Nahla Ben Amor

Fatma Essghaier Larodec

Hélène Fargier

Egalitarian Collective Decision Making under Qualitative Possibilistic Uncertainty: Principles and Characterization

Keywords: Decision under Uncertainty, Possibility Theory, Collective Choice, Egalitarianism, Timing Effect

This paper raises the question of collective decision making under possibilistic uncertainty; We study four egalitarian decision rules and show that in the context of a possibilistic representation of uncertainty, the use of an egalitarian collective utility function allows to get rid of the Timing Effect. Making a step further, we prove that if both the agents' preferences and the collective ranking of the decisions satisfy Dubois and Prade's axioms (1995), and particularly risk aversion, and Pareto Unanimity, then the egalitarian collective aggregation is compulsory. This result can be seen as an ordinal counterpart of Harsanyi's theorem (1955).

Introduction

The handling of collective decision problem under uncertainty resorts on (i) the identification of a theory of decision making under uncertainty (DMU) that captures the decision makers' behaviour with respect to uncertainty and (ii) the specification of a collective utility function (CUF) as it may be used when the problem is not pervaded with uncertainty. One also needs to precise when the utility of the agents is to be evaluated: before (ex-ante) or after (ex-post) the realisation of the uncertain events. In the first case, the global utility function is a function of the DMU utilities of the different agents; in the second case it is an aggregation, w.r.t. the likelihood of the final states, of the collective utilities of the states. For instance, in the probabilistic framework, the comparison of decisions is based on the expected utility model axiomatized by von Neumann and [START_REF] Morgenstern | Theory of Games and Economic Behavior[END_REF]. When several agents are involved, utilitarianism prescribes the maximization of the sum of the expected utilities (ex-ante) or the expected utility of the sum of the agents' utilities (ex-post) -both actually coincide. An egalitarian approach can be based either on the min of the expected utilities (ex-ante), or on the expected utility of the least satisfied of the agents (ex-post) -but the two approaches can lead to divergent rankings. This phenomenon has been called the "Timing Effect" by [START_REF] Myerson | Utilitarianism, egalitarianism, and the timing effect in social choice problems[END_REF] in the early eighties.

Following [START_REF] Fleming | A cardinal concept of welfare[END_REF], [START_REF] Harsanyi | Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility[END_REF] showed that if the collective preference satisfies von Neumann and Morgenstern's axioms (i.e. when the ex-ante preference is an expected utility) and the preference relations of the agents also satisfy these axioms (i.e. when the ex-post preferences also follow the EU model) then the sole possible collective decision making approach (satisfying Pareto unanimity) is the utilitarian one. Making a step further [START_REF] Myerson | Utilitarianism, egalitarianism, and the timing effect in social choice problems[END_REF] proved that only the choice of an utilitarian social welfare function can reconciliate the ex-ante and ex-post approaches.

These results rely on the assumption that the knowledge of the agents about the consequences of their decisions is rich enough to be modelled by probabilistic lotteries. When the information about uncertainty cannot be quantified in a probabilistic way the topic of possibilistic decision theory is often a natural one to consider [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF][START_REF] Dubois | Making decision in a qualitative setting: from decision under uncertainty to case-based decision[END_REF][START_REF] Giang | A qualitative linear utility theory for spohn's theory of epistemic beliefs[END_REF][START_REF] Dubois | Decision theoretic foundations of qualitative possibility theory[END_REF][START_REF] Dubois | Qualitative decision theory: From savage's axioms to nonmonotonic reasoning[END_REF][START_REF] Dubois | Qualitative decision theory with preference relations and comparative uncertainty: An axiomatic approach[END_REF]. The present paper raises the question of collective decision making under possibilistic uncertainty. The next Section recalls the basic notions on which our work relies (decision under possibilistic uncertainty, collective utility functions, etc.). We then present four egalitarian possibilistic utilities and show that if both the collective preference and the individual preferences do satisfy Dubois and Prade's axioms (1995), and in particular risk aversion, then an egalitarian CUF is mandatory. This theorem can be considered as an ordinal counterpart to Harsanyi's theorem. For space reasons, proofs are omitted; they can be found online at url ftp://ftp.irit.fr/IRIT/ADRIA/PapersFargier/aaai15.pdf.

Background Collective utility functions

Let us consider a multi-agent decision problem defined by a set A = {1, . . . , p} of agents, each agent i ∈ A being supposed to express her preferences on a set of alternatives (say, a set X), by a ranking function or a utility function u i that associates to each element of X a value in a subset of R + (typically in the interval [0, 1]). The problem is then to determine, for each x ∈ X, a collective utility degree that reflects the collective preference.

When this collective preference depends only on the individual utilities of the agents, the collective utility can be obtained by a collective utility function (CUF; for more details about collective utility functions see [START_REF] Moulin | Axioms for Cooperative Decision making[END_REF])) of the form u(x) = f (u 1 (x), . . . , u p (x)). Classical utility theory prescribes that the best decisions are those that maximize the sum of the individual utilities, i.e.:

u(x) = Σ i∈A u i (x)
This function possesses several good properties but fails to ensure equity between agents. The egalitarian approach on the contrary proposes to maximize the satisfaction of the least satisfied agent, i.e. the CUF:

u(x) = min i∈A u i (x)
When the agents are not equally important (e.g. in an administration board, or when the aim is more to aggregate criteria than to satisfy a group), a weight w i can be associated to each i; this yields the use of a weighted sum in the utilitarian case:

u(x) = Σ i∈A w i • u i (x)
or of a weighted minimum in the egalitarian case:

u(x) = min i∈A max((1 -w i ), u i (x))

Multi-agent decision making under risk

In presence of risk, i.e. when the information about the consequences of decisions is probabilistic, a popular criterion to compare decisions is the expected utility model axiomatized by von Neumann and [START_REF] Morgenstern | Theory of Games and Economic Behavior[END_REF]: an elementary decision is modelled by a probabilistic lottery over the set X of its possible outcomes. The preferences of a single decision maker are supposed to be captured by a utility function assigning a numerical value to each outcome. The evaluation of a lottery is performed through the computation of its expected utility (the greater, the better) 1 . When several agents are involved, two approaches are possible, depending on when the utility of the agents is to be evaluated: after or before the realization of the uncertain events. The ex-post approach comes down to a problem of mono-agent decision making under uncertainty (this agent being "the collectivity") by defining the utility function u as a CUF. On the contrary, the ex-ante approach combines the DMU utilities of the different agents with the collective utility function.

In the probabilistic context, utilitarianism comes down to calculate either the expected collective utility (ex-post), or the aggregation of the individual expected utilities (ex-ante). Egalitarianism prescribes to maximize either the expectation 1 Notice that this kind of modelling is orthogonal to the one more recently investigated in Computational Social Choice [START_REF] Konczak | Voting procedures with incomplete preferences[END_REF][START_REF] Mattei | Decision making under uncertainty: Social choice and manipulation[END_REF][START_REF] Lang | Winner determination in voting trees with incomplete preferences and weighted votes[END_REF]; in the lottery framework, the uncertainty pertains to the consequences of the decisions and the agents have well established preferences on these consequences; in the latter works, the preferences of the agents are ill-known and can be represented or elicited in sophisticated ways. Besides, the purpose of these works is generally more related to complexity theory (determine the complexity of sophisticated decision rules or the one of manipulating the issue of the vote, etc.).

of the minimum of the satisfaction degrees or the minimum of the mathematical expectations. The two approaches do not always coincide: this is the so called Timing Effect.

Counter-example 1. Consider two agents 1 and 2, two consequences x1 and x2 and the probabilistic lotteries L1 and L2 given in Figure 1. The expected value of the minimum of the utilities are: 0.7 * min(0.3, 0.5) + 0.3 * min(1, 0.4) = 0.33 f or L1 0.2 * min(0.3, 0.5) + 0.8 * min(1, 0.4) = 0.38 f or L2 So, ex-post, L2

L1. On the contrary, ex-ante, computing the minimum of the expected utilities leads to L1 L2. Indeed: min(0.7 * 0.3 + 0.3 * 1, 0.7 * 0.5 + 0.3 * 0.4) = 0.47 f or L1 min(0.2 * 0.3 + 0.8 * 1, 0.2 * 0.5 + 0.8 * 0.4) = 0.42 f or L2. 0.7 0.3

x 1 : <0.3, 0.5> L 1

x 2 : <1, 0.4> 0.2 0.8

x 1 : <0.3, 0.5> L 2

x 2 : <1, 0.4> Figure 1: Two probabilistic lotteries in a bi-agent context. The probability of x i according to a lottery labels the corresponding edge; each x i is labelled by its vector of utilities.

In 1955, Harsanyi provided a theorem that is often interpreted as a justification of utilitarianism; he showed that if (i) the collective preference satisfies von Neumann and Morgenstern's axioms ( 1944), (ii) the preferences of each agent also satisfy these axioms, and (iii) if two lotteries are indifferent for each agent they are considered as collectively indifferent (Pareto indifference axiom), then the only appropriate collective CUF is the classical utilitarian one. [START_REF] Myerson | Utilitarianism, egalitarianism, and the timing effect in social choice problems[END_REF] proved that, in the probabilistic context, only the use of an affine collective aggregation function overcomes the Timing Effect, and conversely, that any attempt to introduce equity causes a divergence between the ex-post and ex-ante approaches.

(Mono agent) Decision Making under Possibilistic Uncertainty

Harsanyi's and Myerson's results are strongly related to the assumption of a probabilistic uncertainty and are valid only in such a rich and sophisticated context. When the information about uncertainty cannot be quantified in a probabilistic way, the topic of possibilistic decision theory is often a natural one to consider.

The basic building block in possibility theory is the notion of possibility distribution. Let S be a variable whose value is ill-known and Ω its domain. The knowledge about the value of S is encoded by a possibility distribution π : Ω → [0, 1]; given ω ∈ Ω, π(ω) = 1 means that realization of ω is totally possible and π(ω) = 0 means that ω is impossible. It is assumed that π is normalized, i.e. that there exist at least one ω which is totally possible.

From π, one can compute the possibility Π(A) and the necessity N (A) of an event A ⊆ Ω: Π(A) = sup ω∈A π(ω) evaluates to which extent A is consistent with the knowledge represented by π, while

N (A) = 1 -Π( Ā) = 1 - sup ω /
∈A π(ω) corresponds to the extent to which Ā is impossible and thus evaluates at which level A is certain.

Giving up the probabilistic quantification of uncertainty yielded to give up the EU criterion as well. The development of possibilistic decision theory has lead to the proposition and the characterization of a series of possibilistic counterparts of the EU criterion [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF][START_REF] Dubois | Making decision in a qualitative setting: from decision under uncertainty to case-based decision[END_REF][START_REF] Giang | A qualitative linear utility theory for spohn's theory of epistemic beliefs[END_REF][START_REF] Dubois | Decision theoretic foundations of qualitative possibility theory[END_REF][START_REF] Dubois | Qualitative decision theory with preference relations and comparative uncertainty: An axiomatic approach[END_REF]. Following [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] a one stage decision is modelled by a (simple) possibilistic lottery, i.e. a normalized possibility distribution over a finite set of outcomes X. In a finite setting, a possibilistic lottery L can be written L = λ 1 /x 1 , . . . , λ n /x n where λ j = π L (x j ) is the possibility of getting outcome x j when choosing decision L; this degree are also denoted by

L[x j ].
A compound possibilistic lottery is a normalized possibility distribution over a set of (simple or compound) lotteries. We shall denote such a lottery L = λ 1 /L 1 , . . . , λ m /L m , λ i being the possibility of getting lottery L i according to L.

The possibility π i,j of getting consequence x j from L i depends on the possibility λ i of getting L i and on the possibility λ j i of getting x j from L i (for the sake of simplicity, we assume that the L i 's are simple lotteries; the principle extends to the general case); in other words, π i,j = min(λ i , λ j i ). The possibility of getting x j from L = λ 1 /L 1 , . . . , λ m /L m is simply the max, over all the L i 's, of the π i,j . In decision theory, a compound lottery is generally assumed to be indifferent (according to the DM's preference) to the simple lottery defined by:

Reduction(L) = max i=1,m min(λ 1 i , λ i )/x 1 , . . . , max i=1,m min(λ n i , λ i )/x n Example 1.
The following figure provides an example of a possibilistic compound lottery with its reduction: (Dubois and Prade 1995) then proposed two global utilities for evaluating any simple lottery:

U -(L) = min x∈X max(n(L[x]), u(x)) U + (L) = max x∈X min(L[x], u(x))
where n is an order reversing function (e.g. n(x) = (1x)). Pessimistic utility U -(L) estimates to what extent it is certain (i.e. necessary according to a measure N ) that L is good. Its optimistic counterpart, U + (L) estimates to what extent it is possible that L is good. They extend to any kind of possibilistic lottery, by considering that the utility of a compound lottery is simply the one of its reduction.

To the best of our knowledge, the question of multi-agent decision making under possibilistic uncertainty has never been studied. Beyond the proposition of egalitarian utilities that suits possibilistic knowledge, we show in the present paper that they do not necessarily suffer from the Timing Effect, and we provide a representation theorem that can be viewed as an ordinal counterpart of Harsanyi's one.

Egalitarian collective decision making under possibilistic uncertainty

In the more qualitative, ordinal, case of possibilistic lotteries, four egalitarian utilities can be proposed (Ben Amor, Essghaier, and Fargier 2014) -two pessimistic utilities and two optimistic ones2 : 2in

Definition 1.

U -min ante (L) = min i=1,p max(1 -wi, min x∈X max(ui(x), 1 -L[x])) U -min post (L) = min x∈X max(1 -L[x], min i=1,p max(1 -wi, ui(x))) U + min ante (L) = min i=1,p max(1 -wi, max x∈X min(ui(x), L[x])) U + min post (L) = max x∈X min(L[x], min i=1,p max(1 -wi, ui(x)))
Example 2. As a matter of fact, consider two agents, the first agent being less important than the second one (w1 = 0.6, w2 = 1), and the simple lotteries L1, L2 on X = {x1, x2, x3} depicted3 in Figure 3. We have:

U -min ante (L1) = min( max(1 -0.6, min max(1 -1, 0.8), max(1 -0.9, 0.1)), max(1 -1, min max(1 -1, 0.1), max(1 -0.9, 0.8))) = 0.1 U -min post (L1) = min( max(1 -1, min(max(1 -0.6, 0.8), max(1 -1, 0.1))), max(1 -0.9, min(max(1 -0.6, 0.1), max(1 -1, 0.8)))) = 0.1 U + min
ante (L1) = min( max(1 -0.6, max(min(1, 0.8), min(0.9, 0.1))), max(1 -1, max(min(1, 0.1), min(0.9, 0.8)))) = 0.8 U + min post (L1) = max( min(1, min(max(1 -0.6, 0.8), max(1 -1, 0.1))), min(0.9, min(max(1 -0.6, 0.1), max(1 -1, 0.8)))) = 0.4

U -min ante (L 2 ) = U -min post (L 2 ) = U + min ante (L 2 ) = U + min post (L 2 ) = 0.8.
L 2 1 x 3 : <0.8, 0.8> L 1 1 0.9

x 1 : <0.8, 0.1>

x 2 : <0.1, 0.8> It is obviously possible to define in the same way a series of utilitarian possibilistic utilities (U -sum ante , U +sum ante , etc.) and a series of max-oriented ones (U -max post , U + max post , etc. ). The present paper prefers to focus in details on the egalitarian ones, that look more coherent, and also more appealing from an ethical point of view. First of all, it appears that the coincidence between ex-post and ex-ante approaches does not imply utilitarianism. It is indeed easy to show that: Proposition 1. U -min ante (L) = U -min post (L). We shall thus simply use the notation U -min . Such a coincidence does not happen in the "optimistic" case; with U + min post a lottery is good as soon as there exists a possible outcome satisfying all the agents; with U + min ante , a lottery is good when each agent forecasts an outcome that is good for her (but it is not necessarily the same one for all): it may happen that U + min post (L) < U + min ante (L), as shown by Counterexample 2. It holds that:

Proposition 2. U + min post (L) ≤ U + min ante (L).

Axioms for collective qualitative decision making

Let us now propose an axiomatization of the pessimistic egalitarian utility (we denote it U -min = U -min post = U -min ante ). Consider a set A of p agents, a finite set of consequences X, a possibilistic scale V , the set of possibilistic lotteries L obtained from V and X. The preference profile 1 , . . . , p gathers the preference relations i of each agent i on L. denotes the collective preference on L.

Let us denote x the "constant" lottery leading to consequence x for sure (s.t. L[y] = 0 for each y = x ; e.g. L 2 in Figure 3): constant lotteries and elements of X are identified. In the same way, let Y be the lottery that represents a subset Y of X (it provides the possibility degree 1 to each y ∈ Y , and 0 otherwise). First of all, we formulate a continuity axiom on the consequences:

Axiom C (Continuity on X): ∀x, y ∈ X, ∀B ⊆ A, ∃z ∈ X such that: z ∼ i x if i ∈ B and z ∼ j y if i / ∈ B.
This axiom requires that there exists a z in X that is indifferent to x for the agents in B and indifferent to y for the others. When two agents are involved, Axiom C says that if x and y are two elements of X, then X contains a z corresponding to the vector of satisfaction x i , y j . More generally, this axiom requires the set of lotteries to be rich enough to contain all the constant acts corresponding to all the vectors of satisfaction (in a sense, C deals more with L than with ). This implies in particular that X contains a consequence x * that is ideal for all the agents, and a consequence x * anti-ideal for all the agents. When the set of consequences X is too small, it is harmless to extend and enrich it in order to obtain all the z that we need: in the following, Axiom C is supposed by construction (in Harsanyi's paper it is implicit: X is identified with the set of utility vectors).

We now introduce the axiom of Pareto unanimity, that is essential for collective choice:

Axiom P (Pareto Unanimity):

If ∀i ∈ A, L i L , then L L .
Because we are rather interested in a cautious way of decision making than in an adventurous one, the next axioms are those proposed by [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] (their interpretation is detailed in the literature, hence we refrain to comment them further) . We write them below for any relation ; in the following, these axioms will apply to and to the i 's:

Axiom 1: on L is an equivalence relation (i.e. is complete and transitive). Axiom 2 (Certainty equivalence): ∀Y ⊆ X, ∃x ∈ Y s.t.

x and Y are equivalent for .

Axiom 3 (Risk aversion): If ∀x ∈ X, L[x] ≤ L [x] (L is more specific than L ), then L L .
Axiom 4 (Weak independence): If L and L are equivalents, then λ/L, µ/L and λ/L , µ/L are also equivalents, for any λ, µ s.t. max(λ, µ) = 1. Axiom 5 (Lottery reduction): For any (compound lottery) L, L ∼ Reduction(L).

Axiom 6 (Continuity of L): [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] show that the preference relation on L defined by U -do satisfy Axioms 1-6 and that, reciprocally, if Axioms 1-6 are satisfied by some relation , then the simple lotteries are ranked as if they where be evaluated by their pessimistic utilities. Technically, the satisfaction of these axioms allows the definition of an ordered scale U , an utility function u : X → U , an order reversing function

If ∀x ∈ X, L [x] ≤ L[x] then ∃λ s.t. L ∼ 1/L, λ/X .
n : V → U such that L L iff U -(L) ≥ U -(L ).
This axiomatization is a qualitative counterpart to von Neumann and Morgenstern's characterization of expected utility.

Properties of possibilistic collective utility functions

We now study the four decision rules in light of the axioms and show that they are consistent, namely obeyed by the pessimistic egalitarian collective utility (U -min ). Consider: a set L of possibilistic lotteries built from a set X and a scale V ; a set u i , i ∈ A of utility functions on X taking their values in [0, 1]; and a weight vector w ∈ [0, 1] p (where w i is the weight of agent i). It holds that: Proposition 3. The relations and i defined by:

L L iff U -min (L) ≥ U -min (L ), L i L iff U - i (L) ≥ U - i (L )
satisfy Axioms 1-6, as well as the Pareto unanimity axiom.

The satisfaction of Axioms 1-6 by the i 's is obvious (U - i is by definition a pessimistic utility). Their satisfaction by is also straightforward: U -min is clearly a pessimistic DMU utility based on the utility function u(x) = min i=1,p max(u i (x), (1 -w i )).

The satisfaction of Pareto Unanimity is also easy to prove. Suppose that L i L , for all i. By definition,

L i L iff U - i (L) ≥ U - i (
L ); L i L for each i implies that the aggregation by the weighted minimum of the U - i 's for L is greater or equal to the one given to L (this aggregation is non decreasing); then U -min (L) ≥ U -min (L ). In other words, L L . Axiom P is thus satisfied.

Generally, we believe that all the ex-ante possibilistic aggregations, and in particular U + min ante , satisfy P, be they egaliriarian or not (e.g. so do U -sum ante , U -max ante , etc.) -simply because the CUFs are non decreasing. In our egalitarian context: Proposition 4. The relations and i defined by:

L L iff U + min ante (L) ≥ U + min ante (L ), L i L iff U + i (L) ≥ U + i (L )
satisfy Pareto Unanimity.

The problem is that U + min ante may fail to satisisfy weak independence, as shown by the following counter-example: Counter-example 2. Consider two equally important agents i.e.

(w1 = w2 = 1), and the three lotteries depicted in Figure 4. Let L and L be the lotteries defined by: L = 1/L1, 0.9/L3 , L = 1/L2, 0.9/L3 . U + min ante (L1) = U + min ante (L2) = 0.5: L1 and L2 are indifferent

U +min ante (L) = U +min ante (Reduction(L)) = U +min ante ( 0.5/x1, 1/x2 ) = 0.5 U +m min ante (L ) = U +min ante (Reduction(L )) = U +min
ante ( 1/x1, 0.9/x2 ) = 0.6. Then, U + min ante (L ) > U + min ante (L), which contradicts the axiom of weak independence.

Concerning U + min post , news are very bad, since it even fails to satisfy Pareto Unanimity:

Counter-example 3. Consider the two lotteries of Figure 3 on X = {x1, x2, x3} and suppose now that the two agents are equally important i.e. (w1 = w2 = 1). We get for agent 1:

U + (L1) = max(min(1, 0.8), min(0.9, 0.1)) = 0.8 U + (L2) = min(1, 0.8) = 0.8
and for agent 2:

U + (L1) = max(min(1, 0.1), min(0.9, 0.8)) = 0.8 U + (L2) = min(1, 0.8) = 0.8 while:

U + min post (L1) = max(min(1, min(0.8, 0.1), min(0.9, min(0.1, 0.8)))) = 0.1 U + min post (L2) = max(min(1, min(0.8, 0.8))) = 0.8

Hence L1 ∼1 L2, L1 ∼2 L2 but L2 L1, which contradicts Pareto Unanimity.

A representation theorem for U -min

Let us now show that the relation that satisfies both Continuity, Pareto Unanimity and the axioms of pessimistic utility can be captured by U -min -thus providing a counterpart of Harsanyi's theorem that allows (weighted) equity. For the sake of brevity (and also because they look less interesting: they suffer from the Timing Effect and can violates important axioms), we let U + min ante and U + min post for further research. We first sketch the proof in the general case, allowing more or less important agents: this provides a characterization of U -min in its full generality. We impose equity between agents in a second step -and rule out the weights.

We have seen in the previous Section that the set of axioms is consistent since satisfied by U -min . Let us now go the reverse way. Consider a relation on L (built on X and V ) that satisfies Axioms 1 to 6, a set of relations i on the same L that also satisfy these axioms, and suppose that the axioms of Pareto unanimity and continuity of X hold.

First of all, since Axioms 1-6 are satisfied by and i , these relations can be represented by pessimistic utilitiesthis is Dubois and Prade's theorem of representation.

Let us now consider for any agent i ∈ A, the set i = {x ∈ X : ∀y, x i y} of the best consequences according to i (this set cannot be empty because the i 's are preorders). Thanks to Axiom C, there exists a consequence x * that belongs to all the i 's. By Pareto unanimity, x * y, ∀y ∈ X. In the same way, there exists a x * such that y x * , ∀y ∈ X.

Thanks to Axiom C, we can define the constant act x i for any agent i: Definition 2. For any x ∈ X and any agent i, let x i be the constant lottery s.t. x i ∼ i x and x i ∼ j x * for each j = i.

x i will be identified with the utility of x according to agent i: the influence of the other agents is neutralized (they get their best outcome, which behaves as a neutral element in the pessimistic approach).

Let ∆ i = {x i , x ∈ X}. (x * ) i and (x * ) i belong to ∆ i by definition. The union of the ∆ i 's, that is to say ∆ = {x i : x ∈ X, i ∈ A}, plays an important role in our proof -it allows the construction of a common evaluation scale. ∆ is naturally ordered by and each ∆ i is ordered by i . By construction, we have: Proposition 5. ∀x ∈ X, (x * ) i i x i (x * ) i . Moreover, we can show that Proposition 6. ∀x i , (x * ) i x i (x * ) i (x * ) i is one of the best consequences for i and (x * ) i is one of her worst ones. It may happen that one of the x i be indifferent w.r.t. to (x * ) i : i prefers x i to (x * ) i , but the collectivity does not; this is due to the fact that agent i is not so important, so the elements of X that are bad for her (e.g. (x * ) i ) are considered as not so bad for the collectivity.

Let us denote B i = {x i ∈ ∆ i : x i ∼ (x * ) i } the set of the elements of ∆ i that are indifferent to (x * ) i according to the collectivity, and this even if agent i makes a difference; the elements of B i form an equivalence class according tobut, again, not necessarily according to i . Let m i denotes the best of the elements of B i (according to i )4 . It reflects the importance of the agent: the greatest m i , the lower the importance of i. Formally5 : Definition 3. For any i ∈ A, let m i = argmax i {x i :

x i ∼ (x * ) i } be the discount degree of i. Lemma 1. ∀x ∈ X, i = 1, p, x i ∼ max i (m i , x i ). Lemma 2. ∀x ∈ X, x ∼ argmin {x i : i ∈ A}.
From Lemmas 1 and 2 we get:

Corollary 1. x ∼ argmin {max i (m i , x i ) : i ∈ A}.
In order to show that a relation satisfying Axioms 1-6 is a pessimistic utility, [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] built the scale U = {[x] : x ∈ X} where [x] is the equivalence class of x according to . U is totally ordered by and these authors set u(x) = [x]. Here, we use the set ∆ = {x i : x ∈ X, i ∈ A}, partially ordered by the relation defined by: Definition 4. x i y i iff x i i y i

x i y j iff x i i m i , y j j m j and x i y j , for all i = j.

This relation is a partial preoder (it is reflexive and transitive) but if x i ≺ m i and i = j, x i and y j are not comparable: neither x i y j nor y j x i hold, but this is harmeless. What is important is that (i) the restriction of to each ∆ i is a preorder (on ∆ i , = i ) and (ii) that any x i that is as least as good as m i (according to i) is comparable to any x j that is as least as good as m j (according to j) . Properties (i) and (ii) ensure that v(x) = min {max (m i , x i ) : i ∈ A} exists. Then from Corollary 1 and Definition 4 it follows that: x ∼ v(x). Let k be the agent for which the min is reached in the expression of v(x): v(x) = max (m k x k ) belongs to ∆ k and is such that v(x) k m k . Hence v(x) and v(y) are comparable w.r.t. , whatever x, y. This allows us to write:

Lemma 3. x y iff argmin {max (m i , x i ) : i ∈ A} argmin {max (m i , y i ) : i ∈ A}.
Because working with a partial preorder is not so convenient, we shall use any complete preorder on ∆ such that x y =⇒ x y (there always exists one). Then we get:

Lemma 4. x y iff argmin {max (m i , x i ) : i ∈ A} argmin {max (m i , y i ) : i ∈ A}.
Since satisfies Axioms 1-6, Dubois and Prade's result applies: there exists an order reversing function n s.t.: n ext (v), m i and x i , u(x) belong to ∆. In order to get a total order, we consider the equivalence classes of ∆, i.e. the set U ext = {[x] : x ∈ X} where [x] is the equivalence class of x w.r.t. . Because x = argmin {x i : i ∈ A} (Lemma 2) U ext contains the equivalence classe of each x ∈ X to , in particular, the equivalence classe [x i ] of each x i ; U ext is ordered by and is equipped with a maximal and a minimal elements ([x * ] and [x * ], respectively).

L L iff min x∈X max (n(L[x]), u(x)) min x∈X max (n(L [x]), u(x)). Let us denote u(x) = argmin {max (m i , x i ) : i ∈ A} and n ext (v) = u(n(v)) (n(v) is an element of ∆).
Setting

u i (x) = [x i ], nw i = [m i ] and n(v) = [n ext (v)], we get: L L iff min x∈X max (n(L[x]), min i∈A max (ui(x), nwi)) min x∈X max (n(L[x]), min i∈A max (ui(x), nwi)).
Hence the main result of this paper: Theorem 1. If the collective preference and individual preference relations satisfy Axioms 1-6, Pareto unanimity (P) and the axiom of continuity of X (C) then there exists a scale U ext totally ordered by , a distribution of weights nw : A → U ext , a series of functions u i : X → U ext , i = 1, n and an order reversing function n : V → U ext s.t. for each couple of lotteries L and L :

L L iff min x∈X max(n(L[x]), min i∈A max(nw i , u i (x))) min x∈X max(n(L [x]), min i∈A max(nw i , u i (x))).
We can add two axioms that leads to pure egalitarianism.

Axiom E: ∀i, j, (x * ) i ∼ (x * ) j Axiom PW: ∀i, if x i y then x i y i By E, the dissatisfaction of one agent has no more power than the one of another agent. A direct consequence is that the agents have the same discount degree. By PW, each agent has some power (she makes the decision at least when every other one is totally happy with both x and y). It implies m i ∼ i x * , for each i. Because all i share the same discount m i , Pareto unanimity implies that m i ∼ x * .

This provides a characterization of the full egalitarian CUF:

Theorem 2. If the collective preference and individual preference relations satisfy Axioms 1-6, P, C, E and PW then there exists a scale U ext totally ordered by , a series of functions u i : X → U ext and an order reversing function n : V → U ext such that: L L iff min

Conclusion

In conclusion, not only egalitarianism and decision under uncertainty are compatible and can escape the Timing Effect, but egalitarianism is compulsory when the decision is to be made on a possibilistic and cautious basis. This is interpreted as a justification of egalitarianism, just like Harsanyi's theorem can be interpreted as a justification of utilitarianism.

The present work, like the seminal work of Harsanyi, assumes that all the agents share the same knowledge, which is seldom the case. This consideration has been the topic of several works, always in a probabilistic context [START_REF] Harsanyi | Games with incomplete information played by bayesian players[END_REF][START_REF] Hammond | Harsanyi's utilitarian theorem: a simpler proof and some ethical connotations[END_REF]): the ex-ante approach can be compatible with the existence of different quantifications of the uncertainty (while the ex-post approach clearly requires a unique knowledge); the Timing Effect shall not always be understood as a paradox. The next step of our work is to characterize egalitarianism in the context of a non homogeneous qualitative knowledge of the agents.

Another way of seeing Harsanyi' Theorem -and oursis to say that the axioms simply transfer the additive nature of probabilities on the collective utility function and the cautiousness of pessimistic utility into a maximization of the least satisfied agent. The problem has to be also studied without any explicit a priori assumption on the type of knowledge -following [START_REF] Savage | The Foundations of Statistics[END_REF] or [START_REF] Arrow | A Difficulty in the Concept of Social Welfare[END_REF] (see also [START_REF] Hammond | On reconciling Arrows Theory of social choice with Harsanyi's fundamental utilitarianism[END_REF]) ways rather than von Neumann and Morgenstern's. We probably then get that when egalitarianism is required, the collective decision is made as if each one were deciding in a qualitative, ordinal way. The probabilities may exist in the mind of (some) decision makers, but are not fully exploited. This is an exciting topic of further research.

Figure 2 :

 2 Figure 2: A possibilistic compound lottery and its reduction.

Figure 3 :

 3 Figure 3: Two bi-agent possibilisitic lotteries.

Figure 4 :

 4 Figure 4: A counter-example to weak independance.

  By applying Lemma 4, we can write: L L iff min x∈X max (n ext (L[x]), u(x)) min x∈X max (n ext (L [x]), u(x)).

In our notation system the first exponent indicates the attitude with respect to uncertainty: optimistic (+) or pessimistic(-); the second one indicates the type of CUF used(min, max, sum, etc.).

 3 The consequences with a possibility degree of 0 are not represented in the drawings.

If |Bi| > 1, mi can be any one of its elements.

In the following, there are many relations (preorders). For the sake of clarity, we indicate for each minimum or maximum operation the preorder it relies on.


M.D Charles-Edouard Luyt
email: charles-edouard.luyt@psl.aphp.fr

Nicolas Bréchot

Pierre Demondion

Tamara Jovanovic

Guillaume Hékimian

Guillaume Lebreton

Ania Nieszkowska

Matthieu Schmidt

Jean-Louis Trouillet

Pascal Leprince

Jean Chastre

Alain Combes

Brain injury during venovenous extracorporeal membrane oxygenation

Keywords: extracorporeal membrane oxygenation, acute lung injury, intracranial brain hemorrhage, cerebral stroke, hypercapnia

The use of venovenous-extracorporeal membrane oxygenation (VV-ECMO) to treat patients with refractory acute respiratory distress syndrome (ARDS) has increased in recent years [START_REF] Luyt | Long-term outcomes of pandemic 2009 influenza A (H1N1)-associated severe acute respiratory distress syndrome[END_REF][START_REF] Combes | Position paper for the organization of extracorporeal membrane oxygenation programs for acute respiratory failure in adult patients[END_REF]. Among complications occurring during ECMO support, neurological injury is important because it may be associated with increased mortality and long-term functional sequelae. In the landmark observational study on VV-ECMO for refractory ARDS complicating H1N1v2009 influenza, most deaths were attributed to cerebral hemorrhage [START_REF] Davies | Extracorporeal Membrane Oxygenation for 2009 Influenza A(H1N1) Acute Respiratory Distress Syndrome[END_REF]. To date, only two studies have specifically investigated neurological complications of ECMO support and both showed high numbers of brain injuries [START_REF] Kasirajan | Risk factors for intracranial hemorrhage in adults on extracorporeal membrane oxygenation[END_REF][START_REF] Mateen | Neurological injury in adults treated with extracorporeal membrane oxygenation[END_REF]. However, those observational studies concerned patients receiving venoarterial-ECMO (VA-ECMO) and most subjects were included after cardiac surgery or cardiac arrest. VV-ECMO neurological complications have mostly been reported as case reports [START_REF] Chow | Outcome in patients with H1N1 influenza and cerebrovascular injury treated with extracorporeal membrane oxygenation[END_REF][START_REF] Guennec | Diffuse cerebral microbleeds after extracorporeal membrane oxygenation support[END_REF].

Because no epidemiological data on neurological complications occurring during VV-ECMO are available, we undertook this retrospective study on information prospectively collected from a large series of VV-ECMO patients to describe the frequency, morbidity, mortality and risk factors of cerebral injury. We then conducted a systematic review of all reported neurological complications in VV-ECMO series.

METHODS

All patients admitted to our intensive care unit (ICU) from 2006 to 2012 who received VV-ECMO support were included. Because circulatory assistance of any type can be complicated by thromboembolic phenomena, particularly cerebral infarction due to embolism, we excluded patients having received a VA-ECMO (peripheral or central), a left ventricle assist device (VAD), a BiVAD or a total artificial heart before, during or after VV-ECMO run.

Thus, patients switched from VA-ECMO to VV-ECMO or from VV-to VA-ECMO were excluded. Clinical information on medical history, clinical and biological parameters at ICU admission and during the stay was collected prospectively. In particular, any events occurring during ECMO support were prospectively recorded in the ICU's database.

We defined clinical neurological complication as any clinical event occurring during ECMO course. It included any clinical sign suggestive of stroke (hemiplegia, mydriasis, anisocoria, asymmetry on clinical examination), but also confusion, delirium, seizures, coma despite sedation withdrawal. Patients were categorized according to the presence or not of brain injury on brain imaging (i.e. no brain damage, ischemic stroke, intracranial bleeding and microbleeds) and groups were compared.

ECMO Circuit

The extracorporeal system consisted of polyvinyl chloride tubing, a membrane oxygenator (QuadroxBioline, Jostra-Maquet, Orléans, France, or Eos ECMO, Sorin, Milan, Italy), a centrifugal pump (Rotaflow, Jostra-Maquet, or Revolution, Sorin), and drainage and reinfusion cannulae (Biomedicus Carmeda, Medtronic, Boulogne-Billancourt, France, or Edwards Lifesciences, Irvine, CA). An oxygen-air blender (Sechrist Industries, Anaheim, CA) was used to ventilate the membrane oxygenator. All patients had two-site cannulation and all cannulae were inserted percutaneously. The drainage cannula was inserted into the femoral vein (extending into the inferior vena cava) and the reinfusion cannula was inserted into the internal jugular vein (extending into the right atrium).

Patient Management under ECMO

All patients had a blood gas analysis in the 2 hours preceding ECMO start (blood gas analysis pre-ECMO) and in the 2 hours following ECMO start (blood gas analysis post-ECMO).

All patients had the same protocol for anticoagulation. A heparin bolus (5000 IU) was injected at ECMO initiation; then all patients were continuously infused with unfractionated heparin. The heparin dose was adapted at least once daily according to activated partial thrombin time (aPTT) value (targeting 1 to 1.5-fold the control value) and clinical tolerance; heparin was stopped when bleeding occurred and restarted once it was controlled. Bleeding leading to heparin withdrawal was defined as any clinical bleeding judged significant by the physician in charge of the patient (at ECMO site, central or arterial lines, tracheal secretions, ear nose and throat) with or without hemodynamic impact, with or without decrease in haemoglobin level. Anticoagulant overdose was defined biologically as an aPTT (patient/control) ratio >2.5, corresponding to an absolute value of ≥80 s.

Membrane oxygenator and circuitry were checked daily by experienced perfusionists and were changed when fibrin deposition or thrombi had deleterious effects on blood oxygenation, platelet count (<20,000/ml) or blood fibrinogen level (<1.5 g/L) decreased significantly, or significant intravascular hemolysis (free plasma haemoglobin> 200 mg/L at 2 time points and no other cause of mechanical hemolysis) appeared. No systematic circuit change was scheduled.

Daily routine neurologic examinations were performed in our patients by the physicians and the nurses in charge of the patient. Physicians performed at least once daily neurologic examination after sedation withdrawal, including Glasgow coma scale calculation, response to verbal orders or pain, tendon reflexes, brainstem reflexes and plantar reflex, eye opening and pupils' examination. Size of pupils and their light reactivity were assessed by the nurse in charge of the patient every 4 hours. Moreover, when an unexpected event occurred (such as seizures, delirium confusion, no awakening after sedation withdrawal…), this was registered in the medical chart. Once a neurologic symptom was observed (including but not restricted to change in neurological exam, mydriasis, anisocoria, seizures, delirium, confusion, coma despite sedation withdrawal…) within 6 hours a cerebral CT scan was performed.

Literature Review

We conducted a systematic MEDLINE-Database literature review through the PubMed search engine with a global search strategy applying prespecified selection and outcome (occurrence of neurological complication and mortality) criteria using the terms ECMO, venovenous-ECMO and extracorporeal oxygenation. Details on methodology and the results of this review are presented in an Online Supplemental Data.

Statistical Analyses

Data are expressed as medians [25 th -75 th percentile interquartile range (IQR)] or means ( standard deviation (SD)), as appropriate. Between-group comparisons were analyzed using Student's t-test or the Mann-Whitney U-test for continuous variables, chi-square test for categorical variables. For dichotomous variables, the median value was chosen for the cut-off to separate the variable into two categories. To examine the univariable impact of patients' clinical characteristics and ICU events on the development of cerebral bleeding, a logisticregression model tested each characteristic. Thereafter, multivariable analysis with a Cox regression model compared the factors that were significant in the univariable analysis. The analysis was run using backward-stepwise variable elimination (with the variable-exit threshold set at P > 0.05). Risk factors achieving P  0.10 in our univariable analysis and parameters previously reported to be strongly associated with intracranial bleeding were entered into the model. Interactions were tested in the model; variables strongly associated with other(s) were not included in the multivariable model. The final model included renal failure on admission, PaO2 change and PaCO2 change. In a sensitivity analysis, lowest platelets count, lowest fibrinogen level and lowest prothrombin time were added to the final model for intracranial bleeding. Two other endpoints were investigated using Cox regression model: ICU mortality without intracranial bleeding, and a composite endpoint; intracranial bleeding or ICU mortality. Analyses were computed with StatView v5.0 (SAS Institute Inc, Cary, NC) and SPSS v11.5 (SPSS Inc, Chicago, IL) software. P < 0.05 defined significance.

Ethics

In accordance with the ethical standards of our hospitals institutional review boards (Committee for the Protection of Human Subjects), informed consent for demographic, physiological and hospital-outcome data analyses was not obtained because this observational study did not modify existing diagnostic or therapeutic strategies.

RESULTS

During the study period, 135 patients required VV-ECMO support. Table 1 reports their baseline characteristics. For 23 of those 135 patients, a cerebral event could not be ruled out because they died before neurological examination could be performed or had preexisting neurological disease. A clinical neurological symptom appeared in 25 (19%) during VV-ECMO support (Figure 1). Three of those 25 were clinically brain dead, confirmed by electroencephalogram, and did not undergo cerebral imaging; therefore, hemorrhagic stroke could not be excluded.

Brain imaging was obtained for the remaining 22 patients (computed-tomography (CT) scan for 21 and magnetic resonance imaging (MRI, performed after ECMO removal) for one), and five other patients without any clinical signs underwent cerebral CT scans (for one of them with traumatic brain injury, the CT scan was obtained for follow-up) (Figure 1).

Cerebral-imaging findings as a function of clinical neurological feature(s) are given in Table 2. Cerebral-imaging sequences for 10 patients were normal; 10 patients had intracranial bleeding, either intraparenchymal hematoma (n = 9) or subarachnoid hemorrhage (n = 1); three patients had ischemic strokes; two patients had diffuse microbleeds and two patients had brain edema (for the latter, the neurological injury preceded ECMO: one suffered traumatic brain injury 7 days before ECMO with cerebral edema before ECMO; the other had prolonged cardiac arrest 2 days pre-ECMO and cerebral edema was attributed to cerebral anoxia).

Thus, 18 (13%, 95% CI 7.3-18.7%) of the 135 patients had a cerebral complication that occurred while on ECMO (three brain deaths without imaging, 10 intracranial bleeds, three ischemic strokes and two diffuse microbleeds). Table 1 gives patients' characteristics as a function of the type of neurological event occurring on ECMO or not. Patients with such complications were older and more frequently had renal failure at ICU admission (especially patients with intracranial bleeding) than those without these events. Other parameters did not differ between groups, although patients with intracranial bleeding tended to have higher mortality and shorter time from ECMO onset to neurological complications.

Patients with Intracranial Bleeding

Because the predominant neurological event was intracranial bleeding, we specifically analyzed those 10 patients to attempt to identify risk factors associated with this complication. Table 4 lists the hemostasis-parameter values 3 days before brain injury. These patients' heparin doses were low, as were their aPPT values and anti-Xa activity, and none had profound thrombopenia or low fibrinogen levels.

Same risk factors were associated with an increased risk of both death and the composite endpoint (Online Supplement, Tables E1 andE2). Among them, PaCO 2 change and lowest prothrombin time during EMCO, but not renal failure, were associated with ICU mortality (Online Supplement, Tables E1 andE2).

DISCUSSION

We report here the largest study to date of neurological complications during VV-ECMO. We found that 25 of our 135 patients (19%) experienced a clinical neurological event, with 18 (13%, 95% CI 7.3-18.7%) of those neurological injuries occurring on VV-ECMO. Cerebral bleeding was the predominant complication, manifesting as coma or mydriasis. Intriguingly, it was not associated with hemostasis disorders or anticoagulant use: platelet counts, prothrombin time, aPPT and fibrinogen levels while on ECMO were similar among patients with and without cerebral hemorrhage, and heparin doses during the 3 days preceding cerebral bleeding were low, without overdose. The only factors independently associated with brain hemorrhage were renal failure at ICU admission and acute PaCO 2 changes at ECMO initiation.

Our results are in accordance with the literature on neurological complications occurring in ECMO patients (Table 5). Kasirajan et al. reported 18.9% intracranial bleeding for their series of 74 VA-ECMO patients. In that study, cerebral bleeding was independently associated with female sex and thrombocytopenia. Mateen et al. retrospectively studied 87 VA-ECMO patients and found a high rate of neurological injury, including stroke, intracranial bleeding and brain death [START_REF] Mateen | Neurological injury in adults treated with extracorporeal membrane oxygenation[END_REF]; the 10 brains examined at autopsy revealed more neurological sequelae than would have been predicted by clinical findings alone. However, that study included VA-ECMO patients after cardiac surgery or cardiac arrest. The authors of the largest study that included 23,951 patients given ECMO support reported 10.9% neurological complications [START_REF] Nasr | Neurologic Complications of Extracorporeal Membrane Oxygenation[END_REF]. Ischemic stroke, cerebral bleeding and seizure frequencies were 4.1%, 3.6% and 4.1%, respectively, but, as underlined by the authors themselves, they mixed adults, children and newborns, and were unable to distinguish patients receiving VV-and/or VA-ECMO [START_REF] Nasr | Neurologic Complications of Extracorporeal Membrane Oxygenation[END_REF]. The cerebral bleeding frequency for VV-ECMO patients observed in the systematic review (see online supplement) was 5%, similar to ours. However, this systematic review has several limitations: first, among the 16 studies reporting neurological complications of ECMO, most of them mixed VV-and VA-ECMO. Thus, the exact cerebral complication frequency of VV-ECMO is difficult to calculate precisely. Another limitation is that most previous studies reported only cerebral bleeding and no other neurological complications, e.g. ischemic stroke. Lastly, unlike our investigation, those studies had not been designed to evaluate factors associated with neurological complications.

The pathophysiology of VA-ECMO-or VV-ECMO-associated brain injury probably differs. In VA-ECMO patients, brain damage could reflect the pre-ECMO clinical context (low blood pressure and cerebral blood flow, hypoxia, acidosis, electrolyte disturbances, and/or hemostasis disorders related to hepatic failure frequently observed in cardiogenic shock), reperfusion injury at ECMO implantation, the ECMO process itself (embolic stroke from the arterial cannula) and/or hemostasis disorders induced by the ECMO circuit. To date, the exact mechanism of brain damage is not fully understood and probably includes a combination of all those factors.

In VV-ECMO patients, a combination of factors also probably leads to neurological complications, even though some of these factors are not strictly the same. Whereas disorders created by the ECMO circuit and the oxygenator (hemolysis, thrombocytopenia, fibrinolysis, acquired von Willebrand syndrome [START_REF] Heilmann | Acquired von Willebrand syndrome in patients with extracorporeal life support (ECLS)[END_REF]) are the same in VA-and VV-ECMO, pre-ECMO factors and ECMO-induced metabolic changes could differ. The pre-ECMO factors include hypoxia, hypercapnia, respiratory acidosis and parameters associated with acute respiratory failure before starting support. VV-ECMO specificities include abrupt O 2 and CO 2 changes at ECMO onset [START_REF] Schmidt | Blood oxygenation and decarboxylation determinants during venovenous ECMO for respiratory failure in adults[END_REF][START_REF] Schmidt | Mechanical ventilation during extracorporeal membrane oxygenation[END_REF]. Because CO 2 is involved in the regulation of cerebral blood flow [START_REF] Lassen | Physiology of cerebral blood flow[END_REF][START_REF] Meng | Regulation of Cerebral Autoregulation by Carbon Dioxide[END_REF], sudden CO 2 -level changes (from hypercapnia to normocapnia or hypocapnia) at ECMO initiation could have induced sudden cerebral blood-flow changes that could have precipitated brain damage. Recently, Muellenbach et al. reported that patients receiving vvECMO treatment were at risk for a decrease in cerebral regional tissue oxygen saturation at ECMO initiation, and that this decreased is linked to PaCO2 change. This could be involved in the pathogenesis of brain injury of ECMO patients [START_REF] Muellenbach | Effects of venovenous extracorporeal membrane oxygenation on cerebral oxygenation in hypercapnic ARDS[END_REF][START_REF] Kredel | Cerebral tissue oxygenation during the initiation of venovenous ECMO[END_REF]. In our study, the acute PaCO 2 change was independently associated with cerebral bleeding and one can hypothesize that these abrupt changes may have facilitated brain hemorrhages. Although we have no definite proof of this association, we think it should be taken into account when initiating VV-ECMO and in our ICU we have now changed our practice by trying to avoid too rapid hypercapnia correction. It could be achieved by starting with a low sweep gas flow and progressively increasing it over time. Even though PaO 2 change was not independently associated with cerebral bleeding, it can be recommended not to correct it too quickly by starting with a low fraction of inspired oxygen and increasing it slowly over time. However, because rapid decrease in PaCO2 leads to vasoconstriction, the relationship between PaCO2 change and cerebral bleeding is difficult to understand. We cannot exclude that ECMO-induced alkalosis have caused microbleeds or excessive local flow with altered brain blood barrier and bleeding.

Surprisingly, hemostasis disorders were not associated with intracranial bleeding herein. However, because lowest prothrombin time while on ECMO was strongly associated with an increased risk of mortality, the lack of association between lowest prothrombin time and intracranial bleeding could be an indirect effect of the increased mortality among patients presenting this risk factor. Moreover, because our patients' platelet function was not explored, we cannot exclude that those with intracranial haemorrhage might have had platelet dysfunction. Our patients received relatively low dose of heparin, but this is explained by our anticoagulation policy; we target an aPTT ratio of 1 to 1. Our study has several limitations. First, this was a single-center study and, although we included many VV-ECMO patients, not many developed neurological events. Thus, our results, particularly the role of rapid gas exchange parameter changes in cerebral bleeding, will have to be confirmed by larger studies. Moreover, because the delivery of ECMO is so variable from centre to centre (devices used, techniques employed; management of anticoagulation and transfusion, thresholds for addressing hypoxemia, and so on), these results would be difficult to extrapolate to other centres. Second, although it is our policy to carefully check hemostasis parameters in patients on VV-ECMO, we did not examine hemostasis, specifically thrombosis and fibrinolysis, with thromboelastogram, D-dimer or other tests [START_REF] Repesse | Recombinant factor VIIa for uncontrollable bleeding in patients with extracorporeal membrane oxygenation: report on 15 cases and literature review[END_REF]. Because it is well-known that ECMO patients have primary hemostasis disorders (e.g., acquired von Willebrand disease [START_REF] Heilmann | Acquired von Willebrand syndrome in patients with extracorporeal life support (ECLS)[END_REF]), we cannot exclude the possibility of unrecognized hemostasis disorders that might explain the high intracranial bleeding frequency. Third, because it was difficult to assess neurological status at ICU admission, it is possible that, at least in some patients, neurological complications may have preceded ECMO and that ECMO aggravated them. However, albeit impossible to demonstrate, it is not likely, since our policy is to carefully check daily for clinical signs of brain involvement, in particular at ICU admission. Fourth, because 23 patients could not be evaluated before dying and because we also cannot exclude that some patients might have had an undetected neurological injury, we may have underestimated the exact frequency of brain injury, as in most studies published to date and reporting neurological outcomes. Fifth, because neuroimaging was not performed in all patients, we could have underestimated the frequency of neurological injury (missing subclinical events). Sixth, we found no direct association between hemostasis disorders and intracranial bleeding, but lowest prothrombin time while on ECMO was strongly associated with ICU mortality. This lack of association between lowest prothrombin time and intracranial bleeding could be an indirect effect of the increased mortality among patients presenting this risk factor. Lastly, we identified 2 conditions associated with cerebral bleeding during ECMO, but their relationships with brain haemorrhage remain unknown: are they causative factors or merely coincidental? Moreover, we explored only the blood gas change at ECMO start, not during the whole ECMO course.

We therefore cannot be sure that rapid change of PaO2/PaCO2 could have occurred after several hours or days and could have play a role in brain injury. Future studies should explore these possibilities.

In conclusion, neurological complications are frequent in patients on VV-ECMO.

Intracranial bleeding, the most frequent event, occurred early during ECMO and was associated with high mortality. The precise roles of sudden O 2 -and CO 2 -level changes need to be more precisely evaluated, not only for the sake of knowledge, but mainly because they might have major implications in the care of patients requiring VV-ECMO.

Extracorporeal Membrane Oxygenation Support for Acute Respiratory Distress

Syndrome. Ann Thorac Surg 100:2059-63. 36. Gray BW, Haft JW, Hirsch JC, Annich GM, Hirschl RB,Bartlett RH (2015)

Extracorporeal life support: experience with 2,000 patients. ASAIO J 61: 2-7. a P < 0.05 for between-group comparisons.

b Organ failure was deemed present when the corresponding Sepsis-related Organ Failure

Assessment score was >2.

c Worst value before or during ECMO.

d Time of neurological event was calculated using the date of cerebral imaging a Because the authors of these studies did not differentiate between VV-and VA-ECMO patients, the exact frequency of neurological complications occurring on VV-ECMO cannot be calculated.

Figure 2

 2 Figure 2 reports the pH, PaCO 2 and PaO 2 changes between just before and after starting

  5, and stop heparin when clinical bleeding occurs. Although hemostasis disorders could have played a role, the mechanism of cerebral bleeding in those patients is probably multifactorial, including vascular injury due to underlying disease severity, disorders created by the ECMO circuit (hemolysis, fibrinolysis…), rapid PaO 2 and PaCO 2 changes and other, as yet unknown, factors. Future studies should investigate the precise mechanisms that could precipitate neurological complications, particularly controllable factors like PaO 2 and PaCO 2 changes.

Figure 1 .

 1 Figure 1. Flow chart of the study. VV-ECMO, venovenous-extracorporeal membrane

Figure 2 .

 2 Figure 2. Gas exchange parameter changes at the start of venovenous-extracorporeal

  Figure 1

Table 1 .

 1 Admission Characteristics, Hemostasis Disorders during VV-ECMO and Patient Outcomes According to Brain Damage

		No Brain	Patients with Brain Damage
		Damage	Intracranial	Microbleeds	Ischemic
	Characteristic	(n = 117)	Bleeding	(n = 2)	Stroke
			(n = 10)		(n = 3)
	Age, yr a	44 ± 14	49 ± 12	54 ± 4	59 ± 6
	Male sex, n (%)	79 (68)	5 (50)	2 (100)	1 (33)
	Body mass index, kg/m 2	26 [24-31]	23 [21-27]	22 [21-22]	26 [25-47]
	McCabe & Jackson	41 (35)	2 (20)	0	1 (33)
	comorbidity score ≥2, n (%)
	ICU-admission SAPS II score	69 ± 14	71 ± 12	61 ± 1	71 ± 9
	H1N1v2009 influenza-related	18 (15)	1 (10)	0	0
	ARDS, n (%)
	Duration of MV before ECMO,	5 [1-10]	8 [4-11]	5 [3-7]	11 [9-17]
	days
	Organ failure at ECMO start b
	Cardiovascular	30 (25)	2 (20)	1 (50)	0
	Hepatic	25 (23)	4 (40)	0	0
	Renal a	48 (41)	8 (80)	0	2 (67)
	Hematological	15 (13)	1 (10)	0	0
	Neurological	94 (80)	8 (80)	1 (50)	3 (100)
	Gas exchange values pre-
	ECMO
	Arterial pH	7.26	7.18 [7.07-	7.16 [7.16-	7.36 [7.32-

Table 3 .

 3 Univariable and Multivariable Analyses of Factors Associated With Intracranial Defined as the post-ECMO pH, PaCO 2 or PaO 2 valuethe pre-ECMO pH, PaCO 2 or PaO 2 value.

	Bleeding (Excluding Microbleeds) on VV-ECMO
	Univariable Analysis	Cox Analysis
	Factor	OR [95% CI]	HR [95% CI]
	Age >46 yr	1.91 [0.51-7.10]
	Female sex	2.02 [0.55-7.41]
	SAPS II score at ICU admission ≥70	1.16 [0.32-4.19]
	Body mass index >26	0.47 [0.1-1.96]
	McCabe & Jackson comorbidity score ≥2	0.46 [0.09-2.24]
	MV duration before ECMO >5 days	1.68 [0.45-6.24]
	Organ failure at ECMO initiation a
	Cardiovascular	0.73 [0.15-3.60]
	Hepatic	2.74 [0.69-10.95]
	Renal	5.80 [1.18-28.47]	6.13 [1.29-28.57]
	Hematological	0.81 [0.09-6.83]
	Neurological	1.05 [0.21-5.25]
	Gas exchange change
	Arterial pH >0.2 b	2.34 [0.61-8.89]
	PaO 2 >50 mmHg b	4.44 [1.16-17.03]
	PaCO 2 <-27 mmHg b	5.95 [1.20-29.52]	6.02 [1.28-28.57]
	Renal replacement therapy	1.26 [0.34-4.68]
	Hemostasis disorders during ECMO
	Platelets <20  10 9 /L	1.31 [0.26-6.62]
	Prothrombin time <30% c , n (%)	0.52 [0.06-4.33]
	Fibrinogen, <1.5 g/L	0.76 [0.15-3.76]

b c Expressed as percentage of the standard value.

Table 4 .

 4 Hemostasis Parameters of Patients on Venovenous-Extracorporeal Membrane Oxygenation with Intracranial Bleeding During the 3 Days Before the Brain Haemorrhage

	Patients with Intracranial
	Parameter	Bleeding (n = 10) a
	aPTT, patient-to-normal value ratio b	1.14 [1.1-1.7]
	Anti-Xa activity, IU/L b	0.1 [0.1-0.1]
	Platelet count,  10 9 /L c	84 [29-135]
	Fibrinogen, g/L c	2.65 [2-5.6]]
	Heparin dose, IU/24 h b	2,750 [0-10,000]
	Abbreviation: aPTT, activated partial thrombin time.

a Values are expressed as median [25 th -75 th interquartile range]. b Highest value. c Lowest value.

Table 5 .

 5 Studies on Neurological Injuries in Patients on ECMO Support Included in the Systematic Review

		Study	No. of	Clinical Outcome		Hemostasis Anticoagulant ECMO-Brain
	Reference Design Patients	Neurological Complications	Deaths	Disorders Use	Injury Interval
	Kolla [22]	Cohort 100: 65 VV-	10 (10%) a :		NR	NR	NR
			ECMO; 11 VA-	2 ischemic strokes	2/2
			ECMO; 34 with	2 cerebral hemorrhages	2/2
			both	6 brain deaths	6/6
	Linden [23] Cohort 17: 8 VV-ECMO;	3 (18%) a :		NR	NR	NR
			7 VA-ECMO; 2	1 cerebral edema	1/1
			with both	1 cerebral bleeding	1/1
				1 ischemic stroke	1/1
	Mols [24]	Cohort 62: all VV-ECMO 2 (3%):
				1 cerebral bleeding	1/1
				1 brain death	1/1
	Hemmila	Cohort 255: 168 VV-	37 (15%) a :		NR	NR	NR
	[25]		ECMO; 47 VA-	14 ischemic strokes	11/14

Acknowledgment:

The authors thank Janet Jacobson for editorial assistance in revising the manuscript and Marguerite Guiguet for statistical assistance.

Conflicts of