Jouini Elyès

Clotilde Ceremade

Napp

Collective Risk Aversion

In this paper we analyze the risk attitude of a group of heterogeneous agents and we develop a theory of comparative collective risk tolerance. In particular, we characterize how shifts in the distribution of individual levels of risk tolerance a¤ect the group's attitude towards risk. In a model with e¢ cient risk-sharing and two agents an increase in the level of risk tolerance of one or of both agents might have an ambiguous impact on the collective level of risk tolerance; the latter increases for some levels of aggregate wealth while it decreases for other levels of aggregate wealth. For more general populations we characterize the e¤ect of …rst-order like shifts (individual levels of risk tolerance more concentrated on high values) and second-order like shifts (more dispersion on individual levels of risk tolerance) on the collective level of risk tolerance. We also evaluate how shifts in the distribution of individual levels of risk tolerance impact the collective level of risk tolerance in a framework with exogenous egalitarian sharing rules. Our results permit to better characterize di¤erences in risk taking behavior between groups and individuals and among groups with di¤erent distributions of risk preferences.

Introduction

Many decisions to undertake risks are made by groups. A priori, one would expect that the theory of comparative risk aversion developed by [START_REF] Pratt | Risk aversion in the small and in the large[END_REF] and [START_REF] Arrow | Aspects of the Theory of Risk-Bearing[END_REF], which characterizes the proclivity of individuals to undertake risks, would easily translate into a theory of group risk taking. Consider, for instance, three individuals, A, B, and C. Suppose that C is more risk averse than B and B is more risk averse than A. Intuition strongly suggests that, when acting together, A and C would be less willing to undertake risks than A and B. Paradoxically, [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF] showed that such intuition is not always correct. For some levels of wealth an increase in the degree of risk aversion of the most risk averse individual in a group may decrease the collective level of risk aversion. 1 [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF] presented this paradoxical result through a numerical example with two individuals and isoelastic preferences. Our objective in this paper is to extend this line of inquiry by establishing precisely the conditions for this phenomenon to occur and, more generally, by evaluating how changes in the distribution of individual preferences a¤ect a group's attitudes towards risk.

To be perfectly clear, the Arrow-Pratt theory of comparative risk aversion does apply to utility functions of groups. So, for example, if a group is more risk averse than another in the Arrow-Pratt sense then this group will also require a larger risk premium to eliminate a fair risk. The interpretation of such comparative statics result, however, is clouded by the following fact. The degree of risk aversion of the group depends upon both the distribution of preferences among the agents and their optimal allocations. So a change in the distribution of preferences impacts the collective level of risk aversion through two channels. A direct one as well as an indirect one due to the fact that changes in the distribution of preferences lead, in turn, to changes in the e¢ cient allocation of wealth. Therefore, if risk is shared e¢ ciently, collective risk aversion has to be determined endogenously.

There is one special case in which the problem greatly simpli…es: given an e¢ cient allocation of wealth, if all individuals in the group have a constant and common absolute cautiousness (the derivative of the reciprocal of absolute risk aversion) -e.g. under CARA or CRRA utility functions with a common level of relative risk aversion -, the group has the same absolute cautiousness [START_REF] Wilson | The theory of syndicates[END_REF]). Comparative statics of risk aversion at the aggregate level is then not di¤erent from comparative statics at the individual level. The assumption of homogeneity in individual preferences, however, does not have empirical support (e.g. [START_REF] Barsky | Preference parameters and behavioral heterogeneity: An experimental approach in the Health and Retirement Study[END_REF]) and, in fact, defeats the purpose of Arrow-Pratt's theory of comparative risk aversion. Therefore, in this paper we tackle the problem of comparing attitudes towards risk among groups composed by individuals with heterogeneous risk preferences.

We show, in the setting of [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF]'s paper, that the collective level of risk tolerance is a wealth share weighted average of the individual levels of risk tolerance and increasing the risk tolerance level of one agent has two e¤ects: an increase of one of the terms of the average but a possible decrease of its relative weight in the average. As a result, there are two possible shapes for the collective risk tolerance as a function of the risk tolerance level of one of the agents: increasing curve or increasing then decreasing curve.

In fact, we establish the possibility of an even more perplexing situation: An increase in the degree of risk tolerance of both members of a couple may decrease their collective degree of risk tolerance. 2 We clearly characterize these di¤erent situations in terms of the size of the aggregate endowment relative to the endowment that corresponds to the fair e¢ cient allocation. We also characterize, for the two-agent case and for more general populations, …rst-order like shifts (individual levels of risk tolerance more concentrated on high values)

that have an unambiguous impact on the collective level of risk tolerance.

Since the key aspect of our analysis is preference heterogeneity we also evaluate how more dispersion on the individual levels of risk tolerance (second-order shifts) a¤ects the collective risk preferences. We show that, for high levels of wealth (relative to the level that corresponds to the fair e¢ cient allocation), more heterogeneity tends to increase collective risk tolerance, while the opposite is true for low levels of wealth.

Finally, we extend our analysis to a framework in which all members of a group receive the same endowment (egalitarian groups). This setup is appropriate to analyze situations in which the members of a group derive utility from a public good and situations in which a private good is simultaneously consumed by many individuals. For example, many goods within a household are simultaneously consumed by all the members of a family. Within this framework, and under very general individual preferences, we establish the impact of …rst-order shifts and second-order shifts on the collective level of risk tolerance.

In addition to the work of [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF], our paper is closely related to the work of [START_REF] Hara | Representative consumer's risk aversion and e¢ cient risk-sharing rules[END_REF], who studied the properties of collective preferences for a given distribution of individual risk preferences. We extend their analysis by evaluating how changes in the distribution of individual preferences a¤ect the collective attitudes towards risk. In this way, our analysis also complements the work of [START_REF] Gollier | Wealth Inequality and Asset Pricing[END_REF][START_REF] Gollier | Whom Should We Believe? Aggregation and Heterogeneous Beliefs[END_REF], who explored how heterogeneity in the initial endowment of wealth and how heterogeneity in beliefs a¤ect a group's attitude towards risk. At a more general level, we believe that our results may shed light into the empirical literature on 'choice shifts', which compares decisions made by groups relative to decisions made by the members of the group in situations of uncertainty (e.g. [START_REF] Baker | Comparing Small-Group and Individual Behavior in Lottery-Choice Experiments[END_REF][START_REF] Shupp | Risk preference di¤erentials of small groups and individuals[END_REF][START_REF] Masclet | Group and individual risk preferences: a lottery-choice experiment with selfemployed and salaried workers[END_REF]), a topic which we further discuss in the conclusion. 3The paper proceeds as follows. In Section 2 we present the model with e¢cient risk sharing and we establish a number of useful results about the e¢ cient allocations of endowments and the collective risk preferences. In Section 3 we brie ‡y evaluate the case of CARA preferences, which serves as a useful benchmark. In Section 4 we analyze the case of isoelastic heterogeneous preferences. After presenting general properties of collective preferences we evaluate shifts in the distribution of individual preferences, …rst in the case of two agents and then under more general populations. In Section 5 we evaluate collective risk preferences for the case of exogenous egalitarian sharing rules, while section 6 concludes. All the proofs are provided in the Appendix.

The model

We consider a standard static model in which a group of heterogeneous agents consume a single good. The endowment per person in the consumption good is de…ned by a random variable x on the probability space ( ; F; P ). Agents have a common belief over the probability space. In order to take into account …nite as well as in…nite sets of agents, the agent space is described by (I; ; Q), where I = [0; 1) and Q is a probability measure on I. Individuals are indexed by i 2 I and we denote by E Q the expectation with respect to Q.

We consider a 'consensus'group à la [START_REF] Samuelson | Social indi¤erence curves[END_REF]. That is, the group acts as if there was a social planner who wants to reach a Pareto e¢ cient allocation of risks and solves the following maximization program

U (x) = max R xidQ(i)=x Z i u i (x i )dQ(i): (1)
where u i is the utility function of agent i, where x i is the consumption of agent i and where i is the weight (e.g. decision power) granted to agent i. The utility function U (x) corresponds to the highest social utility level among all possible endowment distributions across agents.

Throughout the paper, we make the following assumption on the utility functions. Assumption (U) For all i, the utility function u i : [d i ; 1) ! R [ f 1g is assumed to be in…nitely di¤ erentiable on (d i ; 1) with u 0 i > 0 and u 00 i < 0 and satis…es Inada's conditions, i.e. lim x!di u 0 i (x) = 1 and lim x!1 u 0 i (x) = 0: For a given agent i and a given consumption level x, the absolute (resp. relative) risk aversion A i (x) (resp. R i (x)); the absolute (resp. relative) risk tolerance t i (x) (resp. s i (x)) are given by

A i (x) = u 00 i (x) u 0 i (x) ; R i (x) = x u 00 i (x) u 0 i (x) = xA i (x) t i (x) = u 0 i (x) u 00 i (x) = 1 A i (x) ; s i (x) = u 0 i (x) xu 00 i (x) = 1 R i (x) = t i (x) x :
Note that CARA and CRRA utility funtions clearly satisfy assumption (U).

If we denote by v the function de…ned by v(x; i) = u 0 i (x); we will also make the following assumption. Assumption (LSPM) The function v is log-supermodular in (x; i) ; i.e. @ log v @x (x; i) is nondecreasing in i:

Remark that the log-supermodularity of v(x; i) means that A(x; i) = A i (x) is nonincreasing in i or that agent i is less risk averse (and more risk tolerant) than agent j when i j:

We have then the following classical result Proposition 1 Under Assumption (U), there exists a family of functions

(f i ) i2[0;1] such that f i : [d; 1) ! [d i ; 1) with d = R d i dQ(i) is in…nitely di¤ erentiable and increasing R f i (x)dQ(i) = x for all x 2 [d; 1) U (x) = R i u i (f i (x))dQ(i):
We will say that (f i ) i2[0;1] is an e¢ cient sharing rule associated with the maximization program of Eq. ( 1).

We recall the following well known results that relate the collective risk aversion and risk tolerance to the individual ones through the e¢ cient sharing rule.

Proposition 2 [START_REF] Wilson | The theory of syndicates[END_REF][START_REF] Hara | Representative consumer's risk aversion and e¢ cient risk-sharing rules[END_REF]) Let us assume that (U) is satis…ed. Let x be a given aggregate wealth and let (f i ) i2I be the e¢cient sharing rules associated with the maximization program of Eq. ( 1). The collective absolute risk tolerance t(x) = U 0 (x) U 00 (x) and the collective relative risk tolerance s(x) = U 0 (x) xU 00 (x) are given by

t(x) = Z t i (f i (x))dQ(i); s(x) = Z f i (x) x s i (f i (x))dQ(i):
The relative risk tolerance s(x) of the group is then an average of the individual levels of relative risk tolerances s i (f i (x)) weighted by the optimal individual shares of consumption. Analogously, the degree of relative risk aversion of the group is an average of the individual degrees of relative risk aversion.The group is then less risk averse than the most risk averse agent and more risk averse than the least risk-averse one. In terms of the example given in the introduction this implies, in particular, that a group composed by B and C will always be less willing to undertake risks than a group composed by A and B.

It is easy to show that s 0 (x) is positive and then that the collective relative risk aversion R(x) = x U 00 (x) U 0 (x) is decreasing in x: This fact has been underlined by Hara et al. (2007, Proposition 6). They further show (Corollary 7) that R(x) approaches the degree of relative risk aversion of the most (least) risk averse agent as x converges to zero (in…nity).

At this stage we consider very general utility functions and we may assume, without loss of generality, that all the members of the group are granted the same weight (it su¢ ces to replace the utility function u i by i u i , note that the LSPM property is not impacted by this modi…cation). In the next we consider then the equally weighted Pareto optimum. We also assume that there exists an e¢ cient fair allocation. In other words, there exists x such that (x i ) i2I ; with x i = x for all i; is e¢ cient.

The following proposition provides an analysis of how the aggregate consumption x is shared among the agents depending on the position of x relatively to the fair e¢ cient allocation x Proposition 3 Under the Assumptions (U) and (LSPM), we have the following results.

For x

x ; the optimal allocation (x i ) i2I associated to the aggregate wealth x is such that x i

x , for all i, and x i increases with i: Furthermore, if all the utility functions are DARA then t i (x i ) increases with i.

For x

x ; the optimal allocation (x i ) i2I associated to the aggregate wealth x is such that x i x , for all i, and x i decreases with i:

Although of some interest by itself, this Proposition will also play an important role in the analysis that follows.

CARA utility functions

Let us consider constant absolute risk-aversion/tolerance utility functions of the form

u i (x) = i exp( x i ): (2)
We have t i (x) = i and t(x) = R i dQ(i): If the agents are indexed by their absolute levels of risk tolerance we have i = i and the log-supermodularity assumption is satis…ed. We have then t

(x) = E Q h ~ i
and the collective level of risk tolerance does not depend on the wealth allocation among the agents. It is immediate that FSD shifts on the distribution of the individual levels of risk tolerance lead to an increase of the collective level of absolute (and relative) risk tolerance. More heterogeneity, in the sense of shifts in the distribution of preferences that preserve the mean, have no e¤ect on the group's risk tolerance. These results will serve as a useful benchmark.

CRRA utility functions

Let us consider constant relative risk-aversion/tolerance utility functions of the form

u i (x) = 1 1 1 bi x 1 1 b i : (3)
where b i is the level of relative risk tolerance of individual i and 1 bi is his level of relative risk aversion. In such a setting, we have

A i = 1 b i x ; R i = 1 b i ; t i = b i x; and s i = b i :
Since the utility functions are no more de…ned up to a multiplicative constant, we do not assume anymore that the i = 1 for all i: However, we still assume that there exists a wealth level x for which the fair allocation (x i ) i2I with x i = x for all i; is e¢ cient. Note that the existence of such a fair allocation can always be granted through a judicious choice of the weights ( i ) i2I .

The …rst-order conditions for Pareto optimality give then that i (x )

1 1 b i is independent of i:
The Pareto problem can be rewritten as follows

U (x) = max R x i x dQ(i)= x x R 1 1 1 b i xi x 1 1
b i dQ(i); up to a multiplicative constant. If we renormalize the individual consumptions in order to measure them in terms of multiples of x , we are led to analyze the situation where

u i (x) = 1 1 1 b i x 1 1 b i
and all the weights i are equal to 1. Note that with this renormalization, x = 1 corresponds to the e¢ cient fair allocation.

In the next we consider then the equally weighted Pareto optimum. Since the agents di¤er by only one characteristic, namely their level b i of relative risk tolerance, we might index them by this characteristic or we may, in other words, assume that b i = i: For a given function h; we may then write indi¤erently

R h(b i )dQ(i) or R h(b)dQ(b) or E Q h h( b) i :
The level of relative risk-aversion is then decreasing with i and the log-supermodularity condition is immediately satis…ed.

The following Proposition uses these assumptions to characterize precisely the functions de…ning collective preferences and the collective level of risk aversion.

Proposition 4 In a group made of agents with constant but heterogeneous levels of relative risk aversion, we have at the equally weighted Pareto optimum

U (x) = Z b i b i 1 e (1 bi) 1 (x) dQ(i) with (t) Z e bit dQ(i):
The collective degree of relative risk aversion R(x) is given by

R(x) x U 00 (x) U 0 (x) = x R b i d e Q(i) = 1 s(x) : (4)
with d e Q dQ e b 1 (x) R e b i 1 (x) dQ(i) :
As seen in the proof in the Appendix, the Lagrange multiplier of the Pareto optimum problem is given by q = exp( 1 (x)) and q is then the shadow price associated to the constraint P i2I x i = x: We clearly have (0) = 1; which means that 1 (x ) = 0 and q(x ) = 1 for the e¢ cient fair allocation x = 1. Since the agents are risk averse, high levels of aggregate wealth have a low shadow price and low levels of aggregate wealth have high shadow price and we can easily derive that q(x) < 1 for x > x and q(x) > 1 for x < x . This means, in particular, that we have 1 (x) < 0 for x > 1 and 1 (x) > 0 for x < 1.

4.1 A model with two agents [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF] shows that, in a model with two agents, an increase in the level of risk tolerance of one of the agents might have an ambiguous impact on the collective level of risk tolerance. It increases for some levels of aggregate wealth while it decreases for other levels of aggregate wealth. Since this is only stated on a numerical example in [START_REF] Mazzocco | Savings, risk sharing and preferences for risk[END_REF], let us clearly express this result.

Proposition 5 In a model with two agents with b 1 < b 2 ; there exists x 1 such that a small increase of b 2 leads to an increase of the collective level of risk tolerance t(x) for all x x and to a decrease of the collective level of risk tolerance t(x) for all x x:

Recall that after our normalization, x = 1 corresponds to the fair e¢ cient allocation. Proposition 5 means then that an increase of the risk tolerance level of the most risk tolerant agent increases (decreases) the collective level of risk tolerance for levels of wealth above (below) a given threshold that is below the fair e¢ cient allocation. Note that the threshold x depends on b 1 and b 2 : This means that for x above the fair allocation, any increase of b 2 increases the collective risk tolerance. In fact, above the fair allocation, an increase of b 2 also increases the weight granted to b 2 leading to an increase of the collective level of risk tolerance. For a wealth level x below the fair allocation, the impact of an increase of b 2 is less clear. Indeed, we showed in Proposition 3 that for low levels of aggregate wealth the least risk tolerant agent (the most risk averse) has a larger share of the total wealth and an increase of b 2 leads to an increase of the weight granted to b 1 (the share of total wealth of agent 1). The increase of b 2 has then two e¤ects in opposite directions: an increase of one of the terms of the average (namely the greatest one) and an increase of the weight of the smallest one. Since the second e¤ect does not exist for x = 1; the …rst e¤ect continues to dominate for x above a given threshold x 1 while the second e¤ect dominates for x x:

Figures 1 and 2 illustrate the impact of an increase of b 2 :

The next Proposition analyzes more in detail how the collective level of risk tolerance evolves as a function of b 2 :

Proposition 6 In a model with two agents with b 1 < b 2 ; b 1 being given, for

x 1 2 ; the function b 2 ! t x (b 2 ) is increasing on (b 1 ; 1) with lim b2!b1 t x (b 2 ) = xb 1 and t x (b 2 ) b2!1 x 1 2 b 2 : For x < 1 2 ; there exists b (x; b 1 ) > b 1 such that the function b 2 ! t x (b 2 ) is increasing on (b 1 ; b (x; b 1 )) and decreasing on (b (x; b 1 ); 1) with lim b2!b1 t x (b 2 ) = xb 1 and lim b2!1 t x (b 2 ) = xb 1 :
In summary, for (very) low levels of wealth, increasing the risk tolerance of the more risk tolerant agent has an ambiguous impact on the collective attitude towards risk. Proposition 6 characterizes precisely the conditions for this paradoxical result to occur. Figures 3,[START_REF] Bone | Are Groups More (or Less) Consistent Than Individuals?[END_REF]

2 ! t x (b 2 ) (or equivalently of b 2 ! s x (b 2 )):
The asymptotic behavior of t x (b 2 )=b 2 is illustrated in Figures 6 and7.

Propositions 5 and 6 establish the behavior of collective preferences as a function of one of the agent's risk tolerance. Another important question is what happens to the collective level of risk tolerance when both agents become more risk tolerant. In the next Proposition we show that the ambiguous impact disappears when we consider a uniform increase of risk tolerance across the agents, but that for non-uniform increases in risk tolerance the collective degree of risk tolerance may still be lower. Let us illustrate the second point by two extreme situations. For k very small (near to 0), the shifts we are considering are almost of the form (b 1 ; b 2 ) ! (b 1 ; b 2 + ") that have already been considered in Proposition 5. These shifts increase the risk tolerance level of the second agent and also increase its weight for x 1. These shifts lead then to an unambiguous increase of the aggregate level of risk tolerance for x 1. For k near in…nity (and h very small), the shifts we are considering are almost of the form (b 1 ; b 2 ) ! (b 1 + "; b 2 ) and such shifts increase the risk tolerance level of the …rst agent and also increase its weight when x 1: These shifts lead then to an unambiguous increase of the aggregate level of risk tolerance for x 1: The proposition shows that there is a range for k for which the shifts have an unambiguous impact without restrictions on x: However, for small levels of k we cannot conclude that the collective level of risk tolerance is higher. 4We are also interested in the impact of more heterogeneity among our 2 agents. The next result shows that more heterogeneity leads to a higher collective risk tolerance level for high wealth levels (above the fair e¢ cient allocation) and to a lower collective risk tolerance level for low wealth levels (below the fair e¢ cient allocation). This result is very intuitive. We have already seen that the collective level of risk tolerance is near the risk tolerance level of the most risk tolerant agent for high levels of wealth and is near the risk tolerance level of the least risk tolerant level agent for low levels of wealth. More heterogeneity leads to an increase of the risk tolerance level of the most risk tolerant agent and to a decrease of the risk tolerance level of the least risk tolerant agent. This leads then to an increase of the collective level of risk tolerance for high levels of wealth and to a decrease of the collective level of risk tolerance for low levels of wealth. Proposition 8 permits to give a precise meaning to high and low levels of wealth since the fair e¢ cient allocation appears to be the relevant threshold. Figure 8 illustrates this result.

General populations

The di¤erent results of the previous section permit to see that …rst-order stochastic dominance shifts do not guarantee an increase in the collective degree of risk tolerance. In this section we initially consider whether a stronger notion of …rst-order stochastic dominance leads to an unambiguous impact on the group's degree of risk tolerance. Then we evaluate the e¤ect of more heterogeneity within a group. In order to treat the problem in a quite general setting, we consider from now on general populations described by a distribution on I = [0; 1) : To relate the results in this more general setting to those obtained in the 2-agent framework, we will attach a speci…c attention to distributions with a 2-point support.

Since FSD is not a good candidate to obtain comparative static results, let us recall the following de…nition corresponding to a stronger notion of …rst-order dominance.

De…nition 1. Monotone Likelihood Ratio Dominance (MLR). Let P and Q denote two probability measures on I = [0; 1). We say that P dominates Q in the sense of MLR (P < M LR Q) if there exist numbers 0 1 and a nondecreasing function h : [ ; ] ! [0; 1] such that P ([0; )) = Q(( ; 1)) = 0 and dP (i) = h(i)dQ(i):

In other words, an MLR dominated shift for a given probability measure puts less weight for higher values of i. This concept is widely used in the statistical literature and was …rst introduced in the context of portfolio problems by [START_REF] Landsberger | Demand for risky …nancial assets: A portfolio analysis[END_REF]. MLR dominance is stronger than FSD and, in particular, an MLR dominated shift for a given distribution reduces the mean.

When the supports of P and Q are reduced to two points, (b

1 1 ; b 1 2 ) with b 1 1 < b 1 2 for P and (b 2 1 ; b 2 2 ) with b 2 1 < b 2 2 for Q; we necessarily have b 2 1 < b 2 2 b 1 1 < b 1 2 or b 2 1 = b 1 1 and b 2 2 = b 1 2 :
In the …rst case, any average of the risk tolerance levels in the support of Q is smaller than any average of the risk tolerance levels in the support of P and the collective risk tolerance level is higher under P: The (most) interesting case is when both probability measures have the same support. We then have two populations with the same set of possible levels of individual level of risk tolerance b 1 and b 2 but with di¤erent proportions of agents in each category: a proportion p 1 (resp. p 2 = 1 p 1 ) of agents that have an individual level b 1 (resp. b 2 ) of risk tolerance under P and a proportion q 1 (resp. q 2 = 1 q 1 ) of agents that have an individual level b 1 (resp. b 2 ) of risk tolerance under Q: The MLR dominance is characterized in this setting by p2 p1 q2 q1 (or equivalently q 1 p 1 or q 2 p 2 ) Proposition 9 Let us consider two populations. In the …rst one, we have proportions p 1 (resp. p 2 = 1 p 1 ) of agents with an individual level b 1 of risk tolerance (resp. b 2 ): In the second one, we have proportions q 1 (resp. q 2 = 1 q 1 ) of agents with an individual level b 1 of risk tolerance (resp. b 2 ): If we assume that p2 p1 q2 q1 ; then the collective level of risk tolerance is higher in the …rst population.

In summary, when the support of the population is reduced to two points, an MLR dominant shift in the degree of risk tolerance of the members of the group increases the group's risk tolerance, so such shift clearly characterizes the notion of a "more risk tolerant group". The following proposition generalizes the impact of MLR shifts for distributions with more general supports.

Proposition 10 Let us consider two populations characterized by two distributions P and Q of individual levels of risk tolerance. If P < M LR Q then for all x 1; we have t P x t Q x : However, P < M LR Q does not guarantee an increase of the collective level of risk tolerance when x > 1:

MLR provides then a satisfying answer to the impact of shifts for low levels of wealth (when x 1), which corresponds to the case where the unilateral increase of one of the individual levels of risk tolerance failed to guarantee an increase of the aggregate level of risk tolerance. In the following Proposition we show that when the density function (introduced in De…nition 1) h = dP dQ has an exponential growth rate, then we do have an unambiguous impact on collective risk tolerance.

Proposition 11 Let us consider two populations characterized by two distributions P and Q on [0; 1)of individual levels of risk tolerance such that P < M LR Q with dP dQ (b) = exp(kb) for some positive k and : For all x; we have t

P x t Q x :
We have seen in the 2-agent setting (Proposition 8) that more heterogeneity has a clear impact on the collective level of risk tolerance depending on the relative position of the aggregate wealth with respect to the fair e¢ cient allocation. We are now interested in establishing the e¤ect of "more heterogeneity" in the general setting. For this purpose, we introduce the following de…nition.

De…nition 2. Portfolio Dominance (PD). Let Q 1 and Q 2 denote two probability measures on I = [0; 1). We say that

Q 1 dominates Q 2 in the sense of PD (Q 1 < P D Q 2 ) if we have R v(i a)dQ 1 (i) = 0 ) R v(i a
)dQ 2 (i) 0 for any real number a and any nonnegative and nonincreasing function v.

This concept has been introduced in the context of portfolio problems by [START_REF] Landsberger | Demand for risky …nancial assets: A portfolio analysis[END_REF] and further studied by [START_REF] Gollier | A note on portfolio dominance[END_REF]. In the portfolio context it is related to the degree of riskiness of the asset returns. In our context, it is related to the level of individual heterogeneity in relative risk tolerance. In particular, a mean preserving PD dominated shift for a given distribution increases the variance (Jouini and Napp, 2008, Proposition 3).

The following proposition uses this concept to characterize the impact of "more heterogeneity" in the distribution of individual preferences.

Proposition 12 Let us consider two populations respectively characterized by distributions P and Q of individual levels of risk tolerance. If P and Q are symmetric with respect to some b with dQ dP nonincreasing before b and nondecreasing after b then P < P D Q and P < SSD Q and the collective level of risk tolerance t Q

x under Q is higher than (resp. lower than) the aggregate level of risk tolerance t P x under P for x 1 (for x 1):

The intuition is, in essence, the same as that in Proposition 8. At high wealth levels those individuals that have a high tolerance for risk are more representative of the collective level of risk tolerance, but the opposite is true at low wealth levels. More dispersion in the levels of risk tolerance of the group then leads to a higher (lower) level of collective risk tolerance for high (low) wealth levels.

The case of egalitarian groups

It is interesting to analyze the aggregate behavior in a model where all the agents consume the total consumption x: This is the case when x is a public good. This also the case when x is a private good but simultaneously consumed by all the agents in the group. We may consider that both agents in a couple get utility from saving money or from holding consumption goods and consider these goods as owned by the couple and not shared among them through an e¢ cient sharing rule. This is also the setup used in a number of recent experiments that compare the degree of risk aversion of groups with that of individuals (e.g. [START_REF] Shupp | Risk preference di¤erentials of small groups and individuals[END_REF][START_REF] Baker | Comparing Small-Group and Individual Behavior in Lottery-Choice Experiments[END_REF][START_REF] Masclet | Group and individual risk preferences: a lottery-choice experiment with selfemployed and salaried workers[END_REF]) and where the rewards of the group are exogenously divided equally among its members. We may imagine, for example, that such experiments re ‡ect the widely observed regularity of partnerships with equal sharing rules.

In the next, the endowment in the consumption good is de…ned by a random variable x on the probability space ( ; F; P ) and the social utility function is given by

U (x) = Z i u i (x)dQ(i): (5)
where u i is the utility function of agent i and where i is the weight granted to agent i.

We have then

R(x) x U 00 (x) U 0 (x) = x R i u 00 i (x)dQ(i) R i u 0 i (x)dQ(i) = Z R i (x)dP u (i) (6)
where P u is the probability measure de…ned by dP u dQ (i) =

iu 0 i (x) R iu 0 i (x)dQ(i) :
Let us …rst analyze the case with 2 agents and CRRA functions. We have then u

i (x) = 1 1 1 b i x 1 1
b i ; i = 1; 2; and we take i = 1; i = 1; 2; as in the previous section. Equation [START_REF] Clark | Group induced shift toward risk: A critical appraisal[END_REF] can be rewritten as follows

R(x) = 1 b1 x 1 b 1 + 1 b2 x 1 b 2 x 1 b 1 + x 1 b 2 = R 1 x R1 + R 2 x R2 x R1 + x R2
and the aggregate relative risk aversion is a weighted arithmetic average of the individual levels of relative risk aversion. As in the private good case, the collective level of relative risk aversion decreases with x and it approaches the degree of relative risk aversion of the most (least) risk averse agent in the economy as x converges to zero (in…nity). Notice also that the weights are given by the individual marginal utilities and the highest weight is granted to the lowest (highest) individual level relative risk aversion for x > 1(for x < 1).

An increase of the individual level of risk tolerance of the most risk tolerant agent might then have an ambiguous impact. We have the following result

Proposition 13 In a model with a public good, two agents and CRRA functions with b 1 < b 2 ; there exists x 1 such that a small increase of b 2 leads to an increase of the collective level of risk tolerance t(x) for all x

x and to a decrease of the collective level of risk tolerance t(x) for all x x:

In particular, this means that FSD shifts of the individual levels of risk tolerance are not su¢ cient to increase the collective level of risk tolerance. The next result illustrates the impact of a mean preserving spread on the individual levels of risk aversion in a 2-agent setting.

Proposition 14 In a model with a public good, two agents and CRRA functions with b 1 < b 2 (or equivalently R 1 > R 2 ), a shift of the form R 1 h and R 2 + h with h > 0 increases (decreases) the aggregate level of relative risk aversion R(x) for x 1 (for x 1). It increases (decreases) the collective level of risk tolerance t(x) for x 1 (for x 1).

In the next we characterize, in a general distribution setting, the impact of MLR shifts on the collective level of risk tolerance/aversion. Note that the following result is obtained for very general utility functions.

Proposition 15 Let us consider two populations respectively characterized by distributions P 1 and P 2 on (I; ). Under Assumptions (U) and (LSPM) and if P 2 < M LR P 1 then the collective level of risk aversion (risk tolerance) under P 1 is higher (lower) than under P 2 :

The result in Proposition 15 is quite powerful. Under the assumption that the members of the group consume the same endowment, and under weak restrictions on the individual utility functions, if the individual levels of risk aversion are more concentrated on high values (in the sense of MLR dominance) then the collective level of risk aversion will be higher for all endowment levels. In this case MLR dominance provides a clear characterization of comparative collective risk aversion 5 .

The next proposition analyzes the impact of more heterogeneity on the individual levels of risk aversion. For a given distribution P on (I; ) and for a given x; we denote by P x the image measure of P by i !

u 00 i (x) u 0 i (x) :
The measure P x describes the distribution of the individual levels of risk aversion at a given wealth level x:

Proposition 16 Let us consider two populations respectively characterized by distributions P 1 and P 2 on (I; ) and let us assume that (U) is satis…ed. If, for a given x; u 0 i (x) is nondecreasing in i,

u 00 i (x) u 0 i (x)
is decreasing with i6 and P x 2 < P D P x 1 ; then the collective level of risk aversion (risk tolerance), at x; under P 2 is higher (lower) than under P 1 :

In particular, if there exists x such that all the individual marginal utilities u 0 i (x ) are equal, then u 0 i (x) is nondecreasing in i for x x : It su¢ ces then to have that

u 00 i (x) u 0 i (x)
is decreasing with i and P x 2 < P D P x 1 to conclude that the aggregate level of risk aversion, at x; under P 2 is higher than under P 1 : In other words, under weak assumptions on the utility function, less heterogeneity in risk aversion, in the sense of PD dominance, implies a higher level of collective risk aversion when wealth is su¢ ciently low.

Conclusion

Mazzocco (2004) established the counter-intuitive result that an increase in the level of risk tolerance of one of the individuals in a couple may reduce their collective degree of risk tolerance. We studied precisely the conditions for this phenomenon to occur. More generally, we established conditions under which groups with individual levels of risk tolerance more concentrated on high values and groups that are more heterogeneous will display higher risk tolerance, both with e¢ cient risk-sharing and with an exogenous egalitarian sharing rule. Our results permit to better characterize di¤erences in risk taking behavior between groups and individuals and among groups with di¤erent distributions of risk preferences.

It should be possible to design experiments to evaluate if our results are consistent with elicited risk attitudes of groups and individuals. [START_REF] Shupp | Risk preference di¤erentials of small groups and individuals[END_REF] compare the willingness to pay for lotteries of small groups and individuals in a setup similar to that of Section 5. They conclude that, for most lotteries, group choices are signi…cantly di¤erent from the mean of the individual choices (groups tend to be more risk averse than individuals for low-expected-value lotteries but less risk averse than individuals for highexpected-value lotteries). These results are consistent with a large number of studies in social psychology that show "risky" and "cautious" shifts in group risk-taking behavior relative to the mean of the individual choices (see e.g. [START_REF] Clark | Group induced shift toward risk: A critical appraisal[END_REF]). We have seen that there is no reason to believe that the group's willingness to pay (derived from the collective preferences), and more generally the willingness to take risks, should be the same as the mean of the individual members'willingness to pay. In particular, even with CRRA individual preferences, the fact that the group's relative risk aversion decreases with x implies that "cautious" shifts should be more prevalent in low-expected-value (low-stakes) lotteries while "risky" shifts should be more prevalent in high-expected-value (high-stakes) lotteries, precisely what Shupp and Williams (2008) found. 7 It would be interesting to further explore the experimental relevance of our results by proceeding to inter-groups comparisons. For instance, to analyze the di¤erence in risk attitudes between two couples that only di¤er by the risk aversion level of one of the members, e.g. the man.

Appendix

Proof of Proposition 1. Let us denote by ' the function de…ned by '(q) = R (u 0 i ) 1 ( q i )dQ(i): By Inada's conditions and since u i is increasing and strictly concave for all i, ' is well de…ned on (0; 1) and decreasing. Furthermore, from the monotone convergence Theorem we have lim x!0 '(q) = 1 and lim x!1 '(q) = d: Let us then de…ne

f i : [d; 1) ! [d i ; 1) by f i (x) = (u 0 i ) 1 ( ' 1 (x) i ); we clearly have x = R f i (x)dQ(i) and i u 0 i (f i (x)) = ' 1 (x)
for all i and is independent of i: The family (f i (x)) sati…es then the …rst-order conditions of the maximization program de…ned by Eq. ( 1) and since this program is concave we have

U (x) = R i u i (f i (x)
)dQ(i): Proof of Proposition 3. Since the fair allocation x i = x ; i 2 I; is e¢ cient and since we granted the same weight to all the agents, the …rst-order conditions for Pareto optimality give us that u 0 i (x) is independent of i: Since @u @x (x; i) is LSPM, integrating @ @x log u 0 i (x) between x x and x , gives us that u 0 i (x) increases with i: Let us consider the Pareto allocation (x i ) i2I associated to the aggregate wealth x

x : We have that u 0 i (x i ) is independent of i: We also necessarily have x i0 x for some i 0 and then u 0 i0 (x i0 ) u 0 i0 (x ) by concavity of the utility functions. We have then u 0 i (x i ) u 0 i (x ) for all i and consequently x i x for all i: Since u 0 i (x) increases with i and u 0 i (x) is independent of i and since u 0 i (x) is decreasing in x, we have that x i increases with i: By the LSPM property t(x; i) is increasing in i and by the DARA property, it is increasing in x: We have then that t(x i ; i) is increasing in i:

For x x , integrating @ @x log u 0 i (x) between x and x , gives us that u 0 i (x)

decreases with i: The same kind of arguments as above give that x i decreases with i: Proof of Proposition 4. The …rst-order condition gives us the optimal allocation of agent i; x i = q bi , where q is the Lagrange multiplier. Using the resource constraint we obtain R q bi dQ(i) = x or R exp( b i ln q)dQ(i) = x. We obtain that q = exp( 1 (x)) hence x i = e bi 1 (x) . Plugging this back into the utility function we obtain the representative agent's utility.

The degree of relative risk aversion of the representative agent can be derived directly from the expression of the representative agent utility function or using Proposition 2. We have s = R s i xi x dQ(i): Since x i = e bi 1 (x) the result follows.

Proof of Proposition 5. Let us consider b 1 as given and denote by t x (b 2 ) the aggregate risk tolerance level when the risk tolerance level of agent 2 is given by b 2 b 1 : We have

t x (b 2 ) = 1 2 b 1 exp( b 1 a x (b 2 )) + 1 2 b 2 exp( b 2 a x (b 2 ))
where

a x (b 2 ) is the solution of (a x (b 2 ); b 1 ; b 2 ) = x with (a; b 1 ; b 2 ) = 1 2 exp( b 1 a) + 1 2 exp( b 2 a):
We get after computations

d db 2 t x (b 2 ) = 1 2 exp ( (b 1 + b 2 )a x (b 2 )) b 1 exp( b 1 a x (b 2 )) + b 2 exp( b 2 a x (b 2 )) '(a x (b 2 ); b 1 ; b 2 ) with '(a; b 1 ; b 2 ) = b 1 + b 2 exp((b 1 b 2 ) a) a (b 2 b 1 ) b 1 : For x 1; we have a x (b 2 ) 0 and d db2 t x (b 2 ) > 0:
The aggregate level of risk tolerance is an increasing function of b 2 :

Let us now focus on the case x 1: It is easy to check that a x (b 2 ) is always positive for x 2 (0; 1) and decreases with x from 1 to 0: It is also easy to see that '(a; b 1 ; b 2 ) is decreasing in a; positive for a = 0 and converges to 1 when a converges to 1: There exists then a level x < 1 such that

d db2 t x (b 2 ) = 0: For x < x (resp. x > x ), we have d db2 t x (b 2 ) < 0 (resp. d db2 t x (b 2 ) > 0). Proof of Proposition 6. We have t x (b 2 ) = 1 2 b 1 exp( b 1 a x (b 2 )) + 1 2 b 2 exp( b 2 a x (b 2 ))
where a x (b 2 ) is the solution of

1 2 exp( b 1 a x (b 2 )) + 1 2 exp( b 2 a x (b 2 )) = x: We know that d db 2 t x (b 2 ) = '(a x (b 2 ); b 1 ; b 2 ) = b 1 + b 2 exp((b 1 b 2 ) a x (b 2 )) a x (b 2 ) (b 2 b 1 ) b 1 :
We have already seen that '(a; b 1 ; b 2 ) is decreasing in a; positive for a = 0 and converges to 1 when a converges to 1: Let us denote by a(b 1 ; b 2 ) the solution of '(a; b 1 ; b 2 ) = 0: Let us …rst consider the case x b1 : Furthermore, since @' @a is decreasing, @a @b2 has the same sign as @' @b2 : Direct computations give @' @b2 = u ab

1 ab 2 u with u = exp((b 1 b 2 ) a) and a such that b 1 + b 2 u b 1 a(b 2 b 1 ) = 0: Substi- tuting a in @' @b2 gives @' @b2 = (b 1 b 2 ) 1 b 1 1 b 2 b 1 u + b 2 1 + b 2 1 u + b 2 2 u 2 < 0:
d db 2 (a x (b 2 ) a(b 1 ; b 2 )) = @' @b2 @' @a (a ; b 1; b 2 ) @ @b2 @ @a (
d db 2 (a x (b 2 ) a(b 1 ; b 2 )) = b 2 1 a e a b1 + e a b 2 + a (b 2 b 1 )e a b 2 b 1 e a b1 + b 2 e a b 2 b 1 + b 2 e a (b1 b 2 ) (b 2 b 1 )
> 0:

We have then that d db2 a x (b 2 ) > d db2 a(b 1 ; b 2 ) each time these two functions cross. This means that they can cross only once.

For x 1; we have already shown that d db2 t x (b 2 ) is positive for all b 2 and t x (b 2 ) is then increasing.

For 1 x 1 2 ; let us show that there is no crossing between a x (b 2 ) and a(b 1 ; b 2 ). For that purpose let us consider a = a x (b 2 ) = a(b 1 ; b 2 ). We have then b 1 which is impossible since we assumed 2x 1 0: For x 1 2 ; it is easy to check that lim b2!b1 a x (b 2 ) = ln x b1 and a x (b 2 ) b2!1 ln(2x 1) b2

: The limits of t x derive from there. Proof of Proposition 7. It su¢ ces to prove directly the second point. We have

t x (h) = 1 2 (b 1 + kh) exp( (b 1 + kh)a x (h)) + 1 2 (b 2 + h) exp( (b 2 + h)a x (h))
where a x (h) is the solution of (a x (h); For x 1; we have a x (0) 0 and it su¢ ces to remark that @ @a 2 is positive, that @ @a (0; Proof of Proposition 8. We have

t x (h) = 1 2 (b 1 h) exp ( (b 1 h)a x (h)) + 1 2 (b 2 + h) exp ( (b 2 + h)a x (h))
where a x (h) is the solution of (a x (h); b 1 h; b 2 + h) = x: We want to show that for x < 1; we have dtx dh (0) < 0 and for x > 1 we have dtx dh (0) < 0:

We have dax dh (0) = a(exp( ab1) exp( ab2)) b1 exp( ab1)+b2 exp( ab2) where a = a x (0); and

dt x dh (0) = 1
)) = b 1 exp ( 2ab 1 ) + b 2 exp ( 2ab 2 ) (b 2 b 1 ab 2 1 + ab 2 2 ) exp ( a (b 1 + b 2 ))
which has the same sign as `(a) with

`(a) = b 1 exp ( a(b 1 b 2 )) b 2 exp ( a(b 2 b 1 )) + (b 2 b 1 ab 2 1 + ab 2 2 ):
We have lim a!1 `(a) = 1, `(0) = 0 and lim a! 1 `(a) = 1: We also have

`0(a) = (b 1 b 2 )( b 2 b 1 b 1 exp (a(b 2 b 1 )) b 2 exp (a(b 1 b 2 )) > 0:
The function `is then negative on R and positive on R + : Since a > 0 for x < 1 and a < 0 for x > 1 this gives the result. Proof of Proposition 9. We denote by t P x (resp. t Q x ) the aggregate level of risk tolerance in the …rst (resp. second) population when the aggregate wealth is x: We have t P x = p 1 b 1 exp( b 1 a P x ) + p 2 b 2 exp( b 2 a P x ) where a P

x is the solution of

p 1 exp( b 1 a P x ) + p 2 exp( b 2 a P x ) = x:
We have similar formulas for t Q x and we have that t P x t Q x if and only

q 1 exp( b 1 a Q x ) p 1 exp( b 1 a P x ) q 2 exp( b 2 a Q x ) p 2 exp( b 2 a P x )
or equivalently p 1 exp( b

1 a P x ) q 1 exp( b 1 a Q x ) (7) If x 1; we have a P x a Q
x and the result is immediate. Let us now consider the case x 1. Note that exp( b 1 a Q x ) and exp( b 2 a Q x ) correspond to the Pareto optimal allocations x Q 1 and x Q 2 in population Q while exp( b 1 a P x ) and exp( b 2 a P x ) correspond to the Pareto optimal allocation x P 1 and x P 2 in population P: We want to prove that p1 q1 x P

x Q 1 = u 0 2 x Q 2
and since the allocation ( p1 q1 x P 1 ; p2 q2 x P 2 ) is feasible, we necessarily have, by concavity of u 1 and u 2 ; p1 q1 x P 1

x Q 1 : Proof of Proposition 10. We …rst prove that for x 1 an MLR shift is su…cient to increase the aggregate level of risk tolerance. Suppose that P < M LR Q: Since the MLR order is stronger than the FSD order, we have for all nondecreasing function h;

E P h h b i E Q h h b i ; hence Q (t) P (
t) for t 0: Since Q and P are decreasing, then for all x 1, x : However, MLR does not guarantee an increase of the aggregate level of risk tolerance when x > 1 as shown in the next counter-example.

1 P (x) 1 Q (x) : Since d e Q d e P = dQ dP e b( 1 Q (x) 1 P (x)) E P e b 1 P (x) E Q e b 1 Q (x) ,
Let us consider a model in which the distribution of the risk tolerances is given by exp( b)1 b 0 and let us consider a shift obtained through multiplication of the exponential density by 1 + "1 b>b (for some b > 0 and some " > 0) and by renormalization. The function b ! 1 + "1 b>b is nondecreasing and the shift is MLR. The aggregate level of risk tolerance in the initial population is given by t (a+1) 2 = x 2 : After the shift, the aggregate level of risk tolerance is given by

(x) = R 1
t " (x) = R 1 0 b exp( b) exp( a " b)db + " R 1 b b exp( b) exp( a " b)db R 1 0 exp( b)db + " R 1 b exp( b)db = 1 (a"+1) 2 + " 1 a"+1 b exp ( (a " + 1)b ) + 1 (a"+1) 2 exp ( (a " + 1)b ) 1 + " exp ( b ) where a " solves R 1 0 exp( b) exp( a"b)db+" R 1 b exp( b) exp( a"b)db R 1 0 exp( b)db+" R 1 b exp( b)db = x or 1 a"+1 +" 1 a"+1 exp( (a"+1)b ) 1+" exp( b ) = x: We have then 1 a+1 = 1 a"+1 +" 1 a"+1 exp( (a"+1)b ) 1+" exp( b ) : Let us consider the di¤erence t " (x) t(x). It is positively proportional to = 1 2 + " 1 y exp ( y ) + 1 2 exp ( y ) (1 + " exp ( y )) 1 + " 1 exp ( y ) 2
where = a " + 1 and where 1 a+1 has been replaced by its value as a function of : This quantity is of the form " + " 2 and since we want it to be positive for all " > 0 we have to check if is positive.

(x) 1 P (x)) E P h e b 1 P (x) i E Q h e b 1 Q (x) i = exp( kb b( 1 Q (x) 1 P (x))):
To conclude, it is su¢ cient to show that k + ( 1 Q (x)

1 P (x)) > 0: We have E P h e b( 1 Q (x)+k) i = E Q e b 1 Q (x) E Q [exp(kb)]
< x and since . Since e bt is decreasing and convex for t 0, we have by SSD, Q (t) P (t) for all t 0. For x 1;

1 P (x) is characterized by E P h e b 1 P (x) i = x we have k + ( 1 Q (x) 1 P (
E P e b 1 P (x) E Q be b 1 P (x) E Q e b 1 P (x) = E P1 [b] with dP1 dQ = e b 1 P (x) E Q e b 1 P (x) : Let us consider Q 1 de…ned by dQ1 dQ = e b 1 Q (x) E Q e b 1 Q (x) : We have dQ1 dP1 = E Q e b 1 P (x) E Q e b 1 Q (x) e b( 1 Q (x) 1 P (x)) and is decreasing. Hence E P1 [b] E Q1 [b] = E Q be b 1 Q (x) E Q e b 1 Q (x)
: We have then the result for x 1: For x 1; since both distributions are symmetric with respect to b ; we have

E P be b 1 P (x) E P e b 1 P (x) = E P (2b b)e (2b b) 1 P (x) E P e (2b b) 1 P (x)
= 2b

E P be b 1 P (x) E P e b 1 P (x)
and

E Q be b 1 P (x) E Q e b 1 P (x) = 2b E Q be b 1 P (x) E Q e b 1 P (x)
: Since x) is decreasing and positive, and we have by PD that

E P be b 1 P (x) E P e b 1 P (x) E Q be b 1 Q (x) E Q e b 1 Q (x) or E P1 [b] = E Q be b 1 P (x) E Q e b 1 P (x) E P be b 1 P (x) E P e b 1 P (x)
.

Since dQ1 dP1 = E Q e b 1 P (x) E Q e b 1 Q (x) e b( 1 Q (x) 1 P (x))
; it is now increasing, we have then There exists then x < 1 such that @t @h (x; 0) < 0 for x < x and @t @h (x; 0) > 0 for x > x : Proof of Proposition 14. Let us denote by R(x; h) the aggregate level of risk aversion at x when the individual levels of risk aversion are given by R 1 h and R 2 + h, we have R(x; h) = (R1 h) exp( ln x(R1 h))+(R2+h) exp( ln x(R2+h)) exp( ln x(R1 h))+exp( ln x(R2+h))

and @R @h (x; 0) = e 2 ln xR 1 e 2 ln xR 2 +2 ln x(R2 R1)e ln xR 1 e ln xR 2 (e ln xR 1 +e ln xR 2 ) 2 which is clearly negative for x > 1 and positive for x < 1.

Proof of Proposition 15. Let us denote by U (x; 1) and U (x; 2) the social utility functions respectively associated to P 1 and P 2 : We denote respectively by F (i; 1) and F (i; 2) the cumulative distributions of P 1 and P 2 : We have U 0 (x; j) = Z @u @x (x; i)F 0 (i; j)di; j = 1; 2:

By assumption, @u @x (x; i) is log-supermodular. Furthermore, since P 2 < M LR P 1 ; F 0 (i; j) is also LSPM. By Karlin's Theorem U 0 (x; j) is log-supermodular. Therefore, @ ln U 0 (x;j) @x increases with j or in other words U 00 (x; 1) U 0 (x; 1) U 00 (x; 2) U 0 (x; 2)

  and 5 illustrate the di¤erent possible shapes for b

Proposition 8

 8 Let b 1 and b 2 be given with b 1 < b 2 and let us consider a shift of the form b 1 h and b 2 + h with h > 0: The associated level of collective risk tolerance t x (h) increases (resp. decreases) with h for x 1 (for x 1):

1 2 :

 2 The function a x (b 2 ) is clearly decreasing with b 2 and we have lim b2!b1 a x (b 2 ) = ln x b1 and lim b2!1 a x (b 2 ) = ln 2x

  The function a(b 1 ; b 2 ) decreases then with b 2 and we have lim b2!b1 a(b 1 ; b 2 ) = 1 and lim b2!1 a x (b 2 ) = 0: There exists then b 2 such that a x (b 2 ) = a(b 1 ; b 2 ) and d db2 t x (b 2 ) = 0: It is immediate that for a x (b 2 ) > a(b 1 ; b 2 ) we have d db2 t x (b 2 ) < 0 and for a x (b 2 ) < a(b 1 ; b 2 ) we have d db2 t x (b 2 ) > 0: It su¢ ces to show that a x (b 2 ) and a(b 1 ; b 2 ) cross only once to establish the result. Let us consider b 2 such that a x (b 2 ) = a(b 1 ; b 2 ) and let us compute d db2 (a x (b 2 ) a(b 1 ; b 2 )) at b 2 : By de…nition, we have a(b 1 ; b 2 ) = a x (b 2 ) and we denote it by a : Direct computations give

  a ; b 1; b 2 ) = A b 1 e a b1 + b 2 e a b 2 b 1 + b 2 e a (b1 b 2 ) (b 2 b 1 ) : with A = a b 2 1 e a b1 + a b 2 1 e a b 2 + (a b 1 b 2 e a b1 + a b 1 b 2 e a b 2 b 1 e a b1 b 2 e a b 2 )e a (b1 b 2 ) : By de…nition, we have '(a ; b 1 ; b 2 ) = 0. Replacing b 2 e a (b1 b 2 ) by a (b 2 b 1 ) b 1 b 1 ; we get

(b 2 b 1 ) b 1 b 1 b 2

 12 2 a) = x and b 1 + b 2 exp((b 1 b 2 ) a) a (b 2 b 1 ) b 1 = 0 which can be rewritten as follows exp((b 1 b 2 ) a) = 2x exp(b 1 a) 1 and exp((b 1 b 2 ) a) = a which gives 2x exp(b 1 a) 1 = a (b 2 b 1 ) b 1 b 1 b 2 or 2xb 2 exp(b 1 a) = (ab 1 + 1) (b 2 b 1 ) Remark that exp(b 1 a) 1 + b 1 a and we have then 2xb 2 (b 2 b 1 ) or (2x 1) b 2

b 1 ; b 1 + 1 ;

 111 kh) = x: We get after computations d dh t x (0) (a x (0); b 1 ; b 2 ) where (a; b 1 ; b 2 ) = kb 1 exp((b 2 b 1 ) a)+b 2 exp((b 1 b 2 ) a)+b 1 +kb 2 +a (b 2 b 1 ) (kb 2 b 1 ): For x we have a x (0) 0 and it su¢ ces to impose k b1 b2 to have d dh t x (0) 0 for all x.

b 1 ;

 1 b 2 ) = (b 2 b 1 )(b 2 +b 1 )(k 1) and (0; b 1 ; b 2 ) = (k+1) (b 1 + b 2 ) > 0:It su¢ ces to impose k 1 to have @ @a (0; b 1 ; b 2 ) 0 and then (a; b 1 ; b 2 ) 0 for a 0 and hence d dh t x (0) 0:

1 x Q 1 : 1 b 2 : 1 b 1 = x P 2 1b 2 and since p1 q1 1 and p2 q2 1 we have u 0 1 p1 q1 x P 1 u 0 2 p2 q2 x P 2 :

 111211211122 Remark that the allocation ( p1 q1 x P 1 ; p2 q2 x P 2 ) is feasible in population Q: Let us compare u 0 By construction we have x P 1 Since Pareto optimal allocations are characterized by the condition u 0 1

1 x 1

 11 0 b exp( b) exp( ab)db where a solves R 1 0 exp( b) exp( ab)db = x: We obtain a = and t(x) = 1

1 ( 1 (

 11 But = (exp( b ) exp(b )+ b exp(b )) exp( b ) exp(b ) 2 and is positively proportional to exp ( b ) exp (b ) + b exp (b ) : It is easy to remark that for = 1 b 2 and b su¢ ciently large, this quantity is negative. Let us chose then a pair ( ; b ) for which this quantity is negative and let us take " su¢ ciently small such that the quantity itself is negative and let us …nally take x = b exp( (a"+1)b )+ a"+1) 2 exp( (a"+1)b ) 1+" exp( b ): The resulting shift leads then to a decrease of the collective level of risk tolerance at x: Proof of Proposition 11. We just have to consider the case where x 1:

  x)) > 0: Proof of Proposition 12. Assume that P and Q are symmetric with respect to some b with dQ dP nonincreasing before b and nondecreasing after b then P < P D Q and P < SSD Q ( Jouini-Napp, 2008). Let us denote by P and Q the functions respectively de…ned by P (t) = R e bt dP (b) and Q (t) = R e bt dQ(b)

1 P (x) and 1 Q 1 Q (x) 1 P 1 P

 11111 (x) are positive. Furthermore, both Q and P are decreasing and we have then (x) for all x 1: Since b ! e b (x) is decreasing and positive, we have by PD,

1 P(x) is negative for x 1 ;

 11 the function b ! e b 1 P

E 1 b1b 1 b 1 + b 1 b 2 e ln x b 1 + e ln x b 2 e ln x b 2 b 2 b 1 e ln x b 2 + b 2 e ln x b 1 2 1 b 1 1 b 2 ;(b 2 b 1 ) 1 + e ln x 1 b 1 1 b 2 > 0

 111122111211120 P1 [b] E Q1 [b] which gives the result. Proof of Proposition 13. Let us denote by t(x; h) the aggregate level of risk tolerance at x when the individual levels of risk tolerance are given by b 1 and b 2 + h, we have t(x; h) = exp( ln x b1 ) + exp( ln x b2+h ) exp( ln x b1 ) + 1 b2+h exp( ln x b2+h ) (b 2 b 1 ) (ln x) e ln x and we clearly have @t @h (x; 0) > 0 for x > 1: Furthermore, if we denote by L(x) the quantity L(x) = (b 2 b 1 ) (ln x) + b 1 b 2 1 + e ln x we have dL d ln x = and lim x!0 L(x) = 1 and L(1) = 2b 1 b 2 > 0:

Figure 1 & 2 : 1 = 1 and b 0 2 = 3 1 :Figure 3 : 1 15Figure 4 :

 121231314 Figure 1 & 2: These …gures represent, at two di¤erent scales, the collective relative risk tolerance as a function of the total wealth with b 1 = 1 and b 2 = 2 for the blue curve and b 0 1 = 1 and b 0 2 = 3 for the green curve. Both curves converge slowly to the associated level of relative risk tolerance as can be shown on the second …gure. An increase of the risk tolerance level of the most risk tolerant agent leads to an increase (decrease) of the collective level of risk tolerance above (below) a given threshold x 1: With our parameters, we have x = 0:21: When b 2 = 2 and b 0 2 is in the neighborhood of b 2 ; we have x = 0:14: When b 0 2 = 3 and b 2 is in the neighborhood of b 0 2 ; we have x = 0:27:

  27

Figure 5 :

 5 Figure 5: In a setting with two agents with levels of relative risk tolerance b 1 and b 2 ; we represent the aggregate relative risk tolerance s x (b 2 ) as a function of b 2 for b 1 = 1 and for x = 0:9: For b 2 = b 1 = 1; s x is equal to 1. The aggregate relative risk tolerance increases with b 2 and is almost linear.

  28

Figure 6 :

 6 Figure 6: In a setting with two agents with levels of relative risk tolerance b 1 and b 2 ; we represent the ratio t x (b 2 )=b 2 for x = 4 and b 1 = 1: Forb 2 = b 1 = 1 it is immediate that t x (b 2 )=b 2 = t x (b 2 ) = x:The ratio converges to x 0:5 = 3:5 when b 2 converges to 1:

Figure 7 :

 7 Figure 7: In a setting with two agents with levels of relative risk tolerance b 1 and b 2 ; we represent the ratio t x (b 2 )=b 2 for x = 1:1 and b 1 = 1: Forb 2 = b 1 = 1 it is immediate that t x (b 2 )=b 2 = t x (b 2 ) = x:The ratio converges to x 0:5 = 0:6 when b 2 converges to 1:

Figure 8 : 3 4 and b 0 2 = 9 4

 834 Figure 8: In a setting with two agents with levels of relative risk tolerance b 1 and b 2 ; we represent the collective relative risk tolerance as a function of the total wealth with b 1 = 1 and b 2 = 2 for the blue curve and b 0 1 = 3 4 and b 0 2 = 9 4 for the green curve. The shift from (b 1; b 2 ) to (b 0 1; b 0 2 ) is a Mean Preserving Spread as in Proposition 8 and the two curves cross at x = 1.

  Proposition 7 Let b 1 < b 2 be given. 1. Let us consider a uniform increase of the individual levels of risk tolerance of the form b 1 + h and b 2 + h with h > 0: The associated level of collective risk tolerance t

x (h) increases with h for all x: 2. Let k > 0 be given and let us consider an increase of the individual levels of risk tolerance of the form b 1 + kh and b 2 + h with h > 0: The associated level of collective risk tolerance t x (h) increases with h for x 1 if k b1 b2 and increases with h for x 1 if k 1: In particular, t x (h) increases with h for all x if k 2 h b1 b2 ; 1 i :

  = (b 1 exp ( ab 1 ) + b 2 exp ( ab 2 )) (exp ( ab 2 ) exp ( ab 1 ) + ab 1 exp ( ab 1 ) ab 2 exp ( ab 2 )) (a (exp( ab 1 ) exp( ab 2 )) (b 2 1 exp ( ab 1 ) + b 2 2 exp ( ab 2

	2	exp ( ab 1 ) +	1 2	exp ( ab 2 ) +	1 2	ab 1 exp ( ab 1 )	1 2	ab 2 exp ( ab 2 )
		1 2	b 2 1 exp ( ab 1 ) +	1 2	b 2 2 exp ( ab 2 )	a (exp( ab 1 ) exp( ab 2 )) b 1 exp( ab 1 ) + b 2 exp( ab 2 )
	which is of the same sign as g(a) with
	g(a)

  we obtain that d e

						Q d e P is the product
	(modulo a constant) of the decreasing function e	b(	Q	1	(x)	P	1	(x)) with the de-
	creasing function dQ dP (both of them being positive) and is then decreasing and e P M LR e Q which gives E Q h bi E P h bi and t P x t Q x or R P x R Q

The fact that e¢ cient groups may behave in a complex manner is well known. Pratt and Zeckhauser (1989) showed, for example, that a group may be willing to accept a gamble which combines two individually unacceptable lotteries.

We also show, however, that a uniform increase in the degree of risk tolerance of both individuals unambiguously increases the risk tolerance of the group.

In this literature the objective is to elicit the risk attitude of groups as compared to the members of the group. Another strand of related empirical literature evaluates whether, under uncertainty, groups behave in a more consistent manner than individuals (see e.g.[START_REF] Bone | Are Groups More (or Less) Consistent Than Individuals?[END_REF][START_REF] Charness | Individual and group decision making under risk: An experimental study of Bayesian updating and violations of …rst-order stochastic dominance[END_REF]).

This should not be 'too'surprising given our previous results since for k = 0 we are back in the situation in which only b 2 increases.

Note that Proposition 15 can easily be extended for higher order collective preferences towards risk. For example, if we assume that u 00 (x; i) is LSPM in (x; i), then an MLR dominant shift decreases the collective degree of absolute (and relative) prudence.

Note that this last condition is just a little bit stronger than the LSPM condition.

[START_REF] Eliaz | Choice shifts in groups: A decisiontheoretic basis[END_REF] show that risky and cautious shifts in groups can be seen as a failure of expected utility theory.

which gives R 1 (x) R 2 (x): Proof of Proposition 16. Let us denote by U (x; 1) and U (x; 2) the social utility functions respectively associated to P 1 and P 2 : Since @u @x (x; i) is increasing in i and

is decreasing in i; there exists a nonnegative and decreasing function x such that E h u 0 i (x)

for i 2 I: We have then

:

:


Virgile Garcia

Chung Shue Chen
email: cs.chen@alcatel-lucent.com

Yiqing Zhou

Jinglin Shi

Gibbs Sampling Based Distributed OFDMA Resource Allocation

Keywords: Radio resource allocation, power control, interference management, distributed optimization, Gibbs sampling, Metropolis-Hasting

In this paper, we present a distributed resource and power allocation scheme for multiple-resource wireless cellular networks. The global optimization of multi-cell multi-link resource allocation problem is known NP-hard in the general case. We use Gibbs sampling based algorithms to perform a distributed optimization that would lead to the global optimum of the problem. The objective of this paper is to show how to use the Gibbs sampling (GS) algorithm and its variant the Metropolis-Hasting (MH) algorithm. We also propose an enhanced method of the MH algorithm, based on a priori-known target state distribution, which improves the convergence speed without increasing the complexity. Furthermore, we study different temperature cooling strategies and investigate their impact on the network optimization and convergence speed. Simulation results have also shown the effectiveness of the proposed methods.

I. INTRODUCTION

Cellular networks are facing an exponential increase of capacity demand and a key solution to overcome this problem is to multiply the number of access points or base stations (BSs) in an effective way [START_REF]Small cells -what's the big idea[END_REF]. However, the multiplication of BSs, and in particular when a frequency reuse factor of one is used, would yield substantial inter-cell interference and lead to an interference-limited network.

Radio resource management and interference coordination have been extensively studied in the literature [START_REF] Koskie | Signal-to-interference-based power control for wireless networks: A survey, 1992-2005[END_REF], [START_REF] Chen | Round-robin power control for the weighted sum rate maximisation of wireless networks over multiple interfering links[END_REF], trying to provide optimal or near-optimal schemes. In practice, this class of multi-cell or multi-link resource allocation optimization problems is generally non-convex and very hard to solve, especially in a network with a large number of BSs.

Various stochastic optimization methods have been introducedsuch as particle swarm optimization, genetic algorithm, ant colony or tabu search [START_REF] Nawrocki | Understanding UMTS Radio Network Modelling, Planning and Automated Optimisation: Theory and Practice[END_REF]. These approaches generally require a knowledge of the system, and some need several instances of the system, which are only feasible in an off-line, centralized approach. Different to these complex centralized schemes designed to find a global optimal solution, distributed algorithms have drawn more and more attentions recently due to the low complexity and low requirement on backhauling and processing capability. Most of the time, sub-optimal solutions are provided by distributed algorithms, such as those using game theory framework [START_REF] Altman | A survey on networking games in telecommunications[END_REF], often face the issue of converging to equilibria (e.g., Nash equilibrium) that are sub-optimal solutions.

Gibbs sampling (GS) algorithm [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] is stochastic optimization method that can converge to the global optimal solution of non-convex problem. Under some conditions, it can even be applied in a fully distributed manner, with only local decisions. The Metropolis-Hasting (MH) algorithm [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF] is also a widely used MCMC method and has same methodology, which however has a lower complexity but a slower convergence rate.

The first application of GS for wireless networks aimed to minimize co-channel interference and offer optimal frequency band allocation in WiFi networks [START_REF] Kauffmann | Measurement-based self organization of interfering 802.11 wireless access networks[END_REF]. More recently, the technique has been successfully extended to the user association and power control problem [START_REF] Chen | Self-optimization in mobile cellular networks: Power control and user association[END_REF] and the transmit precoding and power efficiency problem [START_REF] Garcia | Self-optimized precoding and power control in cellular networks[END_REF] for cellular networks. However, these works are limited by the use of a single resource for the optimization problem. In [START_REF] Borst | Non-concave utility maximization in locally coupled systems with applications to wireless and wireline networks[END_REF], the Gibbs sampling method is generalized for any arbitrary utility function, by designing an extended interference graph with respect to cellular networks.

Although theoretical backgrounds of the Gibbs sampling algorithm are well known, its application to radio resource allocation is relatively limited so far and previous works only focused on the classical very slow temperature1 cooling and using only one specific algorithm (either GS or MH).

In this paper, considering a large cellular OFDMA network with multiple resources, we apply GS and MH methods for the resource and power allocation optimization and conduct a comparative study. Also, we investigate the performance of different temperature cooling strategies to improve the convergence speed of the classical Gibbs sampling. Moreover, given a prior statistical knowledge of the solution, we propose an enhanced MH algorithm, namely "guided" MH, to speed up the MH convergence while keeping a low complexity. Simulations are carried out to show the performance of the GS, MH and guided-MH algorithms, compared to a deterministic best response (BR) method.

We show that although the logarithmic cooling is optimal in theory, fast cooling strategies such as exponential cooling can be more suitable for the resource and power allocation problem in practice. Also, we show that the guided-MH algorithm allows a faster convergence compared to the classical MH algorithm without increasing its complexity, and may have other interesting applications.

The rest of the paper is organized as follows. In Section II, the system model is presented and the power and resource allocation problem is formulated. In Section III, we develop the Gibbs sampling formulation and its variants, including the proposed guided-MH algorithm and the various temperature cooling strategies studied. In Section IV, we show how to cast the resource and power allocation problem to use the Gibbs sampling algorithms. In Section V, we provide simulation results under OFDMA multi-cell environment to compare the performance of the studied methods. Finally, Section VI contains the conclusion.

II. SYSTEM MODEL A. OFDMA cellular network

We consider the downlink transmission in a classical multi-cell system. Different to the work in [START_REF] Kauffmann | Measurement-based self organization of interfering 802.11 wireless access networks[END_REF]- [START_REF] Garcia | Self-optimized precoding and power control in cellular networks[END_REF] that focused on a single channel per user, we consider in this paper that a user may take multiple orthogonal resources, as in 3GPP-LTE orthogonal frequency-division multiple access (OFDMA).

The power and resource allocation is performed locally at each BS for their respective users. We consider reuse-1 system, i.e., all resource blocks (RBs) are available at each BS. Each user i ∈ U is associated to the BS b i ∈ B that has the smallest path-loss, where U is the set of users and B is the set of BSs. We consider that the total bandwidth is divided into K orthogonal resources, each experiencing an independent and flat Rayleigh fading.

Notice that under 3GPP-LTE, each user can be allocated any number of RBs. However without multi-user MIMO techniques, a RB cannot be allocated to more than one user within a cell. The signal-to-interference-plus-noise ratio (SINR) of a user i ∈ U on the resource k ∈ [1, K] is thus given by:

γ i,k = P bi,k g bi,i,k bj =bi P bj ,k g bj ,i,k + σ 2 i , (1)
where P bi,k is the transmit power of BS b i destined to the user i on the resource k ∈ Z + , g bi,i,k is the channel gain, including fading, between the BS b i and the user i, and σ 2 i is the thermal noise variance, supposed equal for all users here. Let C i,k denote the data rate achievable by the resource k for user i. The total data rate of a user i is given by:

C i = k∈[1,K] log 2 (1 + γ i,k ). (2)
In this paper, we consider the Shannon capacity, yet any other metric can be employed in the following optimization.

B. Resource and power allocation problem formulation

In OFDMA based system like LTE with no multi-user MIMO, orthogonal resources are assigned to users to avoid intra-cell interference. Thus, aiming at inter-cell optimization and for simplification, a single user is focused for each cell, which can use any of the K resources. However, it should be noted that the result can be easily extended to more general case.

We consider that the BSs can adapt their power level on each resource, P b,k , ∀b ∈ B, ∀k ∈ [1, K], independently. We note P b the vector of power allocation of BS b, and P the global matrix of power allocations.

Considering the data rates of the users as the basic metric of interest, a generic optimization problem is formulated as follows:

min U (C);

(3) s.t. :

(4)

P b,k ≥ 0, ∀b ∈ B, ∀k ∈ [1, K], (5)
k∈[1,K] P b,k ≤ P max , ∀b ∈ B, (6)
where C [C 1 , C 2 , ...] is the vector of aggregated data rates of all the users, and U (•) is a chosen utility function that we will minimize. The total transmit power of a BS is limited by P max . With a little abuse of the name, we simply call U (•) the utility function instead of cost function.

A utility function can be used for different objective, e.g., maximizing system capacity, maximizing fairness, minimizing energy consumption, etc.

It has been shown that applying a fairness utility functions does not only help to maintain resource sharing fairness but also favors a system-wide throughput maximization: the global optimization makes a tradeoff between users' selfish and altruistic behavior [START_REF] Garcia | Self-optimized precoding and power control in cellular networks[END_REF]. Meanwhile, it would also bring energy efficiency merit: transmitters are encouraged to lower their transmit power so as to avoid high interference to neighboring users. Therefore, in this paper, the potential delay fairness [START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF] criterion is employed as utility function:

U (C) = i∈U 1 C i , (7)
where 1/C i represents the time required for the transmission of an unitary size. Indeed, the method allows any arbitrary utility function (see e.g., [START_REF] Borst | Non-concave utility maximization in locally coupled systems with applications to wireless and wireline networks[END_REF]).

III. GIBBS SAMPLING OPTIMIZATION

In this study, we focus on the Gibbs sampling and its variant the Metropolis-Hasting algorithm. The key idea of the GS and MH algorithms in brief is to converge, iteratively, to the global optimal solution by probabilistically transiting among the possible states of the system. Based on the interactions of nodes in the graph, the guided random selection of states allows the optimization to avoid getting trapped in local optimal points.

A. Definitions

To begin with, we define a graph G = (U, V), where U is the set of nodes that represent the set of users, and V is the set of edges between the nodes. G is also associated to the following:

• An edge (i, j) between two nodes i and j is set when either i or j affects the performance of the other.

• Each node i ∈ U has a state s i from a finite state set S i . The sets of states may be different for each node.

• The graph G is associated with the global energy metric E, called global energy, which is a function of all the states of the nodes s = [s 1 , s 2 , ...] and will be minimized in the optimization. Note that G is a Gibbs random field if s follows the distribution π(s) = κ exp(-E(s)/T ), where κ is a normalization factor. T is an important parameter called temperature that scales the probability distribution of the system. It can be a fixed constant or be varied with the time. In the latter case, we say annealing.

A centralized processing unit can perform a global sampling and optimization of the problem; however, distributed solution are much preferred. To be able to distribute the GS and MH algorithms, the global energy has to be represented in a certain structural setup [START_REF] Brémaud | Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues[END_REF]: the global energy E has to be derivable from a sum of potential functions, over all the cliques of the graph such that

E(s) = c∈C V (c, s), ( 8
)
where C is the set of all cliques, and V is the potential function, which represents the contribution of c to the global energy, taking into account users' mutual constraints (conflict and interaction). Based on the above notion, we define the local energy E i , ∀i ∈ U, as follows:

E i (s) = c∈C|i∈c V (c, s). (9)
The local energy function of a node i only depends on the states and interactions of nodes in a neighborhood around i, i.e., it does not require a system global knowledge. Performing an iterative stochastic minimization of the local energies, this optimization would lead to the global minimum of the global energy.

B. Classical sampling methods

The basic idea of the sampling process is as follows: iteratively, each node updates its state based on the current evaluation of the decoupled system energy and on the impact of the new (to be chosen) state. The selection of the new state depends on the algorithm used:

1) Gibbs sampler algorithm: The GS algorithm [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] consists of, when the node i updates its state, evaluating all the possible states s ∈ S i and select a new state following a probability distribution. First, we compute the local energy E i (s), ∀s ∈ S i . Secondly, we define the probability distribution based on the local energy of each state. The probability that the state s i ∈ S i will be chosen is given below:

p i (s i ) = exp(-E i (s i )/T ) s∈Si exp(-E i (s)/T ) . ( 10
)
Thirdly, we choose a new state (configuration) to be applied to i based on the Gibbs probability distribution [START_REF] Garcia | Self-optimized precoding and power control in cellular networks[END_REF]. This method requires the computation of the local energy for all s i ∈ S i of a node. Note that the optimization therefore should apply on a reasonable number (size) of possible states.

2) Metropolis-Hasting algorithm: The main difference between the GS method and the MH method is that the latter will only consider a single candidate state during the update process. Consider a node i ∈ U, and assume that it is currently associated with a state s 1 ∈ S i . When it updates its state, it will randomly choose a candidate state s 2 ∈ S i and evaluate its performance. The node i will then select the new state s 2 over s 1 with the following probability:

p i (s 2 ) = min(1, exp( E i (s 1 ) -E i (s 2 ) T )). (11)
In other words, if the candidate state has a lower local energy, it will always be chosen. If its local energy is higher, it will be chosen with a probability, given by [START_REF] Borst | Non-concave utility maximization in locally coupled systems with applications to wireless and wireline networks[END_REF].

The complexity and computation cost of the MH compared to the GS algorithm is divided by the number of possible states for the node, due to the single candidate evaluation of the MH algorithm.However, the convergence time of MH algorithm can also be longer, due to a slower sampling of the whole system, especially when the set of states is large.

C. Proposed guided MH algorithm

To improve the MH selection and speed up its convergence of optimization, we propose a guided version of the MH algorithm. This guided MH method relies on a priori known distribution of the solution: We modify the uniform random candidate state selection to follow a predefined distribution, which is the distribution of the states that is expected to be reached at the end of the optimization. This distribution can be found by offline trials or through some learning processes.

The objective of this modification is to focus the sampling on candidates that have higher probability of being optimal than the other. After the guided selection of a candidate, this algorithm acts similarly as the original MH algorithm. The complexity of this method is similar to the one of the standard MH algorithm, since only one candidate state is evaluated at a time.

Note that all the three methods presented share the same complexity for the evaluation of a given local state, and also require the same exchange in information among neighboring base stations and users for evaluating the states: to determine E i (s) for the optimization, the information to be collected and feedback should include the signal and interference powers acting on the considered node and its neighbors.

The algorithm proposed here is used for our particular resource allocation problem, but could indeed be applied to other similar optimization problems.

D. Annealing and cooling strategies 1) Classical logarithmic cooling:

The temperature parameter T can be chosen to decrease with time to perform a simulated annealing. Given a Gibbs random field, it has been proven that Boltzmann annealing, where the temperature T decreases inversely proportional to the logarithm of time, is the sufficient cooling condition that guarantees the convergence of the system to global optimum as time goes to infinity [START_REF] Brémaud | Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues[END_REF].In brief, at a discrete time n > 1, the temperature is defined as:

T B (n) = T 0 log(n) , (12)
where T 0 is the initial temperature. The Boltzmann annealing may however be very slow to stabilize, since the temperature is decreasing in logarithmic scale. The optimization performed by simulated annealing is guaranteed for static network topology. However, it should be noted that in cellular networks, the system may vary with time, for example due to fast channel fading and/or user arrival and departure. In this paper, we assume that the system is quasi-static.

2) Fast cooling strategies: In the literature, the research on resource allocation using Gibbs sampling methods only focused on the classical logarithmic cooling strategy. We here propose to speed up the convergence, by applying faster cooling strategies:

• The "fast" annealing in [START_REF] Szu | Nonconvex optimization by fast simulated annealing[END_REF] uses a Cauchy distribution and decreases the temperature in a linear manner:

T C (n) = T 0 n , (13)
• The "very fast" annealing reported in [START_REF] Ingber | Very fast simulated re-annealing[END_REF] decreases the temperature in an exponential manner:

T V F (n) = T 0 exp(an) , ( 14
)
where a is a tunable parameter. It should be noted that there does not exist mathematical proofs that the above linear and exponential cooling strategies guarantee to converge to global optimum, under Gibbs sampling setup. Under some sampling setups (modifying the state distribution selection), the linear and exponential cooling strategies have been shown to converge to the optimal solution [START_REF] Ingber | Very fast simulated re-annealing[END_REF]. However, it is not clear how to cast them into distributed optimizations. Advanced studies on their implementability and convergence will be subject of future work. Here, we apply the faster cooling strategies directly into the Gibbs sampling algorithm, similarly as [START_REF] Arora | A Gibbs sampler approach for optimal distributed monitoring of multi-channel wireless networks[END_REF] and we have to rely on simulation results to study its performance.

IV. APPLICATION TO THE RESOURCE AND POWER ALLOCATION PROBLEM A. Graph of interference

First, we define a graph to cast the problem of resource and power allocation into the Gibbs sampling framework. The graph that characterizes the system, as illustrated in Fig. 1, is defined as follows:

• The set of nodes is the set of all users U; • An edge (i, j) is set when the transmit power of BS b i received at j (or the power from b j received at i) is higher than a given threshold θ p . The set of neighbors of i is denoted by N i ; • The state s i of a node i is the power level on the K resources assigned to it, i.e., s i = P bi .

B. Utility function definition

Following [START_REF] Chen | Self-optimization in mobile cellular networks: Power control and user association[END_REF] and the development of [START_REF] Kauffmann | Measurement-based self organization of interfering 802.11 wireless access networks[END_REF], the local energy to be minimized by a node i is defined by the sum of local utilities in its neighborhood:

E i (s) = U i (s) + j∈Ni U j (s), (15)
where U i is the local utility of user i.

In a LTE-like system, one has to perform the optimization over several parallel resources (e.g. OFDMA). As aforementioned, we focus here on a fair approach that minimizes the potential delay of all users, see [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]. As shown in [START_REF] Garcia | Self-optimized precoding and power control in cellular networks[END_REF]), a representation of the global utility ( 7) by ( 8) is not trivial, due to the multiple resources and the logarithmic function [START_REF] Koskie | Signal-to-interference-based power control for wireless networks: A survey, 1992-2005[END_REF].

In [START_REF] Borst | Non-concave utility maximization in locally coupled systems with applications to wireless and wireline networks[END_REF], it has been recently proven that in fact, we can extend the global energy to any utility function, at the price of a more information exchange among nodes: information exchange has to be performed not only among the considered node and its neighbors, but also to its neighbors' neighbors. We can therefore set the global energy to be out utility function [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF] and the local energy [START_REF] Ingber | Very fast simulated re-annealing[END_REF] can be rewritten as:

E i (s) = 1 C i (s) + j∈Ni 1 C j (s) . ( 16
)
We can interpret this local minimization as a trade-off between a user's utility and its neighbors' one. Increasing the transmit power of a given user to improve its data-rate will reduce its neighbors' one, due to the higher level of interference.

C. Sequential state selection

The Gibbs sampling requires the evaluation of the local energy for all possible states of the update node. In the present work, the allocation is performed over several resources. In our multiple resources and power allocation problem, the maximum total number of possible states for a node i is |S i | = L K , where L is the number of power levels possible for a resource.

The cardinality of the set of local states can rapidly become impractical for a large L or K. To overcome this issue, we choose to perform a "sequential" optimization: when a node updates its state, it will take a random permutation of all the possible resources and sequentially determine a new state considering that other resources are fixed. This leads to a maximum number of combination to be evaluated equal to LK, instead of taking a full size of L K .

V. SIMULATION RESULTS

A. Scenarios and simulation parameters

Simulations are carried out to verify the performance of the presented methods and cooling strategies, in solving the resource and power allocation optimization problem.

The system employs a topology with 64 BS placed in classical hexagonal layout. We focus the optimization on a given chunk of K resources blocks. On this chunk, a single user is allocated per station. To avoid border effects, we use a wrap-around model so that locations at the edge of the system are considered adjacent to the opposite edge. The transmit power, on each resource, can be chosen from a set including the 0 value, and a logarithmic partition of the power level from -10 dBm to 43 dBm, with 1 dB step size. Other simulation parameters are summarized in Table I.

In the simulation, one iteration corresponds to the period of time where every node has been updated sequentially but in a random ordering. Here, we assume that no two base stations will conduct a sampling exactly at the same time. This will apply generally to all the algorithms in the comparison. The results are obtained by averaging over 100 independent simulations (random user location and channel condition), each running for 200 iterations.

We aim at comparing the GS, the MH, and the guided-MH using the Boltzmann annealing (labeled as: GS-B, MH-B and guided-MH-B, resp.), as well as using the Cauchy annealing (labeled as: GS-C, MH-C and guided-MH-C, resp.) and using the exponential cooling (labeled as: GS-E, MH-E and guided-MH-E, resp.). Note that the exponential cooling is parametrized so that T (n) = 0.95 n T 0 , which is an empirical choice that fits our model [START_REF] Ingber | Very fast simulated re-annealing[END_REF], [START_REF] Arora | A Gibbs sampler approach for optimal distributed monitoring of multi-channel wireless networks[END_REF].

Additionally, the performance of a deterministic Best-Response (BR) algorithm is also shown for comparison: the BR algorithm acts similarly as the GS one, except that it always chooses the state that minimizes the local energy. This method is sub-optimal and may converge to a local minimum of the system. This local minimum is a generalized Nash equilibrium (GNE) [START_REF] Facchinei | Generalized Nash equilibrium problems[END_REF].

B. Initial temperature setting

As a first step, and to efficiently use the Gibbs sampling and their variants, one should properly choose the initial temperature T 0 . Except by studying the thermodynamic structure of the global system [START_REF] Andresen | On lumped models for thermodynamic properties of simulated annealing problems[END_REF], there is no theoretical way to optimally determine the temperature. Fig. 2 shows the performances obtained by the GS algorithm using the logarithmic (i.e., Boltzmann: GS-B) and exponential (GS-E) cooling for some values of T 0 , in comparison to the BR. Note that the results shown are particularly smooth due to the averaging over 100 independent simulations. For a single simulation, the stochastic methods progress with ups and downs before converging to a stable point value.

It is observed that when T 0 is too low (i.e., T 0 < 0.001), the randomness in the probabilistic sampling is not sufficient and the optimization acts similarly to the BR approach. On the other hand, when T 0 is too high, the convergence would take a much longer time and may not be favorable for practical usage in wireless networks, especially under the Boltzmann cooling strategy (GS-B). Besides, we can see that the different cooling strategies have different ideal initial temperature. Since the exponential cooling strategy reduces the temperature much faster than the Boltzmann one, it tends to allow higher T 0 while still converging fast enough. The result of the linear cooling strategy (GS-C) is not plotted in Fig. 2 for the readability. It follows the same trend as the exponential cooling one and its final result can be found in Fig. 3 later.

One can also see that the choice of T 0 and cooling strategies should be adapted for different scenarios, to balance convergence time and accuracy of the solution. It is still an open problem to derive optimal T 0 in analytic form. However, the present results are valid for scenarios where the variables and possible states are in a similar system setup.

C. Cooling strategies

We now evaluate the performance of the various sampling methods and their cooling strategies. Based on previous result and some experimental investigation, we set the initial temperature to be T 0 = 0.001 for the Boltzmann cooling strategy and T 0 = 0.1 for the Cauchy and exponential ones. These choices are motivated by the performance obtained after a reasonable number of iterations.

Fig. 3 shows the global energy utility [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF], as the iterations of the optimization go on. Note that this global energy is equivalent to the sum of delays experienced by all users in the system. It is observed that the BR method converges most rapidly (in about 40 iterations). However, its final result is not optimal. In the stochastic algorithms, in particular the GS, their performances would depend on the cooling strategy. The initial temperature has been chosen so that a convergence can be reached in a finite (relatively short) time. To do so, the Boltzmann cooling strategy requires a low initial temperature and its performance is limited by the lack of randomness in the stochastic process. The final result of the GS-B strategy is thus close to the BR one but can still outperform it (but slightly). The exponential and Cauchy cooling strategies, using a higher T 0 , are able to converge to better final solutions. Their performances during the initial iterations appear not favorable but soon or later outperform (in about 100 to 150 iterations). One can see that when looking at about 100 to 200 iterations, the GS-E outperforms all the other methods.

D. Design of a guided MH algorithm

In the simulation study, we see that the resulting OFDMA resource block and power allocation follows a particular distribution empirically as follows.

First, users use on average only 35% of their available resource blocks, and only focus on efficient channels (of low interference) to transmit data. This can be assimilated to a dynamic frequency reuse scheme to avoid high level of interference.

Second, the distribution of the power allocated to the used resource blocks follows a log-normal distribution, centered at 26 dBm and with a standard deviation 9.5, as shown in Fig. 4. We can see a particularly high intensity at 42 dBm due to the defined per-BS total power limitation: when a node updates its use of resource blocks, it is limited by the total power of the station (here, 43 dBm). In many cases, at least at least one resource block is already used with some power, so, the chosen maximum power is usually less than 43 dBm. 42 dBm is therefore often a limit when some other resources are allocated with low power levels.

The distribution obtained here is system dependent, but is based on 100 independent realizations of the 64-cells hexagonal scenario. In a non-regular topology, each cell may define its state distribution according to its particular environment. We expect that the knowledge of the distribution can be obtained by the system through learning procedures or with off-line simulation results. When the actual optimal distribution of each realization cannot be determined in practical system, one may use average scenario results. Note that the estimated distribution might not be the optimal choice for every realization, which can reduce the convergence speed compared to the optimal distribution.

Given the above distribution (Fig. 4), instead of using uniform distribution in MH sampling when choosing a candidate state, we adopt the aforementioned distribution to improve the MH sampling for its convergence and optimality. We will show the simulation result of the guided-MH scheme in next section for comparison. Notice that the following results are shown to demonstrate the effectiveness of the method, however, the principle can be applied generally to different probabilistic algorithms.

E. Gibbs sampling vs. MH sampling

We compare the performance of the different sampling methods studied in this paper. Fig. 5 shows the global energy of the system by the GS, MH and guided-MH algorithms, with the Boltzmann and exponential cooling strategies. The result of linear cooling strategy is not displayed for readability of the figure.

Comparing the above stochastic sampling methods, we see that the GS converges faster than the MH and the guided-MH. The MH and guided-MH methods may achieve similar final solution but would take a (much) longer time. On the other hand, it is observed that they still outperform the BR in a limited time (after about 140 iterations) under the exponential cooling. It is worth recalling that the computation requirement of GS is higher than that of MH due to the evaluation of a node's entire state space.

Comparing the performance of guided-MH and MH, we see that the guided-MH which uses the predefined distribution can improve the exponential cooling scheme (guided-MH-E vs. MH-E) before a final convergence. This is due to the fact that when the temperature of the system is high, the effect of guided sampling would be more significant since it will avoid the random sampling spend too much time on "likely not good" states. When the temperature is low (e.g., at convergence or using a low T 0 ), the MH and guided-MH methods show very similar performance. Recall that the complexity of the guided-MH is not increased compared to classical MH, the guided sampling could be a very good option in some applications.

VI. CONCLUSION

In this paper, we investigated the application of Gibbs sampling method and its variants for performing network optimization in a large and practical OFDMA system. We studied the impacts of sampling methods and temperature strategies on the global performance and proposed strategies of initial parameter setting. As expected, the initial temperature and cooling strategy has a fundamental role in the optimization process. Their impact could be even more important than the sampling selection method to use. Exponential cooling is the most promising cooling strategy to obtain an efficient fast optimization, while the logarithmic cooling may require a long convergence time. The MH provides a slower optimization than GS, due to a limited selection of the states. However, its lower complexity allows larger system and requires less information exchange in each iteration. We also proposed an improvement of the MH algorithm, namely guided-MH, which can converge more rapidly than the classical MH algorithm. Since the guided MH algorithm has a low computation cost, it could be considered as an alternative of cost-effective method.

Fig. 1 :

 1 Fig. 1: An illustration of the interference graph: the set of nodes is the set of the users and the edges are represented by the green lines, transmitted signal is denoted by blue solid line, and interference is denoted by blue dashed line.

Fig. 2 :

 2 Fig. 2: Performance comparison for various initial temperatures.

Fig. 3 :Fig. 4 :

 34 Fig.3: Global energy of the system using different cooling strategies for the Gibbs sampling and the Metropolis-Hasting.

Fig. 5 :

 5 Fig.5: Global energy of the system comparing the sampling methods for the logarithmic and exponential cooling strategies.

TABLE I :

 I Simulation parameters

Note that in GS and MH algorithms, a key parameter called temperature will tune the state sampling and transition and impact the speed and optimality of the convergence.

ACKNOWLEDGEMENT This work has been partially funded by the Beijing Natural Science Foundation Major Project (4110001), the National Science and Technology Major Project (2012ZX03003005-004) and the 2010 Main Direction Program of Knowledge Innovation of Chinese Academy of Sciences. The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318115. Part of the work presented in this paper has been carried out at LINCS (www.lincs.fr).