Amar Prakash Azad
email: amarazad@soe.ucsc.edu

John Musacchio
email: johnm@soe.ucsc.edu

Unilateral Altruism in Network Routing Games with Atomic Players

We study a routing game in which one of the players unilaterally acts altruistically by taking into consideration the latency cost of other players as well as his own. By not playing selfishly, a player can not only improve the other players' equilibrium utility but also improve his own equilibrium utility. To quantify the effect, we define a metric called the Value of Unilateral Altruism (VoU) to be the ratio of the equilibrium utility of the altruistic user to the equilibrium utility he would have received in Nash equilibrium if he were selfish. We show by example that the VoU, in a game with nonlinear latency functions and atomic players, can be arbitrarily large. Since the Nash equilibrium social welfare of this example is arbitrarily far from social optimum, this example also has a Price of Anarchy (PoA) that is unbounded. The example is driven by there being a small number of players since the same example with non-atomic players yields a Nash equilibrium that is fully efficient.

I. INTRODUCTION

The topic of network routing games has been studied extensively in recent years (e.g. [START_REF] Orda | Competitive routing in multiuser communication networks[END_REF]- [START_REF] Kameda | Nash equilibria in load balancing in distributed computer systems[END_REF]). Some of these investigations consider network games for which there are so many users that the traffic of each user can be considered negligible -the players for such games are said to be non atomic. Other work considers routing games for which each player is the source of a significant amount of the network's traffic. These players are said to be atomic. The analysis of atomic routing games can be more complex than that of non atomic games since the routing choice of a single player can affect the delays seen by others.

The Price of Anarchy (PoA) [START_REF] Koutsoupias | Worst-case equilibria[END_REF] is one popular measure of the efficiency of equilibria for such games. It is defined as the ratio of the maximum social welfare that could be achieved if a social planner selected the strategies of all players versus the lowest social welfare achieved in Nash equilibrium. The literature shows that in many cases the PoA can be arbitrarily large, although there are several important classes of examples in which the PoA can be bounded. (For example, Roughgarden [START_REF] Roughgarden | Selfish routing and the price of anarchy[END_REF], shows that routing games with non-atomic users and affine latency functions have a PoA of no more than 4/3.)

Naturally if players were not selfish, but instead all were striving to maximize social welfare, the equilibrium welfare could be much more efficient. However, an interesting question is what would happen if just one player chose not to be selfish and instead takes actions that consider the welfare of other players -in other words if he behaves altruistically. In this work we investigate the effects of unilateral altruism. We find by way of example that the effect of unilateral altruism can be arbitrarily large.

A selfish user chooses a path (or a set of paths) that minimizes his total cost without considering that their selection may degrade the performance of the other users in the system. However in practice players may not play like this. Experimental studies [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF]- [START_REF] Ledyard | Public Goods: A Survey of Experimental Research[END_REF] have shown that even in simple games in a controlled environment, players do not always act selfishly, for example they can show reciprocity, act out of spite, or be altruistic. Several explanations have been considered for such behaviour of players. Fehr [START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF], [START_REF] Gintis | Moral sentiments and Material Interests[END_REF] argues that players act out of a perception of fairness and thus they consider a utility function that captures the costs of other players. Models of altruism and spite are discussed in [START_REF] Levine | Modeling altruism and spitefulness in experiment[END_REF], [START_REF] Ledyard | Public Goods: A Survey of Experimental Research[END_REF], [START_REF] Chen | Altruism, Selfishness, and Spite in Traffic Routing[END_REF], [START_REF] Sharma | Stackelberg thresholds in network routing games or the value of altruism[END_REF] etc. Most of the related models are discussed by Sobel in [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF]. The fact that the social welfare observed in experimental trials is often better than that predicted by finding the Nash equilibrium of a model with selfish players is one reason to consider a model in which players act out of altruism. Since the players often interact over a longer period than that modeled in a one-shot game, players have a reason to play more cooperatively.

In this work, we focus on altruistic behaviour with atomic users. We consider a simple model where the perceived cost of a user is a linear combination of the other user's cost, as we proposed in [START_REF] Azad | Routing games : From Egoism to Altruism[END_REF]. We show that unilaterally altruism drastically increases the altruistic player's equilibrium utility. It is obvious that if all players are altruistic then the equilibrium utility can be improved, however it is surprising to observe a large improvement by a single player's altruistic action. However, it needs to be emphasized that the improvement only happens if the other players change their strategies in accordance with the belief that the altruistic player will truly be altruistic. In some sense, the altruistic player has to "credibly commit" to being altruistic.

In section II we present the system model and key definitions. We present an example to motivate the problem by showing very high price of anarchy in section III. Section IV analyzes a two user selfish game. In section V, we introduce and analyze the altruistic routing game. In section VI, we extend our model to a more general topology with n nodes and investigate the impact of altruism. Finally, we summarize with concluding remarks in section VII. We skip several details and proofs in the extended abstract due to space limitation, these details are made available at [START_REF] Azad | Unilateral Altruism in Network Routing Games with Atomic Players[END_REF].

II. BASIC DEFINITIONS AND SYSTEM MODEL

We consider a network G = (V, L), where a set of nodes V are connected by set of directed links L ⊆ V × V. We denote v s ∈ S ⊆ V, v d ∈ D ⊆ V distinct nodes that are called source nodes and destination nodes, respectively. A route R ∈ R is a directed path with distinct nodes that connects a source and a destination. For every link l ∈ L and a route R, we write l ∈ r whenever l is a part of route R.

A set I = {1, 2, ..., I} of users share the network (V, L). We consider that each source node v s is associated with a user i, which ships a throughput demand of average data rate r i to a common destination associated with node v d . User i splits its fixed demand r i among the paths (routes) connecting the source to the destination, so as to optimize some individual performance objective. We assume that there exist at least one route from v s to v d .

Let x i l denote the expected flow that user i sends on link l. The vector of flows x i = (x i l ) l∈L that user i sends onto each path is called the routing strategy of user i. The set of strategies of user i that satisfy the user's demand and preserve its flow at all nodes is called the strategy space of user i and is denoted by X i , that is:

X i = {x i ∈ R |L| ; l∈Out(v) x i l = l∈In(v) x i l + r i v , v ∈ V},
where r i s = r i , r i d = -r i and r i v = 0 for v = v s , v d and Out(v) and In(v) the set of out-going and in-coming links, respectively. The system flow configuration x = (x 1 , ..., x I ) is called a routing strategy profile and takes values in the product strategy space X = ⊗ i∈I X i .

Every link l ∈ L is associated with a separable latency cost function T l (x l ) : R → R which is assumed to be piecewise differentiable, convex, increasing, and T l (x l ) ≥ 0 for every x l > 0. The queueing network interpretation is that T l (x l ) is the delay incurred by traffic crossing link l, or alternatively it can be read as the cost per unit of traffic when the load in link l is x l .

Latency function: We consider the following latency function types in this paper: A : Affine: T l (x l ) = ax l + b, where a, b ≥ 0. E : Elbow function: We define the function with offset o in parametric form as T l (x l ) = max o, L δ (x -r) + L . Unless stated otherwise, we use o = 0, 0 < δ ≪ 1, refer to Fig. 2. The E type latency function posses several features that makes in mathematically tractable, e.g., piece-wise linearity, convexity. It turns out that a steep slope of elbow function causes users to react for even small flow changes which plays a key role in the latter sections. This is why, we designed the elbow function in a way so that the per-unit cost of traffic can increase very rapidly. This is also not unlike the delay in most queueing models such as M/M/1 delay function.

Load Balancing (LB) Network: We introduced a network topology called the "load balancing network" [START_REF] Azad | Routing games : From Egoism to Altruism[END_REF], where two users located one at each of the two nodes, called source nodes, deliver their demand to a common destination node as pictured in Fig. 1(a).

In this paper, we associate E type latency with l 1 , l 2 while we associate A type latency with l 12 , l 21 . One of the reasons of such modeling is that, typically source-destination links are standard queueing network links with limited capacity (approximated by E type). The interconnecting/cross links are meant for balancing skewed loads on the local links. Thus, it is of high capacity but are expensive by provision (approximated by type A).

User Cost: Let J i (x) be the cost of user i under routing strategy profile x. The total cost to user i is the sum of the separable link costs, given as

J i (x) = l x i l T l (x l ), for each i. (1)
The objective of each user i is to find an admissible routing strategy x i ∈ X i so as to minimize his cost. Later we consider altruistic users, where the objective function takes into account of other user's cost also.

A. Nash equilibrium

Each user in this framework minimizes his own cost functions which leads to the concept of Nash equilibrium. The minimization problem here depends on the routing decision of other users, i.e., their routing strategy

x -i = (x 1 , ..., x i-1 , x i+1 , ...x I ), Definition 2.1: A vector xi , i = 1,
x i ∈X i J i (x 1 , x2 , ..., x i , ..., xI ), i = 1, 2, ..., I, (2)
where X i is the routing strategy space of user i. Social Optimum: A centralized or coordinated solution is referred to as social optimum 1 , which can be expressed as:

x OP T = arg min x∈X i J i (x). (3)
Price of Anarchy (PoA): A well known way to quantify the equilibrium inefficiency of a non-cooperative game (in the sense of [START_REF] Koutsoupias | Worst-case equilibria[END_REF]). The Price of Anarchy is defined ( [START_REF] Roughgarden | How bad is selfish routing[END_REF], [START_REF] Koutsoupias | Worst-case equilibria[END_REF]) as the ratio of worst case cost at Nash equilibrium when users are selfish to the cost at social optimum. Therefore,

P oA = sup x∈x SE i J i (x) i J i (x OP T ) . ( 4
)
where x SE denotes the set of equilibrium flow configurations when users are "selfish".

III. MOTIVATING EXAMPLE

In this section we illustrate the inefficiency of noncooperative equilibrium with the help of a simple example and point out the key reason for the inefficiency to motivate the detailed investigation. We also note that the inefficiency appears only for the atomic users.

Consider a LB network (in Fig. 1(a)) with two selfish users, u 1 and u 2 , at source nodes 1 and 2 respectively, having identical demand rates r 1 = r 2 = r, which is to be dispatched to the destination node 3. Since, link l 1 is the most direct way for user 1 to reach the destination, we refer to link l 1 his "local link". Similarly, we refer to link l 2 as user 2's local link. We refer to links l 12 and l 21 as "cross" links.

Example: Let the latency function be as shown in Fig. 2, which is expressed as: We can easily observe that the social optimum is achieved when users use only local links. This is so because any positive flow on cross links add additional latency while retaining the total latency (due to flow) on links l 1 and l 2 remains unchanged. However, when users are selfish, each user optimize its own utility. We illustrate by way of iteratively finding the best responses of the two selfish players why they end up putting a large fraction of their traffic on the cross links, which is the cause of inefficiency.

T l (x) = max(0, L 2 (x)), for l = l 1 , l 2 c, for l = l 12 , l 21 . (5)
Let the initial flow be {r, r}, and the corresponding initial cost of users are {L, L} = {0.1, 0.1}. Let u 1 pushes δ amount of flow to l 2 to bring his local link flow below the knee (see Fig. 2 ), which results in T 1 (.) = 0. On the other hand, the total flow in l 2 raised to r+δ which results in

T 2 (.) = L+ L δ δ = 2L. The updated cost of u 1 is J 1 = δc + δ • 2L = 0.
12, which is much lower than previous cost 0.1. However, the cost of u 2 has increased from 0.1 to (1 -10 -3 ) ≈ 0.2, almost doubled. In turn u 2 will reciprocate by pushing 2δ unit of flow to l 1 to bring his local link latency below the knee, which reduces his cost drastically to 2δ • (2L + c) ≈ 0.003, but raises the cost of u 1 to (1 -δ)2L + δc ≈ 0.2. Due to this raise in cost, again u 1 will push 2δ amount of flow to l 2 in order to bring his local link latency below the knee. This process will keep repeating till the flow configuration reaches to x SE ≈ {0.5, 0.5}, after which none of the users benefit by deviating even a small amount of traffic from their local link. In other words, x SE is the Nash equilibrium. The total cost incurred by any user at x SE is ≈ 0.5 • (0.1 + 1) = 0.55, which is much higher (≈ 50 times) than the cost at social optimum (= 0.1), i.e., P oA ≈ 50.

We demonstrated by a simple LB Network example that selfish routing with atomic users can be significantly inefficient. In the following, we show that it can be even worse and can be arbitrarily large for atomic users. We also show that this is not the case with non-atomic users, suggesting that it is the ability of a user to chase the latency of the links with his strategy that drives the example.

IV. TWO USER SYMMETRIC GAME

In this section we establish results showing the inefficiency of selfish routing games by analyzing the two user LB network game. In the later sections, we construct and analyze the model in which users are altruistic, which exposes our key findings.

We consider a sequence of LB network examples indexed by m ≥ 1. Consider a LB Network of three nodes as shown in Fig. 1(a) with the E type and A type latency functions as follows:

T l (x) = max 0, L δm (x -r) + L , for l 1 , l 2 , c m , for l 12 , l 21 . (6)
1 Existence of minima is due to compact strategy set and piece-wise continuous cost function. It is unique because of potential function structure.

where c m > L > δ m > 0 and c m < r L δm for all m . The sequence δ m decreases and tends to zero such that the gradient L δm → ∞ as m → ∞. Similarly, c m is an increasing sequence such that latency of cross link can be made very high for sufficiently large m. One example of such sequence can be δ m = δ m and c m = c m . It turns out to be important that the gradient L δm be larger than the cross-link latency c m , but the cross-link latency c m be bigger than the latency of the local links when the flow is r, which is L.

Using (1), the cost of users are given by,

J i (x) = x i i T i (x i ) + xi i (T ij (x i i ) + T j (x j )), for i, j = 1, 2.(7) where, xi i = r i -x i i
We often denote T li (.) by T i (.) for the ease of notation.

A. Selfish Routing Game with Atomic Users

Lemma 4.1: Consider two atomic users of identical demands on a load balancing network (Fig. 1(a)) at node 1 and node 2 with the associated latency given by [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF]. In the two user selfish routing game, there exists a unique Nash equilibrium, given by

x SE = {x 1 * 1 , x 2 * 2 } = {r/2 + ζ m , r/2 + ζ m } , (8)
J SE i = rL + (r/2 -ζ m )c m , for i = 1, 2. ( 9
)
where

ζ m = 1 2 ( cm L/δm
) is the ratio of unit latency of cross links to the marginal latency of local links. By abuse of notation, above we denoted x SE as the equilibrium flow.

The proof of Lemma 4.1 uses techniques of Non Zero Sum Games (NZSG) [START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF]. We solve the game by explicitly computing the best responses and finding their intersection (refer [START_REF] Azad | Unilateral Altruism in Network Routing Games with Atomic Players[END_REF]).

Proposition 4.1: The Price of Anarchy of two atomic users on LB network (Fig. 1(a)) with the associated elbow latency function of [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF] and with identical demands is given by

P oA = 1 + ( r 2 -ζ m )c m rL . ( 10
)
The PoA can be arbitrarily large for large enough m.

The proof is based on Lemma 4.1 (see [START_REF] Azad | Unilateral Altruism in Network Routing Games with Atomic Players[END_REF] for details).

B. Selfish Routing Game with Non-Atomic Users

We consider large number of selfish users at each nodes. The demand of each user is assumed very small such that choice of a single user has negligible impact on the system performance. The latency observed by a user is the sum of the link latency that a user traverse to reach to the destination node. Each user would choose a route so as to minimize its latency at equilibrium. Proposition 4.2: The Price of Anarchy for the routing game on the LB network with non-atomic selfish users and associated latency given by ( 6) and with identical demands is equal to 1 for any m. The bounded PoA is achieved because the user's use only their local links. This is because the latency observed by a user via a cross link is larger as compared to the case when he chose his local link. However, such direct intuition is not possible in case of atomic users. The proof of Lemma 4.2 is based on variational inequality at Wardrop equilibrium in the similar lines of [START_REF] Roughgarden | How bad is selfish routing[END_REF] (see [START_REF] Azad | Unilateral Altruism in Network Routing Games with Atomic Players[END_REF]).

Interestingly, the social welfare for selfish routing games with atomic users is unbounded whereas in similar settings it corresponds to social optimum for non-atomic users.

V. ALTRUISTIC ROUTING GAME

In this section, we consider routing games when users are not entirely selfish, but rather altruistic. Each user minimizes the perceived cost, a weighted sum of his own cost and other users' cost (we proposed this model in [START_REF] Azad | Routing games : From Egoism to Altruism[END_REF]). The weight coefficient is called degree of Cooperation (DoC).

Definition 5.1: Let α i k be the degree of cooperation of user i with user k. The perceived cost is a convex combination of the users' cost from the set I as follows:

Ĵi (x, α i ) = k∈I α i k J k (x); k α i k = 1, i = 1, ...|I|. (11)
We can express the DoC vector of user i as

-→ α i = (α i 1 , • • • , α i |I| ).
Based on the DoC, we can view the following properties for user i,

• Non-cooperative: if α i k = 0, for all k = i. Denote the DoC of a selfish user by

e i = {0, • • • , e i i , • • • , 0}
, where e i i = 1. • Altruistic: User i is fully cooperative with all users and does not care for his benefits, i.e., α i i = 0. • Equally-cooperative: if α i j = 1 |I| , ∀j ∈ I, user i is equally cooperative with each user. Note that, although a user cooperates with others, he minimizes his own perceived cost. Therefore, this formulation allows us to analyze the game using the conventional techniques of non cooperative game theory, e.g., Nash equilibrium, etc.

A. Unilateral Altruism

We call user i unilaterally altruistic, if he cooperates with other users irrespective of whether the other users are cooperative or non cooperative with user i. More precisely, let all users j = i be selfish and user i be unilaterally altruistic with degree of cooperation vector α i , then

α = { e 1 , • • • , α i , • • • , e |I| }.
The perceived cost by user i and user j = i can be expressed as Ĵi (x,

α i ) = k∈I α i k J k (x), Ĵj (x, e j ) = J j (x). (12)
Let x SE denote the set of Nash equilibrium flow vectors for the selfish routing game. Then, the best selfish Nash equilibrium flow can be given by x SE = arg min x∈x SE Ĵi (x, e i ). Similarly, let x AL αi be the set of Nash equilibrium flows2 (using ( 12)) when user i is unilaterally altruistic with DoC α i and all the other users are selfish. Recall that, an altruistic player truly commits his strategy to be altruistic such that the other players can change their strategy in accordance to this belief.

Definition 5.2 (VoU):

We quantify the benefit of unilateral altruism by the term Value of Unilateral Altruism (VoU), which is defined as the ratio of the best selfish equilibrium cost to his best equilibrium cost when he is unilaterally altruistic. Mathematically, we can express

V oU (i) = J i (x SE ) inf ( αi,x AL α i ) J i (x AL αi ) (13)
Notice that the equilibrium is obtained based on minimization of perceived cost but the V oU is related to the actual cost incurred by the user. By VoU we identify the bound on the improvement that a user may achieve by becoming altruistic (if altruism is beneficial for the user). Note that, when there exists unique equilibrium with an altruistic user, the V oU corresponds to the α i that minimizes the equilibrium cost. However, when there exist multiple equilibria, the V oU corresponds to the best equilibria at its corresponding α i . By definition, we have x SE ⊆ x AL αi , therefore the feasible values of V oU (i) must be ≥ 1. The case of V oU (i) = 1 indicates, either no improvement or degradation with altruism as compared to the selfish situation. The more interesting is the case of V oU (i) > 1, which corresponds to the situation when a unilaterally altruistic user improves his cost as compared to he being selfish. This is nonintuitive, as an altruistic user is generally expected to incur more cost. In the following, we illustrate that one can achieve drastic improvement by being unilaterally altruistic.

Proposition 5.1: Suppose user 1 is altruistic in a two user routing game on a LB Network (Fig. 1(a)) with the associated latency given by ( 6) and with identical demand. The "Value of unilateral altruism" of user 1, V oU (1), is arbitrarily large for sufficiently large m.

Below we provide a brief sketch of the proof (refer [START_REF] Azad | Unilateral Altruism in Network Routing Games with Atomic Players[END_REF] for details). The selfish users incur unbounded cost at equilibrium (unique), see Lemma 4.1. Therefore, a bounded equilibrium cost of user 1 with his unilateral altruism implies the proposition. However, there can be multiple equilibria with altruism but due to unbounded J SE 1 (.), the V oU (1) still remains unbounded even if there exist other better equilibrium. Using these arguments we show V oU (1) is unbounded. In fact, we show that (x 1 * 1 , x 2 * 2 ) = (r, r -δ) is an equilibrium with unilateral altruism for a set of α ∈ Γ. The users cost is unbounded for this equilibrium, which concludes the proof.

Note that by showing unilateral altruism, not only the altruistic user is able to improve his cost by keeping it bounded, rather the other selfish user also enjoy his improved cost which also remains bounded. This way, even the selfish user also get drastic benefit due to altruistic user's action.

VI. VOU FOR N PLAYERS

From the discussion of previous sections, one might think that the benefit of altruism (the unbounded VoU) is restricted to two user scenario and it may disappear for a larger number of users. In this spirit we investigate for n user game in this section. We are able to construct a LB type network of n + 1 nodes, shown in Figs. 1(a), on which the property of unbounded V oU is retained when a user is unilaterally altruistic.

Consider that n users (a user at each node) have identical demands to be dispatched to a common destination node (marked as n+ 1 th node). We compute the selfish routing game performance and subsequently study the performance with an altruistic user. It is intuitive that if all users turn altruistic then the efficiency of selfish equilibrium can be improved. In the earlier sections we studied the game with two users. However, it is more complex to study for n users. We chose a symmetric network, which is more tractable. We find that even in case of n users, the benefit of a single user's altruism is significant. We formally establish our findings with the help from the following lemma and proposition.

Lemma 6.1: The PoA for the n player selfish routing game on a LB network (Fig. 3) with the latency function as in [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF], and with identical demands is arbitrarily large for sufficiently large m. The proof follows by first showing unique equilibrium by transforming the network into a two node network with parallel

1 2 n + 1 l 1 l n n l 12 • • • r 1 r n r 2 l 21 l j1 l 1j
Fig. 3. A LB type network with n source nodes and a destination node. links. Exploiting the symmetric structure, we obtain the NEP for selfish users and also for the social optimum. Proposition 6.1: Consider a user (say user 1) being unilaterally altruistic while other n users are selfish in a LB network (Fig. 3), where all other users are selfish users with identical demands and latency as in [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF]. The "Value of unilateral altruism" of the altruistic user, V oU (1), is arbitrarily large for sufficiently large m. The proof follows the similar techniques as that in proposition 5.1 and using lemma 6.1.

VII. CONCLUDING REMARKS AND PERSPECTIVE

Our investigation in this work reveals the benefits of unilateral altruism in a network routing game. We showed with the help of a simple example with two users that if one user becomes altruistic, his equilibrium utility drastically improves. An improvement by unilateral altruism is itself non-intuitive. Moreover, the improvement by altruism is drastic, which is quite surprising. To quantify this we proposed a metric, "Value of Unilateral Altruism". Latter, we also show that the benefit of unilateral altruism is visible for a larger number of players.

APPENDIX

A. Proof for Lemma 4.1

Proof: The Nash equilibrium for the selfish game corresponds to the solution to the following minimization problem

P 2 : x i * = arg min x i ∈X i J i (x i , x -i ), ∀ i = 1, 2. (14)
The cost of user 1 and 2 in the above equation can be expressed as follows:

J 1 (x) = x 1 1 T 1 (x 1 ) + x1 1 (T 12 (x 1 1 ) + T 2 (x 2 )), (15) J 2 (x) = x 2 2 T 2 (x 2 ) + x2 2 (T 21 (x 2 2 ) + T 1 (x 1 )). ( 16
)
where xi i = r -x i i for i = 1, 2. The compactness of the strategy set X with the concavity and continuity of the objective function J i (.) imply the existence of a Nash equilibrium (see [START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF]Thm. 4.4]). A formal proof of existence is provided in [START_REF] Azad | Routing games : From Egoism to Altruism[END_REF]. We provide the proof of unique Nash equilibrium for a general n node LB network (n user) in Lemma 6.1. Therefore, we conclude that there exist unique equilibrium for the problem P 1 , as it is a special case of Lemma 6.1 with n = 2. Therefore, if we can compute a Nash equilibrium, then that is the only Nash equilibrium. We can observe that a symmetric Nash equilibrium exist due the symmetric structure of P 1 . In the following, we compute the symmetric Nash equilibrium.

1) Computation of symmetric Nash equilibrium: We compute the best response for each player to compute the equilibrium. The best responses can be given by the following expressions

BR 1 (x 2
2 ) = arg min

0≤x 1 * ≤r1 J 1 (x 1 , x 2 ) = max{min{0, a * }, r}, BR 2 (x 1 1 ) = arg min 0≤x 2 * ≤r2 J 2 (x 1 , x 2 ) = max{min{0, b * }, r}. (17)
where (with some abuse of notation 3 ) we denote the internal solutions by a * and b * . By the internal solution, we mean 0 < a * (or b * ) < r when a * ⇒ ∂J1(x)

x 1 1 = 0 and b * ⇒ ∂J2(x)

x 2 2 = 0. Equating the above and doing some simple calculus we obtain,

a * (x 2 2 ) = F (x 2 ) + (c m + rg(x 2 )) G(x) , b * (x 1 1 ) = F (x 1 ) + (c m + rg(x 1 )) G(x) (18)
where,

F (x 2 ) = T 2 (x 2 ) -T 1 (x 1 ) = -F (x 1 ), g 1 (x 1 ) = ∂T l 1 (x1) ∂x1 , g 2 (x 2 ) = ∂T2(x2) ∂x2 , and G(x) = g 1 (x 1 ) + g 2 (x 2 ). The gradient of T 1 (x) at r is g 1 (r) = L/δ m .
Exploiting the symmetry we note x 1 = x 2 = r, g(x 1 ) = g(x 2 ) = g(r), and F (x 1 ) = F (x 2 ) = 0 at equilibrium. Therefore, the equilibrium flow can be given by

x 1 * 1 = x 2 * 2 = c m 2g(r) + rg(r) 2g(r) = r 2 + ζ m ( 19
)
where ζ m = 1 2 cm L/δm . The Nash equilibrium flow for selfish users is expressed as

x SE = { r 2 +ζ m , r 2 +ζ m }. Note that ζ m ≥ 0 for large enough m. 3 a * is represented in short for a * (x 2
2 ) and b * in short for b * (x 1 1 ).

2) Using ( 19), the equilibrium cost can be given by

J SE 1 = r 2 + ζ m T 1 (x 1 ) + r 2 -ζ m T 12 r 2 -ζ m +T 2 (x 2 ) = rL + r 2 -ζ m c m . ( 20
)
This concludes the proof. ⋄

B. Proof for Proposition 4.1

Proof: Recall that the definition of price of anarchy from ( 4)

P oA = sup x∈x SE i J i (x) i J i (x OP T ) . ( 21
)
One can easily observe that at social optimum (OPT) only local links are used. Therefore, the user cost J OP T i at social optima is given as

J OP T 1 = rT 1 (r) = rL. ( 22
)
Therefore, the price of anarchy for any m can be given by

P oA = rL + r 2 -ζ m c m rL = 1 + ( r 2 -ζ m )c m rL . ( 23
)
Note that c m is an increasing sequence and ζ m is a decreasing sequence. Hence, P oA can be arbitrary large for large enough m. ⋄

C. Proof of Lemma 4.2

Proof: Consider non-atomic type users at source node 1 and source node 2 with a total demand rate of r each to be dispatched to the destination node 3.

Let p ∈ P denote a path that a user can take from a source node to destination node and T p (.) denote the cumulative latency observed for the path. The concept of road traffic wardrop equilibrium, originally introduced by Wardrop [START_REF] Wardrop | Some theoretical aspects of road traffic research[END_REF], which is the non-cooperative equilibrium for non-atomic users in network traffic, is defined in terms of variational inequality as

T p (x) -A ≥ 0; (T p (x) -A)x i p = 0, for p ∈ P, (24)
where A = min p∈P T p (x).

The Wardrop equilibrium falls into the category of potential games with an infinite number of users and can be expressed by a single convex optimization [START_REF] Altman | A survey on networking games in telecommunications[END_REF]. Hence, there exist a unique equilibrium.

There are two possible paths from source node 1 to destination node: a) through l 1 ; and, b) through l 3 → l 2 . Path Similarly, from source node 2 following are the feasible paths: c) through l 2 ; and, d) through l 4 → l 1 . Using (24), we can easily see that at equilibrium the set P = {a, c}, i.e., only local links are used. Therefore, the equilibrium flow is given by x * = {x 1 * , x 2 * } = {r, r} and the latency cost experienced by a user is T 1 (.) = L and total cost of users from a source node equals r k=1 T 1 (.) = rL. Clearly, the user's cost is bounded and equal to the social optimum at Wardrop equilibrium. ⋄ D. Proof of Proposition 5.1

Proof: In an altruistic routing game, users minimize their perceived cost ( Ĵi (.)). It is interesting to note that although the perceived cost is minimized non-cooperatively, the effect of altruism appears due the coupling through the cost. Therefore, the Nash equilibrium of this two user game can be expressed by the following equivalent problem:

P 2 : x i = arg min x∈X Ĵi (x), ∀ i = 1, 2. (25)
where the perceived cost of users (when user 1 is altruistic) can be expressed as

Ĵ1 (x) = (1 -β)J 1 (x) + βJ 2 (x), Ĵ2 (x) = J 2 (x). (26)
Existence of an equilibrium is direct from the compact strategy set X and concave and continuity of the cost function Ĵi (x) [START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF]. In this altruistic equilibrium, we are interested in an equilibrium which can improve the user's cost as the selfish equilibrium cost is very large. It turns out that the cost becomes unbounded or large only when a significant amount of flow is routed thorough cross link, which is the main reason of inefficiency. Therefore, a bounded cost can be achieved only when one can restrict almost all the flow to one's local link. In this spirit an altruistic game can achieve the equilibrium flow to be (x 1 1 , x 2 2 ) = (r, r -δ) for at least some β, because it can result in bounded cost.

We call x i i = r as "load taker" strategy because user i do not reciprocate (sends back flow) to any flow sent to him, i.e., takes all the load. Similarly, x i i = r -δ is the so called "delta load taker" strategy as it only pushes enough flow to the other link so that his local link gets to the knee, provided the other player doesn't push back. We aim to show that (x 1 1 , x 2 2 ) = (load taker, delta load taker) = (r, r -δ) is an equilibrium strategy. we use the best response approach to show the equilibrium for large enough m. We user δ and δ m alternatively in the proof for ease of notation.

Computation of BR 2 (x 1 1 ) : The cost of u 2 for a load taker strategy of user 1 is given by (let denote p = x 2 2 and p = r-x 2 2 ),

J 2 (p) = p[T 12 (p) + T 1 (r + p)] if p ≤ r -δ, pT 2 (p) + p[T 12 (p) + T 1 (r + p)] if p > r -δ. (27)
Notice that the J 2 (p) has discontinuity only at x 2 2 = r -δ. The marginal cost can be given as

∂J 2 (p) ∂p = A(p) < 0 if p ≤ r -δ, B m (p) > 0 if p > r -δ, (28)
Using (27), we have

A(p) = -p L δ m -[d m + L δm (p) + L] < 0, and B m (p) = (2p-r)L δm -d m > (r -2δ m ) L δm -d m .
We note that B m (p) > 0 for sufficiently large m. Hence, BR 2 (r) = r -δ.

Computation of BR 1 (x 2 2 ): The cost of u 1 for a delta load taker strategy of u 2 is given by (let denote x 1 1 = p and x 2 2 = δ = r -δ),

Ĵ1 (p) = (1 -β)J 1 (x) + βJ 2 (x) = (1 -β)[pT 1 (p + δ) + p(T 12 (p) + T 2 (p + δ))] +β[ δT 2 (p + δ) + δ(T 21 (δ) + T 1 (p + δ))] (29) Noting that Ĵ1 (p) is strictly convex in p ∈ [0, r], if ∂ Ĵ1 ∂p < 0 at p = r then ∂ Ĵ1 ∂p < 0 for p ∈ [0, r]. Therefore, we examine if ∂ Ĵ1
∂p < 0 holds true at p = r. Taking the derivative of Ĵ1 (p), we have

∂ Ĵ1 (p) ∂p = (1 -β)[p(g 1 (p + δ) + g 2 (p + δ)) + T 1 (p + δ) -T 2 (p + δ) -(rg 2 (p + δ) + T 12 (p))] + β[δg 1 (p + δ)) -( δ)g 2 (p + δ)] (30)
At p = r -(less than and close to r), we have where the set Γ turns out to be a non empty set, given by ( 1 2 , 1]. Therefore, we can state that BR 1 (r -δ) = r, for β ∈ Γ.

g 1 (p + δ) = g 2 ( δ + p) = L δm , T 1 (p+δ) = L+(p+δ -r) L δm ,
Therefore, together it constitute the Nash equilibrium. Denoting the equilibrium flow by xAL , and cost of user 1 as J 1 (x AL ), we have xAL = (r, r -δ), for β ∈ Γ, (32)

J 1 (x AL ) = rT 1 (r + δ) = 2rL. ( 33
)
The set of equilibria is denoted by x AL when there exist multiple equilibria. Denote x AL ∈ x AL such that

J 1 (x AL ) < J 1 (x), for any x ∈ x AL . (34)
Computation of V oU : Using ( 19) and (33), and the definition 5.2, we can express

V oU (1) = J i (x SE ) inf ( αi,x AL α i ) J i (x AL αi ) > J 1 (x SE ) J 1 (x AL ) = rL + (r/2 -ζ m )c m 2rL . ( 35
)
The last inequality is from (51). The quantity rL+(r/2-ζm)cm 2rL can be arbitrarily large for large m. This concludes the proof. ⋄ E. Proof of Lemma 6.1

Proof: In order to compute PoA, we need to compute the worst equilibrium cost for selfish user and the cost at social optimum.

To do so, We go in the following steps: 1. We show that there exist unique Nash equilibrium for selfish users. 2. Since there can be only one equilibrium, we exploit the symmetric property and compute the Nash equilibrium.

Obviously, this can be used for worst equilibrium cost of PoA computation. 3. Finally, we note that the social optimum is unique, the PoA follows by directly comparing the ration of selfish equilibrium and social optimum. In the following, we go step by step.

We allowed arbitrary number of node and a user associated with each user, as shown in Figs. [START_REF] Kameda | Nash equilibria in load balancing in distributed computer systems[END_REF]. The Nash equilibrium for the selfish game corresponds to the solution to the following minimization problem

P 1 : x i * = arg min x∈X J i (x), ∀ i = 1, • • • , n. ( 36
)
The cost of user a i in the above equation can be expressed as:

J i (x) = x i i T i (x i ) + xi i (T ij (x i i ) + T j (x j )), for i = j = 1, • • • , n. (37)
Where, xi i = r i -x i i . We show the existence of unique Nash equilibrium by transforming our problem into parallel link network problem with arbitrary number of users, for which it known to exist unique Nash equilibrium (see [START_REF] Orda | Competitive routing in multiuser communication networks[END_REF]Theorem 1.]).

Consider a network consisting of two nodes: a and b, and of I parallel directed links, all from node a to node b. There are i users, i = 1, • • • , I, having node a as the source node and node b as the destination node. The total demand rate for user i is r i .

Let x i l be the rate at which user i sends over the link l, (l = 1, • • • , I, which satisfy the positive flow constraint, x i l ≥ 0, ∀l and l x i l = r i . User i determines (x i 1 , • • • , x i I ) so as to minimize his cost J i (x) where x = (x i l , l ∈ (1, • • • , I), i ∈ I). Define x l = i x i l . We assume that J(x) is the sum of the link cost functions: J i (x) = l J l i (x), where J p i (x) are expressed in terms of the cost J l i (x), defined in the original problem, as follows. For any l and any i,

J i i (x) = J i i (x i i , x i ) J j i (x) = J ij i (x i j , x j ) + J j i (x i j , x j ), ∀ j = i.
recall that J i i represent cost of user i in its local link i and the corresponding flow is x i i . We note that the cost function of the original problem satisfies the following assumptions (as that in [START_REF] Orda | Competitive routing in multiuser communication networks[END_REF]): G1. J i (x) is the sum of the link costs incurred in each link, i.e.,

J i (x) = l J l i (x)
G2. J l i (x) is continuous (piecewise continuous with our elbow latency function, which suffice the condition) functions whose range is the nonnegative quadrant and their image is [0, ∞]. G3. J l i are convex functions in the rate sent by user i over link l. G4. Whenever finite, J l i is continuously differentiable in the flow sent by user i to link l. The marginal cost is denoted by

K l i (x) = ∂J l i (x) ∂x l i . G5.
If not all users have finite cost and one user user has infinite cost then it can change its own flow o make this cost finite. This ensures that at any equilibrium users have finite cost if it is feasible. The following assumption is also satisfied by J i (x), which is very important for uniqueness ( [START_REF] Orda | Competitive routing in multiuser communication networks[END_REF]):

Π 1 K l i (.
) is function of two arguments: i) the total flow on the link l, x l and ii) the flow of user i on link l, x l i . K l i (x l i , x i ) is strictly increasing in both of its arguments. 1) Uniqueness of Nash equilibrium: If the cost of original problem with LB network satisfy the assumption (Π 1 ), it follows that the costs for the new routing problem also do satisfy the assumption (Π 1 ). The new routing problem has a unique Nash equilibrium under the assumptions (Π 1 ) (see Theorem 1 in [START_REF] Orda | Competitive routing in multiuser communication networks[END_REF]). By identifying the decision variable x in the new routing problem with the decision variable x of the original problem with LB network, we see that the minimization problem faced by each user is same in both the cases, and therefore we conclude that user optimum in our problem is also unique. Note that the although we considered a symmetric network, the proof holds good for non symmetric LB network also.

2) Computation of equilibrium: Since all the users have identical demands and facing identical network conditions, the users will have identical strategy at equilibrium. Such equilibrium are often called as symmetric equilibrium. Let x i i be the equilibrium strategy of user i, the flow in his local link, and x j i is the flow in a cross link to user j. Let us compute the best response of user i for the strategy of user j = i. Let x i i be user i's best response for the strategy of user j be x j j . Note that we consider that all the other user's have identical strategy(x j j ). Therefore, the best response of user i is given by

BR i (x j j , n) = arg min xi∈X J i (x). (38)
The cost of user i is given by,

J i (x i i , x j j ) = x i [T i (x j i + (r -x j j ))] + j =i xi i n -1 [c m ( xi i n -1 ) + T j (x j j + (r -x i i ) n -1 ) + (n -2)(r -x k k ) n -1 )].
where

k = i, j. The derivative of J i (.) is (denote p = x i i ) ∂J i (p, x j j ) ∂p = pg i (p + r -x j j ) + T i (p + r -x j j ) -p g j x j j + (r -p) n -1 + r -x k k n -1 -c m + T j x j j + (n -2)x k k + p n -1 , = p[g i (x i ) + g j (x j )] + [T i (x i ) -T j (x j )] -(rg j (x j ) + c m ).
Due to symmetric structure, we note that x i = x (j =i) = r. Thus, we have T i (x i ) -T j (x j ) = 0 and g i (x i ) = g j (x j ) = g. Therefore,

BR i (x j j ) = c m + rg 2g = 1 2 r + c m L/δ m = r/2 + ζ m .
Since all the users are symmetric, the equilibrium strategy of any user is same. Therefore, the equilibrium strategy and the equilibrium cost of a user i can be given by

x i SE i = r/2 + ζ m , J SE i = rL + r 2 -ζ m (c m ) . (39)
One can directly observe that at social optimum (OPT) only local links are used. Therefore, the user cost J OP T i at social optima is given as

J OP T i = rT i (r) = rL, ∀ i. ( 40
)
3) Computation of PoA: Therefore the price of anarchy can be expressed as

P oA = sup m i rL + ( r 2 -ζ m ) (c m ) i rL = 1 + sup m ( r 2 -ζ m ) (c m ) rL . (41)
Evidently, the PoA can be arbitrarily large for sufficiently large m. This concludes the proof. ⋄

where the perceived cost of users (when user i is altruistic) can be given by Ĵi

(x) = n (k=1) α i k J k (x). (43)
Existence of an equilibrium is direct from the compact strategy set X and concave and continuity of the cost function Ĵi (x) [START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF]. Following the approach of proposition 5.1, we show that the load taker strategy of altruistic user and delta load taker strategy of other users constitutes a Nash equilibrium. Since this equilibrium yields a bounded cost for altruistic user, V oU is be arbitrarily large for large m. Let index altruistic user by i and selfish users by j( = i) = 1, • • • , n. We compute the best response for both type users.

Computation of BR j (x i i = r) : The cost of a selfish user (u j ) for a load taker strategy of user i is given by (let denote p = x i j , p = r -x i j = x j j , and x k j = q, k = i, j), (44)

Noting the discontinuity only at x j j = r -δ, the marginal cost can be given as

∂J j (p) ∂ p = A 1 (p) < 0 if p ≥ r -δ, B 1 m (p) > 0 if p < r -δ, (45)
Using (44), we have A 1 (p) = -p L δ m -pT i (r + (n -1)p) < 0, and B 1 m (p) = (2 p-r)L δm + (n -2) L δm -c m -(T i (r + (n -1)p) -T j (p)).We note that B 1 m (p) > 0 for sufficiently large m. Hence, the best response is a delta load taker strategy, i.e., BR j (r) = r -δ.

Computation of BR i (x j j = r -δ): The cost of the altruistic user (u i ) for a delta load taker strategy of u j is given by (let denote x i i = p and x j j = r -δ = δ), Ĵi Therefore, together it constitute the Nash equilibrium. Denoting the equilibrium flow by xAL , and cost of user i as J i (x AL ), we have xAL = (r, r -δ), for β ∈ Γ, (49) J i (x AL ) = rT i (r + δ) = rL.

(50)

The set of equilibria is denoted by x AL when there exist multiple equilibria. Denote x AL ∈ x AL such that J i (x AL ) < J i (x), for any x ∈ x AL .

(51)

Computation of V oU : Using ( 19) and (50), and the definition 5.2, we can express

V oU (i) = J i (x SE ) inf ( αi,x AL α i ) J i (x AL αi ) > J 1 (x SE ) J i (x AL ) = rL + (r/2 -ζ m )c m 2rL . ( 52
)
The last inequality is from (51). The quantity rL+(r/2-ζm)cm 2rL can be arbitrarily large for large m. This concludes the proof. ⋄

Fig. 1 .

 1 Fig. 1. Network topology and Latency function

Fig. 2 .

 2 Fig. 2. Link latency function: L 2 (x) = L δ (xr) + L, for rδ ≤ x and the parameters are: r = 1, L = 0.1, δ = 10 -3 , g = L δ = 100.

  and T 2 ( δ+ p) = L + (r -δ -p) L δm . Plugging the values in (30), we obtain ∂ Ĵ2 (p) ∂p < 0, for β ∈ Γ (31)

F. Proof of Proposition 6. 1 Proof:

 1 The equivalent problem for the n user altruistic routing game considered can be expressed as (similar as in proposition 5.1)P n : x i = arg min x∈X Ĵi (x), ∀ i = 1, • • • , n.

J

  j (p) = p[T j (p) + (n -2)qT k (p) + (n -2)qT jk (q) +pT i (r + (n -1)p) + pT ji (p)]Since the latency function is discontinuous, the cost function can be re-expressed as followsJ j (p)= i (r + (n -1)p) + c m ] + (n -2)qc m if p ≥ r -δ, pT j (p) + (n -2)qT k (p) + (n -2)qc m +p[T i (r + (n -1)p) + T ji (p)] if p > r -δ.

  (p) = (1 -β)J i (x) + (n -1) β (n -1) J j (x) = (1 -β)[pT i (p + (n -1)δ) + (n -1) p n -1 (T ij ( p n -1 ) +T j ( p n -1 + δ))] + β[ δT j ( p n -1 + δ) +δ(T ji (δ) + T i (p + δ(n -1)))]. (46) Noting that Ĵi (p) is convex in p ∈ [0, r], if ∂ Ĵi ∂p < 0 at p = r - (less than and close to r) then ∂ Ĵi ∂p < 0 for p ∈ [0, r]. Therefore, below we examine if ∂ Ĵi ∂p < 0 holds true at p = r -. The derivative of Ĵi (p) is ∂ Ĵi (p) ∂p = (1 -β)[p(g i (p + (n -1)δ) + g j ( p n -1 + δ)) +T i (p + (n -1)δ) -T j ( p n -1 + δ) -(rg j + β[δg i (p + δ(n -1))) -( δ)g j ( p n -1 + δ)]. (47)Plugging the values of g(.) and T (.) in (47), we obtain∂ Ĵi (p) ∂p < 0, for β ∈ Γ(48)where the set Γ = ( 1 n , 1]. Hence, we can state that BR i (r-δ) = r, for β ∈ Γ.

  2, ..., I is called a Nash equilibrium if for each user i, xi minimizes the cost function given that other users' routing decisions are xj , j = i. In other words, J i (x 1 , x2 , ..., xI ) = min

Existence of equilibrium with unilateral altruism is guaranteed since the strategy set is compact and continuous, and the cost function is concave[START_REF] Basar | Dynamic Noncooperative Game Theory[END_REF].


Ismael Galván
email: ism.galvan@gmail.com

Ultraviolet-blue plumage colouration can be perceived as an indicator of fluctuating asymmetry by Blue Tits (Cyanistes caeruleus)

Keywords: communication, developmental stability, structural colouration, visual perception Zusammenfassung Ultraviolettblaue Gefiederfärbung als Indikator für variable Asymmetrie bei Blaumeisen Cyanistes cearuleus z. B. erzeugt durch Mikrostrukturen in Federn) ist oftmals ein Zeichen für Qualität, wobei aber der Signalgehalt dieses Merkmals relativ unbekannt ist. Eine Hypothese besagt, dass die Bildung der strukturellen Färbung abhängig ist von einer stabilen Entwicklung, da Variation in der Entwicklung die genaue Federstruktur, welche die strukturellen Farben generiert, beeinflussen kann. Mittels der Analyse der Beziehung zwischen Kronenfärbung, berechnet mit Hilfe eines Modells zur visuellen Wahrnehmung bei Vögeln, und der variablen Asymmetrie erwartet, konnte eine negative Korrelation zwischen der ultraviolett-blauen Zusammenhang zwischen Bruterfolg und der ultraviolett-blauen Färbung, jedoch nicht hinsichtlich einer Konditionsabhängigkeit dieses Merkmals. Die Ergebnisse zeigen, dass der Signalgehalt der strukturellen Färbung bei Blaumeisen Informationen Wahrnehmung bei Vögeln berechnet wurden, deutet dies darauf hin, dass Blaumeisen Unterschiede

The expression of structural plumage colours (i.e. those generated by feather microstructures) often acts as a signal of quality, but the signal content of these traits remains relatively unknown. However, a hypothesis posits that the production of structural colouration depends on developmental stability because variation in developmental programs can affect the precise feather structures that generate structural colour. Here I test this hypothesis by exploring the relationship between crown colouration, calculated by using a model of avian visual perception, and fluctuating asymmetry (FA) of tail feathers in blue tits (Cyanistes caeruleus). As predicted, there was a negative correlation between the ultraviolet (UV)-blue reflectance of the crown patch and FA of feathers. There was indirect evidence for a relationship between fledging success and UV-blue colour, but not for a condition dependence of this trait. These results suggest that the signal content of the structural colouration of blue tits may be information regarding individuals' developmental stability. As the colour components of the crown colouration were calculated with an avian visual perception model, this indicates that blue tits can perceive differences in FA through variation in the expression of structural colour.

Introduction

Structural plumage colouration is generated by the optical effects of light waves interacting with the matrix of keratin and air of feathers, and is responsible for the blue and ultraviolet (UV) hues of several bird species [START_REF] Prum | Anatomy, physics, and evolution of structural colours[END_REF]. These colour traits are often involved in processes of mate choice [START_REF] Hill | Female mate choice for ornamental coloration[END_REF]. Plumage signals produced by pigment-based colours (e.g., melanins or carotenoids) are generally thought to be the product of trade-offs between allocating pigments to that signal vs. physiological maintenance. Theory predicts that only high quality individuals will be able to afford large concentrations of pigments to signalling (e.g. [START_REF] Alonso-Alvarez | The oxidation handicap hypothesis and the carotenoid allocation trade-off[END_REF][START_REF] Galván | An intracellular antioxidant determines the expression of a melanin-based signal in a bird[END_REF][START_REF] Mougeot | Honest sexual signalling mediated by parasite and testosterone effects on oxidative balance[END_REF]. The signal content of structural colours is controversial however, as the feather nanostructures that produce these colours may not be constructed from any limited nutrients, and thus no developmental constraints may act on these structures [START_REF] Prum | Anatomy, physics, and evolution of structural colours[END_REF].

Recent studies suggest that the expression of structural plumage colours may be constrained by diet, condition, developmental stability and environmental stressors. First, [START_REF] Peters | The condition-dependent development of carotenoid-based and structural plumage in nestling blue tits: males and females differ[END_REF] showed that the expression of structural plumage colouration depends on the availability of protein necessary to produce the structural components of feathers (i.e. keratin), and other studies have shown that the production of some types of structural plumage colouration is related to nutritional condition [START_REF] Keyser | Condition-dependent variation in the blue-ultraviolet coloration of a structurally-based plumage ornament[END_REF][START_REF] Doucet | Structural plumage coloration, male body size, and condition in the blueblack grassquit[END_REF][START_REF] Mcgraw | Different colors reveal different information: how nutritional stress affects the expression of melanin-and structurally based ornamental plumage[END_REF][START_REF] Johnsen | Plumage colour in nestling blue tits: sexual dichromatism, condition-dependence and genetic effects[END_REF]. Second, the production of melanins is limited to high quality individuals that are able to afford the antioxidant costs of these pigments [START_REF] Galván | Plumage coloration can be perceived as a multiple condition-dependent signal by Great Tits Parus major[END_REF]Alonso-Alvarez 2008, 2009) or to gain access to melanin precursors [START_REF] Poston | Dietary amino acids influence plumage traits and immune responses of male house sparrows, Passer domesticus, but not as expected[END_REF]. As the presence of melanin layers in feathers are necessary to produce structural colours [START_REF] Shawkey | Significance of a basal melanin layer to production of noniridescent structural plumage color: evidence from an amelanotic Steller's jay (Cyanocitta stelleri)[END_REF], the costs of producing melanins may indeed represent costs for producing structural colouration. Lastly, some analyses suggest that structural colour expression depends upon the nanoscale regularity of the spongy layer of keratin in feather barbs [START_REF] Shawkey | Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour[END_REF], and similar results linking colour expression to nanostructure variables related to developmental instability have been reported in insects [START_REF] Kemp | Stress-mediated covariance between nano-structural architecture and ultraviolet butterfly coloration[END_REF]). These latter results suggest that only individuals able to mount a proper developmental program or to resist the effects of environmental stressors can express structural colouration at a high degree [START_REF] Shawkey | Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour[END_REF][START_REF] Kemp | Stress-mediated covariance between nano-structural architecture and ultraviolet butterfly coloration[END_REF]. Thus, the honesty of structural plumage colouration may be mediated by developmental constraints, which may be generated by individual differences in either the expression of genes that control developmental stability [START_REF] Fitzpatrick | Colour schemes for birds: structural coloration and signals of quality in feathers[END_REF][START_REF] Andersson | Morphology of UV reflectance in a whistling-thrush: implications for the study of structural colour signalling in birds[END_REF][START_REF] Örnborg | Seasonal changes in a ultraviolet structural colour signal in blue tits, Parus caeruleus[END_REF][START_REF] Shawkey | Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour[END_REF][START_REF] Siefferman | UV-blue structural coloration and competition for nestboxes in male eastern bluebirds[END_REF] or in the capacity to afford costs associated with environmental stressors [START_REF] Kemp | Stress-mediated covariance between nano-structural architecture and ultraviolet butterfly coloration[END_REF].

My aim here is to examine the relationship between characteristics of a structurally produced plumage colour and developmental stability, as measured by fluctuating asymmetry (FA) of tail feathers, in the blue tit (Cyanistes caeruleus) [START_REF] Fitzpatrick | Colour schemes for birds: structural coloration and signals of quality in feathers[END_REF][START_REF] Andersson | Morphology of UV reflectance in a whistling-thrush: implications for the study of structural colour signalling in birds[END_REF][START_REF] Örnborg | Seasonal changes in a ultraviolet structural colour signal in blue tits, Parus caeruleus[END_REF][START_REF] Shawkey | Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour[END_REF][START_REF] Siefferman | UV-blue structural coloration and competition for nestboxes in male eastern bluebirds[END_REF]. FA is the deviation from the symmetry of a bilateral trait as a consequence of the incapacity of an organism to perform a developmental program, so FA of tail feathers can be used as a proxy for the developmental stability experienced by birds [START_REF] Møller | Asymmetry, developmental stability, and evolution[END_REF]. I used tail feathers because FA has been measured in this trait in several species of birds, which has been shown to reflect phenotypic quality [START_REF] Møller | Sexual ornament size and the cost of fluctuating asymmetry[END_REF][START_REF] Møller | Patterns of fluctuating asymmetry in avian feather ornaments: implications for models of sexual selection[END_REF][START_REF] Aparicio | Why do some traits show higher fluctuating asymmetry than others? A test of hypotheses with tail feathers of birds[END_REF], and is a non-destructive and easy method to estimate FA in live birds.

The blue tit is a small passerine bird that presents a structural blue crown plumage patch that is sexually dimorphic regarding UV reflectance [START_REF] Andersson | Ultraviolet sexual dimorphism and assortative mating in blue tits[END_REF][START_REF] Hunt | Blue tits are ultraviolet tits[END_REF]. A large body of evidence exists to support a role of crown UV reflectance as an honest signal in intraspecific communication [START_REF] Alonso-Alvarez | Ultraviolet reflectance affects male-male interactions in the blue tit (Parus caeruleus)[END_REF][START_REF] Poesel | Territorial responses of male blue tits, Cyanistes caeruleus, to UV-manipulated neighbours[END_REF]). In particular, male UV-reflectance predicts annual survival and reproductive success [START_REF] Sheldon | Ultraviolet colour variation influences blue tit sex ratios[END_REF][START_REF] Delhey | Paternity analysis reveals opposing selection pressures on crown coloration in the blue tit (Parus caeruleus)[END_REF][START_REF] Griffith | Correlations between ultraviolet coloration, overwinter survival and offspring sex ratio in the blue tit[END_REF], and females mated to more UV reflective males produce more sons and provide more parental care and nest defense [START_REF] Sheldon | Ultraviolet colour variation influences blue tit sex ratios[END_REF][START_REF] Griffith | Correlations between ultraviolet coloration, overwinter survival and offspring sex ratio in the blue tit[END_REF][START_REF] Limbourg | Female blue tits adjust parental effort to manipulated male UV attractiveness[END_REF][START_REF] Johnsen | Male sexual attractiveness and parental effort in blue tits: an experimental test of the differential allocation hypothesis[END_REF][START_REF] Korsten | Primary sex ratio adjustment to experimentally reduced male UV attractivenes in blue tits[END_REF].

Furthermore, there is some evidence that female crown UV reflectance may also be a signal used by males when choosing mates [START_REF] Andersson | Ultraviolet sexual dimorphism and assortative mating in blue tits[END_REF][START_REF] Hunt | Blue tits are ultraviolet tits[END_REF]but see Delhey et al. 2007). However, the mechanisms mediating the honesty of blue tit crown colouration remain unknown [START_REF] Peters | The condition-dependent development of carotenoid-based and structural plumage in nestling blue tits: males and females differ[END_REF]). I predicted that, if the structural plumage colour of blue tits reflects the capacity of birds to perform a tight developmental control of feather elements and thus the developmental stability of birds, high quality individuals (i.e. those presenting high UV-reflectance in the crown patch, see above) should present low FA levels in their feathers. UV colouration and FA levels should thus be negatively correlated. I use current models of avian colour vision [START_REF] Vorobyev | Tetrachromacy, oil droplets and bird plumage colours[END_REF][START_REF] Endler | Comparing entire colour patterns as birds see them[END_REF] to estimate crown colour components that are perceived by blue tits and thus have the potential to be significant in intraspecific signalling.

Methods

The study was carried out in May-June 2007 in a deciduous forest of Pyrennean Oak Quercus pyrenaica in Miraflores de la Sierra, central Spain (40º 49" N, 03º 46" W). Frequent checks of nest-boxes allowed me to determine the date of clutch initiation and clutch size for all breeding pairs. A total of 34 blue tits (16 males and 18 females, representing 15 adults, 15 yearlings and four individuals of unknown age) were captured at the nest-boxes on day six (hatching day = day 0), and classified as yearlings or older according to plumage [START_REF] Jenni | Moult and Ageing of European Passerines[END_REF].

Individuals were uniquely marked with metallic rings, weighed with a portable electronic balance and their tarsus length measured with a digital calliper. The fifth pair of rectrix feathers was plucked in each bird and stored in a bag until measurements were made. The length of the feathers was measured by putting them straight on a piece of polystyrene and marking with two entomological pins the extreme points of the feathers, that is, the base of the calamus and the end of the rachis. The distance between pins was then measured with a digital calliper. All feathers were measured twice, but both measurements were made on different days to avoid unconscious bias in determining feather length. FA was calculated as the absolute value of the difference between the average length of the left and the right rectrix feathers (│L-R│), as this value was not correlated with trait size [START_REF] Møller | Asymmetry, developmental stability, and evolution[END_REF]; see Results).

On day 15, the nestlings were weighed and their tarsus length measured. After day 15, the nest-boxes were inspected shortly after all nestlings had fledged to search for dead nestlings or cues of predation, but none were found. Thus, the number of fledglings on day 15 divided by clutch size was used as an estimate of fledging success (e.g. [START_REF] Sanz | Female hematozoan infection reduces hatching success but not fledging success in Pied Flycatchers Ficedula hypoleuca[END_REF].

Reflectance spectrometry

Plumage reflectance measurements were taken from the crown patch of blue tits. Data were collected in the field using an Ocean Optics USB2000 spectrophotometer (range 250-800 nm) with ultraviolet (deuterium) and visible (tungsten-halogen) lamps and a bifurcated 400 micrometer fiber-optic probe (Dunedin, Florida, USA). The fiber optic probe both provided illumination and obtained light reflected from the sample, and had a reading area of ~1 mm 2 .

The measurements were taken at a 90º angle to the sample. The spectrometer measured reflectance in 0.36 nm increments. All measurements were relative to a white "Spectralon" tablet (WS-1-SS, Ocean Optics, Dunedin, Florida, USA), and the system was frequently calibrated.

The spectral curves were generated by using OOIBase software. Three readings were obtained for each individual. The probe was removed between each measurement, but each replicate reading was taken from the same point on the crown. The results of reflectance measurements are shown in Fig. 1.

Visual modeling

Colour vision in diurnal birds is dependent upon four types of retinal cone cells which are sensitive to either very short (VS, ultraviolet (UV)), short (SWS, blue), medium (MWS, green) or long (LWS, red) wavelengths [START_REF] Maier | Colour vision in the passeriform bird, Leiothrix lutea: Correlation of visual pigment absorbency and oil droplet transmission with spectral sensitivity[END_REF][START_REF] Cuthill | Color Perception[END_REF]. Spectral data were reduced into four quantal cone catches [START_REF] Vorobyev | Tetrachromacy, oil droplets and bird plumage colours[END_REF]) that quantify the amount of light captured by each of the single cones. By using the SPEC package [START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF] implemented in R (R Development Core Team 2004), the four quantum cone catches (Q i ) were calculated by multiplying cone sensitivities by the reflectance spectrum, the irradiance spectrum and the transmission spectrum of the ocular media, as expressed by the formula:

) ( ) ( ) ( ) ( ) ( ) ( i i i R d O I S R Q , where
indicates wavelength (nm),

) ( i R is the sensitivity (nm) of cone type i, ) ( S
is the percent of light reflected from a patch compared to a white standard,

) ( I is the irradiance spectrum (µmol•m -2 •s -1 •nm -1
) and ) ( O is the transmittance spectrum of the ocular media.

As cone catches change with varying illumination,

) ( I

, and this is discounted by animals through colour constancy, the von Kreis algorithm was used to normalise i Q by the quantum cone catch for the irradiance spectrum. The irradiance spectral data used were obtained from a standard forest shade illumination provided by the SPEC package [START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF], as this irradiance spectrum is the most appropriate for our case (see habitat description above), and an achromatic adapting background [START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF]. I used the ocular media transmittance of the blue tit [START_REF] Hart | Visual pigments, oil droplets, ocular media and cone photoreceptor distribution in two species of passerine bird: the blue tit (Parus caeruleus L.) and the blackbird (Turdus merula L.)[END_REF]. Each cone catch was then standardized by dividing them by the sum of the four cone catches, and three of the standardized cone catches were divided by a fourth (that corresponding to LWS in this case) and their natural logarithm was then calculated. Thus, three log-contrasts were calculated to break the unit-sum of the four standardized cone catches [START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF]Galván et al. 2009;[START_REF] Galván | Plumage coloration can be perceived as a multiple condition-dependent signal by Great Tits Parus major[END_REF]). In addition, these log-contrasts permit considering an opponency model of colour vision where the exact opponency mechanisms are unspecified, as perception of chromatic variation occurs through a physiological mechanism that contrasts the relative outputs of the different cone types [START_REF] Endler | Comparing entire colour patterns as birds see them[END_REF][START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF].

Following [START_REF] Hadfield | Strong environmental determination of a carotenoid-based plumage trait is not mediated by carotenoid availability[END_REF], a principal components analysis (PCA) was performed on the three log-contrast colour variables in order to treat them as a multivariate response in a multivariate "animal model". The eigendecomposition showed an axis (PC1) that explained 75.7% of phenotypic variation (eigenvalue = 2.27; mean ± SE: 0.00 ± 0.16; range: -1.73-2.94; n = 38)). The factor loadings for PC1 were high and positive for the first two logcontrasts (0.92 and 0.98 for UVS-and SWS-log contrasts, respectively) and lower for the MWS log-contrast (0.67), indicating that it is determined by the two first cone catches vs. LWS. Thus, high PC1 scores indicate relatively more reflectance at short (i.e. UV-blue) wavelengths as compared to long (i.e. red) wavelengths.

As achromatic variation in birds is thought to not be perceived through the four single cones, but by double cones [START_REF] Campenhausen | Spectral sensitivity of the accessory optic system of the pigeon[END_REF][START_REF] Osorio | Visual ecology and perception of coloration patterns by domestic chicks[END_REF], I calculated the double cone quantum catches from data on double cone spectral sensitivity of blue tits following the same procedure for the four single cones. The double cone quantum catch was used as a "brightness index" (mean ± SE: 0.05 ± 0.00; range: 0.01-0.11; n = 38). This brightness index was negatively correlated with PC1 scores in the blue tits (Pearson correlation test: r = -0.57, n = 34, P < 0.0001).

Analyses of FA Some preliminary analyses were performed to determine if the distribution of L-R values was due to FA or to any other type of asymmetry [START_REF] Van Dongen | The statistical analysis of fluctuating asymmetry: REML estimation of a mixed regression model[END_REF]. First, I tested whether the distribution of L-R values differed from a normal distribution through a Shapiro-Wilks test, and whether the mean value significantly differed from zero through a one-sample t-test. To estimate the magnitude of FA in relation to measurement error and directional asymmetry (DA), a mixed model was performed in which random effects were estimated with a restricted maximum-likelihood (REML) approach [START_REF] Van Dongen | The statistical analysis of fluctuating asymmetry: REML estimation of a mixed regression model[END_REF]. In this model, trait size was a function of side (fixed effect) + individual (random effect) + side nested in individual (random effect) + measurement error (random effect). FA and DA are represented by the random side effect and the fixed side effect, respectively (e.g. [START_REF] Stige | Individual fluctuating asymmetry in pied flycatchers (Ficedula hypoleuca) persists across moults, but is not heritable and not related to fitness[END_REF][START_REF] Stige | Fitness, developmental instability, and the ontogeny of fluctuating asymmetry in Daphnia magna[END_REF]. In addition, I tested whether the skewness and kurtosis of the distribution of L-R values significantly differed from zero according to [START_REF] Zar | Biostatistical Analysis[END_REF]. To test whether FA depended on trait size, I performed a Pearson correlation test on standardized (mean = 0, SD = 1) │L-R│ values and standardized mean values of feather length for each bird.

Statistical analyses

To test for a relationship between crown colouration and FA, General Linear Models (GLM) were performed with each of the colour parameters (i.e. PC1 and brightness index) as a dependent variable. The saturated models included sex and age as fixed factors and FA as a covariate. The interaction between sex and FA was also included because crown colouration is a sexually dichromatic trait in blue tits (e.g. [START_REF] Andersson | Ultraviolet sexual dimorphism and assortative mating in blue tits[END_REF][START_REF] Delhey | Paternity analysis reveals opposing selection pressures on crown coloration in the blue tit (Parus caeruleus)[END_REF][START_REF] Delhey | Seasonal changes in blue tit crown color: do they signal individual quality?[END_REF], which suggests that trait expression may be differentially constrained in males and females [START_REF] Delhey | Paternity analysis reveals opposing selection pressures on crown coloration in the blue tit (Parus caeruleus)[END_REF] and thus FA may have different effects on crown colouration in both sexes.

The interaction between age and FA was not included because there were no a priori reasons to test for differences in the relationship between FA and crown colour between age classes. Similar models were performed to test for a relationship between crown colouration and body condition, including this latter variable as a covariate. Body condition was expressed as body mass divided by the cube of tarsus length [START_REF] Galván | Plumage coloration can be perceived as a multiple condition-dependent signal by Great Tits Parus major[END_REF], and the same analyses were repeated using the residuals of body mass regressed on tarsus length as an alternative measurement of condition. Although the measurements of condition taken for the present study may not reflect the condition of birds at moult (i.e. end of summer), changes in body mass between winter and spring are small and probably not related to changes in body condition in blue tits [START_REF] Delhey | Seasonal changes in blue tit crown color: do they signal individual quality?[END_REF], so that body condition in spring is likely to reflect condition when the feathers were developed like in the closely related great tit (Parus major) [START_REF] Gosler | Assessing the heritability of body condition in birds: a challange exemplified by the great tit Parus major L. (Aves)[END_REF].

To test for a relationship between crown colouration and different parameters of breeding success, laying date (log-transformed), nestling mass, nestling condition (calculated as in the parents) and fledging success (arcsine-square-root-transformed) were included as dependent variables in separate GLMs with sex, age and the interaction between sex and the colour parameters.

In all cases, inspections of the distribution of residuals confirmed that the normality assumption was fulfilled. The presence of outliers was determined on the basis of Cook"s distances greater than 2 and leverages greater than 2p/n, where p is the number of parameters in the model and n is the sample size [START_REF] Crawley | GLIM for ecologists[END_REF].

Results

Measurement error and asymmetry distribution

│L-R│ values were not significantly correlated with trait size (r = -0.24, n = 34, P = 0.177), so that these untransformed absolute values were used as an estimate of FA in subsequent analyses (mean FA ± SE: 0.29 ± 0.04 cm; range: 0.00-0.72 cm; n = 34). The distribution of L-R values did not differ significantly from a normal distribution (W = 0.96, P = 0.328), and the mean of this distribution did not differ significantly from zero (t = 1.10, df = 33, P = 0.277).

Although the measurement error was large (Z = 5.83, P < 0.0001), FA was significantly related to the variance in tail feather length (Z = 5.73, P < 0.0001), while DA was not (F 1,66 = 0.02, P = 0.893). The distribution of L-R values was symmetrical and mesokurtic as derived from a skewness (g 1 = -0.108) and a kurtosis (g 2 = -0.435) not significantly different from zero (Z g1 = -1.02, P = 0.308; Z g2 = 0.76, P = 0.447). More specifically, although the kurtosis was negative, the distribution was not platykurtic because the one-tailed P-value was still not significant (P = 0.224). Therefore, the tail feather length of blue tits presented size-independent FA and not other types of asymmetry, after controlling for measurement error.

Relationship between crown colouration and tail FA

Age was not significantly correlated with colour PC1 (F 1,25 = 1.48, P = 0.235), and was therefore removed from the model. The same was done with the interaction between sex and FA (F 1,30 = 1.23, P = 0.276). The model obtained explained 35.4% of variance in colour PC1 scores and included sex (F 1,31 = 14.61, P = 0.001) and FA (F 1,31 = 4.78, P = 0.036), indicating that males presented more relative UV reflectance than females (least squares mean ± SE, males: 0.69 ± 0.20, females: -0.39 ± 0.19; Fig. 1) and that colour PC1 was negatively correlated with FA (b = -1.29, t = -2.19, P = 0.036; Fig. 2). There was an adult male that presented a very high PC1 score and may thus represent an outlier (see Fig. 2), but a Cook's distance lower than 2 (0.46) and a leverage lower than 2p/n (0.09; 2p/n = 0.18) indicated that it was not. In any case, the effect of FA remained close to significance when this point was removed (F 1,30 = 3.91, P = 0.057).

In contrast, only age was significantly related to the brightness index of blue tits (F 1,28 = 5.71, P = 0.024; least squares mean ± SE, yearlings: 0.05 ± 0.00, adults: 0.04 ± 0.00). Neither FA (F 1,26 = 0.02, P = 0.881), sex (F 1,27 = 0.46, P = 0.504) or their interaction (F 1,25 = 0.20, P = 0.657) contributed to explain variance in the brightness index.

Relationship between crown colouration and body condition

Body condition measured as body mass divided by the cube of tarsus length was not significantly related to colour PC1 (F 1,29 = 0.73, P = 0.399) after controlling for the effects of sex (F 1,29 = 10.02, P = 0.004; least squares mean ± SE: males: 0.59 ± 0.23, females: -0.38 ± 0.20) and age (F 1,29 = 3.34, P = 0.078). The interaction between sex and condition was not significant either (F 1,28 = 0.03, P = 0.860). Similarly, brightness index did not depend on the body condition of birds (F 1,29 = 0.92, P = 0.344), and only age had an effect (F 1,31 = 10.32, P = 0.003; least squares mean ± SE: yearlings: 0.06 ± 0.00, adults: 0.03 ± 0.00; sex: F 1,30 = 1.71, P = 0.200; sex x condition: F 1,28 = 0.05, P = 0.820). The same results were obtained when the residuals of tarsus length regressed against body mass were used as an estimate of body condition (results not shown).

Relationship between crown colouration and breeding success

There were no significant relationships between the colour parameters and laying date, nestling body mass, nestling condition or fledging success (results not shown), with the exception of an effect of brightness index on fledging success that was close to significance (b = -7.28, F 1,30 = 3.63, P = 0.066) after controlling for the effects of sex (F 1,29 = 0.01, P = 0.903), age (F 1,27 = 0.11, P = 0.744) and the interaction between sex and brightness index (F 1,28 = 0.00, P = 0.952), which indicates that fledging success tended to decrease with increasing the crown brightness of the parents.

Discussion

As predicted, the relative UV-blue reflectance of the crown plumage of blue tits was negatively related to the FA of tail feathers. This suggests that the developmental stability experienced by birds is reflected in the colour properties of feathers. As plumage colouration was measured through a model of avian colour vision, this result indicates that blue tits have the potential to perceive the developmental stability of conspecifics through the evaluation of UV-blue colour.

Thus, the present study provides some correlational evidence for the hypothesis that structural colours signal the capacity of individuals to perform a tight developmental control of feather elements [START_REF] Fitzpatrick | Colour schemes for birds: structural coloration and signals of quality in feathers[END_REF][START_REF] Andersson | Morphology of UV reflectance in a whistling-thrush: implications for the study of structural colour signalling in birds[END_REF][START_REF] Örnborg | Seasonal changes in a ultraviolet structural colour signal in blue tits, Parus caeruleus[END_REF][START_REF] Shawkey | Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour[END_REF][START_REF] Siefferman | UV-blue structural coloration and competition for nestboxes in male eastern bluebirds[END_REF], as FA measures the capacity of individuals to perform a developmental program of morphological structures [START_REF] Møller | Asymmetry, developmental stability, and evolution[END_REF].

On the other hand, I found weak evidence for condition dependence of the blue tit crown colouration. Although many types of biological signals exist [START_REF] Hasson | Towards a general theory of biological signaling[END_REF], most of our current understanding on honest signalling comes from the Handicap Principle [START_REF] Zahavi | Mate selection -A selection for a handicap[END_REF], which establishes that variation in the expression of the signals is due to inter-individual differences in the capacity to afford certain costs. These costs are generated by environmental influences, and thus the expression of handicap signals is strongly environmentally dependent, which means that a relationship between signal expression and individual condition must exist [START_REF] Zahavi | Mate selection -A selection for a handicap[END_REF][START_REF] Grafen | Biological signals as handicaps[END_REF]). In the case of the structural colour of blue tits, [START_REF] Johnsen | Plumage colour in nestling blue tits: sexual dichromatism, condition-dependence and genetic effects[END_REF] and [START_REF] Peters | The condition-dependent development of carotenoid-based and structural plumage in nestling blue tits: males and females differ[END_REF] found a positive correlation between UV chroma (a physical measurement of UV relative reflectance) and body condition in nestlings, which led other authors to assume that a similar relationship between structural colouration and condition might exist in adults [START_REF] Limbourg | Female blue tits adjust parental effort to manipulated male UV attractiveness[END_REF]), but to my knowledge there are no published reports of relationships with body condition in adult blue tits. It is thus possible that the lack of correlation between crown UV-blue reflectance and body condition in the present study represents a general pattern and the crown colouration is not a condition-dependent trait in blue tits.

However, the results presented here support the view of this trait as a signal of individual quality, as fledging success tended to increase with an index of plumage brightness as might be perceived by birds, which is in turn positively related to UV-blue reflectance (see Methods), independently of sex. This is in accordance to several previous studies showing fitness benefits for the more reflectant males [START_REF] Sheldon | Ultraviolet colour variation influences blue tit sex ratios[END_REF][START_REF] Delhey | Paternity analysis reveals opposing selection pressures on crown coloration in the blue tit (Parus caeruleus)[END_REF][START_REF] Griffith | Correlations between ultraviolet coloration, overwinter survival and offspring sex ratio in the blue tit[END_REF][START_REF] Limbourg | Female blue tits adjust parental effort to manipulated male UV attractiveness[END_REF][START_REF] Johnsen | Male sexual attractiveness and parental effort in blue tits: an experimental test of the differential allocation hypothesis[END_REF][START_REF] Korsten | Primary sex ratio adjustment to experimentally reduced male UV attractivenes in blue tits[END_REF]) and mutual mate choice based on this trait [START_REF] Andersson | Ultraviolet sexual dimorphism and assortative mating in blue tits[END_REF][START_REF] Hunt | Blue tits are ultraviolet tits[END_REF]. Therefore, the colour properties of the blue tit crown patch may signal quality through other mechanisms different from costs, i.e. by signalling 'good genes'.

FA is produced by differences in the capacity of genotypes to produce a phenotype under certain environmental conditions [START_REF] Møller | Bilateral symmetry and sexual selection: a meta-analysis[END_REF]. This measurement of developmental stability has an important role in sexual selection, probably because it is correlated with different individual attributes [START_REF] Møller | Bilateral symmetry and sexual selection: a meta-analysis[END_REF], and is also negatively related to several components of fitness [START_REF] Møller | Developmental stability and fitness: a review[END_REF]. The present study shows that blue tits can perceive differences in FA by the observed variability in UV-blue reflectance of the crown plumage patch, and that this colour trait is not condition-dependent. This suggests that the signal content of the blue tit structural colouration may be the capacity of the genotype to produce a certain phenotype (i.e. crown colour). To my knowledge, this is the first time that a measurement of developmental stability is related to the expression of structural plumage colouration as might be perceived by birds. Future studies should investigate the structural components of feathers that are more dependent upon the capacity of individuals to perform a developmental program and how these components affect the expression of structural colour traits that can be perceived by birds.

Fig. 1

 1 Fig. 1 Reflectance curves (± SE) for the crown plumage patch of male (solid symbols) and female (open symbols) Blue Tits.

Fig. 2

 2 Fig. 2 Relationship between the scores of crown colour PC1 (relative UV reflectance as might be perceived by birds) and fluctuating asymmetry of tail feathers (absolute values of the difference between the left and the right fifth rectrix feathers) in Blue Tits. The upper point of the figure is not an outlier on the basis of statistical criteria. The line is the regression line.

Figure 2

 2 Figure 2

Acknowledgments

The Consejería de Medio Ambiente y Ordenación del Territorio of Comunidad de Madrid provided me with the licenses required to carry out the study.