ARKANSAS AUCTIONEER CORE EXAM SET QUESTIONS AND
SOLUTIONS 2025/2026 GRADED A+
✔✔Distinguish between private and common values. - ✔✔Common: A bidder's
valuation for the item is affected by the information that other bidders have. With pure
common values, inherent value of item is the same for everybody (e.g. diamonds or
some real estate property).
Private: the valuation is not affected by the information of other bidders. Only the bidder
knows how much he values the item.
✔✔Describe the proxy bid mechanism. - ✔✔A bidder submits the maximum amount
they are willing to pay for a property and the system enters bids on their behalf as
necessary in pre-determined increments until they reach the max amount entered.
For example, imagine a foreclosed house has a minimum (reserve) price of $10, you bid
in increments of $100 and you make your maximum acceptable bid $500. The computer
algorithm will make your first bid $110. If someone bids $150, the computer algorithm
will bid $250 for you (increase from the current highest bid by $100). If no one bids
higher and the auction ends, you end up paying $250, i.e. the second-highest bid ($150)
and the pre-defined increment ($100). Hence, it resembles a second-price auction.
✔✔Explain, in words, what is the difference between saying that the bidding strategy
β∗(·)is
(i) a (symmetric) BNE and
(ii) that it is a (symmetric) BNE in weakly dominant strategies. - ✔✔(i) symmetric BNE:
you have no incentive to deviate from your strategy as long as the other bidders are
following the same strategy and have the same valuations.
(ii) symmetric BNE in weakly dominant strategies: an even stronger concept, saying that
you have no incentive to deviate from your strategy REGARDLESS of how other
bidders are bidding.
Hence, a BNE in weakly dominant strategies is a SUBSET of symmetric BNEs.
✔✔Suppose that the bidding strategy β∗(·) is a weakly dominant strategy for all bidders
in a given auction. Show that all bidders bidding according to β∗(·) is then a (symmetric)
BNE. - ✔✔Firstly, if it's a BNE in weakly dominant strategies, then by definition is also
has to be a symmetric BNE. Why?
If β∗(vi) is a weakly dominant strategy for all bidders in a given auction, then for all i and
any vi it is true that:
E[ui(β∗(vi), ̃β(·); vi)] ≥ E[ui(b, β
̃ (·); vi)]
,Where ̃β(·) is any bidding strategy the rival may follow. Here we see that the rival's and
your bidding strategies may be different in a WEAKLY dominated BNE.
Now, if we set β
̃ (·) =β∗(·) (because in a symmetric BNE, you and your rivals follow the
same strategy), we get that for all i and any vi it is true that:
E[ui(β∗(vi), β∗(·); vi) ] ≥ E[ui(b, β∗(·); vi) ]
**This is the definition of a symmetric BNE!
✔✔In class, we argued that bidding-your-valuation is the unique SYMMETRIC BNE of
SPA when bidders have SIPV. However, this is not the unique BNE in a SPA. In fact,
there are other asymmetric BNE that involve bidders using weakly dominated
strategies.
Suppose that bidders' valuations are independent and identically distributed
Uniform[0,1] (important!). Show that bidder 1 bidding $1 regardless of her valuation
(i.e.,β1(v1) = 1) and all the other bidders bidding $0 regardless of their valuations (i.e.,
βj(vj) = 0 for j= 2,...,n) is also a BNE. - ✔✔If Bidder 1 deviates and bids $0 while others
are also bidding $0, she wins with probability 1/n (n being the number of bidders, and
this is decided by a coin toss). And gets a payoff of vi - 0 if she wins or 0 if she loses.
Since there is a probability of losing, there is no incentive to deviate.
If Bidder 1 deviates and bids $2 (higher), she also always wins and gets the same
payoff vi - 0 (because this is SPA and second highest bid will always be 0 if the rival's
follow their strategy).
---> we also need to show that Bidder 2 (the one bidding $0) has no incentive to deviate
when Bidder 1 is always bidding $1.
If bidder 2 were to deviate and bid $1, he would now tie with bidder 1 and with
probability 1/2 he obtains a payoff of 0 (since we are dealing with a Uniform [0,1]
distribution and the maximum valuation he can have is $1).
If bidder 2 decides to deviate and bid above $1, the expected payoff will always be
negative, because the maximum valuation is 1.
✔✔Suppose you have a single, one-of-a-kind item to sell to a single agent with a private
valuation which is drawn from an Uniform[0,100] distribution.
The agent knows her value but you don't.
(a) If you were to run a FPA to sell the item, how much do you anticipate the (single)
agent would bid? How much are your revenues?
, (b) If you were to run a SPA, how much would the (single) agent bid? How much are
your revenues? Hint:If there is only one bid submitted in a SPA, we consider that the
second highest bid is, effectively, $0. - ✔✔(a) If I know that I have no competition, I
would simply bid $0. Therefore, the seller's revenues will be $0.
(b) It is still a weakly dominant strategy to bid truthfully. The bidder will hence bid her
valuation, she will win and pay $0 (the actual 2nd highest bid). The seller will receive $0.
You could theoretically bid any sum, but you have no incentive to deviate from the
weakly dominant strategy of bidding your valuation, since bidding anything above it
leaves you with the same outcome, so you're indifferent.
✔✔Suppose you post a price, p≥0, that the agent can either take or leave.
a) What is the probability that the agent would buy the item given that the price is p?
b) Write the expression for your expected revenues (as a seller) as a function of p.
c) What price should you set to maximize your revenues? - ✔✔a) The agent will take
the offer whenever her valuation is greater than or equal to the price.
Probability of buying item = Pr(V ≥ p).
= 1 - Pr(V≤p)
= 1 - F(p) --> i.e. probability that V will have a value less than or equal to p.
= 1 - p/100
This is because it's uniformly distributed U[0,100].
You can also say 100-p.
E.g. if p = 1, then there is a 99% area of the uniform distribution to the right of p=1
where the valuation can be.
E.g. If p = 99, then there is only 1% area of the uniform distribution to the right of p=99
where the valuation can lie so that it's above p. We see that the higher the price, the
lower the probability (area of distribution) that the bidder's valuation will be found in that
area.
b) price p x the probability that the transaction occurs. Hence (100 - p)p = 100p - p^2
c) To optimize this problem, we take first derivative of expected revenues and set to 0
and solve.
100 - p^2 -- > 100 - 2p = 0
p* = 50
✔✔What does it mean for values to be private information? - ✔✔That values are private
information means that each bidder knows his or her valuation but this information is not
known by other bidders (hence, the information is private and not public).
SOLUTIONS 2025/2026 GRADED A+
✔✔Distinguish between private and common values. - ✔✔Common: A bidder's
valuation for the item is affected by the information that other bidders have. With pure
common values, inherent value of item is the same for everybody (e.g. diamonds or
some real estate property).
Private: the valuation is not affected by the information of other bidders. Only the bidder
knows how much he values the item.
✔✔Describe the proxy bid mechanism. - ✔✔A bidder submits the maximum amount
they are willing to pay for a property and the system enters bids on their behalf as
necessary in pre-determined increments until they reach the max amount entered.
For example, imagine a foreclosed house has a minimum (reserve) price of $10, you bid
in increments of $100 and you make your maximum acceptable bid $500. The computer
algorithm will make your first bid $110. If someone bids $150, the computer algorithm
will bid $250 for you (increase from the current highest bid by $100). If no one bids
higher and the auction ends, you end up paying $250, i.e. the second-highest bid ($150)
and the pre-defined increment ($100). Hence, it resembles a second-price auction.
✔✔Explain, in words, what is the difference between saying that the bidding strategy
β∗(·)is
(i) a (symmetric) BNE and
(ii) that it is a (symmetric) BNE in weakly dominant strategies. - ✔✔(i) symmetric BNE:
you have no incentive to deviate from your strategy as long as the other bidders are
following the same strategy and have the same valuations.
(ii) symmetric BNE in weakly dominant strategies: an even stronger concept, saying that
you have no incentive to deviate from your strategy REGARDLESS of how other
bidders are bidding.
Hence, a BNE in weakly dominant strategies is a SUBSET of symmetric BNEs.
✔✔Suppose that the bidding strategy β∗(·) is a weakly dominant strategy for all bidders
in a given auction. Show that all bidders bidding according to β∗(·) is then a (symmetric)
BNE. - ✔✔Firstly, if it's a BNE in weakly dominant strategies, then by definition is also
has to be a symmetric BNE. Why?
If β∗(vi) is a weakly dominant strategy for all bidders in a given auction, then for all i and
any vi it is true that:
E[ui(β∗(vi), ̃β(·); vi)] ≥ E[ui(b, β
̃ (·); vi)]
,Where ̃β(·) is any bidding strategy the rival may follow. Here we see that the rival's and
your bidding strategies may be different in a WEAKLY dominated BNE.
Now, if we set β
̃ (·) =β∗(·) (because in a symmetric BNE, you and your rivals follow the
same strategy), we get that for all i and any vi it is true that:
E[ui(β∗(vi), β∗(·); vi) ] ≥ E[ui(b, β∗(·); vi) ]
**This is the definition of a symmetric BNE!
✔✔In class, we argued that bidding-your-valuation is the unique SYMMETRIC BNE of
SPA when bidders have SIPV. However, this is not the unique BNE in a SPA. In fact,
there are other asymmetric BNE that involve bidders using weakly dominated
strategies.
Suppose that bidders' valuations are independent and identically distributed
Uniform[0,1] (important!). Show that bidder 1 bidding $1 regardless of her valuation
(i.e.,β1(v1) = 1) and all the other bidders bidding $0 regardless of their valuations (i.e.,
βj(vj) = 0 for j= 2,...,n) is also a BNE. - ✔✔If Bidder 1 deviates and bids $0 while others
are also bidding $0, she wins with probability 1/n (n being the number of bidders, and
this is decided by a coin toss). And gets a payoff of vi - 0 if she wins or 0 if she loses.
Since there is a probability of losing, there is no incentive to deviate.
If Bidder 1 deviates and bids $2 (higher), she also always wins and gets the same
payoff vi - 0 (because this is SPA and second highest bid will always be 0 if the rival's
follow their strategy).
---> we also need to show that Bidder 2 (the one bidding $0) has no incentive to deviate
when Bidder 1 is always bidding $1.
If bidder 2 were to deviate and bid $1, he would now tie with bidder 1 and with
probability 1/2 he obtains a payoff of 0 (since we are dealing with a Uniform [0,1]
distribution and the maximum valuation he can have is $1).
If bidder 2 decides to deviate and bid above $1, the expected payoff will always be
negative, because the maximum valuation is 1.
✔✔Suppose you have a single, one-of-a-kind item to sell to a single agent with a private
valuation which is drawn from an Uniform[0,100] distribution.
The agent knows her value but you don't.
(a) If you were to run a FPA to sell the item, how much do you anticipate the (single)
agent would bid? How much are your revenues?
, (b) If you were to run a SPA, how much would the (single) agent bid? How much are
your revenues? Hint:If there is only one bid submitted in a SPA, we consider that the
second highest bid is, effectively, $0. - ✔✔(a) If I know that I have no competition, I
would simply bid $0. Therefore, the seller's revenues will be $0.
(b) It is still a weakly dominant strategy to bid truthfully. The bidder will hence bid her
valuation, she will win and pay $0 (the actual 2nd highest bid). The seller will receive $0.
You could theoretically bid any sum, but you have no incentive to deviate from the
weakly dominant strategy of bidding your valuation, since bidding anything above it
leaves you with the same outcome, so you're indifferent.
✔✔Suppose you post a price, p≥0, that the agent can either take or leave.
a) What is the probability that the agent would buy the item given that the price is p?
b) Write the expression for your expected revenues (as a seller) as a function of p.
c) What price should you set to maximize your revenues? - ✔✔a) The agent will take
the offer whenever her valuation is greater than or equal to the price.
Probability of buying item = Pr(V ≥ p).
= 1 - Pr(V≤p)
= 1 - F(p) --> i.e. probability that V will have a value less than or equal to p.
= 1 - p/100
This is because it's uniformly distributed U[0,100].
You can also say 100-p.
E.g. if p = 1, then there is a 99% area of the uniform distribution to the right of p=1
where the valuation can be.
E.g. If p = 99, then there is only 1% area of the uniform distribution to the right of p=99
where the valuation can lie so that it's above p. We see that the higher the price, the
lower the probability (area of distribution) that the bidder's valuation will be found in that
area.
b) price p x the probability that the transaction occurs. Hence (100 - p)p = 100p - p^2
c) To optimize this problem, we take first derivative of expected revenues and set to 0
and solve.
100 - p^2 -- > 100 - 2p = 0
p* = 50
✔✔What does it mean for values to be private information? - ✔✔That values are private
information means that each bidder knows his or her valuation but this information is not
known by other bidders (hence, the information is private and not public).
