Bidding Problems
Several competitive situations involve bidding for contracts,
tenders, licenses, etc. All the bidding problems may be classified into
two groups:
- Auction Bids - In case of auction bids, the bids are open.
- Closed Bids - In case of closed bids, each bidder submits his bid
in a closed envelope.
1. Auction Bids or Open Bids
 Example
A chair and a table worth Rs. 80 and Rs.120 are to be auctioned at
a public sale. There are only two bidders - Vinay and Manish. Vinay
and Manish have Rs. 100 and Rs. 130 respectively. What should be their
strategies, if each bidder is interested in maximizing his return ?
Assume that both the bidders have complete information about each other's
money position.
Solution.
Suppose the bid increases successively by the amount Rs.
D. It should be noted that at any bid, each player has a
option to increase the bid or to leave the opponent's bid stand.
Suppose Manish has bid of Rs. x on the chair.
Case I
If Vinay allows Manish to win the chair for Rs. x then Manish will
have only (130 - x) rupees for bidding on the table. Thus, he can not
make his bid for the table more than Rs. 130 - x. Hence, Vinay will
definitely win the table in Rs. (130 - x + D).
Thus, Vinay's gain when he allows Manish to win the chair for Rs. x
is
Rs. [120 - (130 - x + D)]
= Rs. (x - D - 10)
Case II
To the contrary, if Vinay bids Rs. x + D
for the chair and Manish allows him to win at this bid, then Vinay's
gain is
Rs. [80 - (x + D)]
= 80 - x - D
Now since Vinay wants to maximize his return, he should bid x + D
for the chair provided
80 - x - D ³
x - D - 10
= 2x £ 90
= x £ 45
Thus, till x £ 45, Vinay should bid
for chair. When x > 45, he should allow Manish to win the chair for
that bid.
Likewise, in the two cases, Manish's gains are
[120 - (100 -y) - D] and [80 - (y + D)]
Where y is the Vinay's bid for the chair
Thus, Manish should bid y + D for the
chair provided
80 - (y + D) ³
120 - (100 -y) - D
= y £ 30
Obviously, Vinay will take the chair in Rs. 30 because he can increase
his bid without any loss upto Rs. 45, Manish will take the table in
Rs. (100 - 30) = 70 because Vinay, after winning the chair for Rs. 30
cannot increase his bid for the table more than Rs. 70. Thus, Manish
will get the table for Rs. 70. The gain of Vinay is Rs. (80 - 30) =
Rs. 50, and of Manish is Rs. (120 - 70) = Rs. 50.
2. Closed Bids
Example
A joystick and a keyboard worth Rs 80. and Rs. 100 are to be bid simultaneously
by two bidders A and B. Both have intention of devoting a total sum
of Rs. 110 for the items. If each uses a minimax criterion, find the
resulting bids.
Solution.
Since the bids are to be made simultaneously, they are closed bids.
Suppose P and Q are A's best bids for the joystick
and the keyboard respectively. A's best bids are those which give the
same profit to A on both the items. If t is the total profit associated
with a successful bid, then
2t = (80 - P) + (100 - Q)
2t = 180 -(P + Q)
2t = 180 -110
2t = 70
t = 35
P = 80 - t
= 80 - 35
= 45
and
Q =100 - t
= 100 - 35
= 65.
Thus, optimal bids for A are Rs. 45 for joystick, and Rs. 65 for keyboard.
The optimal bids for B will be same as A's optimal bids.
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