1. An Adjustment Scheme for a Buyer-Seller Game
Harri Ehtamo
Kimmo Berg
Mitri Kitti
Systems Analysis Laboratory
Helsinki University of Technology

2. Mechanism design - revelation of truth is costly
Nonlinear pricing
Design of tariffs and contracts
Auction design
Taxation
Public good (Groves mechanism, 1973)
Bargaining

3. A buyer-seller game Seller: (x, t) = t – c(x)
Buyer: U(x, t) = V(x) - t
max U(x, t(x)) (IC)
V(x) - t(x) = 0 (IR)
x0

4. Solution by a linear tariff: t = x + 
V´(x) =  = c´(x)
V(x) = x +  = t
Linear tariff: t = t + c´(x)(x - x)

5. The linear tariff:
 = const.
c(x)+d
V(x) t
U = const. d x

6. Use production cost for pricing:
t = c(x) + d nonlinear pricing
t = t + c´(x)(x - x) linear pricing
( x , t ) optimal bundle

7. Incomplete information – Bayesian Nash equilibrium
N buyer types: I = {1, ... ,N}

8. The constraints:
(IR)
(IC)

9. Two types H , L :
Optimality conditions:

10. Figure 1: An example of a two buyer case.

11. Assumptions and propositions
Assumption 1: The single crossing property:
Proposition 1: The single crossing property implies that the optimal amounts in the bundles are nondecreasing in type.
Proposition 2: Under the single crossing property, the optimal prices are:

12. Assumption 2: No bunching:
Proposition 3: Without bunching, the first-order optimality conditions are

13. Bayesian Nash equilibrium by adjustment
N buyer groups
pi fraction of group iI, known
k=1,2, ... updating periods

14. Adjustment using linear tariffs
Exploration step:
Increase of bundles (xi,ti), iI

15. Experimentation step: i = L,H

16. Figure 2: Illustration of two iterations.

17. Figure 3: The Method.

18. Figure 4: The limit process.

19. Table 1: A two-type case.

20. Table 2: A four-type case.