Question

You are competing for a contract in a second-price auction. The cost for you to fulfill the contract is $10m. Check if each of the following statements is true. If not, change the statement to make it correct.

1. If with probability 0.4, the lowest cost of your competitors is $6m and with probability 0.6, it is $15m. Then your expected profit from the auction is $1.4m.

2. (4 points) If one of your competitors can fulfill the contract at cost $7m, then you are going to lose at least $3m.

3. (4 points) If 30% of chance that one of your competitors has a cost below $6m, 30% of chance that all of them have a cost above $16m, and 40% of chance all of them have a cost somewhere between $8m and $11m. Then your expected profit is at least $1.8m.

4. (4 points) If all your competitors inflate their ask prices above their costs by 10%, then you will be worse if you do not do so.

Answer 1

Theory: In second price auction you pay the price of lowest bidder in the market excluding you , that means,

• if someone bids less than your bid -> you will get lower bid amount -> you make loss
• if everyone bids more than your bid -> you will get higher bid amount -> you make profit   

statement (1) is true
statement (2) is true
statement (3) is false : Then your expected profit is at least -$1.8m
statement (4) is false : then you will be better off irrespective of your decision to increase/decrease price

Explaination :

1. Expected Price for winning the contract = 0.4* $6m + 0.6 *15m= 2.4+9=$11.4m
Expected Profit=11.4-10=$1.4m  

2. Minimum cost in auction is atleast $7m , hence the company is set to loose atlease ($10m-$7m)=$3m

3. Taking the worst case scenerio and assuming contract is taken every time , Expected price for the contract :

• 30% of chance that one of your competitors has a cost below $6m (maximum loss = 6-10 = -4) [assuming that the cometitor prices the contract at $6m]
• 30% of chance that all of them have a cost above $16m (minimum profit=16-10=6) [assuming that the cometitor prices the contract at $16m]
• 40% of chance all of them have a cost somewhere between $8m and $11m(maximum loss = 8-10 = -2) [assuming that the cometitor prices the contract at $8m]

Expected profit/loss=0.3*(-4)+0.3*6+0.4*(-2)=-1.2+1.8-2.4=$-1.8

4. In this context your profit/loss doesn't depends on how much price you set but on your competitors price. If all your competitors increase their prices then you losses will decrease and profits will increase irrespective of your decision to inflate the price.