Continuous-Time Random Matching Darrell Duffie Lei Qiao Yeneng Sun - - PowerPoint PPT Presentation

Continuous-Time Random Matching

Darrell Duffie Lei Qiao Yeneng Sun Stanford S.U.F.E. N.U.S.

Bernoulli Lecture Stochastic Dynamical Models in Mathematical Finance, Econometrics, and Actuarial Sciences EPFL, May 2017

Duffie-Qiao-Sun Continuous-Time Random Matching 1

Random matching markets

n1 s1 b2 s2 s3 b3 b4 n2 n3 b1

Duffie-Qiao-Sun Continuous-Time Random Matching 2

Reliance on continuous-time independent matching

◮ Many researchers have appealed to a “law of large numbers” for continuous-time

independent random matching among an atomless measure space of agents.

◮ Based on this, the fraction ptk at time t of agents of any type k is presumed to evolve

deterministically, almost surely, with naturally conjectured dynamics.

◮ The optimal strategy of each agent, given the path of pt, is then much easier to solve

than in a finite-agent model with random population dynamics [Boylan (1994)].

◮ Assuming this works, the equilibrium evolution of pt can be analyzed. ◮ But there has been no result justifying the proposed application of the law of large

numbers and conjectured dynamics. [Gilboa and Matsui (1992) have an example based

• n finitely-additive measures.]

Duffie-Qiao-Sun Continuous-Time Random Matching 3

Research areas relying on continuous-time random matching

◮ Monetary theory. Hellwig (1976), Diamond-Yellin (1990), Diamond (1993), Trejos-Wright (1995), Shi

(1997), Zhou (1997), Postel-Vinay-Robin (2002), Moscarini (2005).

◮ Labor markets. Pissarides (1985), Hosios (1990), Mortensen-Pissarides (1994), Acemoglu-Shimer (1999),

Shimer (2005), Flinn (2006), Kiyotaki-Lagos (2007).

◮ Over-the-counter financial markets. Duffie-Gˆ

arleanu-Pedersen (2005), Weill (2008), Vayanos-Wang (2007), Vayanos-Weill (2008), Weill (2008), Lagos-Rocheteau (2009), Hugonnier-Lester-Weill (2014), Lester, Rocheteau, Weill (2015), ¨ Usl¨ u (2016).

◮ Biology (genetics, molecular dynamics, epidemiology). Hardy-Weinberg (1908), Crow-Kimura (1970),

Eigen (1971), Shashahani (1978), Schuster-Sigmund (1983), Bomze (1983).

◮ Game theory. Mortensen (1982), Foster-Young (1990), Binmore-Samuelson (1999),

Battalio-Samuelson-Van Huycjk (2001), Burdzy-Frankel-Pauzner (2001), Bena¨ ım-Weibull (2003), Currarini-Jackson-Pin (2009), Hofbauer-Sandholm (2007).

◮ Social learning. B¨

• rgers (1997), Hopkins (1999), Duffie-Manso (2007), Duffie-Malamud-Manso (2009).

Duffie-Qiao-Sun Continuous-Time Random Matching 4

Parameters of the most basic model

◮ Type space S = {1, . . . , K}. ◮ Initial cross-sectional distribution p0 ∈ ∆(S) of agent types. ◮ For each pair (k, ℓ) of types:

• Mutation intensity ηkℓ.
• Matching intensity θkℓ : ∆(S) → R+, continuous, satisfying the balance identity

pkθkℓ(p) = pℓθℓk(p).

• Match-induced type probability distribution γkℓ ∈ ∆(S).

Duffie-Qiao-Sun Continuous-Time Random Matching 5

Mutation, matching, and match-induced type changes

1 2 3 4 5 1 2 3 4 5 t t + δ 1 2 3 4 5 1 2 3 4 5 t t + δ mutation match, type change ≃ δηkℓ ≃ δθ(pt)kj γkjℓ

Duffie-Qiao-Sun Continuous-Time Random Matching 6

Key solution processes

For a probability space (Ω, F, P), atomless agent space (I, I, λ), and measurability on I × Ω × R+ to be specified:

◮ Agent type α(i, ω, t), for α : I × Ω × R+ → S. ◮ Latest counterparty π(i, ω, t), for π : I × Ω × R+ → I. ◮ Cross-sectional type distribution p : Ω × R+ → ∆(S). That is,

p(ω, t)k = λ({i ∈ I : α(i, ω, t) = k}) is the fraction of agents of type k.

Duffie-Qiao-Sun Continuous-Time Random Matching 7

Evolution of the cross-sectional distribution p p pt of agent types

t buyers sellers inactive Existence of a model with independence conditions under which ˙ pt = ptR(pt) almost surely, where R(pt) is also the agent-level Markov-chain infinitesimal generator: R(pt)kℓ = ηkℓ + K

j=1 θkj(pt)γkjℓ

R(pt)kk = − K

ℓ=k Rkℓ(pt).

Duffie-Qiao-Sun Continuous-Time Random Matching 8

A Fubini extension

A probability space (I × Ω, W, Q) extending the usual product space (I × Ω, I ⊗ F, λ × P) is a Fubini extension if, for any real-valued integrable function f on I × Ω,

• I×Ω

fdQ =

• I
• Ω

f(i, ω) dP(ω)

• dλ(i) =
• Ω
• I

f(i, ω) dλ(i)

• dP(ω).

Such a Fubini extension is denoted (I × Ω, I ⊠ F, λ ⊠ P).

Duffie-Qiao-Sun Continuous-Time Random Matching 9

Sun’s exact law of large numbers

Suppose a measurable f : (I × Ω, I ⊠ F, λ ⊠ P) → R is essentially pairwise independent. Sun (2006) provides an existence result. That is, for almost every agent i, the agent-level random variables f(i) = f(i, · ) and f(j) are independent for almost every agent j. The cross-sectional distribution G of f at x ∈ R in state ω is G(x, ω) = λ({i : f(i, ω) ≤ x}). Proposition (Sun, 2006) For P-almost every ω, G(x, ω) =

• I

P(f(i) ≤ x) dλ(i). In particular, if the probability distribution F of f(i) does not depend on i, then the cross-sectional distribution G is equal to F almost surely.

Duffie-Qiao-Sun Continuous-Time Random Matching 10

Random matching

◮ A random matching π : I × Ω → I assigns a unique agent π(i) to agent i, with

π(π(i))) = i. If π(i) = i, agent i is not matched.

◮ Let g(i) = α(π(i)) be the type of the agent to whom i is matched. (If i is not matched,

let g(i) = J.) π(j) = i g(j) = blue g(i) = red π(i) = j

Duffie-Qiao-Sun Continuous-Time Random Matching 11

Independent random matching with given probabilities

◮ Given: A measurable α : I → S with distribution p ∈ ∆(S) and matching probabilities

(qkℓ) satisfying pkqkℓ = pℓqℓk.

◮ A random matching π is said to be independent with parameters (p, q) if the counterparty

type g is I ⊠ F-measurable and essentially pairwise independent with P(g(i) = ℓ) = qα(i),ℓ λ−a.e.

◮ In this case, the exact law of large numbers implies, for any k and ℓ, that

λ({i : α(i) = k, g(i) = ℓ}) = pkqkℓ a.s. Proposition (Duffie, Qiao, and Sun, 2015) For any given (p, q), there exists an independent matching π.

Duffie-Qiao-Sun Continuous-Time Random Matching 12

Loeb transfer of hyperfinite model

1 2 3 4 5 1 2 3 4 5 t t + δ 1 2 3 4 5 1 2 3 4 5 t t + δ mutation match, type change ≃ δηkℓ ≃ δθ(pt)kj γkjℓ

Duffie-Qiao-Sun Continuous-Time Random Matching 13

Continuous-time random matching

Theorem For any parameters (p0, η, θ, γ), there exists a continuous-time system (α, π) of agent type and last-counterparty processes such that

◮ The agent type process α and last-counterparty type process g = α ◦ π are measurable

with respect to (I ⊠ F) ⊗ B(R+) and essentially pairwise independent.

◮ The cross-sectional type distribution process {pt : t ≥ 0} satisfies ˙

pt = ptR(pt) almost surely.

◮ The agent-level type processes {α(i) : i ∈ I} are Markov chains with infinitesimal

generator {R(pt) : t ≥ 0}.

Duffie-Qiao-Sun Continuous-Time Random Matching 14

Stationary case

Proposition For any (η, θ, γ), there is an initial type distribution p0 such that the continuous-time system (α, π) associated with parameters (p0, η, θ, γ) has constant cross-sectional type distribution pt = p0. If the initial agent types {α0(i) : i ∈ I} are essentially pairwise independent with probability distribution p0, then the probability distribution of the agent type αt(i) is also constant and equal to p0, for λ-a.e. agent.

Duffie-Qiao-Sun Continuous-Time Random Matching 15

With enduring match probability ξ ξ ξ(pt)

1 2 3 4 5 2 4 2 4 2 4 t s T match breakup ξ(pt)bg breakup intensity β(ps)bg

Duffie-Qiao-Sun Continuous-Time Random Matching 16

Further generality

◮ When agents of types k and ℓ form an enduring match at time t, their new types are

drawn with a given joint probability distribution σ(pt)kℓ ∈ ∆(S × S).

◮ While enduringly matched, the mutation parameters of an agent may depend on both the

agent’s own type and the counterparty’s type.

◮ Time-dependent parameters (ηt, θt, γt, ξt, βt, σt), subject to continuity. ◮ The agent type space can be infinite, for example S = Z+ or S = [0, 1]m.

Duffie-Qiao-Sun Continuous-Time Random Matching 17