A few questions to warm up and check the basics, before the exam-type questions below. They are reshuffled each time, so you can keep retrying.
The following are exam-type questions in the style of the examination paper, with marks at the rates used in the papers. A question totalling fewer than 25 marks would, in the examination, be combined with further parts, often one of the examinable proofs, to make a full 25-mark question. Attempt them in full before reading the worked solutions. Some parts use expected values of ordered uniform samples; each question states the facts you may use, and their derivations are in the order statistics appendix of the textbook.
In class we ran a sealed-bid second-price auction. Suppose three bidders have private values \(v_1 = 10\), \(v_2 = 7\) and \(v_3 = 4\); the highest bidder wins and pays the second-highest bid.
(a) Explain the rules of a second-price (Vickrey) auction. [3]
(b) If all bidders bid truthfully, determine the winner, the price paid and the winner's payoff. [3]
(c) Show, for bidder 1, that bidding the true value \(v_1 = 10\) is a weakly dominant strategy, by considering bids above and below 10 and showing neither can do better whatever the others bid. [7]
(d) Explain why truthful bidding being a weakly dominant strategy is a desirable property of the second-price auction. [4]
(e) If the auction were instead first-price (the winner pays their own bid), would bidding your true value still be weakly dominant? Briefly explain. [2]
(a) Provide definitions for the following terms:
(b) Consider a sealed-bid second-price (Vickrey) auction for a single item, where the highest bidder wins and pays the second-highest bid.
(i) Show that bidding one's value \(b = v\) is a weakly dominant strategy. [7]
(ii) Hence state the Bayesian Nash equilibrium. [2]
(c) Two bidders have values drawn independently from the uniform distribution on \([0, 1]\) and bid truthfully. You may use that for two such values, \(\mathbb{E}[\min(v_1, v_2)] = \tfrac{1}{3}\) and \(\mathbb{E}[\max(v_1, v_2)] = \tfrac{2}{3}\).
(i) Compute the seller's expected revenue. [3]
(ii) Compute the expected payoff to the winning bidder. [3]
(iii) State, without further calculation, what the revenue equivalence theorem says the seller's expected revenue would be in a first-price auction for these bidders. [3]
(a) State the symmetric Bayesian Nash equilibrium bidding function for \(N\) bidders with values uniform on \([0, 1]\) in a first-price auction. [3]
(b) Take two such bidders, each bidding \(b(v) = v/2\). You may use that for two independent values \(v_1, v_2\) uniform on \([0, 1]\), \(\mathbb{P}(v_1 \leq z) = z\) for \(z \in [0, 1]\), \(\mathbb{E}[\min(v_1, v_2)] = \tfrac{1}{3}\) and \(\mathbb{E}[\max(v_1, v_2)] = \tfrac{2}{3}\).
(i) Compute the seller's expected revenue. [3]
(ii) Compute the expected payoff of a bidder who knows their own value is \(v\), taking the expectation over the other bidder's value. [4]
(iii) Compute the seller's expected revenue in the corresponding second-price auction and state the revenue equivalence result. [3]
(c) Explain the intuition behind revenue equivalence. [5]
(d) State the two conditions on the auctions that the revenue equivalence theorem requires, and confirm that they hold here. [3]
A single item is sold by sealed-bid first-price auction to \(N\) bidders. The values \(v_1, \dots, v_N\) are independent, each uniform on \([0, 1]\). The highest bidder wins and pays their own bid. We look for a symmetric equilibrium in which every bidder uses the same strictly increasing, differentiable bidding function \(b\) with \(b(0) = 0\). Throughout you may use that a value uniform on \([0, 1]\) satisfies \(\mathbb{P}(v_i \leq z) = z\) for \(z \in [0, 1]\), and that the expected value of the \(k\)-th smallest of \(N\) independent such values is \(\dfrac{k}{N + 1}\).
(a) Suppose every bidder other than bidder 1 uses \(b\). If bidder 1 has value \(v\) and submits the bid \(b(z)\), that is bids as though their value were \(z\), show that the probability they win is \(z^{N-1}\), and hence that their expected payoff is \(U(z) = z^{N-1}\bigl(v - b(z)\bigr)\). [6]
(b) In equilibrium the optimal choice is \(z = v\) for every \(v\). Use this to derive the differential equation \(\dfrac{d}{dv}\bigl[v^{N-1} b(v)\bigr] = (N-1) v^{N-1}\), and solve it with \(b(0) = 0\) to obtain \(b(v) = \dfrac{N-1}{N}\, v\). [10]
(c) Show that the seller's expected revenue is \(\dfrac{N-1}{N+1}\). [5]
(d) State the revenue equivalence theorem, and verify it for these bidders by computing the seller's expected revenue in the corresponding second-price auction. [4]
These exercises are entirely optional: they are extra practice, not extra content. The best way to use an AI assistant is to be its examiner, not its student: checking work is exactly the skill the examination rewards, and the skill you will need to use these tools well. Each exercise gives a transcript in the style of a widely used free AI chatbot answering one of the example questions above. Every transcript contains at least one error. Mark it as an examiner would: find each error, explain what goes wrong, and award a mark out of the marks of the original question. A marker's report for each transcript is on the solutions page.
A student pastes Question 1(c) into a free AI chatbot, which replies:
Claim: bidding the true value v1 = 10 is a strictly dominant strategy.
Suppose bidder 1 bids 12 instead. If the highest rival bid is 11,
bidder 1 wins and pays 11, making a loss of 1. So bidding higher is
worse.
Suppose bidder 1 bids 8 instead. If the highest rival bid is 9, bidder
1 loses and gets 0, whereas bidding 10 would have won for a profit of
1. So bidding lower is worse.
Since bidding higher is worse and bidding lower is worse, bidding
exactly 10 is strictly dominant. ∎
A student pastes Question 3(b) into a free AI chatbot, which replies:
First-price revenue: the winner is the bidder with the higher value and
pays their own bid, so the seller's expected revenue is
E[max(v1, v2)] = 2/3.
Second-price revenue: the winner pays the second-highest value, so the
expected revenue is E[min(v1, v2)] = 1/3.
Conclusion: the first-price auction raises twice the revenue of the
second-price auction, which is why sellers generally prefer first-price
auctions. 💰
You do not need any of this to follow the topic, but the following may help if you would like more background:
Worked solutions to the example questions →
Download the full quiz as a PDF (all questions, no solutions) →