Roger B. Myerson Prize Lecture

330

price $90 if and only if the buyer reports her weak type. It may seem unfair to 

have trading only at a price that is so favorable to the seller, but honest partici-

pation by both individuals would be an equilibrium if the mediator planned 

to use their reported information according to the plan shown Figure 7. 

The simple split-the-difference plan also has two other reporting equilibria. 

In one equilibrium, it is the buyer who is expected to always report her strong 

type, and in response the seller is expected to be honest. The outcomes of 

this equilibrium are the same as the incentive-compatible mechanism shown 

in Figure 8 (the mirror image of Figure 7). In the other equilibrium, both 

individuals have a positive probability of lying, each claiming to be strong 

with probability 0.6 when he or she is weak. For each possible combination 

of types, the conditional probability of trade and the conditionally expected 

price if trade occurs under this randomized equilibrium is the same as the 

incentive-compatible plan shown in Figure 9. 

Seller

’

s value

$20 [s]

$100 [w]

[s]

$80

0, *

0, *

[w]

$0

1, $10

1, $10

 P(trade), E(price if trade)

Buyer

’

s value

Figure 8. Incentive-compatible mechanism equivalent to an equilibrium of split-the-

difference.

Seller

’

s value

$20 [s]

$100 [w]

[s]

$80

0, *

0.4, $90

[w]

$0

0.4, $10

0.64, $50

Buyer

’

s value

 P(trade), E(price if trade)

Figure 9. Incentive-compatible mechanism equivalent to an equilibrium of split-the-

difference.

These plans in Figures 7–9 may not seem very fair or efficient, as one fa-

vors the seller, one favors the buyer, and the third yields a low probability of 

trading. But the general point is that, for any equilibrium of the individuals’ 

reporting strategies under any mediation plan, we can find an incentive-com-

patible mechanism that has the same outcomes for all types. This result is the 

revelation principle, and it allows us to extend our results from incentive-

compatible mechanisms to all possible equilibria of all possible mechanisms. 

It allows us to say, for this example, that no equilibrium of any mediation 

plan, incentive-compatible or not, could have a lower probability of ex-post 

allocative inefficiency than the honest equilibrium of the incentive-compati-

ble mechanism in Figure 4.

3.5 General nonsymmetric mechanisms

Our analysis here has been simplified by focusing only on a class of media-

tion plans that treat the buyer and seller symmetrically. Without this symme-

try, the general class of mediation plans for this example has eight variables, 

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331

instead of just two. For each j in {0,80} and each k in {20,100}, we could let 

q

j,k

 denote the conditional probability of trade when the seller’s value is j and 

the buyer’s value is k, and we could let y

j,k

 denote the conditionally expected 

price if trade occurs when their values are j and k. These variables must sat-

isfy the probability constraints:

0 < q

0,20

 < 1, 0 < q

0,100

 < 1, 0 < q

80,20

 < 1, 0 < q

80,100

 < 1;

the participation constraints:

0.5q

80,20

(y

80,20

 – 80) + 0.5q

80,100

(y

80,100

 – 80) > 0,

0.5q

0,20

(y

0,20

 – 0) + 0.5q

0,100

(y

0,100

 – 0) > 0,

0.5q

80,20

(20 – y

80,20

) + 0.5q

0,20

(20 – y

0,20

) > 0,

0.5q

80,100

(100 – y

80,100

) + 0.5q

0,100

(100 – y

0,100

) > 0;

and the informational incentive constraints:

0.5q

80,20

(y

80,20

 – 80) + 0.5q

80,100

(y

80,100

 – 80) > 0.5q

0,20

(y

0,20

 – 80) + 0.5q

0,100

(y

0,100

 – 80), 

0.5q

0,20

(y

0,20

 – 0) + 0.5q

0,100

(y

0,100

 – 0) > 0.5q

0,20

(y

80,20

 – 0) + 0.5q

80,100

(y

80,100

 – 0),

0.5q

80,20

(20 – y

80,20

) + 0.5q

0,20

(20 – y

0,20

) > 0.5q

80,100

(20 – y

80,100

) + 0.5q

0,100

(20 – y

0,100

),

0.5q

80,100

(100–y

80,100

) + 0.5q

0,100

(100–y

0,100

) > 0.5q

80,20

(100–y

80,20

) + 0.5q

0,20

(100–y

0,20

).

These constraints look very complicated, but they are actually not hard 

to analyze. They can be linearized by the substitution x

j,k

 = q

j,k

 y

j,k

, and then 

optimal mechanism can be found by linear programming. In particular, the 

mechanisms that we have studied above in Figures 4–6 remain incentive-

efficient in this more general class of nonsymmetric mechanisms.

3.6 Incentive constraints as a source of Coase’s transactions costs

Coase (1960) argued that, if there were no transactions costs then unre-

stricted free trade of property rights and resources could achieve allocative 

efficiency, regardless of the initial allocation of property rights. Thus, transac-

tions costs are important for understanding the problems of allocative inef-

ficiency, and we need an analytical theory of where transactions costs come 

from. In situations like this example, the informational incentive constraints 

can be viewed as a source of Coasean transactions costs. The participation 

constraints represent the initial allocation of property rights that each can 

hold without trading (Samuelson 1985). We have shown that ex-post alloca-

tive efficiency cannot be guaranteed by any incentive-compatible mechanism 

when the initial owner is the person whose value may be 0 or 80. But if the 

initial ownership right was assigned instead to the individual whose value 

may be 20 or 100, so that the roles of seller and buyer are reversed, then it is 

easy for a mediator to guarantee that the object ends up with the individual 

who values it most. An incentive-compatible mechanism that would achieve 

ex-post efficiency in this case is shown in Figure 10 (where the mediator just 

proposes the price $50, and they trade if both accept it). Thus, the theory of 

mechanism design gives us an analytical framework where the initial alloca-

tion of property rights can affect the probability of achieving an outcome 

that is allocatively efficient.

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