One buyer, one seller

In an auction, usually only one side has private information: the seller knows her reserve price, or does not care. Many negotiations, though, are bilateral: one buyer and one seller. The buyer has a maximum willingness to pay $v$ in mind, the seller a minimum acceptable price (cost) $c$, and neither knows the other’s number. Whenever $v > c$, a trade would create $v – c$ in value; the trouble is that both have reasons to exaggerate. The buyer says she does not want it that much; the seller says his costs are high.

Is there any rule, however clever, that guarantees a trade whenever $v > c$?

The theorem is often misread as “negotiation is doomed”. That is not what it says. It says that as long as both sides hold private information and no outsider subsidises the deal, every rule must give up trade in some cases where trade would have been worthwhile. This loss of efficiency is not caused by poor negotiating skill but by the information structure itself; skill and rules can only make the loss as small as possible. Note also the overlap condition: if the buyer’s value is known always to exceed the seller’s cost, a fixed price solves the problem.

The derivation: virtual surplus

The proof has a computable core. Using the incentive compatibility and individual rationality constraints, the expectation of “what the buyer pays minus what the seller receives” in any mechanism satisfying conditions 1 and 2 can be written as an expression that depends only on the trading rule. For uniform distributions it is

$$\mathbb{E}\big[\,\big((2v – 1) – 2c\big)\cdot \mathbf{1}\{\text{trade}\}\big],$$

where $2v – 1$ is the buyer’s “virtual value” and $2c$ the seller’s “virtual cost”. They are more “pessimistic” than the true value and cost because the mechanism must leave each party holding private information an information rent to make them tell the truth. Budget balance requires this expectation to be at least zero.

# Bilateral trade, v and c independent uniform on [0, 1]; d = v - c has density 1 - d on
# [0, 1]. For the rule "trade iff d >= t": gains = int_t^1 d(1-d), and expected virtual
# surplus = int_t^1 (2d - 1)(1 - d), which must be >= 0 for a BIC, interim IR, budget-
# balanced mechanism (Myerson-Satterthwaite). Then check the Chatterjee-Samuelson
# linear equilibrium: ask 2c/3 + 1/4, bid 2v/3 + 1/12, price = average when bid >= ask.
def integrals(t, steps=200000):
    h = (1 - t) / steps
    ds = [t + (k + 0.5) * h for k in range(steps)]
    return sum(d * (1 - d) for d in ds) * h, sum((2 * d - 1) * (1 - d) for d in ds) * h

s0, vs0 = integrals(0.0)
print(f"trade iff v > c: gains {s0:.5f} (1/6), virtual surplus {vs0:+.5f}: infeasible")
lo, hi = 0.0, 0.5
for _ in range(40):
    mid = (lo + hi) / 2
    lo, hi = (mid, hi) if integrals(mid, 20000)[1] < 0 else (lo, mid)
s1 = integrals(hi)[0]
print(f"smallest feasible t = {hi:.5f}; gains {s1:.6f} (9/64 = {9 / 64:.6f}),"
      f" {s1 / s0:.3%} of first-best")
pts = [(i + 0.5) / 1000 for i in range(1000)]
ask, bid = (lambda c: 2 * c / 3 + 0.25), (lambda v: 2 * v / 3 + 1 / 12)
offers = [k / 500 for k in range(501)]
def by(v):                      # buyer's best bid against the seller's linear ask
    return max(offers, key=lambda b: sum(v - (b + ask(c)) / 2
                                         for c in pts if b >= ask(c)))
def se(c):                      # seller's best ask against the buyer's linear bid
    return max(offers, key=lambda a: sum((a + bid(v)) / 2 - c
                                         for v in pts if bid(v) >= a))
print("best replies (formula): buyer " +
      " ".join(f"v={v}: {by(v):.3f} ({bid(v):.3f})" for v in (.4, .7, 1)))
print("                       seller " +
      " ".join(f"c={c}: {se(c):.3f} ({ask(c):.3f})" for c in (0, .3, .6)))
# trade iff v > c: gains 0.16667 (1/6), virtual surplus -0.16667: infeasible
# smallest feasible t = 0.25000; gains 0.140625 (9/64 = 0.140625), 84.375% of first-best
# best replies (formula): buyer v=0.4: 0.350 (0.350) v=0.7: 0.550 (0.550) v=1: 0.750 (0.750)
#                        seller c=0: 0.250 (0.250) c=0.3: 0.450 (0.450) c=0.6: 0.650 (0.650)

(a) The expected gains from efficient trade are 0.16667, that is $1/6$; but the expected virtual surplus is $-1/6$, negative, so no mechanism satisfying the other three conditions can “trade whenever $v > c$”.

(b) The smallest threshold that brings virtual surplus up to exactly zero is $t = 1/4$. By hand: $\int_{1/4}^1 (2d-1)(1-d)\,\mathrm{d}d = \big[\tfrac{3}{2}d^2 – d – \tfrac{2}{3}d^3\big]_{1/4}^1 = -\tfrac16 – \big(\tfrac{3}{32} – \tfrac14 – \tfrac{1}{96}\big) = 0$. The expected gains are then

$$\int_{1/4}^{1} d(1-d)\,\mathrm{d}d = \Big[\tfrac{d^2}{2} – \tfrac{d^3}{3}\Big]_{1/4}^{1} = \tfrac16 – \tfrac{5}{192} = \tfrac{27}{192} = \tfrac{9}{64},$$

which is $(9/64)/(1/6) = 54/64 = 84.375\,\%$ of the first-best. The loss, $5/192 \approx 0.026$, comes entirely from cases with $0 < v – c < 1/4$, trades that “should have happened”.

(c) When the other side uses the linear strategy, the numerically computed best price agrees exactly with the formula, so the strategies form an equilibrium. The trading condition $2v/3 + 1/12 \ge 2c/3 + 1/4$ simplifies to exactly $v \ge c + 1/4$: a simple rule, “each names a price and they meet in the middle”, happens to achieve the best result possible under any rule (Chatterjee and Samuelson 1983; Myerson and Satterthwaite 1983).

What this means for negotiation

Translated back to the negotiating table, the theorem has several implications.

First, when both sides conceal their bottom lines, deadlock and missed deals are normal and not necessarily anyone’s fault. In the example above, trades along a band of width about $1/4$ are given up: when the buyer’s value exceeds the seller’s cost by less than 0.25, the two sides’ exaggerations cancel out the entire potential gain.

Second, the information structure matters more than eloquence. Making one side’s bottom line verifiable (for example by a public cost audit or an independent valuation), or turning a bilateral problem into a competition among many (bringing in more buyers or sellers), can narrow the band. The latter links to the auctions of Chapter 2: the fuller the competition, the smaller the rents that private information earns.

Third, an outside subsidy can buy back efficiency. “Budget balance” is the key condition: if a third party is willing to cover a deficit (a government in public procurement, a platform on its marketplace), efficient trade can be achieved. The Clarke mechanism of Chapter 1 belongs to the same family of problems: truthful reporting and efficient decisions can be had together, at a cost to the budget.

Chapter summary

The Myerson–Satterthwaite theorem shows that when buyer and seller both hold private information and their value ranges overlap, no mechanism can be at once Bayesian incentive compatible, interim individually rational, ex post efficient and budget balanced. In the uniform example, the best mechanism trades only when $v \ge c + 1/4$, realising $9/64 \div 1/6 = 84.375\,\%$ of the first-best gains; the linear equilibrium of the Chatterjee–Samuelson double auction attains this bound exactly, though the auction also has many worse equilibria. The loss of efficiency comes from the information structure, not from a lack of skill; the ways to reduce it are to make information verifiable, to bring in competition, or to have an outsider bear the deficit. The next chapter leaves mechanisms behind and turns to how two people facing each other divide a surplus.