1. 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

2. 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

3. On the Computational Complexity of Markets Rob Axtell Center on Social and Economic Dynamics, Brookings Institution External Faculty Member, Santa Fe Institute raxtell@brookings.edu www.brookings.edu/es/dynamics

4. 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

5. Herbert Simon on Computational 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

6. 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, lim n  f(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

7. 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): n 2 –more efficient (Quicksort): n log(n) Conversion to a decision problem: –is the list sorted?

8. 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

9. Complexity Hierarchy P NP

10. Complexity Hierarchy P NP PSPACE

11. Complexity Hierarchy P NP PSPACE EXP

12. Complexity Hierarchy P NP PSPACE EXP NEXP

13. 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

14. Function Problem Hierarchy FP FNP

15. Function Problem Hierarchy FP PPAD PPA FNP Open questions: PPAD = PPA PPA = FNP PPAD = FP

16. 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

17. 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

18. Complexity of k-lateral Exchange: Analytical Results Allocations:

19. Exchange algorithm: Complexity of k-lateral Exchange: Analytical Results

20. Allocations: Exchange algorithm: Evolution equation: x(t+1) = T(x(t)) Pseudo-contraction: Complexity of k-lateral Exchange: Analytical Results

21. Allocations: Exchange algorithm: Since T() is conservative, dominant eigenvalue = 1 Convergence is controlled by subdominant eigenvalue Evolution equation: x(t+1) = T(x(t)) Pseudo-contraction: Complexity of k-lateral Exchange: Analytical Results

22. Convergence of x(t) to x eq is exponentially fast as long as the subdominant eigenvalue < 1 for all t Complexity of k-lateral Exchange: Analytical Results, continued

23. Convergence of x(t) to x eq is exponentially fast as long as the subdominant eigenvalue < 1 for all t In particular, the amount of time, , required to compute an  approximation to equilibrium is given by

24. Complexity of k-lateral Exchange: Analytical Results, continued Convergence of x(t) to x eq is exponentially fast as long as the subdominant eigenvalue < 1 for all t In particular, the amount of time, , required to compute an  approximation to equilibrium is given by Since (AN) 2 multiplies are necessary for each iteration the amount of time needed to compute an  approximation of equilibrium is  (AN) 2, and thus k-lateral exchange processes are in FP

25. 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

26. 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 Complexity of Bilateral Exchange: Analytical Results

27. 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) Complexity of Bilateral Exchange: Analytical Results

28. 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 2 k agents, Complexity of Bilateral Exchange: Analytical Results for all k

29. 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 2 k agents, This is exactly what the bilateral exchange model does! Complexity of Bilateral Exchange: Analytical Results for all k

30. Bilateral Exchange Models Common:  Population of agents with heterogeneous preferences and endowments  Topology of interaction  Exchange rules

31. 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

32. Complexity of Bilateral Exchange: Dependence on Number of Agents Computational Results Implication: Number of interactions/agent is independent of the size of the economy Interactions  Agents

33. Interactions  N 2 Complexity of Bilateral Exchange: Dependence on Number of Goods, Computational Results Implication: Bilateral exchange much more efficient than the Walrasian mechanism

34. Comparison: Walrasian vs. k-lateral Exchange Equilibria Lesson: Price heterogeneity IMPROVES market performance

35. 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

36. Complexity of Decentralized Markets When 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 (e N /A has same complexity as e N )

37. Complexity of Decentralized Markets When 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