European Economic Review

Transcription

1 European Eonom Revew ) Contents lsts avalable at Seneret European Eonom Revew journal homepage: wwwelseverom/loate/euroeorev Equlbrum vanaton patterns n nomplete and heterogeneous networks Wllam Nelson, Yanheng Xao epartment of Eonoms, Unversty of Tennessee, Knoxvlle, TN , Unted States a r t l e n f o a b s t r a t Artle hstory: Reeved 2 July 207 Aepted 2 Marh 208 Avalable onlne 30 Marh 208 JEL Classfaton: I2 I8 85 C72 Keywords: Contagon network Vanaton Free-rdng Targeted poly Unversal mandate Under-vanaton s a usual onern of dsease ontrol studes, but ths paper employs a smultaneous-move game n a three-agent ontagon network to show that t s only one of the three neffent patterns When the network struture s nomplete or ndvdual haratersts are heterogeneous, there exst new types of Nash equlbrum outomes wth ether the rght number but wrong set of people gettng vanated or too many vanatons, and these equlbra are robust to standard refnements Whle untargeted poles an orret the standard under-vanaton problem, targeted poles are more palatable for orretng the new neffenes Unversal mandates an never mprove on any Nash equlbra 208 Elsever BV All rghts reserved Introduton When t omes to vanatons, the prmary fous of the general publ, the medal professon, and aadem researhers tends toward the problem of too few vanatons 2 For eonomsts, under-vanaton s a straghtforward example of freerdng The ndvdual reevng the vanaton bears the ost of the vanaton, but by keepng hm from gettng the dsease, the benefts go not just to hm but also to others wth whom he nterats 3 Free-rdng provdes one explanaton for Correspondng author E-mal addresses: wnelson@utkedu W Nelson), yxao@utkedu Y Xao) We are grateful to the edtor, assoate edtor, and two anonymous revewers as well as all of the followng for helpful omments: Sott Glpatr, Matt Harrs, Georg Shaur, Kyle Woodward, Bran Rogers, partpants of the 207 Mdwest Eonom Theory Conferene, the 207 Sngapore Eonom Revew Conferene, and the 207 Eonoms Graduate Students Conferene, and semnar partpants at Oho Unversty and the Unversty of Tennessee, Knoxvlle Remanng errors are our responsblty 2 For nstane, n 205 the journal Vane publshed a speal ssue on the top of vane hestany, the tendeny of both patents and aregvers to beome hestant about vanatons see Hkler et al, 205 ) 3 That vanatons have a publ-good omponent and lead to free-rdng behavor was already well-known n publ fnane textbooks Stgltz, 988 ), but Brto et al 99) were among the frst to provde a formal treatment Ther paper looks at tax and subsdy shemes to get the soally-optmal number of vanatons, and t was bult upon by Geoffard and Phlpson 997) and Frans 997) Ward 204) and Whte 207) fnd empral evdene of vanaton externaltes usng data on nfluenza vanes / 208 Elsever BV All rghts reserved

2 W Nelson, Y Xao / European Eonom Revew ) why too few ndvduals reeve vanatons, but some reent eonom researh has moved nto other fators that mght exaerbate ths effet, wth ndvduals not reevng vanatons beause of msnformaton or behavoral bases 4 Ths paper fouses on a dfferent problem We show that n nomplete and heterogeneous ontagon networks, not only an too few people get vanated, but also the rght number but the wrong set of people ould get vanated, and the possblty even arses that too many people get vanated As wth free-rdng, the ntuton s straghtforward Suppose the soally-optmal vanaton pattern dtates that one partular ndvdual should be among those vanated, perhaps she has the lowest vanaton ost or the hghest lkelhood to spread the dsease, but n equlbrum, that person does not get vanated Her forgong the vane leads others wth smaller network externaltes to get vanated as the best response, and sometmes t takes more than one person to make up for the mssng vanaton These new results show the mportane not just of nentvzng vanatons, but also of targetng the nentves to the rght ndvduals n nomplete and heterogeneous networks The typal large, stohast network models apture stuatons where dseases spread through random strangers To analyze those models, researhers run smulatons to fnd both the soally-optmal vanaton rate and the equlbrum vanaton rate n the presene of free-rdng see Brtton et al, 2007 or Mller and Hyman, 2007 for examples from the mathemats lterature; see Goyal and Vger, 205 for an example from the eonoms lterature) 5 In ontrast, we adopt small, determnst networks to desrbe stuatons where dseases spread among members who expet to nterat Though the networks are small, eah member an be treated as a representatve agent from a well-defned group We then expltly solve a non-ooperatve game for the soal optma and all pure-strategy Nash equlbra 6 These depend on the struture of the network, the ost of vanaton, the probabltes that the ndvduals beome nfeted from outsde the network, and the nterpersonal transmsson probabltes for the dsease n the network Consstent wth the goal of demonstratng that the problem exsts, we establsh t n the smplest possble ases Threeperson networks prove suffent We begn wth a baselne ase, a omplete network n whh three dental agents all have dret onnetons to eah other The only devatons from the soally optmal vanaton pattern arse when too few ndvduals get vanated n an equlbrum, whh s the standard free-rdng result We then ntrodue asymmetry n the network struture by removng a lnk, reatng a star network n whh two perpheral ndvduals are onneted only through a entral one Vanaton for the entral player generates a larger external beneft than vanaton for a perpheral player, and ths asymmetry n the externaltes leads to two new equlbrum outomes In ths 3-person star network, no one should get vanated when vanatons are prohbtvely expensve As the osts ome down, though, the entral player should be the frst to get vanated beause he an spread the dsease dretly to two ndvduals whereas the perpheral players an spread to just one person However, we show that there exst equlbra n whh one of the perpheral players gets vanated nstead of the entral one Thus, the rght number but the wrong set of people get vanated Moreover, there also exst equlbra n whh both of the perpheral players get vanated whle the entral player does not They do so beause gettng vanated s the best response to the entral player remanng unvanated, whle the entral player s hoe s the best response to both perpheral players gettng vanated Under suh rumstanes, too many people get vanated These new patterns of wrong-vanaton and over-vanaton arse n ases where there are multple equlbra, wth the best equlbrum beng the effent one n whh the entral player takes the vane However, we show that exept on a set of measure zero the neffent equlbra are tremblng-hand perfet, and therefore survve the most ommon equlbrum refnement for normal-form games Moreover, the same patterns an arse when the network s omplete, but the ndvdual parameters dffer Fnally, we show that n a star network n whh the entral player s a healthy arrer, that s, one who an transmt the dsease but suffers no ll effets of t herself, the new patterns unquely emerge beause a healthy arrer has no nentve to vanate All of ths suggests that a suessful pro-vane program must target the agent wth the largest externalty We onsder two targeted poles for the 3-person star network, a targeted subsdy whh subsdzes the vane for the entral agent, and a targeted fne whh penalzes her for falure to vanate Both poles, when admnstered approprately, an aheve the soal optmum, and they avod the problems of too many or the wrong set of vanatons by makng vanaton a domnant strategy for the entral player We go on to show that untargeted poles an also aheve the soal optmum, but they do so n a muh less-straghtforward way They address the problem of free-rdng wth a small subsdy or fne, but they address the problem of too many or the wrong set of vanatons by rewardng nonvanaton e, a negatve untargeted fne) or taxng vanaton e, a negatve vanaton subsdy) These poles work by makng nonvanaton a domnant strategy for the perpheral players, n whh ase the entral player best-responds by vanatng However, poles nentvzng nonvanaton are unlkely to be palatable, ether to polymakers or the publ at large Surprsngly, 4 Some reent expermental evdene nludes Ibuka et al 204), whh fnds evdene of free-rdng but also reeny bas, and Bronhett et al 205), whh looks at attenton, plannng, and follow-through for vanaton behavor 5 Rao et al 2007) looks at a dfferent queston onernng vanatons and networks, fndng evdene of peer effets n vanaton behavor 6 Mxed-strategy equlbra also exst, but we do not pursue them n ths paper for purposes of brevty, but the results are smple to summarze As wth the game of hken, eah ndvdual prefers others to vanate so he an free rde If ndvduals annot oordnate, they must randomze between vanatng and not Then all possble outomes arse wth some probablty, nludng the rght set of vanatons, too few, too many, and the rght number but the wrong set Furthermore, the mxed-strategy equlbrum has the same total payoff as the alloaton under unversal mandates beause ndvduals are ndfferent between vanatng and not Also, the total payoff n the mxed equlbrum s smaller than n pure ones

3 76 W Nelson, Y Xao / European Eonom Revew ) we also fnd that the straghtforward poly of a unversal vanaton mandate not only does worse than a targeted poly, but t also an never do better than no poly at all In essene, the requrements of a Nash equlbrum leave too lttle room for mprovement for a oarse poly lke a unversal mandate to work well Real-world poles nlude both subsdes and penaltes, and many of them target spef, hgh-externalty groups 7 For example, Calforna Senate Bll 205) mandates that after July 206, all hldren n shool from kndergarten through 2th grade, all nomng ollege students, and all hosptal staff get ertan vanatons 8 All these ndvduals provde lnks between otherwse unrelated ndvduals Chldren n shool provde lnks between parents who would otherwse never nterat, and on-ampus ollege students provde lnks between households n dfferent ommuntes Hosptal workers an spread dseases from patent to patent Also, many unverstes requre nternatonal students to get vanated before arrval, and these students fae dfferent dsease exposures than others nsde the ollege network Furthermore, effetve July 207 all sex workers n Germany must use ondoms BGBl, 206 ), whh s agan a mandate targetng entral players n the network that requres preventve atons aganst sexually transmtted dseases Sex workers oupy entral postons n a star network wth ther lents on the perphery, and the poly targets the sex workers The analyss n ths paper hghlghts an unrealzed beneft from these targeted poles, whh arses from avodng neffent behavor from the wrong people gettng vanated n equlbrum The paper has substantal dfferenes from the exstng lterature, all stemmng from ts explt fous on ndvdual behavor rather than the spread of the dsease 9 We use a stat game n whh all players smultaneously hoose whether to get vanated Ths game dffers from Frans 997) n whh the game s dynam, and soety eventually reahes the soal optmum after enough people beome nfeted In our stat game the dsease s not yet present n the network, and so t best fts a stuaton where onneted ndvduals make vanaton desons n advane, as wth measles vanatons, juvenle HPV vanatons, and early-season flu vanatons 0 The paper proeeds as follows Seton 2 presents the general model and dentfes ondtons for a soal optmum and a Nash equlbrum Seton 3 explores a 3-person homogeneous omplete network to show that only under-vanaton an arse Seton 4 ntrodues asymmetry through the network struture to examne a 3-person star network and fnds that neffenes an arse from too few, too many, or the wrong person gettng vanated and also establshes when these equlbra are tremblng-hand perfet Seton 5 shows that the same patterns an arse n omplete and star networks wth heterogeneous payoffs and sometmes these patterns an be unque Seton 6 looks at poly mplatons of nentve programs usng both targeted and untargeted subsdes for vanatng or penaltes for not vanatng Seton 7 onludes 2 The model: a ontagon network Consder an n -person network wth members N = {,, n } Connetons among members are b-dretonal For two dretly onneted players and j, f athes the dsease frst then an nfet j, and f j athes t frst j an nfet Indvduals an only ath the dsease one Player an ontrat the dsease from outsde the network wth probablty β 0, ), whh s the hannel of the ntal nvason of the ontagon When an ndvdual s not nfeted from outsde, he stll faes a probablty t 0, ) that an nfeted player transmts the dsease to hm through a dret onneton n the network An nfeton mposes a ost of 0 on the nfeted ndvdual Indvduals an get vanated to protet themselves from the nfeton, and the vane s assumed to be perfetly effetve, that s, a vanated ndvdual annot ontrat the nfeton whatsoever 2 However, a vane s not free but osts 0 for eah person Obvously, f > for every player, nobody gets vanated beause the ost of the vanaton outweghs the ost of gettng ll In the game, the n players smultaneously hoose whether to get vanated to mnmze ther expeted osts The probablty that ndvdual ontrats the dsease depends on who gets vanated Let denote the probablty that player S gets nfeted gven that the subset S N of the n players gets vanated The perfet effetveness of the vane means that S = 0 whenever S 7 Lawler 207) uses Canadan data to show that both government reommendatons and government mandates have postve mpats on the propensty to vanate 8 Sne 203, Msssspp and West Vrgna mandate vanatons for K-2 students and reommend vanatons for ollege students, but the mandate does not apply to hosptal staff See shool mmunzaton requrements for all states at the webste of Natonal Conferene of State Legslatures 207) 9 For example, Geoffard and Phlpson 997) look at poles for the omplete eradaton of ontagous dsease, sometmes alled herd mmunty, and Galeott and Rogers 203) look at dsease spread when there are two groups and dfferent mathng protools Chen and Toxvaerd 204) analyze the mpats of heterogenety and strateg nteraton on neffenes n a 3-person dynam model Goyal and Vger 205) look at the spread of dsease when agents an mmunze or protet themselves n a large random network, and Andrews and Bauh 206) do the same usng an agent-based model 0 We onsder only dseases spread by human-to-human ontat and not vetor-borne dseases For those, see Gersovtz and Hammer 2005) The settng n ths seton an be used for general, n -person networks, but n the subsequent setons, we restrt analyss to the ase of n = 3, and eah player an be a representatve agent from a well-defned group, suh as dotors and patents, teahers who move between shools and ther students, or foregn travelers and non-travelers 2 Ths assumpton smplfes alulatons but makes no dfferene to the qualtatve results To norporate mperfet vanes, we an add an effetveness parameter to represent the vane mathng rate

4 W Nelson, Y Xao / European Eonom Revew ) We defne S as the soally optmal vanaton set that mnmzes the network s total expeted ost For a vanated agent the expeted ost s smply beause vanes are 00% effetve For an unvanated ndvdual, the expeted ost s S, whh s the probablty of nfeton multpled by the llness ost Total expeted ost sums the expeted osts over the set of ndvduals n the network In ases where total expeted ost remans the same whether player vanates or not, we assume that S nludes Now onsder an ndvdual s nentve to get vanated when the vanaton set s S In a nonooperatve game, ndvdual onduts a ost-beneft analyss by omparng the vanaton ost and the beneft of gettng vanated The beneft omes from not athng the dsease and equals the expeted ost of ontratng the dsease S gven other players vanaton set S, where S denotes the subset S wth removed Assume ndvduals are rsk-neutral If, < S, agent s expeted ost from remanng unvanated exeeds the ost of gettng vanated, and so hooses to vanate If > S, then hooses to reman unvanated beause vanaton s too expensve If = S, then s ndfferent between these two, and we assume that agents elet to vanate whenever they are ndfferent Based on the above deson rules, a pure-strategy Nash equlbrum of the game onssts of a subset S N of vanated ndvduals suh that everyone n S weakly prefers to vanate, and everyone not n S weakly prefers not to In other words, n a pure-strategy equlbrum, no vanated player would be better off forgong vanaton, and no unvanated player would be better off gettng vanated Mathematally, the frst ondton means that f S then S so that s expeted osts of gong unvanated and possbly gettng the dsease exeed the ost of gettng vanated The seond ondton means that f S then > S so that for the ost of vanaton exeeds the expeted ost of athng the dsease 3 Symmetr struture: a omplete network We begn wth the benhmark ase of a 3-person homogeneous omplete network n whh everyone nterats wth everyone else ompleteness), and everyone has the same exogenous parameters, namely the nfeton probablty β, the transmsson probablty t, the vanaton ost, and the dsease ost We refer to the ase of dental exogenous parameters as homogenety We an fnd omplete networks n households, lassrooms, and apartments Indvduals nterat n small groups n these plaes, so they know the members Moreover, though we expltly refer to members as ndvduals, they an be representatve agents from exlusve groups For example, at a ommunty gatherng, eah member s a household representatve Connetons exst among households, but only the representatves nterat The omplete network serves as a good benhmark for two reasons The frst s that t s entrely symmetr n that every player has the same number of onnetons and, n the absene of vanatons, the same probablty of llness The seond s that, as we wll show, Nash equlbrum behavor an only devate from the soal optmum n one dreton, wth too few vanatons The ntuton behnd the result s straghtforward Suppose that the vanaton ost s low enough for t to be soally optmal to vanate two people but too hgh to vanate all three Those two vanatons beneft not only the people who get them but also the other one by elmnatng hs rsk of gettng nfeted from members n the network The soal optmum takes ths external beneft nto aount, but ndvduals do not when dedng whether to get vanated n a nonooperatve equlbrum If the soal beneft of vanaton makes t optmal for two players to get vanated but the prvate beneft does not, we end up wth only one vanated, e, free-rdng To establsh the exstene of free-rdng n equlbrum, we ompare the soally-optmal vanaton set S to the purestrategy Nash equlbrum vanaton set S If # S < # S, we have free-rdng, where # S denotes the number of elements of S The symmetry of both the network struture and the external nfeton probabltes means that only the number of players n S, and not whh ones they are, matters for determnng soal osts Vanated players have zero probablty of nfeton, but pay the vanaton ost An unvanated player does not pay but has expeted ost when the set S S of players get vanated We now go on to ompute for dfferent numbers of players n S S Frst onsder the ase n whh s the only unvanated player, that s, S = { j, k } and # S = 2 Then an only ontrat the dsease by external nfeton, so = β If only one of the other two gets vanated, say j, so that # S =, an ontrat the dsease by external nfeton or through a one-step transmsson from the other unvanated player k ) Thus = β + β) βt The frst term s the probablty that ontrats the external nfeton hmself β) The seond term s the probablty that he does not get the external nfeton β) tmes the probablty that the other unvanated player k gets t β) tmes the probablty that k transmts the dsease to hm t ) Fnally, f # S = 0, e, none of the others get vanated, an ontrat the dsease by external nfeton or nterpersonal transmsson from j by one step j ) or two steps j k ) Hene, s probablty of ontratng the dsease when # S = 0 s = β + 2 β) 2 βt + t 2 t 3 ) + β) β 2 2 t t 2 ) The frst term s the exogenous nfeton β) The seond term aounts for when and k are not exogenously nfeted β) 2 ) but j s β), may get nfeted from j through j by one step t ) or j k by two steps t 2 ) but not both

5 78 W Nelson, Y Xao / European Eonom Revew ) Fg Ineffent outomes n a 3-person homogeneous omplete network wth β = 05, t = 8, = The horzontal axs represents the vanaton ost from 0 to, and the numbered shadngs represent the number of vanatons, wth the top shadngs above the whte mdlne orrespondng to the soal optmum and the bottom shadngs below orrespondng to the Nash equlbrum arker olors represent more vanatons than lghter ones t 3 ) Sne j an be any of the other two players, multply ths probablty by 2 The thrd term aounts for when s not exogenously nfeted β) but both j and k are β 2 ), may get nfeted through j or k by one step 2 t ) but not both t 2 ) Note that n a 3-person homogeneous omplete network, { } < < < 3 The soal ost arsng when the players n S get the vane s 3 f # S = S = f # S = p f # S = ) 3 f # S = 0 The frst lemma shows the soal optma for varous vanaton osts Lemma Assume that βt < 2/3 In a 3-person homogeneous omplete network, the soally-optmal set of vanated ndvduals s gven by [ {, 2, 3 } f 0, p { S j, k } f = p, 2 p f 2 p, 3 2 2) f 3 p 2, Proof In appendx The lemma ontans some ntutve results, but also a less-ntutve one As ntuton would suggest when the normalzed vanaton ost / s very low everyone should get vanated, and when t s very hgh, no vanatons are effent As the ost of vanaton rses, fewer people should get the vane n the soal optmum The less-ntutve result s that vanaton may be effent even when the vane osts more than the llness / > ) beause one vanaton redues the probablty that the dsease spreads n the network, and the soal optmum nternalzes ths external beneft 4 The ondton βt < 2/3 guarantees that 3 2 > 2, so S = an be soally optmal It mples that the dsease s not too ommunable, but removng t does not affet the qualtatve result 5 The next lemma shows the Nash equlbra for varous vanaton osts Lemma 2 In a 3-person homogeneous omplete network, the pure-strategy equlbrum set of vanated ndvduals s gven by [ {, 2, 3 } f 0, p { S j, k } f = p, p f p, 3) f p, Proof In appendx Juxtaposng the two lemmas allows us to dentfy neffent outomes, that s, values of / for whh the Nash outome S dffers from the soally-optmal outome S Fg llustrates the neffent outomes wth a numeral example by omparng the sze of vanaton sets n the soal optmum and the pure-strategy equlbrum 3 = β + 2 β) 2 βt + t 2 t 3 ) + β) β 2 2 t t 2 ) = + β) β2 t β) [ t) t) 2 + β [ 2 t) 2 > beause > t) > t) 2 4 For example, when β = 0 7 and t = 0 8, the soal optmum has everyone vanated when / 07, two people vanated when 07 < / 036, one person vanated when 036 < / 38, and no one vanated when / > 38 5 If βt 2/3, the nequalty may not hold so Then f, 3 2 2, S = { j, k } ; and f 3 2 2,, S = Note , 2

6 W Nelson, Y Xao / European Eonom Revew ) The man result from ths seton of the paper onfrms the onventonal thnkng, fndng that n general, too few members may get vanated beause ndvduals do not take the external beneft of gettng vanated nto onsderaton Proposton Assume that βt < 2/3 In a 3-person homogeneous omplete network, too few ndvduals get vanated n a pure-strategy equlbrum when the vanaton ost, 2, 3 2 Otherwse, the orret number of ndvduals get vanated Proof In appendx Proposton s the result one would expet from the tradtonal theory of publ goods when there are free rders Furthermore, t emphaszes that ndvduals wth full nformaton may delberately delne to get vanated These are not new results, though In the next two setons, we add asymmetres to the network to fnd soures of neffeny dfferent from free-rdng 4 Asymmetr struture: a star network In a 3-person star network, player dretly onnets to both players 2 and 3, but players 2 and 3 do not dretly onnet to eah other Player 2 an stll pass a dsease to player 3, and ve versa, but now the transmsson must go through the entral player Central players an be dotors, nurses, teahers, flght attendants, nternatonal travelers, and people wth multple sexual partners 6 In partular, a dotor examnes many patents whle these patents do not dretly nterat Furthermore, eah player an be a representatve agent from exlusve groups When a patent goes to a hosptal, he expets to meet some dotor from that hosptal Smlarly, a dotor expets some patent from nearby ommuntes 7 In ths seton, we show that free-rdng s not the only potental soure of neffeny n the ontagon network It s possble for the rght number but the wrong set of people to get vanated n equlbrum, and t s also possble for too many people to get vanated n equlbrum Assume homogenety, as n the benhmark model, that all players have the same exogenous nfeton probablty β, the same transmsson probablty t, the same vanaton ost, and the same dsease ost The game s asymmetr, though, beause of the network struture: the entral player has two onnetons, but the perpheral players 2 and 3 have only one eah Ths asymmetry affets who should get vanated n the soal optmum If only one person s vanated, t should be player beause all dsease transmsson must go through her There exsts an equlbrum, though, where player hooses not to get vanated, but players 2 and 3 get vanated themselves as the best response, resultng n an neffently large number of vanatons Frst, onsder the entral player s probablty of ontratng the dsease As before, f gets vanated, her probablty of nfeton s 0 If remans unvanated, the probablty depends on what the perpheral players 2 and 3 do If both get vanated, then an only ath the dsease from outsde the network, whh ours wth probablty β If one of the two gets vanated, s probablty of nfeton s p { } = β + β) βt, whh s the same as n the omplete network when one person gets vanated If none of the perpheral players get vanated, besdes external nfeton, an ontrat the dsease by -step transmsson from 2 or 3, but not both, so s probablty of nfeton s p = β + β)2 βt β2 t 2 ) where the term 2 βt β 2 t 2 reflets that nfeton an be transmtted to from two people 2 βt ) but annot be transmtted from both β 2 t 2 ) Note that p { } < p { 2, 3 } < p { } < p 8 Next, onsder a perpheral player s probablty of ontratng the dsease, = 2, 3 If he vanates hmself, hs probablty of nfeton s 0, and f player gets vanated, hs probablty of nfeton s β If does not get vanated, but the other perpheral player j does, then the possble transmsson exsts between and Player an ontrat the dsease ether from outsde the network or from player, for a probablty of = β + β) βt, whh s the same as n a omplete network wth one vanaton Fnally, f nether nor j gets vanated, then besdes external nfeton, he an ontrat the dsease by -step transmsson from ) or by 2-step transmsson from j j ), so hs probablty of nfeton s = β + β) [ βt + β) βt 2 where the term n brakets represents that an get nfeted from by one step ) βt ) or from j by two steps j ), ondtonal on that s not exogenously nfeted β) βt 2 ) Note that { } < { } < < < p 9 6 Though we defne the entral player as the one who has the most onnetons to faltate our analyss, solely omparng players degree of onnetons an be problemat n a omplex network We an stll alulate every player s nfeton probabltes under all vanaton sets to reveal whose vanaton generates the largest external beneft However, suh alulaton an beome exponentally omplated n an n -person network, sne eah player faes 2 n dfferent vanaton sets See Jakson 2008) for alternatve entralty measures 7 espte the fat that networks may be larger than the ones we onsder and onsequently network struture mght also be more dffult to observe, we emphasze the postve aspets of our results rather than ther normatve aspets The postve aspets hghlght the potental over-vanaton n a Nash alloaton, whle the normatve aspets emphasze who should frst vanate Seton 6 goes on to llustrate normatve onlusons from these results) Regardless, n a omplex network, we usually an fnd a entral subgroup to demonstrate the relevane of postve aspets of our qualtatve results 8 p = β + β)2 βt β2 t 2 ) = p { } + β) βt βt) > p { } 9 = β + β) [ βt + β) βt 2 = + β) 2 βt 2 > and p = β + β)2 βt β2 t 2 ) = + β) βt t ) >

7 80 W Nelson, Y Xao / European Eonom Revew ) Summng over the three members of the network yelds expeted soal osts: 3 f # S = β f # S = 2 S = + 2 β f S = { } + 2 f S = or { 3 } p + 2 f # S = 0 The next lemma shows the optmal alloaton and the equlbrum alloaton of vanaton programs for dfferent values of the vanaton ost, holdng β and t onstant Lemma 3 In a 3-person homogeneous star network, the soally optmal set of vanated ndvduals s gven by {, 2, 3 } f [0 S, β = { } f β, p + 2 β) f p + 2 β) ) 5), Proof In appendx When the vanaton ost s huge, no one gets vanated When the ost falls enough for one ndvdual to get the vane, the soal optmum assgns t to player beause dong so removes any possblty of the dsease spreadng between players 2 and 3 f one of them gets sk Vanatng player 3 nstead of player rases the possblty that the dsease spreads between and 2 Thus, vanatng a perpheral player nstead of the entral one nreases expeted soal ost by 2 β) = 2 β) βt > 0 Also, as wth the benhmark ase of the omplete network, t s possble that vanaton s effent even when t osts more than the llness 20 The next lemma dentfes the pure-strategy Nash equlbra for dfferent values of the vanaton ost In some ases, there are multple equlbra 4) Lemma 4 In a 3-person homogeneous star network, the pure-strategy equlbrum set of vanated ndvduals s gven by {, 2, 3 } f [ 0, β { } or { 2, 3 } f β, p S = { } or or { 3 } f p, p { } f p, p ) f p, Proof In appendx 6) When the normalzed vanaton ost / s below the probablty of external nfeton β everyone gets vanated n equlbrum When t s hgher than the entral player s probablty of nfeton wth none vanated, p, nobody gets vanated For values of / n between there s an equlbrum n whh only the entral player takes the vane But, when / s at the lower end of ths nterval, there are two Nash equlbra, the soally-effent one n whh gets vanated and the soally-neffent one where both 2 and 3 get vanated nstead In ths range, β,, there exsts an equlbrum n whh too many people get vanated Smlarly, when / s n the mddle of the nterval there agan exst two types of Nash equlbra, an effent one vanatng player and an neffent one vanatng ether player 2 or 3 nstead In the range,, then, there exsts an equlbrum n whh the rght number but the wrong set of people get vanated Fg 2 llustrates ths for a star network n whh β = 0 05, t = 0 8, and = We generalze the results from ths example n the followng proposton Proposton 2 In a 3-person homogeneous star network, a) there exsts a pure-strategy equlbrum n whh too few ndvduals get vanated when p, p + 2 β) ; b) there exst pure-strategy equlbra n whh the rght number but the wrong set of ndvduals get vanated when, ; and ) there exsts a pure-strategy equlbrum n whh too many people get vanated when β, Proof In appendx Proposton 2 s the most novel fndng of the paper It shows that under an asymmetr network struture, for some range of the vanaton ost t s soally optmal to vanate only the entral player, but there exst pure-strategy equ- 20 For example, when β = 0 4 and t = 0 6, the soal optmum presrbes that player get vanated when 04 < / 045 The reason for ths s player s vanaton reates a postve externalty for players 2 and 3, and ths externalty lfts the entre beneft of the vanaton beyond the ost of a sngle llness

8 W Nelson, Y Xao / European Eonom Revew ) Fg 2 Ineffent outomes n a 3-person homogeneous star network wth β = 05, t = 8, = Same settngs as n Fg exept that the shadngs hghlght the members n S and S ; the orange shadng wth {2} also nludes {3} lbra vanatng one or both of the two perpheral players nstead Furthermore, the proposton shows that the asymmetres n a 3-person star network are suffent to generate three patterns of neffent vane alloatons and not just the usual one of too few vanatons Comparng Propostons and 2 shows that the network struture has an mpat on the ndene of free rdng In a omplete network free rdng ours when the equlbrum vanaton set s one person smaller than the soally optmal set, and ths an our when there are ether two or one soally-optmal vanatons 2 Lemma 3 shows that two vanatons are never soally optmal n a star network, and three vanatons are only soally optmal when the ost of the vanaton s lower than the external nfeton probablty β Thus, the only nstane of free rdng n the star network arses when player s supposed to get vanated, but nobody takes the vane Beause the exstene of multple equlbra drves the new soures of neffeny n a vanaton network, t s worth explorng whether these neffent equlbra are robust to standard refnements The next proposton shows that, exept at the rght endponts of the relevant ntervals, all of the equlbra are tremblng-hand perfet Proposton 3 In a 3-person homogeneous star network, a) S = { } s tremblng-hand perfet when β, p ; b) S = or {3} s tremblng-hand perfet when ) S = { 2, 3 } s tremblng-hand perfet when β, Proof In appendx, ) ; and At the rght endpont of eah of the ntervals n the proposton, the ndvduals n the equlbrum vanaton set are ndfferent between vanatng and not For the ase of S = { 2, 3 }, when / =, f player remans unvanated, players 2 and 3 are ndfferent between vanatng and not A tremble for player onssts of a small probablty of vanatng, and that small probablty tps players 2 and 3 to strtly prefer to reman unvanated Thus, the neffent Nash equlbrum s not robust to trembles at that endpont To the left of the endpont, though, players 2 and 3 strtly prefer vanatng to not vanatng, and a suffently small tremble by player annot overome the strt preferene Consequently, exept on a set of measure zero all of the Nash equlbra are also tremblng-hand perfet equlbra 5 Asymmetr payoffs n omplete and star networks Seton 4 ntrodued asymmetry n the network struture, but the standard way to brng asymmetry nto a game s through ether the move tmng or the payoffs It s hard to magne how a sequental vanaton game would play out, though, beause no one an ommt to no vanaton There are usually opportuntes to vanate later, n whh ase the hoe s between vanate now or do not vanate yet, but not between vanate now or never vanate 22 Therefore ths seton explores the varous hannels of payoff-based asymmetres that gve rse to the new types of neffent outomes, and also establshes senaros n whh these outomes result n a unque Nash equlbrum 2 The equlbrum vanaton set s possble to be two members smaller than the soally optmal set See the proof for Proposton 22 One an stll address the queston of whether makng the game sequental has any mpat on the results The outome depends on who moves frst If there s a Nash equlbrum n whh ) the frst mover does not have a domnant strategy to vanate, and ) some but not all players get vanated, then the frst mover wll hoose not to get vanated If the frst mover s the one wth the largest postve externaltes, suh as the entral player n the star network, then lettng that player move frst guarantees an neffent outome On the other hand, havng that player move last guarantees that the ndvdual generatng the greatest external beneft takes the vane

9 82 W Nelson, Y Xao / European Eonom Revew ) Begn wth a 3-person heterogeneous omplete network by allowng ndvduals to have dfferent dsease osts, but the same exogenous suseptblty, transmsson probablty, and vanaton ost Ths settng reflets that people wth poor health, suh as senors and hldren, are more lkely to suffer a severe skness when nfeted wth a ontagous dsease than healthy adults Wth ths llness severty heterogenety, the soal optmum frst alloates a vanaton to the hgh-ost ndvdual whenever the vanaton ost s low enough for at least one person to get vanated As the next proposton shows, though, there exst neffent alloatons n whh others get the vane nstead Proposton 4 In a 3-person omplete network wth > 2 = 3, there exsts a pure-strategy equlbrum n whh the rght number but the wrong set of ndvduals get vanated when β, p 2 { 3 } 2 p 2 { 3 }, p 2 2, gven the exstene of suh ntervals Proof In appendx In the frst nterval two ndvduals take the vane n both the soal optmum and the Nash equlbrum, and the neffeny arses when the hgh dsease-ost agent s not among the vanated In the seond nterval only one agent takes the vane, and the neffeny arses when agent s not the one who takes t Unlke wth the star network, there s no ssue wth too many vanatons here beause n a omplete network there exst soally-optmal alloatons wth two vanatons, but n a star network there do not Beause the rato between the dsease ost and the vanaton ost determnes the equlbra and the soal optma, a smlar outome arses n the same network wth heterogeneous vanaton osts In the soal optmum the ndvdual wth the lowest vanaton ost takes the vane frst, assumng that the ost s low enough for anyone to vanate However, there exsts equlbrum n whh others get vanated nstead Suh vanaton ost asymmetry aptures dfferenes n nsurane status, travel ost, and phlosophal belefs Proposton 5 In a 3-person omplete network wth < 2 = 3, there exsts a pure-strategy equlbrum n whh the rght number but the wrong set of ndvduals get vanated when, 2 β, p 2 { 3 } or, 2 p 2 { 3 }, p 2, gven the exstene of suh ntervals Proof In appendx In the frst nterval two agents take the vane and n the seond nterval one agent does, and the neffenes arse beause the low-ost agent remans unvanated Whle the asymmetry n Proposton 2 omes from the shape of the network, the asymmetres n Propostons 4 and 5 nvolve ndvdual osts The asymmetres an also ome from the transmsson probabltes and the external nfeton probabltes Consder heterogeneous transmsson probabltes frst Let t denote the probablty that the dsease s transmtted to player by one of the other two players, and assume that agent s transmsson probablty s the hghest Suh transmsson heterogenety mght reflet that agent s a hghly suseptble ndvdual who s at a hgh rsk of beomng nfeted, suh as senors, hldren, and pregnant women The transmsson probablty dfferenes reate an nentve for the soal planner to vanate the ndvdual wth the greatest transmsson probablty As the next proposton demonstrates, there exst Nash equlbra n whh someone other than the most-suseptble person gets vanated Proposton 6 In a 3-person omplete network wth t > t 2 = t 3, there exsts a Nash equlbrum n whh the wrong person gets vanated when p, p 2, gven the exstene of suh an nterval Proof In appendx The neffeny arses beause ether player 2 or 3 gets vanated n equlbrum but soal ost s lower when the hghly-suseptble player gets vanated nstead Note that when two players get vanated transmsson between agents beomes mpossble, and so the proposton only onerns the ase of one vanaton To omplete the sope of asymmetry soures, suppose that the agents have dfferent external nfeton probabltes, wth agent beng most suseptble to nfeton from outsde the network Thus, agent s the most lkely ndvdual to ntrodue the dsease to the network, and the soal optmum alloates a vanaton to hm whenever the vanaton ost s suffently low Suh heterogenety ould arse beause, for example, agent travels more than the others or omes nto ontat wth more strangers than the others The next result shows that neffenes an arse n ths settng as well Proposton 7 In a 3-person omplete network { wth β > } β 2 = β 3, there exsts a pure-strategy equlbrum n whh too many ndvduals get vanated when max β, 2 p 2 { } β 2, p 3, and the rght number but the wrong set of ndvduals get vanated when p, p 2, gven the exstene of suh ntervals Proof In appendx Ths proposton shows that both types of neffeny, ether from the wrong set of vanatons or from too many vanatons, an our n a omplete network

10 W Nelson, Y Xao / European Eonom Revew ) Propostons 2 and 4 7 demonstrate that any soure of asymmetry n the network, whether from dfferenes n the parameters or from the shape of the network tself, an lead to Nash equlbra wth too many or the wrong set of vanatons In all of these ases the neffenes arse beause there are multple Nash equlbra, and the norret alloatons an be avoded f the network selets a more effent equlbrum We fnsh ths seton wth an example n whh there are not multple equlbra and the unque equlbrum vanaton alloatons dffer from the soal optmum by havng ether too many or the wrong agents vanate To do ths, we ombne two types of asymmetry Consder a 3-person star network n whh the entral player s a healthy arrer who remans healthy even after beng nfeted, so her dsease ost s zero but she stll an transmt the dsease to others 23 The healthy arrer does not have any nentve to vanate, so no Nash outome nludes nludes her n the equlbrum vanaton set However, the soal planner prefers to vanate the arrer beause of her entral poston n the star network Beause of ths, the only Nash equlbrum outomes are neffent, as shown n the next proposton 24 Proposton 8 In a 3-person star network wth = 0 <, = β for > β, there exst unque neffent equlbrum outomes a) that too many ndvduals get vanated when β, p 2 { 3 } ; and b) that the rght number but the wrong set of ndvduals get vanated when p 2 { 3 }, p 2 Proof In appendx 6 Targeted and untargeted vanaton poles Seton 3 shows that when the network s omplete and the parameters homogeneous, neffenes arse only from the famlar free-rdng problem, wth too few ndvduals takng the vane Seton 4 shows, however, that when the network s not omplete, neffenes an arse wth ether the wrong ndvduals or too many ndvduals gettng vanated Seton 5 demonstrates that the same patterns an arse when the network s omplete, but the parameters refletng dsease or vanaton osts or transmsson probabltes or exogenous suseptbltes are heterogeneous In all ases, the neffenes arse beause ether the network struture or the parameter values make t least soally ostly when one partular ndvdual takes the vane, but that ndvdual goes unvanated n the Nash equlbrum Below we refer to ths ndvdual as the hgh-externalty ndvdual, refletng that n the star network vanatng the entral player provdes larger postve externaltes than vanatng ether of the two perpheral players Ths suggests that targetng the entral player ould be a useful poly feature, and we explore targeted and untargeted poles n ths seton In partular, we fous on the homogeneous star network 25 Analyzng a targeted poly requres prese defntons of ts objetves beause there may exst two or more Nash equlbra but only one neffent equlbrum for some parameter values Therefore, we defne the worst equlbrum S as the neffent Nash alloaton wth the entral player / S when multple Nash outomes exst Sne player hooses not to vanate n S, t must be the ase that > p, that s, the vanaton ost s greater than hs expeted llness ost gven S members of S gettng vanated However, a polymaker must motvate player to get vanated beause the effent equlbrum requres vanatng the entral player Two straghtforward optons are avalable to revert player s hoe: subsdzng her vanaton or penalzng her for not gettng vanated To be spef, the subsdy s must be suffent to motvate her to vanate anyway, or s p ; and the penalty f also must be suffent to prevent her from forgong the vanaton, or p S S + f Motvatng the entral player s not suffent to justfy the poly, though, and the soal planner must also demonstrate that the poly nurs smaller ost than beneft to the network In partular, the admnstratve ost g to mplement the poly must be no greater than the soal ost savngs, or g S { } 26 Therefore, we defne a poly as a potental soal mprovement f the Nash equlbrum wth the poly generates a weakly smaller expeted soal loss than the worst Nash equlbrum wthout the poly Next, we use these rtera to analyze the subsdy frst and then the fne A targeted subsdy s onsttutes a potental soal mprovement from the undesrable equlbrum alloaton S f s p S and g S { } That s, the subsdy s large enough to ndue player s vanaton but nurs smaller admnstratve ost than the ost savng by that vanaton 23 Typhod Mary, who was the frst person n the US dentfed as a healthy arrer of the typhod fever pathogen, nfeted 5 people but lved for nearly three deades n solaton Also, for the human papllomavrus HPV), whh ontans more than 00 types of sexually transmtted vruses, some nfeted people show no symptoms, but others are dagnosed wth aner 24 When the number of members nreases, we an obtan ths outome wth homogeneous vanaton osts In an n-person star network n > 3) wth the entral player beng a healthy arrer, holdng other parameters dental, there exsts a unque neffent outome that too many ndvduals get n vanated n a pure-strategy equlbrum for / n 2 β, β + β) βt, whh only requres β) t > n 2 25 Results smlar to those n ths seton hold for the heterogeneous omplete networks and the star network wth a healthy arrer 26 The government budget does not nlude the spendng of the subsdy or the revenue of the fne beause subsdes and fnes are transfers wthn the network

11 84 W Nelson, Y Xao / European Eonom Revew ) Fg 3 Poly omparsons n a 3-person homogeneous star network wth β = 05, t = 8, = The y -axs s the expeted soal ost, and the x -axs s the vanaton ost Proposton 9 In a 3-person homogeneous star network, a subsdy s to the entral player that nurs an admnstratve ost g onsttutes a potental soal mprovement f [ a) s p and g p + 2 β) when b) s and g 2 β) when ) s β and g β when β, Proof In appendx p, p + 2 β) ; ;, Proposton 9 s best understood by juxtaposng t wth Proposton 2 The regon n ondton a) orresponds to too few vanatons, that s, where the soally effent alloaton s for player to get vanated but n the unque Nash equlbrum, nobody gets t Condton b) onerns ost levels for whh the rght number but wrong set of ndvduals get vanated, and ondton ) looks at the regon n whh too many ndvduals get vanated Proposton 9 shows that n all three ases a targeted subsdy an overome the soure of neffeny throughout the regon n whh t ours A targeted nonvanaton fne works n the same way A targeted fne f onsttutes a potental soal mprovement from S f f p S and g S { } That s, the fne s large enough to make gettng the vane more attratve than forgong t but nurs smaller admnstratve ost than the ost savng by that vanaton Proposton 0 In a 3-person homogeneous star network, a fne f to the entral player that nurs an admnstratve ost g onsttutes a potental soal mprovement f [ a) f p and g p + 2 β) when b) f and g 2 β) when ) f β and g β when β, Proof In appendx p, p + 2 β) ; ;, Fg 3 llustrates the results of Propostons 9 and 0 usng the same parameters as n Fg 2 The dashed magenta lne shows the worst Nash equlbrum outome for varous values of the vanaton ost 27 The sold blue lne shows the Nash 27 The fgure s onstruted so that when two or more lnes onde exatly, the dashed dark gray lne s always drawn on top and the sold blak lne s on the bottom, so f the blak or lght gray lnes are hdden, they onde wth the dark gray one

12 W Nelson, Y Xao / European Eonom Revew ) equlbrum outome wth a targeted subsdy or fne, where t optmzes to be zero when the vanaton ost s ether very low or very hgh, and t ondes wth the soal optmum The dstane between the magenta and blue lnes demonstrates the upper bound of the admnstratve ost that an justfy the targeted poly Fnally, the sold blak lne shows the soal ost from a unversal mandate where all members get vanated The fgure does not show the best Nash equlbrum outome beause that lne would always onde wth one of the other ones already shown It ondes wth the sold blue lne from zero to the rght-most vertal porton of the dashed magenta lne, and t ondes wth the magenta lne from there on The best Nash equlbrum has the entral player gettng the vane untl osts reah a level n whh no one gets vanated A targeted subsdy or penalty an orret player s falure to nternalze the externaltes Our poly analyss shows that both a targeted subsdy and a targeted fne an aheve the soally-optmal alloaton of vanatons It s worth notng, though, that a subsdy an do ths through a Pareto mprovement, whle a fne annot Player prefers equlbra where others get vanated nstead of her, whle the perpheral players prefer the equlbrum where only player vanates A subsdy that ndues player to vanate nstead of one or more perpheral players an make everyone better off A fne, n ontrast, makes the perpheral players better off by movng them to a preferred alloaton, but even though she does not pay the fne n equlbrum, t makes player worse off by movng her to a lesspreferred one Both targeted poles avod the problem of too many or the wrong vanatons by makng vanaton a domnant strategy for the entral player, n whh ase the two perpheral players best-respond by remanng unvanated unless / β For untargeted poles, though, any poly that makes vanaton a domnant strategy for player also makes t a domnant strategy for the two perpheral players Beause of ths, untargeted vanaton poles allow for the same multple-equlbrum neffenes that arse wthout a poly Fg 2 llustrates how ths works for a subsdy An untargeted subsdy s effetvely lowers the vanaton ost from to s and therefore an be shown n the graph by a leftward movement The problems of vanatng too many or the wrong people arse when s )/ β,, where s s the subsdy, and there are multple equlbra throughout ths regon The only way to get a unque Nash equlbrum wth only player gettng vanated s to have a negatve subsdy, or a tax, on the vane A tax would make vanaton more expensve, not less, and s qute ounterntutve The more ntutve noton of a postve subsdy mprovng soal welfare arses when the vanaton ost s beyond p so that the Nash equlbrum has no vanatons, but the effent outome presrbes a vane to the entral player An approprate untargeted subsdy an redue the vanaton ost to between and p, n whh ase the entral player vanates n the unque Nash equlbrum An untargeted fne for remanng unvanated works smlarly A fne of f rases the ost of remanng unvanated from S to + f, and the ndvdual hooses to reman unvanated f > + f, whh s somorph to f > So an S S S untargeted fne works the same way as an equal-szed untargeted subsdy As wth the subsdy, a postve untargeted fne an mprove the stuaton n whh there are too few vanatons, but the fne must be negatve to avod stuatons wth too many or the wrong vanatons Ths negatve fne would be a reward for those who reman unvanated, whh s agan ounterntutve Proposton In a 3-person homogeneous star network, assumng the admnstratve ost g s suffently small, the followng poles onsttute potental soal mprovements: a) an untargeted vanaton subsdy or nonvanaton fne when p, p + 2 β ) ; b) an untargeted vanaton tax or nonvanaton reward when β, Proof In appendx Part a) governs stuatons n whh nobody gets vanated even though t s soally optmal for the entral player to take the vane The untargeted vanaton subsdy or untargeted nonvanaton fne works by redung the effetve vanaton ost enough for player to get vanated n equlbrum but not enough for the other, less effent equlbra to arse Part b) governs stuatons n whh ether too many or the wrong people take the vane, and the poles work by makng nonvanaton a domnant strategy for the perpheral players 2 and 3 and ndung player to best-respond by vanatng Stark ontrasts arse between the targeted and untargeted poles The targeted poles avod stuatons of too many or the wrong vanatons by makng vanaton a domnant strategy for player, and the poles do so through straghtforward vanaton subsdes or nonvanaton fnes Untargeted poles avod these same outomes by makng nonvanaton a domnant strategy for the perpheral players, and they do so through ounterntutve vanaton taxes or nonvanaton rewards To address the new neffent patterns dentfed n ths paper, then, targeted poles have easy-to-understand and poltally feasble parameters, whle untargeted poles would be dffult for agenes to explan and sell to the publ One poly that we brefly mentoned n Fg 3 s a unversal mandate, whh an be regarded as an untargeted fne large enough to ndue everyone to vanate The fgure shows that the unversal mandate an never do better than the worst Nash equlbrum That t does worse than an approprate targeted poly s no surprse beause the unversal mandate obtans more vanatons than that are soally optmal However, that t does no better than no poly at all may be a surprse to some The next proposton shows that suh relatonshp dsovered n the homogeneous star network s ompletely general, overng any network wth any amount of heterogenety