Diffusion of information and phase transition in the Fisher market

Diffusion of information and phase transition in theFisher market

Nikos Salamanos* and Michalis VazirgiannisAthens University of Economics and BusinessDepartment of Informatics76, Patission Str./ GR10434 Athens, GreeceE-mail: [email protected]: [email protected]*Corresponding author

Abstract: In this paper, we study the effect of diffusion on the evolution ofa market consisting of two infinitely divisible goods and buyers with constantelasticity of substitution utility functions. In consecutive time periods, thebuyers’ preferences depend on the actions taken by their neighboursin thenetwork. We investigate the properties of the long time states, where amarket state is defined by the market equilibrium prices and goods allocation.The experimental results demonstrate that the long time states are sensitiveto initial conditions and exhibit the following patterns. Homogeneous: themarket prices of the two goods are equal and the buyers split their budgetequally, amongst the goods. Heterogeneous: the buyers’ bids on the twogoods differ. Periodic: the buyers’ bids oscillate with stable oscillation width.Moreover, we present the critical values where a phase transition occursbetween homogeneous, heterogeneous and periodic states.

Keywords: diffusion; cascading behavior; Fisher market; social networks;phase transition; dynamic systems.

Reference to this paper should be made as follows: Salamanos, N. andVazirgiannis, M. (2012) ‘Diffusion of information and phase transitionin the Fisher market’, Int. J. Knowledge and Learning, Vol. 8, Nos.3/4, pp.259–281. The original publication is available at Interscience viadoi:10.1504/IJKL.2012.051678

Biographical notes: Nikos Salamanos is a Ph.D. student at the AthensUniversity of Economics and Business (AUEB), Department of Informatics(Greece). He received his B.Sc. in Physics from the National Universityof Athens, Department of Physics and a M.Sc. in ‘Logic and Theoryof Algorithms and Computation’ from the National University of Athens,Department of Mathematics. His research is focused on diffusion andimitation models in social networks and agent-based computational economics(ACE).

Michalis Vazirgiannis has conducted research in Frauhofer (Germany),INRIA/Paris, IBM and Max Planck Institut für Informatik (Germany). He isrecipient of the Marie Curie Intra-European and the ERCIM fellowships andof the French DIGITEO Chair grant His research interests include web graphanalysis & evolution monitoring, distributed machine learning algorithms,dimensionality reduction, and aspects of text retrieval and mining. Currently,he holds a Professor position at AUEB (Greece). He has participated in

https://doi.org/10.1504/IJKL.2012.051678

2 N. Salamanos and M. Vazirgiannis

international research projects in the area of web mining. He has chairedandparticipated in many programme committees of international conferencesinthe areas of data bases, data mining/machine learning and the Web.

1 Introduction

The role of social influence in the formation of beliefs, customs and trends has beenstudied extensively the last decades. The flow of information plays an integral rolein numerous real networks and markets. The spread of rumoursand the customers’feedback and reviews affect in various ways the success of a new product or technology.The individuals’ preferences are formed not only by their personal needs but also bywhat society considers as valuable or important. Authorityentities as mass media, blogs,and fashion magazines, inform the individuals about the market trends, the quality ofthe products, the reselling prices and affect the market share of the products. Theyare several real world markets where by default the market share of the products iscorrelated with the social beliefs on them. For instance, inmovie industry the successof a new film depends on the feedback that the media provide to community. Moreover,when an individual has to decide which foreign language to learn, the crucial factor forhis decision is the spread of language in the population.

The focus of this paper is concentrated on the effect of social influence on thelong time behaviour of a market when the market prices and theallocation of goods tobuyers are equilibrated by the law of supply and demand. We proposed a diffusion ofinformation model where the main assumption is that the buyers do not have inherentpreferences for the goods in the market. The agents have limited memory of their pastactions and their goods’ valuations are affected by the social environment.

We investigate the market evolution where by the term evolution we refer to the pathtrajectory of market state. A market state is defined by the market prices and the marketshare of the goods. We assume that the buyers are organised bya social network whichis considered as self-reference system with regard to the market trends that are formedamongst the individuals.

The proposed model is an extension of the diffusion model presented inSalamanos & Konstantopoulos(2011), where the valuation of an agent for a givengood was a linear function of the density (in terms of quantity) of the goodin the agent’s neighbourhood and the density of the good in the population. InSalamanos & Konstantopoulos(2011), the agents did not have memory of their pastactions and their goods valuations depend on the current state of the market. In thispaper, we extend the diffusion model by applying a control mechanism which canaccelerate or slow down the spread of influence amongst the agents. In this case theagents actually have one period memory.

In the scenario considered, there is aFisher marketconsisting of buyers and twoinfinitely divisible goods. The Fisher market was first defined by Irving Fisher in 1891.It is a special case of the exchange market and the agents havea distinct role, theyare either buyers or sellers. The buyers’ utilities areconstant elasticity of substitution(CES) utility functions. In consecutive time periods, the consumption plan of a buyer isaffected by the following:

Diffusion of Information and Phase Transition in the FisherMarket 3

a the quantity of the goods that the buyer consumed in the previous period

b the rate of diffusion, an idiosyncratic term of buyer

c the density of the goods at his neighbourhood.

The diffusion model is based on the discrete time version of the classical model ofdiffusion of substances across a membrane between adjacentcells. We study the modelexperimentally and we analyse the distribution of the market prices and the market shareof the goods, when the configuration of the network is a complete graph, Erdös-Rényi(ER) and scale-free (SF). The long time states exhibit threemain patterns:

a homogeneous, a stable state where the market prices of two goods are equalandthe buyers split their money equally amongst the goods

b heterogeneous, a stable state where the buyers’ bids on the two goods differ

c periodic, a dynamic state where the market prices and the buyers’ bidsoscillatewith stable oscillation width.

The proposed diffusion model despite its simplicity exhibits rich behaviour. Thecontrol parameters of the system are the rate of substitution of buyers’ utility functions,the rate of diffusion and the degree distribution of social network. For a specificrange of the control parameters there is a phase transition between homogeneous,heterogeneous and periodic states. According to our knowledge, this is the first studywhich demonstrates phase transition phenomena in the context of diffusion in markets.

The model is sensitive to initial conditions. We analyse an almost homogeneouspopulation where the agents have almost equal initial valuations. We observe that evena very small difference in the initial values (less than10−6) can lead the market toheterogeneous and periodic states.

The paper is organised as follows. Section2 covers the related work. In Section3,we present the notion of the market equilibrium and in Section 4 we present our model.Sections5 and6 cover the experimental analysis and Section7 concludes the paper.

2 Related Work

The role of social influence in the formation of beliefs, customs and trends has beenstudied extensively in the last decades. The literature in the area of social networks isvast and covers different disciplines as Sociology, Economics, Computer Science andPhysics (Jackson 2008, Nisan et al. 2007, Easley & Kleinberg 2010).

Our work is inspired by the pioneer model ofFollmer (1974) where he studiedthe Random Economiesand Ising Economieswith endogenous preferences and byHorst & Scheinkman(2006) which examined random economies with infinite numberof agents and endogenous preferences in the presence of local and global interactions.

Kakade et al.(2004) proposed a model of aFisher marketwith linear utilities wherethe buyers and sellers form a social graph. They defined thegraphical Fisher modelwhere each buyer can have a transaction only with the sellersthat he is connectedwith, in the network. The social graph is generated by a random process, actually avariation of thepreferential attachment(Barabasi & Albert 1999, Albert & Barabasi


2002). In this model the prices of the goods are seller specific andthey are specified bya market equilibrium condition where supply equals demand.They analyzed their modeltheoretically and experimentally and they studied the distribution of the market prices.

Ceyhan et al.(2011) studied the effect ofherdingon the market share of the goodsof a market withm goods. In consecutive time periods a new buyer enters the marketand he chooses the good that maximizes his utility. The buyer’s utility is a function ofthe market share of the goods, the inherent quality of the goods and a noise term.

There are several models of diffusion in social networks from Marketing toComputer Science and Physics. One of the first diffusion models is the Bass Modelfrom Frank Bass.Bass(1969) proposed a differential equation to describe the diffusionof a new durable good and it is actually a macroscopic description of the diffusionprocess. Durable are the goods that cannot be completely consumed in one use but yieldutility over the time. Examples of durable products includecars, furniture, jewelery andsport equipment. The Bass Model is one of the most widely applied and cited modelsin Marketing and Management science and has been voted as oneof the Top 10 mostinfluential papers published in the 50-year history of Management Science (Bass 2004).

Oh et al.(2008) present a thoroughly empirical study of the You Tube socialgraphanalyzing a data set consisting of users and video information which have been collectedfrom You Tube. The authors studied the social structure of You Tube along with severalstatistical measures. Moreover, they proposed a diffusionmodel for the user generatedcontent in You Tube using the Bass model as a baseline.

Backstrom et al.(2006) studied the growth and evolution of explicit groups inLiveJournal and DBLP, andAcemoglu et al.(2011) investigated thelinear thresholdmodelon deterministic social topologies and heterogeneous population in terms of theagents’ threshold values. In the threshold model we assume that the agents’ socialaffiliations form a social network and each agenti is characterized by a threshold valueϕi. In an iterative process an agent adopts a new behaviorA if the fraction of hisneighbors who have already adopted the behaviorA is larger thanϕi. For a detailedanalysis of thethreshold modelsand thecascading behaviorseeEasley & Kleinberg(2010), Chap. 19 andNisan et al.(2007), Chap. 24.

Korosoglou et al.(2010) proposed a model of reaction diffusion and the spread ofinfection in a lattice, Erdös-Rényi and Scale-free graphs.The reaction diffusion modelis actually a spreading infection model where an infected agent could infect all hisneighbors in the network. The basic model is calledSIRsusceptible–infected–removed.In SIR model, when an agent becomes infected, remains infected and cannot recover.

Finally, we have to mention the major progress which has beenconducted in the areaof algorithmic market equilibria. Several algorithms havebeen proposed that computethe market equilibria in polynomial time (seeNisan et al. 2007, Chap. 5).Devanur et al.(2008) presented the DPSV algorithm, the first polynomial algorithm to compute themarket equilibria for the linear case of Fisher model.Fleischer et al.(2008) proposed anew formulation of the market equilibrium problem as a convex optimization problemusing indirect utility functions. The algorithm inFleischer et al.(2008) approximates themarket equilibrium for the general case of exchange markets. It computes an(1 + ǫ)-approximate market equilibrium, where the demand is at most(1 + ǫ) times the supply.In 2009, Zhang proposed theproportional response dynamics, a utility based distributeddynamics, and he proved that for the case of the Fisher Marketwith constant elasticityof substitution (CES) utility functions, the algorithm converges to the market equilibria(Wu & Zhang 2007, Zhang 2009).


3 Preliminaries

In this paper we analyze the evolution of a Fisher market and the effect of socialinfluence on the market prices and the market share of the goods.

A Fisher market, from Irving Fisher, is a special case of the exchange market. Anexchange market is an economy without production and consists of n agents andNperfectly divisible goods. Each agent has a goods endowmentand acts as a trader, heis selling his endowment and consumes other goods. The overall quantity of a good inthe market is the sum of agents’ endowments. Moreover, each agent has a vector withhis individual preferences for the goods in the market. A Fisher market is a special caseof the exchange market and the agents have a distinct role - they are either buyers orsellers.

In an exchange or Fisher market, a vector of prices constitutes amarket equilibrium(or market clearing prices) when the following two conditions hold:

a the marketclears, i.e. all the goods have been sold and all the money has beenspent

b each buyer’s basket of goods is optimal, i.e. it maximizes the buyer’s utility.

The existence of the market equilibrium in the exchange market has been proved byArrow & Debreu (1954). Moreover, if (x, p) is a market equilibrium thenx is Paretoefficient, wherex is the goods allocation andp is the vector of prices [first theoremof welfare economics, seeVarian (1992), p.326]. We recall fromVarian (1992) that anallocationx is weakly Pareto efficientallocation, if there is no feasible allocationx′

such that all agents strictly preferx′ to x. An allocationx is strongly Pareto efficientallocation, if there is no feasible allocationx′ such that all agents weakly preferx′ tox. It can be proved that if the agents’ preferences are continuous and monotonic thenan allocation is strongly Pareto efficient, if and only if it is weakly Pareto efficient.

A price discovery mechanism which leads to a market equilibrium is the Walrasiantâtonnement(Walras 1874). The tâtonnement is an agent-based iterative process wherethe sellers announce the prices of their goods and the buyersrespond to those pricesdefining the demand on the goods. The buyers are so calledprice takers. In each stepthe sellers have to adjust their prices according to the excess demand on their goodsand if the demand exceeds their supply, they increase the prices, or else they reduce theprices. We must note that the final transactions take place only when the process hasconverged to equilibrium. This dynamic process converges to equilibrium (the demandis equal to supply) when the market satisfiesgross substitutability, i.e. increasing theprice of some goods while keeping the others fixed will cause an increase in demandfor the goods that we keep the price fixed.

3.0.1 Proportional response dynamics

In the recent years, the problem of computing the market equilibrium has attractedconsiderable attention. Several algorithms have been proposed for several variations ofthe general problem.

In 2009, Zhang proposed theproportional response dynamics(Wu & Zhang 2007,Zhang 2009), an utility based distributed dynamics, and he proved thatit convergesto a market equilibrium for the case of the Fisher Market withConstant Elasticity ofSubstitution (CES) utility functions.


Consider a market withn buyers andN perfectly divisible goods. Each buyeri hasa budgetbi and each goodJ has an endowment equal to a unit. For the rest of the paperwe use lower case letters when we refer to the buyers and capital ones for the goods.

A utility function is homogeneousof degreek if u(αx) = αku(x) for all α > 0. Aconstant elasticity of substitution (CES) utility function is a homogeneous function ofdegree one and it is of the form

ui(xi1, ..., xiN ) =

(

∑

J

(wiJxiJ )ρ

)1

ρ

where−∞ < ρ ≤ 1 andρ 6= 0.

The parameterσ =1

1− ρis called the elasticity of substitutionof the utility

function. The CES functions contain several well known utility functions depending onthe value of parameterρ [seeArrow et al. (1961), Varian (1992, 2006)]. For instance,when ρ tends to 1 then the CES function tends to linear and the goods are perfectsubstitutes; whenρ tends to0, then the goods are complementary.

Algorithm 1 Proportional response dynamics

1: Initialization:

2: Each buyer splits his budget evenly among all the goods for which wiJ > 0

3: for t = 1, 2, . . . , t do

4: Buyer i places the bidbiJ(t) =uiJ(t− 1)

ui(t− 1)bi on goodJ

5: The price of goodJ is pJ(t) =∑

i

biJ(t)

6: Buyer i consumesxiJ(t) =biJ(t)

pJ(t)fraction of goodJ

7: Next iteration8: Repeat steps 4-6

9: end for

The proportional response dynamics (Algorithm1) is an iterative process, where ineach iteration, the buyers place a bid on the goods proportionally to the utility theyreceived in the previous iteration.Zhang (2009) proposed the following CES utilityfunction:

ui(xi1, ..., xiN ) =N∑

J=1

uiJ =N∑

J=1

(wiJxiJ)ρi (1)

where:0 < ρi ≤ 1, wiJ ≥ 0 andxiJ is the amount of goodJ allocated to useri.Zhang (2009) proved that the proportional response dynamics convergesto an ǫ -

approximate market equilibrium for the Fisher market with CES utilities. We recall fromZhang(2009) that the bid vector{biJ}i along with the price vectorp = {pJ =

∑

i biJ}Jis an ǫ-approximate equilibrium if for each good(1− ǫ)p∗J ≤ pJ ≤ (1 + ǫ)p∗J and foreach buyer,ui ≥ (1− ǫ)umax

i , where{p∗J} is the equilibrium prices andumaxi is the

maximum utility of i given the price vectorp.The computational steps of proportional response dynamicsdepend on the value of

the parameterρ. Specifically:


a whenρM = 1, thenO(W 3/ǫ2)

b whenρM < 1, thenO

(

log(Wǫ)

(1− ρM )2

)

where,ρM = maxiρi andW = N1

(minwiJ>0wiJ )(minibi).

We must note two important observations with respect to the final allocation of thegoods to buyers. First, when the utilities are linear (ρ = 1), all goods in the buyer’sbasket are equivalent in terms of thebang per buck(uiJ/bij), i.e. utility per unit moneyspent. In other words, even if the quantities of the goods that the algorithm assigns toa given buyer are not equal, the bang per buck values are. Second, there is not a lowerbound in terms of how small the quantities that each buyer consumes can be. In otherwords, the algorithm could assign to a buyer a very small fraction of the goods.

4 The diffusion Model

In the area of Microeconomics the famous work ofArrow & Debreu (1954), ensuresthat under specific assumptions for the buyers’ utility function, any exchange markethas an equilibrium state, where the market clears and the allocation of the goods tobuyers is Pareto efficient. The main assumption is that the buyers have inherent andconstant valuations for the goods and there is no information flow or social influenceamong them. Moreover, in the recent years, several algorithms have been proposedwhich compute the market equilibria of an exchange or Fishermarket in polynomialtime. Thus, the notion of stability is applied to the market equilibrium problem, butthose algorithms lack a clear economic interpretation.

In many real world cases it might be difficult for the buyers topredefine a valuationfor the goods without any external information. For instance, consider the case wherea company introduces a new product which is based on a revolutionary technology. Itis natural to assume that there will be a delay in the customers’ response to the newtechnology that will depend on the information that the buyers will receive from thecommunication media. Moreover, the assumption of inherentand constant preferencesimplies that the buyers have infinite memory of their past actions.

Motivated by the previous analysis, our approach for an agent-based market modelcould be summarized in the following general rules:

1 There is a market withn buyers andN goods. The market is equilibrated by aprice discovery mechanism.

2 The buyers of the market are also the agents of a social network. They have socialaffiliations which form a social graph.

• The agents are considered naive, with limited memory. Theyhave limitedcomputational capabilities and they are not able to solve a complexoptimization problem.

3 The social network is a communication medium.

4 The communication medium is the factor for information flowamong the agents,which is determined by the agents’behavioral model.


5 The behavioral model consists of acommunication protocol, which an agentchooses to communicate with, as well as anadaptive rule, of how an agent updateshis strategy.

• The agent’s adaptive rule is deterministic. Any non determinism isencompassed in the communication model or is expressed by a noise term inthe agent’s strategy.

6 By the agents’ interaction, a market evolution process takes place. It is a repeatedtime process and is actually a non linear recursive function.

4.1 The model

In this paper we consider the following scenario. We assume that a setA, |A| = n,of agents are connected by a social network. The social network is represented by anundirected graphG(A,E) with n nodes andE = {(i, j)|i 6= j ∈ A} edges. Each nodeof the graph corresponds to an agent, each edge(i, j) implies thati andj are neighborsin the network.

In consecutive time periodst = 1, 2, . . . , T , the agents participate in a market. Eachagent of the network is considered as a potential buyer with cash endowmentbi, whichremains constant during the periods. There is a setS, |S| = N , of infinitely divisiblegoods available in the market. Each goodJ is associated with one seller and each sellerhas only one good, i.e. seller and good coincide. Each sellerJ has an endowmentgJ ofthe goodJ . The set of the goods and the quantity of the goods are considered constantduring the periods. In every period, a Fisher market is formed by the agents’ preferences.At the end of each period, the agents are able to partially observe the state of the marketwhich is defined by the market prices and the goods allocation. Moreover, the agentsinteract with the society locally. There is information flowbetween the agents, which isrelated to the current state of the market and the social dynamic affects the valuation ofthe agents for the goods in the market.

In a given period, the agents that participate in the market are called active butotherwiseinactive. Inactive are the agents that, in a given period, do not participate inthe market i.e. they assign zero valuation in every product.In this paper we considerthe case where the inactive agents are able to recover in the next period. Thus, in everyperiod, the active and the inactive agents are engaged in thediffusion phase. Similarly,we define the active and inactive goods of the market. In a given period, inactive goodsare those for which the demand is zero. Unlike inactive buyers, inactive goods do notrecover, but on the contrary, they are excluded from the market.

The utility function of a agent, with respect to the amount ofgoods that heconsumes, is defined as:

uti(x

ti1, ..., x

tiN ′) =

N ′

∑

J=1

utiJ =

N ′

∑

J=1

(wtiJx

tiJ)

ρi (2)

where:

a N ′ = |St| is the set of active goods

b xti1, ..., x

tiN ′ is the bundle of goods of agenti, in the periodt


c wtiJ > 0 represents thevaluation of the buyeri for a unit of the goodJ , in the

period t

d 0 < ρi ≤ 1.

Algorithm 2 The Model

1: There is a market consisting ofN goods and a population ofn agents/buyers.

2: Initialization:

3: Initially, each agent is informed about a subset of the goods

4: Each agenti assigns a random valuation/weightwiJ to the goods that he is aware

of.

5: for t = 1, 2, . . . , T do6: Each agenti declares hisstrategy - a consumption plan, for the periodt. The

agent’s strategy is the vectorwti = (wt

i1, . . . , wtiJ , . . . , w

tiN ), wherewt

iJ ≥ 0

7: A Fisher market is formed by the agents’ strategies

8: A Market Equilibrium Mechanism, as an inherent price discovery process,

outputs:

9: The market clearing pricespt = {ptJ}J

10: The equilibrium allocationxti = {xt

iJ}J for each agenti, the bundle of goods that

the agenti consumes in the periodt

11: The current state of the market is(pt, {xti}i)

12: Diffusion:

13: A diffusion process takes place and the agents respond to thecurrent state of the

market

14: Each agent interacts with his neighbors and defines his strategywt+1

i for the next

period t+ 1, according to the adaptive rule [equation (3)]

15: end for

4.1.1 The adaptive rule

The timing of the process is presented in details in Algorithm 2. In the first period, theagents assign a positive weight to the goods that they are aware of. In a given periodt+ 1 we defined the valuationwt+1

iJ of the buyeri for a unit of the goodJ as:

wt+1

iJ = xtiJ + βi(d

tiJ − xt

iJ) (3)

where:

• 0 ≤ βi ≤ 1 is thediffusion rate, an idiosyncratic term of the agenti

• dtiJ is thedensityof the goodJ at thei′s neighborhood, in the periodt. It is equalto the overall quantity of the goodJ that the neighbors of i have bought, dividedby the degree ofi


• xtiJ is the quantity of the goodJ that the agenti consumes in the periodt

WhendtJ > xtiJ , the neighbors ofi possess a large amount of the good and there is

a positive feedback on thei′s valuation for that good. In other words, the agents followthe trends of the local community where they belong.

The case whereβi = 0, ∀i, corresponds to a market where the agents follow onlytheir own taste or they are isolated without neighbors. In this casewt+1

iJ = xtiJ , the

valuation for the periodt+ 1 is equal to the quantity that the agenti consumesin the periodt. If βi = 1, ∀i then wt+1

iJ (J) = dtJ and the model is similar with themodel presented inSalamanos & Konstantopoulos(2011), where the agents do not havememory of their own actions and their strategy for the next period was based only onthe consumption plan of their neighbors in the previous period.

The adaptive rule [equation (3)] is a discrete time version of the diffusion ofsubstances across a membrane between adjacent cells. Consider two cellsi andj and asubstanceA with concentrationci andcj in the celli andj respectively. The cells couldcommunicate across the cell membrane. The gain or the loss ofthe celli is DA(cj − ci),whereDA is the diffusion constant. In our model we consider a given agent and hisneighborhood as two cells both with volume one and the goodJ corresponds to thesubstance. ThedtJ andxt

iJ values are the concentrations of the substanceJ in the twocells.

Note that the equation (3) is a recursive function on thewti values. In a given

period t, the market equilibrium mechanism takes as input the agents’ budgets{bi}iand strategies{wt

i}i and outputs the market pricespt = {ptJ}J and the bundle of goods{xt

iJ}J that each agent consumes (see Algorithm2, Steps 6–10). Thus, thextiJ is a

function of the agents’ strategies{wti}i in the periodt.

Defining the agent’s valuation for a given goodJ as a function of thextiJ we imply

that the agent is also affected by the market price of the goodJ . When the competitionfor a good is large, then the price of the good increases and even if a given agenthad initially a large valuation for the good, he could have ended up consuming a smallamount. In our model the agent’s preference for a given good is a function of thequantity that he has finally received and not his initial desire. For example, consider anagent who is fond of luxury products and is willing to spend his vacations in a luxuryhotel. Let us assume that due to the large competition for thehotel, the market allocationis very small and the agent in question is allowed to stay onlyone hour per day to thehotel. According to our model, in the next period, the agent’s valuation decreases. Inother words, the agent adapts himself to the current competition of the market.

5 Simulation setup

The simulation was implemented in MATLAB. In order for the problem to be tractable,we simulate a market with two goods and we study the model in the light of severalconfigurations of the control parameters and the social graph.

5.1 Critical parameters and initialization

In the first period we initialize the values ofρi andwiJ as follows. Consider the simplecase of a market with two goods,A andB, and with a homogeneous population ofn


buyers, where every buyer has initial valuation equal to onefor both goods. Moreover,the buyers have the same budget, utility function and diffusion rate, i.e. for everyi,bi = b, ρi = ρ, andβi = β. Obviously, in every period the equilibrium prices of bothgoods are equal to

∑

i

bi/2 and the goods will be equally distributed among the agents.

Moreover, if the social network is an undirected graph, thenall the agents will observethe same density for both goods. Thus, for every buyer we havewt

iA = wtiB . In this case,

the diffusion model will produce the same outcome during theperiods and from thefirst period the system will reach a stable and symmetric state. Now we assume that dueto an external shock the agents’ characteristics slightly shift from the initial conditionsand the agents differ in terms of the initial valuation values and the rate of substitution.We define an almost homogeneous population of agents as follows: In each simulationinstance, we first select aρ andβ from the interval(0, 1]. Then, we generate uniformlyat random 50ρi values from the interval[ρ− 10−6, ρ]. These are theρi values of the50 buyers. We define that for every buyerβi = β. Finally, in all simulation instances,the initial valuationswiJ are uniformly distributed on the interval[1− 10−6, 1].

5.2 Market equilibria mechanism

The market mechanism in our model is represented by the proportional responsedynamics (Algorithm1). The iterations of the proportional response dynamics aresetto 1,000. In order to verify that the outcome of the proportional response dynamicsapproximates the market equilibrium, we compute the standard deviation of the marketprices as well as the buyers’ utility values based on the last50 iterations of theproportional response algorithm. Generally, in all instances and in every period, thestandard deviation was less than5 · 10−3. In our setting, the proportional responsedynamics approximates the market clearing prices very fastand the 1,000 iterations weresufficient despite the theoretic bound.

5.3 Social network configuration

We simulate the model for the following social graph configurations.

• Complete graphwhere each pair of agents is connected by an edge. The completegraph corresponds to the extreme case where all the agents ofthe network areneighbors and the information flow between the agents is maximized.

• Erdös-Rényi (ER)is a simple random graph model, where the edges of the graphare constructed by connecting each pair of nodes randomly with probability p andindependent from every other pair of nodes (Erdös & Rényi 1959). Consequentlythe total number of edges is a random variable with the expectation valueE(n) =

pN(N − 1)

2. If p >

ln(n)

nalmost surely the graph is connected. In the experiments

we setp = 0.2.

• Scale-free (SF)are graphs whose the degree distribution follows a power lawP (k) ∼ k−γ , where the parameterγ controls the broadness of the distribution(Barabasi & Albert 1999, Albert & Barabasi 2002). The degree distribution isindependent of the graph size, the path lengths are short andhighly clustered. Scale-free graphs describe many real world networks. A very important characteristic of


SF is the condition ofpreferential attachment, i.e., the most linked nodes gain moreedges in the growth of the network. In the experiments we useda MATLAB routinepublished in MATLAB website (George 2007). The routine is an implementationof the Barabasi Albert (BA) model (Barabasi & Albert 1999, Albert & Barabasi2002), and the frequency of degrees distribution forms a power-law curve, with anexponent between−2 and−3.

6 Analysis

In this section we present the experimental evaluation of the model. We concentrate theanalysis in a market with two goods{A,B} and n = 50 agents/buyers and we studythe properties of the long time state of the market. A market state consists of the marketpricespt = {ptJ}J and the allocation of the goods to buyers{xt

i}i, wherexti = {xt

iJ}J .We assume that each good has endowment equal to a unit and the agents’ budget remainconstant during the periods. Thus, in every period, the total amount of money is constantand equal to the

∑

i

bi, wherebi is the budget of agenti. The price of each good is

equal to the summation of the agents’ bid i.e.ptA =∑

i

btiA and ptB =∑

i

btiB . Thus,

we haveptB =∑

i

bi − ptA and it suffices to study the evolution of the market price of

one of the goods.

In all the experiments the agents’ budget is set tobi =1

n=

1

50= 0.02. We study

the market prise only of the goodA as ptB =∑

i

bi − ptA = 1− ptA. Similarly, we

concentrate our attention to the agents’ bids on the goodA asbtiB = 0.02− btiA.We study the evolution of the market when:

a the diffusion rate of each agent is zero,βi = 0, ∀i. In this case there is not adiffusion effect

b a market whereβi > 0, ∀i.

The critical parameters of the model are theρ, β and the degree distribution of thenodes of the graph. We execute several experiments for a variety of ρ and β values.A pair (ρ, β) along with the social graph correspond to asimulation instance. In otherwords, a simulation instance is an experiment where we newlydefine initialization, theagents parameters and the social graph (see5.1). In this paper we execute 280 simulationinstances.

The experimental results demonstrate that the long time state of the market couldbe classified by the following patterns.Homogeneous, where the goods equally sharethe market, i.e., the bids of each buyer on each of the goods are equal.Heterogeneous,where the buyers’ bids on the two goods differ.Periodic, where the buyers’ bids oscillatewith period 2. In order to identify the periodic states, we compute the differenceLiA =bt+1

iA − btiA of the agents bids and the standard deviation (std) of the absolute difference|LiA|.

We observe that in every simulation instance there is a critical periodtcrit wherethe system converges to the long time state. In order to classify the long time states of


the market we use the following criteria. Recall that the total amount of money in themarket is one andbi = 1/n:

• Homogeneous: A stable state where the goods have equal prices. In every periodtcrit < t and for every agent we havebtiA = bi/2 (= btiB), ptA = 0.5 (= ptB) andstd(|LiA|) = 0

• Heterogeneous: The buyers’ bids on the two goods differ. In every period andforevery agentbtiA 6= btiB , LiA > 0 andstd(|LiA|) = 0

• Periodic: The agents’ bids oscillate, with period 2, between two extreme values andwith stable oscillation width. In this case,btiA 6= btiB , theLiA becomes sequentiallypositive and negative (oscillation) and thestd(|LiA|) is close to zero (stableoscillation width).

6.1 Evolution without diffusion

In order to gain insights of the properties of the model, we analyze the case whenβ = 0, and we study two markets with 3 and 50 buyers respectively. When β = 0 thenthere is no diffusion effect, the buyers are not affected by their neighbors and the graphstructure is irrelevant. In this case the agents’ valuations are equal to the quantity thatthey consume in the previous period i.e.wt+1

iJ = xtiJ [equation (3)]. In each market, the

agents have budgets equal to1

n, i.e. if n = 3 then bi =

1

3≈ 0.33 and if n = 50 then

bi =1

50= 0.02.

We investigate the long time state of the markets, whenρ = {0.1, 0.2, . . . , 1}. Foreachρ we execute 10 independent simulation instances for 200 timeperiods each. Thus,we perform 100 experiments in total. In each simulation instance we generate, at new,the {ρi}i and{wiJ}i values following the initialization method in5.1.

We compute the market prices and the buyers’ bids, on each of the two goods. InFigure 1 we plot for eachρ the market prices and the buyers’ bids during the last 10periods of each simulation instance. Specifically, for eachρ ∈ {0.1, 0.2, . . . , 1} we plotthe points(ρ, ptJ ) [Figures1(a) & 1(c)] and (ρ, btiJ ) [Figures1(b) & 1(d)].

Figure 1 depicts that in both markets there is a phase transition in terms of themarket prices and the buyers’ bids. The critical point here is when we haveρ =0.5, where we observe a transition between homogeneous and heterogeneous state.Specifically:

• Homogeneous: Whenρ ≤ 0.5, then the market converges to a homogeneous state.In a homogeneous state, the bids of each buyer on the two goodsare equal (btiA =btiB) and as a result, the prices of the two goods are also equal (ptA = ptB). Observein Figure1 that both goods have price 0.5 and each buyer places half of his budgeton each of the goods. Recall that the total amount of money in the market is one.In other words, the goods are sharing the market equally.

• Heterogeneous: When 0.5 < ρ, the market converges to a heterogeneous state. Ina heterogeneous state the buyers’ bid on the two goods differ. For instance, in themarket with 3 buyers and whenρ = 0.6, the goods have the same price 0.5 butthe bids of some buyers are not equal [Figures1(a) & 1(b)]. In order to investigate


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

ρ

Pric

e3 Isolated Buyers

Good AGood B

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

3 Isolated Buyers

ρ

Buy

ers

Bid

s

Good AGood B

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.650 Isolated Buyers

ρ

Pric

e

Good BGood A

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.005

0.01

0.015

0.02

0.02550 Isolated Buyers

ρ

Buy

ers

Bid

s

Good AGood B

(d)

Figure 1: Isolated Buyers

the Periodic states, we compute for eachi the differenceLiA = bt+1

iA − btiA and thestandard deviation of the|LiA|, during the last 10 periods. For eachi, LiA ≈ 0 andstd(|LiA|) < 10−18, thus, the long time states were heterogeneous and not periodic(due to space limitations, we omit the plots).

Finally, observe that for0.8 ≤ ρ and 0.6 ≤ ρ in the market with 3 and 50 buyersrespectively, the buyers are divided in two groups with antisymmetric preferences. Ineach group, the agents place non zero bid only to one of the twogoods [Figures1(b)& 1(d)].

6.2 The effect of diffusion

In the previous section we studied the evolution of a market when the agents’preferences do not depend on the actions taken by their neighbors. In this section weinvestigate the effect of diffusion, for a variety of theρi and theβ values. Recall thattheβ is the diffusion rate [equation (3)] and theρi is the exponent on the agent’s utilityfunction [equation (2)]. Whenρi tends to 1 then the utility tends to linear. The diffusionmodel represents a feedback mechanism which could accelerate or slow down the spreadof influence between the agents.

We simulate a market withn = 50 buyers and two goods{A,B}, and we run eachinstance for 500 periods. We study the market prices and the agents bids during the last


10 periods. We apply the model on three graph families, Complete graph, Erdös-Rényi(ER) and Scale-free (SF).

Recall that, the agents’ budget is set tobi =1

n=

1

50= 0.02 therefore, the total

amount of money in the market is one. Thus, we study the marketprice only of thegoodA as ptB =

∑

i

bi − ptA = 1− ptA. Similarly, we concentrate our attention on the

agents’ bids only for the goodA asbiB = 0.02− biA.The control parameters are set toρ = {0.5, 0.6, . . . , 1} and β = {0.1, 0.2, . . . , 1}.

We execute a simulation instance for each pair of(ρ, β) and for every graph familyindependently, i.e. 180 experiments in total. A pair(ρ, β) along with a graph correspondsto a simulation instance. For a givenρ, we select, at random, 50 values from the interval[ρ− 10−6, ρ] which correspond to the agents’ρi values (see Section5.1). Moreover, weproduce the agents’ initialwi valuations which are uniformly distributed on the interval[1− 10−6, 1] (see Section5.1). Furthermore, for each agent we haveβi = β. In otherwords, the population of agents is almost homogeneous.

Finally, we present the experimental results using three dimensional plots in the samefashion as in abifurcation diagram, i.e., we present the outcome of the model whenthe market has converged to the long time state. Specifically, for each graph family weplot for each agent the data points(ρ, β, ptA), (ρ, β, b

tiA) and (ρ, β, LiA). Due to space

limitations, we omit the plots(ρ, β, std(|LiA|)). The data points correspond to the last10 periods of each simulation instance. We describe the phase transitions and the regionsof the market states following the direction where we increase theρ first and then foreachρ we increase theβ.

6.2.1 Complete graphs

In Figure 2 we present the experimental results for the complete graphs. The standarddeviation of the|LiA| values, for each agent independently, was less than10−6 in allsimulation instances. From Figures2(a) & 2(b) we conclude that for each0.5 < ρ thereis a criticalβ where a phase transition occurs. Specifically:

• No phase transition: When ρ = 0.5 and 0.1 ≤ β ≤ 1, then the market is in thehomogeneous state. Each agenti equally splits his budget (bi = 0.02) between thetwo goods,btiA = 0.01 = bi/2 andLiA = 0 [Figures2(a) & 2(b)]. Also, the marketprice of the goodA is 0.5 [Figure2(c)] which is half of the total amount of moneyin the market, thusptA = ptB .

• Heterogeneous to homogeneous: When0.6 ≤ ρ ≤ 1, then as theβ increases thereis a transformation between the heterogeneous to homogeneous state. The agents’bids on the two goods differ,biA 6= 0.01 and in every instanceLiA = 0 [Figure2(b)]. The critical values(ρ, β) are (0.6, 0.2), (0.7, 0.4), (0.8, 0.6) and (0.9, 0.7)[Figure 2(a)].

Observe in Figure2(a), that whenρ = 0.7 and the diffusion rate lies on the interval[0.1, 0.3], then only a fraction of the agents places non zero bid on the goodA. Inother words, the agents are divided into two groups, the agents who prefer only thegoodA and those who prefer only the goodB. As theβ increases, the agents arestarting to place non zero bids on both goods. Finally, for0.4 ≤ β, the agents splittheir budget equally, among the two goods.


0.5 0.6 0.7 0.8 0.9 10.10.4

0.710

0.005

0.01

0.015

0.02Complete Graph

ρβ

Buy

ers

Bid

−A

(a)

0.5 0.6 0.7 0.8 0.9 10.10.4

0.71

−0.02

−0.01

0

0.01

0.02Complete Graph

ρβ

LiA

(b)

0.5 0.6 0.7 0.8 0.9 10.10.4

0.71

0.48

0.49

0.5

0.51

0.52

Complete Graph

ρβ

Pric

e A

(c)

Figure 2: Diffusion in complete graphs


• Heterogeneous to periodic: Finally, when ρ = 1 and β = 1 there is atransformation from the heterogeneous to periodic state. Recall that when a stateis periodic, then the market price of the goodA oscillates with period 2, betweentwo extreme values. TheLiA = bt+1

iA − btiA takes sequentially positive and negativevalues but the|LiA| is always positive and stable, i.e.std(|LiA|) = 0. The criticalvalues are(ρ, β) = (1, 1). In other words, when the agents’ utilities are linearand the diffusion rate is large, then the market becomes dynamically stable andthe agents’ bids oscillate with stable oscillation width [see Figure2(b), points(1, 1,−0.02) and (1, 1, 0.02)]. Observe that even if the agents’ bids oscillate, themarket prices of the two goods are equal, i.e.,pA = 0.5 [Figure 2(c)].

6.2.2 ER graphs

We repeat the previous analysis for the case where the socialnetwork is represented byan ER graph. In each simulation instance and for every agent,the standard deviationof the |LiA| was less than10−7 which indicates that the long time state patterns werestable. The experimental results depict that there are critical points in which there is atransition between the heterogeneous and homogeneous states and a transition betweenthe heterogeneous and periodic states. Specifically:

• No phase transition: When ρ = 0.5, the market is in the homogeneous state foreveryβ value. The reasoning is the same as that for the complete graphs in 6.2.1.For every agent we havebtiA = 0.01 andLiA = 0 [Figures3(a) & 3(b)].

• Heterogeneous to homogeneous: Whenρ = 0.6, then there is a transition betweenthe heterogeneous and homogeneous state. From Figure3(a) we conclude that forβ = 0.1, the market is in the heterogeneous state. As theβ increases, the marketis in a mixed-phase regime, where a fraction of the population places bids closeto 0.01. At the critical value(ρ, β) = (0.6, 0.6), the market state transforms tohomogeneous state where for each agentbtiA = 0.01 = bi/2.

• Heterogeneous to periodic: When 0.7 ≤ ρ ≤ 1 and 0.1 ≤ β ≤ 0.9, the market isin the heterogeneous state and as theβ increases the heterogeneity also increases.However, for0.7 ≤ ρ ≤ 1 andβ = 1, the market state transforms to periodic state.Moreover, the periodic states have a variety of oscillations width, especially whenβ = 1 andρ takes the values 0.6 and 0.7 [Figure3(b)]. In addition, in Figure3(c)we observe that the points with0.7 ≤ ρ ≤ 1 andβ = 1 have two distinct values inthe z-axis, which implies that the market price of the goodA oscillates during thelast 10 periods of the simulation.


0.5 0.6 0.7 0.8 0.9 10.10.4

0.7110

0.005

0.01

0.015

0.020.02ER Graph

ρβ

Buy

ers

Bid

−A

(a)

0.5 0.6 0.7 0.8 0.9 10.10.4

0.71

−0.02

−0.01

0

0.01

0.02ER Graph

ρβ

LiA

(b)

0.50.6 0.7

0.8 0.91

0.1

0.4

0.7

1

0.46

0.48

0.5

0.52

ρ

ER Graph

β

Pric

e A

(c)

Figure 3: Diffusion in ER Graphs


0.5 0.6 0.7 0.8 0.9 10.10.4

0.710

0.005

0.01

0.015

0.02SF Graph

ρβ

Buy

ers

Bid

−A

(a)

0.5 0.6 0.7 0.8 0.9 110.10.4

0.711

−0.02

−0.01

0

0.01

0.020.02SF Graph

ρβ

LiA

(b)

0.5 0.6 0.7 0.8 0.9 10.1

0.4

0.7

110.4

0.45

0.5

0.55

0.6

SF Graph

ρ

β

Pric

e A

(c)

Figure 4: Diffusion in SF graphs


6.2.3 SF graphs

We conclude the analysis investigating the effect of diffusion in the case where theagents’ social affiliations form a SF graph. The experimental results are presented in theFigure 4. The standard deviation of the|LiA| values were less than10−10 for every i,which indicates that the long time state patterns were stable.

• No phase transition: The result is similar to that for the Complete and ER graphs.For ρ = 0.5 and0.1 ≤ β ≤ 1, the market is in the homogeneous State. For everyiwe havebtiA = 0.01 and the differenceLiA = bt+1

iA − btiA is zero for everyi [Figures4(a) & 4(b)].

• Heterogeneous to periodic: For 0.6 ≤ ρ ≤ 1, there is a transition between theheterogeneous and periodic states. Figure4(a) depicts that forβ = 0.1, the marketis in the heterogeneous state and as theβ increases the heterogeneity in terms of theagents’ bid becomes more intense. Finally, the market transforms to a periodic stateand the critical (ρ, β) values are(0.6, 1), (0.7, 1), (0.8, 0.7), (0.9, 0.6) and (1, 0.7).Moreover, there is a region where the oscillation width in terms of agents’ bidsvaries, for instance whenρ = 0.7 and0.8 ≤ β ≤ 1 [Figure4(b)]. Furthermore, fromFigure4(c), we observe that forβ = 1, the market price of the good A oscillates; theargument is similar to the one described for the ER graphs. Moreover the oscillationwidth is larger than the one in the ER graphs.

7 Discussion and conclusions

In microeconomics, the fundamental property of a market is the existence of a marketequilibrium. Under certain assumption for the buyers’ utility function, the theory ensuresthat there is always a market equilibrium, the competition leads the market to a stablestate, where for each good the supply is equal to demand and the allocation of the goodsto buyers is Pareto efficient.

In this paper our goal was to investigate the evolution of a market when we applya social dynamic to the economic competition. According to our knowledge, this isthe first study which shows evidence of phase transition phenomena in the context ofdiffusion in markets.

Specifically, in this paper we studied the special case in which the consumption planof a buyer is affected by the consumption plan of his neighbors in the previous timeperiod. The proposed diffusion model has a natural economicand social interpretation.Moreover, we assumed that in every time period, the market isequilibrated by a marketequilibrium mechanism. For simplicity, we considered a market with two goods and westudied the properties of the long time states for several configurations of the socialgraph and the control parameters.

The experimental results show that the long time states of the market are correlatedwith the rate of diffusion and the elasticity of substitution of the agents’ utility functions.The long time states can be classified in three patterns

a homogeneouswhere the market prices for both goods are equal and the buyerssplit their money equally amongst the two goods

b heterogeneouswhere the buyers bids for the two goods differ


c periodic where the buyers bids oscillate with stable oscillation width.

The findings could be considered interesting in general. Theexperimental resultsdemonstrate that the model exhibits rich behavior despite it’s simplicity. In order toemphasize on the phase transition phenomena and the correlation between the longtime states and the initial conditions, we studied an almosthomogeneous populationof agents. The initialization method in5.1 is very important for the analysis of themodel. In each simulation instance we generate an almost homogeneous population ofagents where the agents’ initial parameters lie on intervals with only 10−6 intervalwidth. Recall that in a perfect homogeneous population, allagents have exactly the samecharacteristics (ρi, wiJ andβi) and from the first period, the market state is symmetric,stable and the two goods share the market equally, i.e., the market prices and the agents’valuation and bids are equal. Thus, from the first period, themarket is in a “frozen”state. In order to break the symmetry, we assume an external shock which slightlyalters the homogeneity of the population. The experimentalresults show that even if thepopulation is almost homogeneous, the long time states are sensitive to initial conditionsand phase transition phenomena emerge.

Other initializations have been tested, for instance the one where in the first periodonly a small fraction of the agents is informed about the two goods. Specifically, inthe first period, we uniformly at random select a small subsetof the population. Then,the informed agents assign a random valuation to the goods and all the other agentshave initial valuation zero. Also, in this case, the experimental results depict a phasetransition between the three state patterns. Unfortunately, this method leads to a complexanalysis, since we add two free parameters, the size of the initially informed agents andthe degree of those agents. The long time states are highly correlated with the degreeof the initially informed agents. For instance, consider a SF graph and a market wherein the first period, only the agentsi and j are informed about the two goods. In thiscase, the graph has power law degree distribution and the degree of i and j could varysignificantly.

Finally, in a future work we intend to study the market evolution in directed graphsand an extension of the model where the termdtiJ [the density in equation (3)] is definedas non-linear combination of the density of a given good at the agent’s neighbourhoodand the density of that good in a subset of the population.

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Diffusion of information and phase transition in the Fisher market

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