Networks in Economics and Finance Michele Tumminello DSEAS – University of Palermo 05/20/2014 „Infocommunication technologies and the society of future

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

Outline


• 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

Allocation of resources: learning outcomes


• 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

Bipartite Network


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)

The problem of finding a perfect matching


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.

The problem of finding a perfect matching


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.

Do all bipartite networks have a perfect matching?


NO

Constricted Set (of nodes)


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)

Constricted Set and Perfect Matching


SN(S)

If a bipartite networkhas a constricted setthen it does NOT admitperfect matching


Matching theorem (König 1931)


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.


Is the search of perfect matching an optimization process?

What is the utility function?


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.

Prices and Market Clearing


Is a central authority really necessary?

NO



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




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


pa

pb

pc

Prices

Payoff


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?



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


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?



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


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

Is the set of market-clearing prices unique?


NO



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


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



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


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

Theorem: Existence of Market-Clearing Prices


For any set of buyer valuations, there exists a set of market-clearing prices.

Constructing Market Clearing Prices


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

Concepts to take home


• 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

Is there something important that we are missing?


What is the “cost” of sellers?

The Apple Market


“Experiments with Economics Principles”T. Bergstrom and J.H. Miller

McGraw-Hill Companies, Inc. (1996)

The Apple Market


Crowd

The Apple Market


Crowd of buyers and sellers

Terminology


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.

Information Sheets for Participants


Each participant receives an information sheet

Example of information sheet (seller)


7

Example of information sheet (buyer)


8

Sales Contract


Each transaction between a buyer and a seller must be recorded on a sales contract

Trading Rules


•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.

Suggestions for traders


•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.

Class experiment: settings


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.

Round 1


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

Total payoff


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

Round 1


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

Summary of results


Round 1: Trades = 5Total payoff = 90 Average price = 21.2




Competitive Equilibrium

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$)



Price = 10 $Price = 10 $



(BV=40$)



10 $ < Price < 20 $10 $ < Price < 20 $



(BV=40$)






(BV=40$)



20 $ < Price < 30 $20 $ < Price < 30 $



(BV=40$)






(BV=40$)



30 $ < Price < 40 $30 $ < Price < 40 $



(BV=40$)


Competitive equilibrium


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

Competitive Equilibrium VS Experiments


Trades = 4Total payoff = 100 Price = 20

CE:

Round 1:

Trades = 5Total payoff = 90 Average price = 21.2

Round 2:




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);

Synchronous Bazaar (SB)


Traders randomly meet (SYNCHRONOUSLY) and a transaction occurs when mutual gains are possible.

JH Miller and MT, submitted (2013)

Synchronous Bazaar (SB)


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.

Aynchronous Bazaar (AB)


Traders randomly meet (ASYNCHRONOUSLY) and a transaction may occur when mutual gains are possible.

JH Miller and MT, submitted (2013)

Asynchronous Bazaar (AB)


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

Trading probability


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

The effect of Asynchronicity


Decreasing payoff/efficiency 40 low-cost suppliers (SC=10$) 20 high-cost suppliers (SC=30$) 40 low-value demanders


(BV=40$)CE

The effect of Patience


Decreasing payoff/efficiency 40 low-cost suppliers (SC=10$) 20 high-cost suppliers (SC=30$) 40 low-value demanders


(BV=40$)CE



• 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.

A real system:

The network of market members at London Stock Exchange (LSE)


The network of market members at LSE


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.

The role of Market Members at LSE


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.

Market venues at LSE


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

Scientific question


Is the number of transactions occurred between two market members (MMs) consistent with a null hypothesis of random matching of MMs?

We consider the number of transactions between two market members in a given market venue, in a given time window


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

A statistical validation of the number of transaction between two market members


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

Hypergeometric distribution


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

Corrections for multiple hypotheses testing and network construction


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

BP (2005): weekly scale

Original Bonferroni

seller buyer


BP (2005): Number of links at weekly scale


A. Carollo, F. Lillo, R.N. Mantegna, MT, G. Vaglica, manuscript in preparation

Persistence of links: Lagged Mutual Information between Bonferroni networks




• 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.


Thank you for your attention!

Documents

Networks in Economics and Finance Michele Tumminello DSEAS – University of Palermo 05/20/2014 „Infocommunication technologies and the society of future