# 1. Queen Elizabeth has decided

1. Queen Elizabeth has decided to auction off the crown jewels.There are two bidders: the Sultan and the Sheikh. Each willsimultaneously submit a bid in a sealed envelope; the highestbidder will win, and pay what he bid. (This is a “First PriceAuction”.) 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 SecondPrice Auction instead, so the highest bidder still wins, but hepays the amount of the losing player’s bid. The players’ valuationsof the jewels remain the same.

Find all Nash Equilibria of this game.

Answer:

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

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 areshown below.

Sultan Sheikh2 4 6 8 101 0,8-2=6 0,8-4=4 0,8-6=2 0,8-8=0 0,8-10=-23 7-3=4,0 0,8-4=4 0,8-6=2 0,8-8=0 0,8-10=-25 7-5=2,0 7-5=2,0 0,8-6=2 0,8-8=0 0,8-10=-27 7-7=0,0 7-7,0 7-7=0,0 0,8-8=0 0,8-10=-29 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 willwin. He has to pay 6 hundred million pound for the jewel. As itsvalue is 8 hundred million pound to him, the pay off is 8-6=2hundred million pound. Sultan being the looser will have zeropayoff.

———————————————————————————————————————————————————-

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

Sultan Sheikh2 4 61 0,8-2=6^ 0,8-4=4 0,8-6=23 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 3as it will give him highest pay off 4. Indicate it by * sign in thetable.

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

3.Suppose sheikh has decided to bid 6. Then Sultan will bidaything between 1,3 and 5 and loose the auction. So his payoff is0. 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 Sheikhwill go for 2 to get highest pay off 6. Put ^ sign here.

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

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

Now observe the cell where both * and ^ signs are appearing . Itis 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 isNash equilibrium..