Radoslav S. Raykov. Bank of Canada Financial Stability Department. August 2017 Econometric Society Meeting

Transcription

1 Stability and Efficiency in Decentralized Two-Sided Markets with Weak Preferences or How Important Are Weak Preferences for the Efficiency of the Stable Match? Radoslav S. Raykov Bank of Canada Financial Stability Department August 2017 Econometric Society Meeting Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 1 / 14

2 Motivation and Research Question When there are indifferences, stability does not necessarily imply efficiency: m1 : w 1, w 2, w 3, m 1 ; w1 : {m 1, m 2, m 3 }, w 1 m2 : w 1, w 3, w 2, m 2 ; w2 : {m 1, m 2, m 3 }, w 2 m3 : w 2, w 1, w 3, m 3 ; w3 : {m 1, m 2, m 3 }, w 3 ( ) m1 m µ 1 = 2 m 3 w 2 w 3 w 1 }{{} stable + inefficient ( ) m1 m µ 2 = 2 m 3. w 1 w 3 w 2 }{{} stable + efficient Quantitatively, how inefficient is a stable matching on average? Does a large market help a stable one-to-one matching become more efficient? Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 2 / 14

3 Results Large, heterogeneous markets limit (bound) the inefficiency With independent heterogeneous agents, it is harder for preferences to line up so as to permit Pareto-improving exchanges. Metric of inefficiency = the fraction f of agents who improve strictly from exchange With uniform iid preferences, the maximum fraction of strict improvers is bounded by: 1/4 when one side of the market has weak preferences 3/8 when both sides have weak preferences Non-uniform preferences: Can eliminate inefficiency completely using indifferences of the same order as uniform I characterize different weak preference types and their effect Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 3 / 14

4 Setup Large market with heterogeneous, independent agents as in Kojima and Pathak (2009), Kojima and Manea (2010), or Ashlagi, Kanoria and Leshno (2015) First, fix n. Each agent randomly and uniformly selects a preference profile from the list of all possible profiles (uniformity relaxed later) Preference draws are independent across agents Now, let n ; process repeated for each value of n. Initially, only one side of the market has weak preferences (later, both sides) Quantify the maximum inefficiency of the average stable matching: Average := uniformly selected at random from all stable matchings (given realized ) Maximum inefficiency := the maximum number of strict improvers as a fraction of n Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 4 / 14

5 Pareto-Improvement Cycles and Chains Identify inefficient marriages with Pareto-Improvement Cycles and Chains: w 5 m 5 m 3 m 2 m 1 w 1 m 1 m 4 w 4 m 2 m 3 w 2 w 3 w 4 w 3 w 2 Lemma (Ergin and Erdil, 2015) A one-to-one matching is Pareto efficient iff it admits neither Pareto-improvement cycles nor Pareto-improvement chains. Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 5 / 14

6 Necessary and Sufficient Condition for Cycles in a Stable Matching w 4 Strict envy Indifference m 4 m 3 m 2 m 1 w 3 m 3 m 5 w 5 m 2 m 1 w 2 w 1 w 4 w 3 w 2 Necessary Condition for Cycles in a Stable Matching The 2-tuple (m 1, m 2 ) cannot participate in any Pareto-improvement cycle unless m 1 µ(m2 ) m 2 m 1 µ(m1 ) m 2 µ(m 2 ) m1 µ(m 1 ) µ(m 1 ) m2 µ(m 2 ) Corollary. No more than half of the agents in a cycle can improve strictly. Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 6 / 14

7 Decoupling the Average Stable Matching µ from Preferences Problem: Preferences and the outcome µ are not independent. Insight: We can asymptotically decouple preferences from the outcome, so that µ n (m 2 ) = w is independent from m 1 w m 2 Theorem a) As n, for every two fixed men m 1 and m 2 and every fixed woman w, Pr ( µ n (m 2 ) = w m 1 w m 2 ) Pr ( µn (m 2 ) = w ) b) This implies that, for any two arbitrary men m 1, m 2 and arbitrary woman w, as n, Pr ( m 1 µn(m 2 ) m 2 ) Pr ( m1 w m 2 ). Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 7 / 14

8 Bounding the Necessary Condition for Cycles from Above ( ) Pr(m 1, m 2 C) 2 Pr m 1 µ(m2 ) m 2 µ(m2 ) m1 µ(m 1 ) = = 2 Pr(m 1 µ(m2 ) m 2 ) Pr(µ(m 2 ) m1 µ(m 1 )) = }{{} 1/2 Pr(m 1 w m 2 ) := p(n). Hence the (relative) frequency of pairwise indifferences is key. What is this frequency for uniformly selected preferences? How about for non-uniform preferences? Can we design preferences with custom p(n)? Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 8 / 14

9 Preferences To create uniform preferences, use random ordered partitions: m 7 m 2 m 3 m 5 m 4 m 6 m 1 m 8 m 7 {m 2 m 3 m 5 } m 4 {m 6 m 1 } m 8. With uniform choice, the probability of a pairwise indifference is p(n) = # of weak orders where m 1 and m 2 are in the same partition block total # of weak orders T n T n Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 9 / 14

10 Quantifying p(n) for Uniform Preferences Theorem T n = T n 1 ; T n = n 1 ( ) n T i. i i=0 Theorem With uniform weak preferences, as n, p(n) = T n 1 T n ln 2 n 0. Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 10 / 14

11 Relating p(n) to the inefficiency metric We know p(n) 0. Thus the fraction of 2-tuples permitting cycles 0. But our inefficiency metric is the fraction of strict improvers, who are 1-tuples! Recast the problem as an Erdös-Rényi random graph G ( n, p(n) ) : If two men m 1 and m 2 permit a cycle, we ll say they are connected with an edge Asymptotically, connections are independent and occur with probability p(n). The (max) fraction of men who permit cycles are those with degree 1: Pr ( deg(m) 1 ) = 1 Pr ( deg(m) = 0 ) = 1 e np. Max Fraction of Strict Improvers f = 1 e np 2 Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 11 / 14

12 Results With uniform preferences, the max inefficiency is bounded and constant with n When only one side has weak preferences: p(n) ln 2/n 1 e ln 2 f (, ) = = When both men and women have weak preferences: p(n) 2 ln 2/n 1 e 2 ln 2 f (, ) = = These are only upper bounds. In reality, the inefficiency is likely smaller: A large cycle may strictly improve only a single person The same person can be part of many cycles, but only one gets executed Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 12 / 14

13 How About Non-Uniform Random Preferences? Group 1! Partitions Pr m 0 Group 2!Partitions Pr m! Pr =1 Pr = ln 2 could be constant or a function of Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 13 / 14

14 Results for Non-Uniform Preferences Thus we can create symmetric preferences with custom p(n) What preferences generate a maximum inefficiency of at most f? Given f, such preferences should satisfy lim np(n) ln(1 f); n For the np const case, such preferences are generated by weighting probability q = log 2 (1 f) 1 (f [0, 0.5]). f can be dialed arbitrary low by choosing suitable const using q = log 2 (1 f) 1. Such non-uniform preferences have indifferences of the same order as the uniform! p(n) ln 2 n const. versus n are of the same order! Radoslav Raykov (Bank of Canada) How Important Are Weak Preferences? August 2017Econometric Society Meeting 14 / 14