Jaime Iranzo

Javier Román

Angel Sánchez

The spatial ultimatum game revisited

Keywords: evolution, fairness, rationality, game theory, spatial dynamics

], who showed in a specific example that on a two-dimensional setup evolution may lead to strategies with some degree of fairness. Within this spatial framework, we carry out a thorough simulation study and show that the emergence of altruism is a very generic phenomenon whose details depend on the dynamics considered. A very frequent feature is the spontaneous emergence and fixation of quasiempathetic individuals, whose offers are very close to their acceptance thresholds. We present analytical arguments that allow an understanding of our results and give insights on the manner in which local effects in evolution may lead to such non-rational or apparently maladaptive behaviours.

Introduction

The evolution of cooperation and the difference between humans and other primates are two intimately related questions that attract a lot of interest among researchers in fields from biology through economics to psychology. From a biological viewpoint, understanding the emergence of cooperation is crucial to unveil the mechanisms of the evolution of complex organisms through the association of simpler entities [START_REF] Smith | The Major Transitions in Evolution[END_REF]. On the other hand, the question is socio-economically relevant because human societies have arisen mostly due to the readiness of humans to cooperate with non-kin even when the interaction is unlikely to take place again in the future, or in sizeable groups [START_REF] Fehr | The nature of human altruism[END_REF]. This type of behaviour has been pointed out as one of the characteristics that set humans apart from all other animals (Kappeler and van Schaik, 2006). Today, this topic remains a very active field, and different mechanisms are being proposed to understand the evolutionary origin of cooperation [START_REF] Nowak | Five rules for the evolution of cooperation[END_REF].

At the roots of many cooperative behaviors in humans lies altruism [START_REF] Gintis | Strong reciprocity and human sociality[END_REF]. Altruism is defined as the capacity to perform costly acts that confer benefits on others and, hence, it is in principle detrimental for the altruistic individuals. In fact, seemingly altruistic acts can be understood, when looked at in depth, in terms of kin selection [START_REF] Hamilton | The genetical evolution of social behaviour I[END_REF] or reciprocity [START_REF] Trivers | The evolution of reciprocal altruism[END_REF][START_REF] Axelrod | The evolution of cooperation[END_REF], for instance. These interpretations notwithstanding, there are many reports of truly altruistic behavior in humans, in particular in experiments among anonymous subjects that interact only once (see, e.g., [START_REF] Kagel | The Handbook of Experimental Economics[END_REF]; [START_REF] Camerer | Behavioral Game Theory[END_REF]; [START_REF] Fehr | The nature of human altruism[END_REF]; Henrich et al. (2004) for reviews).

A framework that has proven itself particularly suited to explore this issue is the Ultimatum game [START_REF] Güth | An experimental analysis of ultimate bargaining[END_REF]. The Ultimatum game consists of two players deciding how to distribute a reward. One of the players, known as proposer, makes an offer and the other player (the responder ) decides whether to accept it or not. There is only one opportunity to make a deal: If the responder accepts, the reward is distributed between the players according to the proposer's offer, otherwise both players receive nothing. Rational players's behavior is easy to predict: Given that any positive amount should be acceptable for the responder, the proposer will offer such minimum quantity, which will be accepted. On the contrary, a wealth of experimental evidence about the Ultimatum game [START_REF] Camerer | Behavioral Game Theory[END_REF]Henrich et al., 2004) has established clearly that humans are universally altruistic, in so far as subjects from ethnic groups all over the world show some degree of fairness: Indeed, most proposers offer a fair share, mean offers spanning the range from 25 to 57 percent of the amount to be splitted. Interestingly, it is also observed often that unfair offers (typically below 20 percent) are rejected many times, although there is larger variability in this respect than in the proposers' behavior.

A great deal of research has been devoted to understand the origins of this behaviour. One of the factors that has been proposed to play an important role in this issue is culture (see, e.g., [START_REF] Gintis | The hitchhiker's guide to altruism: Gene-culture coevolution and the internalization of norms[END_REF]; [START_REF] Hammerstein | Genetic and Cultural Evolution of Cooperation[END_REF]; [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF]Henrich et al. (2004); [START_REF] Marlowe | More 'altruistic' punishment in larger societies[END_REF]) but other works point out the need to consider the influence of genetic or biological features [START_REF] Wallace | Heritability of ultimatum game responder behavior[END_REF][START_REF] Burnham | High-testosterone men reject low ultimatum game offers[END_REF]. Furthermore, instances of fair behavior related to those arising on the Ultimatum game have been reported among non-human primates [START_REF] Brosnan | Monkeys reject unequal pay[END_REF]Kappeler and van Schaik, 2006) (note, however, that rational-like behavior has also been found in chimpanzees [START_REF] Jensen | Chimpanzees are rational maximizers in an ultimatum game[END_REF]). In fact, the Ultimatum game has indeed very direct biological applications beyond the context of human behaviour. This is the case, e.g., of the division of parental care between the sexes, when mate desertion has a positive effect on the deserting individual's reproductive success [START_REF] Hammerstein | Games and markets: Economic behaviour in humans and other animals[END_REF]. Another situation with Ultimatum game characteristics arises when individuals need to divide a reward or a prey, such as in cooperative hunting. Thus, among chimpanzees dominant males, who did not necessarily participate in the hunt itself, tend to end up with the prey and control its distribution [START_REF] Boesch | Cooperative hunting in chimpanzees[END_REF]. Even outside of the realm of primates, sharing rewards or food may lead to contexts related to the Ultimatum game, such as when fish distribute among patches to feed [START_REF] Milinski | Competitive resource sharing: An experimental test of a learning rule for esss[END_REF]. Therefore, even if culture may have contributed to the appearance of altruistic behaviors in humans, these are likely to have complex causes, in which more biological considerations should enter as well, and more so if we think of non-human animals.

In view of the above comments, explanations of non-rational actions in the Ultimatum game in evolutionary terms are very relevant. Several such mechanisms have been proposed in the past few years. Thus, it has been argued that if selection is rapid, i.e., if reproduction takes place at a faster scale than interaction among individuals, altruistic behavior may be evolutionary selected [START_REF] Sánchez | Altruism may arise from individual selection[END_REF]. Another possible scenario leading to large offers in the Ultimatum game is the existence of empathy (Page andNowak, 2000, 2002), in which players choose as their offer the smallest amount they are willing to accept. Finally, elaborating on ideas proposed by Nowak and May for the Prisoner's Dilemma [START_REF] Nowak | Evolutionary games and spatial chaos[END_REF], it was suggested that the existence of a spatial structure could support the appearance of altruistic behavior. The original proposal was done by Page et al. (2000), who carried out a comparative study of the Ultimatum game on a well-mixed population, a one dimensional system (a ring), and a sketchy analysis of two-dimensional lattices. They concluded that large offer levels, around a 34% of the amount to share, emerged and were stable in the ring and the lattice, while evolution converged to the rational solution on a well mixed population. A similar result was later shown for a generalization of the Ultimatum game to describe collaborations by [START_REF] Killingback | Spatial ultimatum games, collaborations and the evolution of fairness[END_REF]. More recently, the effect of complex (non-spatial) networks of interactions was addressed by [START_REF] Kuperman | The effect of topology on the spatial ultimatum game[END_REF]; [START_REF] Eguíluz | Critical behavior in an evolutionary ultimatum game with social structure[END_REF]; [START_REF] Sinatra | The ultimatum game in complex networks[END_REF], finding again that large offers appeared in the population.

In this paper we revisit the issue of the effect of the spatial structure on the Ultimatum game by focusing on the two dimensional case. We largely expand the only previous work on the subject (Page et al., 2000) by considering different dynamical rules (possibly with noise), including the possibility of empathetic players, and presenting detailed analytical arguments that allow for understanding the mechanisms at work in the evolution of the strategies.

Model

We consider a set of agents playing the spatial Ultimatum game, i.e., they are located on the nodes of two-dimensional square lattices and play the game only with their neighbors. Most results have been obtained for connectivity equal to 4 (players connected to their neighbors north, south, east and west) and size 40x40. For comparison, some other simulations have been performed with network size 10x10 or larger connectivities. We will mention on this different parameters as needed along the paper.

Without loss of generality, the total reward will be hereafter normalized to the unity. The strategy of every player is then characterized by two parameters, p, q ∈ [0, 1]. The value of p indicates the fraction of the reward offered by the player when acting as proposer. On the other side, q denotes the acceptance threshold, i.e. the minimum quantity that the player accepts when acting as responder. We will study two different possibilities for this two parameters: A empathetic setting, in which players offer the minimum amount they are ready to accept, so p = q and strategies are defined by a single parameter, and an independent setting in which both parameters evolve independently. With this notation, an interaction between players i and j, with i taking the role of proposer and j the responder, takes place as follows (ΔΠ O ij and ΔΠ R ji denote the increment of payoff for i and j, respectively):

ΔΠ O ij = 1 -p i if p i ≥ q j 0 ifp i < q j (1) ΔΠ R ji = p i if p i ≥ q j 0 if p i < q j (2)
Every time step, the lattice is swept and every player participates in an Ultimatum game with each of her neighbors. Regarding the assignation of roles (proposer and responder) between the players, two settings have been considered: Non-random (nR), in which the focal agents plays as proposer (and therefore she plays as responder when her neighbors are the focal agents), and Random (R), in which roles are assigned randomly in every encounter, so that the same individual can act twice as proposer or responder. In the latter case, the expectation for the payoff increment of i after a double interaction with j is

E(ΔΠ ij ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 -p i + p j if p i ≥ q j and p j ≥ q i 1 -p i
if p i ≥ q j and p j < q i p j if p i < q j and p j ≥ q i 0 i fp i < q j and p j < q i

(3)

In case non-random roles are assigned, the previous expectation becomes the exact payoff after a double (i, j) interaction. The final payoff is the sum of the increments after playing with all neighbors. Subsequently to the games, players update their strategies. Each individual compares its final payoff with that of its neighbors and modify its parameters (p i , q i ) according to one of the following two update rules. The first one, unconditional imitation (UI), also known as "imitation of the best", stipulates that each player copies the strategy of the neighbor with highest payoff, provided it is greater than the player's. As a result, this is a deterministic rule. This update procedure was first proposed by [START_REF] Nowak | Evolutionary games and spatial chaos[END_REF] and has been widely used in the literature since then. The second rule, proportional imitation, is defined by having each player i select one neighbor j at random. Then, provided that the neighbor's payoff is greater, her strategy will be copied with probability

P ij = Π j -Π i 2 max{k i , k j } (4)
where k i and k j are the number of neighbors of i and j respectively. The denominator takes this form in order to ensure that P ij ≤ 1. This update rule is also known as "replicator rule", since in the thermodynamical limit it results in the replicator equation, as shown by [START_REF] Helbing | Interrelations between stochastic equations for systems with pair interactions[END_REF]; [START_REF] Schlag | Why imitate, and if so, how? A boundedly rational approach to multi-armed bandits[END_REF]. Again, this rule is a common choice in research on evolutionary games. In addition, to address the effect of noise or mistakes, we have considered two scenarios, with or without noise in the copy of parameters (Noise and No-noise, abbreviated as N and nN ). Noise has been implemented by adding to the copied parameters a random value uniformly distributed in [-0.001, 0.001] (not necessarily the same for p and q). The choice of the above rules is due to the fact that they have not been considered in the literature about the spatial Ultimatum game. Indeed, Page et al. used a Moran-like process [START_REF] Moran | The Statistical Processes of Evolutionary Theory[END_REF]Roca et al., 2009b) in which all neighbors try to take over the focal one and succeed with probability proportional to their payoffs. Therefore, it is important to consider additional update rules that can add to our knowledge on this problem as it has been shown that in 2×2 games, they can lead to very different outcomes of the evolution (Roca et al., 2009b,a).

Alternative

Possible settings Abbrev.

Parameters p,q Empathetic / Independent p = q / p,q Role assignment Random / Once prop. The final stage of a time step is that all payoffs are reset to zero. Therefore, there is neither memory nor information about neighbors strategies in the model here presented. Simulations were run for a maximum of 4 • 10 4 generations, stopping before if equilibrium was reached. A typical outcome of single simulations was the convergence of all the population to the same strategy, with all p and q values located within a narrow interval corresponding to the magnitude of noise. In order to obtain frequency distributions for that final strategy averages were taken for 10 4 independent realizations. A summary of the different conditions explored in this work is given in Table 1.

Results: Empathetic players

As stated above, this setting is characterized by the fixed relationship p = q. On a well-mixed population, the constraint that p = q leads to offers close to one half of the total reward (p = 1/2) [START_REF] Page | Empathy leads to fairness[END_REF][START_REF] Sánchez | Altruism may arise from individual selection[END_REF]. In other words, empathy promotes fair behaviors. Our aim here is to contrast this result when players interact no longer with all the population but with a small, highly structured neighborhood.

The main result of our simulation is that for almost all the variations of our model, the population converges towards the well-mixed population behaviour, that is p = 1/2 (Fig. 1b). In particular, this is the case as soon as any source of noise is introduced, either through random role assignment, proportional imitation or noise in the copy of parameters. Due to finite size effects, the final value in small populations (size 10x10) can be rather smaller than 1/2 if no noise is introduced, as will be explained below. Nevertheless, the convergence of the population to a unique offer is a general feature. The only exception to the above outcome takes place for the setting {UI/nR/nN}, when there is no source of stochasticity in evolution. In this case, many different values of parameter p coexist in the population in a stable way (Figure 1a). From a spatial point of view, stable clusters of individuals with the same p are formed that can neither invade nor be invaded by their neighbors. Thus, coexistence is a result of the spatial structure of the population: In fact, we checked that if the UG takes place on random networks (even with homogeneous degree), the coexistence of offers is lost.

In order to understand why random roles, proportional imitation and noise prevent coexistence and lead to the well-mixed result, we will follow Page et al. (2000) and consider a 3x3 cluster (or, alternatively, cross-like configurations) of individuals with a different strategy than that of the surrounding population. By comparing the payoff of every player it is possible to predict (exactly in the deterministic setting and approximately in the stochastic ones) what will happen to the cluster. Let p pop be the offer parameter of the population and p cl that of the players in the cluster. Let us denote by a, b, c, d the players involved in the interaction cluster-environment, with d at the corner of the cluster, c at the lateral of it and a,b outside the cluster in contact with c and d, respectively, as shown in Fig. 2. We will consider several cases separately in what follows, in order to assess the effects of the different types of stochasticity we are studying.

Cluster analysis: Deterministic case {UI/nR/nN}

We will begin by analyzing the deterministic situation, i.e., when there are no sources of noise in the model. First, consider a cluster of players whose p is further from 1/2 than the surrounding population, p cl < p pop ≤ 1/2. When an individual plays with other of the same type two deals are made (as proposer and responder) and both players get payoff equal to 1. However, since p cl < q 1 , distinct players at the interface only accept the deal when the player in the cluster acts as responder. According to Eq. 3, the payoffs for every player are (see Fig. 2a)

Π a = Π b = 3 + (1 -p pop ) ≥ 3.5 Π c = 3 + p pop ≤ 3.5 Π d = 2 + 2p pop ≤ 3
Note that the individual at the centre of the cluster, as well as those inside the surrounding population get a total payoff equal to 4. Evolution by unconditional imitation consist on looking for the neighbor with greatest payoff and imitating it. For players a and b the richest neighbor is that with payoff 4 inside the population, so their strategy will remain the same. Player c is in contact with the centre of the cluster, whose payoff is equal to 4, so it will not change either. Finally, the neighbors of player d are b (richer) and c: Imitation of b will change the strategy of d to the majoritary one. As a result, a 3x3 cluster with p cl < p pop becomes a cross-shaped structure. In the cross cluster, all individuals are connected to the central one, who still has payoff equal to 4. Therefore, the resulting cross cluster is stable.

Always within the deterministic dynamics, consider now the case when individuals in the cluster have greater p than the surrounding population,

p pop < p cl ≤ 1/2. Now the payoffs become Π a = Π b = 3 + p cl ≤ 3.5 Π c = 3 + (1 -p cl ) ≥ 3.5 Π d = 2 + 2(1 -p cl ) ≥ 3
Players a, b and c are in contact with individuals of their same type with payoff 4, so their strategy will not change. Player d best neighbor is c, that also belongs to the cluster. Hence, the 3x3 is a stable configuration. In the same way, it can be shown that a cross structure is also stable. The conclusion of this analysis is that small clusters can neither invade nor be invaded, no matter if their p is greater or lower than that of their surroundings. In consequence, coexistence is the expected outcome in the deterministic setting.

Cluster analysis: Unconditional imitation with noise {UI/nR/N}

We can now turn to the discussion of the effect of noise. When there is noise in the system, there will be no more individuals with exactly the same value of p (note that q is fixed for every player equal to his p). Even if an individual plays with other of the same type, their parameters will still be slightly different and as a result only one deal will be finalized (that in which the player with greater p acts as proposer). On average, when a player interacts with others of the same type half of the deals will be made as proposer and half as responder; hence, the expected payoff will be p i /2 + (1p i )/2 = 1/2. This result is in contrast with the case without noise, where players of the same type received a payoff of 1. This is the main difference between both settings and has strong consequences in the evolution of offers.

In order to apply the same cluster analysis as before, we will approximate the payoff between players of the same type by means of the above expected value. This will give an idea of what will occur on average. Now the total payoff in the centre of the cluster, as well as inside the population, is 2. As before, let us begin with with lower offer than the surroundings, p cl < p pop ≤ 1/2. In the absence of noise, the stable configuration for that cluster is the cross structure. Let it be the starting shape of the cluster, the payoffs for every player are (see Fig 2b)

Π a = 3/2 + (1 -p pop ) ≥ 2 Π c = 1/2 + 3p pop ≤ 2 Π d = 1 + 2(1 -p pop ) ≥ 2
According to these payoffs, player c will imitate a or d and only the central individual in the cluster will resist. This remaining individual will dissapear in the next generation. Results are the same if a square cluster is considered.

Now suppose a single individual with parameter p cl greater than the rest of the population (p pop < p cl ≤ 1/2). Her total payoff will be then 4(1p cl ) ≥ 2 while the payoff of her neighbors will be 4p cl ≤ 2. In consequence, this strategy will spread and give rise to a cross cluster (Fig 2b). Payoffs in the next generation are

Π a = 3/2 + p cl ≤ 2 Π c = 1/2 + 3(1 -p cl ) ≥ 2 Π d = 1 + 2p cl ≤ 2
Both players a and d imitate c and the cluster with greater p continues growing. These results show that a strategy closer to the mean-field solution, p = 1/2, is able to invade the population if noise is present. No coexistence is possible in this setting. The underlying reason is the intrinsic weakening of clusters when its members can not take exactly the same p.

Cluster analysis: Random roles {UI/R/nN}

When roles are randomly assigned, the same player can act twice as proposer or as responder when interacting with the same neighbor. As a result, stochasticity is introduced in the interaction between players of any type. The probability of acting twice as proposer, as well as twice as responder, is 1/4, hence the expected payoff for a given interaction is which coincides with the case when roles are not random. In turn, the particular payoff increment for a concrete interaction can be quite different to this expectation, specially if the players are of different type (note, for instance, that if the player with lower p acts twice as proposer the payoff increment for both players is zero). As a result, the boundaries of a cluster will fluctuate almost randomly. To study the effect of random roles the same cluster analysis than before will be used. Now we will pay attention to a stable cross cluster when it undergoes a boundary perturbation (Fig. 3). The subsequent evolution of the cluster will be computed by means of expected payoffs, so it can be seen as the average behavior of the perturbed cluster.

E(ΔΠ i ) = 1 4 2 ΔΠ O i + 1 2 ΔΠ O i + ΔΠ R i + 1 4 2 ΔΠ R i = ΔΠ O i + ΔΠ R i
Let us consider first the case p cl < p pop ≤ 1/2 (cluster further from 1/2 than the rest of the population). Suppose a new individual is added at one edge of the cluster (position a). The payoff for a neighbor outside the cluster is Π b = 3 + (1p pop ) ≥ 3.5, whilst for the neighbor inside Π c = 2 + 2p i ≤ 3 (payoff for a being lower than both). Thus, player a imitates d and the cluster reverts on average to the cross shape. On the other hand, if one of the arms of the cross is removed (position a) the updated payoff at that position will be Π a = 3 + (1p pop ) ≥ 3.5, whilst for the player previosly in the middle of the cross Π c = 3 + p i ≤ 3.5 and for the remaining arms Π d = 1 + 3p i ≤ 2.5. As a consequence, players c and d will imitate its neighbors outside the cluster and the cluster will dissapear.

In the opposite case, p pop < p cl ≤ 1/2. When the cluster is enlarged at one arm, the player at that position will get Π a = 1 + 3(1p cl ) ≥ 2.5, his neighbor outside the cluster P i b = 3 + p cl ≤ 3.5 and the neighbor inside P i c = 2 + 2(1p cl ) ≥ 3. Provided that p cl > 1/3 player a imitates b and the cross shape is recovered. On the contrary, if an arm is removed (position a) payoffs will be

Π a = Π b = 3 + p cl ≤ 3.5 Π c = 3 + (1 -p cl ) ≥ 3.5 Π d = 1 + 3(1 -p cl ) ≥ 2.5
Since players a and b have neighbors outside the cluster with payoff 4, this T-configuration proves to be stable. However, removal of any of the remaining elements by further perturbations would result in its extinction. Hence, the T-cluster should be considered as metastable.

We have thus seen that a majority of players with strategy close to p = 1/2 is able to invade small clusters of players with lower p. On the other hand, a small cluster of players closer to p = 1/2 can hardly invade a population with lower p and boundary perturbations will make it eventually dissapear. In conclusion, coexistence is not the expected outcome when roles are randomly assigned. Moreover, metastability gives advantage to players with higher p when small clusters of similar size interact. This explains why the most frequent (but not the only one) outcome is fixation of strategies close to p = 1/2.

Cluster analysis: Proportional imitation {P I/nR/nN}

Finally, let us analyze the case in which the source of noise is the choice of the neighbor to copy. Under proportional imitation, a player can adopt the strategy of any of her neighbors with payoff greater than own. As a result, given a certain configuration several transitions will be possible with different probabilities. To make things easier, we will take a qualitative approach and consider only which configurations are reachable from a given one. Note that transitions which require a player to imitate the strategy of a neighbor with lower payoff are not permitted. Payoffs can be calculated for every configuration as before, and moreover they will be exact if neither noise nor random roles are considered.

The possible dynamics for a 3x3 cluster are shown in Fig. 4, where thick dark arrows are used when there is only a unique permitted transition (evolution is deterministic in that cases). Starting with a cluster of players with p cl < p pop ≤ 1/2 the cluster will fluctuate between 3x3 and cross configuration, but eventually will reach T-configuration and subsequent extinction. On the other hand, if p pop < p cl ≤ 1/2 both extinction and spreading can occur.

In conclusion, when proportional imitation operates no coexistence in the long term will be observable. Fixation is prone to occur for strategies with p close to 1/2 since small clusters of players of this type can spread (with certain probability) into a population with lower p but not the opposite.

Size and connectivity effects

In order to shed further light on the spatial Ultimatum game, we have also studied the effect of the system size and the connectivity of the lattice. A summary of the results for simulations with different settings and network sizes is given in Table 2. Finite size effects can be observed in small networks without noise, while the results become size independent when noise is added. This is due to the depletion of individuals with high p at early steps of the evolution: starting with a population with random p's the maximum mean payoff does not correspond to individuals with p = 1/2 but those with p = 1/3 [START_REF] Sinatra | The ultimatum game in complex networks[END_REF] removed in small populations and cannot be recovered if noise is absent. As one can expect, this effect becomes specially relevant as connectivity increases and the real payoffs approach the well-mixed limit (for instance, mean final offer reduces from 0.397 to 0.321 when increasing connectivity from 4 to 8 neighbors, setting {UI/R/nN}). One exception to this rule would be the setting {UI/nR/nN}. Since coexistence is allowed for this setting the final configuration gets fixed early in the evolution. As a result, the mean final offer is close to the best one at initial steps, p ≈ 1/3. Another peculiar feature dependent on size is observed for setting {UI/-R/nN} (Fig. 1c). If the system is large enough a fraction of realizations ends with the fixation of very low offer strategies. Alternatively, for small networks all final fixed offers are distributed around the mean. This phenomenon also manifests itself in the standard deviation of the final p distribution (Table 2), which increases dramatically with system size. The reason for this behavior is not obvious, but we believe that it is related to the survival of low p clusters at initial steps of the evolution, in certain places of a large network and under favourable initial conditions. As it was seen before for this setting, once clusters are big enough their dynamics become dominated by random fluctuations, what gives a chance for fixation of that low p clusters.

Results: Independent thresholds

In the second part of our simulation program, we study the spatial Ultimatum game when players can independently choose their offers and their acceptance thresholds. The rational game theory prediction for this situation is the following: Given that the responder always receives greater payoff by accepting a deal than by rejecting it, the acceptance threshold q will tend to zero and all offers will be accepted. As a consequence, proposers will offer the smallest amount possible. In conclusion, game theory predicts that both parameters, p and q, will tend to zero. This is indeed the observed outcome for the evolutionary Ultimatum game in a well-mixed population (Page et al., 2000). In this section, we will see how this result changes on a spatial setting. As in the previous section for empathetic players, here we will explore the stationary outcomes of the evolution when different evolutionary rules are considered. For this purpose, the comparison with the much simpler case of empathetic players will give insight into the causes behind the observed differences.

Variations on the evolutionary rules give four qualitatively different outcomes for the game. Representative distributions of parameters p and q are depicted in Fig. 5 where we have verified that the simulations have indeed converged to an asymptotically stationary state. A summary of the results for different settings and networks is collected in Table 3. We can classify the simulation results in four different types:

Coexistence of offers: As in the empathetic case, coexistence of markedly different strategies is only observed in the setting {UI/nR/nN} (Fig. 5a). In turn, any source of stochasticity in the evolution (random roles, proportional imitation, noise) leads to an homogeneous final population. Coexistence of offers depends on the formation of stable spatial configurations, and therefore it cannot be observed if the game takes place on random networks (even with homogeneous degree).

Rational (p, q ≈ 0): In a first stage, parameters p and q concentrate around well defined values far from zero. Afterwards, both evolve gradually towards zero (Fig. 5b). This behavior is only observed in the setting {UI/nR/N}, i.e. when noise is introduced into an otherwise deterministic setting. Proportional imitation and random role assignment prevent the population from reaching the rational solution.

Bell-shaped p > 0 with q close to zero: Offer parameter p takes a bell-shape distribution with values greater than zero. On the other side, distribution of the acceptance threshold q takes significant values at zero (Fig. 5c). That means that in some realizations the population fixes the rational value for q. This outcome is typical of small networks when either noise or random roles are absent (excluding the aforementioned settings with rational and coexistence outcomes). Specifically it can be observed in 10x10 systems with settings {UI/R/nN}, {P I/R/nN}, {P I/nR/N} and {P I/nR, nN}. Bell-shaped "quasiempathetic" (p ≈ q > 0): The population takes p and q greater than zero, with q just slightly below p (Fig 5d). The distance between p and q is typically on the order of 0.02, so the final population resembles an empathetic one. This is the most frequent outcome, that appears in large networks (40x40) whenever proportional imitation or random roles are taken into account (with or without noise). It also appears in small networks if noise and random roles are combined (independently of unconditional or proportional imitation).

An analytical argument for the case of quasiempathetic behavior

Among the four different types of behavior summarized above, the most common outcome is quasiempathetic behavior, namely the fixation of a strategy with q very close to p and both greater than zero. Table 3: Average final offers (p), acceptance thresholds (q) and its standard deviations (σ p and σ q respectively) for different settings and system sizes. High dispersion in setting {U I/nR/nN } is due to stable coexistence of several strategies in the population. In all the other settings a unique final offer is reached in every realization. Quasiempathetic outcome in large networks produces similar values for p and q and small standard deviations, while rational outcome in {U I/nR/N } leads to p and q close to zero.

result from rational prediction, we believe it is worth to analyze it in detail. Our explanation of this behavior proceeds in two stages: Firstly, we will deal with the disappearance of rational players in the population after the first time steps. Subsequently, we will discuss the efficiency of a quasiempathetic strategy when evolution is driven by local interactions.

To begin with, we have observed in our simulations the remarkable fact that rational strategies (p and q close to zero) are no more found in the population after the first time steps, despite they can be present in the starting random configuration. Dissapearance of rational strategies takes place not only in small 10x10 networks but also in larger 40x40 ones. One simple reason for this phenomenon can be found by looking at the average payoff of a rational player in a well-mixed population. Assume that the parameters of the players in the population are uniformly distributed in [0, 1], as the initial condition in our simulations. Since the expected payoff for a player with strategy p = q = 0 is equal to the average p in the population, we have that Π (0,0) = 1/2, which in turn coincides with the average payoff in the population. This means that there will be strategies better than the rational one and that it will be suppressed when individuals copy more successful ones. Note that if one considers the values for p and q uniformly distributed in [0, 1/2], the expected payoff for a rational player becomes Π (0,0) = 1/4, while the average payoff in the population remains equal to 1/2. As a result, selective pressure against rational strategy is strong and it will dissapear easily even in large populations.

Interestingly, a straightforward consequence of this phenomenon is that noise is necessary for a rational outcome to occur. This is just because only noise can recover individuals with low p and q later in the evolution. Since changes introduced by noise are very small, evolutionary conditions must be such that gradual evolution through small changes is possible. As we will see later, this is not the case if random roles or proportional imitation are introduced.

Having dealt with the early disappearance of rational players, we will now focus on explaining why quasiempathetic strategies are successful in the spatial UG. Our approach relies on the fact that, when spatial structure is present and connectivity is low, competition among players takes place at a local level. Accordingly, a successful player is one that defeats its neighbors, no matter how optimal his strategy is from a whole population point of view. Consider the following example: a player with q = 0 will accept every offer of his neighbors, whose p's we will assume lower than 1/2 and q > 0. As a result, that player will obtain payoff p in an interaction, while his neighbor will receive 1p > p. This gives a payoff advantage to the neighbor, so if the game took place only between these two players the rational strategy would be beaten. It is clear that this argument loses applicability when the number of neighbors increases and the final payoff becomes less and less dependent on the result of a single interaction, so that in the limit of a large, totally connected population the optimal strategy is indeed the rational one. Nevertheless, for the square lattices here considered, it can be assumed that local competition between pairs of players is relevant. Accordingly, a "one against one" approach should give a good aproximation of the processes that drive evolution in these populations, and we will stick to it in what follows, taking back the issue of multiple neighbors afterwards.

Firstly, assume that q ≤ p ≤ 1/2 for all players in the population. It is an observable fact that this condition becomes fullfilled very soon in the evolution. Let us focus on player i and define the distances a = 1/2p i , b = p iq i and c = q i . neighbor j's strategies can be classified according to the order relationships between their thresholds p j , q j and those of player i (see Fig. 6). We now want to find the optimal combination for p i and q i in order to achieve greater payoff than one's neighbor, provided that players lack any knowledge about their neighbors.

Player i receives lower payoff than her neighbor j in three cases: If both thresholds p j and q j are (1) greater than p i or (2) just between p i and q i , or if (3) p i > p j > q i and q j < q i . Since in the beginning of the game thresholds are randomly assigned according to a uniform distribution, it is reasonable to assume (at initial steps) that the probability of p j and q j belonging to a certain interval is proportional to the length of that interval. Taking into account the above definition of the intervals a, b and c together with the aforementioned constraint q ≤ p ≤ 1/2, the probability that player i loses can be expressed as the function

F (a, b, c) = 4a 2 + 4b 2 + 8bc (5)
On the other hand, player i gets greater payoff than j if (1) both thresholds of j are below q i , (2) p j > p i and p i > q j > q i , or if (3) p j > p i and q j ≤ q i . The same considerations as before for the probability of each combination give the following function for the probability that player i beats his neighbor

G(a, b, c) = 4c 2 + 8ab + 8ac (6)
The best strategy for i must maximize G(a, b, c) and minimize F (a, b, c) subject to the constraints a

+ b + c = 1/2 and 0 ≤ a, b, c ≤ 1/2. By writing b = 1/2 -a -c, function G becomes G(a, c) = 4a -8a 2 + 4c 2 , and it must be maximized in the triangle 0 ≤ a ≤ 1/2 , 0 ≤ c ≤ (1/2 -a).
The solution is a = b = 0, c = 1/2. In a similar fashion, it can be checked that function F reaches its minimum for the same values. The conclusion of this analysis is that optimal strategy in a "one against one" game with the considered assumptions is p = q = 1/2, what resembles the empathetic outcome.

We must now discuss how this result changes if we recall that players interact with more than one neighbor and payoffs are the sum of all that interactions. From a phenomenological point of view, it means that player i can also lose against neighbor j if it performs very poorly in the interaction with the rest of his neighbors. Indeed, this is the case when q i is very high or p i is very low, what results in player i making only one deal in every interaction. It is reasonable to assume that this kind of strategies are undesirable when multiple neighbors are considered, and players that make two deals in the interaction perform better than those leading to only one deal. Admittedly, in a quasiempathetic situation, most individuals will make only one deal per interaction, but we believe that this effect will have less influence than using a strategy that from the very beginning would preclude one of the two exchanges (even for non-quasiempathetic players). Within this assumption, functions F and G are recalculated excluding the two strategies (marked with an asterisk in Fig. 6) for which only one deal is typically accepted. Thus these functions become F (a, b, c) = 4b 2 + 8bc ( 7)

G(a, b, c) = 8ab + 8ac (8)
The solution that maximizes G and minimizes F is a = c = 1/4, b = 0, that corresponds to strategy p = q = 1/4, and once again, quasiempathetic strategies are successful. We stress that we only have explicitly considered interaction between players i and j (local approach), while the effect of having more neighbors is included in the requisite for making typically two deals in every interaction. For comparison with the well-mixed setting, note that this requisite only precludes fixation of strategies that perform very poorly from a global point of view. Although it can seem very simplistic, this approximation is qualitatively consistent with experimental results and provides intuitive explanations to the exceptional observed outcomes. As a final note, observe that local analysis also predict early dissapearance of rational strategies (p = q = 0), as we had already seen above in a well-mixed context.

Discussion of other outcomes

As it has been just explained, two processes take place at the first steps in evolution that determine the fate of the whole population. First, strategies close to the rational one dissapear, as a result of being less fit than the average. Second, quasiempathetic strategies, with q similar to p and both located somewhere between 1/4 and 1/2, take advantage and spread through the population. This is indeed the final outcome in most of the cases, when no further evolution is possible. Nevertheless, differences appear in case unconditional imitation is chosen as the updating rule and roles are not randomly assigned. Finite size effects can also be found under certain circumstances and should be discussed as well Total lack of stochasticity (setting {UI/nR/nN}) results in coexistence of several strategies in the final population. As in the empathetic case, this is due to the formation of spatial clusters that can neither invade nor be invaded, and the same arguments that explained it before are valid here again. Perturbations induced by random role assignment and proportional imitation destroy spatial clusters and make coexistence inviable.

The only setting for which the rational outcome can be observed is {UI/-nR/N}. This setting is the result of adding noise in the copy of parameters into an otherwise deterministic system. In fact, this is the only case in which evolution after fixation of a quasiempathetic behavior can occur. Firstly, it is straighforward to see that once a single strategy has been fixed, noise is necessary in order to let it evolve. Secondly, evolution of quasiempathetic strategies is not easy at all, since it requires a series of gradual processes that are very weakly selected for and, as a result, strongly sensitive to stochastic perturbations. Specifically, evolution from a homogeneous population towards the rational strategy requires the appearance of an individual with q lower than the rest. This individual receives greater payoff than her neighbors and gives rise to a cross cluster (in average, provided that p > 1/5). But the same noise that allowed this individual to appear produces variations inside the cluster that are of the same order of magnitude than the differences between the cluster and the population. As a result, there is not evolution as a cluster, but as individual players that successively appear. This slow, gradual evolution can take place in spatially ordered, almost deterministic systems, but is suppressed as soon as stochastic perturbations (induced by random roles or proportional imitation) are present. It can be therefore concluded that the quasiempathetic outcome is robust against noise, while the coexistence of strategies resulting from deterministic setting {UI/nR/nN} is unstable and evolves to the rational solution if imitation is not exact.

The last alternative outcome consists on a bell-shaped distribution of offers greater than zero with acceptance thresholds close to zero. It only appears in small networks, where it is possible that no quasiempathetic individual exists at the beginning of the simulation. Indeed, we have checked that if a small number of players with p ≈ q ≈ 1/4 are introduced in the starting configuration the standard quasiempathetic distribution is obtained as final outcome. Quasiempathetic outcome is also the natural one in small networks if noise and random roles are combined (settings {UI/R/N} and {P I/R/N}). In those cases random role assignment induces fluctuations that result in slow fixation of strategies. This provides enough time so that noise generates quasiempathetic players.

Regarding the effect of connectivity, results do not suffer major changes when eight neighbors instead of four are considered in the game. Quasiempathetic behavior remains, although the mean final values for p and q decrease slightly (p = 0.309, q = 0.284 in setting {P I/R/N}). This result is in agreement with those observed in different networks (Page et al., 2000;[START_REF] Kuperman | The effect of topology on the spatial ultimatum game[END_REF]. At the same time, as connectivity increases a growing fraction of realizations ends with acceptance threshold close to zero. This deviation from quasiempathetic behavior is easy to explain, provided that higher connectivity reduces local competition (and therefore our local approach above ceases to hold). In any case, connectivity must be considerably higher than eight so that this effect becomes relevant.

Conclusions

We have studied in detail the Ultimatum game on a spatial setting, considering different update rules for the dynamics and introducing noise or randomness effects in several manners. Our simulations show that for a vast majority of settings (combinations of update rules, noise sources and one-or twoparameter strategies) evolution leads to the emergence of altruism, with offers and acceptance thresholds much larger than zero. Thus, empathetic players, who offer their own minimum acceptable amount, almost always evolve to fair offers, with values close to a 50-50 split. The only situation in which this outcome does not arise is when the evolution is fully deterministic, i.e., under unconditional imitation with no noise. Furthermore, when the two parame-ters p and q are let to evolve independently, in most occasions they collapse to very close values for most cases, i.e., quasiempathetic behavior emerges spontaneously. This is a very interesting result because it resembles behaviors often found in experiments, above all regarding the offer (acceptance thresholds are in general significantly smaller). In this respect, let us note that a distribution of offers and thresholds very similar to the experimental ones was found for several combinations of dynamics (mostly for proportional imitation) and noise, but only for small systems. This suggests that the values observed in human populations might have arisen from evolution in small groups where competition is limited. On the other hand, the values we find for p and q in the quasiempathetic case are similar to those found by Page et al. (2000) for a different dynamics although, as we do not have data for the distribution of their values, we can not take the comparison any further. Nevertheless, we believe that the dynamics in that previous work would most likely lead to similar distributions, in view of the role played by noise we have discussed above. In any event, the fact that the mean values are similar speaks in favor of the robustness of the result, in so far as it is only weakly dependent on the dynamics (it changes qualitatively only for special choices of update rules and noise).

Along the paper, we have been able to provide explanations for the simulation results using analytical arguments. Thus, for empathetic players we have shown that using the stability of square and cross clusters one can predict the final outcome of the evolution in the different dynamical scenarios. More importantly, we have also succeeded in developing an approximation to the case of two independent paramters for the strategies. Our results show that the convergence process can be understood in terms of a quick suppression of rational strategies followed, in a local (one or only a few neighbors) context, by the successful emergence of quasiempathetic players. Our approximation predicts that the offers should be between 1/4 and 1/2 of the total amount to share, values which are consistent with those reported in human experiments. This result agrees also with the finding that practically all dynamics lead to the emergence of a unique strategy throughout the population, coexistence being only possible in very few situations. While it is by no means the case that humans have all the same behavior in Ultimatum game experiments, we believe that the range of offers found through research all over the world is not that large as compared to the population. On the other hand, as was discussed in the Introduction, cultural effects could account for the coexistence of different values among cultures, while the individuals' response is very similar within monocultural groups (as is found, for instance, with European and US university students, whose range of variability is much smaller (Henrich et al., 2004).

We believe that the fact that quasiempathetic behavior is the generic outcome of the evolutionary, spatial Ultimatum game, is a relevant and non-trivial one. Indeed, when empathy is imposed on the individuals from the beginning (by setting p = q as a fixed condition), evolution generically leads to fairness with or without a spatial structure, as was shown by [START_REF] Page | Empathy leads to fairness[END_REF]. Therefore, the spatial Ultimatum game does not add anything new. On the contrary, in the case in which the two parameters evolve independently, they first converge to similar levels because of a fast primary evolutionary process and, subsequently, quasiempathetic players retain offers much greater than zero as a consequence of a lack of later evolution. Indeed, if gradual evolution through small changes were effective, as it occurs for unconditional imitation with noise, the population will drift away from a fair split towards the rational solution with almost zero offers. Only this subtle two-stage process leads to the emergence of fairness.

It is also important to put our conclusions in the context of findings on other networks. Our results are in contrast with those obtained by [START_REF] Kuperman | The effect of topology on the spatial ultimatum game[END_REF] on small world networks, as we observe fair behavior while they did not. We believe that the convergence to the rational behavior they observe is due to the destruction of the spatial structure (and hence of clusters) by the long range links. On the other hand, the results on complex networks reported by [START_REF] Sinatra | The ultimatum game in complex networks[END_REF] agree with the ones we are presenting here at least as far as the average values of offers and acceptance thresholds are concerned and for the case of proportional imitation. In their system, however, acceptance thresholds tend to be not so close to offers, and therefore their agents are less quasiempathetic; in any event, the results are qualitatively similar, a remarkable conclusion in view of the differences among the networks studied in both works and of the influence that the type of network has on symmetric 2×2 games (Roca et al., 2009b,a).

To conclude, the picture that is beginning to emerge from our work and the previous ones is that the Ultimatum game on structured population is very robust under changes of the dynamics or the network of connections. The generic result is the appearance of fair behavior, with individuals that are quasiempathetic to some extent, and with little or no variability of strategies within the group. Therefore, as well-mixed populations are hard to find in actual situations, and particularly so within humans, these results suggest that experiments should always show fair behavior because of the structure of the population, be it social or spatial. A word of caveat is in order here regarding the influence of the size of the groups as discussed in the text, that may lead to somewhat different outcomes, including much lower acceptance thresholds.

This must be taken into accout when applying this kind of models to non human animals, particularly primates, where groups are not big and therefore effects as those described could be relevant. This conclusion paves the way to further studies that try to combine biological evolutionary dynamics with cultural influences, as to provide a more complete picture of the emergence of fairness.

Figure 1 :

 1 Figure1: Observed outcomes with empathetic players (p = q). Plots show the final distribution of offers averaged over 10 4 realizations. Two major outcomes are possible: (a) stable coexistence of several offers in the population -only for the deterministic setting-and (b) mean-field behaviour, with p ≈ 1/2. Inset (c) shows a peculiar behaviour that appears in large networks with setting {U I/R/nN } and is characterized by fixation of low p strategies in some realizations.

Figure 2 :

 2 Figure 2: Clusters of players evolve in the absence (a) or presence (b) of noise (unconditional imitation, non-random roles). Coloured players make offers closer to 1/2 than the white ones. Stable coexistence is the expected outcome without noise, while strategies closer to p = 1/2 invade the population if noise is added.

Figure 3 :

 3 Figure 3: Clusters of players evolving under random role assignment suffer continuous boundary perturbations (white thin arrows). In such circumstances, small clusters with p far from 1/2 (white) dissapear, while those closer to 1/2 (coloured) behave as metastable. Dark thick arrows indicate the average progression of a perturbated cluster.

Figure 4 :

 4 Figure 4: Evolution of clusters through proportional imitation (neither noise nor random roles). Coloured players are closer to p = 1/2 than white ones. Arrows indicate permitted trasitions between different configurations. Dark thick arrows are used when there is only one available evolutionary pathway.

Figure 5 :

 5 Figure5: Observed outcomes with independent p and q. Black line corresponds to the distribution of offers p and red line to the acceptance thresholds q, averaged for 10 4 realizations. The four characteristic outcomes are: (a) stable coexistence -only for the deterministic setting-, (b) rational outcome -dotted lines represent distribution at intermediate time steps-, (c) bell-shaped p > 0 with q close to zero -typical of small networks-and (d)"quasiempathetic" behavior -the most frequent outcome.

Figure 6 :

 6 Figure 6: Classification of the possible strategies of neighbor j with respect to the thresholds of considered player i. Each strategy is represented by two open circles connected with a vertical line. Upper circle indicates the value of p and lower circle that of q. Strategies marked with an asterisk are suboptimal, since only one deal per interaction is made.

Table 1 :

 1 Summary of the variations of the model considered in this study.

	-once resp.	R / nR

Table 2 :

 2 . As a result, individuals with p close to 1/2 are usually Average final offers (p) and standard deviations (σ) in the empathetic case p=q. High dispersion in setting {U I/nR/nN } is due to stable coexistence of several strategies in the population. In all the other settings a unique final offer is reached in every realization. See text for an explanation of high dispersion in setting {U I/R/nN } size 40x40.

	setting	size	p	σ	setting	size	p	σ
	{UI/nR/nN} 10x10 0.323 0.127 {P I/nR/nN} 10x10 0.435 0.034
		40x40 0.323 0.127		40x40 0.482 0.008
	{UI/nR/N}	10x10 0.500 0.004 {P I/nR/N}	10x10 0.500 0.001
		40x40 0.500 0.004		40x40 0.500 0.001
	{UI/R/nN}	10x10 0.397 0.064 {P I/R/nN}	10x10 0.434 0.035
		40x40 0.405 0.150		40x40 0.482 0.008
	{UI/R/N}	10x10 0.500 0.004 {P I/R/N}	10x10 0.500 0.005
		40x40 0.500 0.004		40x40 0.500 0.005

Acknowledgments

We are indebted to Susanna C. Manrubia for support and encouragement. We are very thankful to an anonymous reviewer and to José A. Cuesta for helping us finding the correct form of the analytical argument in Sec. 4.1. This work was supported in part by MICINN (Spain) through grants MOSAICO and RESINEE, and by Comunidad de Madrid (Spain) through grant MOD-ELICO. J. I. is supported by a contract from the Consejería de Educación of the Comunidad de Madrid (Spain) and from the European Social Fund.


Helmut Wiesemeyer

Rolf Güsten

Stefan Heyminck

Karl Jacobs

Karl M Menten

David A Neufeld

Miguel Angel

Jürgen Stutzki

M A Requena-Torres

High-resolution absorption spectroscopy of the OH 2Pi 3/2 ground state line

Keywords: ISM: lines and bands, molecules, clouds

Introduction

The first molecule discovered in space was the CH radical (Dunham 1937). At that time, the first predictions were made regarding the abundances of molecules in interstellar space, identifying the hydroxyl radical (OH) as a promising candidate (Swings & Rosenfeld 1937). It took more than two decades until the latter was detected in the ism (Weinreb et al. 1963), in absorption against the supernova remnant Cas A, in the F = 1 -1 and 2 -2 transitions between the hyper-fine structure split Λdouble levels of OH's 2 Π 3/2 , J = 3/2 rotational ground state. Comprehensive models of the gas phase chemistry of diffuse interstellar clouds constructed by van Dishoeck & Black (1986) revealed the importance of the OH radical in the network of reactions leading to the formation of oxygen-bearing molecules. Unfortunately, OH column densities in these objects are difficult to determine from radio lines, due to frequently observed deviations of the underlying level population from LTE (e.g., Neufeld et al. 2002). The highest densities occurring in diffuse clouds amount to ∼10 4 cm -3 (Greaves & Williams 1994), while their mean density is ∼10 2 cm -3 (Snow & McCall 2006;Cox et al. 1988 derived upper limits of a few thousand cm -3 ). Storey et al. (1981) first detected the Λ doublet line from the OH ground state ( 2 Π 3/2 J = 5/2 ← 3/2) towards Sgr B2 near the Galactic centre with the Kuiper Airborne Observatory, but their spectral resolution (250 km s -1 ) proved inadequate to separate the absorption by the line-of-sight clouds from that occurring in Sgr B2 itself. Here we report the detection of one doublet line of this transition (at 2514 GHz) and of its isotopolog 18 OH (at 2495 GHz) with GREAT 1 onboard SOFIA, in absorption towards the giant Hii regions W49N and W51 and the ultracompact Hii region G34.26+0.15. Both lines are inaccessible for Herschel/HIFI. The three observed lines of sight are within a 15 • wide Galactic longitude interval. Those towards W51 and G34.26+0.15 cross the near side of the Carina-Sagittarius spiral arm, with W51 (ℓ = 49. • 5) at a distance of 5.41 (+0.31, -0.28) kpc [START_REF] Sato | [END_REF]) and G34.26+0.15 at ∼2 kpc distance (cf. measurements of G35.10-0.74, which has a comparable radial velocity, Zhang et al. 2009). The line of sight to W49N (ℓ = 43. • 2) first crosses the near side of the Carina-Sagittarius arm, grazes the Crux-Scutum arm, and then again crosses the Carina-Sagittarius arm on its far side, where W49N is located at a distance of (11.4 ± 1.2) kpc (Gwinn et al. 1992).

Observations, data reduction and analysis

The observations reported here were performed with the GREAT receiver (Heyminck et al. 2012) onboard the SOFIA airborne observatory (Young et al. 2012), as part of the basic science programme (flights on 2011 July 26 and November 8). On the first flight the receiver's M and L2 bands were tuned to 2514.317 GHz for the 2 Π 3/2 J = 5/2 ← 3/2 group of OH hyperfine structure lines (in the lower sideband) and to 1837.817 GHz for the 2 Π 1/2 J = 3/2 ← 1/2 lines (in the upper sideband), respectively. On the second flight, the M band was also tuned to the 2494.695 GHz frequency of the 2 Π 3/2 J = 5/2 ← 3/2 transition of 18 OH. Typical DSB receiver temperatures were 4500 K and 2500 K for the M and L2 bands, respectively. Total power subtraction was performed by chopping with an amplitude of 90 ′′ at 1 Hz. The raw data were converted from XFFT spectrometer (Klein et al. 2012) count rates to forward-beam brightness temperatures with the module kalibrate (Guan et al. 2012) as part of the kosma software observing software, analysing the data from the calibration loads and the atmospheric total power and allowing us to fit both the wet component (typically pwv=10-20 µm) and dry content of the atmospheric emission and thus to determine the opacity correction (a few ×0.1). Further processing of the data (conversion to main-beam brightness temperature, with a beam efficiency of 0.58, and averaging with 1/σ 2 rms weighting) was made with the class software. The overall calibration uncertainty does not exceed 20%.

Thanks to the high critical density of the J = 5/2 → 3/2 transition (5.1 × 10 9 cm -3 for a 15 K gas, collision coefficients from Dewangan et al. 1987, Einstein coefficients as given in Tab. 1), we can safely expect almost all OH to be in its ground state at the density of the foreground diffuse clouds along the sight-line. This makes the determination of column densities much more reliable than those derived from the λ 18 cm line. Likewise, the 2 Π 1/2 J = 1/2 level (64 K above the 2 Π 3/2 J = 5/2 level) cannot be substantially populated either, because we failed to detect absorption in the simultaneously observed 2 Π 1/2 J = 3/2 ← 1/2 transition. The spectral profile, in absence of emission, is thus is given by (b) fwhm of Gaussian absorption profile, deconvolved from hfc split. (c) Peak opacity in the strongest hfc. The saturated absorption towards W49N (υ lsr < 25 km s -1 ) provides only a lower opacity limit. (d) Column density per fitted velocity component.

T mb (υ) = T mb,c exp         - N vc i=1 N hfc j=1 τ ij,υ         , (1)
where T mb and T mb,c are the main beam brightness Rayleigh-Jeans temperatures of the spectral profile (here as a function of velocity) and of the continuum (in single-sideband calibration), respectively, and N vc and N hfc are the number of velocity components and hyperfine components, respectively. While uncertainties in the calibration temperature cancel out in the opacity determination, any residual offset of variance σ 2 rms,Tc in the definition of the continuum level leads to an additional uncertainty in the derived opacity of σ rms,τ = σ rms,T c /T c . The absorption spectra suggest σ rms,τ ∼0.1, which is tolerable in view of the substantial opacities. A simultaneous least-squares fit to the line profiles of all velocity components (Eq. 1) with the opacity

τ ij,υ = ln 2 π A E, j c 3 4π∆υ i ν 3 j g u,j g l,j N OH w Λ exp        -4 ln 2 υ -υ 0,i j ∆υ i 2        (2)
yields N OH , the OH column density per velocity component.

Here ∆υ i is the fwhm of the Gaussian component i, g u,j and g l,j are the statistical weights (2F + 1) of the upper and lower level, respectively, of a given hyperfine component j, υ 0,ij is the offset of its velocity from the line-of-sight velocity of the source, and w Λ = 0.5 corrects for the fact that only one doublet line was observed (Fig. 1). Owing to the relatively large number of free parameters (3N vc ), a simulated annealing method (Metropolis algorithm) was used in combination with a downhill simplex method [START_REF] Press | Numerical Recipes[END_REF]). The former assists the minimisation process in escaping from a local minimum, and the latter improves the efficiency of the convergence. The velocity structure in the para-H 2 O 1 11 -0 00 spectrum towards W49N (Sonnentrucker et al. 2010) suggests five velocity components as a strict minimum, while with more components, the procedure would start to fit noise features. The results are summarised in Tab. 2. The OH column density can be expressed by the relationship N OH [cm -2 ] = 7.8 × 10 12 τ max ∆υ fwhm [km s -1 ] as a function of the opacity in the strongest hyperfine component and of the width of the absorption profile of the deconvolved spectrum. The uniqueness of the solution was tested with a Monte Carlo study, yielding the standard deviation of each parameter. 2012). (c) The sensitivity of the OH observations is not sufficient for this velocity interval.

Results

For the spiral arm clouds there is no ambiguity in the fit results (Fig. 2). Towards the three continuum sources the absorption is saturated, i.e., in the (5, 20), (40, 80) and (35, 70) km s -1 velocity intervals for W49N, W51 and G34.26+015, respectively, where the derived main line opacities and column densities are to be considered lower limits. For W49N, this caveat is corroborated by the 18 OH absorption profile (Fig. 3). The 18 OH/ 16 OH abundance ratio is not expected to be affected by chemical fractionation (Langer et al. 1984). The synthesised opacity in the W49N spectrum peaks at τ= 5.7. Assuming for the 18 O/ 16 O ratio the value in the 4 kpc ring (327 ± 32, Wilson & Rood 1994;Polehampton et al. 2005 found no evidence of an abundance gradient with galactocentric distance), the estimated opacity in our 18 OH detection (τ∼0.2) would require the main line opacity to be higher by at least an order of magnitude with respect to that estimated by the absorption profile fit. Unfortunately, in the 30-40 km s -1 velocity range the 18 OH absorption is affected by a telluric ozone feature, and a confirmation by observations of sources with a more favourable velocity is planned to definitely rule out a baseline ripple. In the unsaturated wings of the absorption profile, the sensitivity of the corresponding 18 OH measurement is no longer sufficient to estimate the 18 OH/OH abundance ratio. A two-component fit to the 18 OH absorption (Fig. 3) yields a column density of 4 × 10 13 cm -2 for the whole absorption feature. Although the spiral arm clouds along the sight-line exhibit substantial opacities on the order of unity, the absorption is not saturated and OH column densities can be derived whose accuracy only depends on the signal-to-noise ratio, the quality of the fitted profile, and the assumed continuum level. For a comparison of our OH column densities with those observed for H 2 O (Sonnentrucker et al. 2010) and inferred for H 2 (Godard et al. 2012), the absorption profiles of the clouds towards W49N and W51 are integrated within the velocity intervals of Tab. 3. With the exception of a velocity interval with an abnormally low water abundance, probably due to a spectral baseline problem, the H 2 O/OH ratios are in the range 0.3-1.0. Plume et al. ( 2004) have determined the H 2 O/OH ratio by a comparison of submillimeter H 2 O observations with ground-based radio observations of the 18 cm transitions within the ground rotational state of OH. They thereby estimated a H 2 O/OH ratio of 0.4 at υ lsr =68 km s -1 , in agreement with our measurement of 0.6. The ratio of ∼0.3 measured by Neufeld et al. (2002) towards W51, at υ lsr =6 km s -1 , compares to our value of 0.4 in the υ lsr =(-1,11) km s -1 interval. Their OH column density is compatible with our value (8 × 10 13 cm -2 and 6.4 × 10 13 cm -2 , respectively). Observations of the 2 Π 1/2 ← 2 Π 3/2 cross-band transitions towards Sgr B2 (Polehampton et al. 2005) suggest a H 2 O/OH range of 0.6-1.2.

Generally, discrepancies between different sets of data may be due to nlte effects in the radio lines, different spectral resolutions and definitions of velocity components, and uncertainties in the definition of the respective continuum levels. The chemistry leading to interstellar OH and water has been considered in many theoretical studies over the past thirty years (e.g. Draine et al. 1983; van Dishoeck & Black Fig. 3. 18 OH absorption (top, grey-shaded), a least-squares, twocomponent fit to it (dashed line) and OH absorption towards W49N. The spectra are scaled by the corresponding continuum level, to facilitate a comparison. The insert at the top shows a telluric ozone feature (as observed in total power), where the calibration is more uncertain.

1986; Hollenbach et al. 2009, and references therein) The branching ratio for this process is important in determining the resultant OH/H 2 O ratio and has been studied in two recent ion storage ring experiments (Jensen et al. 2000;Neau et al. 2000): these suggest that ∼74% to 83% of dissociative recombinations lead to OH, with almost all the remainder leading to H 2 O (and less than ∼1% resulting in the production of O). In diffuse or translucent clouds, both neutral molecules are destroyed by photodissociation, which -in the case of H 2 Ois an additional formation process for OH. A second and different pathway may be important in shocks or turbulent dissipation regions, where elevated gas temperatures can drive a series of neutral-neutral reactions with significant energy barriers: O(H 2 , H)OH(H 2 , H)H 2 O. Finally, OH and H 2 O may be produced by means of a grain-surface chemistry, in which O nuclei are hydrogenated on grain surfaces and subsequently photodesorbed. The relative importance of these three pathways will determine the exact H 2 O/OH abundance ratio, but all three predict a close relationship between OH and H 2 O. This relationship is supported by the observations reported here, which indicate a good correspondence between the OH and H 2 O absorption features; detailed modelling, which must await a larger sample of sight-lines, will be needed to interpret observed variations in the H 2 O/OH ratio. We note, however, that the lower end of the observed range of H 2 O/OH ratios (0.3-1.0) is predicted by models for turbulent chemistry (Godard et al. 2009). We note also that the observed distribution of OH is quite different from that of OH + ; the latter is believed to arise primarily in material with a molecular fraction that is too low to permit the efficient production of H 3 O + , whereas the former will arise in clouds with a substantial abundance of H 2 .

Future data of the OH ground state transition and the relatively high precision of the resulting column densities will not only allow us to assess the correlation between the abundances of OH and H 2 O, but also to re-calibrate less accurate OH column densities derived from decades of radio observations.

  (a) Sonnentrucker (2010),(b) Godard et al. (

Fig. 2 .

 2 Fig. 2. Top: OH absorption towards W49N. The velocities of the line-of-sight clouds and of W49N are indicated. The red dashed line is a least-squares fit. Middle: Same for the OH absorption against W51e4. The relative positions and strengths of the hfc splitting are indicated for the υ lsr =7 km s -1 component. Bottom: OH absorption against G34.26+0.15.

Fig. 4 .

 4 Fig. 4. Para-H 2 O absorption (dashed red line; from Sonnentrucker et al. 2010) and OH absorption (solid line), normalised by the respective single sideband continua, towards W49N (top) and W51 (bottom).

Table 1 .

 1 List of the observed 2 Π 3/2 J = 5/2 ← 3/2 16 OH and 18 OH lines

	Transition	Frequency [GHz] (a) A E [s -1 ] (b)
		OH, 2 Π 3/2 , J = 5/2 ← 3/2
	F = 2 -← 2 + 2514.298092	0.0137
	F = 3 -← 2 + 2514.316386	0.1368
	F = 2 -← 1 + 2514.353165	0.1231
		18 OH, 2 Π 3/2 , J = 5/2 ← 3/2
	F = 2 + ← 2 -2494.68092	0.0136
	F = 3 + ← 2 -2494.69507	0.1356
	F = 2 + ← 1 -2494.73421	0.1221
	(a) Varberg & Evenson (1993, σ rms = 30 kHz). The frequencies for
	18 OH were derived by isotope scaling. (b) Pickett et al. (1998).
	Fig. 1. Level diagram (not to scale) for the 2 Π 3/2 OH ground and first ex-
	cited state. The observed 16 OH transitions are indicated by bold arrows
	and labelled with their corresponding frequency. The observed 18 OH
	transitions are drawn as grey arrows.

Table 2 .

 2 Results of the OH line profile fitting. lsr velocity of component. The first and third hyperfine component (hfc) are +2.2 and -4.4 km s -1 , respectively, off the second one.

	Source	υ (a) 0	fwhm (b)	τ (c) max	N (d) OH
	(T mb,c )	[km s -1 ]		[10 14 cm -2 ]
	W49N (12.3 K)	2.9 ± 1.9 12.3 ± 1.0 10.1 ± 1.1 10.1 13.4 ± 0.4 0.9	8.8 ± 4.6
		36.8 ± 3.5 11.8 ± 4.4	1.8	1.6 ± 0.9
		39.1 ± 1.3	2.8 ± 0.5	2.3	0.5 ± 0.1
		60.5 ± 0.3 10.4 ± 3.9	2.5	2.0 ± 0.2
	W51	7.2 ± 0.4	4.6 ± 2.2	2.9	0.66 ± 0.08
	(8 K)	59.4 ± 3.3 16.8 ± 4.5 68.2 ± 1.4 4.6 ± 2.7	5.2 1.9	5.6 ± 2.1
	G34.26+0.15 12.6 ± 0.2	5.7 ± 0.8	2.5	1.12 ± 0.21
	(9 K)	28.3 ± 0.5	6.0 ± 1.6	1.4	0.67 ± 0.09
		55.4 ± 0.4 12.9 ± 1.0	4.8	4.73 ± 1.20
	(a)

Table 3 .

 3 OH and H 2 O abundances and their ratios in line-of-sight clouds towards W49N and W51. Statistical errors are given in brackets.

				W49N υ lsr intervals [km s -1 ]		W51 υ lsr intervals [km s -1 ]
			(30,37)	(37,44)	(44,49)	(49,54)	(54,72)	(-1,11)	(11,16) (c)	(43,50)
	N OH	[10 13 cm -2 ]	7.1(0.5)	11.3(6.3)	1.2(0.2)	1.5(0.3)	18.1(1.3)	6.4(0.9)	-	3.1(0.6)
	N (a) H 2 O N (b) H 2 [OH]/[H 2 ]	[10 13 cm -2 ] [10 20 cm -2 ] [10 -8 ]	2.3(0.1) 5.9(1.9) 12.0(4.0)	6.2(0.6) 6.6(2.0) 17(11)	0.1(0.05) 0.9(0.03) 13.2(2.5)	1.5(0.07) 3.2(0.1) 4.5(0.9)	11.6(0.4) 22.6(3.7) 8.0(1.4)	2.6(0.1) 4.3(0.1) 15(2.1)	0.23(0.05) 0.6(0.03) -	2.5(0.1) 2.6(0.8) 11.9(4.3)
	[H 2 O]/[H 2 ]	[10 -8 ]	3.9(1.3)	9.4(3.0)	1.1(0.6)	4.7(0.3)	5.1(0.9)	6.0(0.3)	3.8(0.9)	9.6(3.0)
	[H 2 O]/[OH]		0.32(0.03) 0.55(0.31) 0.08(0.04) 1.03(0.20) 0.64(0.05) 0.40(0.06)	-	0.81(0.16)

  . Three main pathways to OH have been identified in diffuse and translucent molecular clouds. The first pathway involves an ion-molecule chemistry, initiated by the cosmic-ray ionization of H 2 or H. The resulting H 2 + and H + ions can lead to OH + through the reaction sequences H 2 + (H 2 , H)H 3 + (O, H 2 )OH + or H + (O, H)O + (H 2 , H)OH + . In clouds with a low molecular fraction, the resulting OH + is destroyed primarily by dissociation recombination with electrons. In clouds with a high molecular fraction, however, OH + is rapidly converted to H 3 O + by a series of two H atom abstraction reactions: OH + (H 2 , H)H 2 O + (H 2 , H)H 3 O + . The H 3 O + ion then undergoes dissociative recombination with electrons to form OH or H 2 O.

GREAT is a development by the MPI für Radioastronomie and the KOSMA/Universität zu Köln, in cooperation with the MPI für Sonnensystemforschung and the DLR Institut für Planetenforschung.

Acknowledgements. Based on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy. SOFIA Science Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. We greatfully acknowledge the support by the observatory staff and a helpful referee report.