Information Percolation in Segmented Markets Darrell Duffie, Gustavo Manso, Semyon Malamud Stanford University, U.C. Berkeley, EPFL Probability, Control, and Finance In Honor of Ioannis Karatzas Columbia University, June, 2012 Duffie-Manso-Malamud Information Percolation 1
Figure: An over-the-counter market. Duffie-Manso-Malamud Information Percolation 2
Cusip: 592646-AX-1 5% 4% Markup 3% 2% 1% 0% -1% 1 2 3 4 5 6 7 8 9 10 Day Figure: Transaction price dispersion in muni market. Source: Green, Hollifield, and Sch¨ urhoff (2007). See, also, Goldstein and Hotchkiss (2007). Duffie-Manso-Malamud Information Percolation 3
Figure: Daily trade in the federal funds Market. Source: Bech and Atalay (2012). Duffie-Manso-Malamud Information Percolation 4
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 (2010). Duffie-Manso-Malamud Information Percolation 5
Figure: Many OTC markets are dealer-intermediated. Duffie-Manso-Malamud Information Percolation 6
Model Primitives ◮ Agents: a non-atomic measure space ( G, G , γ ) . ◮ Uncertainty: a probability space (Ω , F , P ) . ◮ An asset has a random payoff X with outcomes H and L . ◮ Agent i is initially endowed with a finite set S i = { s 1 , . . . , s n } of { 0 , 1 } -signals. ◮ Agents have disjoint sets of signals. ◮ The measurable subsets of Ω × G are enriched from the product σ -algebra enough to allow signals to be essentially pairwise X -conditionally independent, and to allow Fubini, and thus the exact law of large numbers (ELLN). (Sun, JET, 2006). Duffie-Manso-Malamud Information Percolation 7
Information Types After observing signals S = { s 1 , . . . , s n } , the logarithm of the likelihood ratio between states X = H and X = L is by Bayes’ rule: n log P ( X = H | s 1 , . . . , s n ) P ( X = L | s 1 , . . . , s n ) = log P ( X = H ) log p i ( s i | H ) � P ( X = L ) + p i ( s i | L ) , i =1 where p i ( s | k ) = P ( s i = s | X = k ) . We say that the “type” θ associated with this set of signals is n log p i ( s i | H ) � θ = p i ( s i | L ) . i =1 Duffie-Manso-Malamud Information Percolation 8
ELLN for Cross-Sectional Type Density ◮ The ELLN implies that, on the event { X = H } , the fraction of agents whose initial type is no larger than some given number y is almost surely � � F H ( y ) = 1 { θ α ≤ y } dγ ( α ) = P ( θ α ≤ y | X = H ) dγ ( α ) , G G where θ α is the initial type of agent α . ◮ On the event { X = L } , the cross-sectional distribution function F L of types is likewise defined and characterized. ◮ We suppose that F H and F L have densities, denoted g H ( · , 0) and g L ( · , 0) respectively. ◮ We write g ( x, 0) for the random variable whose outcome is g H ( x, 0) on the event { X = H } and g L ( x, 0) on the event { X = L } . Duffie-Manso-Malamud Information Percolation 9
Information is Additive in Type Proposition Let S = { s 1 , . . . , s n } and R = { r 1 , . . . , r m } be disjoint sets of signals, with associated types θ and φ . If two agents with types θ and φ reveal their types to each other, then both agents achieve the posterior type θ + φ . This follows from Bayes’ rule, by which log P ( X = H | S, R, θ + φ ) log P ( H = H ) = P ( X = L ) + θ + φ, P ( X = L | S, R, θ + φ ) log P ( X = H | θ + φ ) = P ( X = L | θ + φ ) Duffie-Manso-Malamud Information Percolation 10
Dynamics of Cross-Sectional Density of Types Each period, each agent is matched, with probability λ , to a randomly chosen agent (uniformly distributed). They share their posteriors on X . Duffie and Sun (AAP 2007, JET 2012): With essential-pairwise-independent random matching of agents, � + ∞ g ( x, t + 1) = (1 − λ ) g ( x, t ) + λg ( y, t ) g ( x − y, t ) dy, x ∈ R , a . s . −∞ which can be written more compactly as g ( t + 1) = (1 − λ ) g ( t ) + λg ( t ) ∗ g ( t ) , where ∗ denotes convolution. Duffie-Manso-Malamud Information Percolation 11
Solution of Cross-Sectional Distribution Types ◮ The Fourier transform of g ( · , t ) is � + ∞ 1 e − izx g ( x, t ) dx. ˆ g ( z, t ) = √ 2 π −∞ ◮ From (11), for each z in R , d g 2 ( z, t ) , dt ˆ g ( z, t ) = − λ ˆ g ( z, t ) + λ ˆ (1) ◮ Thus, the differential equation for the transform is solved by ˆ g ( z, 0) ˆ g ( z, t ) = g ( z, 0) . (2) e λt (1 − ˆ g ( z, 0)) + ˆ Duffie-Manso-Malamud Information Percolation 12
Solution of Cross-Sectional Distribution Types Proposition The unique solution of the dynamic equation (11) for the cross-sectional type density is the Wild sum � e − λt (1 − e − λt ) n − 1 g ∗ n ( θ, 0) , g ( θ, t ) = (3) n ≥ 1 where g ∗ n ( · , 0) is the n -fold convolution of g ( · , 0) with itself. The solution (3) is justified by noting that the Fourier transform ˆ g ( z, t ) can be expanded from (2) as � e − λt (1 − e − λt ) n − 1 ˆ g ( z, 0) n , g ( z, t ) = ˆ n ≥ 1 which is the transform of the proposed solution for g ( · , t ) . Duffie-Manso-Malamud Information Percolation 13
Numerical Example ◮ Let λ = 1 and P ( X = H ) = 1 / 2 . ◮ Agent α initially observes s α , with P ( s α = 1 | X = H ) + P ( s α = 1 | X = L ) = 1 . ◮ P ( s α = 1 | X = H ) has a cross-sectional distribution over investors that is uniform over the interval [1 / 2 , 1] . ◮ On the event { X = H } of a high outcome, this initial allocation of signals induces an initial cross-sectional density of f ( p ) = 2 p for the likelihood P ( X = H | s α ) of a high state. Duffie-Manso-Malamud Information Percolation 14
On the event { X = H } , the evolution of the cross-sectional population density of posterior probabilities of the event { X = H } . 4 t = 4 . 0 t = 3 . 0 t = 2 . 0 t = 1 . 0 3.5 t = 0 . 0 3 Population density 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Conditional probability of high outcome. Duffie, Stanford Information Percolation in Large Markets 9
Multi-Agent Meetings The Boltzmann equation for the cross-sectional distribution µ t of types is d dtµ t = − λ µ t + λ µ ∗ m . t 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 = . (4) t µ m − 1 µ m − 1 e ( m − 1) λt (1 − ˆ ) + ˆ 0 0 Duffie-Manso-Malamud Information Percolation 16
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 17
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 18
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 19
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 20
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 21
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 22
Groups of 2 (blue) versus Groups of 3 (red) 1 0.9 0.8 0.7 0.6 y 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 x Duffie-Manso-Malamud Information Percolation 23
Other Extensions ◮ Privately gathered information. ◮ Public information releases (such as tweets or transaction announcements). • Duffie, Malamud, and Manso (2010). ◮ Endogenous search intensity • Duffie, Malamud, and Manso (2009). Duffie-Manso-Malamud Information Percolation 24
A Segmented OTC Market ◮ Agents of class i ∈ { 1 , . . . , M } have matching probability λ i . ◮ Upon meeting, the probability that a class- j agent is selected as a counterparty is κ ij . ◮ At some time T , the economy ends, X is revealed, and the utility realized by an agent of class i for each additional unit of the asset is U i = v i 1 { X = L } + v H 1 { X = H } , for strictly positive v H and v i < v H . Duffie-Manso-Malamud Information Percolation 25
Trade by Seller’s Price Double Auction ◮ If v i = v j , there is no trade (Milgrom and Stokey, 1982; Serrano-Padial, 2008). ◮ Upon a meeting with gains from trade, say v i < v j , the counterparties participate in a seller’s price double auction. ◮ That is, if the buyer’s bid β exceeds the seller’s ask σ , trade occurs at the price σ . ◮ The class of one’s counterparty is common knowledge. Duffie-Manso-Malamud Information Percolation 26
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