MPRA Munic Personal RePEc Arcive Outflow Dynamics in Modeling Oligopoly Markets: Te Case of te Mobile Telecommunications Market in Poland Katarzyna Sznajd-Weron and Rafal Weron and Maja Wloszczowska Hugo Steinaus Center, Wroclaw University of Tecnology 1. September 28 Online at ttp://mpra.ub.uni-muencen.de/1422/ MPRA Paper No. 1422, posted 13. September 28 :41 UTC
Outflow Dynamics in Modeling Oligopoly Markets: Te Case of te Mobile Telecommunications Market in Poland Katarzyna Sznajd-Weron 1, Rafa l Weron 2 and Maja W loszczowska 3 1 Institute of Teoretical Pysics, Wroc law University, pl. Maxa Borna 9, 5-24 Wroc law, Poland 2 Hugo Steinaus Center for Stocastic Metods, Wroc law University of Tecnology, Wyb. Wyspiańskiego 27, 5-37 Wroc law, Poland 3 Institute of Matematics and Computer Science, Wroc law University of Tecnology, Wyb. Wyspiańskiego 27, 5-37 Wroc law, Poland E-mail: kweron@ift.uni.wroc.pl, rafal.weron@pwr.wroc.pl and 126582@student.pwr.wroc.pl Abstract. In tis paper we introduce two models of opinion dynamics in oligopoly markets and apply tem to a situation, were a new entrant callenges two incumbents of te same size. Te models differ in te way te two forces influencing consumer coice (local) social interactions and (global) advertising interact. We study te general beavior of te models using te Mean Field Approac and Monte Carlo simulations and calibrate te models to data from te Polis telecommunications market. For one of te models criticality is observed below a certain critical level of advertising te market approaces a lock-in situation, were one market leader dominates te market and all oter brands disappear. Interestingly, for bot models te best fits to real data are obtained for conformity level p (.3,.4). Tis agrees very well wit te conformity level found by Solomon Asc in is famous social experiment. Keywords: opinion dynamics, outflow dynamics, agent-based model, oligopoly market, advertising, mobile telepony. 1. Introduction Economic and social studies suggest tat te decision of a consumer to purcase a particular product depends not only on price and product quality, but also on social effects and advertising exposure [1, 2]. On one and, te social validation penomenon is a powerful motivation for our actions. People listen to teir family and friends experiences of network operators, price plans and brands of andsets [3, 4]. Te tariffs temselves add to tis clustering effect by giving preferences (including lower prices) to connections witin a network or even witin a group of friends.
Outflow Dynamics in Modeling Oligopoly Markets 2 On te oter and, te willingness to pay for a product is increased by advertisements [5]. Advertising is used to reinforce te brand image as it is used as an indicator of quality and makes services suc as telecommunications more tangible to consumers. Indeed, Turnbull et al. [3] found advertising to be quite an important source of information wen purcasing a mobile pone, but not as important as family and friends. At te same time, mobile telepony is te most eavily advertised business in many countries [6]. It seems tat te operators are aware of te power of advertising. And since tey cannot influence social relations directly, tey are doing te second best ting advertising. But wat affects consumer decisions more? Is it social influence or advertising? Te aim of tis paper is to build a model of te mobile telecommunications market in Poland, were tree major operators (Plus, Era and Orange) figt an on-going battle for domination. To tackle tis problem we introduce two models of opinion dynamics in oligopoly markets, wic differ in te way te two forces influencing consumer coice (local) social interactions and (global) advertising interact. We study te general beavior of te models: steady states in terms of market sares for te entire space of parameters, as well as calibrate tem to real data. Hopefully, te obtained results will add to te understanding of oligopoly market beavior and elp in answering te above question. Te paper is structured as follows. In te next Section we provide a sort overview of te mobile pone market in Poland. In Section 3 we discuss te factors influencing our coice of te mobile operator and introduce two opinion dynamics models. We conclude te Section by presenting Mean Field equations for bot models. In Section 4 we present te obtained results bot in terms of Monte Carlo simulations and Mean Field approximations. Finally, in Section 5 we conclude and discuss possible follow-up researc. 2. Te Market 2.1. Te Mobile Telecommunications Market in Poland Mobile telepony in Poland dates back to 1992, wen PTK (brand Centertel) started offering services in te analog NMT45i system. Te first mobile pones were te bricks (Nokia Cityman 45) and te radiators (Motorola 2, Talkman 9). Teir price exceeded te average annual salary at tat time. In te early 199s mobile pones were te synonym of success in te rapidly developing Polis economy. Second generation services (GSM 9 MHz) were offered in 1996 by two new operators PTC (brand Era) and Polkomtel (brand Plus). Only ten, troug competition and te policy of subsidizing mobile pones ave tey become more popular. Centertel started loosing clients due to worser quality of transmission and bulkier andsets and in 1998 decided to launc a digital system of its own Idea Centertel (rebranded to Orange in 25) working in te GSM 18 MHz system. In late 1998 all tree operators made a next
Outflow Dynamics in Modeling Oligopoly Markets 3 major step forward and introduced prepaid services. Tis move eventually led to mobile pones becoming everyday appliances. In 2 te operators received concessions for new frequencies and since ten all tree ave used bot te 9 and 18 MHz systems. Tis was a great opportunity especially for Idea Centertel, wic as been using te sorter range, city-concentrated GSM 18 MHz system and ad tree times fewer clients tan te two competitors. Te next major tecnological cange introduction of te tird generation UMTS telepony in 24 did not ave a noticeable impact on te market. Even today tere are not too many users of tis system. In Marc 27 a fourt player entered te market P4 (brand Play), but by te end of 27 its market sare was still negligible (2.5% in terms of te number of users and 1.28% in terms of net revenues). During te last 15 years te number of mobile pone users as been steadily increasing. In 27 te number of sold SIM cards exceeded 4 million yielding a market penetration of 18.6%; te penetration by active SIM cards was estimated at 9.9% [7]. Compared to oter European countries tis is not a spectacular result. Te mobile pone market is a typical example of an oligopoly. Entry barriers (including infrastructure and concessions) are extremely ig. As of end 27 tere were four operators in te Polis market [7]: Polkomtel S.A. (brand: Plus, prepaid brands: Simplus and Sami Swoi) Polska Telefonia Cyfrowa Sp. z o.o. (brand: Era, prepaid brands: Tak Tak and Heya) Polska Telefonia Komórkowa Centertel Sp. z o.o. (brand Orange; includes prepaid services) P4 Sp. z o.o. (brand Play; includes prepaid services) Tere were also a few Mobile Virtual Network Operators (MVNO) brands mbank mobile, myavon, WPmobi, Simfonia and Ezo but teir market sare was negligible (.12% in terms of te number of users and.2% in terms of net revenues as of end 27). In tis paper we study te period from te end of year 2, wen te tecnological constraints of te tree major operators were leveled out, till te tird quarter of 27, see Figure 1. We will only consider tese tree players (and generally denote tem later in te text by teir main brands Plus, Era and Orange) as te remaining operators ave not played a visible role in te studied period. For a long time tese tree players ave fougt an on-going battle for domination in te Polis mobile pone market. But despite te competition, tey ave sustained relatively ig prices for teir services and tempted new clients wit ceap andsets instead (similar strategies ave been observed in oter markets as well [8]). Only wen te market as saturated (24-25), te operators ave started attracting te less ric clients. First, by lowering te prices for prepaid connections, next by lowering te prices for subscription (postpaid) services. Te latter move as reversed te dominating
Outflow Dynamics in Modeling Oligopoly Markets 4 SIM Cards Sold (mln) 4 3 2 1 22 24 26 28 Percent of SIM Cards Sold 45 4 35 3 25 2 Plus Era Orange 22 24 26 28 Total Income (bln PLN) 7 6 5 4 3 2 22 24 26 28 Percent of Total Income 5 4 3 2 1 Plus Era Orange 22 24 26 28 Figure 1. Left panels: Te cumulative number of SIM cards sold and te total income (i.e., jointly for all tree major market players) from te first quarter of 21 till te 3rd quarter of 27. A steady upward trend can be observed in bot cases, wit noticeable seasonality in income figures. Rigt panels: Market structure in terms of SIM cards sold and income. Te turmoil around PTC led to a depreciation of te company s leading brand Era and, consequently, te company s financial standing. trend instead of offering ceap andsets te operators ave started luring clients wit ceap connections. Apart from te service-andset price competition, te operators ave always fougt a war on te billboards and in radio, TV and internet commercials. In fact, in te last ten years te tree major mobile market players ave been te most eavily advertised companies in Poland. Nearly every week tey make te top ten list, if not te top tree. Mobile pone ads are visible everywere and it is ard to imagine a new market entry witout an intensive advertising campaign. 2.2. Te turmoil around PTC Writing about te Polis mobile telecommunications market we ave to mention te controversial events related to te dispute over 48% of sares of PTC (owner of te brand Era). Four companies were involved in tis: Elektrim, Elektrim Telekomunikacja, Deutsce Telekom and Vivendi. Te dispute started in 1999, wen Elektrim bougt 15.8% of sares from oter sareolders (Kulczyk Holding and BRE Bank). At te same time, anoter sareolder Deutsce Telekom (DT) was convinced tat it ad te preemtion rigt to tose sares and in December 2 filed a case in te Arbitrage Court
Outflow Dynamics in Modeling Oligopoly Markets 5 in Vienna. Legal actions ended in November 24. In te meantime, te sareolder structure canged and te sentence did not urt Electrim but a Frenc company Vivendi. In 1999 Vivendi bougt 49% of sares of Elektrim Telekomunikacja (ET), a company to wic Elektrim transferred its PTC sares in 2. In 21 Vivendi bougt anoter 2% of sares of ET gaining control not only over ET but also over PTC. However, te Arbitrage Court ruled tat te transfer of sares from Elektrim to ET was illegal. Te Provincial Court in Warsaw (S ad Rejonowy w Warszawie) recognized tis decision and 48% of PTC sares returned to Elektrim. And ten all te fuss began. In February 25 te Provincial Court in Warsaw made canges in te National Company Register (Krajowy Rejestr S adowy, KRS), including a new board of directors and a new supervisory board appointed by Elektrim in cooperation wit DT. ET and Vivendi did not accept tis, wic resulted in a blockade of PTC s eadquarters by te former management. Vivendi also filed a complaint to te Polis government, backed by a bilateral agreement between France and Poland on observance of international investments. After some time te new board of directors gained control over PTC s offices and Elektrim started negotiations wit DT and Vivendi on te sale of PTC s sares. However, in August 25 te roles canged: te District Court in Warsaw (S ad Okrȩgowy w Warszawie; playing te role of te court of appeal for te Provincial Court) reversed te decision of te lower instance and sent te case back to te Provincial Court in Warsaw. In November 25 te Provincial Court made canges in KRS in favor of ET. Tis time te Elektrim and DT appointed board blocked PTC s eadquarters. A number of managing directors resigned. PTC started to ave liquidity problems. It seems tat te case was finally resolved in Marc 26, wen te Warsaw Court of Appeal (S ad Apelacyjny w Warszawie) upeld te sentence of te Arbitrage Court in Vienna from November 24. Te situation of PTC stabilized, altoug courts of different instances were still processing te case for two more years. During tis turmoil Era lost its pole position in te mobile pone market in Poland (Fig. 1). Tis appened despite te fact tat advertising remained ig, bot in terms of advertising expenditures and Gross Rating Points (Fig. 2). Some autors [9] consider te Gross Rating Points (GRP), i.e. te percentage number of targeted viewers contacted times te average number of contacts per targeted viewer, to be muc more representative of te actual advertising campaign. In our case, owever, te advertising expenditures and GRP numbers yield a very similar picture. Also te correlation between tese two variables is very ig and for individual companies ranges from ρ =.92 to.95 in te studied period. A Marc 26 survey performed by PBS (www.pbsdga.pl), a Polis market researc company, seds some ligt on te discord between PTC s market sare and advertising expenditures. Nearly 4% of respondents (and over 55% of corporate respondents) declare tat news of an uncertain future of te company would make tem cange te operator. Also over 3% of respondents would cange te operator if serious carges were brougt against te company s management.
Outflow Dynamics in Modeling Oligopoly Markets 6 Total GRP (x 1) 5 4 3 2 1 22 24 26 28 Percent of Total GRP 8 6 4 2 Plus ERA Orange 22 24 26 28 Total Expenditures (mln PLN) 2 15 1 5 22 24 26 28 Percent of Total Expenditures 8 6 4 2 Plus ERA Orange 22 24 26 28 Figure 2. Left panels: Quarterly Gross Rating Points (GRP) and advertising expenditures (jointly for all tree major market players) for te same period as in Figure 1. Te processes are not identical, owever, contrary to some reports [9] tey are not tat different (te correlation for individual companies ranges from ρ =.92 to.95). Rigt panels: Market structure in terms of GRP and advertising expenditures. 3. Te Model 3.1. Factors Influencing Our Coice of te Mobile Operator We generally prefer to make decisions consciously. In an attempt to make te best coice possible we study te operators offers: special deals, price plans, offered andsets, loyalty programs, network coverage and quality of transmission. Especially subscription clients spend long ours on studying and weigting tese factors. Quite often, owever, te offers are very similar wen we take all factors into account and it is very ard to make a decision based only on simple statistical comparisons. Economic and social studies suggest tat te decision of a consumer to purcase a particular product depends not only on price and product quality, but also on social effects and advertising exposure [1, 2]. Moreover, links in a social network and te stream of advertisements may lead to common actions of large groups of consumers and induce correlation between consumers decisions [1]. On one and, te social validation penomenon is a powerful motivation for our actions. People listen to teir family and friends experiences of network operators, price plans and brands of andsets. Turnbull et al. [3] found family and friends to be te main source of information used by UK consumers wen purcasing a mobile pone. A similar effect was observed in te Tai market [4]. Note tat tariffs add to
Outflow Dynamics in Modeling Oligopoly Markets 7 tis clustering effect by giving preferences (including lower prices) to connections witin a network or even witin a group of friends. On te oter and, according to Becker and Murpy [5] te willingness to pay for a product is increased by advertisements. Tis implies tat wen a positive news arrive at a consumer, tey raise is/er perception of utility of tat brand. Advertising is used to reinforce te brand image as it is used as an indicator of quality and makes services suc as telecommunications more tangible to consumers. Indeed, Turnbull et al. [3] found advertising to be quite an important source of information wen purcasing a mobile pone in te UK (but not as important as family and friends). At te same time, mobile telepony is te most eavily advertised business in many countries, see Section 2 and Ref. [6]. It seems tat te operators are aware of te power of advertising. And since tey cannot influence social relations directly, tey are doing te second best ting advertising. 3.2. Building te Model From te previous Section we know tat social influence plays a major role in selecting a mobile pone company. But can we be more precise and say wic type of influence is more and wic is less important for te decision making process? In te early 195s Solomon Asc reported an ingenious series of experiments on social influence (for a fascinating review see Levine [11]). Asc used several measures of social influence but ere we recall only two of is results [12], wic elp us answer te question Wic aspect of te influence of a majority is more important te size of te majority or its unanimity? : (i) Asc found tat minority participants confronting a unanimous majority were often capable of resisting social influence. Wen a subject was confronted wit only a single individual wo contradicted is answers, e was swayed little: e continued to answer independently and correctly in nearly all trials. Wen te opposition was increased to two, te pressure became substantial: minority subjects now accepted te wrong answer 13.6 % of te time. Under te pressure of a majority of tree, te subjects errors jumped to 32 %. But furter increases in te size of te majority apparently did not increase te weigt of te pressure substantially. Clearly te size of te opposition is important only up to a point. (ii) Asc also found tat e could increase independence dramatically te presence of a social supporter reduced te total number of yielding responses from 32% to 5.5% [12]. Te power of social support was furter demonstrated in a study sowing tat participants were far more independent wen tey were opposed by a sevenperson majority and ad a partner tan wen tey were opposed by a tree-person majority and did not ave a partner [11]. A number of later experiments sowed tat an individual wo breaks te unanimity principle reduces social pressure of te group dramatically [13]. Tis observation was recently expressed in a simple one dimensional USDF ( United we Stand, Divided we
Outflow Dynamics in Modeling Oligopoly Markets 8 Fall ) model of opinion formation [14]. Te model was later renamed te Sznajd model by Stauffer et al. [15] and generalized on a two dimensional square lattice. Te crucial difference between te Sznajd model and oter Ising-type models of opinion dynamics [16, 17, 18] is tat information flows outward from te center nodes to te surrounding neigborood (so called outflow dynamics [19]) and not te oter way around. Isingtype models wit outflow dynamics ave been successfully applied in marketing, finance and politics; for reviews see [2, 21, 22, 23, 24]. Te aim of tis paper is to build a model of te mobile telecommunications market in Poland, were tree major operators (Plus, Era and Orange) figt an on-going battle for domination. Te market is represented by a two dimensional L L lattice wit periodic boundary conditions. Eac site of te lattice is occupied independently by an individual (a customer), wo is caracterized by a variable S i = 1, 2, 3, i = 1,...,L 2, tat represents is/er mobile operator. In eac time step ( τ) one of te L 2 customers is selected randomly (random sequential updating) and ten togeter wit is/er tree neigbors forms a 2 2 panel. We measure te time so tat te speed of all processes remains constant wen L, and tus one update takes time τ = 1/L 2. Two forces influencing consumer coice are considered: (local) social interactions and (global) advertising. Some empirical studies [3] suggest tat of te two forces social interactions play a greater role. Following te unanimity principle discovered by social scientists we assume tat only an unanimous panel (all four customers in te panel use te same mobile network) persuades all eigt of its nearest neigbors to switc te operator to te one favored by te panel. In lack of unanimity te eigt neigboring individuals make teir coice based on advertisements: independently wit probability 1 p eac of tem switces te operator. Te coice of te operator is determined by te relative advertising level (a kind of an external field [25, 26, 27]). Wit probability 1 e/se cooses company 1 (say, Plus), wit probability 2 company 2 (Era) and wit probability 3 = 1 1 2 company 3 (Orange). Obviously, te customer may switc to te same operator, i.e., stay wit te original one. In tis model conformity (social influence) comes in first, ten advertising affects te customers. Hence, we will call te model Conformity First (CF). A different idea of incorporating advertising in a duopoly setting was proposed by te social psycologist Andrzej Nowak at te GIACS summer scool Applications of Complex Systems to Social Sciences [28]. He suggested to generalize te USDF model by flipping eac up opinion down (S i = 1 S i = 1) wit probability γ and eac down opinion up (S i = 1 S i = 1) wit probability β as a result of global effects like advertising troug mass media. Te traditional conformity rule was applied to eac of te neigbors (of te panel) independently wit probability α. In tis model (called Nowak-Sznajd by Wo loszyn et al. [28]) conformity influences individuals parallel to advertising. We borrow Nowak s idea to construct our second model and call it CAP (Conformity and Advertising Parallel). In te CAP model, for eac of te te eigt nearest neigbors of te selected panel we independently apply eiter te social (wit probability p) or advertising (wit
Outflow Dynamics in Modeling Oligopoly Markets 9 probability 1 p) forces. All neigbors selected to be influenced by social interactions switc te operator to te one favored by te panel, but only if te panel is unanimous. In lack of unanimity te selected neigbors do not perform any action. Te remaining neigbors (selected to be influenced by advertising) are subdued to te same rules as in te CF model. Namely, te coice of te operator is determined by te relative advertising level: wit probability 1 te individual cooses company 1, wit probability 2 company 2 and wit probability 3 = 1 1 2 company 3. Note, tat in contrast to te CF model, in te CAP model an unanimous panel does not guarantee conformity. 3.3. Mean Field Approac Let us denote by N 1 (t), N 2 (t) and N 3 (t) te number at time t of Plus s, Era s and Orange s customers, respectively. Furter, define by c 1 (t) = N 1(t), c L 2 2 (t) = N 2(t) and L 2 c 3 (t) = N 3(t) te respective market sares (concentrations). Of course, te normalization L 2 condition is fulfilled, i.e. t c 1 (t) + c 2 (t) + c 3 (t) = 1. We measure te time so tat te speed of all processes remains constant wen L, and tus one update takes time τ = 1/L 2. Eac basic time step t = 1 consists of L 2 updatings, ence te balance (evolution) equation takes te form: N s (t + 1) N s (t) = L 2 (incremental canges decremental canges), (1) were s = 1, 2, 3. Te incremental and decremental canges are (sligtly) different in te two models. For instance, in te CAP model te number N 1 (t) of Plus s (operator #1) clients can increase in time due to te following events: an Era client (operator #2) follows Plus s advertising te probability of suc an event is equal to (1 p) 1 c 2 (t); an Orange client (operator #3) follows Plus s advertising te probability of suc an event is equal to (1 p) 1 c 3 (t); an Era client subdues to te unanimous panel of operator #1 te probability of suc an event is equal to pc 1 (t) 4 c 2 (t); an Orange client subdues to te unanimous panel of operator #1 te probability of suc an event is equal to pc 1 (t) 4 c 3 (t). Similarly, te number N 1 (t) of Plus s clients can decrease in time due to te following events: a Plus client follows Era s advertising te probability of suc an event is equal to (1 p) 2 c 1 (t); a Plus client follows Orange s advertising te probability of suc an event is equal to (1 p) 3 c 1 (t); a Plus client subdues to te unanimous panel of operator #2 te probability of suc an event is equal to pc 2 (t) 4 c 1 (t);
Outflow Dynamics in Modeling Oligopoly Markets 1 a Plus client subdues to te unanimous panel of operator #3 te probability of suc an event is equal to pc 3 (t) 4 c 1 (t). Now, we can put down te balance equation for te first operator: N 1 (t + 1) N 1 (t) = L 2 [(1 p) 1 c 2 (t) + (1 p) 1 c 3 (t) + pc 1 (t) 4 c 2 (t) + pc 1 (t) 4 c 3 (t) (1 p) 2 c 1 (t) (1 p) 3 c 1 (t) pc 2 (t) 4 c 1 (t) pc 3 (t) 4 c 1 (t)]. (2) Dividing bot sides of te equation by L 2 and denoting c s (t) by c s and c s (t + 1) by c s, for s = 1, 2, 3, after simple algebraic transformations we obtain: c 1 c 1 = (1 p) 1 (c 2 + c 3 ) c 1 (1 p)( 2 + 3 ) + pc 1 c 2 (c 3 1 c 3 2) + pc 1 c 3 (c 3 1 c 3 3). (3) Applying te normalization conditions (c 2 +c 3 = 1 c 1 and 2 + 3 = 1 1 ) we obtain te complete set of evolution equations for te CAP model: c 1 c 1 = (1 p)( 1 c 1 ) + pc 1 [ c2 (c 3 1 c 3 2) + c 3 (c 3 1 c 3 3) ], c 2 c 2 = (1 p)( 2 c 2 ) + pc 2 [ c1 (c 3 2 c 3 1) + c 3 (c 3 2 c 3 3) ], c 3 c 3 = (1 p)( 3 c 3 ) + pc 3 [ c1 (c 3 3 c 3 1) + c 2 (c 3 3 c 3 2) ]. (4) Analogously, we can derive te evolution equations for te CF model: c 1 c 1 = (1 p)( 1 c 1 ) + c 1 [ c2 (c 3 1 c 3 2) + c 3 (c 3 1 c 3 3) ], c 2 c 2 = (1 p)( 2 c 2 ) + c 2 [ c1 (c 3 2 c 3 1) + c 3 (c 3 2 c 3 3) ], c 3 c 3 = (1 p)( 3 c 3 ) + c 3 [ c1 (c 3 3 c 3 1) + c 2 (c 3 3 c 3 2) ]. (5) Note, tat for p = 1 te CAP and CF evolution equations are identical and tat te differences increase wit decreasing p. Note also, tat te sets of evolution equations (4) and (5) can be easily solved numerically, see te discussion in Section 4. 3.4. Te Model of te Polis Telecommunications Market (2-27) Recall from Section 2 tat back in year 2 (beginning of te study period) tree mobile operators were offering teir services in te Polis market: a new entrant (Orange; until 25 under te brand Idea Centertel) was callenging two incumbents (Plus and Era). Te market sares of te two incumbents were nearly te same in terms of income (ca. 4% eac; see te bottom rigt panel in Figure 1), but not in terms of te number of SIM cards sold. Following W loszczowska [29], in tis study we use income as te measure of market sare. Tere are two major reasons for tis. First, te publised data concern te number of SIM cards sold, not te actual number of users of a given network. Some of te cards are not active and some clients are using more tan one SIM card (private, business, internet). Second, different categories of clients use te pones wit different intensities. Tis could be accounted for by assigning more lattice sites to
Outflow Dynamics in Modeling Oligopoly Markets 11 some clients, but for te sake of parsimony we ave decided not to execute tis option. Consequently, in our model one lattice node represents an average (in terms of income) client. Te market state were a new entrant callenges two incumbents of te same size allows us to make furter simplifications. We limit te ric parameter space (p, s,c s ()) to te situation were te initial concentrations of clients and te levels of advertisement of Plus and Era are equal, i.e. c 1 () = c 2 () c and 1 = 2, respectively. 4. Te Results For te sets of rules governing te beavior of mobile pone users (see Section 3.2), we study te system via Monte Carlo (MC) simulations and compare te final steady states wit tose obtained from te Mean Field Approac (MFA), see equations (4) and (5). 4.1. Monte Carlo Simulations We present results for lattice size 1 1, altoug we ave performed simulations for oter L s. In all simulations we took initially c of Plus customers, c of Era customers and 1 2c of Orange customers randomly distributed on te lattice. Te level of advertisement is te same for Plus and Era and constant trougout te simulations. For Orange te level is equal to 1 2. During te simulations, we ave observed tat after a transient ( termalization ) time t T te concentrations (market sares) reac a final level around wic tey fluctuate. Due to tese fluctuations, we define te final value of concentration as te mean value over a time interval T: = 1 T t T + T t=t T c(t). (6) For bot models (CAP and CF), te final steady state was reaced in all performed simulations independently of te parameters (p,) and te initial conditions (c ). Moreover, averaging over different T s (1, 5, 1) did not influence te results. For eac set of parameters (p,,c ) we performed 1 3 independent Monte Carlo simulations and calculated by averaging over all samples. Te final steady concentration results are presented in Figures 3, 4, 5 and 6. For te CAP and CF models two qualitatively different regimes are observed depending on conformity level p. For p.7 te final steady state does not depend on te initial concentration c, i.e. it is a function of only two parameters: = f(p,). In te low conformity regime (p.7) substantial differences between te studied models can be seen. For te CAP model (see te left panel in Figure 3), te dependence between final concentration and advertising level is almost linear, but generally = f(p,) is an S-saped function of. Deviations from linear dependence increase wit p. A different beavior is observed for te CF model (see te left panel in Figure
Outflow Dynamics in Modeling Oligopoly Markets 12.5.33 p=.2 p=.4 p=.6 p=.7.5.33 p=.2 p=.4 p=.6 p=.7 c MC MFA.33.5.33.5 Figure 3. MC and MFA results for te CAP model and p.7 p=.7 p=.86.5.5.5 c.5.5 c.5 p=.92 p=.98.5.5.5 c.5.5 c.5 Figure 4. MC results for te CAP model and p.7.5.33 p=.2 p=.4 p=.6 p=.7 Simulations.5.33 p=.2 p=.4 p=.6 p=.7 MFA MC MFA.33.5.33.5 Figure 5. MC and MFA results for te CF model and p.7
Outflow Dynamics in Modeling Oligopoly Markets 13 p=.7 p=.86.5.5.5 c.5.5 c.5 p=.92 p=.96.5.5.5 c.5.5 c.5 Figure 6. MC results for te CF model and p.7 5). A critical value c of te advertising level exists: = for < c, > for > c. (7) In te ig conformity regime (p.7), te final steady state depends not only on conformity and advertising, but also on initial concentration c. Neverteless, in te CAP model te dependence between and is still an S-saped function and no critical value of exists (see Figure 4). Tis is in sarp contrast to te CF model for wic c exists (see Figure 6). For p.7 te dependence between te final and initial concentration grows wit conformity level in bot models. Tis is an understandable result, because for ig values of p interactions between individuals dominate over te external field (advertising). On te oter and, for ig values of p te dependence between and sould decrease. In te limit (p = 1) only interactions between customers exist and tere is no external field, tus no dependence on is expected. Tis is confirmed by MC results (see Figures 4 and 6). 4.2. Mean Field Results In Section 3.3 we ave derived general sets of evolution equations describing ow concentrations cange in time for bot models. Tese sets of equations can be easily solved numerically. Recall from Section 3.4, tat we limit te parameter space (p, s,c s ()) to te situation were te initial concentrations of clients and te levels of advertisement of Plus and Era are equal, i.e. c 1 () = c 2 () c and 1 = 2,
Outflow Dynamics in Modeling Oligopoly Markets 14 p=.7 p=.86.5.5.5 c.5.5 c.5 p=.92 p=.98.5.5.5 c.5.5 c.5 Figure 7. MFA results for te CAP model and p.7 respectively. Ten te MFA equations (4) and (5) lead to te final fixed point = f(c,p,) independently of te parameters (c,p,). In agreement wit MC simulations, te MFA equations lead to two regimes depending on conformity level p: For p.7 te final steady state does not depend on initial concentration c, i.e. it is function of only two parameters = f(p,), see Figures 3 and 5. For p.7 te final steady state depends not only on conformity and advertising level, but also on initial concentration c. In tis regime dependence between te final and initial concentration grows wit conformity level p. On te oter and, dependence between and decreases and in te limit (p = 1) tere is no dependence on, see Figures 7 and 8. For te CAP model, MFA and MC results are in a ig agreement bot in te low (see Figure 3) and ig conformity regime (Figures 4 and 7). Te dependence between final concentration and advertising level is an S-saped function of and wit decreasing p it approaces a linear function: =. Tis similarity indicates a mean field caracter, i.e. lack of spacial fluctuations, of te CAP model. On te contrary, for te CF model MFA results are qualitatively different from tose obtained by Monte Carlo simulations, especially in te low conformity regime. Most importantly, tere is no critical value of witin te Mean Field Approac. Monte Carlo results sow tat, generally, te relationsip between market sare and advertising level is less straigtforward for te CP model due to stronger clusterization caused by a greater role of social validation. Fluctuations witin te CP model were able to produce a critical pase transition in terms of advertising. Below te critical
Outflow Dynamics in Modeling Oligopoly Markets 15 p=.7 p=.86.5.5.5 c.5.5 c.5 p=.92 p=.98.5.5.5 c.5.5 c.5 Figure 8. MFA results for te CF model and p.7 value c (p) (of incumbents advertising) only te entrant survives. Te existence of fluctuations is te reason wy MFA for te CP model does not give as compatible results as for te CAP model. It sould be noticed tat altoug a critical value of advertising does not exist witin te MFA, MFA results still give more complicated dependences between market sare and advertising for te CP model tan for te CAP model. 4.3. Fitting to real data Up till now we ave been investigating general features and differences between CAP and CP models. We ave found tat generally te CF model is more interesting from a statistical pysics point of view. Now we would like to ceck wic model, if any, better describes real data from te Polis telecommunications market. Te results mentioned below refer to computer simulations for a 1 1 square lattice wit periodic boundary conditions. Te initial concentrations c s (),s = 1, 2, 3, were set to te market sares of Plus, Era and Orange, respectively, in terms of income for te end of year 2 (see Section 3.4). Te levels of advertising s (t),s = 1, 2, 3, were set to te respective percentages of total advertising expenditures for te 27 consecutive quarters (t = 1,..., 27), see te bottom rigt panel in Figure 2, wit one important cange. Namely, due to te turmoil around Era (see Section 2.2), Era s advertising levels 2 (t) were decreased in te years 25-26 (t = 17,..., 24) by 1%, a value tat was found to best represent te inefficiency of advertising in te turmoil period [29]. Finally, te time scale ad to be set. We ave decided to take suc a number of time
Outflow Dynamics in Modeling Oligopoly Markets 16 Percent of Total Income 45 4 35 3 25 Plus 22 24 26 28 Percent of Total Income 45 4 35 3 25 Era 22 24 26 28 Percent of Total Income 35 3 25 2 Orange 22 24 26 28 Data CF Model 95% Bounds CAP Model 95% Bounds Figure 9. Market sares in terms of income for te tree operators (Plus, ERA and Orange) and te model generated 95% bounds for conformity level p =.4. Te 95% bounds are obtained as te 97.5% quantile line (upper bound) and te 2.5% quantile line (lower bound) based on 1 simulated trajectories of eac model (CF and CAP). steps τ as to allow eac client to cange operator once every two years (on average): 1 1 7.75 = 4219. Te rationale for tis comes from te fact tat in Poland te 8 2 standard agreement for subscription customers concerns a two year period. To simplify calculations te number 4219 was replaced by 4212, wic is divisible by 27. Simulation results are presented in Figure 9. Te market sares in terms of income for te tree operators (Plus, ERA and Orange) and te CF and CAP model generated 95% bounds are displayed. Te 95% bounds are obtained as te 97.5% quantile line (upper bound) and te 2.5% quantile line (lower bound) based on 1 simulated trajectories of eac model. Te conformity level was set to p =.4. Tis coice is not accidental. For bot models te best fits to real data were obtained for conformity level p (.3,.4). Interestingly, tis agrees very well wit te conformity level found by Asc in is famous social experiment [12]. Looking at te plots we can conclude tat sligtly better fits were obtained for te CF model, partly because te bounds were wider. However, we are fitting teoretical models to only one set of data and, ence, we cannot definitely claim tat te CF model describes reality better tan te CAP model. Obviously, tests wit oter empirical data sets are needed.
Outflow Dynamics in Modeling Oligopoly Markets 17 5. Conclusions In tis paper we ave introduced two models of opinion dynamics in an oligopoly market. We ave applied tem to a situation, were a new entrant callenges two incumbents of te same size. Two forces influencing consumer coice were considered: (local) social interactions and (global) advertising. In te Conformity First (CF) model, conformity but only in te case of unanimity comes in first, ten advertising affects te customers. A different idea of incorporating advertising, borrowed from [28], was used to build te Conformity and Advertising Parallel (CAP) model, were conformity influences individuals parallel to advertising. We ave studied te general beavior of te models: steady states in terms of market sares for te entire space of parameters, as well as calibrating models to real data. For studying te steady states we ave used bot te Mean Field Approac and Monte Carlo Simulations. Tis allowed us to determine te final steady state in terms of market sares as a function of initial market sares c, te level of conformity p and te level of advertising, i.e. = f(c,p,). It occurred tat bot tecniques gave very similar results for te CAP model, wic indicated te lack of spacial fluctuation witin tis model. Moreover, te dependence between te final value of market sare and te level of advertising was relatively simple and for small values of conformity level p it approaced a linear function =. On te contrary, for te CP model MFA gave qualitatively different results from MC simulations. Most importantly, tere was no critical value of advertising level witin te Mean Field Approac, wile suc a criticality was observed for Monte Carlo simulations. Similar results were obtained earlier for a duopoly market [26]. Existence of te critical value is very interesting from te social point of view below a certain critical level of te market approaces a lock-in situation, were one market leader dominates te market and all oter brands disappear. Groot [1] associates suc a market wit a situation wen consumers tend to copy te beavior of teir neigbors (ig conformity level) and want te best value for teir money, wile an open market arises wen consumers ignore wat teir friends buy (low conformity level). In our models te dependence between final market sares and te level of advertising becomes less straigtforward wit increasing conformity. For low conformity levels advertising becomes te main force of market canges. Furtermore, Robertson and Gatignon [3] point out tat incumbents ave an advantage over new entrants, but firms witout a responsive defense strategy may forfeit tat advantage (tis appens in te CF model below te critical advertising level). We ave also calibrated te models to real data from te Polis telecommunications market. Sligtly better fits were obtained for te CF model. However, we are fitting teoretical models to only one set of data and, ence, we cannot definitely claim tat te CF model describes reality better tan te CAP model. Obviously, more empirical tests wit real data are needed. Interestingly, for bot models te best fits to real data were obtained for conformity level p (.3,.4). Tis agrees very well wit te
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