1. 2-Nash and Leontief Economy
Jugal Garg

2. 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)

3. 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?

4. 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

5. 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?

6. Leontief and Symmetric 2-Nash
Consider a positive U
LCP solves the Leontief economy
Proof on board
Leontief  symmetric Nash equilibrium

7. 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

8. 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