Information Percolation in Segmented Markets Darrell Duffie Semyon - PowerPoint PPT Presentation

Information Percolation in Segmented Markets Darrell Duffie Semyon Malamud Gustavo Manso Stanford EPFL Lausanne MIT Texas Monetary Conference - December 2009 Duffie, Malamud, and Manso Information Percolation 1

Information Transmission in Markets Informational Role of Prices: Hayek (1945), Grossman (1976), Grossman and Stiglitz (1981). ◮ Centralized Exchanges: • Wilson (1977), Townsend (1978), Milgrom (1981), Vives (1993), Pesendorfer and Swinkels (1997), and Reny and Perry (2006). ◮ Over-the-Counter Markets: • Wolinsky (1990), Blouin and Serrano (2002), Golosov, Lorenzoni, and Tsyvinski (2009). • Duffie and Manso (2007), Duffie, Giroux, and Manso (2008), Duffie, Malamud, and Manso (2009). Duffie, Malamud, and Manso Information Percolation 2

Contributions of Today’s Paper 1. tractable model of information diffusion in over-the-counter markets with investor segmentation by preferences, initial information, and connectivity. 2. double auction with common values. 3. effects of information and connectivity on profits: • more informed/connected investors attain higher expected profits than less informed/connected investors if they can disguise trades. • more informed/connected investors may not attain higher expected profits than less informed/ connected investors if characteristics are commonly observed. Duffie, Malamud, and Manso Information Percolation 3

Outline of the Talk Information Percolation 1 Segmented Markets 2 Double Auction 3 Connectedness and Information 4 Duffie, Malamud, and Manso Information Percolation 4

Outline of the Talk Information Percolation 1 Segmented Markets 2 Double Auction 3 Connectedness and Information 4 Duffie, Malamud, and Manso Information Percolation 5

Model Primitives Duffie and Manso (2007) and Duffie, Giroux, and Manso (2010): ◮ Continuum of agents ◮ Two possible states of nature Y ∈ { 0 , 1 } . ◮ Each agent is initially endowed with signals S = { s 1 , . . . , s n } s.t. P ( s i = 1 | Y = 1 ) ≥ P ( s i = 1 | Y = 0 ) ◮ For every pair agents, their initial signals are Y -conditionally independent ◮ Random matching, intensity λ . Duffie, Malamud, and Manso Information Percolation 6

Initial Information Endowment After observing signals signals S = { s 1 , . . . , s n } , the logarithm of the likelihood ratio between states Y = 0 and Y = 1 is by Bayes’ rule: n log P ( Y = 0 | s 1 , . . . , s n ) P ( Y = 1 | s 1 , . . . , s n ) = log P ( Y = 0 ) log P ( s i | Y = 0 ) � P ( Y = 1 ) + P ( s i | Y = 1 ) . i = 1 We say that the “type” θ associated with this set of signals is n log P ( s i | Y = 0 ) � θ = P ( s i | Y = 1 ) . i = 1 Duffie, Malamud, and Manso Information Percolation 7

What Happens in a Meeting? ◮ Upon meeting, agents participate in a double auction. ◮ If bids are strictly increasing in the type associated with the signals agents have collected, then bids reveal type. Duffie, Malamud, and Manso Information Percolation 8

Information is Additive in Type Space Proposition: Let S = { s 1 , . . . , s n } and R = { r 1 , . . . , r m } be independent sets of signals, with associated types θ and φ . If two agents with types θ and φ reveal their their types to each other, then both agents achieve the posterior type θ + φ . This follows from Bayes’ rule, by which log P ( Y = 0 | S , R , θ + φ ) log P ( Y = 0 ) = P ( Y = 1 ) + θ + φ, P ( Y = 1 | S , R , θ + φ ) log P ( Y = 0 | θ + φ ) = P ( Y = 1 | θ + φ ) Duffie, Malamud, and Manso Information Percolation 9

Information is Additive in Type Space Proposition: Let S = { s 1 , . . . , s n } and R = { r 1 , . . . , r m } be independent sets of signals, with associated types θ and φ . If two agents with types θ and φ reveal their their types to each other, then both agents achieve the posterior type θ + φ . This follows from Bayes’ rule, by which log P ( Y = 0 | S , R , θ + φ ) log P ( Y = 0 ) = P ( Y = 1 ) + θ + φ, P ( Y = 1 | S , R , θ + φ ) log P ( Y = 0 | θ + φ ) = P ( Y = 1 | θ + φ ) By induction, this property holds for all subsequent meetings. Duffie, Malamud, and Manso Information Percolation 9

Solution for Cross-Sectional Distribution of Information The Boltzmann equation for the cross-sectional distribution µ t of types is d dt µ t = − λ µ t + λ µ t ∗ µ t . with a given initial distribution of types µ 0 . Duffie, Malamud, and Manso Information Percolation 10

Solution for Cross-Sectional Distribution of Information The Boltzmann equation for the cross-sectional distribution µ t of types is d dt µ t = − λ µ t + λ µ t ∗ µ t . with a given initial distribution of types µ 0 . Proposition: The unique solution of (10) is the Wild sum e − λ t ( 1 − e − λ t ) n − 1 µ ∗ n � µ t = 0 . n ≥ 1 Duffie, Malamud, and Manso Information Percolation 10

Proof of Wild Summation Taking the Fourier transform ϕ ( · , t ) of µ t of the Boltzmann equation d dt µ t = − λ µ t + λ µ t ∗ µ t . we obtain the following ODE d µ 2 dt ˆ µ t = − λ ˆ µ t + λ ˆ t . whose solution is ˆ µ 0 µ t = ˆ . e λ t ( 1 − ˆ µ 0 ) + ˆ µ 0 This solution can be expanded as e − λ t ( 1 − e − λ t ) n − 1 ˆ µ n � µ t = ˆ 0 , n ≥ 1 which is the Fourier transform of the Wild sum (10). Duffie, Malamud, and Manso Information Percolation 11

Multi-Agent Meetings The Boltzmann equation for the cross-sectional distribution µ t of types is d dt µ t = − λ µ t + λ µ ∗ m . t Taking the Fourier transform, we obtain the ODE, d µ m dt ˆ µ t = − λ ˆ µ t + λ ˆ t . whose solution satisfies µ m − 1 ˆ µ m − 1 0 ˆ = . (1) t µ m − 1 µ m − 1 e ( m − 1 ) λ t ( 1 − ˆ ) + ˆ 0 0 Duffie, Malamud, and Manso Information Percolation 12

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 13

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 14

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 15

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 16

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 17

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 18

Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 Population mass 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Current posterior Duffie, Malamud, and Manso Information Percolation 19

New Private Information Suppose that, independently across agents as above, each agent receives, at Poisson mean arrival rate ρ , a new private set of signals whose type outcome y is distributed according to a probability measure ν . Then the evolution equation is extended to d dt µ t = − ( λ + ρ ) µ t + λ µ t ∗ µ t + ρ µ t ∗ ν. Taking Fourier transforms, we obtain the following ODE d µ 2 dt ˆ µ t = − ( λ + ρ ) ˆ µ t + λ ˆ t + ρ ˆ µ t ˆ ν. whose solution satisfies µ 0 ˆ ˆ µ t = e ( λ + ρ ( 1 − ˆ ν )) t ( 1 − ˆ µ 0 ) + ˆ µ 0 Duffie, Malamud, and Manso Information Percolation 20

Other Extensions ◮ Public information releases • Duffie, Malamud, and Manso (2010). ◮ Endogenous search intensity • Duffie, Malamud, and Manso (2009). Duffie, Malamud, and Manso Information Percolation 21

Outline of the Talk Information Percolation 1 Segmented Markets 2 Double Auction 3 Connectedness and Information 4 Duffie, Malamud, and Manso Information Percolation 22

Model Primitives Same as the previous model except that: ◮ N classes of investors. ◮ Agent of class i has matching intensity λ i . ◮ Upon meeting, the probability that a class- j agent is selected as a counterparty is κ ij . Duffie, Malamud, and Manso Information Percolation 23

Evolution of Type Distribution The evolution equation is given by: N d i ∈ { 1 , . . . , N } , � dt ψ it = − λ i ψ it + λ i ψ it ∗ κ ij ψ jt , j = 1 Taking Fourier transforms we obtain: N d ψ it = − λ i ˆ ˆ ψ it + λ i ˆ � κ ij ˆ i ∈ { 1 , . . . , N } , ψ it ψ jt , dt j = 1 Duffie, Malamud, and Manso Information Percolation 24