Bidding Problems: Game Theory

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: Bidding Strategies

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 Bidding 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. Δ. 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 + Δ). Thus, Vinay's gain when he allows Manish to win the chair for Rs. x is

Rs. [120 - (130 - x + Δ)]
= Rs. (x - Δ- 10)

Case II

To the contrary, if Vinay bids Rs. x + Δ for the chair and Manish allows him to win at this bid, then Vinay's gain is
Rs. [80 - (x + Δ)]
= 80 - x - Δ

Now since Vinay wants to maximize his return, he should bid x + Δ for the chair provided
80 - x - Δ ≥ x - Δ - 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) - Δ] and [80 - (y + Δ)]
Where y is the Vinay's bid for the chair

Thus, Manish should bid y + Δ for the chair provided

80 - (y + Δ) ≥ 120 - (100 -y) - Δ
= 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.