**Homework number 3**

**1.
**__Strategy Proof __

We have a tree
T. Each player is located in a node in the tree T, which is its private
information. We like to select one node X. The cost of a player is the distance
(in edges) from his location to X. Build a strategy proof algorithm (without
money) to select X, which is not a dictatorship.

2.
** Sequential Mechanism:** Assume that we
have two Strategy Proof mechanisms, A and B over the same set of players. Would
running sequentially, first mechanism A and then mechanism B maintain the
strategy proof property?

HINT : Consider selling two identical products by
running sequentially two second price auctions.

CORRECTION:
The question should be: Is the joint action, where every agent bids its true
value in both mechanisms a Nash Equilibrium.

3.
__Market Clearing Price____:__ Assume
that we have *n* products and *n* buyers in a matching market. Each
buyer has a non-negative valuation for each product, and each buyer will buy
exactly one product. Assume that every buyer has a higher valuation for product
*a* then product *b*. Does this implies that in ANY market clearing
prices the price of product *a* is higher than
the price of product *b*.

__ __

__The
homework is due June 7, 2010__