Exposure problem in package bidding
A Socratic walk-through of the exposure problem — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why can a bidder who wins one of the two licences it needs end up worse off than one who wins nothing?
An auction ends. One operator has won a licence; another has won none. The winner is worse off than the loser, and not because it overpaid in any ordinary sense. It paid less than the licence was worth to the business it was trying to build, and still lost money.
That is not an accident of bad judgement. A companion piece here works out how the rules of a single-item auction decide whether bidders state their values honestly, and ends by pointing at exactly this case: what happens when many things are sold at once. This is that continuation. The difference is not that the bidders are less honest. It is that the thing being sold is no longer the thing being valued.
Reasoning it through
REASONING #Suppose an operator needs two adjacent frequency blocks, A and B, to build a network with usable bandwidth. Put stipulated numbers on it, purely to make the arithmetic visible: the pair is worth 100 to this firm. Either block on its own is worth 20 — not nothing, but not a business.
The regulator sells A and B as separate lots. Ask the obvious question: what should the operator bid on lot A?
Its value for A is not a number. It is 20 if it fails to get B, and 80 if it does — and it will not know which until the auction closes. That is the whole problem in one line: separate lots require a bidder to name a price for something whose worth to it is not yet determined.
Now play it out against two rivals, S1 wanting only A and S2 wanting only B, each willing to go to 30 for their single block. Our operator bids 31 on each. If it wins both it pays 62 for something worth 100 — a gain of 38, and the efficient outcome besides, since 100 beats the 60 the two small firms would realise between them. But suppose S2's value was really 40, unknown to anyone. Then the operator wins A at 31, loses B, and holds a block worth 20 that cost it 31. Payoff: minus 11. S1, who won nothing, is at zero. The winner is the loser.
What, then, is the operator's safe bid? Twenty. Cap every bid at the stand-alone value and no fragment can hurt you, because you never pay more than the fragment is worth. The synergy — the 60 of extra value that exists only in the pair — simply cannot be bid. So the rational response to exposure is to underbid, by an amount equal to the complementarity.
Follow that to the allocation. With the operator capped at 20, S1 and S2 win at around 21 each. Two firms hold two blocks worth 60 between them, and the arrangement worth 100 never happens. Nobody cheated, nobody blundered, and the auction produced the wrong answer — with revenue lower too, so the seller is not compensated for the inefficiency.
So what is the actual fault? Not dishonesty, not timidity. The unit of sale is finer than the unit of value. And it bites in only one direction: it requires superadditive values, where the whole exceeds the sum of the parts. A bidder wanting substitutes — any two blocks will do — faces no exposure, because losing one lot makes the other more valuable, not less.
Is that diagnosis testable, or a story that fits? It predicts that the shortfall between a bidder's bid on a single lot and its share of the package value should scale with the degree of complementarity, and fall to nothing when the lots are substitutes — independently of budgets, experience and the number of rivals. Laboratory auctions, where the experimenter assigns values so the complementarity is known rather than inferred, report broadly this pattern.
The refuting observation is correspondingly clean. If bidders facing strong, known complementarities bid their full marginal package value on separate lots — accepting the fragment risk — then exposure is not what suppresses bids, and the underbidding seen in the field is about capital or nerve instead. Equally, if allowing all-or-nothing package bids left efficiency unchanged in exactly the cases where values are most superadditive, the mechanism would be wrong about its own cause.
The analogy
THE ANALOGY #You want a pair of shoes, and the left and the right are being auctioned in separate rooms at the same time. Bid boldly in both and you may walk out with a pair for a good price — or with one left shoe, having paid real money for something you cannot wear. Bid only what a single shoe is worth to you and you will never be stuck, and you will also never get the pair.
a lone shoe is worth almost exactly nothing and can be thrown away, whereas a single spectrum licence retains genuine stand-alone value and carries build-out obligations you cannot walk away from; and shoes come up for sale again next week, while a spectrum band may be auctioned once in a decade, which is what makes the fragment unrecoverable.
Clarifying the model
THE MODEL #The obvious fix is to let bidders bid on the package: one all-or-nothing bid of up to 100 for "A and B together", which cannot leave anyone holding a fragment. That is what combinatorial and clock auctions do, and it is why regulators moved toward them after two decades of the simpler simultaneous-ascending format. But the fix is a trade, and two costs are worth naming.
The first is the mirror image of exposure, the threshold or free-rider problem. To beat a package bid of 100, S1 and S2 must jointly raise, and each would rather the other put up the increase — so small bidders who collectively outvalue the package can fail to assemble against it, and the format that removes one coordination failure introduces another. The second is computational: choosing the revenue-maximising set of non-overlapping package bids is a hard combinatorial problem, and the number of expressible packages explodes with the number of lots.
One thing this is not: a problem of aggregating preferences across people, as voting rules are. The difficulty here is internal to a single bidder, which cannot express what it wants in the language the auction offers — a language problem, not a fairness problem, which is why the remedy is a richer bid rather than a fairer count.
A picture of it
THE PICTURE #How to readStart at the parallelogram, which carries the bidder's own values, and take the first diamond — everything depends on which auction rule applies. The right-hand route is the package format: a single indivisible bid, with no path from it to a negative payoff. The left-hand route is separate lots, where the second diamond is the bidder's real choice and the third is chance rather than choice. The red node is the fragment outcome, and the dotted back-edge is the point of the picture: because it can be foreseen, it reaches back and turns the earlier answer into "No", so the profitable branch is never taken.
What became clearer
WHAT CLEARED #Exposure is not a pricing error but a mismatch of units. When value lives in a combination and bids must be made on the pieces, a bidder is asked to price something whose worth depends on an outcome not yet known, and the only bid safe under every outcome is the one that ignores the combination entirely. Rational bidders therefore shade down to stand-alone values, packages that should be assembled are not, and the auction returns an allocation nobody wanted, with the seller collecting less besides. Letting the bid describe the package removes that trap and installs a different one, in which small bidders must coordinate to clear a threshold.
Key terms
TERMS #| Term | What it means |
|---|---|
| Exposure problem | the risk of winning part of a needed combination at a price justified only by the whole, which drives bids down to stand-alone values. |
| Superadditive values | values for which a set of items is worth more than the sum of its members; the precondition for exposure. |
Every term the collection defines is gathered in the glossary.