networks in economics and finance michele tumminello dseas – university of palermo 05/20/2014...
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Networks in Economics and Finance
Michele TumminelloDSEAS – University of Palermo
05/20/2014
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
International School on Network ScienceMay 19-23, 2014
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Outline
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
• Networks and allocation of resources: the concept of clearing prices.
• Networks to describe a simple market: the theory of competitive equilibrium
• Networks to describe preferential trading patterns at London Stock Exchange
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Allocation of resources: learning outcomes
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
• To understand the concept of perfect matching in bipartite networks
• To understand how the price of goods serves to decentralize the market
• To understand the concept of market-clearing prices
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Bipartite Network
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Type 1 Type 2
There are two types of nodesand
nodes of the same type are NOTconnected in a bipartite network
There are two types of nodesand
nodes of the same type are NOTconnected in a bipartite network
Easley and Kleinberg, Networks Crowds and Markets (2010)
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The problem of finding a perfect matching
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Type 1 Type 2
Problem: Assigning each student a room that he/she would be happy to accept
Allocation of resources
Perfect Matching: It is an assignment of nodes on the left to nodes on the right, insuch a way that
(i)each node is connected by an edge to the node it is assigned to,
(ii) no two nodes on the left are assigned to the same node on the right.
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The problem of finding a perfect matching
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Type 1 Type 2
Problem: Assigning each student a room that he/she would be happy to accept
Allocation of resources
Perfect Matching: It is an assignment of nodes on the left to nodes on the right, insuch a way that
(i)each node is connected by an edge to the node it is assigned to,
(ii) no two nodes on the left are assigned to the same node on the right.
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Do all bipartite networks have a perfect matching?
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
NO
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Constricted Set (of nodes)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
SN(S)
S is any set ofnodes of a given type(students in this case)
N(S) is the set of allthe neighbors of nodesbelonging to S(rooms in this case)
We say that S isa constricted setif S contains (strictly)more nodes (3 in thiscase) than N(S) does(2 in this case)
Easley and Kleinberg, NCM (2010)
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Constricted Set and Perfect Matching
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
SN(S)
If a bipartite networkhas a constricted setthen it does NOT admitperfect matching
Easley and Kleinberg, NCM (2010)
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Matching theorem (König 1931)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
If a bipartite network (with equal numbers of nodes on the left and right) has no perfect matching, then it must contain a constricted set.
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„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Is the search of perfect matching an optimization process?
What is the utility function?
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„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Suppose that agents not only give a preference, but also indicate a value for each preference, quantifying their degree of happiness. The result is a weighted bipartite network
Type 1 Type 2
Allocation of resources7
45
8
6
8
9
715
5
What is the “optimal” way toallocate resources in this case?
It is assumed that any missing link corresponds to a link with weight 0
Optimal Assignment: The matching that maximizes the sum of weights.
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Is a central authority really necessary?
NO
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
pa
pb
pc
Prices
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Payoff
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
If a buyer buys a house at price p and her evaluation of the house
was v then her payoff is
Payoff = v - pPayoff = v - p
What happens if p>v?
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
pa=2
pb=1
pc=0
Prices
Suppose that eachbuyer chooses the house(s) that maximizes her payoff
v-p
These prices don’t help to find perfect matching!These prices don’t help to find perfect matching! Is house “a” too cheap?
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
pa=5
pb=2
pc=0
Prices
Suppose that eachbuyer chooses the house(s) that maximizes her payoff
v-p
The prices {5,2,0} are Market-Clearing Prices
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Is the set of market-clearing prices unique?
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
NO
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
pa=3
pb=1
pc=0
Prices
Suppose that eachbuyer chooses the houses that maximize her payoff
v-p
The prices {3,1,0} are Market-Clearing Prices… Sellers can choose the buyer
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Prices and Market Clearing
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
aa
bb
cc
xx
yy
zz
Houses
Sellers Buyers
Valuations (v)
a=12, b=4, c=2
a=8, b=7, c=6
a=7, b=5, c=2
Easley and Kleinberg, NCM (2010)
pa=3
pb=1
pc=0
Prices
Suppose that eachbuyer chooses the houses that maximize her payoff
v-p
The prices {3,1,0} are Market-Clearing Prices… Sellers can choose the buyer
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Theorem: Existence of Market-Clearing Prices
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
For any set of buyer valuations, there exists a set of market-clearing prices.
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Constructing Market Clearing Prices
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
1. Set all the prices equal to 0
2. Construct the preferred-seller graph and check whether there is a perfect matching
3. If there is a perfect matching then we are done
4. If not, find a constricted set of buyers, S, and their neighbors N(S)
5. Each seller in N(S) synchronously raises her price by 1 unit. If necessary we reduce the prices of all the houses of the same amount so that the smallest price is 0
6. Go to step 2
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Concepts to take home
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
• Nodes of bipartite networks are of two types, and each edge connects a node of one type to a node of the other type
• Perfect matching in a bipartite network is used to allocate resources
• Prices can replace the “central authority” in the process of finding the optimal allocation of resources (market-clearing prices)
• Single-item auction can be seen as a particular case of matching market
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Is there something important that we are missing?
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
What is the “cost” of sellers?
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The Apple Market
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
“Experiments with Economics Principles”T. Bergstrom and J.H. Miller
McGraw-Hill Companies, Inc. (1996)
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The Apple Market
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Crowd
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The Apple Market
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Crowd of buyers and sellers
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Terminology
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Transaction: it is a deal between a buyer and a seller,consummated in the form of a filled-in sales contract which is delivered to the market manager.
Round: a round of trading begins when the market manager declares trades to be open and ends when transactions cease.
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Information Sheets for Participants
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Each participant receives an information sheet
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Example of information sheet (seller)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
7
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Example of information sheet (buyer)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
8
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Sales Contract
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Each transaction between a buyer and a seller must be recorded on a sales contract
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Trading Rules
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
•A participant cannot buy or sell more than 1 bushel of apples in a round.
•A participant does not have to make a trade. It is better to make no trade than to trade at a loss.
•Each pair of traders should turn in only one sales contract for their transaction.
•A participant should return to her seat after she has traded and turned in a sales contract.
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Suggestions for traders
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
•You don’t have to deal with the first person you encounter. Different people have different Buyer’s Value and Seller’s cost. So, be ready to shop around!
•You want “Buy low” and “Sell high”, in order to make greater profits.
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Class experiment: settings
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Session 1. 15 participants:
• SL = 4 low-cost
suppliers (SC=10$),
• SH = 2 high-cost
suppliers (SC=30$),
• BL = 6 low-value
demanders (BV=20$),
• BH = 3 high-value
demanders (BV=40$).
value=20
value=40
cost=10
cost=30
BL=6
BH=3
SL=4
SH=2
The weight of links, that is the number of trades between two groups of participants, depends on the outcome of the experiment.The weight of links, that is the number of trades between two groups of participants, depends on the outcome of the experiment.
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Round 1
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
value=20
value=40
cost=10
cost=30
BL=6
BH=3
SL=4
SH=2
1
2
2
Total number of trades = (sum of the weights of links) = 5Total number of trades = (sum of the weights of links) = 5
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Total payoff
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
The total payoff is the sum of the individual payoffs of buyers and sellers:
Total payoff = payoff buyer, i payoff seller, i trades i
Total payoff = value(buyeri) - price(i) price(i) - cost(selleri) trades i
Total payoff = value(buyeri) - cost(selleri) trades i
The total payoff is the sum of the value of all buyers minus the cost of all the sellers that have traded in the round.
The total payoff does NOT depend on the transaction pricesThe total payoff does NOT depend on the transaction prices
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Round 1
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
value=20
value=40
cost=10
cost=30
BL=6
BH=3
SL=4
SH=2
1
2
2
Total payoff= (20-10) + (20-10) + (40-10) + (40-10) + (40-30) = 90Total payoff= (20-10) + (20-10) + (40-10) + (40-10) + (40-30) = 90
Average price = (20 + 15 + 20 + 20 + 31)/5 = 21.2Average price = (20 + 15 + 20 + 20 + 31)/5 = 21.2
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Summary of results
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Round 1: Trades = 5Total payoff = 90 Average price = 21.2
Round 2: Trades = 5Total payoff = 90 Average price = 25.0
Round 3: Trades = 6Total payoff = 80 Average price = 23.5
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„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Competitive Equilibrium
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Supply and demand curves
Price smaller than 10 $Price smaller than 10 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
Price = 10 $Price = 10 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
10 $ < Price < 20 $10 $ < Price < 20 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
Price = 20 $Price = 20 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
20 $ < Price < 30 $20 $ < Price < 30 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
Price = 30 $Price = 30 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Supply and demand curves
30 $ < Price < 40 $30 $ < Price < 40 $
4 low-cost suppliers (SC=10$) 2 high-cost suppliers (SC=30$) 6 low-value demanders
(BV=20$) 3 high-value demanders
(BV=40$)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Competitive equilibrium
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Trades = 4Total payoff = 100 Price = 20
value=20
value=40
cost=10
cost=30
BL=6
BH=3
SL=4
SH=2
0
1
3
Competitive Equilibrium maximizes the total payoff and minimizes the number of tradesCompetitive Equilibrium maximizes the total payoff and minimizes the number of trades
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Competitive Equilibrium VS Experiments
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Trades = 4Total payoff = 100 Price = 20
CE:
Round 1:
Trades = 5Total payoff = 90 Average price = 21.2
Round 2:
Trades = 5Total payoff = 90 Average price = 25.0
Trades = 6Total payoff = 80 Average price = 23.5
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„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Deviations from competitive equilibrium are likely to occur in the experiments, in terms of both trading configurations and transaction prices
Bergstrom and Kwok, J. Econ. Educ. (2005);
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Synchronous Bazaar (SB)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Traders randomly meet (SYNCHRONOUSLY) and a transaction occurs when mutual gains are possible.
JH Miller and MT, submitted (2013)
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Synchronous Bazaar (SB)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
If BH BL SH SL then :
p(nLH ) =
SLnLH
SHBH nLH
SH SL BH
H(nLH ,BH ,SL ,SH SL )
nLL Min BL ,SL nLH ; nHH BH nLH ; nHL 0.
vL
vH
cL
cH
BL
BH
SL
SH
nLH
nHH
nLL
If BH BL SH SL then :
p(nLH ) =
BHnLH
BLSL nLH
BH BL SL
H(nLH ,SL ,BH ,BH BL )
nLL SL nLH ; nHH Min SH ,BH nLH ; nHL 0.
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Aynchronous Bazaar (AB)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Traders randomly meet (ASYNCHRONOUSLY) and a transaction may occur when mutual gains are possible.
JH Miller and MT, submitted (2013)
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Asynchronous Bazaar (AB)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
vL
vH
cL
cH
BL
BH
SL
SH
nLH
nHH
nLL
Ct1 nLH nHHnLL nHL
:configuration after t +1 trades
s 2; b 2; M s SL SH ; M b BL BH .
Normalization
Trading probability
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Trading probability
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
This model is obtained in the case of very limited patience. Then the trading probability is
This model is obtained in the case of infinite patience, that is
P1 1 1 P(ps pb ) 1 P(ps pb ) (v c)2
2(v cmin )(vmax c)s(v c)
AB1
AB
P limQ
1 1 P(ps pb ) Q s(v c)
where s(x) is the step function
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The effect of Asynchronicity
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Decreasing payoff/efficiency 40 low-cost suppliers (SC=10$) 20 high-cost suppliers (SC=30$) 40 low-value demanders
(BV=20$) 20 high-value demanders
(BV=40$)CE
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The effect of Patience
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Decreasing payoff/efficiency 40 low-cost suppliers (SC=10$) 20 high-cost suppliers (SC=30$) 40 low-value demanders
(BV=20$) 20 high-value demanders
(BV=40$)CE
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Concepts to take home
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
• Patience of agents (ABinfinity) tends to reduce the efficiency of the market, because it favors extramarginal traders.
• Impatience of agents (AB1) introduces a selection mechanism that favors trading between agents with higher difference between the buyer value and seller cost, and, therefore, increases the probability that CE is attained.
• Synchronous pairing (SB) disfavors extramarginal traders, and, therefore, increases the probability that CE is attained.
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A real system:
The network of market members at London Stock Exchange (LSE)
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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The network of market members at LSE
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
We have access to a version of the Rebuild Order Bookof the London Stock Exchange with a codified identityof Market Members for the years 2004-2006.
The data includes electronic transactions and off-book transactions (dealers’ market). Here we will focus on a highly liquid stock: BP.
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The role of Market Members at LSE
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Market members (MMs) are not single investors.
A MM may act on behalf of many different investors.
A MM may act as an intermediary and/or can do client trading and/or
proprietary trading.
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Market venues at LSE
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Market members (MMs) can execute their trades by submitting their orders in the electronic book in an anonymous form or can execute a transaction in the
dealers' off-book venue by directly interacting with other (known) MMs. Recorded transactions are anonymous in both cases.
Electronic book
Off-bookvenue
MM
MMMM
MM MMMM
MM
MM
MM
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Scientific question
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Is the number of transactions occurred between two market members (MMs) consistent with a null hypothesis of random matching of MMs?
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We consider the number of transactions between two market members in a given market venue, in a given time window
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Market member i acted as a buyer in Nib transactions
Market member j acted as a seller in Njs transactions
During the selected time period and in the considered venue the twomarket members did Nijbs transactionsbetween them and all market members didNt transactions
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A statistical validation of the number of transaction between two market members
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
What is the probability that the Nijbs buy-sell transactionsoccurred by chance?
Suppose there are NtNt transaction records in the investigated timewindow. Suppose we are interested to compare the occurrence of two given states B and S of two market members MMi and MMj. Market member MMi is buying Nib times whereas market member MMj is selling Njs times. Let us call Nijbs the buy-sell transactions between them.
NtNt
Total # of transactions
NjsNjsNibNib
NijbsNijbs
# of buying transactions of MMi
# of sellingtransactions
of MMj
# of B-S transactions
between the two MMs
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Hypergeometric distribution
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Hypergeometric distribution:
P(X |N,NA ,NB )
NAX
N NANB X
N
NB
X x P(x |N,NA ,NB ) NA NBN
Expected number of co-occurrences (transactions):
p-value associated with a detection of co-occurrences ≥ X:
p iX
Min(NA , NB )
NAi
N NANB i
N
NB
In the present application to MMs we set NA=NibNib, NB=NjsNjs, X=NijbsNijbs, and N=NtNt
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Corrections for multiple hypotheses testing and network construction
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
We set a directed link between two market members if the associated p-value is below a given statistical threshold.
Bonferroni threshold = B =0.01
T,
where T is the total number of tested hypotheses.
False Discovery Rate (FDR) threshold = F = k B,
where k is the largest integer such that there are k
p - values smaller than k B in the system
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BP (2005): weekly scale
Original Bonferroni
seller buyer
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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BP (2005): Number of links at weekly scale
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
A. Carollo, F. Lillo, R.N. Mantegna, MT, G. Vaglica, manuscript in preparation
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Persistence of links: Lagged Mutual Information between Bonferroni networks
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
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Concepts to take home
„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
• The off-book market at LSE shows a networked structure of preferential trading patterns.
• Such a structure is rather persistent.
• The electronic market does not show any structure of preferential trading patterns.
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„Infocommunication technologies and the society of future (FuturICT.hu)” TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Thank you for your attention!