Patient Players Fill the Core: A Conspicuously Simple Solution to the Empty Core Problem

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Patient Players Fill the Core A Conspicuously
Simple Solution to the Empty Core Problem

• Joshua S. Gans and Michael D. Ryall
• University of Melbourne
• Annual Australian Economic Theory Workshop,
  February 2007

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Problem of the Empty Core

• Core theory looks for allocations that are stable
  in the face of blocking coalitions
• The Empty Core Problem is that there may be no
  such allocation
• Responses
• Cant exist in reality
• Outside the game arrangements (integration or
  contracts)

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Standard Example

• Three identical agents 1, 2 and 3
• The core is empty

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Is there a prediction?

• There can be complete instability
• Coase
• If my argument is to be taken seriously, it has
  to be shown that the grand coalition is stable.
  It might be argued that no firm would withdraw
  from this arrangement to enter into a two-party
  agreement since this would set in motion a
  process which would lead ultimately to lower
  profits.
• Folk theorem-like argument players would realise
  not agreeing to an efficient allocation would
  lead to outcomes with a lower payoff.
• Why deviate and create 100 if this too will be
  unstable?

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Our Model

• Model what happens in deviations carefully
• Do not presume a deviating coalition will stick
• When there is an empty core, there is always
  another partner available so there is uncertainty
  regarding whether a coalition will stick
• Efficient outcomes
• A non-empty core is a sufficient but not
  necessary condition to predict a grand coalition
  forming
• In a repeated game, the grand coalition forming
  is always an equilibrium with sufficiently
  patient players
• Assumptions
• Restrict attention to 3 player case
• Usual coalitional game assumptions with single
  player coalitional value set equal to 0.

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Stage Game

• Demands Each agent, i, simultaneously announces
  a value demand with r the set of
  demands.
• Feasibility search for coalitions, G, such that
• Arms-Length Consistency amongst feasible
  coalitions rule out Pareto dominated ones

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Stage Game

• Payoff agent i receives
• Intuition amongst feasible coalitions, each has
  an equal chance of forming.
• But if grand coalition is feasible it will form.

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Definitions

• Marginal product
• Von Neumann-Morgenstern demand
• Non-empty core
• Nash equilibrium unilateral deviations in value
  demands
• Strong Nash equilibrium group deviations in
  value demand that make all members of a group
  strictly better off.
• Coalition-Proof Nash equilibrium group
  deviations in value demands with credible
  coalitions

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Nash equilibrium

• If the core is non-empty, then any allocation in
  the core is also a NE of the stage game.
• If the core is empty, then
• (Grand Coalition forms) If for some i,
  then the unique NE involves ,
  and
• (Each two agent coalition forms with equal
  probability) If for all i, then
  the unique NE involves for all i.

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Proof (Example)

• In our standard example,
• So the unique NE involves each player stating
  their VNM demand of 50.
• If one player increases their demand, all of
  their coalitions are non-feasible so they get 0
  rather than 33.33
• If one player decreases their demand to enable
  the grand coalition, they get 20 rather than
  33.33.

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Robustness

• All the NE are coalition proof NE as well
• If two agents deviated,
• by increasing their demands they rule all
  coalitions out and so earn 0.
• by decreasing their demands to allow the grand
  coalition, they can do no better
• The third agent with a VNM demand can block the
  others from a certain agreement.
• Strong Nash qualified extensions

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Characterising Equilibria Outcomes

• If the core is non-empty,
• If the core is empty,
• With an empty core, you are guaranteed at least
  your marginal contribution.

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Repeated Game

• For all possible value functions, there exists d
  such that for all d?d, 1), there exist supergame
  strategies that are Coalition-Proof Subgame
  Perfect with the grand coalition forming in each
  period.
• If for all i, then for all d,
  there exist supergame strategies that are
  Coalition-Proof Subgame Perfect with the grand
  coalition forming in each period.

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Proof (Example)

• Note that the VNM stage game in every period is a
  Nash equilibrium (and it is coalition proof).
  Hence, it is a sustainable punishment.
• Can therefore sustain demands of each player
  being greater than 33.33 (say 40)
• Need to set up strategies that give enough value
  to agents so that they cannot deviate profitably
  and guarantee a two player outcome.
• Suppose that two agents received only 33.3. Then
  one or the other could deviate and have their two
  agent coalition form with certainty.
• However, one agent could receive only 33.3 as by
  increasing their demand, they could not do
  better.
• Thus, additional constraint is that any two
  agents jointly receive more than their
  coalitional value.
• Core-like blocking conditions continue to hold.

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Conclusions

• Provide a non-cooperative means of broadening the
  applicability of core theory
• Economic implications
• Empty core implies that agents receive more than
  their marginal products
• Empty core allocations that are efficient are
  constrained