Object-oriented Bayesian networks for a decision support system for antitrust enforcement

The Annals of Applie d Statistics 2013, V ol. 7, No. 2, 714–738 DOI: 10.1214 /12-A OAS625 c  Institute of Mathematical Statistics , 2 013 OBJECT-ORIENTED BA YESIAN NETW ORKS F OR A DECISION SUPPOR T SYSTE M F OR ANTITRUST ENF OR C EMENT 1 By Julia Mor tera, P a ola Vicard a nd Cecilia Vergari Universit` a R oma T r e, Universit` a R oma T r e and Universit` a di Bolo gna W e study an economic decision problem where the actors are tw o firms and t he A ntitrust Authority whose m ain task is to monitor and preven t firms’ potential an ti-comp etitive behaviour and its effect on the mark et. The A ntitrust Authority’s decision pro cess is mo delled using a Ba yesian netw ork where b oth the relational structure and the parameters of the model are estimated from a data set provided by the A uthority itself. A num b er of economic v ariables that influence this d ecision pro cess are also included in the model. W e analyse how monitoring by the Antitrust Authority affects firms’ strategies ab out coop eration. Firms’ strategies are mo delled as a rep eated prisoner’s dilemma using ob ject-orien t ed Bay esian net w ork s. W e show h ow the integ ration of firms’ decis ion pro cess and external market information can b e m o delled in th is wa y . V arious decision scenarios and strategies are illustrated. 1. In tro d uction. Firms in man y cases ha ve incen tiv es to coop erate (col- lude) to increase their profits. The p ossib ilit y for firms to collude do es not dep end solely on their decision b ut also on external circumstances. First of all, firms need to comply with antit rust la ws. If the Antit rust Au thorit y (AA) finds negativ e an ti-competitiv e effects, resulting from firms’ co op era- tiv e b eha viour , it m ay interv ene to pr even t the firms from m erging. The AAs decision pro cess is mo delled here b y u sing a Ba yesia n netw ork (BN) or Probabilistic Exp ert S y s tem (PES) [Co we ll et al. ( 1999 )] estimated from real d ata. A BN is a graphical mo del that enco des the p robabilistic relationships among the v ariables of interest allo wing f or the application of fast general-purp ose algorithms to compute inferences. Often go v ernm en ts may find negativ e anti-c omp etitiv e effects resulting from a merger. As a consequ ence, the decisio n by fi rms to coop erate is Received Novem b er 2011; revised September 2012. 1 Supp orted by a PRIN07 Italian Researc h Grant. Sections 3.1 , 4.1 and 4.3 are due to C. V ergari, the remaining sections are d u e to J. Mortera and P . Vicard. Key wor ds and phr ases. Antitrust A uthority , Ba yesi an netw orks, mergers, mo del inte- gration, prisoner’s dilemma, rep eated games. This is an electronic reprint o f the o riginal article published by the Institute of Mathematical Statistics in The Annals of Applie d St atistics , 2013, V ol. 7, No. 2, 714 –738 . This r eprint differs from the orig ina l in pag ination and t y po graphic detail. 1 2 J. MOR TERA, P . V ICARD AND C. VERGARI actually affected b y the decision p ro cess of the AA. The AA may start an in vesti gation either b ecause t wo fi rms mak e a formal request to merge (explicit collusion) or b ecause th e authorit y susp ects that t wo fir ms are implicitly coll uding. In wh at follo ws the term merger will b e used for b oth explicit and implicit collusion. W e also study ho w the AAs monitoring affects firms’ strategies ab out co- op eration. F or this pur p ose, the firms’ set of p oten tial strategies are mo d elled in turn as a rep eated prisoner’s dilemma u sing ob ject-orien ted Ba y esian net- w orks (OOBNs) [Koller and Pfeffer ( 1997 ), Bangsø and W uillemin ( 20 00 )]. OOBNs are a recen t extension of BNs whic h allo w for a hierarc hical d efini- tion and construction of a BN. T hey pro vid e a compact and intuitiv e repr e- sen tation of th e rep eated prison er ’s dilemma (PD). F urthermore, thanks to the mo dularit y and flexibilit y of this approac h, v arious sou r ces of uncertaint y within the game and generalizat ions of th e rep eated prisoner’s dilemma can b e analysed. W e us e th e PD as a naiv e representati on of firms’ economic in teraction, the f o cus of this p ap er b eing that of analysing the ev olution of firms’ b ehaviour according to v arious external scenarios. F or th eoretical asp ects on su b optimal s trategies in Ba yesian games see, for example, Y oung and Smith ( 1992 ). W e presen t t w o differen t net works: the fi rst mo dels the AAs decision pro- cess, and the s econd represents the b eha viour of the t wo fi rms in a du op oly . OOBNs giv e the graphical framew ork to int egrate these t wo net works and to represen t their time evo lution. Both the graph ical s tructure and the as- so ciated probability tables of AAs decision pro cess n et wo rk are estimated from a r eal data set. As a r esult, w e obtain the estimat ed probability that AA in tervenes to preven t antico mp etitiv e b eh aviour of a merger. F or v arious economic sectors (mark ets of inte r est) we stud y the s en sitivit y of co op era- tiv e outcomes with r esp ect to f actors s uc h as geographical size, marke t share, Herfindahl–Hirsc hman Index (HHI) v ariation, v er tical effec ts, the presence of entry barr iers and buyer p o wer. Th e global OOBN mo del w h ic h inte- grates the AAs decision pro cess with a duop oly mo del is u sed to obtain the optimal d ecision in ligh t of a s er ies of inte resting s cenarios that could o ccur in practice. The outline of th e pap er is as f ollo ws. In Section 2 w e b riefly d escrib e the merger con trol p roblem. W e illustrate the BN for the AAs decision pr o cess estimated from th e data and show its use in v arious scenarios in S ection 3 . A brief introdu ction to the prisoner’s dilemma is illustr ated in Section 4.1 follo w ed b y the Bay esian net w ork represen tation of the PD in Section 4.2 . After in tro ducing the rep eated prisoner’s dilemma in Section 4.3 , in Section 4.4 w e s h o w how this can b e repr esen ted as an OOBN. In S ection 5 we show ho w w e in tegrate the PD net wo r k with the AA net wo r k obtaining a general purp ose global represen tation of the problem, and in Section 5.1 w e app ly this to sev eral decision scenarios. Finally , in Section 6 w e d ra w conclusions and discuss further dev elopments. DECISION SUPPOR T SYSTEM 3 Fig. 1. (a) Pictorial r epr esentat ion of the AA de ci si on pr o c ess and Fi rms’ b ehavior in a Duop oly. (b) Corr esp onding r epr esentat i on for a r ep e ate d sc enario. 2. The merger con trol problem. The AA studies the impact of a merger on the mark et an d its consequences on so cial welfare. Hence, the AAs de- cision affec ts the d ynamics in firms’ eco nomic in teraction as w ell as the corresp ondin g equilibrium outcome. When c ho osing b et w een co op erating or defecting, firms tak e the AAs decision pro cess int o accoun t, b oth when they formally request to merge and in the case of implicit collusion. In our setup, th e actors are as follo ws: the Ant itrust Authority and the t wo merging firms, termed Firm 1 and Firm2 (the duop olists). Figure 1 (a) sho w s a pictorial repr esen tation of th e effects of AAs con trol activit y on Firms’ b ehavio ur. The t wo rounded recta ngles, AA and Duop oly , represent the AAs decision pro cess and the Firms’ merging strategy , r esp ectiv ely . The AAs decision pro cess is mo delled b y a Ba yesia n n et work learned from r eal data (see Sections 3.2 and 3.3 ). The du op oly is mo delled as a P D usin g a Ba y esian net w ork for decision making (see Section 4 ). The tw o net works are then int egrated giving rise to a global mo del, w here b oth the AAs d ecision pro cess and the du op oly are represen ted b y OOBNs. Figure 1 (a) represent s a single stage (v ertical slice) of the ov erall mo d el. Th e merger p roblem, as well as AAs activit y , ev olve in time. Figure 1 (b) giv es a graphical representa tion of th e decision pro cess dyn amics. Details on these net works are give n in Sections 4 and 5 . 3. An titrust Authorit y’s decision pro cess. 3.1. Curr ent pr actic e. The p rimary task of the AA is to enforce the an- titrust la w whic h prohibits an ticomp etitiv e b eha viour , so as to p r ev ent a reduction in so cial w elfare. 2 In p articular, the AA is resp ons ib le for de- tecting the follo w ing: (a) agreeme n ts restricting comp etition; (b ) abu ses of dominan t p ositions; (c) merger op erations in volving the creation or strength- ening of dominan t p ositions in w ays that eliminate or substanti ally reduce comp etition. 2 F or details on the Italian antitrust la w and AA s tasks see: http://www .agcm.it/ en . 4 J. MOR TERA, P . V ICARD AND C. VERGARI T able 1 Description of the variables in the AA network V ariable States Description Y ears { 1991–1 996, 1997–2000 , Reference p erio ds 2001–20 03 } A TEC O Mining, fo o d & beverage Relev ant m arket Man ufacture, etc. (see Figure 3 ) Geo size { Sub-national, n ational, Size of the relev ant market supra-national } Buyer p ow er { Y es, No } Presence (Y es) of comp etitive pressure on the merging p arties Entry b arriers { Y es, No } Presence (Y es) of entry barriers HHI va riations { 0, (0 , 100), [100, 500), V ariati on in mark et [500, 1000), ≥ 1000 } concentrati on index P ost market share { < 20%, [20%–40%], > 40% } P ost-merger market share V ertical effects { Y es, No } Presence (Y es) of vertical effects AA interv entio n { 0, 1 } No (0)/Y es (1) Once the Authorit y has receiv ed a complain t or h as collected informa- tion on p ossible inte r ference with comp etition, a p reliminary examination is carried out and if there are alleged violations of the Antitrust la w, the AA carries out a full in vestig ation. T he la w requires th at wh enev er the p o- ten tially merging firm s exhibit sale reve n u es in excess of certain predefined thresholds, the merger op eration must b e notified to the authorit y in ad- v ance. The thresholds are up dated annually acco rding to the deflator index for gross domestic pro duct. Decisions on a merger are based on a case by case examination and, to our kn o wledge, currently , no sp ecific mo d els are u sed. Th e la w also do es not giv e an y sp ecific thresholds for relev ant v ariables, suc h as mark et share or a mark et concen tration index. 3.1.1. The data. The data we use were collect ed b y the Italian Anti trust Authorit y and concern all the cases examined from 1991 to 2003. T his data set consists of 6920 observ ations. Based on this data set, La No ce et al. ( 2006 ) dev elop ed a logit mo del to analyse the impact of differen t factors on the Authorit y decision. F ollo wing La No ce et al. ( 2006 ), w e consider rele- v an t m ark ets affected by the merger as elemen tary units of analysis. These mark ets are denoted by the IST A T (Italian National In s titute of Statistics) economic activit y co d e A TECO. T able 1 describ es the v ariables in the data set that were used to estimate the AA net w ork. The Herfin dahl–Hirsc hm an Index, HHI, is defi ned as the sum of the squares of n firms’ mark et share, P n i α i 2 , where α i denotes DECISION SUPPOR T SYSTEM 5 Fig. 2. L o gic al c onstr aints f or AA network estimation. firm i ’s mark et share and P n i α i = 100. Increase in the HHI indicate s a decrease in comp etition and an increase in marke t p o we r. V ertical effects refer to the anti comp etitiv e effects that a vertica l merger could imply , that is, the p ossibility to raise entry barriers b y inp ut foreclosure or b y customer foreclosure. The estimatio n (learning) pro cess of a Ba ye sian net wo r k consists of t wo phases: the graphical stru cture estimatio n and the conditional probabilit y table estimation. These will b e illus tr ated in turn. 3.2. Estimation of the network’s gr aph i c al structur e. The graphical s truc- ture of the AA net work represen ting th e AA decision pr o cess is obtained by a com bin ation of sub ject-matter kno wledge, pro vided by a domain exp ert, and the information in th e d ata. The Ne c essary Path Condition (NPC) algorithm [Stec k ( 200 1 )] imple- men ted in Hu gin is used to estimate the graph ical stru ctur e of the net wo r k. The NPC is a constraint- based algorithm recursiv ely testing marginal and conditional asso ciation b et wee n categorica l v ariables. The NPC algorithm allo ws the user to choose the m ost suitable among indep endence equiv alen t mo dels. The NPC algorithm tak es in to accoun t logical constr aints, su c h as presence/absence of a link or assignment/ban of a sp ecific dir ection b et w een v ariables. The logica l constrain ts we implemen ted h ere are sh o wn in Figure 2 . These imply that if there is a relation b et w een tw o v ariables in differen t b o xes, it must h a ve the same direction as that in Figure 2 . F ur thermore, if t wo v ariables b elong to the same b ox, their asso ciation (if it exists) can b e in any one of the t wo p ossible directions. F or example, if no de AA Interven tion 3 is connected with an y of the other v ariables, th e direction has to b e from these in to AA Interven tion no de (AA decision logically d ep ends on the v alues of the other v ariables). Th is means that arrows from AA Interven tion to any other v ariable are logically prohib ited. The reference p erio d (no de Years ) is not influenced b y any of the other v ariables in the mo del. The dep enden ce stru cture—based on the logical constraints giv en in Fig- ure 2 —learn t from the data is shown in Figure 3 . The main d ep endence relationships estimated from the data are as follo ws: 3 Here w e indicate nodes in teletype . 6 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 3. AA network showing the dep endencies of AA Intervent ion on the r elevant vari- ables describing the market and the mar ginal pr ob abilities of the variables. (i) The market of in terest ( ATEC O ) can dep end on Year : an economic sector could b e more r elev ant and w orth in vesti gating du ring one of the three reference p erio ds (note that the president of the AA change d in 1997 and from 2001 I talian currency Lira w as replaced b y the Euro). (ii) AA Interven tion dep ends directly on HHI Variation , Vertical Effects , P ost M arket Sh are , Geo Size and Entr y Barrie rs . F urthermore, the relev an t market ( ATEC O ) do es not affect AAs decision ( AA Interventi on ) directly but only thr ough the relev ant features of the m arket and of the merg- ing firms ( HHI Variat ion , Vert ical Effect s , Pos t M arket Sh are , Geo S ize and Entry Barriers ). These r esu lts are consisten t with those in Bergman, Jak obsson and Razo ( 2005 ) and La No ce et al. ( 2006 ). (iii) The Herfindahl–Hirsc hman concen tration index v ariation ( HHI Variatio n ) d ep ends on all the v ariables that logically precede it or are on an equal fo oting (as shown in Figure 2 ), wh ereas Post Mark et Share dep end s only on Entry Barriers , Geo Size and A TECO . An explanation of this could b e that when a mark et sector is c haracterised b y ent ry barriers (b ecause of p aten ts or incr easing returns to scale) w e exp ect that this m ark et DECISION SUPPOR T SYSTEM 7 ma y b e comp osed of a f ew firms with high mark et shares, th us influ encing Post Market Share and a relev an t H HI Variation . Man y other conditional indep endencies can b e read off the AA net wo rk in Figure 3 , but for brevit y they will not b e presen ted h ere. 3.3. Estimation of the pr ob ability tables. T o complete the construction of our mo d el, w e estimate the conditional probabilit y distributions of the v ariables from the data. Th e EM-algorithm [Dempster, Laird and Ru bin ( 1977 )] is used for learning the p robabilities. The net work in Figure 3 exhibits a complex asso ciation str u cture among the v ariables. F or example, no d e HHI vari ation has sev en paren ts. Its con- ditional p r obabilit y table h as 17 × 3 3 × 2 3 × 5 = 18, 360 entries corresp onding to the state sp ace of its parent v ariables: ATECO , Post Ma rket Share , Ye ars , Geo Size , Entry Barrie rs , Buye r P ower , Vert ical E ffects , as well as HHI Variatio n ’s s tate sp ace. Many of these com bin ations are not represen ted in the data set, although they cannot b e considered imp ossible ex ante . In fact, according to Bergman, Jak obss on and Razo ( 2005 ), if a threshold for relev an t v ariables—lik e p ost mark et sh are—can b e detected in AAs legal practice, this thresh old may v ary according to other v ariables, s u c h as buy er p o wer and entry barriers. Therefore, no v ariable lev el com bin ations can in principle b e ru led out. So, in ord er to av oid that certain p ossible configurations in the conditional probability tables ha v e zero pr obabilit y , w e set noninformative nonzero prior probabilities. Figure 3 displays the marginal probab ilities 4 estimated from our data. Note, for example, that the probability of an AA inte rv ention is only 0 . 018 9 , whic h could b e d ue to the fact th at in most case s, 74 . 38%, the p ost market share is less than 20% and ent ry barr iers and vertic al effects are absent (with probabilit y 0 . 979 3 and 0 . 926 8, resp.), HHI ind ex is less than 100 in 87 . 85% of the cases and only in 15 . 38% the geographical size is sup ra-national. 3.4. Using the network. Once the mo del has b een estimated, w e can ad- dress a n um b er of qu estions ab out the AAs d ecision pr o cess. V arious p ossible scenarios can b e examined by inserting and propagating the appropr iate ev- idence throughout the net work. W e illustrate three hyp othetical scenarios. Sc enario A. What is the probability of an AA interv en tion in a merger request w hen there are en try barriers in the mark et? This scenario is rep- resen ted in Figure 4 (a ). The p osterior pr ob ab ility of an A A Inte rvention increases from 0 . 018 9 to 0 . 5790 when the evidence Entry Barr iers = Y es is inserted and propagated throughout the net wo r k. Sc enario B. How would the probabilit y ob tained in S cenario A change if the Herfin dahl–Hirsc hm an concen tration index v ariation ( HHI variat ion ) 4 In all figures probabilities are exp ressed as p ercentages. 8 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 4. Sc enarios (a) , (b) and (c) giving mar ginal p osterior pr ob abilities for the AA network. is in the class [10 0 , 500)? Note in Figure 4 (b) that the p robabilit y of AA Interven tion no w increases to 0 . 7 741. The n et wo rk can b e used not only f or direct r easoning ab out the p roba- bilit y of AA Interventio n , bu t also for reasoning ab out p ossible “causes” of a giv en AA decision. Sc enario C. A question ab out comp etition auth orities’ b ehavio ur that has b een rarely addressed in the literature is ab out the t yp e of mergers that are t ypically p r ohibited [Bergman, Jakobsson and Razo ( 2005 )]. O ur net work can b e used for this pur p ose. Su pp ose that the AA decides to interv ene in a fi rm’s merger r equest. What are the most plausible reasons of this decision? Figure 4 (c) giv es the p osterior pr obabilities giv en the evidence that AA Inter vention is equal to one. On comparing Figures 3 and 4 (c) we see that: • The probabilit y of entry barr iers increases from 0.0207 to 0.6367; • The probabilit y of ve rtical effects increases from 0.0732 to 0.4536 . T his is an in teresting result, since, although there is common agreemen t ab out the relev ance of v ertical effects f or AAs decision on a mer ger request, it is DECISION SUPPOR T SYSTEM 9 T able 2 Payoff matrix for the prisoner’s dilemma Firm2 C D Firm1 C a, a c, d D d, c b, b con trov ersial whether v ertical effects influence th e market negativ ely b y foreclosing comp etitors or p ositiv ely b y reducing transaction costs. Here w e find that the pr esence of vertic al effects is m u c h more probable for those fir ms where AA decides to in terv ene. La No ce et al. ( 2006 ) foun d similar results. • The pr obabilit y of p ost marke t s h are less than 20% decreases fr om 0.7438 to 0.0922, wh ereas the probabilit y of p ost market share greater than 40% increases from 0.077 7 to 0.700 6. • The HHI ind ex decreases in the fir st tw o classes and increases in the last three classes. Note that when evidence is propagated in the netw ork, all marginal proba- bilit y tables are u p dated acc ordingly . 4. Duop oly represen tation. 4.1. The prisoner’s dilemma. The prisoner’s dilemma [Flo o d ( 1958 )] de- scrib es co op eration b y rational agen ts. The PD is a 2-pla ye r s ymmetric game where the t wo pla y ers ha ve th e same rˆ ole and ha v e th e same set of p oten- tial strategies termed c o op er ate C and defe ct D . The PD is a s imultaneous game where the pla yers choose just once and sim ultaneously and the unique equilibrium 5 is the pair of strategie s ( D , D ). Pla y ers ’ pa y offs are suc h that defect is a dominan t strategy , that is, a strateg y that is preferred by eac h pla yer indep end en tly of his/her riv al. The problem is that th is strategy is in- efficien t since b oth pla yers wo u ld gain more if they co op erated and adopted the ( C , C ) strategy . Th e source of the dilemma lies in the fact th at eac h pla yer has an incen tiv e to d efect if the riv al play er coop erates, so that an agreemen t to co op erate w ould not b e credible. Sim ultaneous games, su ch as the PD, are commonly r ep resen ted in either the n ormal or the extensiv e f orm. I n the normal f orm repr esen tation, the PD can b e describ ed by the pa yo ff matrix in T able 2 . The t wo firms, Firm1 and Firm2, h a ve t wo a v ailable strategie s : co op erate C or d efect D . The pa yoffs 5 An equilibrium is a strategy pair such that no play er can improv e his p osition by unilaterally c han ging his decision. I n other wo rds, it is a situation in whic h all pla yers choose mutual b est resp onses. 10 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 5. (a) T r e e r epr esentation of the simultane ous duop oly game. (b) Corr esp onding Bayesian network r epr esentation. need to b e such that d > a > b ≥ c and 2 a > ( c + d ) > 2 b , so th at ( C, C ) maximises pla yers’ join t p a yo ff. Give n that b < a , the strategy pair ( D , D ) is strictly w orse than ( C, C ). In the extensiv e form the game is repr esen ted by a tree. Figure 5 (a) sho ws the tree r epresen tation (equiv alent to T able 2 ) of the simulta n eous duop oly game. Firm1 mo v es firs t and chooses either C or D , Firm2 mo v es second but without kno wing wh at Firm1 d id. A symmetric duop oly , s u c h as a mark et with tw o symmetric p rofit-maxi- mising firms in mutual comp etition, can b e mo d elled as a PD. The d uop oly profit is the gain of eac h of the sellers in this market. Supp ose the tw o firms pro duce iden tical go o ds, incurring constant margi- nal costs, and th ey comp ete setting their prices. Since consu mers will buy from the fir m charging the lo we st price, fi rms ha v e an incen tiv e to u n dercut their price to conquer the mark et (nonco op erativ e or defect strategy). At equilibrium firms w ill set th e competitiv e p rice (the market p rice under DECISION SUPPOR T SYSTEM 11 p erfect comp etition whic h is equal to fi rm’s marginal cost of pro d u ction), gaining duop oly profit b = 0. T his result is often called a paradox, since there are just t wo fir ms in the market and still the p er f ectly comp etitiv e strategy yields zero profit. Ho wev er, if fir ms d ecide to co op erate and set the monop oly p rice, they can share p ositiv e monop oly profits. The monop oly profit is alw a ys greate r than t wice the duop oly profit, 2 a > 2 b . In most mark ets, from a consum er ’s p oin t of view, goo ds are not id en tical. This giv es fir ms the abilit y to raise the p rice ab o v e the m arginal cost of pro du ction without losing their customers to comp etitors. In a symmetric duop oly with pro du ct d ifferen tiation fir ms pro duce and sell different iated go o ds (imp erfect su bstitutes). As long as pro duct different iation is not to o large, firms face a P D: if they coop er ate, they could share monop oly profit, but they ha ve incen tiv e to defect if the riv al co op erates. Ho wev er, when go o ds are imp erfect substitutes, fi rms mak e p ositiv e duop oly profit, b > 0, under the nonco op erativ e s tr ategy p air ( D , D ). This du op oly profi t is smaller than half the monop oly profit, b < a , so that the co op erative strategy C is sup er ior for eac h fir m sin gly . 4.2. The prisoner’s dilemma network. Ba ye sian n et works for d ecision supp ort sys tems can incorp orate b oth decision no des and u tilit y no des [Jensen ( 2001 )], giving r ise to an influ ence diagram (ID) representa tion. IDs w ere extended b y Lauritzen and Nilsson ( 2001 ) to allo w for limited in formation decision p roblems (LIMIDs). A different approac h to r ep resen t and solv e games u sing grap h ical m o dels wa s initially prop osed b y Smith ( 1996 ) and later by La Mura ( 2000 ), Kearns, Littman and S ingh ( 2001 ) and K oller and Milc h ( 2003 ). The one stage PD b eing a symmetric game can b e represent ed b y the ID netw ork in Figure 5 (b). The simulta neit y of the game is implemen ted b y represen tin g Firm1 as a random v ariable (o v al no de) and Firm2 as the decision mak er (rectangular n o de) ha ving tw o p ossible actions: defect D and co op erate C . Firm2’s decision is in fluenced by Firm1. Firm1’s asso ciated prior probabilit y d istribution represen ts Firm2’s s ub jectiv e opinion ab out Firm1’s b ehavio ur. Random v ariable Firm1 has t wo states, defect (coded as 0) an d co op erate (cod ed as 1), w ith u niform p rior pr ob ab ilities indicating Firm2’s ignorance ab out Firm1’s c h oice. Firm2 could assign differen t pr ior probabilities b ased on h is/her prior kn o wledge ab out Firm1’s b eha viour . T able 3 sho ws Firms2’s u tilit y [no d e Firm2’s ut ility U2 in Figure 5 (b)] based on Firm1 and Firm 2’s actions. Thanks to game symmetry , T able 3 is equiv alen t to the normal form pa yoff matrix give n in T able 2 . Once th e n etw ork is compiled, the optimal decision for Firm2 is automati- cally computed b y maximising exp ected utilit y . Since the game is symmetric, Firm2’s optimal strategy coi ncides with Firm1’s optimal strategy and this pair of strategies constitutes a Nash equilibrium. Thus, in the ID represen- tation the c hoice of Firm2 as decision mak er is without loss of generalit y . 12 J. MOR TERA, P . V ICARD AND C. VERGARI T able 3 Firm2’s utility U2 c onditional on Firm1 and Firm2 ’s actions Firm1 Defect (0) Coop erate (1) Firm2 Defect (0) Coop erate (1) Defect (0) Co op erate (1) U2 b c d a In what follo ws w e alw a ys consider Firm2 as the decision maker. The prior p robabilit y distrib u tion on the r andom v ariable Firm1 r eflects Firm2’s sub jectiv e opinion on the type of riv al pla yer h e/she is pla ying against. 4.3. R ep e ate d prisoner ’s dilemma. Since firms interact more th an once, w e need to consider the rep eated ve rsion of the PD. In rep eated games, pla yers’ actions are observ ed at th e end of eac h p erio d and their o verall pay off is the sum of the pa y offs in eac h stage d iscoun ted by a factor δ ∈ [0 , 1]. Th u s, pla yers ma y condition th eir pla y on the opp onen ts past pla y . Here we assume that firms nev er forget previous mo ves and other information acquired, in other w ord s, w e assume that firms ha ve p erfect recall. The rep eated P D analyzes how threats and pr omises ab out fu ture b e- ha viour can affect and impro ve current b eha viour. When the time h orizon is in definite fir ms may decide to adopt a co op erativ e strategy wh ere the discoun t facto r δ represents uncertain ty ab out the n u mb er of s tages faced b y firms. This uncertain t y is usually not mo delled within the game itself. In S ection 5 we illustrate ho w to incorp orate this un certain t y in the m erger con trol p roblem. 4.4. OOBN for r ep e ate d priso ner’s dilemma. Generalising the tree rep- resen tation in Figure 5 (a) to rep eated ga mes is b oth compu tationally and graphically demand ing. The game tree grows exp onentia lly with the n u m- b er of stages. F or example, Figure 6 (a) sho ws th e tree representa tion of a t wo- stage PD. OOBNs are particularly w ell su ited for an app licatio n area su ch as th e present b ecause the similarit y b et ween net work elemen ts (the stage s of th e game) can b e exploited in a mo dular and flexible constru ction. Ob ject- orien ted Ba y esian net wo rks ha ve a hierarc h ical structure where a no de itself can represent a (ob ject-orien ted) net wo rk cont aining sev eral instanc es of other generic classes of n et works. Instances ha ve interface input and output no des as well as ordinary no des. In stances of a particular class h a ve ident ical conditional p robabilit y tables for n oninput no des. I nstances are conn ected b y arr o ws fr om output no d es into inpu t no des. These arro ws, as well as those from ord inary no des to inp ut n o des, r epresen t iden tity links, whereas arro ws b et we en t w o ordinary nod es or an output no de and an ordinary no de repr esen t p robabilistic dep enden ce. Th e grap h ical simplicit y automat- DECISION SUPPOR T SYSTEM 13 Fig. 6. (a) T r e e r epr esentation of the two-stage duop oly game. (b) Corr esp onding OOBN r epr esentat ion. ically pro duces computational efficiency . As a r esult, incr easingly complex net works can b e constructed by simp ly adding new ob jects whic h p erform differen t tasks. Since w e assum e p erfect r ecall, Hu gin 6 v ersion 6.9 soft w are, whic h au- tomatica lly implements th e fact that at ev ery stage the decision m ak er re- calls all previous decisions, is used to build the net works. This implies that eac h d ecision dep ends on the decisions tak en in all previous stages, so ev en though the graphical representat ion do es not implicitly repr esen t this, in the jun ction tree construction [Co w ell et al. ( 1999 )] these d ep endencies are explicitly considered. In wh at follo ws w e ind icate an instance in b old . Fig- ure 6 (b) sh ows the O OBN t wo-sta ge r ep eated game that co rresp ond s to the tree representat ion in Figure 6 (a). Each roun d ed rectangle r epresen ts an in stance termed Duop oly and mo dels a stage of the rep eated game . In order to sp ecify the links b et ween successiv e stages (instances), Figure 5 (b) 6 www.h ugin.com . 14 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 7. C l ass network for r ep e ate d PD with asso ciate d mar ginal prior pr ob ability tables. (whic h repr esen ts eac h Duop oly instance) needs to b e generalised as sho wn in Figure 7 . The no d e Firm 1 ∗ mo dels the b eha viour of Firm1 in th e next stage. In eac h stage the game can either con tinue or terminate. F irm1 and Firm1 ∗ no w need to b e giv en thr ee states: defect (0), coop erate (1) and stop (2). Since in a rep eated game every stage dep en ds on the actions tak en in the pre- vious stages, Firm1 ∗ is logically dep end en t on Firm2 . Uncertain ty ab out the existence of further stages is mo delled b y addin g a new rand om n o de stop? . No de st op? has tw o states, { 0 , 1 } according to wh ether the game con tinues or stops and has a Bernoulli distribution Bin(1 , 1 − delta ). The parameter no de delta is th e p robabilit y th at the game con tinues P ( stop? = 0). No de delta has a u niform prior distribution o v er a plausible set of v alues. In the firs t s tage, to ensure that the game starts, Firm1 can only c h o ose b et w een defect and co op erate. T able 4 giv es the conditional probabilit y dis- tribution of Fi rm1 ∗ giv en stop? and Firm2 . It shows that if the game stops ( stop? = 1), Fi rm1 ∗ stops with certain t y , else Fi rm1 ∗ co op erates or defects according to Firm2 ’s decision. This implemen ts the tit for tat (T FT ) s tr at- T able 4 Conditional pr ob ability table for Firm1 ∗ given stop? and Firm2 Stop? No (0) Y es (1) Firm2 Defect (0) Coop erate (1) Defect (0) Coop erate (1) Defect (0) 1 0 0 0 Coop erate (1) 0 1 0 0 Stop (2) 0 0 1 1 DECISION SUPPOR T SYSTEM 15 Fig. 8. Gener alise d r ep e ate d PD network r epr esenting various str ate gies and i nc omplete information. egy in whic h Firm1 b egins by co op erating and coop erates as long as Firm2 co op erates, and defects otherw ise. V ariatio ns on this strategy will b e sh o wn in Section 4.4.1 . 4.4.1. Other str ate gies. Exp erimen tal results sho w that p eople, contrary to stand ard p rescriptions of game theory , may co op erate more f requen tly than exp ected [And reoni and Miller ( 1993 )]. An explanation b ehind this empirical evidence is pro vid ed b y the theoretical mo dels of Kreps and Wilson ( 1982 ) and Kr eps et al. ( 1982 ). Figure 7 can b e mo dified to provide a general class netw ork that explicitly incorp orates a set of p otenti al strategies for Firm1 other than TFT. This net work is displa yed in Figure 8 . T he net work can, for example, mo del a rep eated PD with in complete information, that is, where there is u ncertain ty ab out the t yp e of riv al that a firm is going to face. The conditional probabilit y distribution of Firm1 ∗ reflects Firm2’s uncertain ty ab out its opp onent. If Firm 2 b eliev es Firm1 to b e “altruistic”, it can exp ect Firm1 to co op erate, with probabilit y α D > 0, even if it defected in the pr evious stage. On th e other hand, if Firm2 b eliev es Firm1 to b e “egoisti c”, then it exp ects Firm1 to co op erate, with probabilit y α C < 1 , ev en if it co op erated in th e p revious stage. Additional n o des, F irm1 ∗ |D and Fi rm1 ∗ |C , h aving Bernoulli distribu - tions with parameter n o des alpha D and al pha C are added to the net w ork of Figure 7 . No de Fi rm1 ∗ tak es v alue 2 if the game stops in the curr ent stage, wh er eas if the game con tin ues ( stop? = 0), the v alue of Firm 1 ∗ de- p ends on that of Firm2 . If Firm2 defects (co op erates), Firm1 ∗ is Firm1 ∗ |D ( Firm1 ∗ |C ), with alpha D ( alpha C ) b eing the probabilit y that Firm1 will co op erate in the next s tage giv en that Firm2 defected (co op erated) in the previous stage. T he conditional pr ob ab ility distribution of Firm1 ∗ is th us 16 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 9. Inte gr ate d AA-duop oly mer ger stage game. defined by the logical expression if ( sto p == 1 , 2 , i f ( Firm2 == 0 , Firm 1 ∗ |D , Firm1 ∗ |C )). 7 Firm1 ∗ represent s Firm2’s sub jectiv e opinions ab out Firm1’s b ehavi our in eac h single stage of the rep eated game. This mod el can also incorp orate a large set of strategies, including T FT , and it can mo del s cenarios where the probabilit y that the game con tinues dep end s on external factors. An illustrativ e example is given in Section 5 . 5. Global net wo r k. Th an k s to th e m o dularit y and flexibility of OOBNs, it is p ossible to integrate the AA an d the Duop oly net works, giving rise to a u nique ov erall OOBN represen tation of the p r oblem [Figure 1 (a)]. An expanded representa tion of this mod el is s h o wn in Figure 9 . 7 The function if ( A, x, y ) takes v alue x if condition A is satisfied, otherwise y . DECISION SUPPOR T SYSTEM 17 Fig. 10. OOBN r epr esenting a thr e e-stage r ep e ate d mer ger game with unc ertainty ab out the numb er of stages. The Duop oly net work (the b ottom n et wo rk) in Figure 9 is s imilar to the net work in Figure 8 except that the u n certain t y ab out the next stage stop? is no w identified w ith AA I nterventi on in the A A net wo rk (the top net work in Figure 9 ) r epresen ting AAs decision p ro cess. The AA d ecision pro cess is usually d ynamic; it can c hange o v er time due to c hanges in the antit rust la w as w ell as c hanges in mark et conditions. W e are th u s intereste d in the rep eated v ersion of the mo d el in Figure 9 . Figure 10 represents the global mo del (Figure 9 ) rep eated four times for a thr ee-stage merger game with u ncertain ty on the num b er of stages. In general, an O OBN with n + 1 instances mo d els a game rep eated n times with uncertaint y ab out the successiv e stage. In th is mo del, the AAs d ecision pro cess is represen ted by th e same instance in eac h p eriod . Th is is ju stified b y assumin g that, ev en if the AA decides not to in tervene, it con tinues monitoring firms’ b ehavio ur in successiv e stages. 5.1. Firms’ str ate gy. W e n o w stud y the sens itivit y of co op erativ e b e- ha viour with resp ect to t w o sets of utilities and all the f actors that might directly or ind irectly influence the AAs d ecision. W e consider b oth the TFT strategy and a more general s tr ategy . The TFT strategy can b e implemen ted using the glo bal net work b y setting Firm1 ∗ = 1 in stage Duop oly 1 and Firm1 ∗ |C = 1, Firm1 ∗ |D = 0 in all other stages. 18 J. MOR TERA, P . V ICARD AND C. VERGARI T able 5 Firm2’s utility U2 for a market with p erfe ct substitutability Firm1 Defect (0 ) Coop erate (1) Firm2 Defect (0) Co op erate (1) Defect (0) Coop erate (1) U2 0 − 10 150 100 5.1.1. TFT str ate gy: Perfe ct substitutability. T able 5 sho ws an example of Firm2’s utilit y for a mark et with p erfect s ubstitutable go o ds. Figures 11 , 12 and 13 sh o w the marginal pr obabilities f or a selectio n of random v ariables and the exp ected utilities f or the d ecision no des in the first stage AA 1 and Duop oly 1 . Fig. 11. Mar ginal pr ob abil ities and optimal de cision in the first stage AA 1 and Duop oly 1 , under p erfe ct substitutability, when Fi rm1 pl ays TFT. DECISION SUPPOR T SYSTEM 19 Fig. 12. Mar ginal pr ob abil ities and optimal de cision in the first stage AA 1 and Duop oly 1 , under p erfe ct substitutability, when Firm1 plays TFT, Entry Barrier s = Y es and HHI Variatio n > = 1000. When no evidence ab out the v ariables in the market is inserted in the net work (Figure 11 ) Firm2’s optimal decision is to co op erate (1), ha ving exp ected utilit y equal to 443 .40 (while defect has exp ected utilit y equal to 385.47). This could b e in part du e to the small pr obabilit y of an AA in terven tion, 0.0189. Figure 12 sho w s the case where there are entry barriers in the mark et of in terest ( Entry Barriers = Y es) and the merger causes the HHI v ariation to b e in the last class ( HH I Variat ion > = 1000). The resulting p robabilit y of AA in terven tion sho ots up to 0.94 35 and Firm 2’s optimal decision is to defect with exp ected utilit y of 394.72, against 350.93 for co op erating. This strategy still remains optimal (although with a smaller gap b et wee n the exp ected utilities) when based only on the p resence of en try barriers . 20 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 13. Mar ginal pr ob abil ities and optimal de cision in the first stage AA 1 and Duop oly 1 , under p erfe ct substitutability, when Firm1 plays TFT, Entry Barriers = Y es, HHI Variatio n > = 1000 and Buyer Power = Y es. Figure 13 sh o ws the case where, as b efore, there are entry barriers, the HHI v ariation is ≥ 1000, and customers exert comp etitiv e p ressure on the merging parties ( Buyer Power = Y es). The probabilit y of AA in terv en tion decreases from 0.943 5 to 0.2915 and Firm 2’s optimal decision is to co op erate, ha ving exp ected utilit y of 416.1 4. It is in teresting to n ote that bu y er p o we r is able to counterbalance th e effect of b oth en try barriers an d a large HHI v ariation. 5.1.2. TFT str ate gy: Imp erfe ct substitutability. W e n o w use Firm2’s util- it y for a market with imp erfect sub stitutable goo ds giv en in T able 6 . Figure 14 sho ws r esu lts when evidence ab out the market is n ot a v ailable. Firm2 ’s optimal decision is to co op er ate (1), having exp ected utilit y equal DECISION SUPPOR T SYSTEM 21 T able 6 Firm2’s utility U2 for a market with imp erfe ct substitutability Firm1 Defect Co op erate Firm2 Defect Coop erate Defect Coop e rate U2 100 5 0 160 150 to 601.33 (while defect has exp ected u tilit y equal to 513 .21). Aga in, this is most plausibly due to the small pr obabilit y of an AA in terven tion. When Entr y Barriers = Y es and HHI Varia tion > = 1000 , Firm2’s ex- p ected u tility to co op erate or to defect is almost equal, although th e prob- abilit y of AA int erv entio n is close to 1 (Figure 15 ). Fig. 14. Mar ginal pr ob abil ities and optimal de cision in the first stage AA 1 and Duop oly 1 , under i mp erfe ct substitutability, when Fi rm1 pl ays TFT. 22 J. MOR TERA, P . V ICARD AND C. VERGARI Fig. 15. Mar ginal pr ob abilities and optimal de cision in the first stage AA 1 and Du op oly 1 , under imp erfe ct substitutability, when Firm1 plays TFT, Entry Barriers = Y es and HHI Variation > = 1000. F urthermore, in con tr ast to p erfect su bstitutabilit y , account ing for the presence of en try barriers alo ne is not su fficien t to mo d ify the optimal deci- sion from co op erate to d efect. The main r eason b eing that when the firms’ pro du cts are imp erfect s ubstitutes, the set of utilities reflects the f act that the defect strategy do es not corresp ond to such a strong pu nishment, so that a firm can contin ue to co op er ate even if ther e is high r isk that the game migh t stop. 5.1.3. Inc omplete information. Assu me that Firm2 has incomplete in- formation ab out the t yp e of riv al it is going to f ace. This is a reasonable scenario, as firms are likel y to b e uncertain ab out their r iv als’ costs and b enefits from co op eration. T able 7 shows Firm2’s exp ected u tilit y in case of p erf ect substitutabil- it y (based on Firm2’s utilit y given in T able 5 ) for different probability DECISION SUPPOR T SYSTEM 23 T able 7 Firm2’s exp e cte d utili ty for differ ent values of α C and α D , without evidenc e, with evidenc e E 1 and E 2 , for likeliho o d evidenc e and for the TFT str ate gy Without evidence With evidence E 1 With evidence E 2 α C α D E [ u ( D )] E [ u ( C )] E [ u ( D ) | E 1 ] E [ u ( C ) | E 1 ] E [ u ( D ) | E 2 ] E [ u ( C ) | E 2 ] 1 0 . 25 337 388 329 339 322 298 0.8 0 . 25 286 316 278 277 271 24 5 0.6 0 . 25 238 250 228 219 220 19 3 0.4 0 . 25 203 193 190 170 180 152 1 0 . 2 332 388 326 339 321 298 0.8 0 . 2 281 316 275 277 270 245 0.6 0 . 2 231 247 225 217 219 19 3 0.4 0 . 2 192 188 183 167 177 149 1 0 . 1 321 388 321 339 320 298 0.8 0 . 1 270 316 270 277 269 245 0.6 0 . 1 219 243 219 215 218 19 3 0.4 0 . 1 172 179 171 159 170 14 3 Likel ihoo d 280 313 273 275 268 243 TFT 385 443 390 394 395 353 v alues of α C and α D (no des alpha C and alp ha D in Figure 9 ). Th ree t yp es of information ab out the r elev an t market are considered : n o evi- dence, eviden ce E 1 = { Post Market Share ≥ 40%, Entry Barriers = Y es and Bu yer Power = Y es } and evidence E 2 = { Entry Ba rriers = Y es and HHI Variatio n ∈ [500–1000] } . The optimal decision yielding the highest ex- p ected utilit y for eac h scenario is italic ised. The second last row of T able 7 give s the resu lts w hen inserting a uniform lik eliho o d function for α C > 0 . 5 and α D < 0 . 5. I n th is case, Firm2’s optimal decision is to coop erate under no evidence and E 1 . Wh er eas, for E 2 , w hen the probabilit y of AA int erv entio n is close to one, E [ u ( D ) | E 2 ] > E [ u ( C ) | E 2 ], so Firm2’s optimal d ecision is to defect. These results coi ncide with those obtained u sing the TFT strategy shown in the last ro w of T able 7 . Reca ll that the TFT s tr ategy corresp ond s to sett ing α C = 1 and α D = 0 in all Duop oly instances. No w, supp ose Firm2 b eliev es that its riv al co op erates—with probabil- it y α C = 0 . 8—if Firm2 co op erates; and co op erates—with probabilit y α D = 0 . 25—ev en if Firm2 defects. This is imp lemen ted in the net work insert- ing and propagating evidence alpha C = 0.8 and a lpha D = 0.25 in eac h Duop oly instance. As w e can see in T able 7 , Firm 2’s exp ected utilit y to co op erate, E [ u ( C )] = 316, is greater than to defect, E [ u ( D )] = 286. Int ro- ducing evidence E 1 in AA 1 , the tw o decisions b ecome almost utilit y equiv- 24 J. MOR TERA, P . V ICARD AND C. VERGARI alen t. Whereas, under the T FT strategy , E 1 yields an optimal decision to co op erate E [ u ( C ) | E 1 ] = 394, whereas E [ u ( D ) | E 1 ] = 390. Recall that when information ab out the relev an t mark et is not tak en into accoun t, the p robabilit y of AA in terven tion is 0.0189. If th e probability that Firm1 coop erates wh en Firm2 defects is very small ( α D = 0 . 1), then its op- timal decision is to co op erate, ev en for sm all v alues of α C . On the other hand, w h en α D ≥ 0 . 2, defecting is Firm2’s b est c h oice for α C = 0 . 4, yield- ing a differen t b eha viour from that obtained us ing the TFT str ategy . Ho w- ev er, using evidence E 1 , when the prob ab ility of AA in terven tion is 0.51 4, E [ u ( D ) | E 1 ] > E [ u ( C ) | E 1 ] ev en when Firm1 is sligh tly altruistic, α D ≤ 0 . 2 and α C ≤ 0 . 6. F urthermore, if α D = 0 . 25, then E [ u ( D ) | E 1 ] > E [ u ( C ) | E 1 ] also for α C ≤ 0 . 8. I f the TFT strateg y is adop ted, Firm2 optimally coop erates b oth under no evidence and E 1 , whereas for E 2 the asso ciated probabilit y of AA interv en tion is v ery large, so that Firm 2’s optimal decisio n is to d efect for all v alues of α C and α D considered here. While the examples sh o wn here are merely illustrativ e, the n u m b er of questions and different strategies that can b e analysed is clearly huge and increases with the n umb er of s tages consid ered. 6. Conclusion. When the an titrust authorit y starts an inv estigation, the t wo p oten tially merging fir ms are likely to r epresen t a relev an t share of the mark et, hence, they migh t affect the price of the go o d s traded. In con trast, the d ecisions of other firms inside the mark et, but outside the merged en- tit y , can b e assumed to b e irrelev ant. In circum s tances suc h as these, a P D duop oly mo del is a reasonable r epresen tation. F rom an economic p ersp ectiv e, the metho d ology w e present can b e seen as a u seful decision supp ort system. It mo d els and inte grates the different uncertain ty sources deriving from a riv al comp etitor and from the economic en viron m en t. F urthermore, the mod el can b e up dated as w e consider new cases, changes in market conditions or new ant itrust r egulations. Th e em- phasis in this pap er is to sho w the p oten tialit y of OOBNs in the analysis of duop oly mark ets with external un certain t y . F or the sak e of simplicit y , the duop oly is represented b y a rather naiv e game theoretic mo d el; in future studies we wish to implemen t a more complex in teraction mo del b et we en firms. As is standard in ind ustrial organization, the fir m is seen as a s ingle de- cision making un it; generalisatio n s of our OOBN to mo del firms’ in ternal organizatio n could also b e considered. Indeed, a fi r m’s top and middle man - agemen t ma y hav e differen t ob jectiv es fr om its o wner. An appropr iate BN could b e built to mo del these int errelationships and incorp orate them in to a more general OOBN mo d el. This would yield a more complete and realistic picture of firms’ coop erativ e b ehavio u r. 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