2-Nash and Leontief Economy

Jugal Garg

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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)

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Leontief Economy

• m agents, n goods• Agent i has endowment

• Equilibrium prices and allocation?

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Pairing Leontief Economy

• n agents and n goods• W = I (

• utility of agent i • price of good j • –

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Pairing Leontief Utility

• Market clearing: •

• Consider the following LCP:

– How to get prices?

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Leontief and Symmetric 2-Nash

• Consider a positive U• LCP solves the Leontief economy– Proof on board

• Leontief symmetric Nash equilibrium

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2-Nash and Leontief

• Consider • All entries in (A,B) are positive.• 2-Nash two-group pairing Leontief economy – Proof on board

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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