GEK1544 The Mathematics of Games

Suggested Solutions to Tutorial 10

In the following questions, the payoff is defined by

$ (valuation of the article)− $ (price to be paid )

in case the player wins the bid, and is set to be zero if the player loses the bid, or if theauction is inclusive (two persons bid the same price).

1. In a first-price sealed-bid auction three bidders are involved with valuations of thearticle given by

V1 > V2 > V3 (arranged in this order) .

The bidding prices are B1 for the first bidder (with valuation V1), likewise, B2 and B3 .In any Nash equilibrium, demonstrate the following.

(a) Show that the two highest bids are the same.

(b) Moreover, show that the highest bid is at least V2 .

Suggested Solution. (a) Suppose not. Consider the case where

B1 > maximum {B2 , B3} .

In this situation, the first bidder can lower the bidding price a bit and still wins the auction,while increases the payoff, contradicting the definition of Nash equilibrium. Likewise,consider for the cases

B2 > maximum {B1 , B3}

andB3 > maximum {B1 , B2} ,

again the winning bidder can lower the bidding price a bit and still wins the auction,while increases the payoff. We conclude that in any Nash equilibrium, two of the highestbids are the same.

(b) Suppose not, that is

V2 > maximum {B1 , B2 , B3} .

Base on (a), as nobody wins the auction, everyone’s payoff is zero. But then the secondbidder can win the auction by bidding a price that is just below V2, and still make apositive payoff (no matter how small, it is better than zero). Hence we must have

V2 ≤ maximum {B1 , B2 , B3} .

2. In a second-price sealed-bid auction three bidders are involved with valuations of thearticle given by

V1 > V2 > V3 (arranged in this order) .

The bidding prices are B1 for the first bidder (with valuation V1), likewise, B2 and B3 .

(a) Show that the bidding strategies

(2.1) (B1, B2, B3) = (V2, V1, 0)

is a Nash equilibrium .

(b) Show that in any bidding strategies

(B1, B2, B3)

with B2 6= V2 , bidder 2 won’t lose out if she/he changes the bidding price from B2 to V2 .

Suggested Solution. (a) We have the following cases.

(i) For the first player , the “ claimed equilibrium (2.1) ” payoff = 0 , as the player losesthe auction . Suppose the player changes according to :

B1 → V1 + “ a bit ” =⇒ the first player wins and pays V1

=⇒ zero pay-off = before ;

B1 → a new bid ≤ V1 =⇒ no difference ; the first player still loses

=⇒ zero payoff = before .

(ii) For the second player , the “ claimed equilibrium (2.1) ” payoff = V2 − V2 = 0 , asthe player wins the auction and pays the second highest bid . Suppose the player changesaccording to :

B2 → V1 + “ a bit ” =⇒ no difference ; the second player still wins

=⇒ payoff = V2 − V2 = 0 = before ;

B2 → a new bid B2 which satisfies V2 < B2 < V1

=⇒ no difference ; the second player still wins

=⇒ pay-off = V2 − V2 = 0 = before ;

B2 → B2 ≤ V2 =⇒ the second player does not win =⇒ pay-off = 0 = before .

(iii) For the third player, any changes in strategy only make a difference if B3 → B3 > V1 ,in which the third player wins the auction, and pays the second highest bid, which is nowV1 . Thus the payoff is V3 − V1 < 0 − worse off.

Hence anyone changes with other not changing , the payoff cannot be improved. It isindeed a Nash equilibrium.

(b) We have four cases based on the diagram :

V2 > max {B1 , B3}Win−−−− + −−−−−−−−−Loss

V2 ≤ max {B1 , B3}

* Suppose bidder two wins the auction and

V2 > max {B1 , B3} =⇒ payoff for bidder 2 = V2 −max {B1 , B3} .

In this case if bidder 2 changes the bidding price from B2 to V2, bidder 2 still wins theauction and

new payoff of bidder 2 = V2 −max {B1 , B3} ,

which is the same as before (i.e., do not lose out) .

* Suppose bidder 2 wins the auction and

V2 ≤ max {B1 , B3} =⇒ payoff of bidder 2 = V2 −max {B1 , B3} ≤ 0 .

In this case if bidder 2 changes the bidding price from B2 to V2, bidder 2 either loses theauction or gets a draw (i.e., the auction does not have a winner). In this case

new payoff of bidder 2 = 0 ,

which is the same or better than before (again do not lose out) .

* Suppose bidder 2 loses the auction or the auction has no conclusion (payoff = 0), and

V2 ≤ max {B1 , B3} .

In this case if bidder 2 changes the bidding price from B2 to V2, bidder 2 either loses theauction or gets a draw. In this case

new payoff of bidder 2 = 0 ,

which is the same as before (again won’t lose out) .

* Suppose bidder 2 loses the auction or the auction has no conclusion (payoff = 0), and

V2 > max {B1 , B3} .

If bidder 2 changes the bidding price from B2 to V2, bidder 2 wins the auction. In thiscase

new payoff of bidder 2 = V2 −max {B1 , B3} > 0 ,

which is better than before .

Considering all the cases, bidder 2 won’t lose out if she/he changes the bidding price fromB2 to V2 .