Cooperation with Network Monitoring

Transcription

1 Cooperaion wih Nework Monioring Alexander Wolizky Microsof Research and Sanford Universiy November 2011 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 second-order 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 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 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, 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 local public good, like a well, school, or road, in a small village. 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 o he public good is publicly observable, bu i would probably be ideal o assume insead ha each household s conribuion is only observed by a subse of he oher households; 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 long-disance 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 nework-like insiuions like rading coaliions Greif 1989, 1993, rade fairs Milgrom, Noh, 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 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 2

3 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 acion or abou heir informaion abou ohers informaion abou acions, and so on, and who herefore mus make predicions ha are robus o his higher-order informaion. 1 A firs observaion is ha for any given specificaion of players higher-order 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 ha 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 her 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 higher-order 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 on-pah hisory whenever anoher player cooperaes more a any on-pah 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 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 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 This resul is hen used o show ha cooperaion in he provision of pure public goods where he marginal benefi of cooperaion is independen of group size is ypically greaer in large groups, while cooperaion in he provision of divisible public goods where he marginal benefi of cooperaion is inversely proporional o group size is ypically greaer in small groups, as well as ha making monioring more uncerain 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 graph-heoreic propery of cenraliy wih he game-heoreic 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 beer-conneced 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 srand sudies one-sho games of public good provision in neworks Balleser, Calvó-Armengol, 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 free-riding 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 free-riding, and ha adding links o a nework decreases average maximum cooperaion by increasing free-riding. 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 Pecorino 1999 shows ha wih perfec monioring public good provision is easier in large groups, because shirking and hus causing everyone else o sop cooperaing 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 assumpion of perfec monioring, and find an ambiguous relaionship beween group size and maximum cooperaion. 4

5 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, 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 informaion-sharing insiuions are no always necessary. The curren paper conribues o his lieraure by deermining he maximum level of cooperaion in a prisoner s dilemma-like 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 informaion-sharing 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, Rodriguez-Barraquer, and Tan, 2011 and risk-sharing Ambrus, Möbius, and Szeidl, 2010; Bramoullé and Kranon, 2007b; Bloch, Genico, and Ray, The predicions of his paper enumeraed above can be suggesively compared o some early empirical resuls in his lieraure, 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, Rodriguez-Barraquer, and Tan 2011 find ha favor-exchange 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 5

6 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 on-pah 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 on-pah, and hen noing ha cooperaion is more appealing on-pah han off-pah since off-pah 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 on-pah, as oherwise hey could be asked o cooperae slighly more. argumen as in Ellison, his implies ha players weakly prefer o shirk off-pah. By essenially he same Hence, he key conribuion of his paper is showing ha grim rigger sraegies provide he sronges possible incenives for robus cooperaion on-pah, no ha hey provide incenives for shirking off-pah. 3 The mos closely relaed paper is conemporaneous and independen work by Ali and Miller Ali and Miller sudy a nework game in which links beween players are recognized according o a Poisson process so he underlying nework is fixed over ime. 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 on-pah incenive consrains imply slack off-pah incenive consrains. The mos imporan difference 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, 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 firs-bes payoffs are no aainable. I make no aemp o characerize he enire se of sequenial equilibria, or 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. 4 Of he many privae monioring folk heorem papers, he mos relaed are probably Ben-Porah and Kahneman 1996 and Renaul and Tomala 1998, which assume a fixed monioring nework. 6

7 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. Appendix A conains omied examples, and Appendix B conains omied proofs. 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 f i,j 0 = 0, f i,j is non-decreasing, 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 non-decreasing 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, higher-order informaion y = y i, i N, y i, Y i is drawn independenly from a probabiliy disribuion π y L, where he Y i 7

8 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 ou-neighbors 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 higher-order 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 higher-order 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. σ i, specifies a probabiliy disribuion over period acions as a funcion of h i. A behavior sraegy of player i s, 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 consan, see Secion 5. For random maching, by changing he higher-order 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 higher-order informaion is irrelevan alhough echnically higher-order 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. 5 As o wheher players observe heir realized sage-game 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. 8

9 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 τ= δτ E [u i σ j h τ j ] n is well-defined; ha is, lim s s τ= δτ E [u i σ j h τ j L exiss. 6 This echnical resricion ensures ha players coninuaion payoffs are well-defined, condiional on any pas realized monioring nework. Fixing a descripion of he model oher han he higher-order 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 higher-order 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 higher-order informaion srucure. Definiion 1 Player i s maximum cooperaion wih higher-order informaion srucure Y, π is x i Y, π sup 1 δ δ E [ ] σ i h i. σ Σ P BE Y,π Player i s robus maximum cooperaion is =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 susainable in PBE for a given sage game, discoun facor, and monioring echnology. differenly, i is he highes level of cooperaion ha an ouside observer who does no know he higher-order informaion srucure can be sure is susainable. j=1 Pu This seems reasonable for applicaions like local public good provision or long-range 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 higher-order informaion free if σ i h i does no depend on y i,τ 1 τ=0 for all i N. A higher-order informaion free sraegy profile can naurally be viewed as a sraegy profile in he game corresponding o any higher-order informaion srucure Y, π. So he following definiion makes sense. 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. 7 j=1 However, I have implicily assumpion ha he higher-order informaion srucure is common knowledge among j=1 ] L he players. Bu relaxing his would no affec he resuls. 9

10 Definiion 2 For any player i N and level of cooperaion x i, a higher-order 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 higher-order 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 higher-order 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 firs-bes level for every i N. Leing f j,i lef-derivaive 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. j i denoe he This condiion can be checked easily using he formula for x i n i=1 given by Theorem 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 higher-order informaion srucure, maximum cooperaion may be susained by complicaed sraegies ha seem non-robus. Secion 3.2 hen presens he main heoreical resul. 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 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. 10

11 1 probabiliy 1/2 2 probabiliy 1 3 Figure 1: An Example where Complex Sraegies are Opimal 3.1 Opimaliy of Complex Sraegies wih Imperfec Higher-Order Informaion This secion shows by example ha for some higher-order informaion srucures a player s maximum level of cooperaion canno be susained in saionary grim rigger sraegies. example here and defer he deails o Appendix A. 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 Appendix A. I now skech a sraegy profile in which player 1 s maximum cooperaion equals Player 1 always plays x 1 = on-pah. Players 2 and 3 each have wo on-pah 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 odd-numbered periods, player 2 plays x H 2 period- 1 acion, and oherwise plays each of x H 2 and xl 2 even-numbered periods, player 2 plays x H 2 acion, and oherwise plays each of x H 2 and xl 2 player 1 s acion in even-numbered 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 on-pah 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 Appendix A, 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 11

12 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 2 ; bu player 3 s expecaion of his own acion in period + 2 afer seeing player 2 play x H 2 x H 2 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 2 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 on-pah 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. 3.2 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 off-pah 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 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 on-pah hisory. Finally, grim rigger sraegy profiles are clearly higher-order informaion free. 12

13 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 = D τ,, 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. The se 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. ha Furhermore, x i n i=1 is he componen-wise greaes vecor x i n i=1 such 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 lef-hand side of 1 is he cos o player i of conforming o σ ; and he righ-hand 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 highes vecor of acions ha equalizes he cos and benefi of cooperaion for each player. x i n i=1, as discussed in foonoe 20 in Appendix B. In addiion, i is easy o compue he vecor 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 higher-order informaion srucure. However, if one shows ha 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, 9 so Theorem 1 follows. Hence, he following key lemma 9 I is no diffi cul o show ha if a grim rigger sraegy profile susains each player s maximum cooperaion x i wih some higher-order informaion srucure hen i robusly susains x i. See Appendix A. 13

14 implies Theorem 1. Lemma 1 The grim rigger sraegy profile wih on-pah acions given by 1 susains each player s maximum cooperaion wih perfec higher-order informaion. Lemma 1 is also of ineres in is own righ, as i shows ha grim rigger sraegies maximize cooperaion when higher-order informaion is perfec. For example, a sraighforward consequence of Lemma 1 is ha grim rigger sraegies always maximize cooperaion for fixed monioring neworks, as wih fixed monioring neworks who observes whom is always common knowledge. 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 on-pah hisory, because he firs player is more likely o benefi from his increased cooperaion when she conforms han when she deviaes. 10 Thus, here is a kind of sraegic complemenariy beween he acions of any wo players a any wo on-pah hisories. This suggess he following proof of Lemma 1: Define a funcion φ ha maps he vecor of all players on-pah acions a every on-pah hisory, x, o he vecor of he highes acions ha each player is willing o ake a each on-pah hisory when acions a all oher on-pah hisories are as in x, and players shirk a off-pah hisories. Le X be an acion greaer han any on-pah PBE acion, and le X be he vecor of on-pah acions X. By complemenariy among on-pah acions, ieraing φ on X yields a sequence of vecors of on-pah 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 on-pah acions given by he greaes fixed poin of φ is a PBE. 11 The problem wih his proof and here mus be a problem, because he proof does no menion perfec higher-order informaion is ha, while he highes acion ha a player is willing o ake a any on-pah hisory is non-decreasing in every oher player s on-pah acions, i is decreasing in her own fuure on-pah 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 10 This observaion relies on he assumpion of nework monioring, since oherwise a deviaion by he firs player may make some on-pah hisories more likely. 11 For his las sep, one migh be concerned ha grim rigger sraegies do no saisfy off-pah incenive consrains, as a player migh wan o cooperae off-pah in order o slow he conagion of defecing, as in Kandori 1992 and Ellison As discussed above, his problem does no arise wih coninuous acions and payoffs. 14

15 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 sage-game 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 on-pah 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 sage-game 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 higher-order 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 well-moniored in a sense ha leads o sharp comparaive saics resuls. In paricular, assume hroughou his secion: 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 12 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 Appendix A. 15

16 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 higher-order 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 higher-order 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. 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 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 righ-hand side of 1 when f x j = 1 for all j N, which is independen of he choice of i N by equal monioring. for games wih equal monioring is he following: The comparaive saics resul I 16

17 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 higher 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 higher. 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. 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 one 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 17

18 are more likely o observe which oher players are all irrelevan. 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 per-capia 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 > 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. 18

19 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 non-decreasing 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 each 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 non-decreasing 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 > Monioring on a Circle Monioring is on a circle if he players are arranged in a fixed circle and here exiss an ineger d 1 such ha l i,j, = 1 if and only if he disance beween i and j is a mos d. I is a sraighforward consequence of Corollary 2 ha robus maximum cooperaion is increasing in group size wih monioring on a circle and pure public goods. Proposiion 3 Wih monioring on a circle and pure public goods, if n > n hen x n x n, wih sric inequaliy if x n > 0. Finally, Corollary 3 implies ha robus maximum cooperaion is decreasing in group size wih monioring on a circle and divisible public goods. 19

20 Proposiion 4 Wih monioring on a circle and divisible public goods, if n > n hen x n x n, wih sric inequaliy if d < n /2 and x n > The Effec of Uncerain Monioring on Global Public Good Provision This secion provides a resul comparing monioring echnologies in erms of he maximum level of robus global public good provision hey suppor, for a fixed group size. As discussed in he previous subsecion, a monioring echnology suppors greaer robus maximum cooperaion in global public good provision if and only if i has greaer effecive conagiousness, =0 δ E [#D ], where he parameer n is omied because i is held fixed in his subsecion. I compare less cerain monioring, where i is likely ha eiher a large or small fracion abou he populaion finds ou abou a deviaion, wih more cerain monioring, where i is likely ha an inermediae fracion of he populaion finds ou abou i, in he sense of second-order sochasic dominance. fairly broad condiions, more cerain monioring suppors greaer robus maximum cooperaion. Under The analysis of his subsecion relies on he following assumpion, which saes ha he disribuion over #D + 1 depends only on #D. There exiss a family of funcions {g k : {0,..., n} [0, 1]} n k=1 #D = k, Pr #D + 1 = k = g k k, for all, k, and k. wih such ha, whenever This assumpion is saisfied by random maching, for example, bu no by monioring on a circle, because wih monioring on a circle he disribuion of #D + 1 depends on he ideniies of he of he members of D. Given a probabiliy mass funcion g k, define he corresponding disribuion funcion G k k k s=0 g k s. Recall ha a disribuion G k sricly second-order sochasically dominaes G k if n s=0 η s g k s > n s=0 η s g k s for all increasing and sricly concave funcions η : R R. The following resul compares monioring under { g k } n k=1 and {g k} n k=1. Theorem 3 Suppose ha Gk k and G k k are decreasing and sricly convex in k for k {0,..., k } and k {0,..., n}, and ha G k sricly second-order sochasically dominaes G k for k {1,..., n 1}. Then robus maximum cooperaion is sricly greaer under a monioring echnology corresponding o { g k } n k=1 han under a monioring echnology corresponding o {g k } n k=1. The inuiion for Theorem 3 is fairly simple: If G k sricly second-order sochasically dominaes G k for all k, hen under G k i is more likely ha an inermediae number of players find ou abou an 20

21 iniial deviaion each period. Since G k k and G k k are decreasing and convex in k, he expeced number of players who find ou abou he deviaion wihin periods increases in more quickly when i is more likely ha an inermediae number of players find ou abou he deviaion each period. Hence, =0 δ E [#D ] is sricly higher under a monioring echnology corresponding o { g k } n k=1 han under a monioring echnology corresponding o {g k } n k=1, and he heorem hen follows from Theorem 2. 5 Fixed Monioring Neworks This secion sudies boh global and local public good provision wih nework monioring when he monioring nework is fixed over ime. Tha is, hroughou his secion I make he following assumpion on he deerminisic monioring echnology. Fixed Undireced Monioring Nework: There exiss a nework L = l i,j i,j N N such ha l i,j, = l i,j = l j,i for all. I also assume ha he sage game saisfies one of he following wo properies, where N i is he se of player i s neighbors in L. Global Public Goods: u i x = j i f x j x i. Local Public Goods: u i x = j Ni f x j x i. The exensions of all of he resuls in his secion o direced neworks is sraighforward. discuss below where he assumpion of global or local public goods can be relaxed. Secion 5.1 inroduces a new definiion of cenraliy in neworks, and uses Theorem 1 o show ha more cenral players have greaer robus maximum cooperaion. Secion 5.2 shows ha cenraliy can also be used o deermine when one nework dominaes anoher in erms of supporing cooperaion. Finally, Secion 5.3 remarks on he sabiliy of monioring neworks, emphasizing differences beween he cases of global and local public goods. I 5.1 Cenraliy and Robus Maximum Cooperaion Theorem 1 provides a general characerizaion of players robus maximum cooperaion as a funcion of he discoun facor and benefi funcions. Here, I provide a parial ordering cenraliy of players in erms of heir nework characerisics under which higher players have greaer robus 21