral, so that she seeks to maximize her expected revenue. There is one buyer in this model. The parameters <span class="math-container">$v_A$</span> and <span class="math-container">$v_B$</span> indicate the buyer's willingness to pay for the two goods. These parameters are known to the buyer but not known to the seller. The seller's belief about these two parameters is given by the uniform distribution <span class="math-container">$F$</span> over the unit square <span class="math-container">$[0,1]^2$</span>. Note that we assume here that <span class="math-container">$v_A$</span> and <span class="math-container">$v_B$</span> are stochastically independent. We assume that the seller quotes three prices: <span class="math-container">$p_A, p_B$</span>, and <span class="math-container">$p_{A B}$</span>. The interpretation is that the buyer can buy good <span class="math-container">$A$</span> at price <span class="math-container">$p_A$</span>, good <span class="math-container">$B$</span> at price <span class="math-container">$p_B$</span>, or goods <span class="math-container">$A$</span> and <span class="math-container">$B$</span> at price <span class="math-container">$p_{A B}$</span>. We assume that the seller cannot stop the buyer from buying goods <span class="math-container">$A$</span> and <span class="math-container">$B$</span> at price <span class="math-container">$p_A+p_B$</span>, so the price <span class="math-container">$p_{A B}$</span>, if it is to have any effect, has to satisfy <span class="math-container">$p_{A B} \leq p_A+p_B$</span>. What is the optimal choice of <span class="math-container">$p_A, p_B$</span> and <span class="math-container">$p_{A B}$</span>?</p>

CC BY-SA 4.0
2023-02-27T07:38:20.893Z
Please pick a more descriptive title.
Rodrigo de Azevedo
<mechanism-design>
Book: Introduction to Mechanism Design, Problem Outlined on Page 28
Student
# Book: Introduction to Mechanism Design, Problem Outlined on Page 28

(Asked by: username_0 on 2023-02-26)

Suppose a seller has two distinct indivisible goods, good $A$ and good $B$, for sale. For simplicity, we assume that the seller values the goods at zero and is risk-neutral, so that she seeks to maximize her expected revenue. There is one buyer in this model. The parameters $v\_A$ and $v\_B$ indicate the buyer's willingness to pay for the two goods. These parameters are known to the buyer but not known to the seller. The seller's belief about these two parameters is given by the uniform distribution $F$ over the unit square $[0,1]^2$. Note that we assume here that $v\_A$ and $v\_B$ are stochastically independent. We assume that the seller quotes three prices: $p\_A, p\_B$, and $p\_{A B}$. The interpretation is that the buyer can buy good $A$ at price $p\_A$, good $B$ at price $p\_B$, or goods $A$ and $B$ at price $p\_{A B}$. We assume that the seller cannot stop the buyer from buying goods $A$ and $B$ at price $p\_A+p\_B$, so the price $p\_{A B}$, if it is to have any effect, has to satisfy $p\_{A B} \leq p\_A+p\_B$. What is the optimal choice of $p\_A, p\_B$ and $p\_{A B}$?

**Tags:** mechanism-design
> **Comments:**
> **username_1**: Please pick a more descriptive title.  

---

## 1 answers

### Answer by username_2 on 2023-02-26. Score: 3 | Accepted answer

Perhaps there is an elegant solution, but the problem is simple enough that one can reason graphically. In the case where
$$
0 \leq p\_A,p\_B \leq 1
$$
and
$$
p\_{AB} \leq p\_A + p\_B
$$
we have something like this:

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

In the $(v\_A,v\_B)$ coordinate system the green area shows where the buyer achieves the highest surplus by buying $A$ only, so where
$$
v\_A + v\_B - p\_{AB} \leq v\_A - p\_A \text{ and } 0 \leq v\_A - p\_A.
$$
Blue area shows where buyer buys $B$ only, and in the red area they buy both products. Since the probability distributions are independent and uniform, the probabilibities are proportional to these areas, thus you can get them as not too complicated formulas of $p\_A,p\_B,p\_{AB}$. Then you can sketch the expected profit
$$
P(A \text{ only}) p\_A + P(B \text{ only}) p\_B + P(A \text{ and } B) p\_{AB},
$$
and maximize this w.r.t. the prices. The problem is that these geometric formulas are only valid when the above constraints hold, otherwise you have to rethink them, and there could potentially be many cases; though my guess is that the real solution lies within the above constraints.

> **Comments:**
> **username_0**: Your answer is actually as far as I got. I find $P(A$ only $)=1-p_A, P(B$ only $)=1-p_B$, and $p(A$ and $B)=1-\frac{\left(P_{A B}\right)^2}{2}, \mathrm{I}$ believe. However, I am not sure if this expected profit is correct since there is some overlap with these probabilities. For example, you can't buy A and B and then buy A individually.  
> **username_2**: @AaronZheng $P(A \text{ only}) \neq 1 - p_A$. That is equal to the total area to the right of the green line. The green rectangle - defined by the intersections of the green and red lines - is smaller than this.  
> **username_2**: "*there is some overlap with these probabilities*" The three events described above are mutually exclusive, notice the words "only" in the descriptions. The inequalities $$ v_A + v_B - p_{AB} \leq v_A - p_A  \text{ and } 0 \leq v_A - p_A $$ iny my answer are about the choices of "$A$ only" and "$A$ and $B$" being mutually exclusive.  
> **username_0**: Letting $p(A \text { and } B)=p\left(v_A+v_B-p_{A B} \geq v_A-p_A \wedge v_A+v_B-p_{A B} \geq v_B-p_B \wedge v_A+v_B \geq p_{A B}\right)$, and a few lines of code, we obtain that $p_A,p_B=0.66$ and $p_{AB}=0.862$! This answer is given in Borger's book on page 28.  
> **username_2**: @AaronZheng If you have a reference for the problem (Book X, problem Y), would you please include that at the top of your question? This will help other users find it in search.  

---

