Question

Game theory:

Analyze the second price" auction.
There are two bidders and one object being sold by auction. Each bidder
knows what the object is worth to him, but not what it is worth to the other
bidder. In other words, the object is worth v1 to bidder 1 and v2 to bidder 2.
Bidder 1 knows v1 but not v2, while bidder 2 knows v2 but not v1.
In the sealed bid second price auction, each bidder privately submits a bid
to the auctioneer. Let b1 be the bid submitted by bidder 1 and let b2 be the bid
submitted by bidder 2. The bidder submitting the highest bid wins the auction.
Rather than paying what he bid, the winning bidder pays the bid submitted by
the other bidder. Thus, if bi > bj , then bidder i's payo is vi ? bj and bidder
j's payo is zero. If they submit the same bid, then bidder 1 wins.

1. Can bidder i be worse o bidding bi > vi than bidding bi = vi?
2. Show that bidding higher than the bidder's valuation can never increase
his payoff. In other words, show that bidder i is never worse o bidding
bi = vi than bidding any bi > vi. ?
3. Can bidder j be worse o bidding bi < vi than bidding bi = vi?
4. Show that bidding lower than one's valuation can never increase a bidder's
valuation. In other words, show that bidder i is never worse o bidding
bi = vi than bidding any bi < vi. ?
5. What bid would you advise these bidders to submit?

Answer 1

In this case, there are two bidders :

Bidder 1 Bidder 2

Object Valuation /

( Reservation Price) V₁ V₂

Bid Value: b₁ b₂

Reservation price is the maximum price that an individual is ready to pay for a good.

A. Yes, definitely bidder i will always worse off if he submits a bid b_i > v_i because v_i is the reservation price or the maximum bid that he proposes for the auction and if he bid more than that then he will necessary worse off. So, he always tries to bid at that point where v_i = b_i.

B. Bidder i will never worse off if he bids where v_i = b_i. rather than b_i > v_i it means he never increases his bid more than v_i because if he bids more than his reservation price which is v_i then it means that he pays more than he values the bid, in that case, he will surely worse off. Hence he will always bid at v_i = b_i.

C. Yes, definitely bidder j will always better off if he submits a bid b_j< v_j because vjis the reservation price or the maximum bid that he proposes for the auction and if he bid less than that of his reservation price then he will necessary better off. Even if he bids at the point where b_j = v_j then still he will not worse off.

D. Bidder i will never worse off if he bids where v_i = b_i. rather than b_i < v_i it means he never increases his bid more than v_i because if he bids more than his reservation price which is v_i then it means that he pays more than he values the bid, in that case, he will surely worse off. Hence he will always bid at v_i = b_i. and if bids less than his reservation price than he still does not worse off. But he always pays equivalent to the reservation price.

E. The bid that I would suggest is to bid the reservation price that is V_i at that price he will not worse off. And that is the only solution to this game.