Question

In the following auction:

• There are three bidders, and there are two identical items for sale.

• Reservation prices are drawn independently from a uniform distribution on [0, 1].

• The high and middle bidders each get one of the items and pay their own bids. The low bidder pays nothing.

Suppose your opponent (who is not very strategic) makes the mistake of using the strategy “bid half your reservation price”. What is your optimal strategy against him? What is his optimal response to your optimal response? Explain why this shows there is no Nash equilibrium where the other player uses the “bid half your reservation price” strategy.

Answer 1

There are three bidders say b1, b2, and b3.

The first two highest bidders get the items by paying their own bidding amount.

The opponent strategy is to “bid half your reservation price”.

By going through the opponent strategy it is clear that my any reservation price bid would cost me much more. Because items are identical and out of three bidders two will get the items. So in that case lets b1 reservation price is 10 then opponent would bid 5. So in this situation, if I bid 5.1 then too I can get the item and can save 4.9. But as soon as I bid 5.1 or say (5+e) then opponent bid becomes (5+e)/2. This is a not ending game. Hence there can not be any Nash Equilibrium.