Some good trades cannot be made

Haggling that ends at the largest product followed two people dividing a sum they could both see. Offers alternate, delay costs both, and Rubinstein’s theorem picks out one outcome, reached at once: no time is wasted and nothing is left on the table. That essay ended on the question its model assumed away. What if the size of the pie is private — if the seller knows what the object is worth to her, the buyer knows what it is worth to him, and each knows the other’s value only as a spread of possibilities?

The answer is one of the sharpest impossibility theorems in economics. Roger Myerson and Mark Satterthwaite proved in 1983 that no procedure for trading between one buyer and one seller with private values can guarantee that every mutually beneficial trade takes place, so long as both join voluntarily, neither can gain by misreporting, and no outsider pays to make up a shortfall. Some trades that would make both better off are always lost, and they are lost not through bad luck or stubbornness but because no rule can be both honest and efficient.

The figure shows the theorem for the simplest case, with both values spread evenly between 0 and 1. A trade is worth making anywhere above the diagonal, and it would realise an expected surplus of $\mathbb{E}[(b - s)^+] = 1/6$. The best that any procedure can do is to trade when $b - s \ge 1/4$, the green triangle, which realises $9/64$ — that is $27/32$, or 84.4%, of the possible surplus. The red band, trades worth less than a quarter, is lost.

What a procedure has to satisfy #

Any procedure — haggling, an auction, a broker, a computer that takes sealed reports and announces a price — can be described by what it does with the two values. A theorem from the 1970s, the revelation principle, says that whatever the procedure, there is an equivalent one in which each side simply reports its value and the procedure decides whether they trade and what money changes hands, arranged so that reporting the truth is each side’s best move. So the question becomes: what trading rules can be paired with payments that make truth-telling pay, leave each side no worse off for taking part, and balance — the buyer’s payment equal to what the seller receives?

The constraint that does the damage is the first. A seller who reports a higher value than her true one can expect a higher price, at the cost of trading less often. To stop her lying, the procedure must make the price she receives rise with her report in a precisely calibrated way — and it must do the same, mirrored, for the buyer. Those calibrated payments are forced, up to a constant, once the trading rule is fixed; this is the same uniqueness that makes two quite different auctions earn the same revenue. And once the payments are forced, so is the budget.

There is one honest procedure that makes every good trade, and it shows where the money goes. Let each side report a value; if the buyer’s report exceeds the seller’s, they trade, the buyer pays the seller’s report and the seller receives the buyer’s. Neither side’s own report affects the price it faces, only whether the trade happens, and the trade happens exactly when it benefits the reporter — so telling the truth is each side’s best move whatever the other does. This is the Vickrey–Clarke–Groves idea applied to a single trade. Its flaw is visible in one line: the seller receives $b$ and the buyer pays $s$, and the difference $b - s$ has to come from somewhere. Averaged over all the trades it makes, the shortfall is $\mathbb{E}[(b - s)^+] = 1/6$ — the entire surplus. A broker running this procedure would have to give away, on average, every bit of value the trade creates.

The budget of every rule #

For the uniform case the computation is short. With the payments forced by truth-telling and the least-favoured types of each side given exactly nothing, the money left over equals the average, over the trades the rule makes, of

$$\varphi_b(b) - \varphi_s(s) = (2b - 1) - 2s,$$

where $\varphi_b$ and $\varphi_s$ are the virtual values: what the buyer can be charged once he must be given a reason not to understate, and what the seller must be paid once she must be given a reason not to overstate. For the rule “trade when $b - s \ge c$” the integral comes out in closed form as $(1 - c)^2 (4c - 1)/6$, which the figure checks against a direct integration on a fine grid.

At $c = 0$, the efficient rule, the money left over is $-1/6$. A procedure that makes every good trade must be subsidised by exactly a sixth — as much, on average, as the entire surplus it creates. Raising $c$ drops the trades that cost the most to make honest, and the budget first reaches nought at $c = 1/4$. Any rule that balances must stop at least that far from the diagonal, and since the surplus $(1 - c)^2 (1 + 2c)/6$ falls as $c$ rises, $c = 1/4$ is the best. That is the theorem for this case, in two curves.

Which trades pay for themselves #

The virtual values make the shape of the answer visible. A trade adds to the budget if $2b - 1 > 2s$, which is the green triangle above $b = s + 1/2$: a buyer who values the object highly, meeting a seller who values it little, can be charged enough and paid enough to leave money over. Everything between that line and the diagonal drains the budget. Trading only in the green triangle would be self-financing and absurdly timid. The best balanced rule takes in red trades in order of how little they drain — the ones farthest from the diagonal first — until the drain uses up the green triangle’s surplus exactly. With uniform values that happens at the line $b = s + 1/4$, and every trade between it and the diagonal is one that honesty cannot afford.

The reason honesty is expensive is that each side’s private information is worth something to it. A seller whose value is low knows she could pretend it is higher and extract a better price; to stop her, the procedure must give her, in expectation, as much as she could have gained by pretending. That gift is an information rent, and the virtual value is what is left of a trader’s value after the rent is paid. Efficient trade needs both rents paid in full, and between them they cost more than the trade is worth.

Every penny of the surplus is rent #

The budget identity has a striking consequence for the best procedure. The surplus a procedure creates is split three ways: what the seller gains, what the buyer gains, and what is left over. At $c = 1/4$ the third part is nought, so the whole of the 9/64 goes to the two traders — and both of them are paid, on average, exactly their information rent.

The rent can be computed directly. A seller with value $s$ trades under the best rule whenever $b \ge s + 1/4$, which happens with probability $3/4 - s$, and truthfulness requires that her expected gain fall as her value rises at exactly that rate: her gain is $\int_s^{3/4} (3/4 - t)\,dt = (3/4 - s)^2/2$. Averaged over sellers that is $27/384$, and the buyer’s rent is the same by symmetry. Together they come to $54/384 = 9/64$, the whole surplus. In the best honest procedure nothing is left for efficiency to buy: every trade it makes is paid for in rents, and the trades it refuses are the ones whose rents would cost more than they create.

It is the same logic, in a harder setting, as the rule that one cuts and the other chooses: a procedure that is safe against each side’s private knowledge has to give each side a guaranteed share, and the guarantee is what it costs. With a divisible cake the guarantee can be paid in cake without waste, because a cut can always be moved a little. With one indivisible object it cannot, and the waste is the lost trades.

A double auction finds the line #

The best possible procedure is not an exotic piece of engineering. Kalyan Chatterjee and William Samuelson studied in 1983 the most obvious procedure there is: each side names a price, and if the buyer’s bid is at least the seller’s ask they trade at the average of the two.

Nobody in a double auction tells the truth. The seller asks $2s/3 + 1/4$, more than her value whenever $s < 3/4$; the buyer bids $2b/3 + 1/12$, less than his whenever $b > 1/4$. Against each other these strategies are an equilibrium: the figure’s dots are each side’s best reply to the other’s line, found by trying every price, and every one lands on its line. The bid reaches the ask when $2b/3 + 1/12 \ge 2s/3 + 1/4$, that is when $b - s \ge 1/4$ — exactly the boundary of the best procedure.

Two examples make the shading concrete. A seller who values the car at 0.3 asks 0.45; a buyer who values it at 0.5 bids about 0.42; the bid falls short and they part, though a sale at 0.4 would have made both better off. A seller at 0.1 asks about 0.32, a buyer at 0.6 bids about 0.48, and they trade at the average, 0.4. Each side shades because the price is the average of the two names: raising an ask by a little raises the price by half as much whenever a sale happens, and costs only the sales that the raise prevents.

So the double auction realises 27/32 of the surplus, and Myerson and Satterthwaite’s theorem says that nothing voluntary and self-financing can do better. The strategic shading that looks like an inefficiency to be engineered away is, for uniform values, exactly as much inefficiency as the information requires. That is a strong conclusion about a familiar thing: the lost trades are not caused by haggling, and no cleverer haggling could recover them.

Simpler and worse, larger and better #

A broker who posts one price and lets each side accept or refuse is the bluntest procedure of all. It is honest by construction, since neither side’s report changes the price, and with values spread evenly the best price is a half, realising $p(1 - p)/2 = 1/8$, or 75% of the surplus. A badly placed price is much worse: a quarter or three quarters realises 56%. The double auction beats the best posted price by nine percentage points because it lets the price respond to both values, and the theorem says the extra responsiveness is worth exactly that much and no more.

The impossibility is a fact about small markets. With many buyers and many sellers, R. Preston McAfee’s double auction of 1992 ranks the bids and asks, reads a price from the first pair that should not trade, and drops at most one trade that should. Telling the truth is then each trader’s best move whatever the others do, the auction never needs outside money, and its loss is one trade out of many: with one buyer and one seller it trades nothing at all, with two of each it realises 77%, with eight 97.5%, and with sixty-four of each it loses about a twentieth of one per cent. Aldo Rustichini, Mark Satterthwaite and Steven Williams showed in 1994 that the plain double auction’s loss shrinks even faster, like one over the square of the number of traders. The information rent each trader can extract shrinks as competitors crowd round its value, which is the private-information version of the core shrinking in a large market.

A family of impossibilities #

Myerson and Satterthwaite’s theorem belongs with a small number of results that say honesty and efficiency cannot be had together when information is private. Allan Gibbard and Mark Satterthwaite — the same Satterthwaite, eight years earlier — showed that every voting rule with three or more possible winners either is a dictatorship or gives some voter a lie that pays. Alvin Roth showed that no rule for forming stable pairs can make every participant’s truth safe: no stable rule is safe from a lie. And in auctions with many objects, the procedure that makes truth-telling safe for every bidder — the generalised Vickrey auction — can earn almost nothing, which is why practical auctions use prices that the bidders raise in rounds instead.

What distinguishes the trading theorem is that its impossibility has a size. Voting impossibilities say that some lie pays somewhere; this one says that 5/32 of the surplus is lost, no more and no less, for uniform values, and it identifies the procedure that loses only that much. It is an impossibility with a price tag, and the price can be computed for any distribution of values. That makes it closer in spirit to the core’s guarantee of a split nobody can walk away from than to a pure no-go theorem: it says exactly how much cooperation private information allows.

What the picture cannot show #

Every figure here assumes values spread evenly and independently between 0 and 1, and the numbers belong to that assumption. The theorem does not: for any continuous distributions whose ranges overlap, efficient trade with voluntary participation, truth-telling and a balanced budget is impossible. What changes with the distributions is the size of the loss, the best threshold and whether a simple double auction achieves the optimum — for uniform values it does, and for most others the best procedure has a more complicated shape.

Nor do the figures show what happens if the conditions are loosened. If the ranges of the two values do not overlap — the buyer certainly values the object more — a fixed price between them is efficient and there is no theorem. If someone outside will pay the subsidy, efficient trade is possible at a cost of a sixth. If the seller owns only a share of the object and the buyer the rest — a partnership being dissolved — Peter Cramton, Robert Gibbons and Paul Klemperer showed in 1987 that efficient dissolution can be possible after all, because each side’s outside position softens the rents. The lost trades are the price of a particular combination of honesty, voluntariness and self-reliance.

Still open: haggling over time with hidden values #

What happens when the two sides bargain over time, as in Rubinstein’s game, but with private values? Offers can now reveal information gradually, and a seller who holds out signals a high value, which a buyer who waits can exploit. The theory has many equilibria and few sharp predictions: in some models delay replaces the lost trades as the cost of private information, in others both appear, and which features of real bargaining pick out which equilibria is not settled. Experiments with the double auction itself show that people trade more often than the Chatterjee–Samuelson equilibrium predicts — they shade less than the theory says they should — and whether that reflects other equilibria, bounded reasoning or something else about how people bargain is studied and disputed.

Honesty has a price, and here it is #

When each side knows only its own value, some trades that would benefit both cannot be made by any procedure that is voluntary, honest and self-financing. With values spread evenly the efficient rule needs a subsidy of exactly a sixth; the budget of the rule “trade when $b - s \ge c$” is $(1 - c)^2(4c - 1)/6$, which balances first at $c = 1/4$, so the best procedure realises 9/64, or 27/32 of the surplus. Two people naming prices and splitting the difference reach that bound by shading their prices to $2s/3 + 1/4$ and $2b/3 + 1/12$; a posted price reaches 75%; and with many traders on each side the loss falls towards nought, so the impossibility is the cost of being one buyer facing one seller.