Question

1. Queen Elizabeth has decided to auction off the crown jewels. There are two bidders: the Sultan and the Sheikh. Each will simultaneously submit a bid in a sealed envelope; the highest bidder will win, and pay what he bid. (This is a “First Price Auction”.) The Sultan is only allowed to bid an odd number: 1, 3, 5, 7, or 9. The Sheikh is only allowed to bid an even number: 2, 4, 6, 8, or 10. The Sultan places a value of 8 on the crown jewels, while the Sheikh values them at 7.Now suppose the game is a Second Price Auction instead, so the highest bidder still wins, but he pays the amount of the losing player’s bid. The players’ valuations of the jewels remain the same.

Find all Nash Equilibria of this game.

Answer 1

It is a problem of auction for crown jewels of Queen Elizabeth. Two bidders are Sultan of Brunei and Sheikh of Abu Dhabi. sultan has valued the jewel as 7 million pounds. Valuation of Sheikh is 8 million pound. Both of them will submit bid simulataeously. Sultan will bid odd numbers and Sheikh in even number. It will vary from 1 to 10. Highest bidder will win and has to pay bid amount. Thus pay off for a bidder is value of jewel to him minus the amount paid.

Here Sultan can bid in 5 different ways like 1,3,5,7 and 9. Sheikh can bid 2,4,6,8 and 10. Their game table with pay off are shown below.

Sultan   Sheikh
2   4   6   8   10
1   0,8-2=6   0,8-4=4   0,8-6=2   0,8-8=0   0,8-10=-2
3   7-3=4,0   0,8-4=4   0,8-6=2   0,8-8=0   0,8-10=-2
5   7-5=2,0   7-5=2,0   0,8-6=2   0,8-8=0   0,8-10=-2
7   7-7=0,0   7-7,0   7-7=0,0   0,8-8=0   0,8-10=-2
9   7-9=-2,0   7-9=-2,0   7-9=-2,0   7-9=-2,0   0,8-10=-2

Above pay off matrix clearly indicates the calculation process. Suppose Sultan has bid 5 and Sheikh has bid for 6. Then Sheikh will win. He has to pay 6 hundred million pound for the jewel. As its value is 8 hundred million pound to him, the pay off is 8-6=2 hundred million pound. Sultan being the looser will have zero payoff.

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(b) For ascertaining Nash equilibrium, first eliminate row 7,9 as they are dominated by other three bids 1,3 and 5. Simlarly eliminate column bids 8 and 10 as they are dominated by bids 2,4 and 6. After their elimination, it is a 3x3 pay off table as shown below:

Sultan   Sheikh
2   4   6
1   0,8-2=6^   0,8-4=4   0,8-6=2
3   7-3=4*,0   0,8-4=4   0,8-6=2^
5   7-5=2,0   7-5=2,0   0*,8-6=2^

Now compare the figures. It is explaied below:

1. Suppose sheikh has decided to bid 2. Then Sultan will bid 3 as it will give him highest pay off 4. Indicate it by * sign in the table.

2.Suppose sheikh has decided to bid 4. Then Sultan will bid 5 as it will give him highest pay off 2. Indicate it by * sign in the table.

3.Suppose sheikh has decided to bid 6. Then Sultan will bid aything between 1,3 and 5 and loose the auction. So his payoff is 0. Put * sign against all 0 value in cell (1,3),(2,3) and cell (3,3).

4. Now assume that Sultan has decide to bid for 1. Then Sheikh will go for 2 to get highest pay off 6. Put ^ sign here.

5. 4. Now assume that Sultan has decide to bid for 3. Then Sheikh will go for 4 to get highest pay off 4. Put ^ sign here.

6. Finally assume that Sultan has decide to bid for 5. Then Sheikh will go for 6 to get highest pay off 2. Put ^ sign here.

Now observe the cell where both * and ^ signs are appearing . It is cell (3,3). So Seikh will bid 6 and Sultan will bid 5. Ultimately Sheikh will win with 2 hudred crore pound pay off. It is Nash equilibrium..