1. 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 the
auction is inclusive (two persons bid the same price).
1. In a first-price sealed-bid auction three bidders are involved with valuations of the
article 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 }
and
B3 > 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 highest
bids 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 second
bidder can win the auction by bidding a price that is just below V2 , and still make a
positive payoff (no matter how small, it is better than zero). Hence we must have
V2 ≤ maximum {B1 , B2 , B3 } .

2. 2. In a second-price sealed-bid auction three bidders are involved with valuations of the
article 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 = 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 loses
the 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 , as
the player wins the auction and pays the second highest bid . Suppose the player changes
according 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 now
V1 . Thus the payoff is V3 − V1 < 0 − worse off.
Hence anyone changes with other not changing , the payoff cannot be improved. It is
indeed a Nash equilibrium.

3. (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 the
auction 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 the
auction 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 the
auction 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 this
case
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 from
B2 to V2 .