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2-Nash and Leontief Economy. Jugal Garg. 2-Nash. Recall the LCP formulation (A, B): payoff matrices Every non-zero solution of above LCP is in one-to-one correspondence with Nash equilibrium of (A,B). Leontief Economy. m agents, n goods Agent i has endowment

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Presentation Transcript
2 nash
2-Nash
  • Recall the LCP formulation
    • (A, B): payoff matrices

Every non-zero solution of above LCP is in one-to-one correspondence with Nash equilibrium of (A,B)

leontief economy
Leontief Economy
  • m agents, n goods
  • Agent i has endowment
  • Equilibrium prices and allocation?
pairing leontief economy
Pairing Leontief Economy
  • n agents and n goods
  • W = I (
  • utility of agent i
  • price of good j
pairing leontief utility
Pairing Leontief Utility
  • Market clearing:
  • Consider the following LCP:
    • How to get prices?
leontief and symmetric 2 nash
Leontief and Symmetric 2-Nash
  • Consider a positive U
  • LCP solves the Leontief economy
    • Proof on board
  • Leontief  symmetric Nash equilibrium
2 nash and leontief
2-Nash and Leontief
  • Consider
  • All entries in (A,B) are positive.
  • 2-Nash  two-group pairing Leontief economy
    • Proof on board
conclusion
Conclusion
  • 2-Nash is no harder than Leontief economy
  • NP-hard:
    • more than one equilibrium
    • Equilibrium with positive price of a given set of goods