Second-price sealed-bid auction

MAS

A single item is up for auction. Each buyer submits a (private) bid $b_i\ge 0$simultaneously in a sealed envelope. The item is given to the highest bidder who pays the second-highest bid.

• Every bidder attaches a private value $v_i>0$ to the item.

• The winner is selected uniformly at random from the k highest bidders.

• What is the utility?

• Is there a nash equilibrium? Is it unique?