Harri Ehtamo

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An Adjustment Schemefor a Buyer-Seller Game

• Harri Ehtamo
• Kimmo Berg
• Mitri Kitti

Systems Analysis Laboratory Helsinki University
of Technology www.sal.tkk.fi
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Mechanism design - revelation oftruth is costly

• Nonlinear pricing
• Design of tariffs and contracts
• Auction design
• Taxation
• Public good (Groves mechanism, 1973)
• Bargaining

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A buyer-seller game

• Seller ?(x, t) t c(x)
• Buyer U(x, t) V(x) - t

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Solution by a linear tariff t ?x ?

• V(x) ? c(x)
• V(x) ?x ? t
• Linear tariff t t c(x)(x - x)

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The linear tariff
? const.
c(x)d
V(x)
U const.
t
d
x
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Use production cost for pricing

• t c(x) d nonlinear pricing
• t t c(x)(x - x) linear pricing
• ( x , t ) optimal bundle

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Incomplete information Bayesian Nash
equilibrium

• N buyer types I 1, ... ,N

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The constraints
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• Two types H , L
• Optimality conditions

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Figure 1 An example of a two buyer case.
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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

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

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Bayesian Nash equilibriumby adjustment

• N buyer groups
• pi fraction of group i?I, known
• k1,2, ... updating periods

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Adjustment using linear tariffs

• Exploration step
• Increase of bundles (xi,ti), ?i?I

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• Experimentation step i L,H

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Figure 2 Illustration of two iterations.
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Figure 3 The Method.
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Figure 4 The limit process.
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Table 1 A two-type case.
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Table 2 A four-type case.