Fair and Efficient Allocation in a Finite Local Public Goods Economy

1
Fair and Efficient Allocation in a Finite Local
Public Goods Economy

• Stuart McDonald
• Social and Information Systems Laboratory SISL
• California Institute of Technology

2
Motivation

• The market for internet services
• Global Characteristics
• Quality of access determined by the capacity of
  local infrastructure
• Local infrastructure subjected to congestion and
  rivalry
• Local Characteristics
• Consumption (once material is downloaded) is
  non-rivalrous
• Local public good subject to congestion
• The internet access is available at r locations
  (ISPs). Access is controlled by local monopolists
  who have limited capacity

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Motivation

• Formulates an approach for valuing ISPs which is
  based on Shapley value
• The services provided by an ISP to customers are
  subject to an agreement or contract of limited
  duration, the ISP can be viewed as a coalition
• Shapley value is used to value the benefits
  derived from customers joining ISP
• Comparing the Shapley value against joining fees
  and congestion costs derives a fair price for
  services offered

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Outline

• Economy
• Consists of
• A set of consumers ? 1, 2, , N (finite in
  size)
• Consumers choose to consume quantities of a
  private good and a public good
• The public good is available for consumption at r
  locations
• Each of the r locations has a capacity
  constraint, leading to congestion in public good
  consumption

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Outline

• The problem
• Each location funds the availability of the
  public good using taxes
• e.g. proportional income taxes and poll taxes
• Each consumer chooses a location based on two
  factors
• The level of congestion experienced when
  consuming the public good
• The amount of taxation they have to pay as
  resident of a location
• Tax revenue and congestion are connected, i.e.
  tax revenue determines local capacity

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Outline

• Difficulty
• Finite number of consumers
• Non-cooperative setting
• Payoffs will be non-continuous, therefore
  standard fixed point arguments dont work
• Cooperative setting
• Coalitions very sensitive to taxation structure,
  e.g. income taxes lead to non-anonymity
• Non-existence of core-like solutions

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Motivation
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Valuing Jurisdictions

• Situation
• Public good is provided a jurisdiction to each of
  their constituent consumers
• This is subject to an agreement or contract,
  where consumers pay some type of tax in return
  for the public good
• Contracts viewed as cooperative agreement to
  defray the cost of service provision
• Each jurisdiction is viewed as coalitions of
  consumers

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Valuing Jurisdictions

• Shapley value
• Provides information about the contribution each
  user must pay to receive access to the public
  good
• Provides two bits of information
• Each Shapley value provides the individual
  valuation of the quality of access to the public
  good
• Valuation of public good per jurisdiction when
  summing across members
• This approach has been used in the internal phone
  billing rates
• Billera, Heath and Raanan (1978) Internal Billing
  Rates a Novel Application of Non-Atomic Game
  Theory. Operations Research 26, 956965
• Shapley value will exist even when there is no
  core

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Valuing Jurisdictions

• Recall
• A set of consumers ? 1, 2, , N and a
  collection of r ISPs
• For each ISP, Gk ? ? such that Gk ? Gk ? and
  ?k Gk ?
• For the partition (G1,,Gr)
• The value of services provided by an ISP Gk is
  given by v(Gk, G) v(Gk, (G1,,Gr)), with
• v(Gk, G) 0 whenever Gk ?

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Valuing Jurisdictions

• The Shapley value for an (N, r) game is given
  by
• Bolger (1993) The value for games with N players
  and r alternatives. International Journal of Game
  Theory 29, 319--334

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Valuing Jurisdictions

• To provide a sketch of how to derive out the vN-M
  utility function
• Define a new value function vGk,G(T) where
  vGk,G (T) 1 if T ? Gk and zero otherwise
• Hence vGk,G forms a basis function over the space
  (Gk,G)

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Valuing Jurisdictions

• Roth (1977)
• The Shapley Value as von Neuman-Morgenstern
  Utility function. Econometrica 45, 657--644
• Bolger (1993) has shown that under Axioms 1-3 and
  k-efficiency, then
• Roths conditions can be adapted to the (N, r)
  game and used to establish

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Valuing Jurisdictions

• By summing across coalitions it can be seen that
• Make substitution for the utility to get the
  equality

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Tiebout Economy

• Expected utility function
• Utility function for the public good, which is
  given by the Shapley value
• Cost in terms of the private good
• Poll tax
• Congestion cost function

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Tiebout Economy

• A free mobility equilibrium with congestion is a
  feasible allocation ? such that
• for any k and l 1? k, l ? r
• for any k, 1? k ? r and any (?k, ?)?G

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Tiebout Economy

• It can be shown that if the congestion function
  ?i are common for all i ? ?, then a Nash
  equilibrium exists for this economy
• The ?i represent a time preference for waiting
• The presence of a congestion function is
  important because
• Its absence results in groups of different sizes
  because people move to the cheapest jurisdiction
• But what happens is that as jurisdictions get
  larger then congestion makes people switch groups

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Tiebout Economy

• The proof relies on the existence potential
  function for this economy
• Once this potential function is shown to exist,
  then there exists a pure Nash equilibrium for the
  game
• The proof is based on Konishi, Weber and LeBreton
  (1997)

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Tiebout Economy

• Existence of a Nash equilibrium does not
  necessarily deliver a Pareto-optimal outcome
• A strong equilibrium leads to notion of
  renegotiation proofness within the implicit
  contractual agreement between a jurisdiction and
  its members
• As the outcome is also Pareto-optimal, then
  Shapley value will provide a fair valuation for
  the public good provided by the jurisdiction

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Tiebout Economy

• It can be shown that a strong equilibrium exists
  if a Tiebout economy is balanced
• It is argued that the appropriate fair price
  will reflect each users marginal valuation of the
  services acquired from any ISP, and at a
  Pareto-optimum this cannot be improved
• The Shapley value will therefore reflect the
  allocation at the strong equilibrium
• Under common congestion functions the strong
  equilibrium is within the set of Nash equilibrium

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Conclusion

• What I have done
• Examined the finite local public goods economy
  using a cooperative game theoretic approach
• Based on this model, a Shapley value is derived
  for each member of a jurisdiction
• Its equivalence to the von Neuman-Morgenstern
  utility function
• Looked at existence of equilibrium allocations in
  the economy
• Via the equivalence of cooperative and
  non-cooperative potential games
• Show existence of a Strong Nash equilibrium
• Pareto optimum emerges as a result of consumers
  exercising their outside option

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Conclusion

• Possible extension
• The connection between the queue and the Shapley
  value
• If the Shapley value should embed some
  information about the queue in terms of the
  marginal of each agent to the level of congestion
• Ojective Apply this approach to pricing
  congestion in queuing problems