Cooperation with Network Monitoring


 Deborah Blake
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1 Cooperaion wih Nework Monioring Alexander Wolizky Microsof Research and Sanford Universiy February 2012 Absrac This paper sudies he maximum level of cooperaion ha can be susained in perfec Bayesian equilibrium in repeaed games wih nework monioring, where players observe each oher s acions eiher perfecly or no a all. The foundaional resul is ha he maximum level of cooperaion can be robusly susained in grim rigger sraegies. If players are equally well moniored, comparaive saics on he maximum level of cooperaion are highly racable and depend on he monioring echnology only hrough a simple saisic, is eff ecive conagiousness. Typically, cooperaion in he provision of pure public goods is greaer in larger groups, while cooperaion in he provision of divisible public goods is greaer in smaller groups, and making monioring less uncerain in he secondorder sochasic dominance sense increases cooperaion. For fixed monioring neworks, a new noion of nework cenraliy is developed, which deermines which players cooperae more in a given nework, as well as which neworks suppor greaer cooperaion. 1 Inroducion How can groups susain as much cooperaion as possible? Should hey rely exclusively on punishing individuals who are caugh shirking, or should hey also reward hose who are caugh working? This paper was previously circulaed under he ile, Repeaed Public Good Provision. I hank he edior, Marco Oaviani, and four anonymous referees for helpful commens. I hank my advisors, Daron Acemoglu, Glenn Ellison, and Muhame Yildiz, for deailed commens and suggesions and for exensive advice and suppor; and hank Nageeb Ali, Abhiji Banerjee, Alessandro Bonai, Gabriel Carroll, Ma Jackson, Anon Koloilin, Parag Pahak, Larry Samuelson, Juuso Toikka, Iván Werning, and seminar paricipans a MIT and he 2010 Sony Brook Inernaional Conference on Game Theory for addiional helpful commens. I hank he NSF for financial suppor. 1
2 Relaedly, wha kinds of groups can susain he mos cooperaion? Large ones or small ones? Ones where who observes whom in he group is known, or where i is uncerain? These are fundamenal quesions in he social sciences Olson, 1965; Osrom, 1990; Coleman, 1990; Punam, 1993; Greif, 2006). In economics, exising work on he heory of repeaed games provides a framework for answering hese quesions when individuals can perfecly observe each oher s acions e.g., Abreu, 1988), bu provides much less explici answers in he more realisic case where monioring is imperfec. This weakness is paricularly acue in seings where public signals are no very informaive abou each individual s acions and high qualiy bu dispersed privae signals are he basis for cooperaion. Consider, for example, he problem of mainaining a school in a small village in he developing world. Every year, say, differen villagers mus conribue differen inpus o running he school: some provide money, some provide labor o mainain he building, some voluneer in oher capaciies, ec. These inpus are no publicly observable, and differen villagers observe each oher s acions wih differen probabiliies. The overall qualiy of he school is very hard o observe direcly, and indeed one migh no be able o infer much abou i unil one sees how well he sudens do years down he road, by which ime he enire sysem of providing educaion in he village may have changed. This problem was sudied heoreically and empirically using daa on schools and wells in rural Kenya) by Miguel and Gugery 2005), under he assumpion ha each household s conribuion is publicly observable, bu his assumpion is ofen unrealisic; for example, Miguel and Gugery emphasize he imporance of ehnic divisions in he villages hey sudy, so a naural assumpion would be ha a household is more likely o be moniored by households from he same ehnic group. A second example is he problem of cooperaion in longdisance rade, argued by Greif and ohers o be an essenial hurdle o he developmen of he modern economy. Here, a key issue is ofen how sharing informaion hrough neworklike insiuions like rading coaliions Greif 1989, 1993), rade fairs Milgrom, Norh, and Weingas, 1990), and merchan guilds Greif, Milgrom, and Weingas, 1994) faciliaes cooperaion. Thus, i is cerainly plausible ha local, privae monioring plays a larger role han public monioring in susaining cooperaion in many ineresing economic examples, and very lile is known abou how cooperaion is bes susained under his sor of monioring. This paper sudies cooperaion in repeaed games wih nework monioring, where in every period a nework is independenly drawn from a possibly degenerae) known disribuion, and players perfecly observe he acions of heir neighbors bu observe nohing abou any oher player s acion. The model covers boh monioring on a fixed nework as when a household s acions are 2
3 always observed by is geographic neighbors, or by households in he same ehnic group), and random maching as when raders randomly mee in a large marke). Each player s acion is simply her level of cooperaion, in ha higher acions are privaely cosly bu benefi ohers. The goal is o characerize he maximum level of cooperaion ha can be susained robusly in equilibrium, in ha i can be susained for any informaion ha players may have abou who has moniored whom in he pas. This robusness crierion capures he perspecive of an ouside observer, who knows wha informaion players have abou each oher s acions, bu no wha informaion players have abou each oher s informaion abou acions or abou heir informaion abou ohers informaion abou acions, and so on), and who herefore mus make predicions ha are robus o his higherorder informaion. 1 A firs observaion is ha for any given specificaion of players higherorder informaion, he sraegies ha susain he maximum level of cooperaion can depend on players privae informaion in complicaed ways ha involve a mix of rewards and punishmens, and deermining he maximum level of cooperaion appears inracable. In conras, my main heoreical resul is ha he robus maximum level of cooperaion is always susained by simple grim rigger sraegies, where each player cooperaes a a fixed level unless she ever observes anoher player fail o cooperae a his prescribed level, in which case she sops cooperaing forever. Thus, robus cooperaion is maximized hrough sraegies ha involve punishmens bu no rewards. In addiion, grim rigger sraegies also maximize cooperaion when players have perfec knowledge of who observed whom in he pas as is he case when he monioring nework is fixed over ime, for example); ineresingly, i is when players have less higherorder informaion ha more complicaed sraegies can do beer han grim rigger. A rough inuiion for hese resuls is ha when players know who observed whom in he pas here is a kind of sraegic complemenariy in which a player is willing o cooperae more a any onpah hisory whenever anoher player cooperaes more a any onpah hisory, because wih nework monioring and grim rigger sraegies shirking makes every onpah hisory less likely; bu his sraegic complemenariy breaks down when players can disagree abou who has observed whom. This resul abou how groups can bes susain cooperaion has implicaions for wha groups can susain he mos cooperaion. For hese more applied resuls, I focus on wo imporan special cases of nework monioring: equal monioring, where in expecaion players are moniored equally 1 There are of course oher kinds of robusness one could be ineresed in, and sraegies ha are robus in one sense can be fragile in ohers. See he conclusion of he paper for discussion. 3
4 well ; and fixed monioring neworks, where he monioring nework is fixed over ime. Wih equal monioring, I show ha he effeciveness of a monioring echnology in supporing cooperaion is compleely deermined by one simple saisic, is eff ecive conagiousness, which is defined as δ E [number of players who learn abou a deviaion wihin periods]. =0 This resul formalizes he simple idea ha more cooperaion can be susained if news abou a deviaion spreads hroughou he nework more quickly. I implies ha cooperaion in he provision of pure public goods where he marginal benefi of cooperaion is independen of group size) is increasing in group size if he expeced number of players who learn abou a deviaion is increasing in group size, while cooperaion in he provision of divisible public goods where he marginal benefi of cooperaion is inversely proporional o group size) is increasing in group size if he expeced fracion of players who learn abou a deviaion is increasing in group size. Hence, cooperaion in he provision of pure public goods ends o be greaer in larger groups, while cooperaion in he provision of divisible public goods ends o be greaer in smaller groups. In addiion, making monioring more uncerain in a cerain sense reduces cooperaion. Wih fixed neworks, I develop a new noion of nework cenraliy ha deermines boh which players cooperae more in a given nework and which neworks suppor more cooperaion overall, hus linking he graphheoreic propery of cenraliy wih he gameheoreic propery of robus maximum cooperaion. For example, adding links o he monioring nework necessarily increases all players robus maximum cooperaion, which formalizes he idea ha individuals in beerconneced groups cooperae more. The resuls of his paper may bear on quesions in several fields of economics. Firs, a lieraure in public economics sudies he effec of group size and srucure on he maximum equilibrium level of public good provision. One srand of his lieraure sudies repeaed games, bu characerizes maximum cooperaion only wih perfec monioring. Papers in his srand have found few unambiguous relaionships beween group size and srucure and maximum cooperaion. 2 A second 2 Pecorino 1999) shows ha wih perfec monioring public good provision is easier in large groups, because shirking and hus causing everyone else o sar shirking is more cosly in large groups. Haag and Lagunoff 2007) show ha wih heerogeneous discoun facors and a resricion o saionary sraegies, maximum cooperaion is increasing in group size. Bendor and Mookherjee 1987) consider imperfec public monioring, and presen numerical evidence evidence suggesing ha higher payoffs can be susained in small groups when aenion is resriced o rigger sraegies. In a second paper, Bendor and Mookherjee 1990) allow for nework srucure bu reurn o he 4
5 srand sudies onesho games of public good provision in neworks Balleser, CalvóArmengol, and Zenou, 2006; Bramoullé and Kranon, 2007a; Bramoullé, Kranon, and D Amours, 2011), where he nework deermines local payoff ineracions and, in paricular, incenives for freeriding raher han monioring. These papers find ha more cenral players measured by Bonacich cenraliy or a modificaion hereof) cooperae less and receive higher payoffs, due o freeriding, and ha adding links o a nework decreases average maximum cooperaion, by increasing freeriding. In conras, my model, which combines elemens from boh srands of he lieraure, makes he following predicions, which are made precise laer: 1. Cooperaion in he provision of pure public goods is greaer in larger groups, while cooperaion in he provision of divisible public goods is greaer in smaller groups. 2. Less uncerain monioring increases cooperaion. 3. More cenral players cooperae more unlike in he public goods in neworks lieraure) bu sill receive higher payoffs wih local public goods like in ha lieraure). 4. Adding links o a monioring nework increases all players cooperaion. Second, several seminal papers in insiuional economics sudy he role of differen insiuions in sharing informaion abou pas behavior o faciliae rade Greif, 1989, 1993; Milgrom, Norh, and Weingas, 1990; Greif, Milgrom, and Weingas, 1994). Ellison 1994) noes ha he models underlying hese sudies resemble a prisoner s dilemma, and shows ha cooperaion is susainable in he prisoner s dilemma wih random maching for suffi cienly paien players, which suggess ha informaionsharing insiuions are no always necessary. The curren paper conribues o his lieraure by deermining he maximum level of cooperaion in a prisoner s dilemmalike game a any fixed discoun facor for any nework monioring echnology. Thus, i allows one o deermine he exacly how much more cooperaion can be susained in he presence of a given informaionsharing insiuion. Third, a young and very acive lieraure in developmen economics sudies he impac of nework srucure on differen kinds of cooperaion, such as favor exchange Karlan e al, 2009; Jackson, RodriguezBarraquer, and Tan, 2011) and risksharing Ambrus, Möbius, and Szeidl, 2010; Bramoullé and Kranon, 2007b; Bloch, Genico, and Ray, 2008). The predicions of his paper enumeraed above can be suggesively compared o some early empirical resuls in his lieraure, assumpion of perfec monioring, and find an ambiguous relaionship beween group size and maximum cooperaion. 5
6 alhough clearly much empirical work remains o be done. For example, Karlan e al 2009) find ha indirec nework connecions beween individuals in Peruvian shanyowns suppor lending and borrowing, consisen wih my finding ha more cenral players cooperae more. More subly, Jackson, RodriguezBarraquer, and Tan 2011) find ha favorexchange neworks in rural India exhibi high suppor, he propery ha linked players share a leas one common neighbor. While i seems naural ha suppor which is he key deerminan of cooperaion in Jackson, Rodriguez Barraquer, and Tan s model) should be correlaed wih robus maximum cooperaion in my model, I leave sudying he precise empirical relaionship beween he wo conceps for fuure research. A few final commens on relaed lieraure: I should be noed ha he aforemenioned paper of Ellison 1994), along wih much of he relaed lieraure e.g., Kandori, 1992; Deb, 2009; Takahashi, 2010) focuses on he case of suffi cienly high discoun facors and does no characerize effi cien equilibria a fixed discoun facors, unlike my paper. In addiion, a key concern in hese papers is ensuring ha players do no cooperae off he equilibrium pah. The issue is ha grim rigger sraegies may provide such srong incenives o cooperae onpah ha players prefer o cooperae even afer observing a deviaion. Ellison resolves his problem by inroducing a relening version of grim rigger sraegies ailored o make players indifferen beween cooperaing and shirking onpah, and hen noing ha cooperaion is more appealing onpah han offpah since offpah a leas one opponen is already shirking). This issue does no arise in my analysis because, wih coninuous acion spaces, players mus be jus indifferen beween cooperaing and shirking onpah in he mos cooperaive equilibrium, as oherwise hey could be asked o cooperae slighly more. By essenially he same argumen as in Ellison, his implies ha players weakly prefer o shirk offpah. Hence, he key conribuion of his paper is showing ha grim rigger sraegies provide he sronges possible incenives for robus) cooperaion onpah, no ha hey provide incenives for shirking offpah. 3 The mos closely relaed paper is conemporaneous and independen work by Ali and Miller 2011). Ali and Miller sudy a nework game in which links beween players are recognized according o a Poisson process. When a link is recognized, he linked players play a prisoner s dilemma wih variable sakes, and can also make ransfers o each oher. Like my model, Ali and Miller s feaures smooh acions and payoffs, so ha, wih grim rigger sraegies, binding onpah incenive consrains imply slack offpah incenive consrains. The mos imporan difference 3 Anoher difference is ha i is imporan for he curren paper ha in each period he monioring nework is observed afer acions are chosen, whereas his iming does no maer in mos papers on communiy enforcemen. 6
7 beween Ali and Miller s paper and mine is ha hey do no show ha grim rigger sraegies always maximize cooperaion in heir model. Ali and Miller also do no emphasize sraegic complemenariy or robusness o higherorder informaion. They do however discuss nework formaion and comparisons among neworks, developing insighs ha are complemenary o mine. Finally, his paper is relaed more broadly o he sudy of repeaed games wih privae monioring. Mos papers in his lieraure sudy much more general models han mine, and eiher prove folk heorems or sudy robusness o small deviaions from public monioring Mailah and Morris, 2002, 2006; Sugaya and Takahashi, 2011). 4 However, o my knowledge his is he firs paper ha characerizes even a single poin on he Pareo fronier of he se of perfec Bayesian equilibrium payoffs in a repeaed game wih imperfec privae monioring where firsbes payoffs are no aainable. I make no aemp o characerize he enire se of perfec Bayesian equilibria, or any large subse hereof. Insead, I use he sraegic complemenariy discussed above o derive an upper bound on each player s maximum cooperaion, and hen show ha his bound can be aained wih grim rigger sraegies. I would be ineresing o see if similar indirec approaches, perhaps also based on sraegic complemenariy, can be useful in oher classes of repeaed games wih privae monioring of applied ineres. The paper proceeds as follows: Secion 2 describes he model. Secion 3 presens he key resul ha maximum cooperaion is robusly susained in grim rigger sraegies. Secion 4 derives comparaive saics in games wih equal monioring. Secion 5 sudies games wih fixed monioring neworks. Secion 6 concludes and discusses direcions for fuure research. Major omied proofs and examples are in he appendix, and minor ones are in he online appendix. 2 Model There is a se N = {1,..., n} of players. In every period N = {0, 1,...}, every player i simulaneously chooses an acion level of cooperaion, conribuion ) x i R +. The players have common discoun facor δ 0, 1). If he players choose acions x = x 1,..., x n ) in period, player i s period payoff is u i x) = f i,j x j ) x i, j i where he funcions f i,j : R + R + saisfy 4 Of he many privae monioring folk heorem papers, he mos relaed are probably BenPorah and Kahneman 1996) and Renaul and Tomala 1998), which assume a fixed monioring nework. 7
8 f i,j 0) = 0, f i,j is nondecreasing, and f i,j is eiher sricly concave or idenically 0. ) ) lim x1 j i f i,j x 1 ) x 1 = lim x1 j i f j,i x 1 ) x 1 =. The assumpion ha f i,j is nondecreasing for all i j is essenial for inerpreing x j as player j s level of cooperaion. Noe ha he sage game is a prisoner s dilemma, in ha playing x i = 0 shirking ) is a dominan sraegy for player i in he sage game. The second assumpion saes ha he cos of cooperaion becomes infiniely greaer han he benefi for suffi cienly high levels of cooperaion. Concaviy and he assumpion ha u i x) is separable in x 1,..., x n ) play imporan roles in he analysis, and are discussed below. Every period, a monioring nework L = l i,j, ) i,j N N, l i,j, {0, 1}, is drawn independenly from a fixed probabiliy disribuion µ on {0, 1} n2. In addiion, higherorder informaion y = y i, ) i N, y i, Y i is drawn independenly from a probabiliy disribuion π y L ), where he Y i are arbirary finie ses. A he end of period, player i observes h i, = {z i,1,,..., z i,n,, y i, }, where z i,j, = x j, if l i,j, = 1, and z i,j, = if l i,j, = 0. Tha is, player i observes he acion of each of her ouneighbors and also observes he signal y i,, which may conain informaion abou who observes whom in period as well as informaion abou ohers informaion abou who observes whom, and so on). 5 The special case of perfec higherorder informaion is when y i, = L wih probabiliy 1 for all i N; his is he case where who observes whom is common knowledge while monioring of acions remains privae). Assume ha Pr l i,i = 1) = 1 for all i N; ha is, here is perfec recall. A repeaed game wih such a monioring srucure has nework monioring, he disribuion µ is he monioring echnology, and he pair Y = Y 1... Y n, π) is he higherorder informaion srucure. Le h i h i,0, h i,1,..., h i, 1 ) be player i s privae hisory a ime 1, and denoe he null hisory a he beginning of he game by h 0 = h 0 i for all i. A behavior) sraegy of player i s, σ i, specifies a probabiliy disribuion over period acions as a funcion of h i. Many imporan repeaed games have nework monioring, including random maching as in Kandori 1992) and Ellison 1994)) and monioring on a fixed nework where L is deerminisic and 5 As o wheher players observe heir realized sagegame payoffs, noe ha f i,j x j) can be inerpreed as player i s expeced benefi from player j s acion, and player i may only benefi from player j s acion when l i,j, = 1. However, some combinaions of assumpions on f i,j and µ are no consisen wih his inerpreaion, such as monioring on a fixed nework wih global public goods, where Pr l i,j, = 1) = 0 bu f i,j 0 for some i, j. An alernaive inerpreaion is required in hese cases: for example, he infinie ime horizon could be replaced wih an uncerain finie horizon wihou discouning, wih payoffs revealed a he end of he game and δ viewed as he probabiliy of he game s coninuing. The former inerpreaion is appropriae for he longdisance rade example, while he alernaive inerpreaion is appropriae for he school example. 8
9 consan, see Secion 5). For random maching, by changing he higherorder informaion srucure he model can allow for he case where players learn nohing abou who maches wih whom ouside heir own maches Y i = for all i), he case where who maches wih whom is common knowledge y i, = L wih probabiliy 1 for all i), or any inermediae case. For monioring on a fixed nework, however, players already know who maches wih whom, so higherorder informaion is irrelevan alhough echnically higherorder informaion could sill ac as a correlaing device in his case). To fix ideas, noe ha a repeaed game in which players observe he acions of heir neighbors on a random graph ha is deermined in period 0 and hen fixed for he duraion of play does no have nework monioring, because he monioring nework is no drawn independenly every period e.g., player i observes player j s acion in period 1 wih probabiliy 1 if she observes i in period 0, bu she does no observe player j s acion wih probabiliy 1 in period 0). Throughou, I sudy weak perfec Bayesian equilibria PBE) of his model wih he propery ha, for every player i, ime, and monioring nework L, for <, he sum )) ) ] n is welldefined; ha is, lim s s τ= δτ E [u i σ j h τ j L exiss. 6 j=1 τ= δτ E )) n [u i σ j h τ j This echnical resricion ensures ha players coninuaion payoffs are welldefined, condiional on any pas realized monioring nework. Fixing a descripion of he model oher han he higherorder informaion ) srucure ha is, a uple N, f i,j ) i,j N N, δ, µ le Σ P BE Y, π) be he se of PBE sraegy profiles when he higherorder informaion srucure is Y, π). Player i s level of cooperaion under sraegy profile σ is defined o be 1 δ) =0 δ E [ σ i h i )]. The main objec of ineres is he highes level of cooperaion for each player ha can be susained in PBE for any higherorder informaion srucure. Definiion 1 Player i s maximum cooperaion wih higherorder informaion srucure Y, π) is x i Y, π) Player i s robus maximum cooperaion is sup 1 δ) σ Σ P BE Y,π) δ E [ )] σ i h i. =0 x i inf Y,π) x i Y, π). Player i s robus maximum cooperaion is he highes level of cooperaion ha is sure o be j=1 ) ] L susainable in PBE for a given sage game, discoun facor, and monioring echnology. Pu 6 Recall ha a weak perfec Bayesian equilibrium is a sraegy profile and belief sysem in which, for every player i and privae hisory h i, player i s coninuaion sraegy is opimal given her beliefs abou he vecor of privae hisories ) h N j, and hese beliefs are updaed using Bayes rule whenever possible. j=1 9
10 differenly, i is he highes level of cooperaion ha an ouside observer who does no know he higherorder informaion srucure can be sure is susainable. This seems reasonable for applicaions like local public good provision or longrange rade, where i seems much more palaable o make assumpions only abou he probabiliy ha players observe each oher s acions he monioring echnology), raher han also making assumpions abou wha players observe abou each oher s observaions, wha hey observe abou wha ohers observe abou his, and so on. 7 One more definiion: a sraegy profile σ is higherorder informaion free if σ i h i ) does no depend on y i,τ ) 1 τ=0 for all i N. A higherorder informaion free sraegy profile can naurally be viewed as a sraegy profile in he game corresponding o any higherorder informaion srucure Y, π). So he following definiion makes sense. Definiion 2 For any player i N and level of cooperaion x i, a higherorder informaion free sraegy profile σ robusly susains x i if x i = 1 δ) =0 δ E [ σ i h i )] and σ ΣP BE Y, π) for every higherorder informaion srucure Y, π). This definiion is demanding, in ha a sraegy profile can robusly susain a level of cooperaion only if i is a PBE for any higherorder informaion srucure. However, my main heoreical resul Theorem 1) shows ha here exiss a grim rigger sraegy profile ha robusly susains all players robus maximum cooperaion simulaneously and he applied analysis in Secions 4 and 5 hen focuses on his equilibrium). The resuling equilibrium is paricularly imporan when i is also he PBE ha maximizes social welfare. This is he primary case of ineres in he lieraure on public good provision, where he focus is on providing incenives for suffi cien cooperaion, raher han on avoiding providing incenives for excessive cooperaion. For example, he grim rigger sraegy profile ha simulaneously robusly susains each player s maximum cooperaion also maximizes uiliarian social welfare if x i is below he firsbes level LindahlSamuelson benchmark) for every i N. Leing f j,i denoe he lefderivaive of f j,i which exiss by concaviy of f j,i ), his suffi cien condiion is f j,i x i ) 1 for all i N. This condiion can be checked easily using he formula for x i )n i=1 j i given by Theorem 1.8 As consequence, when his condiion holds, all of he comparaive saics on robus maximum cooper 7 However, I have implicily assumpion ha he higherorder informaion srucure is common knowledge among he players. Bu relaxing his would no affec he resuls. 8 I would of course be desirable o characerize he enire se of payoffs ha can be robusly susained in PBE, or a leas he enire Pareo fronier of his se, raher han only he equilibrium ha robusly susains maximum a 10
11 aion derived below are also comparaive saics on ineffi ciency relaive o he LindahlSamuelson benchmark. I also presen some quaniaive examples of he relaionship beween ineffi ciency and nework srucure in Secions 4 and 5. Before beginning he analysis, le me remark briefly on he moivaion for sudying his model. The model is inended o capure he essenial feaures of cooperaion in seings like hose discussed in he inroducion. Consider again he example of mainaining a school in a small village. In his seing, i is naural o hink ha villagers someimes observe each oher s conribuions quie accuraely bu someimes do no observe hem a all e.g., a villager migh usually know how hard her friends have been working on he school, and migh occasionally see someone else working, or learn ha someone else has conribued money), and ha i is very hard o observe he school s overall qualiy e.g., because school qualiy migh be bes measured by sudens labor marke oucomes in he disan fuure). This suggess ha repeaed game models wih possibly imperfec) pure public monioring are no wellsuied for sudying cooperaion in his seing. My model insead makes he opposie assumpion of pure nework monioring, and his leads o predicions ha are very differen from hose ha would emerge wih imperfec public monioring; for example, none of he four predicions enumeraed in he inroducion have been made in he lieraure on repeaed games wih imperfec public monioring, and hose predicions ha relae a player s locaion in a monioring nework o her maximum cooperaion canno possibly be made in such models. I will become clear ha my model is also very racable: given a monioring echnology, i is easy o calculae each player s robus maximum cooperaion. Of course, allowing players o access boh nework monioring and noisy public monioring which is cerainly more realisic han eiher pure public monioring or pure nework monioring remains a very ineresing direcion for fuure research. I discuss his possibiliy furher in he conclusion. 3 Characerizaion of Robus Maximum Cooperaion This secion presens he main heoreical resul of he paper, which shows ha all players robus maximum cooperaion can be robusly susained in grim rigger sraegies. To furher moivae he focus on robusness, Secion 3.1 presens an example showing ha, wih a given higherorder informaion srucure, maximum cooperaion may be susained by complicaed sraegies ha seem cooperaion. However, his problem appears inracable, jus as i seems inracable in general repeaed games wih imperfec privae monioring for fixed δ, raher han in he δ 1 limi). 11
12 1 probabiliy 1/2 2 probabiliy 1 3 Figure 1: An Example where Complex Sraegies are Opimal nonrobus. Secion 3.2 hen presens he main heoreical resul. 3.1 Opimaliy of Complex Sraegies wih Imperfec HigherOrder Informaion This secion shows by example ha for some higherorder informaion srucures a player s maximum level of cooperaion canno be susained in saionary) grim rigger sraegies. example here and defer he deails o he appendix. There are hree players, arranged as in Figure 1. probabiliy 1/2 and is never observed by player 3. Player 1 observes nohing. I skech he Player 1 is observed by player 2 wih Players 2 and 3 always observe each oher. The realized monioring nework drawn independenly every period) is unobserved; in paricular, player 3 does no observe when player 2 observes player 1 and when ) ) he does no formally, Y i = for all i). For each player i, u i x j ) 3 j=1 = j i xj x i, and δ = 1/2. I is sraighforward o show ha player 1 s maximum cooperaion in grim rigger sraegies equals 0.25 see he appendix). I now skech a sraegy profile in which player 1 s maximum cooperaion equals Player 1 always plays x 1 = onpah. Players 2 and 3 each have wo onpah acions, denoed x L 2, xh 2, xl 3, and xh 3, wih xl 2 < xh 2 and xl 3 < xh 3. Player 2 plays x 2 = x H 2 subsequen oddnumbered periods, player 2 plays x H 2 period 1 acion, and oherwise plays each of x H 2 and xl 2 evennumbered periods, player 2 plays x H 2 acion, and oherwise plays each of x H 2 and xl 2 player 1 s acion in evennumbered period, he hen plays x H and + 2. Finally, player 3 plays x 3 = x H 3 in period 0. A wih probabiliy 1 if he observed player 1 s wih probabiliy 1/2. A subsequen wih probabiliy 1 if he observed player 1 s period 2 wih probabiliy 1/2. Thus, if player 2 observes wih probabiliy 1 in boh periods in period 0, and in every period 1 he plays x H 3 if player 2 played xh 2 in period 1, and plays xl 3 if player 2 played xl 2 in period 1. If any player i observes a deviaion from his specificaion of onpah play i.e., if any player deviaes herself; if player 2 observes x or observes player 3 failing o ake her prescribed acion; or if player 3 observes x 2 / { x L 2, } xh 2 ), she hen plays xi = 0 in all subsequen periods. In he 12
13 appendix, I specify x L 2, xh 2, xl 3, and xh 3, and verify ha he resuling sraegy profile is a PBE. Why can sraegies of his form susain greaer maximum cooperaion by player 1 han grim rigger sraegies can? The key is ha he difference beween player 1 s expecaion of player 3 s average fuure cooperaion when player 1 cooperaes and when player 1 shirks, condiional on he even ha player 2 observes player 1 which is he only even ha maers for player 1 s incenives), is larger han wih grim rigger sraegies. To undersand his, consider wha happens afer period 2 sees player 1 play in period 1, for odd. Condiional on his even, player 1 s expecaion of player 3 s acion in boh periods + 1 and + 2 equals x H 3 ; bu player 3 s expecaion of his own acion in period + 2 afer seeing player 2 play x H 2 x H 3 in period is less han, because he is no sure ha player 2 observed player 1 in period 1. Indeed, if player 3 were sure ha player 2 had observed player 1 in period 1, he would no be willing o play x H 3 would have o play x H 3 as he in period + 2 in addiion o + 1). Thus, he disagreemen beween player 1 s expecaion of player 3 s average fuure cooperaion condiional on player 2 observing player 1) and player 3 s uncondiional) expecaion of his own average fuure cooperaion improves player 1 s incenive o cooperae wihou causing player 3 o shirk. Noe ha all his example direcly proves is ha player 1 s maximum cooperaion is no susainable in grim rigger sraegies. However, i is no hard o show ha any sraegies ha susain more cooperaion han is possible wih grim rigger mus involve rewards, in ha onpah acions mus someimes increase from one period o he nex. This observaion places a lower bound on how complicaed he sraegies ha do susain player 1 s maximum cooperaion in he example mus be, even hough acually compuing hese sraegies seems inracable Robus Opimaliy of Grim Trigger Sraegies This secion shows ha all players robus maximum cooperaion can be robusly susained in grim rigger sraegies, defined as follows. Definiion 3 A sraegy profile σ is a grim rigger sraegy profile if here exis acions x i ) n i=1 such ha σ i h i ) = xi if z i,j,τ {x j, } for all z i,j,τ h i,τ and all τ <, and σ i h i ) = 0 oherwise. In a grim rigger sraegy profile player i s acion a an offpah hisory h i does no depend on he ideniy of he iniial deviaor. In paricular, by perfec recall, player i plays x i = 0 in every 9 I is also rivial o modify his example o show ha a player s payoff need no be maximized by grim rigger sraegies: simply add a fourh player, observed by no one, who only values player 1 s conribuions. 13
14 period following a deviaion by player i herself. Also, if a grim rigger sraegy profile σ susains each player s robus maximum cooperaion, hen under σ each player i plays x i a every onpah hisory. Finally, grim rigger sraegy profiles are clearly higherorder informaion free. Nex, I inroduce an imporan piece of noaion: define D τ,, i) recursively by D τ,, i) = if τ < D,, i) = {i} D τ + 1,, i) = {j : z j,k,τ = x k,τ for some k D τ,, i)} if τ. Tha is, D τ,, i) is he se of players in period τ who have observed a player who has observed a player who has observed... player i since ime. By perfec recall, D τ + 1,, i) D τ,, i) for all τ,, and i. The se D τ,, i) is imporan because j D τ,, i) is a necessary condiion for player j s ime τ hisory o vary wih player i s acions a imes afer. In paricular, if players are using grim rigger sraegies and player i shirks a ime, hen D τ,, i) is he se of players who shirk a ime τ. Noe ha he probabiliy disribuion of D τ,, i) is he same as he probabiliy disribuion of D τ, i) D τ, 0, i), for all i and τ. I now sae he main heoreical resul of he paper. Theorem 1 There is a grim rigger sraegy profile σ ha robusly susains each player s robus maximum cooperaion. Furhermore, he vecor of players robus maximum cooperaion x i )n i=1 is he componenwise) greaes vecor x i ) n i=1 such ha x i = 1 δ) δ Pr j D, i)) f i,j x j ) for all i N. 1) j i =0 Given ha grim rigger sraegies susain each player s robus maximum cooperaion, equaion 1) is almos immediae: he lefhand side of 1) is he cos o player i of conforming o σ ; and he righhand side of 1) is he benefi o player i of conforming o σ, which is ha, if player i deviaed, she would lose her benefi from player j s cooperaion whenever j D, i). Thus, 1) saes ha he vecor of robus maximum cooperaion is he greaes vecor of acions ha equalizes he cos and benefi of cooperaion for each player. he vecor x i )n i=1, as discussed in foonoe 21 in he appendix. In addiion, i is easy o compue Thus, he subsance of Theorem 1 is showing ha grim rigger sraegies susain each player s robus maximum cooperaion. As shown above, grim rigger sraegies do no susain each player s maximum cooperaion wih every higherorder informaion srucure. 14 However, if one shows ha
15 a grim rigger sraegy profile σ susains each player i s maximum cooperaion x i wih some higherorder informaion srucure, hen his implies ha boh x i x i by definiion of x i ) and x i x i because σ mus robusly susain x i ), 10 so Theorem 1 follows. implies Theorem 1. Hence, he following key lemma Lemma 1 The grim rigger sraegy profile wih onpah acions given by 1) susains each player s maximum cooperaion wih perfec higherorder informaion. Lemma 1 is also of ineres in is own righ, as i shows ha grim rigger sraegies maximize cooperaion when higherorder informaion is perfec. For example, Lemma 1 implies ha grim rigger sraegies always maximize cooperaion for fixed monioring neworks, as wih fixed monioring neworks who observes whom is always common knowledge. Since grim rigger sraegies are higherorder informaion free, Lemma 1 also implies ha each player s maximum cooperaion wih perfec higherorder informaion is weakly less han her maximum cooperaion wih any oher higherorder informaion srucure. The key idea behind Lemma 1 is ha a player is willing o cooperae weakly) more a any onpah hisory if any oher player cooperaes more a any onpah hisory, because he firs player is more likely o benefi from his increased cooperaion when she conforms han when she deviaes. 11 Thus, here is a kind of sraegic complemenariy beween he acions of any wo players a any wo onpah hisories. This suggess he following proof of Lemma 1: Define a funcion φ ha maps he vecor of all players onpah acions a every onpah hisory, x, o he vecor of he highes acions ha each player is willing o ake a each onpah hisory when acions a all oher onpah hisories are as in x, and players shirk a offpah hisories. Le X be an acion greaer han any onpah PBE acion, and le X be he vecor of onpah acions X. By complemenariy among onpah acions, ieraing φ on X yields a sequence of vecors of onpah acions ha are all consan across periods and weakly greaer han he greaes fixed poin of φ, and his sequence converges monoonically o he greaes fixed poin of φ. Therefore, he greaes fixed poin of φ is consan across periods, and i provides an upper bound on each player s maximum cooperaion. Finally, verify ha he grim rigger sraegy profile wih onpah acions given by he greaes fixed 10 I is no diffi cul o show ha if a grim rigger sraegy profile susains a player s maximum cooperaion x i wih some higherorder informaion srucure hen i robusly susains x i. See he appendix. 11 This observaion relies on he assumpion of nework monioring, since oherwise a deviaion by he firs player may make some onpah hisories more likely. 15
16 poin of φ is a PBE. 12 The problem wih his proof and here mus be a problem, because he proof does no menion perfec higherorder informaion) is ha, while he highes acion ha a player is willing o ake a any onpah hisory is nondecreasing in every oher player s onpah acions, i is decreasing in her own fuure onpah acions. Tha is, a player is no willing o cooperae as much oday when she knows ha she will be asked o cooperae more omorrow. Hence, he funcion φ as defined in he previous paragraph is no isoone, and hus may no have a greaes fixed poin. This problem ) may be addressed by working no wih players sagegame acions σ i h i, bu raher wih heir coninuaion acions X i 1 δ) τ δτ σ i h τ i ). Indeed, i can be shown ha E [ Xi h ] i δ τ [ Pr j D τ,, i) \D τ 1,, i)) f i,j E X τ j h i, j D τ,, i) \D τ 1,, i) ]), j i τ= for every player i and onpah hisory h i. The inuiion for his inequaliy is ha, if player i shirks a ime, hen player j sars shirking a ime τ wih probabiliy Pr j D τ,, i) \D τ 1,, i)), ]) and his yields los benefis of a leas f i,j E [X j τ h i, j D τ,, i) \D τ 1,, i) o player i. This inequaliy yields an upper bound on player i s expeced coninuaion acion, E [ X i h i], in erms of her expecaion of oher players coninuaion acions only. This raises he possibiliy ha he funcion φ could be isoone when defined in erms of coninuaion acions Xi, raher han sagegame acions. For an approach along hese lines o work, however, one mus be able o ] [ ] express E [X j τ j D τ,, i) \D τ 1,, i) in erms of E Xj τ hτ j for player j s privae hisories h τ j. Wih perfec higherorder informaion bu no oherwise), E [ X τ j j D τ,, i) \D τ 1,, i) ] = E [ E [ X τ j h τ j ] j D τ,, i) \D τ 1,, i) ], so such an approach is possible Equal Monioring This secion imposes he assumpion ha all players acions are equally wellmoniored in a sense ha leads o sharp comparaive saics resuls. In paricular, assume hroughou his secion: 12 For his las sep, one migh be concerned ha grim rigger sraegies do no saisfy offpah incenive consrains, as a player migh wan o cooperae offpah in order o slow he conagion of defecing, as in Kandori 1992) and Ellison 1994). As discussed in he inroducion, his problem does no arise wih coninuous acions and payoffs. 13 The assumpions ha payoffs are concave and separable are also necessary. Wihou concaviy, PBE acions could be scaled up indefiniely. Wihou separabiliy, higher cooperaion may be susained when players ake urns cooperaing see Example A1 in he online appendix). 16
17 Parallel Benefi Funcions: There exiss a funcion f : R + R + and scalars α i,j R + such ha f i,j x) = α i,j f x) for all i, j N and all x R +. Equal Monioring: for all i, j N. =0 δ k i Pr k D, i)) α i,k = =0 δ k j Pr k D, j)) α j,k Parallel benefi funcions imply ha he imporance of player j s cooperaion o player i may be summarized by a real number α i,j. Wih his assumpion, equal monioring saes ha he expeced discouned number of players who may be influenced by player i s acion, weighed by he imporance of heir acions o player i, is he same for all i N. To help inerpre hese assumpions, noe ha if α i,j is consan across players i and j hen, for generic discoun facors δ, equal monioring holds if and only if E [#D, i)] = E [#D, j)] for all i, j N and N; ha is, if and only if he expeced number of players who find ou abou shirking by player i wihin periods is he same for all i N. Secion 4.1 derives a simple and general formula for comparaive saics on robus maximum cooperaion under equal monioring. Secions 4.2 and 4.3 apply his formula o he leading special case of global) public good provision, where α i,j = α for all i j; ha is, where all players value each oher s acions equally. Secion 4.2 sudies he effec of group size on public good provision, and Secion 4.3 considers he effec of uncerainy in monioring on public good provision. Finally, he higherorder informaion srucure plays no role in his secion or he following one, because hese secions sudy comparaive saics on players maximum robus cooperaion, which is independen of he higherorder informaion srucure by definiion. 4.1 Comparaive Saics Under Equal Monioring The secion derives a formula for comparaive saics on robus maximum cooperaion under equal monioring. The firs sep is noing ha each player s robus maximum cooperaion is he same under equal monioring proof in appendix). Corollary 1 Wih equal monioring, x i = x j for all i, j N. Thus, under equal monioring each player has he same robus maximum cooperaion x. wish o characerize when x is higher in one game han anoher, when boh games saisfy equal monioring and have he same underlying benefi funcion f. Formally, a game wih equal monioring Γ = N, α i,j ) i,j N N ), δ, µ is a model saisfying he assumpions of Secion 2 as well as equal I 17
18 monioring. For any game wih equal monioring Γ, le x Γ) be he robus maximum cooperaion in Γ, and le B Γ) 1 δ) δ Pr j D, i)) α i,j j i =0 be player i s benefi of cooperaion i.e., he righhand side of 1)) when f x j ) = 1 for all j N, which is independen of he choice of i N by equal monioring. The comparaive saics resul for games wih equal monioring is he following: Theorem 2 Le Γ and Γ be wo games wih equal monioring. B Γ), wih sric inequaliy if B Γ ) > B Γ) and x Γ ) > 0. Then x Γ ) x Γ) if B Γ ) Proof. Since x i = x for all i N, 1) may be rewrien as x = 1 δ) δ Pr j D, i)) α i,j f x ) = B Γ) f x ). j i =0 Hence, x Γ) is he greaes zero of he concave funcion B Γ) f x) x. If B Γ ) B Γ), hen B Γ ) f x Γ)) x Γ) B Γ) f x Γ)) x Γ) = 0, which implies ha x Γ ) x Γ). If B Γ ) > B Γ) and x Γ ) > 0, hen eiher x Γ) = 0 in which case x Γ ) > x Γ) rivially) or x Γ) > 0, in which case B Γ ) f x Γ)) x Γ) > B Γ) f x Γ)) x Γ) = 0, which implies ha x Γ ) > x Γ). Theorem 2 gives a complee characerizaion of when x Γ) is greaer or less han x Γ ), for any wo games wih equal monioring Γ and Γ. In paricular, robus maximum cooperaion is greaer when he expeced discouned number of players who may be influenced by a player s acion, weighed by he imporance of heir acions o ha player, is greaer. For example, in he case of global public good provision where all players value all oher players acions equally), robus maximum cooperaion is greaer when he ses D, i) are likely o be larger; while if each player only values he acions of a subse of he oher players her geographic neighbors, her rading parners, ec.), hen robus maximum cooperaion is greaer when he inersecion of he ses D, i) and he se of players whose acions player i values is likely o be larger. Hence, Theorem 2 characerizes how differen monioring echnologies susain differen kinds of cooperaive behaviors. 4.2 The Effec of Group Size on Global Public Good Provision This secion uses Theorem 2 o analyze he effec of group size on robus maximum cooperaion in he leading special case of global public good provision, where α i,j = α for all i j. 18
19 In he case of global) public good provision, B Γ) = α δ E [#D, i)] 1). =0 Thus, for public goods, all he informaion needed o deermine wheher changing he game increases or decreases he robus) maximum per capia level of public good provision is conained in he produc of wo erms: he marginal benefi o each player of public good provision, α, and 1/ 1 δ) less han) he effecive conagiousness of he monioring echnology, =0 δ E [#D, i)]. Informaion such as group size, higher momens of he disribuion of #D, i), and which players are more likely o observe which oher players are no direcly relevan. In paricular, he single number =0 δ E [#D, i)] he effecive conagiousness compleely deermines he effeciveness of a monioring echnology in supporing public good provision. This finding ha comparaive saics on he percapia level of public good provision are deermined by he produc of he marginal benefi of he public good o each player and he effecive conagiousness of he monioring echnology yields useful inuiions abou he effec of group size on he per capia level of public good provision. In paricular, index a game Γ by is group size, n, and wrie α n) for he corresponding marginal benefi of conribuions and =0 δ E [#D, n)] for he effecive conagiousness I use his simpler noaion for he remainder of his secion). Normally, one would expec α n) o be decreasing in n a larger populaion reduces player i s benefi from player j s conribuion o he public good) and =0 δ E [#D, n)] o be increasing in n a larger populaion makes i more likely ha player i s acion is observed by more individuals), yielding a radeoff beween he marginal benefi of conribuions and he effecive conagiousness. Consider again he example of consrucing a local infrasrucure projec, like a well. In his case, α n) is likely o be decreasing and concave: since each individual uses he well only occasionally, here are few exernaliies among he firs few individuals, bu evenually i sars o becomes difficul o find imes when he well is available, waer shorages become a problem, ec.. Similarly, =0 δ E [#D, n)] is likely o be increasing, and may be concave if here are congesion effecs in monioring. Thus, i seems likely ha in ypical applicaions α n) =0 δ E [#D, n)] 1), and herefore per capia public good provision, is maximized a an inermediae value of n. Theorem 2 yields paricularly simple comparaive saics for he leading cases of pure public goods α n) = 1) and divisible public goods α n) = 1/n), which are useful in examples below. Corollary 2 Wih pure public goods α n) = 1), if E [#D, n )] E [#D, n)] for all hen x n ) x n), wih sric inequaliy if E [#D, n )] > E [#D, n)] for some 1 and x n ) > 19
20 0. Wih pure public goods, x n) is increasing unless monioring degrades so quickly as n increases ha he expeced number of players who find ou abou a deviaion wihin periods is decreasing in n, for some. This suggess ha x n) is increasing in n in many applicaions. Corollary 3 Wih divisible public goods α n) = 1/n), if E [#D, n )] 1) /n E [#D, n)] 1) /n for all hen x n ) x n), wih sric inequaliy if E [#D, n )] 1) /n > E [#D, n)] 1) /n for some 1 and x n ) > 0. Wih divisible public goods, x n) is increasing only if he expeced fracion of players oher han he deviaor herself) who find ou abou a deviaion wihin periods is nondecreasing in n, for all. This suggess ha, wih divisible public goods, x n) is decreasing in many applicaions. The following wo examples demonsrae he usefulness of Theorem 2 and Corollaries 2 and 3. An earlier version of his paper available upon reques) conains addiional examples Random Maching Monioring is random maching if in each period every player is linked wih one oher player a random, and l i,j, = l j,i, for all i, j N and all. This is possible only if n is even. I can be show ha, wih random maching, E [#D, n)] is nondecreasing in n and is increasing in n for = 2. Therefore, Corollary 2 implies ha, wih pure public goods, robus maximum cooperaion is increasing in group size. Proposiion 1 Wih random maching and pure public goods, if n > n hen x n ) x n), wih sric inequaliy if x n ) > 0. However, i can also be shown ha =0 δ E [#D, n )] 1) /n < =0 δ E [#D, n)] 1) /n whenever n > n, n and n are suffi cienly large, and δ < 1/2. In his case, Theorem 2 implies ha, wih divisible public goods, robus maximum cooperaion is decreasing in group size. Proposiion 2 Wih random maching and divisible public goods, if δ < 1 2 hen, for any γ > 0, here exiss N such ha x n ) x n) if n > 1 + γ) n N, wih sric inequaliy if x n ) > 0. 20
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