CS535 HW3

Problem 4 §

Let agents 1 (Alice) and 2 (Bob). The optimal case is when $1$ gets $a$ and $2$ gets $b$ if $\alpha >\beta$, and vice versa, thus maximum welfare for efficient allocation is

$$\underset{\text{ by agent 2 }}{\underbrace{2}}+\underset{\text{ by agent 1 }}{\underbrace{\text{max}(\alpha ,\beta )}}$$

$1$ will demand both initially when $p_a=p_b=0$ but will drop out once

$$\begin{array}{rl}2-(p_a+p_b)& <\text{max}(\alpha -p_a,\beta -p_b)\end{array}$$

Case $\alpha -p_a>\beta -p_b$. Then

$$\begin{array}{rlr}{p}_a+p_b& >2-\alpha +p_a& \\ p_b& >2-\alpha & \text{as }1<2-\alpha <2\text{ since }\alpha \in (0,1)\\ p_b& >1& \end{array}$$

And $1$ will not demand $b$ anymore. This will lead to two cases:

• If $p_a<p_b$ auction continues since both chooses $a$. $1$ will drop out eventually, as $p_a$ will be raised until $p_a>\alpha$ while $2$ still demands $a$ as long. Total welfare is $2$.
• If $p_b<p_a$ then $1$ will drop out, $2$ will get $b$ with total welfare $2$. Same can be shown for $\alpha -p_a<\beta -p_b$.

Problem 5 §

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