# Exercise 6

To create an analog of the Vickrey Auction, we also have sealed bids.

* It would be useful that all winners are going to pay the same
* The players with the $k$ highest bids, should all receive items
* All winning players should pay the $k+1$ highest bid

We have 3 players and 2 goods

Player 1: $\quad v_1 = 10 \quad b_1 = 10$
Player 2: $\quad v_2 = 7 \quad b_2 = 7$
Player 3: $\quad v_3 = 3 \quad b_3 = 3$

# Exercise 7

To create an analog for the Vickrey Auction, we also want to maximize social welfare.
Differently than before, our valuations are representing a cost, and not a surplus to the society, so we want to minimize the cost, by selecting the lowest bid.

We could also reformulate this, and say the minimization of social cost, is the minimization over all valuations

We are going to choose the payment

$u_i = v_i(a) - p_i$

$u_i = v_i \times x(b_i) - p_i$

$v_i(a) = \begin{cases}
0 \qquad \text{if }b_i \neq \max b\\
v_i \quad\text{if } b_i = \max b
\end{cases}$

$z > y$