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

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

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

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

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

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

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

The constraints: (IR) (IC)

Two types H , L : Optimality conditions:

Figure 1: An example of a two buyer case.

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:

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

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

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

Experimentation step: i = L,H

Figure 2: Illustration of two iterations.

Figure 3: The Method.

Figure 4: The limit process.

Table 1: A two-type case.

Table 2: A four-type case.