Guillaume Cheikbossian
email: guillaume.cheikbossian@umontpellier.fr

Evolutionarily stable in-group altruism in intergroup con ‡ict

Keywords: Indirect Evolutionary Approach, Evolutionary Stability, Groups, Altruism, Con- ‡icts JEL classi…cation: C73, D64, D74

We provide an evolutionary explanation for the well-established evidence of the existence of in-group favoritism in intergroup con ‡ict. Using a model of group contest, we show that the larger the number of groups competing against one another or the larger the degree of complementarity between individual e¤orts, the more likely group members are altruistic towards their teammates under preference evolution.

Introduction

Since ancient times, a number of economic, social and political activities involve groups that are in opposition to one another. Intergroup con ‡icts have thus been extensively studied within di¤erent disciplines. For example, in social psychology, social con ‡icts are the main focus of the social identity theory (Tajfel and Turner, 1979). A central feature of this theory is the recognition that humans have a tendency to discriminate between "in-group" and "out-group" members even though "groups" are not formed according to some intrinsic characteristics or preferences but by random assignment. Furthermore, it is recognized that a stronger con ‡ict with an "out-group" is likely to reinforce group cohesion by strengthening in-group favoritism (see Tajfel, 1982). More recently, social psychologists have conducted experimental studies in the laboratory which con…rm that, in general, inter-group competition improves intra-group coordination in simple team games (see, e.g., Bornstein et al., 2002; Bornstein, 2003). There is also an important literature on human evolutionary biology that explains, often using dynamic games and simulations, that non-kin altruism towards members of one's own group together with out-group hostility -what is called "parochial" altruism -is a powerful force of the evolution of behavior in human societies (see, e.g., Choi and Bowles, 2007; Lehmann and Feldman, 2008; and Rush; 2014 or Glowaki et al. 2017, for a survey). As to the economic literature, the analysis of group con ‡icts is principally based on contest or rent-seeking games between groups (Katz et al. 1990 and Nitzan, 1991). [START_REF] Abbink | Intergroup Con ‡ict and Intra-Group punishment in an Experimental Contest Game[END_REF] A number of relatively recent experimental studies aim at testing this type of games and generally conclude that subjects over contribute to group e¤ort compared to the theoretical predictions (see, e.g., Abbink et al., 2010, 2012; Ahn, et al., 2011; and Sheremeta, 2018, for a survey).

In this paper, we provide an evolutionary and theoretical analysis of the emergence and stability of in-group favoritism in intergroup con ‡ict by using a model of group contest. More precisely, there is large population of players who are randomly matched in groups that compete for an exogenous prize. We think of local (common-access) resources, each of which being targeted by a certain number of groups. The probability of winning the contest of a certain group is given by a Contest Success Function (CSF) that depends on the group members'e¤orts relative to members'e¤orts of competing groups. We consider that each group has two members and that the CSF has the standard ratio-form such that each group's probability of winning the prize is equal to the proportion of its collective e¤ort out of the sum of collective e¤orts by all groups involved in the contest. However, in contrast to most analysis on group contests that assume a 'summation technology'with perfect substitutability between individual e¤orts, we consider that the e¤ective level of group e¤ortits impact function -is given by a technology featuring a varying degree of complementarity between individual e¤orts. Indeed, as …rst pointed out by Alchian and Demsetz (1972), team or group production exists to the extent that it can exploit complementarities of inputs and this might be especially the case in a context of competition between groups (see also Kolmar and Rommeswinkel, 2013; and Brookins et al., 2015).

Another crucial feature of the present analysis is that each player has a utility -his/her "preference type" -that depends not only about his/her own material payo¤ but also on that of his/her teammate. We …rst remain agnostic as to whether this concern is altruistic or spiteful, that is to whether each member puts a positive or negative weigh on the payo¤ of his/her teammate when deciding his/her contribution to collective e¤ort. We then characterize the (pure strategy) Nash equilibria of this group contest game when group members have heterogeneous "other-regarding" preferences within and across groups. Next, we use the indirect evolutionary approach pioneered by Güth and Yaari (1992) to endogenize players' preferences. That is, evolution does not play directly at the level of strategies but indirectly at the level of preferences while players act rationally. In other words, preferences determine the players' actions which in turn determine (individual) material payo¤ -or "…tness"and ultimately the evolutionary survival of preferences. [START_REF] Abbink | Parochial Altruism in Inter-group Con ‡icts[END_REF] Thus, applying the concept of evolutionary stability (see Maynard Smith, 1982) to preferences -rather than to strategiesallows us to endogenize attitudes towards one's partner.

It is worth pointing out that a change in the preference type and thus in the equilibrium action of one player has strategic implications in that it induces a change in equilibrium actions of other players. The di¢ culty of the present analysis stems from the fact that there is a dual level of strategic interactions between players. The …rst one occurs within groups. Each group member decides on his/her contribution to group e¤ort according to his/her preferences given the preference type and the resulting action of his/her teammate. The second level of strategic interactions occurs across groups. A change in the preference type of one or both members of the same group is passed on group e¤ort which in turn induces a change in the winning probabilities for all groups. The evolutionary success of a certain preference type is the product of this dual level of strategic interactions. And we are particularly interested by the impact of the existence of within-group complementarities and by the intensity of the competition between groups, as measured by the number of competing groups, on the evolutionary selection process of preferences.

Using a notion of local evolutionary stability (Alger and Weibull, 2010), we can explicitly determine the evolutionarily stable degree to which a group member cares about the material payo¤ of his/her teammate. Clearly, a given group would be more successful in the group contest with more in-group altruism. But in the process of evolution, the question arises of whether a certain degree of altruism within groups is immune against "mutant" members with lower degrees of altruism. In fact, we show that preference evolution can equally result in altruism or spite within groups in intergroup con ‡ict. Yet, we show that the larger the degree of complementarity between individual e¤orts or the larger the number of opponents, the more likely group members are altruistic towards each other. Furthermore, a further increase in the degree of complementarity between partners' e¤orts or in the intensity of competition usually tends to reinforces group cohesion in that it increases altruism towards one's partner.

While there are several evolutionary analysis of con ‡icts between single players,3 there are very few theoretical analysis applying the evolutionary approach to group con ‡icts. In fact, we are aware of only two papers. The …rst one is due to Eaton et al. (2011) who consider a production and con ‡ict model with a large population of players. In each period, players are randomly matched to form groups of two members and each group competes for a common access resource with just one other group. After appropriating some of the common resource, the members of each group can spend some processing e¤orts to produce a consumption good. While the model is speci…c, they cannot prove the existence of evolutionarily stable preferences and have to rely on numerical simulations for endogenizing the preference parameters. The numerical results show that the evolutionary stable parameter on the payo¤ of one's teammate is positive -thus featuring intra-group altruism -while that on the payo¤s of the out-group members is negative -thus featuring intergroup spitefulness. [START_REF] Alchian | Production, Information Costs, and Economic Organization[END_REF] The other theoretical analysis is due to Konrad and Morath (2012). They consider a …nite population of players with two groups of equal size competing for a prize (or a rent). Each player cannot observe the preference types of other players and, thus, they introduce the concept of robust beliefs such that any player with a certain preference type believes that all other players are of the same type (and have the same robust beliefs). This assumption greatly simpli…es the analysis since it eliminates all strategic e¤ects on behaviors of a change in preference types -i.e. a change in the weighs attributed to others'payo¤s. Konrad and Morath (2012) then characterize the set of evolutionarily stable preferences, which involve a linear combination of in-group favoritism and out-group spite with the two traits being perfect substitutes. Indeed, the utility of each player is itself given by a linear combination of material payo¤s of all players (including out-group members), while each player exerts only one level of e¤ort. And this last can be increased by either more in-group altruism or more spiteful behavior towards the out-group.

Our contribution is that we demonstrate the existence and obtain an analytical solution for the evolutionary stable preference parameters in a simple model featuring strategic interactions within and across groups. Furthermore, we explicitly show that as stronger adversity, as measured by the number of groups competing for the same prize, reinforces in-group favoritism when group partners perform complementary tasks.

The Framework 2.1 A simple group contest game

We consider a large population of players. In each period, players are randomly matched to form groups of two members playing a group contest game for a common access resource Y normalized to 2. As in Eaton et al. (2011), group contests are localized and isolated from each other in that each contest involves a certain number n of groups. The justi…cation is that the common access resources are situated in geographically identi…ed places.

Let (i; j); for i = f1; 2g and j = f1; 2; :::; ng, denote member i of group j. e ij 2 R + is the amount of e¤ort expended by player (i; j) and e j = (e 1j ; e 2j ) 2 R 2 + is the vector of e¤orts in group j: The e¤ort of group j depends on group members'e¤orts according to an impact function G j : R 2 + ! R + , which has the CES form, that is

G j (e j ) = e 1j + e 2j 1
; for j = f1; 2; :::; ng ;

where 2 f( 1; 0) [ (0; 1]g measures the degree of complementarity between individual e¤orts. The elasticity of substitution is 1=(1 ). Thus, the lower , the lower is the elasticity of substitution or the higher is the degree of complementarity between individual e¤orts within groups. For = 1, we have perfect substitutability between individual e¤orts and Eq. ( 1) becomes the standard 'summation technology', i.e. G j (e j ) = P i e ij . For ! 1, we have perfect complementarity, i.e. G j (e j ) = M infe ij g (referred as to the 'weakest-link'function). Finally, for < 0 and e ij = 0, the impact function of group j is not well-de…ned. Hence we will take the limit of (1) as e ij ! 0, which means G j (e j ) = 0 in this case. [START_REF] Alger | Kinship, Incentives and Evolution[END_REF] Group e¤orts determine the division of the prize. The share allocated to group j is given by a contest success function p j : R n + ! [0; 1], which has the ratio-form [START_REF] Alger | Homo Moralis -Preference Evolution under Incomplete Information and Assortative Matching[END_REF] p j (e j ; e j ) =

8 > > > < > > > : G j (e j ) P n k=1 G k (e k ) if P n k=1 G k (e k ) 6 = 0; 1 n otherwise, (2)
where e j = ((e 11 ; e 21 ); :::; (e 1j 1 ; e 2j 1 ); (e 1j+1 ; e 2j+1 ); :::; (e 1n ; e 2n )).

Each member in each group receives an equal amount of the share of the prize of total value equal to 2 -implying that each group member receives a share of a prize of value Y =2 = 1. In addition, it is assumed that each player incurs a constant marginal cost of e¤ort. The material payo¤ to member (i; j), for i = f1; 2g and j = f1; 2; :::; ng ; is given by an additively separable function ij : R + [0; 1] ! R + , that is ij (e j ; e j ) = p j (e j ; e j ) e ij .

(3)

Preference interdependence

Let consider that each player has a utility that depend not only on his/her own material payo¤ but also on that of his/her teammate. The utility of member (i; j), for i = f1; 2g and j = f1; 2; :::; ng ; is also given by an additively separable function

V ij : R 2 + ! R + , that is V ij (e j ; e j ) = ij (e j ; e j ) + ij ij (e j ; e j ) , (4)
where the subscript ij stands for the member other than i in team j: ij 2 ( 1; R], with R 1 and for i = f1; 2g and j = f1; 2; :::ng, is the utility weight given by member i in group j to the material payo¤ of his/her teammate -with positive values representing "altruism" and negative values representing "spite". [START_REF] Bester | Is altruism Evolutionarily Stable[END_REF] Let 2 ( 1; R] 2n be the vector of preference parameters in a localized contest that involves n groups of two members.

In contrast to Eaton et al. (2011) and Konrad and Morath (2012), the utility of a player does not depend -presumably negatively -on the material payo¤s of out-group members. Again, in a simple rent-seeking game, this is not a restrictive assumption since players exert just one level of e¤ort and this last could increase with either more in-group altruism or more spiteful behavior towards out-group members. Hence, player (i; j), for i = f1; 2g and j = f1; 2; :::; ng chooses his e¤ort level e ij so as to maximize his utility given by ( 4), which can be rewritten with (3) as

V ij (e j ; e j ) = (1 + ij )p j (e j ; e j ) (e ij + ij e ij ): (5)
We have the following Lemma. [START_REF] Bornstein | The E¤ect of Intergroup Competition on Group Coordination: An Experimental Study[END_REF] Lemma 1: (i) Given 2 ( 1; R] 2n , there exists a pure-strategy Nash equilibrium. In equilibrium, the e¤ort of player (i; j) is characterized for i = f1; 2g and j = f1; 2; :::; ng by the following …rst-order condition

(1 + ij ) p j (e j ; e j ) [1 p j (e j ; e j )] e 1 ij (e ij + e ij )
1 with equality for e ij > 0;

(ii) In any equilibrium, a given group j is either fully active i.e. e ij > 0 for i = f1; 2g or fully inactive i.e. e ij = 0 for i = f1; 2g.

The proof consists in three steps. We …rst show that V ij (e j ; e j ) is strictly concave in e ij so that the …rst-order condition ( 6) is necessary and su¢ cient for characterizing the bestresponse function of player (i; j). Next, we show that there cannot exist an equilibrium in which a corner solution holds for a member of one group while an interior solution holds for the other member of the same group (property (ii)). This important property allows us to reduce the group contest to a lottery contest between individual players with heterogeneous preferences and then use the existence theorems of Cornes and Hartley (2005). It is also worth pointing out that when < 0, there are multiple equilibria since if player (i; j) chooses e ij = 0, then G j (e j ) = 0 and thus player ( i; j) cannot do better than choosing e ij = 0.

In other words, the members of a given group can "coordinate" on participating or not participating to the group contest when < 0 (see also, Kolmar and Rommeswinkel, 2013). [START_REF] Bornstein | Intergroup Con ‡ict: Individual, Group and Collective Interests[END_REF] Suppose now that all players in the population except one player, whom we will call the mutant, have a certain degree of altruism 2 ( 1; R] towards their teammates. Without loss of generality, let assume that the mutant is player (1; 1) -i.e. the member 1 of group 1 -and let m 2 ( 1; R] be his/her degree of altruism/spitefulness. These weights are exogenous to the players.

The mutant with the preference parameter m chooses e 11 so as to maximize his/her utility given by V 11 (e

Finally, the utility of all other players are symmetric since they all belong to a group where the two members have the same preference parameter . Thus, player (i; j), for i = f1; 2g and j = f2; 3; :::; ng, chooses e ij to maximize his/her utility given by V ij (e j ; e j ) = (1 + )p j (e j ; e j ) (e ij + e ij ).

There are thus three distinct equilibrium e¤ort levels that must satisfy the …rst order conditions for the two members of group 1 and the …rst-order condition for one member of any other group than group 1.

We have the following Proposition. (i) There exists a unique interior pure strategy Nash equilibrium with e ij > 0 for all i and j if and only if

( m ; ) 2 1 (n 2)(1 + ) (n 1) ; (8)
(ii) If (8) holds, equilibrium e¤ort levels are given by

e 11 ( m ; ) = 2 1 (n 1)(1 + )(1 + m ) 1 1 [(n 1) [ ( m ; )] 1 2 1 (n 2)(1 + )] ( m ; ) [2 1 (1 + ) + (n 1) [ ( m ; )] 1 ] 2 ; e 21 ( m ; ) = 2 1 (n 1)(1 + ) 2 1 [(n 1) [ ( m ; )] 1 2 1 (n 2)(1 + )] ( m ; ) [2 1 (1 + ) + (n 1) [ ( m ; )] 1 ] 2 ; e ij ( m ; ) = 2 1 (n 1) (1 + ) 2 [ ( m ; )] 1 2[2 1 (1 + ) + (n 1) [ ( m ; )] 1
] 2 for i = f1; 2g and j = f2; :::; ng : (9) As shown in the proof of this Proposition, there are two types of equilibrium (excluding those in which group members "coordinate" on not participating when < 0): either all groups are fully active or group 1 is the only inactive group. Observe that the inequality ( 8) is always veri…ed for m = and that its left-hand term is increasing in m . Therefore, the inequality is always veri…ed for m . If however, given and n, m is su¢ ciently small relative to so that the inequality ( 8) is reversed then, in equilibrium, the group with the mutant member is fully inactive because the utility weight given by the mutant to the material payo¤ of his/her partner is relatively too small (and potentially negative).

Suppose that (8) holds, then the material payo¤ -or …tness -of player (i; j) for i = f1; 2g and j = f1; 2; 3; :::

; ng, is ij ( m ; ) = p j e j ( m ; ) ; e j ( m ; ) e ij ( m ; ) . ( 10
)
3

Evolutionarily Stable Preferences

To study the evolutionarily stability of altruism or spitefulness, we employ the indirect evolutionary approach pioneered by Güth and Yaari (1992). All players choose e¤ort levels that maximize their utility and evolution pressure ensures the survival of preferences'parameters that induce equilibrium behavior providing the highest levels of material payo¤.

Initially, all players have the same preference parameter and the question is whether this preference parameter is immune against invading mutant players with a preference parameter m . Suppose a mutant is selected to play a n-group contest game as member 1 of group 1. The preference parameter is said to be evolutionarily stable if 11 ( ; ) > 11 ( m ; ) for all m 6 = . In other words, a degree of altruism is evolutionary stable if 11 ( m ; ) reaches its unique global maximum in m = . Unfortunately, we will not be able to verify global stability or that the above inequality holds for all m 6 = . Hence, following Alger and Weibull (2010), we use the concept of local stability.

De…nition 1 (Alger and Weibull, 2010): A necessary and su¢ cient condition for a degree of altruism/spitefulness 2 ( 1; R] to be locally evolutionarily stable is (i)-(ii) where

(i) @ 11 ( m ; ) @ m j m= = 0; (ii) @ 2 11 ( m ; ) @ 2 m j m= < 0:
In other words, is locally evolutionarily stable if and only if 11 ( m ; ) has a strict local maximum in m = .

We have the following Proposition.

Proposition 2: If , then there exists a unique locally evolutionarily stable preference parameter 2 ( 1; R] given by

= (n 2 2)(1 ) n (n 2 + 2)(1 ) n(1 2 ) : (11)
and where

= 2n 3 + 3n 2 6n + 4 n p 4n 3 (n 1) + 17n 2 20n + 12 4(n 1) 2 ; ( 12
)
which is increasing in n over 5 3 p 2; 1 .

Recall that the lower , the greater is the complementarity between group members'contributions. Thus, there may not exist stable evolutionary preferences if individual contributions are too substitutable. The limit value of is nevertheless increasing in the number of competing groups. It reaches a minimum in n = 2 in which case j n=2 = 5 3 p 2 0:76 and approaches 1 as n is going to in…nity.

From this Proposition, we have the following Corollary.

Corollary 1: (i) T 0 for S ^ where ^ = (n 2 n 2) =(n 2 2); (ii) ^ is increasing in n over [0; 1).
Corollary 1 states that can be positive -featuring altruism -or negative -featuring spite -depending on the degree of complementarity between individual e¤orts within groups and on the number of groups competing against each other.10 Indeed, a crucial feature of the contest game is that individual …tness is determined by the probability of success of the group and this last has the characteristics of a public good. Thus, increasing one's own contribution to group e¤ort above the level corresponding to sel…sh behavior -as a result for example of altruism -can be advantageous vis-à-vis the players of the other groups but can also be detrimental vis-à-vis one's teammate because it could induce him/her to freeride on one's own additional e¤ort. Therefore, it is not clear whether altruism or spiteful behavior is more successful in terms of individual …tness. This depends on the technology for aggregating individual e¤orts into group e¤ort and on the number of groups competing against each other for the resource. Yet, if the degree of the complementarity between individual e¤orts is su¢ ciently strong -i.e.

^ -then evolution leads towards intra-group altruism. Furthermore, the requirement about the degree of complementarity becomes less stringent as the number of competing groups increases. Indeed ^ is increasing in n, reaches a minimum in n = 2 in which case ^ j n=2 = 0, and approaches 1 as the number of groups competing against each other is going to in…nity. As a result, preference evolution leads to intra-group altruism for any < 0 independently of the number of competing groups and, in the limit, when the number of competing groups becomes very large, preference evolution leads to intra-group altruism for any 1: We can also determine how changes with the degree of complementarity between individual e¤orts. The derivative of with respect to is given by

@ @ = 2n 2 (n 1) [(n 2 + 2)(1 ) n(1 2 )] 2 : (13)
We thus have the following Proposition.

Proposition 3: The higher the degree of complementarity between individual e¤orts -i.e. the lower -the lower is the degree of spitefulness or the higher is the degree of altruismi.e. the higher is .

As already stated, the degree of complementarity must be su¢ ciently strong -i.e. ^for a positive degree of in-group altruism to be (locally) evolutionarily stable. In any case, an increase in the degree of complementarity between partners'e¤orts reinforces group cohesion in that it decreases spitefulness or increases altruism towards one's partner. Intuitively, higher degrees of complementarity make group e¤ort and in turn the probability of success of a given group more sensitive to individual e¤ort. As a result, "free-riding" on the contribution of one's teammate becomes detrimental to group success and in turn to individual …tness as the degree of complementarity increases.

There remains the question as to whether a larger number of competing groups increases the degree of intra-group altruism (or decreases the degree of intra-group spitefulness). Calculating the derivative of with respect to n, we obtain

@ @n = 2(1 ) [2(2n 1) + (n 2 4n + 2)] [(n 2 + 2)(1 ) n(1 2 )] 2 : (14)
One can observe that this derivative is always positive for any > 0 while its sign is ambiguous for < 0. We can thus state the following Proposition.

Proposition 4: (i) When > 0, the evolutionarily stable preference parameter is increasing in n and becomes positive from n ñ where ñ = )]; (ii) When < 0, the evolutionarily stable preference parameter is not monotonous in n, but it is always positive.

h 1 + p 9 8 (2 ) i = [2 (1
When the degree of complementarity between individual e¤orts is relatively low -i.e. when > 0 -preference evolution can lead to spitefulness within groups if few groups compete against each other. However, as the number of groups increases, it leads to the emergence of in-group altruism and this behavioral trait becomes increasingly strong as the number of competing groups keeps going up (property (i) of Proposition 4). This result echoes that of an increase in the degree of complementarity between individual e¤orts. When the intensity of the con ‡ict between groups increases, the success of one's own group is becoming more and more decisive for the evolutionary success of a group member. In other words, the intensity of the con ‡ict helps solving the tendency to "free-riding" -or sel…shness -by developing in-group favoritism. When the degree of within-group complementarities is relatively large -i.e. when < 0 -preference evolution leads to in-group altruism independently of the number of groups. Yet, in this case, a larger number of competing groups can increase as well as decrease the evolutionary stable degree of altruism within groups (property (ii) of Proposition 4). It remains that < 0 always leads to a stronger degree of in-group altruism than > 0 for any given number of competing groups (Proposition 3).

Finally, we can characterize equilibrium levels of e¤orts and of individual …tness when players have locally evolutionarily stable preferences. When all players in society have the same preference parameter , they all exert the same level of individual e¤ort i.e. -using ( 9) -e = (1 + )(n 1)=2n 2 . Substituting given by (11) into this expression, we obtain e =

(1 ) (n 1) 2 n [(n 2 + 2)( 1) n( 12 )] :

Furthermore, all groups have the same probability of success i.e. 1=n, so that each player's material payo¤ induced by evolutionarily stable preferences is = (1=n) e or 1) n( 12 )] :

= n + 1 n [(n 2 + 2)(
It can be easily veri…ed that an increase in the degree of complementarity between individual e¤orts -i.e. a decrease in -unambiguously decreases material payo¤. Indeed, the lower the higher is the evolutionarily stable level of in-group altruism and thus the higher the level of individual e¤ort. As a result, the con ‡ict becomes more severe. A larger number of competing groups has an ambiguous impact on the evolutionarily stable preference parameter and thus on equilibrium levels of e¤orts. Yet, a larger number of groups also unambiguously decreases individual payo¤s simply because a larger number of players are involved in the con ‡ict for a resource of a …xed amount.

Conclusion

The inclination of people to pull together in face of a common enemy appears to be a universal trait of human behavior and this is con…rmed by several experimental studies in economics or social psychology. In this paper, we provide an evolutionary foundation for the emergence and stability of 'parochial' altruism in a context where several groups compete against each other for an exogenous resource. It is shown that both a strong degree of complementarity between individual e¤orts and a large number of competing groups reinforce in-group favoritism. Indeed, the success of a group crucially depends on its ability to contain the tendency to free-riding or sel…shness. There is thus an evolutionary pressure towards in-group altruism because it increases the probability of success of the group in the con ‡ict, which in turn increases individual …tness. The downside of this behavior is that it makes the con ‡ict between groups more severe and potentially more destructive. A fundamental reason for this is that there is a …xed amount of resources at the origin of the con ‡ict. Thus, it would be interesting to try to extend the analysis to a model where players have some endowments -for example units of time -that can be used for production or for rent-seeking. However, the present analysis shows that the existence and characterization of evolutionarily stable preferences can be very challenging even in a very simple framework.

Appendix

Proof of Lemma 1

The proof of this Lemma is conducted in three steps. The …rst one is to show that the …rst-order conditions given by ( 6) are necessary and su¢ cient for maximization.

The …rst derivative of V ij (e j ; e j ), given by ( 5), with respect to e ij is given by @V ij (e j ; e j ) @e ij = (1 + ij ) @p j (e j ; e j ) @G j (e j ) @G j (e j ) @e ij 1:

(A1)

Using ( 1) and ( 2), we have @p j (e j ; e j ) @G j (e j ) =

P k6 =j G k (e k ) ( P n l=1 G l (e l )) 2 ; (A2) and @G j (e j ) @e ij = e ij + e ij 1 1 e 1 ij = G j (e j ) e 1 ij e ij + e ij : (A3)
Thus (A1) can be rewritten as

(1 + ij ) p j (e j ; e j ) h P k6 =j p k (e k ; e k ) i e 1 ij e ij + e ij 1: (A4)
As a result, the …rst-order condition for i = f1; 2g and j = f1; 2; :::; ng is given by ( 6) since p j (e j ; e j ) + P k6 =j p k (e k ; e k ) = 1. The second derivative of V ij (e j ; e j ) with respect to e ij is given by

@ 2 V ij (e j ; e j ) @e 2 ij = (1 + ij ) 8 > > > > > < > > > > > :
[1 2p j (e j ; e j )] @p j (e j ; e j ) @G j (e 1 ) @G j (e j ) @e ij From (A2) and (A3), we also have @p j (e j ; e j ) @G j (e j ) @G j (e j ) @e ij = p j (e j ; e j ) [1 p j (e j ; e j )] e (A7) which is always negative for i = f1; 2gand j = f1; 2; :::; ng since 1.

As a result V ij (e j ; e j ), for i = f1; 2gand j = f1; 2; :::; ng, is strictly concave and continuous in e ij for 2 ( 1; 0) or 2 (0; 1]. Thus, the …rst-order condition given by ( 6) is both necessary and su¢ cient for characterizing the best-response function of player (i; j) for i = f1; 2gand j = f1; 2; :::; ng :

The second step of the proof is to show that if a group participates to the contest, then its two members produce positive levels of e¤ort. In other words, a corner solution for player (i; j)that is e ij = 0 -and an interior solution for player ( i; j) -that is e ij > 0 -cannot be mutual best responses for the two players of group j. Indeed, suppose …rst that 2 (0; 1] with e ij = 0 and e ij > 0, then we would have p j (e j ; e j ) > 0 but the denominator of the LHT of ( 6) would tend to 0, so that the LHT would approach in…nity. Hence, (6) cannot be satis…ed for e ij = 0 and e ij > 0 when 2 (0; 1]. Now suppose that 2 ( 1; 0) with e ij = 0; then G j (e j ) = 0 and thus V ij (e j ; e j ) is strictly decreasing in e ij so that player ( i; j) cannot do better than e ij = 0 in this case. To conclude, if a group participates to the contest it fully participates with both members being active (see also Kolmar and Rommeswinkel, 2003). Furthermore, there cannot exist an equilibrium where all groups do not participate to the contest for 2 (0; 1]. Indeed, if all other groups do not enter the contest, the members of group j could win the prize with probability 1 in return for an arbitrarily small e¤ort (exerted by both group members). As a result for 2 (0; 1], at least one group is fully active. If 2 ( 1; 0), e ij = 0 and e ij = 0 are mutually best responses independently of the behavior of other groups. Hence, there may exist an equilibrium in which none the n groups participate to the contest.

The …nal step for demonstrating the existence of a pure strategy Nash equilibrium characterized by (6), is to reduce the group contest to a contest among individual players [START_REF] Corchón | The theory of Contests: A Survey[END_REF] and then use Theorem 1 of Cornes and Hartley (2005). Suppose that there are m active groups and consider the system of 2m …rst-order conditions (6) holding with equality. This system can be rewritten as (with (A1), (A2) and (A3))

P k6 =j G k G 1 j P l2M G l 2 = e 1 ij (1 + ij ) ; (A8)
for i = f1; 2gand j being an element of the set of active groups denoted by M . For a given group j, the LHS of (A8) is the same for i = f1; 2gand hence e 1j (1 + 2j ) 1=(1 ) = (1 + 1j ) 1=(1 ) e 2j . The aggregate output of group j 2 M can thus be written as a function of the e¤ort of player 1, that is

G j = " (1 + 1j ) 1 + (1 + 2j ) 1 (1 + 1j ) 1 # 1 e 1j 1j ( 1j ; 2j )e 1j : (A9)
We thus have e 1j = [ 1j ( 1j ; 2j )] 1 G j .

The system of of 2m …rst-order conditions (A8) can thus be reduced to a system of m equations in G j for j = f1; 2; :::; mg, that is

P k6 =j G k P l2M G l 2 = 1 j ( 1j ; 2j ) ; (A10) where j ( 1j ; 2j ) = (1 + 1j ) [ 1j ( 1j ; 2j )] 1 = h (1 + 1j ) 1 + (1 + 2j ) 1 i 1 : (A11)
In other words, the system of …rst-order conditions (A4) for the group contest with heterogeneous (other-regarding) preferences can be reduced to a system that is induced by a lottery contest of m individual players choosing G j with heterogeneous (and constant) marginal costs given by the RHT of (A10). Applying Theorem 1 of Cornes and Hartley (2005), we can conclude that there exists a pure strategy Nash equilibrium G j for j 2 M . In turn, e ij = [ 1j ( 1j ; 2j )] 1 G j for i = f1; 2g and j 2 M satisfying ( 6) constitute an equilibrium in the contest between groups.

Proof of Proposition 1

(i) From the proof of Lemma 1, we know that if a group participates to the contest, then its two members produce positive levels of e¤ort. Furthermore, the group contest with heterogeneous "other-regarding" preferences can be reduced to a lottery contest between individual players choosing G j with heterogeneous marginal costs. Let G = P k G k . From (A10), the …rst-order condition for player j can be rewritten as

G G j G 2 1 j ( 1j ; 2j ) 0: (A12)
It is non-positive at G j = 0 for j ( 1j ; 2j ) G. Thus player/group j is fully inactive if j ( 1j ; 2j ) G. If, however j ( 1j ; 2j ) > G, then player j is active and thus G j = G (G 2 = j ( 1j ; 2j )). Again, let M be the set of the m active players in equilibrium. We have G = P j2M G j , and hence G = m 1 P j2M (1= j ( 1j ; 2j ))

. (A13)

Suppose now that all group members have the same preference parameter except group member

(1; 1) with the preference parameter m . We thus have (from (A11))

1 ( ) = 2 1 (1 + ) for all j 6 = 1 and 1 ( m ; ) = h (1 + m ) 1 + (1 + ) 1 i 1
. We …rst show that there cannot exist an equilibrium in which player (i.e. group) 1 with marginal cost 1= 1 ( m ; ) is active and some x > 1 -but not all i.e. x < n 1 -other players (i.e. groups) with the common marginal cost 1= 1 ( ) are inactive. If it were the case, we would have

G = n x 1 [1= 1 ( m ; )] + (n x 1) [1= 1 ( )] = (n x 1) 1 ( m ; ) 1 ( ) (n x 1) 1 ( m ; ) + 1 ( ) : (A14)
But for a player j 6 = 1 to be inactive, we must also have 1 ( ) G which is in contradiction with (A14). As a result, there are two possibilities. All groups j 6 = 1 are either fully active or fully inactive. However, there cannot exist an equilibrium where all these groups are fully inactive while group 1 is fully active. Indeed, in that case, we would have that the LHT of (A12) is strictly negative for any positive level of G 1 > 0 so that this group would not play its best response. As a consequence, if player (i.e. group) 1 is active, then all players (groups) are active in equilibrium.

( m ; ) = (1 + m ) 1 + (1 + ) 1 : (A25)
Then, using (A11) and (A13), (A14) can be rewritten has e S h 2(n 1) [ (:)] (A26)

1 2 1 (1 + ) (n 2) i (1 + m ) 1 1 = (1 + ) [ (:)]
The …rst-order condition (A18) can also be rewritten as Using (A21) and (A25), (A28) becomes

2 1 (n 1)(1 + m ) 2 1 [ (:)] 1 e S = h [ (:)] 1 e 11 + 2 1 (n 1)(1 + m ) 1 1 e S i 2 (A29)
Substituting e S given by (A26) into (A29), we …nd after some tedious rearrangements, the equilibrium level of e¤ort e 11 ( m ; ) by the mutant -agent 1 of group 1 -given in (9). The equilibrium level of e¤ort e 21 ( m ; ) by the mutant's partner is obtained by using (A21). Finally, the common equilibrium level of e¤ort for any player i = f1; 2g of group j, for j = f2; 3; ::ng, that is e ij ( m ; ) in ( 9) is obtained by substituting e 11 ( m ; ) into (A26).

Proof of Proposition 2

We …rst determine condition (i) of De…nition 1. We have @ 11 @ m = @p 1 e 1 ; e 1 @G 1 (e 1 ) @G 1 (e 1 ) @e 11 @e 11 @ m + @G 1 (e 1 ) @e 21 @e 21 @ m + X j6 =1

@p 1 e 1 ; e 1 @G j (e j ) @G j (e j ) @e 1j @e 1j @ m + @G j (e j ) @e 2j @e 2j @ m @e 11 @ m . (A30)

Using the …rst-order conditions for the e¤ort levels of the two members of group 1, that is (A1) with 11 = m and 21 = , we have

@ 11 @ m = m 1 + m @e 11 @ m + 1 1 + @e 21 @ m + X j6 =1
@p 1 e 1 ; e 1 @G j (e j ) @G j (e j ) @e 1j @e 1j @ m + @G j (e j ) @e 2j @e 2j @ m : (A31)

We have

@p 1 e 1 ; e 1 @G j (e j ) = G 1 (e 1 ) h G 1 (e 1 ) + P j6 =1 G j (e j ) i 2 for j 6 = 1: (A32)
From (A3), we also have @G j (e j ) @e ij = (e ij ) + (e ij )

1 (e ij ) 1 : (A33)

Since e ij = e S for i = f1; 2g and j 6 = 1, we have G j (e j ) = G S (e S ) and @G j (e j )=@e ij = 2 (1 )= for i = f1; 2g and j 6 = 1 and thus (A31) reduces to

@ 11 @ m = m 1 + m @e 11 @ m + 1 1 + @e 21 @ m 2 1= (n 1)G 1 (e 1 ) [G 1 (e 1 ) + (n 1)G S (e S )] 2 @e S @ m : (A34)
We now evaluate this expression at m = . Recall …rst that e 21 = [(1 + )=(1 + m )]

We have

@ 3 (:) @ m = 2 1= (n 1) 8 > > > > > > > > > > < > > > > > > > > > > :
@p 1 (:) @ m @e S @ m (:) + p 1 (:) (:) @ 2 e S @ 2 m +p 1 (:) @e S @ m 2 6 6 6 6 4 @ (:) @G 1 (:) @G 1 (:) @e 11 @e 11 @ m + @G 1 (:) @e 21 @e 21 @ m + @ (:) @G S (:) @G S (:) @e S @e S @ m 3 7 7 7 7 5

9 > > > > > > > > > > = > > > > > > > > > > ; : (A53)
We have

@ (:) @G S (:) = (n 1) @ (:) @G 1 (:) = n 1 [G 1 (:) + (n 1)G S (:)] 2 : (A54)
Furthermore, from the …rst-order condition (A30), we have @p 1 (:)=@ m = @e 11 =@ m . Thus (A53) becomes Thus (A44) becomes

@ 3 @ m = 2 1= (n 1) 8 > > > > > > > > > > < > > > > > > > > > > :
@e 11 @ m @e S @ m (:) + p 1 (:) (:) @ 2 e S @ 2 m +p 1 (:) @e S @ m @ (:) @G 1 (:)

2 6 6 6 6 4
@G 1 (:) @e 11 @e 11 @ m + @G 1 (:) @e 21 @e 21 @ m +(n 1) @G S (:) @e S @e S @ m 3 7 7 7 7 5

9 > > > > > > > > > > = > > > > > > > > > > ; : (A55)
In a symmetric equilibrium, we have

p 1 (:) j m= = 1 n ; (:) j m= = 1 2 1 ne ; @ (:) @G 1 (:) j m= = 1 [nG(e )] 2 = 1 h 2 1 ne i 2 ;
@G S (:) @e S j m= = 2 1 ; @G 1 (:) @e 11 j m= = @G 1 (:) @e 21 j m= = 2

1 : (A56)
As a result, we have

@ 3 (:) @ m j m= = (n 1) ne 8 > > > > > > > > > < > > > > > > > > > : @e 11 @ m @e S @ m j m= + 1 n @ 2 e S @ 2 m j m= 1 2n 2 e @e S @ m j m= 2 6 6 6 4
@e 11 @ m j m= + @e 21 @ m j m= +2(n 1) @e S @ m j m=

3 7 7 7 5 9 > > > > > > > > > = > > > > > > > > > ; : (A57)
From (A35), we have @e 21 @ m j m= = e (1

)(1 + ) + @e 11 @ m j m= : (A58)

Substituting (A58) into (A57), we obtain

@ 3 (:) @ m j m= = (n 1) ne @e S @ m j m= 8 > > > > < > > > > : 1 1 n 2 e @e 11 @ m j m= + 1 2n 2 (1 )(1 + ) (n 1) n 2 e @e S @ m j m= 9 > > > > = > > > > ;
(n 1) n 2 e @ 2 e S @ 2 m j m= :

Substituting e given by (A40) into (A59), one obtain

@ 3 (:) @ m j m= = 2n (1 + ) @e S @ m j m= 8 > > > > < > > > > : (1 + )(n 1) 2 (1 + )(n 1) @e 11 @ m j m= + 1 2n 2 (1 )(1 + ) 2 (1 + ) @e S @ m j m= 9 > > > > = > > > > ; 2 (1 + )
@ 2 e S @ 2 m j m= :

(A60) Substituting (A38) and (A43) into (A60), we obtain after some trivial (but tedious) calculations @ 3 (:) @ m j m= = (n 1)(n 2) (A61)

The second derivative of 11 ( m ; ) with respect to m evaluated at m = is given by the sum of (A51) and (A61). One can observe that the sign of this expression is always negative for any 0. For 0, one can easily show that @ 2 11 =@ 2 m j m= = 0 as three roots in , that is

Proposition 1 :

 1 Let ( m ; ) = (1 + m ) 1 + (1 + ) 1 where m 2 ( 1; R] is the preference parameter of the mutant and where 2 ( 1; R] is the incumbent preference parameter, then:

e 1 ije 2 ij( 1 ) e ij e ij e ij + e ij 2 9

 1212 ij + e ij p j (e j ; e j ) [1 p j (e j ; e j )] e

1 e 11 :

 11

( 1 + 11 (e 11 +

 11111 m ) (n 1)G 1 (e 1 ) G S (e S ) e 1 e 21 ) [G 1 (e 1 ) + (n 1)G S (e S )] 2 1 = 0: (A27) Since G S (e S ) = 2 1= e S and G 1 (e 1 ) = (e 11 + e 21 ) 1= , (A27) can be rewritten as 2 1 (n 1)(1 + m ) (e 11 + e 21 )

  1 ; e 1 ) = (1 + m )p 1 (e 1 ; e 1 ) (e 11 + m e 21 ):(5) Member 2 of group 1, with the preference parameter , chooses e 21 so as to maximize his/her utility given by V

21

(e 1 ; e 1 ) = (1 + )p 1 (e 1 ; e 1 ) (e 21 + e 11 ).

  ) and the fact that G j (e j ) = G S (e S ) = 2 1= e S for j = f2; 3; :::; ng and that p 1 (e 1 ; e 1 ) and p S (e 1 ; e 1 ) have the same numerator, (A22) becomes e S 2G 1 (e 1 ) (n 1)e 1 21 2 1= (n 2) (e 11 + e 21 ) = G 1 (e 1 ) (e 11 + e 21 ) : (A23) Since G 1 (e 1 ) = (e 11 + e 21 ) 1= , (A23) can be rewritten as

	Thus, using (A18) and (A19), we have
				e 21 =	1 + 1 + m	1	1	e 11 :	(A21)
	With (A19) and (A20), we have
	2e S p 1 (e 1 ; e 1 ) (n 1)e 1 21 = [p 1 (e 1 ; e 1 ) + (n 2)p S (e 1 ; e 1 )] (e 11 + e 21 ) :	(A22)
	Using (1) and (2e S	h	2(n 1)e 1 21	2 1= (n 2) (e 11 + e 21 ) ( 1)=	i	= (e 11 + e 21 ) :	(A24)

Let

(:)

  We now calculate the derivative of 3 (:) with respect to m . From (A44), 3 (:) can be rewritten

	as
		3 (:) =		2 1= (n 1)p 1 (e 1 ; e S ) (e 1 ; e S )	@e S @ m
				where
	(e 1 ; e S ) =	1 G 1 (e 1 ) + (n 1)G S (e S )	:
						1	1	e 11 . Therefore, the derivative of e 21 with respect
	to m is thus given by
	@e 21 @ m	=	1 + 1 + m	1	1		(1	e 11 )(1 + m )	+	@e 11 @ m	;	(A35)
	and hence,
	@e 21 @ m	j m= =	(1		e )(1 + )	+	@e 11 m	j m= ;	(A36)

since at m = , we have e 11 = e 21 = e S = e :

  It can be easily veri…ed that 1 and 3 are strictly larger than 1 for any n 2. As a result, condition (ii) of de…nition is veri…ed if and only if

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1 CEE-M Working Papers / Contact : laurent.garnier@inra.fr  RePEc https://ideas.repec.org/s/hal/wpceem.html  HAL https://halshs.archives-ouvertes.fr/CEE-M-WP/ WP 2019 -13

For a review of the literature on group contests, see Gar…nkel and Skaperdas (2007, Section 7), Corchón (2007, Section 4.2), and Konrad (2009, Chapter 6).

This approach has been further developed by, among others,[START_REF] Bester | Is altruism Evolutionarily Stable[END_REF] and[START_REF] Sethi | Preference Evolution and Reciprocity[END_REF]. More recently, the literature on the evolution of preferences has focused on the informational structure in general settings.Heiftez et al. (2007aHeiftez et al. ( , 2007b) ) show that in a large class of games, agents with "biased" preferences, such as altruism, spite, reciprocity or fairness, may actually be more successful in terms of material payo¤ even though agents' preferences are not fully observable. The reason is that a bias in a player's objective function may be favorable by changing other players'equilibrium actions. However, if preferences are completely unobservable, payo¤-maximization is evolutionary stable (see, e.g.,[START_REF] Ok | On the Evolution of Individualistic Preferences: An Incomplete Information Scenario[END_REF][START_REF] Dekel | Evolution of Preferences[END_REF].[START_REF] Alger | Homo Moralis -Preference Evolution under Incomplete Information and Assortative Matching[END_REF] show that, under incomplete information, sel…shness will indeed prevail if there is no assortative matching at all but that, with some degree of assortativity, preference evolution leads to a certain degree of kantian morality.

The evolutionary analysis of contests con ‡ict between single players started with Scha¤er (1988) who adapted the notion of an Evolutionary Stable Strategy (ESS) by Maynard[START_REF] Smith | Evolution and the Theory of Games[END_REF] to a …nite population of players. Following Scha¤er (1988),[START_REF] Hehenkamp | Evolutionary Equilibrium in Tullock Contests: Spite and Overdissipation[END_REF] compare behaviors induced by an ESS to behaviors in Nash equilibrium and show that an ESS involves spiteful e¤orts in the contest between individuals. This in turn involves overdissipation of the rent compared to Nash equilibrium. Yet, in an in…nite population, ESS behavior coincides to Nash equilibrium behavior. Finally,[START_REF] Leininger | Evolutionarily Stable Preferences in Contests[END_REF] applies the indirect evolutionary approach to contests between single players and shows that it gives rise to spiteful preferences that induce the same aggressive behavior than in an ESS.

In the second part of their paper, they analyze a rent-seeking game for a private prize and for a public prize with interdependent preferences. However they do not try to endogenize the preference parameters of the players in this part of the analysis.

Note that (1) is also discontinuous at = 0. This case is excluded from our analysis.

For an axiomatization of group contest success functions, seeMunster (2009). And for a rather complete analysis of group contests -but without other-regarding preferences -where the impact function is given by a CES technology, see[START_REF] Kolmar | Contests with Group-Speci…c Public Goods and Complementarities in E¤orts[END_REF].

We exclude the case ij 1 since then it would prevent positive levels of e¤orts. However, degrees of altruism ij 1 may exist, for instance, between biological parents or couples.

All the proofs are given in the Appendix.

Thus, there also exists the corner equilibrium -that we ignore -with e ij = 0 for all i and j:

Since ^ < for any n 2, 0 can indeed be locally evolutionarily stable. One can also easily verify that given by (11) is also strictly greater than 1.

See also[START_REF] Brookins | Optimal Sorting in Group Contests with Complementarities[END_REF] in a model of group contest with CES impact functions and heterogeneous and convex cost functions.

Since the mathematical expressions are very long, we only write the values of these expressions evaluated at m = . We also used the Mathematical software to be sure not to have made any errors in the computation of these derivatives.

Now, suppose player 1 is fully inactive while some x > 1 -but not all i.e. x < n 1 -other players are also fully inactive. In this case, we would have G = (n x 2) 1 ( ) (n x 1) : (A15)

But again for a player j 6 = 1 to be inactive, we must also have 1 ( ) G which is in contradiction with (A15). Thus when player 1 is fully inactive, All groups j 6 = 1 are either fully active or fully inactive. As stated in footnote 9, we ignore the equilibrium where all groups are fully inactive.

Thus, suppose that all players j 6 = 1 are fully active (still with player 1 being active). In this case, G is given by (A15) with x = 0. This is an equilibrium if 1 ( m ; ) G, or

To conclude there are two types of equilibrium. If (A16) holds, group 1 is fully inactive while all other groups are active. If (A16) does not hold, then all groups are fully active. Let de…ne

(1 + ), the necessary and su¢ cient condition for all groups being active is given by ( 8).

(ii) Suppose (8) holds. If ij = for i = f1; 2g and j = f2; :::; ng, then …rst-order conditions given by (6) are symmetric for all j 6 = 1 and thus all players -except the members of group 1 -exert the same level of individual e¤ort that we denote e ij = e S for i = f1; 2g and j = f2; 3; :::; ng : As a result p j (e j ; e j ) = p S (e 1 ; e 1 ) for all j = f2; 3; :::; ng and where e 1 = ((e S ; e S ); :::; (e S ; e S ))

| {z } n 1 times

:

The two members of group 1 di¤er -in terms of (other-regarding) preferences -and thus will exert di¤erent levels of e¤orts in equilibrium. This gives rise to a probability of success for group 1, i.e. p 1 (e 1 ; e 1 ), that di¤ers from p S (e 1 ; e 1 ).

From these observations, there are three distinct equilibrium levels of e¤ort that must satisfy the following …rst-order conditions, for all players (i; j), for i = f1; 2gand j = f2; :::; ng.

Also using (9), the derivative of e S with respect to m is given (after tedious calculations and rearrangements) by @e S @ m = 2

and hence @e S @ m j m= = (n 2)(n 1) 4n 3 : (A38)

Substituting (A38) and (A36) into (A34), we have

Using (9), we also have

Substituting (A40) into (A39), one obtain

Also, after long and tedious calculations, we obtain @e 11 @ m = 2

+ ) + (n 1) [ (:)]

1 i 3 where

Therefore, we have @e 11 @ m j m= = (n 1) (n 1) 2 (n 2)(2 ) (n 1) ( n 3) 4( 1)n 3 : (A43)

Substituting into (A41) and setting it to 0 yields a unique value for given by (11). One can also easily verify that 1, since this inequality reduces to n =( 1), which holds for any 2 f( 1; 0) [ (0; 1]g.

We now verify condition (ii) of De…nition 1. (A34) can be rewritten as @ 11 @ m = 1 (:) + 2 (:) + 3 (:) where

We have

Now, let evaluate this expression at m = . We have

We also have @ 2 (:)

Using (A35), the second derivative of e 21 with respect to m is given by

Substituting (A48) into (A47) and evaluating this last expression at m = , we obtain @ 2 (:) @ m j m= = 1 1 + 2 ( 1)(1 + ) @e 11 @ m j m= + (2 )e ( 1) 2 (1 + ) 2 + @ 2 e 11 @ 2 m j m= :

(A49) Recalling that e = (1 + )(n 1)=2n 2 , using (A46) and (A49) we obtain @ ( 1 (:

)(1 + )

What remains to be done is the calculation of the second derivative of e 11 and of e S with respect to m . Calculating the derivative of @e 11 =@ m given by (A42) with respect to m and evaluating the resulting expression at m = , we obtain 12 @ 2 e 11 @ 2 m j m= = (n 1) [n 3 5n 2 + 10n 6 2 (2n 3 6n 2 + 8n 3)] 8( 1) (1 + ) n 4 : (A62)

Similarly, calculating the derivative of @e S =@ m given by (A37) with respect to m and evaluating the resulting expression at m = , we obtain @ 2 e S @ 2 m j m= = (n 1) [3n 2 10n + 6 2 (2n 2 6n + 3)] 8( 1) (1 + ) n 4 : (A63)

Substituting (A62) into (A51) to obtain [@ ( 1 (:) + 2 (:)) =@ m ] j m= and (A63) into (A61) to obtain [@ 3 (:)=@ m ] j m= and adding the two terms, we obtain @ 2 11 =@ 2 m j m= (see (A44)), that is (A64)

Substituting by given by (11) into this expression, we obtain @ 2


Robert Chreiky

Gilles Delmaire

Matthieu Puigt

Gilles Roussel

Antoine Abche

INFORMED SPLIT GRADIENT NON-NEGATIVE MATRIX FACTORIZATION USING HUBER COST FUNCTION FOR SOURCE APPORTIONMENT

Keywords: Informed source separation, Non-negative matrix factorization, Robustness, Split gradient, Source apportionment, Huber cost function

est destinée au dépôt

Informed split gradient non-negative matrix factorization using Huber cost function for source apportionment

INTRODUCTION

Source apportionment aims to find particulate matter sources and their emissions in the ambiant air. In practice, a device is trapping particulate matter on filters, which are then analyzed by chemical experts. They actually estimate the masses of each chemical species present in the filters, which are converted into concentrations and provide a n × m non-negative data matrix X, where n is the number of samples and m the number of species. The data matrix may be approximated by a linear mixture model also called Receptor Model in source apportionment, i.e.,

X ≈ G • F, (1)
where G is the n × p contribution matrix gathering the emissions of all the sources over time and F is the p × m profile matrix, where a profile is a source signature involving all the chemical species proportions (in ng/ng). The non-negativity property of the factors usually leads to tackle the approximated factorization in Eq. (1) with NMF methods which generally carry out an alternating minimization between factors of the Frobenius norm of the residuals [1]. However, even a small number of deviating data points completely alter the quality of the approximation. Indeed, the Frobenius cost function is well-suited for Gaussian noise while degrading in the presence of atypical observations. As a result, robust NMF methods are devised to withstand a certain proportion of corrupted data points called outliers. In source apportionment, outliers may result from an additional contamination on the filter, or from a measurement error.

Two main strategies are used to handle outliers in NMF. A first class of methods [2][START_REF] Rapin | Sparse and non-negative BSS for noisy data[END_REF][START_REF] Févotte | Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization[END_REF] proposes to rewrite the matrix X as the sum of a low rank matrix L which can be factorized according to Eq. (1), and a sparse matrix S which should contain the outliers available in the collected data. The sparsity of S is reached by using special norms favoring this property. The second type of strategy suggests the use of a robust cost function between parts of Eq. (1), e.g., the 1 -2 [START_REF] Hamza | Reconstruction of reflectance spectra using robust nonnegative matrix factorization[END_REF] or 2,1 [START_REF] Kong | Robust non-negative matrix factorization using l 21 -norm[END_REF] norms, the Huber function [START_REF] Du | Robust non-negative matrix factorization via half-quadratic minimization[END_REF], or a parametric divergence [START_REF] Cichocki | Generalized alpha-beta divergences and their application to robust nonnegative matrix factorization[END_REF]. While the latter is asymmetric and implicitly assumes large residuals to be positive, the other cost functions generally behave as a Frobenius norm for small residuals while providing less penalization to large residuals than the Frobenius norm.

Moreover, in source apportionment, the definition of the profile matrix imposes that rows of F sum to one. It should be noticed that sum-to-one constraints are also met in remote sensing [START_REF] Févotte | Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization[END_REF][START_REF] Chouzenoux | Algorithme primal-dual de points intérieurs pour l'estimation pénalisée des cartes d'abondances en imagerie hyperspectrale[END_REF][START_REF] Ma | A signal processing perspective on hyperspectral unmixing[END_REF], but they apply along the columns of the second matrix factor. As a result, each column of the data matrix may be viewed as a convex combination of the columnscalled endmembers-of the first matrix factor. The geometric point of view (see the left plot of Fig. 1) shows that the column vectors of X belong to a non-negative polytope spanned by the endmembers. Searching for the mixing matrix thus consists of splitting the whole problem into independent subproblems related to the search of one column together with the sum-to-one constraint of this column.

In source apportionment, the issue is different since in this case, the sum-to-one constraints apply to the rows of the profile matrix F . The right plot of Fig. 1 illustrates that each row of the data matrix is only a non-negative linear combination of the profile row vectors, so that it belongs to a simplicial cone in the non-negative orthant. The only noticeable property (shown on the right plot of Fig. 1) is that each row of the profile matrix lives in a non-negative polytope spanned by the usual orthonormal basis vectors. Therefore, it is impossible to split the NMF problem into independent sub-problems. As a source apportionment needs some specific methods.

Moreover, blind NMF methods for source apportionment may result in solutions without physical meaning [START_REF] Viana | Intercomparison of receptor models for PM source apportionment: Case study in an industrial area[END_REF], especially when some source profiles are geometrically close to each other. To overcome these drawbacks, we recently proposed some informed NMF approaches [START_REF] Limem | Non-negative matrix factorization under equality constraints-a study of industrial source identification[END_REF][START_REF] Limem | Bound constrained weighted NMF for industrial source apportionment[END_REF][START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF][START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF] which take into account some prior information-e.g., some set values in the profile matrix provided by chemical experts-together with the row sum-to-one property. In [START_REF] Limem | Non-negative matrix factorization under equality constraints-a study of industrial source identification[END_REF][START_REF] Limem | Bound constrained weighted NMF for industrial source apportionment[END_REF], the constraints were alternatingly satisfied along iterations and only the limit matrix was fulfilling both of them. We recently proposed a new parameterization [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF][START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF] which fulfills both constraints at the same time, and we then derived informed NMF methods using the split gradient strategy [START_REF] Lantéri | Split gradient method for nonnegative matrix factorization[END_REF]. While the approach in [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF] is based on the Frobenius norm-and is then sensitive to outliers-the one in [START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF] is using a robust αβ-divergence, which requires the parameters α and β to be properly tuned. In this paper, we propose a novel robust NMF approachextending [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF]-which is based on the Huber cost function [START_REF] Huber | Robust estimation of a location parameter[END_REF] and which does not need to some parameter tuning.

This paper is organized as follows. Sections 2 and 3 briefly recall the parameterization proposed in [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF] and the Huber function, respectively. We derive the update rules of the proposed informed NMF method in Section 4 and we investigate its performance in Section 5. The conclusion and the perspectives of this work are discussed in Section 6.

PARAMETERIZATION OF F

In source apportionment, prior information often concerns the profile matrix 1 , for which some entries are provided by ex-1 Please note that it is also possible to inform G [START_REF] Plouvin | Enhanced NMF initialization using a physical model for pollution source apportionment[END_REF]. perts. Let Ω be a p × m binary matrix which shows the presence/absence of constraints on each entry of F , i.e.,

Ω ij = 1 if F ij is known, 0 otherwise. (2)
Conversely, we define Ω 1 pm -Ω, where 1 pm is the p × m matrix of ones. Defining Φ and ∆F , the p × m matrices of set and free parts of F , respectively, we derive [START_REF] Limem | Bound constrained weighted NMF for industrial source apportionment[END_REF]

F = Ω • Φ + Ω • ∆F , (3)
where • stands for the Hadamard product. It should be noticed that updating F consists of updating its free part ∆F . However, at this stage, the sum-to-one constraint is not satisfied. For that purpose, we extended in [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF] the idea proposed in [START_REF] Lantéri | Split gradient method for nonnegative matrix factorization[END_REF] which consists in normalizing the constrained ∆F by replacing it by an unconstrained p × m matrix Z [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF][START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF], i.e.,

F = Ω • Φ + Ω • Z (Ω • Z) • 1 mm • (1 pm -Φ • 1 mm ), (4)
where / stands for the elementwise division. As an example, if we assume that F is a 1 × 4 profile matrix and Φ [0, 0.2, 0, 0.3], then the maximum free part value is 1 pm -Φ • 1 mm = [0.5, 0.5, 0.5, 0.5] and Eq. ( 4) reads

F = Z 11 • 0.5 Z 11 + Z 13 , 0.2, Z 13 • 0.5 Z 11 + Z 13 , 0.3 .
In the next sections, source apportionment is addressed with the above parameterization combined with the Huber cost function.

HUBER COST FUNCTION

Data are often corrupted by outliers so that quadratic cost functions-i.e., using the 2 norm-become inappropriate. Robust cost functions are designed to provide less penalization to large entries of E, the matrix of residuals defined as

E X -G • F. (5)
On the other side, the 1 norm may be used to achieve robustness but-because of its sparsity promoting behaviourit tends to overweight near-zero residuals. As a consequence, intermediate solutions may be found to reduce the weight of small residuals. One of the widely used techniques is the M-estimators [START_REF] Huber | Robust Statistics[END_REF]-defined as a generalization of maximum likelihood estimators-of the form min

G,F i,j ρ(E ij ), (6)
where ρ(.) is a loss function, e.g., 1 -2 [START_REF] Hamza | Reconstruction of reflectance spectra using robust nonnegative matrix factorization[END_REF] or 2,1 [START_REF] Kong | Robust non-negative matrix factorization using l 21 -norm[END_REF] cost functions. Another popular choice of M-estimators is the Huber function which is based on the derivable connection between the 2 and 1 norms, i.e.,

ρ huber (e) = e 2 for |e| < c, 2c|e| -c 2 otherwise, ( 7
)
where c is the cutoff parameter of the Huber cost function, which divides the domain between the quadratic and linear loss functions. Moreover, in source apportionment, individual standard deviations are available, thus yielding weighted residuals, i.e.,

E σ,ij E ij σ ij . ( 8
)
The Weighted Huber-NMF problem is thus defined as:

min G,F 0 n i=1 m j=1 ρ huber (E σ,ij ). (9)
It should be noticed that ρ huber (.) behaves as an 2 norm for small weighted residuals while it is linear for large residuals. The function is convex and it has a bounded influence function-defined as the differentiation of the cost functionso that the influence of a single deviating point is minimized.

PROPOSED INFORMED NMF METHOD

In this section, we aim to solve Eq. ( 9) subject to the parameterization (4), i.e., min G,F 0 J (G, F )

s.t. F = Ω • Φ + Ω•Z (Ω•Z)•1mm • (1 pm -Φ • 1 mm ), (10) where
J (G, F ) n i=1 m j=1 ρ huber (E σ,ij ).

Update rules for F

We now aim to find the update rules of the profile matrix F which are consistent with the KKT conditions. This is done by differentiating J in Eq. ( 10) with respect to F for both expressions of ρ huber (.). For that purpose, we firstly define the following weight matrices W 1 and W 2 :

W 1,ij 1 σ 2 ij , W 2,ij 1 if (GF -X)ij σij < c, c•σij |GF -X|ij otherwise.
(11) W 2 should be seen as an adaptive re-weighting matrix, depending on small/large weigthed residuals. In the case of a single data point labelled as an outlier, the weight in W 2 is proportional to the inverse of residual value and is below 1.

Noticing that W 2 = 1 nm for the 2 part of the cost function, the differentiation of J for a single entry F rj reads

∂J ∂F rj = G ir (GF -X) ij W 1ij . ( 12
)
The differentiation of J for the 1 part yields

∂J ∂F rj = n i=1 c σ ij • G ir • sign((GF -X) ij ). (13)
Then, using the definition of the sign function, an equivalent expression is derived, i.e.,

∂J ∂F rj = n i=1 c • σ ij σ 2 ij • G ir • (GF -X) ij |GF -X| ij . ( 14
)
We lastly derive from Eq. ( 11)

∂J ∂F rj = G ir (GF -X) ij W 1ij W 2ij . (15)
Equations ( 12) and ( 15) yield a unified matrix expression, i.e.,

∂J ∂F = G T [W • (GF -X)], (16)
where

W W 1 • W 2 . ( 17
)
Let us emphasize again that the differentiation of J has a similar expression for both the quadratic and the linear parts of the Huber function, except that W is reweighted for large residuals, as explained when we introduced W 2 . At this stage, we still need to express the differentiation of ∂Fij ∂Z ik based on the parameterization (4) where the (i, j)th entry of F only depends on the i-th row of Z and ∀k ∈ {1 . . . m}. We thus obtain

∂F ij ∂Z ik = (1 - m l=1 Φ il ) Ω ij m l=1 Ω il Z il δ jk - Ω ik Z ij m l=1 Ω il Z il , (18)
where δ jk is the Kronecker index which is equal to 1 when j = k and 0 otherwise. For the sake of readability, we define B as a p × m matrix of scale factors, i.e.,

B ik 1 -l Φ il ( m l=1 Z il Ω il ) . ( 19
)
Using the chain rule and Eqs. ( 18) and ( 19), the matrix form of ∂J ∂Z reads [14]

∂J ∂Z = B •   ∂J ∂F • Ω -Ω • ( ∂J ∂F • F • Ω) • 1 mm 1 pm -Φ • 1 mm   .
(20) The first term (respectively, second) of the difference in the bracket of Eq. ( 20) corresponds to the first term (respectively, second) of the difference in Eq. [START_REF] Plouvin | Enhanced NMF initialization using a physical model for pollution source apportionment[END_REF].

The third KKT condition has to be fulfilled to get a stationary point with respect to Z, i.e., Z • ∂J ∂Z = 0. Equivalently and according to Eq. ( 4), it may be written as

F • Ω • ∂J ∂Z = 0. (21)
A stationary point means that F = F k = F k+1 , and using Eqs. ( 20) and ( 21), and letting U -∂J ∂F (see Eq. ( 16)), the new update rules of F read

F k+1 = Ω • Φ + Ω • F k • U [U • Ω • F k ] • 1 mm • (1 pm -Φ • 1 mm ).
(22) Moreover, we need to make sure that the updated rules of F in Eq. ( 22) remain positive which implies that U is positive at every iteration. In [START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF], we proved the following theorem:

Theorem 1. Every U s of the form U s = U + S • 1 mm checks the third KKT condition [START_REF] Févotte | Algorithms for nonnegative matrix factorization with the beta-divergence[END_REF].

As a consequence, a shift equal to the sum of the negative part of -∂J ∂F was proposed in [START_REF] Chreiky | Split gradient method for informed nonnegative matrix factorization[END_REF]. Applying the same idea to Eq. ( 22), we derive

U s = G T (W •X)-G T (W •(GF ))+[G T (W •(GF ))]1 mm . (23)

Update rules for G

Due to the application used in this paper, the contribution matrix G is unconstrained. Once again, the weight W should be taken into account in the updated rules. Due to symmetry in the expression of the weighted Huber cost function, the derivation with respect to G remains similar to Eq. ( 16) and reads

∂J ∂G = [W • (GF -X)] • F (24)
Expression ( 24) is similar to the differentiation of the quadratic loss function in [START_REF] Ho | Non negative matrix factorization algorithms and applications[END_REF], except that we use the weight matrix [START_REF] Huber | Robust estimation of a location parameter[END_REF]. As a consequence, the update rules for G read [20]

G k+1 = G k • (W • (X)) F T (W • (GF ))F T , (25)
Please notice that the update of the weight matrix should be made prior to the computation of F and G, which implies that the cutoff parameter should be carefully chosen.

Choice of the cutoff parameter

The cutoff parameter is an adaptive parameter depending on the residuals. In [START_REF] Du | Robust non-negative matrix factorization via half-quadratic minimization[END_REF], it is chosen as the median of the residuals. In our framework, it relies on weighted residuals. While it could also be chosen as the median-i.e., half of the data are seen as outliers-we consider that this choice prevents to modify the matrix factors in a sufficient way along iterations and as a consequence, we propose a different strategy. We also think that at the beginning of the NMF iterations, all the data should be processed in the quadratic part of the Huber function, resulting that c should go to the maximum value of the data. Conversely, at the end of the iterations, we feel that c should tend to an intermediate value, enabling to point only the outlying points. As a consequence, we propose an heuristic choice for c based on the quantile function with a probability value linearly depending on the current iteration number k, i.e.,

c = quantile(E σ , 1 - k Itermax * 0.1) ( 26
)
where Itermax is the maximum number of iterations. This choice implicitly states that a maximum of 10 percent of the data may be labelled as outlying data, which was found to be a good trade-off in preliminary tests.

Algorithm

The algorithm may be outlined in the following form:

Algorithm 1 Weighted Huber Informed NMF description Reading of X, Φ and Ω Initialization of F and G while the stopping rule is not fulfilled do Compute E and the weighted residuals E σ Compute c according to Eq. ( 26) Update W according to Eqs. ( 11) and ( 17) Update F according to Eq. ( 22) with Eq. ( 23) Update G at fixed F according to Eq. (25) end while

SIMULATIONS

In this section, we are investigating the performance of the newly proposed method-named Huber SG-CWNMF for Split-Gradient Constrained Weighted NMF using Huber cost function-in several tests with and without outliers. The collected data matrix X is made of 50 samples and 7 species. Each entry in X has a known variance σ ij provided by chemical experts. The collected samples are possibly corrupted by outliers located at random locations and tuned by a gain which is always greater than one in order to simulate a pollutant contamination. Then a uniform noise is added, ranging in the interval [-min(λσ ij ; x ij ); λσ ij ] while taking into consideration that the data is always positive. As a result, λ is related to an input Signal-to-Noise Ratio (SNR) without any outliers. Moreover, the profile matrix F consists of three partially correlated industrial sources for which 9 entries are known and provided in Table 1. In the simulation, the initial F is consistent with the provided constraints and the initial G is computed with a quadratic program. The performance criterion is the Mixing-Error Ratio (MER) expressed in dB, computed over each column of G and averaged.

In addition to the proposed method, several methods are tested. They are divided into two categories, i.e., the blind and the informed ones. In the first category, the tested methods which are all based on β-divergence, are the original unweighted β-NMF [START_REF] Févotte | Algorithms for nonnegative matrix factorization with the beta-divergence[END_REF], the weighted β-NMF [START_REF] Ho | Non negative matrix factorization algorithms and applications[END_REF] and the robust NMF (rNMF) [START_REF] Févotte | Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization[END_REF] in which the data matrix is split into a low rank matrix satisfying Eq. ( 1) and a sparse matrix containing the outliers. Two other informed approaches based on the β-divergence are experimented (tested with β = 0.8 in this paper), i.e., our previous β-CWNMF with Residuals (β-CWNMF-R) and without residuals (β-CWNMF) [START_REF] Limem | Bound constrained weighted NMF for industrial source apportionment[END_REF]. In addition, we test a weighted version of the Split Gradient approach developped by Lantéri et al. [START_REF] Lantéri | Split gradient method for nonnegative matrix factorization[END_REF], which involves a sum-to-1 parameterization, and our previous proposed αβ-SG-CWNMF with α = β = 0.6 [START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF]. All these methods are compared for several input SNR with the newly proposed one, based on the weighted Huber cost function and for 200k iterations. Finally, the upper bound of the performance is computed with the Huber cost function by assuming F is set to its real value and using the update rules (25) together with the iterative re-weighting process [START_REF] Viana | Intercomparison of receptor models for PM source apportionment: Case study in an industrial area[END_REF]. In a first experiment, each method is tested over 400 simulations with respect to the input SNRs ranging from 15 to 60 dB without and with several outliers. For space considerations, we only plot the case of 5 outliers on Fig. 2. Moreover, in the sake of readbility, we do not show the performance of all the tested methods. First of all, our proposed Huber method outperforms the blind and informed tested methods, for any input SNR interval and for both cases with and without outliers (except our previous αβ-SG-CWNMF [START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF] which behaves very similarly and which is not shown as a consequence). Moreover, the achieved performance is almost equal to the Bound MERs. Interestingly, the rNMF method completely fails over all the tests. This is probably due to the fact that rNMF is not able to split the data matrix into appropriate sparse and low rank matrices. Indeed, in our experiments, outliers are relatively small for high input SNRs and mediumsized for low SNRs. Such a situation is challenging for rNMF which performs well in the presence of larger outliers.

In a second experiment, we investigate the influence of an increasing number of outliers on the behavior of the tested NMF methods, for an input SNR of 40 dB without outliers. Figure 3 shows the achieved MER with respect to the the numbers of outliers (associated with the input SNR with outliers). All the methods provide decreasing MERs when the number of outliers increase. However and similarly to the previous experiment, our proposed Huber SG-CWNMF method outperforms all the tested approaches and provides a performance very close to the Bound, which shows the relevance of this work.

Finally, we experiment different cutoff parameter strategies, either constant or adaptive with linear decrease along iterations. Figure 4 shows the obtained performance on the above simulations with each choice. In fact, the legend specifies the percentage of data points considered in the 1 part of the Huber function at the beginning and at the end of the iterations. The best choice is achieved with the cutoff proposed in Eq. (26) as indicated in Figure 4. Moreover, the usual choice of the median proposed in the literature is here shown to be inappropriate.

CONCLUSION

In this paper, a new robust informed NMF method is proposed for solving the source apportionment problem. It is based on a weighted symmetric Huber cost function which adaptively connects weighted 2 and 1 norms according to the size of the computed residuals. In addition, a specific parameterization and the split gradient framework allow to provide new update rules which take into account set entries and the sumto-one constraint in one factor matrix. This method has been tested for a large range of SNRs and for several outliers. It overcomes all the tested state-of-the-art methods and besides, behaves very close to the bound computed with the Huber regression method and the exact profile matrix.

[1] D. D. Lee and H. S. Seung, "Learning the parts of objects by non negative matrix factorization," Nature, vol. 401, no. 6755, pp. 788-791, 1999.

[2] B. Shen, L. Si, R. Ji, and B. Liu, "Robust nonnegative matrix factorization via l 1 norm regularization," arXiv preprint arXiv:1204.2311, 2012.

Fig. 1 .

 1 Fig. 1. Geometric point of view of the sum-to-one constraint for (left) hyperspectral data, and (right) chemical data.

Fig. 2 .

 2 Fig. 2. MERs provided by the tested NMF methods vs. the input SNR for 5 outliers.

Fig. 3 .

 3 Fig.3. NMF performance vs. the number of outliers.

15-20 20 -

 20 25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 60

Fig. 4 .

 4 Fig. 4. MERs obtained with the Huber SG-CWNMF method vs. the input SNR for different cutoff strategies.

Table 1 .

 1 Positions and values of the constraints used in the informed NMF methods. X means no constraint.

		Fe	Ca	SO 4	Zn	Mg	Al	Cr
	Source 1	0.7	X	X	X	X	X	0
	Source 2	X	0.4	5	0	X	0.075	X
	Source 3	0.4	X	X	0	X	X	0