The Collective Action Problem: Within-Group Cooperation and ...

The Collective Action Problem: Within-Group CooperationandBetween-GroupCompetitioninaRent-SeekingGame ∗Guillaume CHEIKBOSSIANUniversité de Montpellier 1 and GREMAQ, Université de ToulouseManufacture des Tabacs (Bâtiment F)21AlléedeBrienne31000 Toulouse, FranceTel:(33)561128554Fax:(33)561225563E-mail: guillaume.cheikbossian@univ-tlse1.frMay 5, 2006AbstractOlson’s thesis argues that the free-rider problem makes larger groups less effective thansmaller groups in pursuing their interest. In this paper, we challenge this view of groupaction with a very simple contest game that exhibits a bilateral interplay between intergroupinteraction and within-group organization. In a static setting, because of the free-ridingincentives, the larger group is disadvantaged in the competition with its rival. When groupsinteract in a dynamic environment, individuals within each group may be able to achievetacit cooperation. We then show that the larger group has more chance to enforce withingroupcooperation than the smaller group. In accordance with many informal and formalobservations, the larger group can thus produce a higher level of collective action.Keywords: Collective Action, Rent-seeking, Within-group cooperation, Between-groupcompetition, Repeated game.JEL Classification: D72; D74; C72; C73.∗ I thank Léonard Dudley, Patrick Gonzalez, Bruno Jullien, Laurent Linnemer, Nicolas Marceau, Wilfried Sand-Zantman, Etienne Wasmer and seminar participants at the Université de Montpellier, Université de Toulouse,Université de Montréal, UQAM (Montréal) and at the SCSE and Canadian Economic Association 2006 annualconferences for comments and suggestions. This paper was completed while I was visiting the Department ofEconomics at Université de Montreal. I would like to thank the Economics Department for its hospitality duringmy visit in 2005-06. The usual disclaimer applies.1

to be an important resource for interest groups with economic concerns such as labor unions orfarm lobbies. Indeed, it is well recognized, although there is no consensus on the direction of thecausality, that there is a positive relationship between effectiveness and size of unions (see e.g.Checchi and Lucifora, (2002)). In the farming context, scholars also agree that the considerableinfluence of farmers’ unions through the European Common Agricultural Policy is related totheir extraordinary capacity to organize broadly at a national level. For instance, Keeler (1996,134) writes: "In the four most populous EC states the largest farmers’ unions alone claims amembership density much higher than all labor unions...As case studies of agricultural politicsat the national level make clear, this density is one of the more important "secrets" behind thefarm lobby’s disproportionate clout".If the reward of group action is purely public so that a group member’s payoff is unaffectedby the number of members, the "common wisdom" of group action is actually consistent with the"divide-and-conquer" maxim i.e. the smaller the group, the lower is its aggregate effectiveness.But in the labor market and in the farming context as well as in many other circumstances,the collective prize is neither purely public nor purely political or ideological. Quite often, thereward of group action has, at least to some extent, private characteristics so that it can be,to that extent, divided among group members. One just needs to remember that an importantfunction of governments is to create and distribute rents through regulation and these rents canbe divided both among groups and among individuals. One can also think of the various transferssuch as pensions, subsidies or health benefits that are targeted to specific groups. According tothe common wisdom, when groups compete for such transfers or for prizes with private economiccharacteristics, we should observe individuals organized in very narrow interest groups to wardoff the free-rider problem and to have the prize divided by a smaller number of individuals.This is also not really conforming to what we observe in a majority of situations. Even thoughsome studies find that smaller groups receive larger transfers, a lot of empirical works find thecontrary (e.g., Pincus (1975); Becker (1986); Congleton and Shugart;(1990); Kristov, Lindertand McClelland, (1992); for a survey, see Potters and Sloof, (1996)).One then needs to understand, how groups and particularly large groups manage to controlfree-riding and their collective action problem especially in a context where individuals andgroups are motivated by private economic considerations. One can think of various ways bywhich groups can circumvent free-riding. For example, this could be done by implementingcostly incentives schemes, by direct monitoring of member actions or by exerting moral or other3

pressures on them. But these procedures might be very costly to implement especially in largegroups. We argue, in this paper, that it is more likely that members in a group tacitly cooperatein pursuing their interest. By "tacitly", we mean that individuals may well find cooperation tobe a more fruitful individual strategy than free-riding. Axelrod (1984), in his classical book,explains that cooperation can emerge and stabilize in multiple-participants environments as aresult of reciprocal altruism. When agents take account of the probable future response of othersto their own behavior, they may find profitable to cooperate anticipating reciprocal cooperationfrom others. The simplest strategy based on this principle of reciprocal altruism is known asthe tit-for-tat strategy in which each player cooperates in the current period if others playerscooperated in the previous period. If one player defects in the current period, other playerswill respond by defection next period. Obviously, this strategy does not make any sense in aone-shot environment. But clearly, groups and group members do not play only once for all;they rather face repeated interactions with each other in the real world.There is another crucial feature of the collective action problem within groups that we wantto emphasize. Most of the collective action theory deals with actions of a single group. Inreality, a large number of economic and political activities involve several groups which arein opposition to one another. In essence, the willingness to participate to collective actionthrough interest group membership is driven by the opportunity to pursue an interest that isnot universally shared. Therefore, the collective action problem within a group does not onlydepend on internal attributes and on whether the reward is divisible but also on the interactionbetween groups. Hence, if groups are in opposition to one another, the ability of a group toattain its objective crucially depends upon its strength relative to that of its antagonists. Inturn, group confrontation is likely to affect the internal organization of the groups, most notablyin terms of free-riding and cooperative incentives. There are thus double-edged incentives sinceincentives within groups spill over into the intergroup arena, and between-group competitionaffect within-group incentives. 1The questions we try to answer, in this paper, are the following: How does group interactionaffect the ability of the groups to overcome their free-rider problem; hence how does it affectindividual behavior and individual incentive within the groups? In turn, how does individualbehaviors within groups reflect in the competition between groups? For this purpose, we develop1 We borrow the term "double-edged incentives" to Persson and Tabellini (1995) who analyze the bidirectionalinteraction between domestic policy processes and international strategic interactions.4

a simple model that exhibits three important features. (1) Collective action is undertaken in acontext of competition between groups. (2) The contested prize is purely private. (3) Competinggroups have repeated interactions.Concerning the first feature, we model between-group competition as a rent-seeking gamein which group members make efforts or expenditures to increase their share of a contestablerent. There are two groups of different sizes and all individuals have the same valuation for therent which is divisible both among groups and among individuals. Aggregate group spendingdetermines the proportion of the rent allocated to the group. Hence, the group share is a publicgood to the members in a group so that there is a collective action problem within groups.Thereafter, because individual contributions are not observable, this proportion is shared by themembers of the group on an equal division basis.The fact that the contested rent is purely private and divisible is a crucial element of ourframework. This is in line with Olson’s classical analysis where individual benefits of groupmembership are private and decreasing with the number of members. As a result, in the oneshotgame, the larger the group, the lower is the level of collective action because of the free-ridingincentives. In our rent-seeking model, this is reflected in group shares: the group with fewermembers gets, in equilibrium, the larger share of the rent.Concerning the third feature of our analysis, we analyze the repeated interactions betweengroups as an infinitely repeated game in which a simple (tit-for-tat) trigger strategy, as describedabove, is the enforcement mechanism for within-group cooperation. We also assume that everyoneobserves, without errors, the aggregate level of efforts made in the previous period so thatdefection is always detected. It is well-known that in such set-up, the cooperative outcomemaybeachievedifplayersaresufficiently patient (i.e. if the discount parameter is sufficientlylarge). In our framework, however, the threshold value of the discount parameter above whichcooperation within a particular group can be maintained is endogenously determined as a functionof the sizes of the two groups. Moreover, the critical discount factor associated with agroup depends upon the organizational properties of the rival group. This is because payoffsunder defection, cooperation and non-cooperation in a group depend upon the outcome of thecompetition between the two groups and this outcome, in turn, depends upon whether there iscooperation within the rival group or not.We proceed as it follows. We first examine how the critical value of the discount parameter fora given group changes with the number of members in both groups when the status (cooperation5

or non-cooperation) within the rival group is taken as given. We show that the ability of a groupto maintain a cooperative outcome increases as the group grows larger no matter the internalorganization (cooperation or non-cooperation) of the rival group. We then characterize thesubgame perfect Nash equilibria of the game between the two groups of people. In particular,when there is defection and reversion to non-cooperation within one group, the enforcementof cooperation within the other group might become a collective best response. Hence, whenmembers of the rival group do not cooperate, cooperative incentives in the home group maybe enhanced by the possibility that breaking cooperation incites members of the rival group tocooperate.The main result of the paper is that the larger group has more chance to maintain withingroupcooperation. Indeed, if there is an equilibrium where individuals of the smaller group actcooperatively, then there is also one where individuals of the larger group act cooperatively. Inaddition, for sufficiently high levels of discount parameters, we show that cooperation can beachieved only within the larger group. One of the driving elements behind these results is thatinfinite reversion to the non-cooperative outcome is more costly for the larger group than for thesmaller group. This is because the free-rider problem is more severe in the larger group which isreflected in a lower group’s share. This is turn helps to make cooperation less difficult to sustainin this group.Our paper is related to, but differs in its focus from, the large literature on rent-seekingtheory. Following the seminal contributions of Tullock (1967, 1980), this literature focuses on therelation between the magnitude of the rents created by different organizations or political systemsand the amount of resources wasted by agents in the pursuit of those rents. Various aspects ofrent-seeking conflicts have been analyzed. For example, rent-seeking has been studied in contextssuch as uncertainty (Hurley and Shogren (1998)), risk aversion (Hillman and Katz, (1984);Skarpedas and Gan, (1995)), free entry into the rent-seeking market (Pérez-Castrillo and Verdier,(1992)), intergroup migration (Baik and Lee, (1997)), asymmetric valuations (Nti (1999)) ormultistage contests (Gradstein and Konrad, (1999)); see Nitzan (1994) for a survey. But in allthese works, each group is acting as a single agent and there is no room for collective actionproblems. Nitzan (1991) analyzes a contest between several groups in which group membersdecide voluntarily on the extent of their rent-seeking efforts with the probability of success ofa group depending on the sum of efforts of group members. He then shows, for different rulesconcerning the distribution of the rent within groups, that collective rent-seeking contests induce6

McGuire (1974) depend on the linearity of cost functions, an assumption that they argue to beunrealistic. However, their analysis is static and there is no scope for cooperation within groups.We present an alternative explanation of the higher potency of larger groups that relies on theincentives to cooperate within competing groups rather than on a technical assumption on theelasticity of the cost of contributing to group action.We present a simple specification of a rent-seeking game between two groups in Section2. We then characterize the single shot equilibrium when members in both groups act noncooperativelyin rent-seeking. In Section 3, we first define the critical value of the discountparameter above which within-group cooperation is maintained. We then calculate the payoffsunder cooperation, non-cooperation and defection within a group when the members of the rivalgroup act non-cooperatively and when they act cooperatively. In Section 4, we first determinethe critical value of the discount parameter for each group when the status (cooperation andnon-cooperation) within the rival group is taken as given. We then characterize the subgameperfect Nash equilibria of the game between the two groups. Section 5 concludes.2 The ModelConsider a world with two groups A and B, which have n A and n B identical risk neutralmembers, respectively. We assume that the groups are isolated from each other in the sensethat it is prohibitively costly for individuals to move from one group to the other. Such costsmay arise from cultural differences, for example differences of language. The government intendsto adopt a policy that will provide a rent Y . The only role of the government is precisely todivide this pie among the groups in response to the rent-seeking pressures and lobbying activitiesof the members of the groups. Following Katz and Tokatlidu (1996), we assume that a groupgets a share of the pie equal to the ratio of the sum of the expenditures of its members to thetotal expenditures. The share of group j isp j (t) =½Pn ji=1µP nAt ji / t Ai + n BPt Bi ; ifi=1 i=1n A Pi=1t Ai + n BPt Bi > 0i=11/2, otherwisewhere t is the vector of all individual rent-seeking expenditures. Even though the aggregate levelof efforts is perfectly observed, individual contribution is not. We then suppose that the shareof the rent acquired by each group is distributed equally among the group members. We also(1)8

assume that individual preferences are represented by an additively separable utility function.Specifically, individual i in group j, forj = A, B, has the following utilityu ji (t) = p j (t) Yn j− t ji . (2)We first analyze the one-shot equilibrium outcome in which members in both groups act noncooperativelyin rent-seeking. Since there is no equilibrium where nobody invests anything, theoptimal expenditure of agent i in group A is given by the following first-order condition 4nPBt Bkk=1µP nAt Ak + n BPk=1k=1t Bk 2Yn A=1. (3)The corresponding condition for an agent in group B isµ nAPPn At Akk=1t Ak + n BPk=1 k=1t Bk 2Yn B=1. (4)Given that members in each group are identical, it is natural to focus on a symmetric equilibriumin which all members in a group make the same level of effort. Let t A and t B be the commonequilibrium expenditure in group A and group B respectively. The conditions for equilibriumare thenfor members of group A, andn B t B Y(n A t A + n B t B ) 2 =1 (5)n An A t A(n A t A + n B t B ) 2 Yn B=1 (6)4 Note that the marginal return to an additional unit of individual rent-seeking effortaswellastoanadditionalunit of group rent-seeking effort is decreasing in effort, so that second-order conditions will be satisfied.9

for members of group B.The solution to this system of equations isandt NNA =t NNB =n B Y(n A + n B ) 2 (7)n An A Y(n A + n B ) 2 . (8)n BThe double upper-script NN denotes a Non-cooperative behavior within both groups. By convention,we will use the first and the second upper scripts to indicate the behavior of memberswithin groups A and B, respectively. The share allocated to region A is thenp NNA =n Bn A + n B(9)while the share allocated to region B is p NNB= ¡ 1 − p NN ¢A . Hence, the group with fewer membersgets the larger share of the pie. Because the share of a group is a public good to the agents in agroup, there is a free-rider problem which makes the smaller group more successful. In addition,the larger the group, the smaller is its share of the rent. Finally, it is worth pointing out thatnot only the larger group ends up with a lower share of the rent but it must divide this shareamong a larger number of members. The equilibrium utility level of an agent of group A is thenu NNA = n B (n B + n A − 1) Y(n A+ n B ) 2 . (10)n AThe equilibrium utility level of an agent of group B is obtained by permuting n A and n B . Wenote that individual utility is falling in the number of fellow members and increasing in thenumber of members of the rival group.3 Trigger Strategies and Within-Group Cooperation3.1 The Critical Discount ParameterThe objective of this subsection is to define the critical discount parameter above which cooperationcan be maintained within a given group. Consider an infinitely repeated game in which10

individuals adopt a simple (tit-for-tat) trigger strategy to maintain within-group cooperation. 5Hence, each agent in a group makes the cooperative level of effortinthecurrentperiodifallgroup members behaved in a similar manner in the previous period. If any agent defects in thecurrent period, she will trigger the non-cooperative outcome within the group for all subsequentperiods. We assume that all agents observe without errors the aggregate level of efforts at theend of each period which implies that defections are always detected. In addition, all individualsin both groups have the same discount factor 0

there is cooperation within the rival group or non-cooperation. This is because, as explainedearlier, the organizational properties of a group affect individual payoffs within the other groupthrough intergroup competition. It is further assumed that members in a group perfectly observethe aggregate level of efforts of the rival group. Third, it is well-known that there are manyequilibria in this type of infinitely repeated game and we will focus, as is usual, on the Paretooptimal equilibria from the viewpoint of the members in a group.3.2 Cooperation and the Number of Members3.2.1 Cooperation in the Other GroupLet consider that at date t, there is cooperation in rent-seeking within both groups. Let t A andt B be the common individual rent-seeking effort under within-group cooperation in group A andB, respectively. The share allocated to region A isp A =n A t An A t A + n B t B. (13)The optimal individual expenditure of an agent of group A is given by the following first-orderconditionThe corresponding condition for an agent of group B isn B t B2Y =1. (14)(n A t A + n B t B )n A t A2Y =1. (15)(n A t A + n B t B )It is easily checked that the solution of this system is, for j = A, B, t CCjp CCA= pCC B= Y/4n j and thereby=1/2. The double upper script CC indicates that there is cooperation in rentseekingwithin both groups. The individual payoff in each group isu CCj= Y4n j, j = A, B. (16)Because individuals preferences are represented by a utility function that is additively separableand linear in Y and in rent-seeking activities, within-group cooperation makes each group actingas it was a single agent.12

Consider now that in group A, att +1, one agent deviates and contributes nothing while allother agents make the cooperative rent-seeking effort. 6 At that date, members in group B alsocontinue to cooperate in rent-seeking. With one defecting agent, the share allocated to region Aat date t +1is then p DCA=(n A − 1) t CCA/ £ (n A − 1) t CCA+ n Bt CC ¤B =(nA − 1) / (2n A − 1). Theequilibrium utility level of this agent is thereforeu DCA = n A − 12n A − 1Y. (17)n AThe double upper script DC indicates that this agent deviates while all members of the rivalgroup act cooperatively (as well as all fellow members). Note that the larger the group, thelower is the individual payoff of defection. A symmetric expression is obtained for an agentwho defects from the cooperative outcome within group B when members within group A actnon-cooperatively.3.2.2 Non-cooperation in the Other GroupNow consider that at date t, members of group A act cooperatively while members of group Bact non-cooperatively. Let t A be the common equilibrium expenditure in group A. The shareof that group isp A =n A t An A t A + n BPk=1t Bk(18)and the share of group B is 1 − p A . The optimal individual expenditure of an agent in group Ais given by the following first-order conditiont Bkk=1n An B Pµn A t A + n BPk=1The corresponding condition for an agent in group B ist Bk 2Yn A=1. (19)6 We indeed show in a technical Appendix (available upon request) that the agent who defects optimally cutsits contribution to 0 when there is cooperation as well as when there is non-cooperation within the rival group.13

n A t A Yµn A t A + n BP 2=1. (20)n Bt Bkk=1Assuming that all members in a group make the same level of effort, the solution of this systemof equations is thenandt CNA =n Bn A (n B +1) 2 Y (21)t CNB =1n B (n B +1) 2 Y (22)where the double upper script CN indicates C ooperation and N on-cooperation within group Aand B respectively. In equilibrium, the share allocated to group A is thenp CNA = n Bn B +1(23)which is approaching 1 when n B is large. When members of a group make their lobbying effortscooperatively, it is as though this group acts as a single agent. The utility of a representativeagent in group A is then∙u CNA =nBn B +1¸2Yn A. (24)This utility level is increasing in n B . The larger the size of group B, themoresevereisthefree-rider problem in this group and the larger is the share of group A. This in turn implies ahigher individual payoff for members of group A. The utility of a representative agent in groupB is∙u CNB =1n B +1¸2Y. (25)Now, we must characterize the payoff of a defecting agent within group A, att +1,whenallfellow members act cooperatively while all members of group B still act non cooperatively. In14

= £ (n A − 1) t CN ¤ £A / (nA − 1) t CNAthis case the share allocated to group A is: p DNAn B (n A − 1) / [n B (n A − 1) + n A ]. Therefore, payoff under defection isu DNA =+ n Bt CN ¤B =n B (n A − 1) Y. (26)n B (n A − 1) + n A n AThis utility level is decreasing in n A and increasing in n B . The analysis of the rent-seeking gamein which there is cooperation within group B and non-cooperation within group A is symmetric.4 Maintaining Cooperation and Between-Group Competition4.1 Partial AnalysisIn this subsection, we fix the behavior of members within a group (let say group B) andweexamine the difficulty of maintaining cooperation in the other group (group A) as a function ofthe sizes of the two groups. In other words, we first characterize the subgame perfect equilibriawhen each agent in a group plays her best reply to the actions of fellow members with the status(cooperation or non-cooperation) within the rival group taken as given. Thereafter, in the nextsubsection, we will characterize the subgame perfect equilibria when each agent in a group canalso play her best reply of one’s own group to the members’ behavior of the rival group.If there is defection of any agent at date t +1within group A, then members of that grouprevert, from the date t +2, to the non-cooperative outcome forever. Suppose firstthatmemberswithin group B act cooperatively. Using (12), the limit discount factor above which cooperationcan be sustained within group A is defined as followsδ C A = U ADCUADC− U ACC− U ANC(27)where the single upper script C of the discount parameter denotes Cooperation between memberswithin the other group. UACC and UDCAare given by (16) and (17) respectively. UANC is givenby (25) in which, by symmetry of the model, n B has been replaced by n A . (Recall that byconvention, the first upper script indicates the internal organization of group A and the secondupper script that of group B). Therefore, we haveδ C A (n A )= (2n A − 3) (n A +1) 24 £ n 2 A (n A − 1) − 1 ¤ . (28)15

A similar expression can be found for group B by permuting n A and n B .derivative of δ C A with respect to n A,wehaveCalculating the∂δ C A= − (3n A − 2) (n A +1) ¡ n 2 A − 3n A +2 ¢∂n A 4 £ n 2 A (n A − 1) − 1 ¤ 2(29)which is negative for any n A ≥ 2.Suppose now that members within group B act non cooperatively. The limit discount factorabove which cooperation can be sustained in group A is now defined as followsδ N A = U ADNUADN− U ACN− U ANN(30)where the single upper script N of the discount parameter denotes the Non-cooperative outcomewithin the other group. UANN,U ACN and UDNAthen haveδ N A (n A ,n B )=are given by (10), (24), and (26) respectively. We(n A + n B ) 2 [n B (n A − 2) + (n A − 1)](n B +1) 2 [n A (n A − 1) (n A + n B − 1) − n B ] . (31)A symmetric expression can be found for group B by permuting n A and n B .Calculating the derivative of δ N A with respect to n A,wehave 7½ £nA n∂δ N (n A + n B ) B nA (n B +1)(n A − 5) + n 2 B (n A − 4) + 7n B +8 ¤ ¾+An A (n A − 1) 2 ¡+ n B 3n2= −B− 4n B − 3 ¢∂n A (n B +1) 2 [n A (n A − 1) (n A + n B − 1) − n B ] 2 < 0. (32)It is immediate to check that the numerator of this expression is always positive for any n A ≥ 4and n B ≥ 2. Hence, δ N A is decreasing with n A for any n A ≥ 4. Calculating the derivative of δ N Awith respect to n B ,wehave∂δ N A∂n B= −⎡(n A + n B ) ⎣n 2 B¡n4A− 3n 3 A + n2 A +4n A − 3 ¢+n B (n 5 A − 4n4 A +10n3 A − 15n2 A +9n A − 1)+ ¡ ⎦n 5 A − 3n4 A +5n3 A − 6n2 A +3n ¢A(n B +1) 3 [n A (n A − 1) (n A + n B − 1) − n B ] 2 < 0. (33)7 The calculus of the derivative of δ N A with respect to n A and n B is trivial but painful. The details of thesescalculus are given in an appendix available upon request.⎤16

Again, it is immediate to check that the numerator of this expression is positive for any n A ≥ 4and n B ≥ 2. By symmetry, the derivative of δ N B will be negative with respect both to n A andn B for any n B ≥ 4 and n A ≥ 2. We will then consider, from now, that the size of both group isat least equal to 4. Thus, we haveProposition 1 (i) When the cooperative outcome within the rival group is taken as given, thedifficulty of maintaining cooperation in a group depends only on the number of fellow membersand is decreasing in that number. (ii) When the non-cooperative outcome within the rival groupis taken as given, the difficulty of maintaining cooperation in a group is decreasing in the numberof members of both groups.From the one-shot analysis of the previous section, we know that the larger the size of agroup, the more severe is the collective action problem which is reflected in a lower share ofthe rent. Hence, we would presume that the ability to maintain cooperation deteriorates as agroup expands in size. However, in a dynamic setting, according to the above Proposition, thisconjecture is overturned.The difficulty of maintaining cooperation in a group depends on how payoffs under defection,cooperation and non-cooperation evolve has both groups grow larger. The one-period unit ofpayoff is the product of the group’s share of the rent by the per-capita value of the aggregaterent Y . When a group becomes larger, the per-capita value of the prize diminishes but thegroup’s share does not change under within-group cooperation. Indeed, when members in agroup cooperate, the share of that group is either equal to 1/2 (when there is cooperationwithin the rival group) or depends only on the number of members of the rival group (when thenon-cooperative outcome prevails within the rival group as shown by (23)). When members donot cooperate, however, both the per-capita value of the prize and the group’s share decreaseas the group becomes larger. Therefore, the cost of infinite reversion to the non-cooperativeoutcome is increasingly higher with group size regardless of whether members of the rival groupact cooperatively or not. As a result, the ability of a group to maintain cooperation is increasingin its size. In addition, since the share of the group in which there is cooperation is increasing inthe size of the rival group in which there is non-cooperation (because of the free-riding incentiveswithin the rival group), the larger the size of the rival group, the less difficult is to maintaincooperation in the former group.It could also be interesting to rank the different discount parameters. Without any loss ofgenerality and for felicity only, let consider that group A is the larger group i.e. n A ≥ n B .17

By recalling that the upper scripts of the limit discount parameters indicate the organizationalproperties of the rival group, we can establishProposition 2 When the status within the rival group is taken as given, we have the followingranking (for n A ≥ n B ): 0

members behave non-cooperatively while members of the competing group still act cooperatively.Hence, the infinite reversion to the non-cooperative outcome is more costly for the larger groupthan for the smaller group which in turn makes cooperation less difficult to sustain in the largergroup (hence δ C A ≤ δC B ).However, when the non-cooperative outcome prevails in the rival group, cooperation is moredifficult to sustain in the larger group than in the smaller group. This result is less immediate.On the one hand, the relative benefits of cooperation are more important for the smaller groupthan for the larger group when members within the rival group act non-cooperatively. This isbecause, when there is cooperation in a group, its share of the rent is increasing in the size of therival group in which there is non-cooperation (as shown by (23)). On the other hand, the costof infinite reversion to the non-cooperative outcome is lower for the smaller group than for thelarger group. This is because the larger group, when all individuals within the two groups behavenon-cooperatively, gets the lower share of the rent because of the collective action problem.Hence, the relative benefits of cooperation are more important while the relative costs of noncooperationare less important for the smaller group than for the larger group. Consequently,when there is non-cooperation within the rival group, there exists two opposing forces on theability to maintain cooperation. According to the above result, the one-period unit of defection islower than the difference in future payoffs between cooperation and non-cooperation (in presentdiscounted value) for the smaller group than for the larger group (and then δ N B ≤ δN A ).4.2 Equilibrium ConfigurationsWe now consider that members in a group not only can play their best reply to defection bya fellow member (i.e. a non-cooperative level of effort) but they can also play the group’s"pseudo" best reply to the reversion to the non-cooperative outcome within the rival group (i.e.a cooperative level of effort). In other words, when there is defection and reversion to the noncooperativeoutcome within one group, the enforcement of cooperation within the other groupmight become a best response. Hence, defection in a group may trigger both reversion to thenon-cooperative outcome within that group and a change from non-cooperation to cooperationwithin the rival group. Since cooperation within the rival group is always detrimental to aplayer in a group, cooperative incentives may be enhanced by the possibility that breakingcooperation incites members of the rival group to cooperate. However, once there is defectionfrom the cooperative outcome in a group, this group still reverts to the non-cooperative outcome19

forever.For felicity only, let continue to assume that n A ≥ n B .Wefirst analyze the ability to maintaincooperation in the smaller group i.e. group B. Consider that at date t, there is cooperation inrent-seeking activities within both groups. If there is defection of one individual at date t +1in group B, this group from t +2reverts to the non-cooperative outcome forever. This will notchange individual behavior within the rival group. Indeed, reversion to non-cooperation withingroup B increases the relative benefits of cooperation within group A. The relevant discountparameter is therefore the one obtained in the previous subsection i.e. δ C B .Now consider that at date t, the members of group A act non-cooperatively and membersof group B act cooperatively. At t +1, if one individual of group B defects, then this group,from t +2, reverts to the non-cooperative outcome forever. This change in the status of groupB may lead to cooperation within group A from the date t +2.Ifδ

t +2, reverts to the non-cooperative outcome forever. This change in the status of group Amay lead to cooperation within group B from the date t +2. If δδ≥ δN B then reversion to the non-cooperative outcome withingroup A may lead, from the date t +2, to the cooperative outcome within group B. In thatsituation, if a member of group A defects, she will earn u DNAin the current period and u NCAin all future periods. (Again, recall that the first upper script of payoffs indicates the behaviorof members of group A and the second one that of group B). Hence, a necessary conditionfor maintaining cooperation in group A under a trigger strategy with infinite reversion to noncooperationis now given byu DNA +∞Xt=1δ t u NCA≤∞Xt=0δ t u CNA (34)u CNAand uDNAare given by (24) and (26) respectively. u NCAis given by (25) in which n B havebeen replaced by n A . We then obtain the same expression than (30) except that u NNAreplaced by u NCAin the denominator. We then haveis nowδ N0A (n A ,n B )= n B (n A +1) 2 [n B (n A − 2) + (n A − 1)](n B +1) 2 £ n 2 A (n An B − 1) − n B¤ . (35)Since individual payoff within group A is lower when there is cooperation than when there isnon-cooperation within group B (i.e. u NCAδ N B . Thisisbecause,wehave21

n A =λn Bλ=1.25λ=1.5λ=2λ=5λ=10λ=100n B0.000n B40.1260.1000.0690.016-0.001-0.01680.0800.0660.0500.0190.009-0.001200.0360.0300.0230.0090.0050.000Figure 1: The differential between δ N 0A and δN BIn the following, we then consider that in most cases δ N 0A(bold numbers represent a higher level of δN0A )

No matter how members of group B behave, cooperation can always be maintained within groupA because δ ≥ δ C A . Cooperation in this group does not allow to maintain the cooperative outcomewithin group B since δ

since the rent must be shared by a larger number of people. For instance, if members withinboth groups cooperate, each group gets half of the total rent and individual payoff is lower in thelarger group. Now consider the situation in which there is cooperation within the larger groupand non-cooperation within the smaller group. Comparing (24) and (25), individual payoff willbe higher in the larger group than in the smaller group if n A

er that this assumption makes larger groups extremely vulnerable to the free-rider problem. 9In a static setting, notwithstanding a larger number of claimants, the larger group gets a lowershare of the total rent. In a dynamic setting, the predominance of this problem may help thelarger group to achieve within-group cooperation and to produce an optimal level of collectiveaction. In other words, the intrinsic disadvantage of large groups might become an asset thatmakes the threat strategy of infinite reversion to the non-cooperative outcome sufficiently costlyto maintain cooperation. It would be both natural and interesting to extend the analysis byconsidering that the prize for which groups compete has both public and private characteristics.We conjecture that the intrinsic disadvantage of large groups would be attenuated and so theirability to overcome their collective action problem in a dynamic setting.9 In the same type of model with two groups competing for a prize which is purely public and not divisibleamong groups and among individuals, each group would have a probability 1/2 of getting the prize no matterits relative size. As explained by Katz, Nitzan and Rosenberg (1990), in this type of contest game, the free-riderproblem within each group just counterbalances the size of the total prize for the group.25

References[1] Axelrod, R., (1984), The Evolution of Cooperation, New-York, Basic Books.[2] Baik, K.H. and S. Lee, (1997), "Collective rent seeking with endogenous group sizes,"European Journal of Political Economy 13, 121-130[3] Becker, G., (1986), "The Public Interest Hypothesis Revisited: A New Test of Peltzman’sTheory of Regulation," Public Choice 49, 223-34.[4] Chamberlin, J., (1974), ”Provision of Collective Good as a Function of Group Size,” AmericanPolitical Science Review 68, 707-16.[5] Checchi, D. and C. Lucifora, (2002), "Unions and Labor Market Institutions in Europe,"Economic Policy 35, 363-408.[6] Congleton, R.D. and W.F. Shugart, I, (1990), "The Growth of Social Security: ElectoralPush or electoral Pull," Economic Inquiry 28, 109-132.[7] Esteban, J. and D. Ray, (2001), ”Collective Action and the Group Size Paradox,” AmericanPolitical Science Review 95, 663-672.[8] Friedman, J.W., (1971), "A Non-Cooperative Outcome for Supergames" Review of EconomicStudies 38, 1-12.[9] Gradstein, M., and K. Konrad, (1999), "Orchestrating Rent-seeking Contests," EconomicJournal 109, 536-545.[10] Hillman, A. and E. Katz, (1984), Risk-Averse Rent Seekers and the Social Cost of MonopolyPower," Economic Journal 94, 104-110.[11] Hurley, T and J. Shogren, (1998), "Effort Levels in a Cournot Nash Contest with AsymmetricInformation," Journal of Public Economics 69, 195-210.[12] Katz, E., S. Nitzan and J. Rosenberg, (1990), "Rent-seeking for Pure Public Goods," PublicChoice 65, 49-60.[13] Katz, E., and J. Tokatlidu, (1996), "Group Competition for Rents," European Journal ofPolitical Economy 12, 599-607.26

[14] Keeler, J.T., (1996), "Agricultural Power in the European Community: Explaining the Fateof CAP and GATT Negotiations," Comparative Politics 28, 127-149.[15] Kristov, L., P. Lindert and R. McClelland, (1992), "Pressure Groups and Redistribution,"Journal of Public Economics 48, 135-63.[16] McGuire, M., (1974), ”Group Size, Group Homogeneity and the Aggregate Provision of aPure Public Good under Cournot Behavior,” Public Choice 18, 107-26.[17] McMillan, (1979), "Individual Incentives in the Supply of Public Inputs," Journal of PublicEconomics 12, 97-98.[18] Moe, T., (1981), "Toward a Broader View of Interest Groups," The Journal of Politics 43,531-543.[19] Nitzan, S., (1991), "Collective Rent Dissipation," Economic Journal 101, 1522-34.[20] Nitzan, S., (1994), ”Modelling Rent-Seeking Contests,” European Journal of Political Economy10, 41-60.[21] Nti, K.O., (1999), Rent-seeking with Asymmetric Valuations, Public Choice 98, 415-430.[22] Olson, M., (1965), The Logic of Collective Action, Cambridge, MA: Harvard UniversityPress.[23] Pecorino, P., (1998), "Is There a Free-Rider Problem in Lobbying? Endogenous Tariffs,Trigger Strategies, and the Number of Firms," American Economic Review 88, 652-660.[24] Pecorino, P., (1999), "The Effect of Group Size on Public Good Provision in A RepeatedGame Setting," Journal of Public Economics 72, 121-134.[25] Pérez-Castrillo, D. and T. Verdier, (1992), "A General Analysis of Rent-Seeking Games,"Public Choice 73, 335-50.[26] Persson, T., and G. Tabellini, (1995), "Double-Edged Incentives: Institutions and PolicyCoordination," Chapter 38 in Grossman, G. and K. Rogoff (eds.) Handbook of InternationalEconomics, Vol III, Elsevier, North—Holland: Amsterdam.[27] Pincus, J.J., (1975), "Pressure Groups and the Pattern of Tariffs," Journal of PoliticalEconomy 83, 557-577.27

[28] Potters, J. and R. Sloof, (1996), "Interest Groups: A Survey of Empirical Models that Tryto Assess their Influence," European Journal of Political Economy 12, 403-442.[29] Skaperdas, S. and L. Gan, (1995), "Risk-Aversion in Contests," Economic Journal 105,951-962.[30] Tullock, G., (1967), ”The Welfare Costs of Tariffs, Monopolies and Theft,” Western EconomicJournal 5, 224-232.[31] Tullock, G., (1980), ”Efficient Rent-seeking,” in Buchanan, J.M., Tollison, R.D., and G.Tullock (Eds.), Toward a Theory of the Rent-seeking Society, College Station: Texas A&MUniversity Press.[32] Wärneryd, K., (1998), ”Distributional Conflict and Jurisdictional Organization,” Journalof Public Economics 69, 435-50.28

Appendix (Not for publication)6 The optimal level of rent-seeking expenditures under defectionThe purpose of this subsection is to show that when an individual defects from the cooperativeoutcome, she will cut her contribution to 0. We consider the optimal behavior of a defectingagent within group A when there is, within group B, non-cooperation on the one hand andcooperation on the other.Let first consider that there is cooperation within the rival group (i.e. group B). When anagent within group A defects from the cooperative outcome, the share allocated to group A isp DCA = (n A −1)t CCA(n A −1)t CCwhere t CCAA+tDC A+tDC A+n Bt CCBand tCCBare equal to Y/(4n A) and Y/(4n B ) respectively and represent the individualexpenditure level in group A and B when there is cooperation within the two groups. t DCAis theexpenditure level of the agent who defects within group A. This agent optimally chooses t DNAto maximizeu DNA= pDC An AY− t DCA .The optimal expenditure of the defecting agent is given by the first-order condition4n A Y 2=1.[(2n A −1)Y +4n A t DCA ] 2Simplifying the above expression, we get4n A t DCA = £ 1 − 2 √ n A¡√nA − 1 ¢¤ Y.The right-hand term is negative for any n A ≥ 2. Therefore, when there is cooperation withinthe rival group, an agent who defects from the cooperative outcome in a group, optimally cutshis effort level to 0.Let now consider that there is non-cooperation within group B. When an agent within groupA defects from the within-group cooperative outcome, the share allocated to group A isp DNA = (n A −1)t CNA(n A −1)t CNA+tDN A+tDN A+n Bt CNBwhere t CNAand t CNBare given by (21) and (22) in the text and represent the individual expenditurelevel in group A and B respectively when there is cooperation within group A and29

non-cooperation within group B. t DNAis the expenditure level of the agent who defects. Thisagent optimally chooses t DNAto maximizeu DNA= pDN An AY− t DNA .Using (21) and (22), the optimal expenditure of the defecting agent is given by the first-orderconditionn A (n B +1) 2 Y 2=1.[(n A −1)n B Y +n A (n B +1) 2 t DNA +n AY ] 2Simplifying the above expression, we getn A (n B +1) 2 t DNA = £ √nA¡1 −√nA¢(nB +1)+n B¤Y.The right-hand term of the above expression in negative when√ ¡√nA nA − 1 ¢ >n B / (1 + n B ).Asufficient condition for t DNAto be equal to 0 is therefore √ ¡√n A nA − 1 ¢ > 1 which is alwayssatisfied for any n A ≥ 3. Therefore, when the non-cooperative outcome prevails within therival group, the agent who defects from the cooperative outcome in a group optimally cuts hiscontribution to 0.30

7 The discount parameters δ N A and δ N B7.1 The derivative of δ N A with respect to n AThe derivative of δ N A with respect to n A is given bywith∂δ N A∂n A=∂UV −U ∂V∂n A ∂n A(n B +1) 2 [n A (n A −1)(n A +n B −1)−n B ] 2U ≡ (n A + n B ) 2 [n B (n A − 2) + (n A − 1)]V ≡ [n A (n A − 1) (n A + n B − 1) − n B ]We have∂U∂n A=(n A + n B )[2n B (n A − 2) + 2 (n A − 1) + (n A + n B )(n B +1)]∂U∂n A=(n A + n B ) £ 3n A n B +3n A + n 2 B − 3n B − 2 ¤We also have∂V∂n A=(2n A − 1) (n A + n B − 1) + ¡ n 2 A − n ¢A∂V∂n A=3n 2 A +2n An B − 4n A − n B +1Let denote the numerator of ∂δN A∂U∂n Aas follows:∂n AV − U∂n ∂VA=(n A + n B )[X − Y ] withX ≡ £ 3n A n B +3n A + n 2 B − 3n B − 2 ¤£¡ n 2 A − n ¢ ¤A (nA + n B − 1) − n BY ≡ (n A + n B )[n A n B − 2n B + n A − 1] £ 3n 2 A +2n An B − 4n A − n B +1 ¤We now develop the expressions X and Y .X = £ 3n A n B +3n A + n 2 B − 3n B − 2 ¤£ n 3 A + n2 A n B − 2n 2 A − n ¤An B + n A − n BDeveloping again, we haveX =3n 4 A n B +3n 3 A n2 B − 6n3 A n B − 3n 2 A n2 B +3n2 A n B − 3n A n 2 B +3n4 A +3n3 A n B − 6n 3 A − 3n2 A n B +3n 2 A − 3n An B + n 3 A n2 B + n2 A n3 B − 2n2 A n2 B − n An 3 B + n An 2 B − n3 B − 3n3 A n B − 3n 2 A n2 B +6n2 A n B +3n A n 2 B − 3n An B +3n 2 B − 2n3 A − 2n2 A n B +4n 2 A +2n An B − 2n A +2n BSimplifying, we haveX =3n 4 A n B +4n 3 A n2 B − 6n3 A n B + n 2 A n3 B − 8n2 A n2 B − n An 3 B +4n2 A n B + n A n 2 B − 4n An B +3n 4 A −8n 3 A +7n2 A − 2n A − n 3 B +3n2 B +2n BWe now develop the expression Y .Thisgives31

⎡Y =(n A + n B ) ⎣3n 3 A n B +2n 2 A n2 B − 4n2 A n B − n A n 2 B + n An B − 6n 2 A n B−4n A n 2 B +8n An B +2n 2 B − 2n B +3n 3 A +2n2 A n B−4n 2 A − n An B + n A − 3n 2 A − 2n An B +4n A + n B − 1Simplifying, this gives∙ 3n3Y =(n A + n B ) An B +2n 2 A n2 B − 8n2 A n B − 5n A n 2 B +6n ¸An B+3n 3 A − 7n2 A +5n A +2n 2 B − n B − 1Developing the above expression, we haveY =3n 4 A n B +2n 3 A n2 B − 8n3 A n B − 5n 2 A n2 B +6n2 A n B +3n 4 A − 7n3 A +5n2 A +2n An 2 B − n An B −n A +3n 3 A n2 B +2n2 A n3 B − 8n2 A n2 B − 5n An 3 B +6n An 2 B +3n3 A n B − 7n 2 A n B +5n A n B +2n 3 B − n2 B − n BSimplifying, we haveY =3n 4 A n B +5n 3 A n2 B − 5n3 A n B +2n 2 A n3 B − 13n2 A n2 B − n2 A n B − 5n A n 3 B +8n An 2 B +4n An B +3n 4 A − 7n3 A +5n2 A − n A +2n 3 B − n2 B − n BWe now calculate the difference Z ≡ X − YZ = −n 3 A n2 B − n3 A n B − n 2 A n3 B +5n2 A n2 B +5n2 A n B +4n A n 3 B − 7n An 2 B − 8n An B − n 3 A +2n2 A −n A − 3n 3 B +4n2 B +3n BFactorizing, we haveZ = −n A n B£nA (n B +1)(n A − 5) + n 2 B (n A − 4) + 7n B +8 ¤ −n A (n A − 1) 2 −n B¡3n2B− 4n B − 3 ¢The derivative of δ N A with respect to n A is then given by∂δ N A∂n A=(n A +n B )Z(n B +1) 2 [n A (n A −1)(n A +n B −1)−n B ] 2which gives expression (32) in the text.⎤⎦7.2 The derivative of δ N A with respect to n BThe derivative of δ N A with respect to n B is given bywith∂δ N A∂n B=∂UV −U ∂V∂n B ∂n B(n B +1) 4 [n A (n A −1)(n A +n B −1)−n B ] 2U ≡ (n A + n B ) 2 [n B (n A − 2) + (n A − 1)]V ≡ (n B +1) 2 [n A (n A − 1) (n A + n B − 1) − n B ]We have32

∂U∂n B=(n A + n B )[2n B (n A − 2) + 2 (n A − 1) + (n A + n B )(n A − 2)]∂U∂n B=(n A + n B ) £ 3n A n B + n 2 A − 6n B − 2 ¤We also have∂V∂n B=(n B +1) £ 2 ¡ n 2 A − n ¢A (nA + n B − 1) − 2n B +(n B +1) ¡ n 2 A − n A − 1 ¢¤∂V∂n B=(n B +1) £ 2n 3 A +3n2 A n B − 3n A n B − 3n 2 A + n A − 3n B − 1 ¤Let denote the numerator of ∂δN A∂n Bas follows:∂U∂n BV − U ∂V∂n B=(n A + n B )(n B +1)[X − Y ]X =(n B +1) £ 3n A n B + n 2 A − 6n B − 2 ¤ [n A (n A − 1) (n A + n B − 1) − n B ]Y =(n A + n B )[n B (n A − 2) + (n A − 1)] £ 2n 3 A +3n2 A n B − 3n A n B − 3n 2 A + n A − 3n B − 1 ¤We now develop these two expressions. Let start with XX =(n B +1) £ 3n A n B + n 2 A − 6n B − 2 ¤£ n 3 A + n2 A n B − n A n B − 2n 2 A + n ¤A − n BDeveloping the last two brackets, we have⎡3n 4 A n B +3n 3 A n2 B − 3n2 A n2 B − 6n3 A n B +3n 2 A n B − 3n A n 2 B + n5 A + n4 A n BX =(n B +1) ⎣ −n 3 A n B − 2n 4 A + n3 A − n2 A n B − 6n 3 A n B − 6n 2 A n2 B +6n An 2 B +12n2 A n B−6n A n B +6n 2 B − 2n3 A − 2n2 A n B +2n A n B +4n 2 A − 2n A +2n BWe now can simplify. This gives∙ 4n4X =(n B +1) An B +3n 3 A n2 B − 13n3 A n B − 9n 2 A n2 B +12n2 A n B +3n A n 2 B − 4n ¸An B+n 5 A − 2n4 A − n3 A +4n2 A − 2n A +6n 2 B +2n BWe develop again. This gives⎤⎦X =4n 4 A n2 B +3n3 A n3 B − 13n3 A n2 B − 9n2 A n3 B +12n2 A n2 B +3n An 3 B − 4n An 2 B + n5 A n B − 2n 4 A n B −n 3 A n B +4n 2 A n B −2n A n B +6n 3 B +2n2 B +4n4 A n B +3n 3 A n2 B −13n3 A n B −9n 2 A n2 B +12n2 A n B +3n A n 2 B −4n A n B + n 5 A − 2n4 A − n3 A +4n2 A − 2n A +6n 2 B +2n BSimplifying and rearranging the terms, we haveX = n 5 A n B +4n 4 A n2 B +2n4 A n B +3n 3 A n3 B − 10n3 A n2 B − 14n3 A n B − 9n 2 A n3 B +3n2 A n2 B +3n An 3 B −n A n 2 B +16n2 A n B − 6n A n B + n 5 A − 2n4 A − n3 A +4n2 A − 2n A +6n 3 B +2n2 B +6n2 B +2n BWe now develop the expression Y .WehaveY =(n A + n B )[n A n B − 2n B + n A − 1] £ 2n 3 A +3n2 A n B − 3n A n B − 3n 2 A + n A − 3n B − 1 ¤Developing the last two terms in brackets, we have33

⎡Y =(n A + n B ) ⎢⎣2n 4 A n B +3n 3 A n2 B − 3n2 A n2 B − 3n3 A n B + n 2 A n B − 3n A n 2 B − n An B−4n 3 A n B − 6n 2 A n2 B +6n An 2 B +6n2 A n B − 2n A n B +6n 2 B +2n B+2n 4 A +3n3 A n B − 3n 2 A n B − 3n 3 A + n2 A − 3n An B − n A−2n 3 A − 3n2 A n B +3n A n B +3n 2 A − n A +3n B +1Simplifying the term in brackets, this yields∙ 2n4Y =(n A + n B ) An B +3n 3 A n2 B − 4n3 A n B − 9n 2 A n2 B + n2 A n B +3n A n 2 B − 3n ¸An B+2n 4 A − 5n3 A +4n2 A − 2n A +6n 2 B +5n B +1We develop again,Y =2n 5 A n B +3n 4 A n2 B − 4n4 A n B − 9n 3 A n2 B + n3 A n B +3n 2 A n2 B − 3n2 A n B +2n 5 A − 5n4 A +4n3 A −2n 2 A +6n An 2 B +5n An B + n A +2n 4 A n2 B +3n3 A n3 B − 4n3 A n2 B − 9n2 A n3 B + n2 A n2 B +3n An 3 B − 3n An 2 B +2n 4 A n B − 5n 3 A n B +4n 2 A n B − 2n A n B +6n 3 B +5n2 B + n BWe simplify. This givesY =2n 5 A n B +5n 4 A n2 B − 2n4 A n B − 13n 3 A n2 B − 4n3 A n B +3n 3 A n3 B +4n2 A n2 B + n2 A n B − 9n 2 A n3 B +3n A n 3 B +3n An 2 B +3n An B +2n 5 A − 5n4 A +4n3 A − 2n2 A + n A +6n 3 B +5n2 B + n BWe can now calculate the difference X − Y .ThisgivesX −Y = −n 5 A n B −n 4 A n2 B +4n4 A n B +3n 3 A n2 B −10n3 A n B −n 2 A n2 B +15n2 A n B −4n A n 2 B −9n An B −n 5 A +3n4 A − 5n3 A +6n2 A − 3n A +3n 2 B + n BWe can make the following factorizationX − Y = −n 2 ¡B n4A− 3n 3 A + n2 A +4n A − 3 ¢ − n B (n 5 A − 4n4 A +10n3 A − 15n2 A +9n A − 1) −¡n5A− 3n 4 A +5n3 A − 6n2 A +3n A¢⎤⎥⎦The derivative of δ N A with respect to n B is given by∂δ N A∂n B=(n A +n B )(n B +1)[X−Y ](n B +1) 4 [n A (n A −1)(n A +n B −1)−n B ] 2which gives expression (33) in the text.7.3 The comparison between δ N A and δ N BWe haveδ N A =(n A+n B ) 2 [n B (n A −2)+(n A −1)](n B +1) 2 [n A (n A −1)(n A +n B −1)−n B ]The inequality δ N A ≥ δN Breduces toand δ N B =(n A+n B ) 2 [n A (n B −2)+(n B −1)](n A +1) 2 [n B (n B −1)(n A +n B −1)−n A ](n A +1) 2 [n B (n A − 2) + (n A − 1)] [n B (n B − 1) (n A + n B − 1) − n A ]34

≥ (n B +1) 2 [n A (n B − 2) + (n B − 1)] [n A (n A − 1) (n A + n B − 1) − n B ]We develop∙ ¸(n A +1) 2 n2B(n A − 2) (n B − 1) (n A + n B − 1) − n A n B (n A − 2) +n B (n B − 1) (n A − 1) (n A + n B − 1) − n A (n A − 1)∙≥ (n B +1) 2 n2A(n B − 2) (n A − 1) (n A + n B − 1) − n A n B (n B − 2) +n A (n B − 1) (n A − 1) (n A + n B − 1) − n B (n B − 1)We can rewrite the above inequality as it follows: ∆ ≡ (n A + n B − 1) [X + Y ]+Z ≥ 0hiX ≡ (n A +1) 2 n 2 B (n A − 2) (n B − 1) − (n B +1) 2 n 2 A (n B − 2) (n A − 1)Y ≡ (n B − 1) (n A − 1)hn B (n A +1) 2 − n A (n B +1) 2iZ ≡ n B (n B +1) 2 [n A (n B − 2) + (n B − 1)] − n A (n A +1) 2 [n B (n A − 2) + (n A − 1)]We now develop these expressions. We haveX = ¡ n 2 A +2n A +1 ¢ (n A − 2) ¡ n 3 B − ¡B¢ n2 − n2B+2n B +1 ¢ (n B − 2) ¡ n 3 A − ¢n2 AThis givesX = ¡ n 3 A − 3n A − 2 ¢¡ n 3 B − ¡B¢ n2 − n3B− 3n B − 2 ¢¡ n 3 A − ¢n2 ADeveloping, this gives¸X = −n 3 A n2 B − 3n An 3 B +3n An 2 B − 2n3 B +2n2 B + n2 A n3 B +3n3 A n B − 3n 2 A n B +2n 3 A − 2n2 AWe now develop Y .ThisgivesY =(n A n B − n A − n B +1) £ n 2 A n B − n A n 2 B + n ¤B − n AWe develop again and after simplification, we haveY = n 3 A n2 B − n2 A n3 B − n3 A n B + n A n 3 B + n2 A − n2 B + n B − n AThe sum X + Y givesX + Y =2n 3 A n B − 2n A n 3 B +3n An 2 B − 3n2 A n B +2n 3 A − n2 A − 2n3 B + n2 B + n B − n AWecannowdevelopthefollowingexpressionW ≡ (n A + n B − 1) [X + Y ]. ThisgivesW =2n 4 A n B − 2n 2 A n3 B +3n2 A n2 B − 3n3 A n B +2n 4 A − n3 A − 2n An 3 B + n An 2 B + n An B − n 2 A +2n 3 A n2 B − 2n An 4 B +3n An 3 B − 3n2 A n2 B +2n3 A n B − n 2 A n B − 2n 4 B + n3 B + n2 B − n An B − 2n 3 A n B +2n A n 3 B − 3n An 2 B +3n2 A n B − 2n 3 A + n2 A +2n3 B − n2 B − n B + n ASimplifying, we get35

W =2n 4 A n B −2n A n 4 B +2n3 A n2 B −2n2 A n3 B −3n3 A n B +3n A n 3 B +2n2 A n B −2n A n 2 B +2n4 A −2n4 B −3n 3 A +3n3 B + n A − n BWe now develop the expression Z. WehaveZ = ¡ n 3 B +2n2 B + n B¢[nA n B + n B − 2n A − 1] − ¡ n 3 A +2n2 A + n A¢[nA n B + n A − 2n B − 1]Developing again the expression, we haveZ = n A n 4 B + n4 B − 2n An 3 B − n3 B +2n An 3 B +2n3 B − 4n An 2 B − 2n2 B + n An 2 B + n2 B − 2n An B −n B − n 4 A n B − n 4 A +2n3 A n B + n 3 A − 2n3 A n B − 2n 3 A +4n2 A n B +2n 2 A − n2 A n B − n 2 A +2n An B + n ASimplifying, we haveZ = n A n 4 B − n4 A n B +3n 2 A n B − 3n A n 2 B + n4 B − n4 A + n3 B − n3 A + n2 A − n2 B + n A − n BFinally, we show that ∆ ≡ W + Z ≡ (n A + n B − 1) [X + Y ]+Z ≥ 0. Summing W and Z, wehave∆ = n 4 A n B − 2n A n 4 B +2n3 A n2 B − 2n2 A n3 B − 3n3 A n B +3n A n 3 B +5n2 A n B − 5n A n 2 B + n4 A − n4 B −4n 3 A +4n3 B + n2 A − n2 B +2n A − 2n BFactorizing the above expression, we have¡∆ = n A n B n3A− n 3 B¢+2n2An 2 B (n ¡A − n B )−3n A n B n2A− n 2 B¢+5nA n B (n A − n B )+n 3 A (n A − 4)−n 3 B (n B − 4) + n A (n A +2)− n B (n B +2)Factorizing again, we have£∆ = n A n B n2A(n A − 3) − n 2 B (n B − 3) + (n A − n B )(2n A n B +5) ¤ +n 3 A (n A − 4)−n 3 B (n B − 4)+n A (n A +2)− n B (n B +2)This expression is always positive for any n A and n B higher than four and for n A ≥ n B . Wethen have δ N A ≥ δN B . 36