E0-239  ELECTRONIC COMMERCE
January - April 2005

Problem Set # 3

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1. Consider an exchange where a single unit of an item is traded.
There are 4 sellers S1, S2, S3, S4 and 3 buyers B1, B2, and B3. Here are the
bids from the buyers and the asks from the sellers for the single item.
The objective is to maximize the surplus in the exchange.

S1: 10  S2: 12  S3: 14  S4: 16
B1: 8   B2: 12  B3: 18

Find a surplus maximizing allocation for this exchange. If now Vickrey
pricing is used, will the exchange have budget balance?

2. Suppose there are two units of the item, instead of one in the above
example. Repeat the exercise.

3. Suppose the Vickrey auction pricing mechanism is used for English
auctions. Will it generate more revenue, less revenue, or the same
revenue as the classical English auction. Why?

4. Dutch auction and first price sealed bid auction produce the same
average revenue even if all the assumptions of the benchmark model
(independent private values, symmetry, risk-neutal bidders, prices
depend on bids alone) are violated. Is this statement true or false.
Justify.     

5. Consider first price sealed bid auction under the benchmark model.
Let v_i be the valuation of bidder i. Let bidder i conjecture that
all other bidders are using a bidding function B to deicide their bids.
That is, if v_j is the valuation of bidder j, then bidder j will bid
B(v_j). Note that the individual valuations v_j are not known to
bidder i. Assume that B is a monotonically increasing function (which
guarantees that the inverse of B exists). Compute the probability that
bidder i will win with a bid b_i in the above setting.

6. For the benchmark model, it is found that variance of revenue is
lower in English or Vickrey auction than in first price sealed bid
or Dutch auction. How would a seller and a buyer use this information?

7. Assume that there is perfect competition (infinitely many bidders)
and assume the benchmark model. Is this good for the seller or the
buyers. What would be the price the winning bidder will pay?

8. Assume that there are two items {A,B} and three bidders {X,Y,Z}.
X has only one bid (AB, 400). Y has only one bid (AB, 500). Z also has
only one bid (B, 600). What is the revenue maximizing allocation for
this? What problem will this produce when Sandholm's algorithm is used.
How do you modify Sandholm's algorithm to "fix" this problem.

9. Sandholm's algorithm assumes that there is exactly one unit of each
item. If there are multiple units of each item, Sandholm's algorithm
can still be applied by viewing multiple units as multiple items.
Without taking this view, how do you generalize Sandholm's algorithm
for multiple units of different items.

10. Let there be two items {A,B} and three bidders {X,Y,Z}. Let the
values of each bidder for the bundles A, B, and AB be 10,15,25,
respectively. Let the bids be as follows:

Bidder   Bids for subsets

         A  B  AB
X        5  10 12
Y        8  5  12
Z        5  5  10

For the above bids, find the allocation,  payments, revenue to the market,
and revenue to the bidders. Assume XOR bids. Now do the following.                                                            

(a) Change the bids of X to his true values and repeat the computations.
(b) Now change the bids of Y to her true values and repeat.
(c) Finally, change the bids of Z to his true values and repeat.


11. Let there be 3 items {A,B,C} and three bidders {X,Y,Z}. Let the
XOR bids be as follows:

Bidder Bids submitted for Subsets
       A B  C  AB BC AC ABC
X     10 20 30 25 40 35 45
Y     15 25 25 30 35 40 55
Z      5 15 35 20 50 45 50

If GVA is the mechanism used, who will be the winners and what prices
will they pay? What valuations do you think the winners have for the
allocated subsets?