Effcient Dynamic Barter Exchange (2015)

Transcription

1 Effcient Dynamic Barter Exchange (2015) Ross Anderson, Itai Ashlagi, David Gamarnik, and Yash Kanoriay David Walker-Jones February 21, 2016

2 Barter s Literature Review One can envisions scenarios where the distribution of goods may not be optimal, but pricing mechanisms may be deemed inappropriate for reallocation. Kidneys Vacation Time

3 Barter s Literature Review This paper discuses a dynamic world in which individuals are periodically born, and endowed with a good that they find unsatisfactory. While the good held is not desired by the owner, there may be other disgruntled individuals that find it satisfactory.

4 Barter s Literature Review Broadly, the goal of the paper is to consider policy choices of a social planner, and compare the efficiency of certain simple policies in terms of average waiting times.

5 Barter s Literature Review There are a number of papers that address similar, but slightly varied, topics. These papers differ in at least one of the following ways: They are not dynamic in nature. There are two sides to the market. A finite time horizon is used. Matching probability and pool size are inversely related. Individuals vary in an underlying characteristic that effects match probabilities. Feature stochastic exit. Ignore waiting times. Focus on who to match with, not when.

6 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation In each period t: There exists a directed graph inherited from the previous period. A new individual arrives. Each possible directed link between it and each of the predecessors occurs with exogenous and independent probability p. The social planner may reallocate goods according to a set of rules if they see fit. Individuals that are satisfied leave the system, finalizing the directed graph inherited by the next period.

7 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation A Policy is a mapping from the history of exchanges and the state of the marketplace to a set of feasible exchanges involving non overlapping sets of agents.

8 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation A Periodic Markov Policy, is a policy that employs τ homogeneous first order Markov policies in round robin for some τ N, and the market state is irreducible and positive recurrent under the policy. This set of policies have ergodic properties.

9 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation There are two main types of greedy policies: Greedy Cycle Removal: Each time a new individual arrives, if their nodes creates a cycle, and the cycle requires no more than a pre specified number of individuals, the cycle is completed. Greedy Chain Removal: Each time period inherits a bridge node from the previous period. If upon arrival of a new individual a chain that originates from the bridge is created, it is removed. Note: this leaves a bridge node for the next period.

10 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation A Batching Policy with size N waits for N arrivals, and then executes the combination of permitted removals that minimizes the number of remaining individuals.

11 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation The analysis is done in a steady state environment. Affords several luxuries: Minimizing average waiting time is equivalent to minimizing average pool size. In some circumstances, the directed graph with n nodes and a probability p of any directed edge being realized, will be equivalent for analysis purposes, to a directed graph with n nodes and M uniformly distributed directed edges. Can equate average inflows and outflows to find steady state.

12 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation The authors claim that while their model is simple, it possesses two advantages that many similar models do not. It avoids use of a market size parameter. Comparisons between different policies can be made in a single dimension.

13 Specific Formulation Types of Greedy Policies Batching Steady State Analysis Advantages Notation The authors use the following notation: f (p) = O(g(p)) if C < s.t. f (p) < Cg(p) p (0, 1]. f (p) = o(g(p)) if C > 0, p o (0, 1] s.t. f (p) < Cg(p) p (0, p o ]

14 Theorem 3.1 Theorem 3.2 Theorem 3.3 Theorem: In the setting where only cycles of length two may be executed, the greedy policy achieves the optimal average waiting time of log(2)/p 2 + o(1/p 2 ).

15 Theorem 3.1 Theorem 3.2 Theorem 3.3 Theorem: In the setting where cycles of two or three may be executed, the average waiting time under the greedy policy is O(1/p 3/2 ). Furthermore, a constant C < s.t., for any batching policy, the average waiting time is at least 1/(Cp 3/2 ).

16 Theorem 3.1 Theorem 3.2 Theorem 3.3 Theorem: In the chain removal setting, the greedy policy achieves an average wait time of O(1/p). Further, in the setting where chain removal and cycle removal of length two and three are permitted, a constant C s.t. for any policy, the average wait time will be at least 1/(Cp).

17 Figure: Simulations

18 Findings Implications The paper featured the following findings: Greedy strategies are approximately optimal. Cycles of length three and chains each significantly reduce waiting times.

19 Findings Implications The authors discuss several implications: Competition between clearing houses may not hurt consumer welfare. In circumstances of decentralized marketplaces, switching to a centralised system may create large returns.

20 Questions or Comments?