Yesterday

Yesterday

• Walras-Arrow-Debreu equilibria require centralized price determination

• Decentralized exchange models…– Generate market-clearing prices that are path-

dependent– Yield non-Walrasian allocations that may be

Pareto optimal– Produce wealth inequality endogenously

Today

• Code for decentralized market• Computational complexity of markets

– High complexity of Walras-Arrow-Debreu– Low complexity of decentralized markets

• Add production:– Sugarscape ‘production’– Sugarscape exchange

On theComputational Complexity

ofMarkets

Rob AxtellCenter on Social and Economic Dynamics, Brookings Institution

External Faculty Member, Santa Fe Institute

[email protected]/es/dynamics

The Complexity of Exchange

• What is the computational complexity of economic exchange processes?

• First variant: How hard is it for the Walrasian auctioneer to determine p*?

• Second variant: What is the complexity of decentralized (non-Walrasian) markets

• Third variant: What is the complexity of realistic market processes

Herbert Simon onComputational Complexity

in Economics

…the statement that a certain class of problems is ‘exponential’ means that as we increase the number of…components, the maximum time required for solution will rise exponentially….Notice that these are not theorems about the efficiency of particular computational algorithms. They are limits that apply to any algorithms used to solve problems in the domain in question. The theorems warn us that we must not aspire to construct a [better] algorithm.

Bell Journal of Economics, 1978

Complexity Class P: Definition

• Consider a problem to be solved by ‘yes’/’no’• Let n be a measure of the size of the problem• Let f(n) be the amount of (computer) time

required to solve the problem• Typically, limnf(n) = ∞• If f(n) is a polynomial of degree d < ∞, the

problem is soluble in ‘polynomial time’• Let P be defined as the set of problems that can

be solved in polynomial time

P example: Sorting

• Given: A list of objects to sort of length n (e.g., names)

• How much computation to do the sorting?– brute force (Bubblesort): n2

– more efficient (Quicksort): n log(n)• Conversion to a decision problem:

– is the list sorted?

Class NP: Definition

• Assume that the answer to the problem is given, i.e., the answer is provided by an oracle

• Let g(n) be the amount of (computer) time required to check the solution

• If g(n) is a polynomial of degree d < ∞, the problem is nondeterministic polynomial time

• Let NP be the set of problems of nondeterministic polynomial time

• Theorem: P NP

Complexity Hierarchy

P

NP


P

NP

PSPACE


P

NP

PSPACE

EXP


P

NP

PSPACE

EXP

NEXP

Classes FP and FNP

• Consider function problems, i.e. those requiring more than ‘yes’/’no’ solutions

• Let FP be the set of function problems that can be solved in polynomial time

• Let NFP be the set of function problems of nondeterministic polynomial time

• Theorem: FP FNP• Theorem: FP = FNP P = NP

Function Problem Hierarchy

FP

FNP

Function Problem Hierarchy

FP

PPAD

PPA

FNPOpen questions:PPAD = PPAPPA = FNPPPAD = FP

Pure Exchange Economies:Walras-Arrow-Debreu Equilibria• Existence of equilibrium is proved:

– Brouwer fixed-point theorem when aggregate demand is a function

– Kakutani fixed-point theorem when aggregate demand is a correspondence (set-valued)

• Constructive proofs of Brouwer and Kakutani use Sperner’s lemma:– Any admissible coloring of any triangulation of the unit simplex

has a trichromatic triangle (an odd number of them!)

• The Scarf algorithm for computing general equilibrium is a variant of Sperner’s lemma

Complexity of Walras-Arrow-Debreu Exchange

• Theorem (Hirsch, Papadimitriou and Vavasis, 1987): Lower bound on worst case complexity exp(N), i.e, EXPTIME

• Theorem (Papadimitriou, 1994):– Complexity of Sperner reduces to the parity

argument (every finite graph has even number of odd degree nodes)

– Parity requires an exponentially large graph– Thus, Brouwer, Kakutani PPAD FNP

Complexity of k-lateral Exchange:Analytical Results

x = x

1, x

2, K , x

N, x

N + 1, K , x

2 N, x

2 N + 1, K x

AN( )

T

∈ R+

ANAllocations:

x = x

1, x

2, K , x

N, x

N + 1, K , x

2 N, x

2 N + 1, K x

AN( )

T

∈ R+

AN

T ( x ; f ) : R+

AN

→ R+

AN

Allocations:

Exchange algorithm:


x = x

1, x

2, K , x

N, x

N + 1, K , x

2 N, x

2 N + 1, K x

AN( )

T

∈ R+

AN

T ( x ; f ) : R+

AN

→ R+

AN

Allocations:

Exchange algorithm:

Evolution equation: x(t+1) = T(x(t))

x t( ) − xeq

= λt

x 0( ) − xeq

Pseudo-contraction:


x = x

1, x

2, K , x

N, x

N + 1, K , x

2 N, x

2 N + 1, K x

AN( )

T

∈ R+

AN

T ( x ; f ) : R+

AN

→ R+

AN

Allocations:

Exchange algorithm:

x t( ) − xeq

= λt

x 0( ) − xeq

Since T(•) is conservative, dominant eigenvalue = 1Convergence is controlled by subdominant eigenvalue

Evolution equation: x(t+1) = T(x(t))

Pseudo-contraction:


Convergence of x(t) to xeq is exponentially fast as long as the subdominant eigenvalue < 1 for all t

Complexity of k-lateral Exchange:Analytical Results, continued


Convergence of x(t) to xeq is exponentially fast as long as the subdominant eigenvalue < 1 for all tIn particular, the amount of time, t, required to compute an e approximation to equilibrium is given by

e ≡ x τ( ) − xeq

= λτ

x 0( ) − xeq

≡ k λτ

τ ∝ln ε( )

ln λ( )

=

ln1

ε( )

ln1

λ( )


Convergence of x(t) to xeq is exponentially fast as long as the subdominant eigenvalue < 1 for all tIn particular, the amount of time, t, required to compute an e approximation to equilibrium is given by

e ≡ x τ( ) − xeq

= λτ

x 0( ) − xeq

≡ k λτ

τ ∝ln ε( )

ln λ( )

=

ln1

ε( )

ln1

λ( )

Since (AN)2 multiplies are necessary for each iteration the amount of time needed to compute an e approximation of equilibrium is t(AN)2, and thus k-lateral exchange processes are in FP

Tighter bounds by divide-and-conquer:• Divide the agent population into pairs (A/2 pairs) and equilibrate each pair

Complexity of Bilateral Exchange:Analytical Results

Tighter bounds by divide-and-conquer:• Divide the agent population into pairs (A/2 pairs) and equilibrate each pair• This requires a number of exchange interactions proportional to N 2 for each pair, AN 2/2 overall


Tighter bounds by divide-and-conquer:• Divide the agent population into pairs (A/2 pairs) and equilibrate each pair• This requires a number of exchange interactions proportional to N 2 for each pair, AN 2/2 overall• Now match each pair with another and re-equilibrate, another AN 2/4 interactions (exact for identical preferences)


Tighter bounds by divide-and-conquer:• Divide the agent population into pairs (A/2 pairs) and equilibrate each pair• This requires a number of exchange interactions proportional to N 2 for each pair, AN 2/2 overall• Now combine two pairs and re-equilibrate, another AN 2/4 interactions (exact for identical preferences)• Overall, for 2k agents,


AN2

2i

i = 1

k

∑ = AN2 1

2i

i = 1

k

∑ ≤ AN2 for all k

Tighter bounds by divide-and-conquer:• Divide the agent population into pairs (A/2 pairs) and equilibrate each pair• This requires a number of exchange interactions proportional to N 2 for each pair, AN 2/2 overall• Now combine two pairs and re-equilibrate, another AN 2/4 interactions (exact for identical preferences)• Overall, for 2k agents,

• This is exactly what the bilateral exchange model does!


AN2

2i

i = 1

k

∑ = AN2 1

2i

i = 1

k

∑ ≤ AN2 for all k

Bilateral Exchange Models

Common: Population of agents with heterogeneous preferences and endowments Topology of interaction Exchange rules

Bilateral Exchange Models

Common: Population of agents with heterogeneous preferences and endowments Topology of interaction Exchange rules

Example: ‘Soup’ Population of agents, A {10 - 1,000,000} N commodities, N {2 - 20,000} Randomly distributed preferences Randomly distributed initial endowments Random pairings: -Sequential or parallel -Synchronous or asynchronous -Ex post, random graph of interactions Edgeworth barter A bargaining rule

100 1000 10000 100000. 1. ¥ 106Agents

100

1000

10000

100000.

1. ¥ 106

1. ¥ 107

1. ¥ 108Interactions e = 10- 2

N = 2N = 10

N = 50

Complexity of Bilateral Exchange:Dependence on Number of Agents

Computational Results

Implication: Number of interactions/agentis independent of the size of the economy

Interactions Agents

1 10 100 1000 10000Commodities

100

10000

1.¥ 106

1. ¥ 108

1. ¥ 1010Interactions e = 10- 2

A = 10A = 100

A = 1000

Interactions N2

Complexity of Bilateral Exchange:Dependence on Number of Goods,

Computational Results

Implication: Bilateral exchange much moreefficient than the Walrasian mechanism

Comparison:Walrasian vs. k-lateral Exchange

Equilibria

Walras-Arrow-DebreuBilateral Exchange

Price formation global localDynamics one-shot path-dependentExistence via fixed point theoremstrivialStability ambiguous asymptoticAllocations Pareto-efficient Pareto-efficientPrice determination OR problem DAI problemComplexity exponential polynomial (linear)Wealth effect none dispersive

Lesson: Price heterogeneity IMPROVES market performance

Summary

• Walras-Arrow-Debreu equilibria computationally intractable for auctioneer

• Decentralized equilibria are tractable• Non-Walrasian allocations result

– Non-core allocations (no equal treatment)– Inequality endogenously created by market

processes• Permitting agents to make local welfare

improvements breaks the complexity barrier

Complexity of Decentralized MarketsWhen Agents are Strategic

• Old literature: Strategic reallocation of endowments• If agents believe the market can be predicted, it is

rational to act strategically (Izumi [2003])• Limiting case: Each agent announces its prediction

function in advance:– Each agent now must compute Nash equilibrium (FNP)– Complexity of market exponential in agents and goods– Effective parallelization doesn’t alter complexity (eN/A

has same complexity as eN)

Complexity of Decentralized MarketsWhen Agents are Strategic, II

• The actual case is much worse:– Reaction functions not common knowledge– Population of predictors is evolving over time

• Possible to learn rational expectations equilibria?– Agents cannot learn in finite time [Spear 1989]– Agents cannot learn in polynomial time [Board 1994]

• Implications for the ‘efficient markets’ hypothesis– Strategic agents remove arbitrage opportunities

• Price random walks• Market cannot be ‘predicted’

– They also sever all connection to Pareto efficiency

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