The Edgeworth Conjecture with Small Coalitions and Approximate - PowerPoint PPT Presentation

The Edgeworth Conjecture with Small Coalitions and Approximate Equilibria in Large Economies S. Barman F. Echenique Indian Institute of Science Caltech USC Oct 31, 2019

◮ Scope of the “competitive hypothesis,” or validity of price-taking assumption. ◮ New algorithmic “testing” question. Barman-Echenique Edgeworth

Price-taking behavior Barman-Echenique Edgeworth

Francis Ysidro Edgeworth 1884 “. . . the reason why the complex play of competition tends to a simple uniform result – what is arbitrary and indeterminate in contract between individuals becoming extinct in the jostle of competition – is to be sought in a principle which pervades all mathe- matics, the principle of limit, or law of great numbers as it might perhaps be called. ” Barman-Echenique Edgeworth

Competitive hypothesis ◮ Core convergence theorem (Aumann; Debreu-Scarf): in a large economy, where no agent is “unique,” bargaining power dissipates and the outcome of bargaining approximates a Walrasian equilibrium ◮ Competitive prices emerge as terms of trade in bargaining. Barman-Echenique Edgeworth

Competitive hypothesis ◮ Core convergence theorem (Aumann; Debreu-Scarf): in a large economy, where no agent is “unique,” bargaining power dissipates and the outcome of bargaining approximates a Walrasian equilibrium ◮ Competitive prices emerge as terms of trade in bargaining. ◮ Requires coailitions of arbitrary size. Barman-Echenique Edgeworth

Our results – I Coalitions of size � h 2 ℓ � O ε 2 suffice, where: ◮ h is the heterogeneity of the economy ◮ ℓ is the number of goods ◮ ε > 0 approximation factor. ◮ We use the Debreu-Scarf replica model. Barman-Echenique Edgeworth

Our results – II The same ideas give answers to a new algorithmic question. Given an economy E and an allocation x , are there prices p such that ( x , p ) is a Walrasian equilibrium? Barman-Echenique Edgeworth

Our results – II The same ideas give answers to a new algorithmic question. Given an economy E and an allocation x , are there prices p such that ( x , p ) is a Walrasian equilibrium? Contrast with Second Welfare Thm. We provide a poly time algorithm that (under certain sufficient conditions) decides the question. Barman-Echenique Edgeworth

Our results – II The same ideas give answers to a new algorithmic question. Given an economy E and an allocation x , are there prices p such that ( x , p ) is a Walrasian equilibrium? Contrast with Second Welfare Thm. We provide a poly time algorithm that (under certain sufficient conditions) decides the question. Barman-Echenique Edgeworth

Hardness of Walrasian eq. Context: existing hardness results for Walrasian equilibria: ???? Our contribution: finding prices is easy even when finding a W-Eq. is hard. Specifically: ◮ Leontief utilities ◮ Piecewise-linear concave utilities Barman-Echenique Edgeworth

Economies An exchange economy comprises ◮ a set of consumers [ h ] := { 1 , 2 , . . . , h } , ◮ a set of goods, [ ℓ ] := { 1 , 2 , . . . , ℓ } . Each consumer i described by ◮ A utility function u i : R ℓ + �→ R ◮ An endowment vector ω i ∈ R ℓ + . An exchange economy E is a tuple (( u i , ω i )) h i =1 . Barman-Echenique Edgeworth

Assumptions on u i ◮ u i s are continuous and monotone increasing. ◮ utilities are continuously differentiable ◮ and α -strongly concave, with α > 0: u : R ℓ �→ R , is said to be α -strongly concave within a set R ⊂ R ℓ if u ( y ) ≤ u ( x ) + ∇ u ( x ) T ( y − x ) − α 2 � y − x � 2 . ∇ u ( x ) is the gradient of the function u at point x Barman-Echenique Edgeworth

Allocations An allocation in E is h h � � x = ( x i ) h i =1 ∈ R h ℓ st x i = ω i . + i =1 i =1 Barman-Echenique Edgeworth

Utility normalization Utilities are normalized so that u i ( x i ) ∈ [0 , 1) for all consumers i ∈ [ h ] and all allocations ( x i ) i ∈ R h ℓ + . Barman-Echenique Edgeworth

The Core ◮ An allocation in E is x = ( x i ) h i =1 ∈ R h ℓ + , s.t � h i =1 x i = � h i =1 ω i . Barman-Echenique Edgeworth

The Core ◮ An allocation in E is x = ( x i ) h i =1 ∈ R h ℓ + , s.t � h i =1 x i = � h i =1 ω i . ◮ A nonempty subset S ⊆ [ h ] is a coalition . ◮ ( y i ) i ∈ S is an S-allocation if � i ∈ S y i = � i ∈ S ω i . Barman-Echenique Edgeworth

The Core ◮ An allocation in E is x = ( x i ) h i =1 ∈ R h ℓ + , s.t � h i =1 x i = � h i =1 ω i . ◮ A nonempty subset S ⊆ [ h ] is a coalition . ◮ ( y i ) i ∈ S is an S-allocation if � i ∈ S y i = � i ∈ S ω i . ◮ A coalition S blocks the allocation x = ( x i ) h i =1 in E if ∃ an S -allocation ( y i ) i ∈ S s.t u i ( y i ) > u ( x i ) for all i ∈ S . ◮ The core of E is the set of all allocations that are not blocked by any coalition. Barman-Echenique Edgeworth

The κ -core The κ -core of E , for κ ∈ Z + , is the set of allocations that are not blocked by any coalition of cardinality at most κ . Note: ◮ Core: all 2 h coalitions ◮ κ -core: small coalitions � h � ◮ κ -core: few ( ) coalitions κ Barman-Echenique Edgeworth

Equilibrium and approximate equilibrium A Walrasian equilibrium is a pair ( p , x ) ∈ R ℓ + × R h ℓ + s.t 1. p ∈ R ℓ + is a price vector 2. p T x i = p T ω i and, for all bundles y ∈ R ℓ + with the property that u i ( y ) > u i ( x i ), we have p T y i > p T ω i . 3. � h i =1 x i = � h i =1 ω i ( supply equals the demand ). Barman-Echenique Edgeworth

Equilibrium and approximate equilibrium A Walrasian equilibrium is a pair ( p , x ) ∈ R ℓ + × R h ℓ + s.t 1. p ∈ R ℓ + is a price vector 2. p T x i = p T ω i and, for all bundles y ∈ R ℓ + with the property that u i ( y ) > u i ( x i ), we have p T y i > p T ω i . 3. � h i =1 x i = � h i =1 ω i ( supply equals the demand ). i.e x = ( x i ) i ∈ [ h ] ∈ R h ℓ + is an allocation Barman-Echenique Edgeworth

Approximate Walrasian equilibrium A ε -Walrasian equilibrium is a pair ( p , x ) ∈ R ℓ + × R h ℓ + in which p ∈ ∆ and (i) | p T x i − p T ω i | ≤ ε and (ii) for any bundle y ∈ R ℓ + , with the property that u i ( y ) > u i ( x i ), we have p T y > p T ω i − ε/ h . iii) x is an allocation ( supply equals the demand ). Barman-Echenique Edgeworth

Replica economies Let E = (( u i , ω i )) i ∈ [ h ] be an exchange economy. The n-th replica of E , for n ≥ 1, is the exchange economy E n = (( u i , t , ω i , t )) i ∈ [ n ] , t ∈ [ h ] , with nh consumers. In E n the consumers are indexed by ( i , t ), with index i ∈ [ n ] and type t ∈ [ h ], and they satisfy: u i , t = u t and ω i , t = ω t . Barman-Echenique Edgeworth

Equal treatment property An allocation in E n has the equal treatment property if all consumers of the same type are allocated identical bundles. Barman-Echenique Edgeworth

Equal treatment of equals Let E = (( u i , ω i )) i ∈ [ h ] be an exchange economy. Lemma (Equal treatment property) Suppose each u i is strictly monotonic, continuous, and strictly concave. Then, every κ -core allocation of E n satisfies the equal treatment property. Barman-Echenique Edgeworth

Core convergence: Debreu-Scarf (1963) Let E = (( u i , ω i )) i ∈ [ h ] be an exchange economy. Theorem (Debreu-Scarf Core Convergence Theorem) Suppose u i is st. monotonic, cont., and strictly quasiconcave. + is in the core of E n for all n ≥ 1 , If the allocation x ∈ R h ℓ = ⇒ ∃ p ∈ ∆ s.t ( p , x ) is a Walrasian equilibrium. Barman-Echenique Edgeworth

Main result Let E = (( u i , ω i )) i ∈ [ h ] be an exchange economy with h consumers and ℓ goods. Theorem Let ε > 0 . Suppose u i is st. monotonic, C 1 , and α -strongly concave. If the allocation x is in the κ -core of E n , for � λℓ h � + h 2 n ≥ κ ≥ 16 . ε 2 α ε Then ∃ p ∈ ∆ s.t ( p , x ) is an ε -Walrasian equilibrium). Here, λ is the Lipschitz constant of the utilities. Barman-Echenique Edgeworth

Testing Assume black-box access to utilities and their gradients. Barman-Echenique Edgeworth

Testing Let E = (( u i , ω i )) i ∈ [ h ] be an exchange economy. Theorem (Testing Algorithm) Suppose that each u i is monotonic, C 1 , and strongly concave. Then, there exists a polynomial-time algorithm that, given an allocation y in E , decides whether y is an ε -Walrasian allocation. Barman-Echenique Edgeworth

Testing Remark Analogous results are possible without strong concavity: Leontief and PLC utilities, for instance. Barman-Echenique Edgeworth

Ideas in the proof. Barman-Echenique Edgeworth

Approximate Caratheodory Theorem Let x ∈ cvh ( { x 1 , . . . , x K } ) ⊆ R n , ε > 0 and p an integer with 2 ≤ p < ∞ . Let γ = max {� x k � p : 1 ≤ k ≤ K } . Then there is a vector x ′ that is a convex combination of at most 4 p γ 2 ε of the vectors x 1 , . . . , x K such that � x − x ′ � p < ε . See ? . Barman-Echenique Edgeworth

Upper contour sets Let y = ( y i ) i ∈ [ h ] be an allocation. Let � � y ∈ R ℓ V i := + | u i ( y ) ≥ u i ( y i ) be the upper contour set of i at ¯ y . Obs: V i is closed and convex. Barman-Echenique Edgeworth