2-Nash and Leontief Economy Jugal Garg

2-Nash Recall the LCP formulation (A, B): payoff matrices 𝐴𝑦≤1 ; 𝑦≥0 𝐴𝑦≤1 ; 𝑦≥0 𝑥≥0 ; 𝑥 𝑇 𝐵≤1 𝑥 𝑇 𝐴𝑦=0; 𝑥 𝑇 𝐵𝑦=0 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 𝑤 𝑖1 , 𝑤 𝑖2 , …, 𝑤 𝑖𝑛 𝑈 𝑖 𝑥 𝑖 = min 𝑗 { 𝑥 𝑖𝑗 𝑢 𝑖𝑗 } Equilibrium prices and allocation?

Pairing Leontief Economy n agents and n goods W = I ( 𝑤 𝑖𝑖 =1, 𝑤 𝑖𝑗 =0, 𝑖≠𝑗, ∀𝑖,𝑗) 𝑈 𝑖 𝑥 𝑖 = min 𝑗 { 𝑥 𝑖𝑗 𝑢 𝑖𝑗 } 𝛽 𝑖 : utility of agent i 𝑝 𝑗 : price of good j 𝛽 𝑖 = 𝑝 𝑖 𝑗 𝑢 𝑖𝑗 𝑝 𝑗 𝑗 𝑢 𝑖𝑗 𝑝 𝑗 >0 𝛽 𝑖 >0 iff 𝑝 𝑖 >0

Pairing Leontief Utility Market clearing: 𝑖 𝛽 𝑖 𝑢 𝑖𝑗 ≤1, ∀𝑗 𝑝 𝑗 >0⇒ 𝑖 𝛽 𝑖 𝑢 𝑖𝑗 =1 ( 𝑖 𝛽 𝑖 𝑢 𝑖𝑗 −1) 𝑝 𝑗 =0, ∀𝑗 Consider the following LCP: 𝑖 𝛽 𝑖 𝑢 𝑖𝑗 ≤1, 𝛽 𝑗 ≥0 𝛽 𝑗 ( 𝑖 𝛽 𝑖 𝑢 𝑖𝑗 −1)=0, ∀𝑗 How to get prices?

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 Consider 𝑈= 0 𝐵 𝑇 𝐴 0 All entries in (A,B) are positive. 2-Nash  two-group pairing Leontief economy Proof on board

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