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A buyer would pay up to 60 for a house; a seller would let it go for 50. There is a deal to be had, worth 10 to split. Yet each hides their number to bargain harder, and often the deal that should happen simply does not.

Implementation and Myerson-Satterthwaite

Mechanism design asks what outcomes rules can implement. Sometimes the answer is a hard no. When a buyer and a seller each hold private information, Myerson and Satterthwaite proved that no mechanism can always trade when it should, break even, and respect both parties’ incentives. Some efficient trades are simply unreachable.

From earlier lessons

Two quick ones before we start.

A mechanism is individually rational if each party is willing to...

In Nash bargaining, gains from trade exist when the buyer values the good...

Predict, then play

The buyer’s value and the seller’s cost are each uniform on $[0,1]$ , and each shades in the double auction. Move both and see when trade happens, and when a genuinely good deal is missed.

Why did that happen?

Take a guess, then reveal the reason

Each side has an incentive to misrepresent: the buyer wants to look less eager, the seller more reluctant, so each shades. That shading is individually rational, but collectively it means small gains from trade get squeezed out, because neither will move far enough to close a thin margin. There is no clever payment scheme that undoes this while keeping both parties honest and the budget balanced. The loss is baked into two-sided private information.

The math

Buyer value $v_b$ and seller cost $v_s$ are each uniform on $[0,1]$ . The efficient rule trades whenever $v_b \ge v_s$ , with first-best expected gains $\tfrac16$ . In the Chatterjee-Samuelson (1983) linear equilibrium the buyer bids $\tfrac23 v_b + \tfrac1{12}$ and the seller asks $\tfrac23 v_s + \tfrac14$ ; trade happens when the bid meets the ask, which reduces to

v_b \ge v_s + \tfrac14.

So every efficient trade with $0 \le v_b - v_s < \tfrac14$ is missed. The linear mechanism realizes surplus $\tfrac{9}{64} \approx 0.141$ , below the first-best $\tfrac16 \approx 0.167$ .

Definition, Myerson-Satterthwaite and Maskin monotonicity

Myerson-Satterthwaite: for bilateral trade with independent private values drawn from overlapping intervals, no Bayesian incentive-compatible, interim individually rational, budget-balanced mechanism is ex-post efficient. On the positive side, Maskin implementation asks whether a social-choice rule can be made the outcome of every Nash equilibrium of some mechanism. Maskin monotonicity is necessary, and with at least three agents it is sufficient together with no veto power.

Work it out

The 1/4 gap and the lost surplus

The buyer’s bid $\tfrac23 v_b + \tfrac1{12}$ meets the seller’s ask $\tfrac23 v_s + \tfrac14$ exactly when $\tfrac23(v_b - v_s) \ge \tfrac14 - \tfrac1{12} = \tfrac16$ , that is $v_b - v_s \ge \tfrac14$ . Trade requires a gap of at least a quarter.

Now the surplus. First-best trades whenever $v_b \ge v_s$ , giving expected gains $\iint_{v_b \ge v_s} (v_b - v_s)\,dv_b\,dv_s = \tfrac16$ . The linear equilibrium only trades above the $\tfrac14$ gap, capturing $\tfrac{9}{64} \approx 0.141$ . The difference is the deadweight loss forced by two-sided private information, and no mechanism removes it entirely.

Key idea

When both sides hold private information, efficiency, honesty, participation, and budget balance cannot all hold. Rational bargainers shade, thin gains from trade vanish, and some deals that should happen never do.

Check yourself

Check yourselfapply

The buyer values the good at 0.6 and the seller's cost is 0.5. Does the double auction trade?

Check yourselfanalyse

What exactly does the Myerson-Satterthwaite theorem rule out?

Check yourselfbrings back: nash bargaining

How does this square with the Nash bargaining solution, which always splits the surplus efficiently?

Sources

Myerson, R. B., and Satterthwaite, M. A. (1983). Efficient mechanisms for bilateral trading. Journal of Economic Theory, 29(2), 265-281.
Chatterjee, K., and Samuelson, W. (1983). Bargaining under incomplete information. Operations Research, 31(5), 835-851.
Maskin, E. (1999). Nash equilibrium and welfare optimality. Review of Economic Studies, 66(1), 23-38.