Auctions and mechanism design

Some of the most valuable things an African state owns are never priced in a shop: a band of radio spectrum, the licence to drill an offshore block, ninety-one days of the government's credit, a chest of Kenyan tea at Mombasa. Each is handed out by auction, and the rules of that auction decide both who gets the asset and how much the public collects. Choose them badly and you either hand a billion-shilling resource to the wrong firm or leave most of its value on the table. Auction theory treats those rules as the object of design — game theory meeting the state's balance sheet.

The four canonical formats

Start with the simplest problem: one seller, one indivisible object, n bidders, and bidder i privately knows a value vᵢ — the most they would pay. There are four textbook ways to sell it, and every real auction is a variation on one of them.

• English (ascending, open-outcry). The price rises and bidders drop out until one remains, paying roughly the level at which the last rival quit. The weekly Mombasa tea auction, one of the world's largest, runs this way: buyers bid openly, chest by chest, and price discovery is public.
• Dutch (descending). The price starts high and falls until the first bidder calls “mine”, winning at the price they stopped the clock. Fast — hence its use for perishables like cut flowers.
• First-price sealed-bid. Everyone submits one secret bid; the highest wins and pays what it bid. You never see rivals' numbers, so you can win and still regret overpaying.
• Second-price sealed-bid (Vickrey). Everyone submits one secret bid; the highest wins but pays the second-highest bid. That you do not pay your own number is exactly what makes honesty optimal — which we now prove.

Sealed versus open is not the distinction that matters; look at the incentives and the four collapse into two pairs. In a Dutch auction you watch the price fall and fix the single number at which you pounce, knowing only your own value and your beliefs about rivals — the identical decision to a first-price sealed bid. The falling clock reveals nothing before you act, so the two are the same game. Likewise the English auction: with private values you stay in until the price reaches your value, so the sale ends when the second-last bidder quits and the winner pays essentially the second-highest value — the Vickrey outcome. Dutch ≡ first-price; English ≈ second-price.

Vickrey's second-price auction: why truth-telling is optimal

Fix the rules. Each of n bidders privately knows a value vᵢ and submits one sealed bid bᵢ. The highest bid wins; the winner pays the second-highest bid; everyone else pays nothing. Claim: bidding your true value, bᵢ = vᵢ, is weakly dominant — it does at least as well as any other bid, and sometimes strictly better, against every combination of rivals' bids. Take one bidder with value v, and let B be the highest of all the other bids. You never observe B, but you need not: we show truth-telling is best for every value B could take. The engine of the proof is one fact — when you win you pay B, not your own bid — so your bid decides only whether you win, never how much you pay.

Case 1 — you bid above your value (b > v). Compare bidding b with bidding v. If B < v, both bids beat B, so both win and both pay B: the same payoff v − B. If B > b, both lose and earn 0. The two differ only when v < B < b. There the truthful bid loses (0) while the inflated bid wins but pays B > v, a negative payoff v − B. So overbidding changes the outcome only by winning auctions you should have lost, at a price above the object's worth. It never helps and sometimes strictly hurts.

Case 2 — you bid below your value (b < v). If B < b, both win and pay B; if B > v, both lose. They differ only when b < B < v. There the truthful bid wins and pays B for a strictly positive payoff v − B, while the shaded bid loses (0). So underbidding changes the outcome only by discarding auctions you should have won at a profit. It too never helps and sometimes strictly hurts. Combining the cases: for every realisation of B, truth-telling does at least as well as any deviation, and strictly better for some B — which is exactly weak dominance. ∎

Your value v = 100. You pay B (the top rival bid) only when you win.

  B = 70    truth bid 100: win, pay 70 → +30    over bid 120: win, pay 70 → +30    under bid 80: win, pay 70 → +30
  B = 90    truth bid 100: win, pay 90 → +10    over bid 120: win, pay 90 → +10    under bid 80: LOSE   →    0
  B = 110   truth bid 100: LOSE   →    0        over bid 120: win, pay 110 → −10   under bid 80: LOSE   →    0

Truth-telling ties or beats every deviation in every row.
Overbidding only flips the B = 110 row — into a −10 loss.
Underbidding only flips the B = 90 row — discarding a +10 profit.

Revenue equivalence and the winner's curse

The second-price auction has a first-price twin, and comparing them raises a puzzle. In a first-price auction you pay your own bid, so bidding your value guarantees zero surplus — you must shade below it. How far? In the symmetric equilibrium you bid the expected highest rival value conditional on your own being the highest, which for n bidders with uniform values is the fraction (n − 1)/n of your value. You win a thinner margin per unit of value but win in states where a second-price winner would have paid a high second price. The two effects, remarkably, wash out exactly.

That is the point, because the assumptions are strong and every interesting case is where one fails. Risk-averse bidders bid more aggressively in a first-price auction — they dislike losing — so it raises more than a second-price auction, breaking the equivalence. Asymmetric bidders (an incumbent telco versus a hungry entrant), binding budgets, and — above all — values that are not independent and private break it too. Every reason a designer agonises over format is a reason revenue equivalence fails. The deepest is the common-value case.

Common values and the winner's curse

Sometimes the object is worth roughly the same to everyone, but nobody knows that worth exactly. An offshore oil block holds some number of recoverable barrels — fixed, independent of who wins — but each bidder has only a noisy geological estimate. Call the true value V and your estimate a signal centred on V. Winning now carries information you lacked when you bid: you win because your bid was the most aggressive, which usually means your signal was the most optimistic — and the most optimistic signal is, on average, an over-estimate. Conditional on winning, the object is worth less than your estimate implied.

This is the central hazard of mineral- and oil-block licensing rounds across the continent. Where blocks of uncertain reserves are auctioned for a cash bonus, one of two things happens. Naive or desperate bidders fall to the curse, win, overpay, then stall the project, default, or renegotiate — value the state never collects. Or sophisticated bidders shade so heavily that the winning bid is low and the treasury captures little of the resource's worth. The design response is to stop auctioning a fixed price for an uncertain value: production-sharing contracts and royalties make payment rise with realised output, so the state shares the upside and the bidder need not gamble on its own estimate.

Many units at once: the CBK Treasury auctions

So far, one object. But the largest auctions African governments run every week sell many identical units at once. The Central Bank of Kenya auctions 91-, 182- and 364-day Treasury bills and longer bonds, ranking competitive bids by price (equivalently, yield) and filling from the best down until the offer is exhausted. The design choice is the multi-unit cousin of first-versus-second price. Under a discriminatory (multiple-price) auction each winner pays its own bid; under a uniform-price auction every winner pays the marginal clearing price — the lowest accepted bid. Discriminatory pricing punishes high bids, inviting shading and demand reduction by large dealers; uniform pricing is gentler but lets big bidders hold back to depress the clearing price. Which raises more is genuinely ambiguous — revenue equivalence again, at the scale of sovereign financing.

On offer: 100 units. Competitive bids, price per 100 of face value, best first:
  Bidder A   60 units  @ 98.0
  Bidder B   50 units  @ 97.5   ← marginal: only 40 of its 50 are needed
  Bidder C   40 units  @ 97.0   (rejected)
  Bidder D   30 units  @ 96.0   (rejected)

Fill from the top until 100 units are gone: A takes 60, B takes 40 → 100.
Marginal (clearing) price = 97.5, the lowest accepted bid.

Discriminatory / multiple-price — each winner pays its own bid:
  revenue = 60 × 98.0 + 40 × 97.5 = 9 780
Uniform price — every winner pays the clearing price 97.5:
  revenue = 100 × 97.5 = 9 750

Here discriminatory collects more, but paying-your-own-bid punishes
aggressive bids, so bidders shade toward the expected clearing price —
and which format wins on revenue is, in general, ambiguous.

Mechanism design: engineering the rules

Everything so far took the rules as given and solved for behaviour. Mechanism design inverts the question: fix the outcome you want — the object to whoever values it most, truthful revelation, maximal revenue — and design rules so that self-interested play produces it. Inverse game theory. The second-price auction is the model citizen: truth-telling is dominant, so it is dominant-strategy incentive-compatible. And the revelation principle says this focus costs nothing — any outcome any mechanism achieves in equilibrium, with bidders strategically misreporting, can also be achieved by a direct mechanism in which everyone simply reports the truth and truth-telling is an equilibrium. So the designer can search only over honest, direct mechanisms, which is what makes the problem tractable.

The trilemma: revenue, efficiency, and collusion-resistance

In practice the designer cannot have everything, and spectrum auctions — run by the Communications Authority in Kenya, the NCC in Nigeria, and ICASA in South Africa — show why. Efficiency wants the licences with whoever will build the most valuable networks; revenue wants a high price; collusion-resistance wants bidders unable to carve up the bands. These pull apart. A reserve price protects revenue and blocks giveaways, but set too high it shuts out an operator who would genuinely use the spectrum and the band goes unsold — the fate of South Africa's long-delayed rounds. A simultaneous ascending auction gives fine price discovery and tends to be efficient, but its open bids are a coordination device: rivals signal through the trailing digits of bids, split lots, and retaliate on defectors. A sealed first-price format smothers that collusion — a defector's undercut is invisible until the award — but bids blind and can misallocate. On top sits roll-back risk: an award challenged in court and annulled months later. The same tensions govern public procurement, where the enemy is a contractor cartel and a captured official — the second exercise.

Objective:            Efficiency   Revenue     Collusion-resistance
──────────────────────────────────────────────────────────────────────
Simultaneous          high         medium      low
 ascending (SMRA):     open bids let rivals signal via bid amounts,
                       split the bands, and retaliate on defectors
Sealed first-price:    medium       med–high    high
                       a defector's undercut stays hidden until the
                       award, so a cartel cannot police itself
Ascending + strong     high         high        medium
 reserve price:        the reserve guards revenue, but set too high
                       it leaves the band unsold (cf. delayed rounds)

Same trade-offs face Kenya (CA), Nigeria (NCC's 5G clock auction) and
South Africa (ICASA). Add roll-back risk: an award challenged in court
and annulled months later. Design for legal robustness, not just price.