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Computer Science
Mathematics

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Approximate Judgement Aggregation

In this paper we analyze judgement aggregation problems in which a group of agents independently votes on a set of complex propositions that has some interdependency constraint between them(e.g., transitivity when describing preferences). We consider the issue of judgement aggregation from the perspective of approximation. That is, we generalize the previous results by studying approximate judgement aggregation. We relax the main two constraints assumed in the current literature, Consistency and Independence and consider mechanisms that only approximately satisfy these constraints, that is, satisfy them up to a small portion of the inputs. The main question we raise is whether the relaxation of these notions significantly alters the class of satisfying aggregation mechanisms. The recent works for preference aggregation of Kalai, Mossel, and Keller fit into this framework. The main result of this paper is that, as in the case of preference aggregation, in the case of a subclass of a natural class of aggregation problems termed `truth-functional agendas', the set of satisfying aggregation mechanisms does not extend non-trivially when relaxing the constraints. Our proof techniques involve Boolean Fourier transform and analysis of voter influences for voting protocols. The question we raise for Approximate Aggregation can be stated in terms of Property Testing. For instance, as a corollary from our result we get a generalization of the classic result for property testing of linearity of Boolean functions. An updated version (RePEc:huj:dispap:dp574R) is available at http://www.ratio.huji.ac.il/dp_files/dp574R.pdf

Abstract Judgement Aggregation
Judgement Aggregation deals in scenarios in which one should aggregate a set of opinions/judgement done by independent judges or agents to one opinion. For instance, assume a committee needs to decide whether to deploy a suggested network protocol. It is acceptable by all voters that the protocol should be deployed if and only if it stands in two criteria: security (resistance to attacks) and scalability (supports several sizes of a network). We assume that each voter decides his opinion on the two criteria independently and based on this decides on whether to deploy the protocol. The voters cast their votes simultaneously and we assume no strategic behavior on their behalf. Now assume that a third of the voters are convinced that the protocol 1 stands in the two criteria and hence think it should be deployed while the other voters think it should not be deployed but disagree on the criterion that is violated. One third of the voters think it is not secure enough and hence should not be deployed although it is scalable. On the other hand, a third of the voters think it is secure but not scalable enough and hence should not be deployed. Then we have that although a minority of the committee (a third of the voters) thinks the protocol should be deployed, for each criterion separately there is a supporting majority (two thirds) that thinks the protocol passes. This discrepancy between the majority vote on premises (security and scalability) and the majority vote on the conclusion (deploy) was presented by Kornhauser and Sager in 1986 [28] and was later named 'The Doctrinal Paradox'. The deploy-network scenario is an example for a case when the decision is based on several arguments (while the logical connection between the arguments and the decision is commonly accepted) and we need to aggregate the decisions of (many) individuals into one decision. In general, one can think of scenarios in which the accepted opinions do not have this premise-conclusion structure ,for instance the Arovian agenda of preferences. A discrepancy phenomena as described above can happen for a lot of such sets of 'accepted opinions' (e.g., Condorcet Paradox for preference aggregation) and is the subject of a growing body of works in economics, political science, philosophy, law, and other related disciplines. (A survey of this field can be found in [32] and [30]) Abstract aggregation is the field that deals with such problems that arise when aggregating several opinions (constrained to be in some given group, the agenda) to one opinion. It can be formalized in the following way. There is a committee of n individuals (also called voters) that needs to decide on m binary issues (that is, each question has exactly two possible answers True and False 1 ). Each individual holds an opinion which is an answer for each of the issues. We denote the answer of the i th voter for the j th issue by X j i and the vector of all opinions in the committee (called profile) by X ∈ ({0, 1} m ) n (For the ease of presentation we will identify True with 1 and False with 0). Like in the example above, not all opinions are acceptable (one cannot accept a network protocol if it is not secure). We assume a non-empty set X of {0, 1} m called the agenda is given. The opinions in X are called the consistent opinions and only these opinions are held by voters 2 . For instance the conjunction agenda, which is the agenda described in the example, is defined to be the set {000, 010, 100, 111} 3 . A related model (that is more common in the literature) is 'Judgement Aggregation'. In this model the issues are logical propositions over a set of variables and a consistent opinion is an assignment to these variables (so not every combination of truth values for the proposition is achievable). We feel that the logical model is more suitable in cases where one searches for aggregation mechanisms that respect the semantics of the agenda (which is represented by the propositions). Since our work can be stated as impossibility results, we prefer using the more general model. In [16] Dokow and Holzman prove that the two models are equivalent in the sense that each set of consistent opinions can be described using a proposition set (although not uniquely).
An aggregation mechanism is a function that defines for any profile the aggregated opinion (F : ({0, 1} m ) n → {0, 1} m ). In this work we concentrate on two properties of aggregation mechanisms, which were the first to be studied, independence and consistency. Independence states that the aggregated opinion on the j th issue, (F (X)) j depends solely on the opinions on that issue X j . For instance, issue-wise majority that was presented above satisfies independence. Consistency of the aggregation mechanism states that whenever all the members of the committee hold consistent opinions, i.e., X ∈ X n , F returns a consistent opinion as well, i.e., F (X) ∈ X. For instance, for the conjunction agenda issue-wise majority , as can be seen in the deploy-protocol example, does not satisfy consistency. The consistency criterion is commonly accepted as regarding a system of agents as an agent and hence requiring it the same rationality constraint we require from the individual agents. For instance, firms(in which decisions are formed by aggregating opinions of different board members) act as a unit and it matters to the firms's credibility if it can be expected to behave coherently. The independence criterion, while not being uncontroversial, has some appeal in that it guarantees a proposition-wise approach to aggregation. In section 2.1 we discuss further this criterion and claim that it can be justified in many scenarios.
A natural question is to characterize the aggregation mechanisms that satisfy independence and consistency (for a given agenda). For the conjunction agenda (under mild and natural constraint 4 ) the only aggregation mechanisms that satisfy independence and consistency are the oligarchies (The oligarchy of a coalition S returns for each issue True iff all voters in S voted True for that issue). Since, the set of consistent and independent aggregation mechanisms is small and unnatural this characterization theorem is referenced as an impossibility result. Such impossibility results are quite strong. They shows the impossibility of finding any aggregation mechanism that satisfies the conditions. In other words: for each mechanism there will always be some judgement profiles that lead to a breakdown of the mechanism. In this work we deal with this point (and we think we even strengthening the impossibility result) by quantifying the number of profiles that lead to such breakdown.
For the conjunction agenda we prove: Theorem. 5 For any ǫ > 0, n 1, δ < Cn −2 ǫ 4 (for some constant C that does not depend on n), no aggregation mechanism F for the conjunction agenda over n voters satisfies the following five conditions: • F is δ-close to being independent.
I.e., there exists an independent (not necessarily consistent) aggregation mechanism G that returns the same aggregated opinion as F for at least (1 − δ) fraction of the profiles. • F is δ-close to being consistent.
I.e., F returns a consistent result for at least (1 − δ) fraction of the profiles. • F is ǫ-far from returning always False on one of the basic premises.
-F returns True as an aggregated opinion for the first issue for at least at least ǫ fraction of the profiles.
-F returns True as an aggregated opinion for the second issue for at least at least ǫ fraction of the profiles.
• F is ǫ-far from being an oligarchy. I.e., for any oligarchy G of a coalition S, F returns the same aggregated opinion as G for at most (1 − ǫ) fraction of the profiles.
In this paper we prove similar theorems for a family of agendas: truth-functional agendas in which every issue is either a premise or a conclusion defined as conjunction or xor of several premises (up to input & output negation). In a truth-functional agenda the issues are divided into two types: premises and conclusions. Each conclusion j is characterized by a binary function Φ j over the premises and an opinion is consistent if the answers to the conclusion issues are attained by applying the function Φ j on the answers to the premise issues.
For instance the conjunction agenda is a truth-functional agenda with two premises and one conclusion and we mark this by notating the agenda as A, B, A ∧ B . We restrict ourselves to truth-functional agendas in which the conclusions are either conjunction or xor up to negation of inputs and outputs 6 .

Approximate Judgement Aggregation
Lately there is a series of works coping with impossibility results in Social Choice using approximations (e.g., [7] and [22]). The version of approximation we define in this work is studying aggregation mechanisms that are almost consistent and almost independent. We quantify being almost consistent by defining δconsistency of an aggregation mechanism as having a consistent aggregation mechanism G that disagrees with F on at most δ fraction of the inputs 7 . Similarly, we quantify being almost independent by defining δ-independence of an aggregation mechanism as having an independent aggregation mechanism that disagrees with F on at most δ fraction of the inputs. Both terms can be equivalently defined as the failure probability of tests as we show in section 2 and actually the second versions of the definitions are those we use in the analysis since they are easier to work with.
Both definitions use the Hamming distance d X (F, G) = Pr [F (X) = G(X) | X ∈ X n ] which includes an assumption of uniform distribution over the opinions for each voter and that voters draw their opinions independently (Impartial Culture Assumption). This assumption, while certainly unrealistic, is the natural choice in this kind of work and is discussed further in section 2.
Trivially, if F is close to G and G is independent and consistent, F is δ-independent and δ-consistent for δ linear in d(F, G). Our main question is whether there are aggregation mechanisms that are close to being independent and consistent that are not close the (usually small) set of consistent and independent aggregation mechanisms.
In several agendas the set of consistent independent aggregation mechanisms is a very small set(E.g., dictatorship or oligarchies) and hence this question is equivalent to asking whether we can look for aggregation mechanisms that are close to being independent and close to being consistent without collapsing to the known small (perturbed) set of consistent independent aggregation mechanisms.

Connection to Approximate Preference Aggregation
Preceded our work is the works by Kalai [24], Mossel[35], and Keller [25] that proved similar approximate aggregation theorems for preference aggregation. In this agenda the consistent opinions represent the linear orders over a set of candidates {c 1 , c 2 , . . . , c s } and the issues are the s 2 pair-wise comparisons between candidates 8 .Similarly to the doctrinal paradox, the Condorcet Paradox [10] shows that issue-wise majority might lead to an inconsistent result for preference aggregation as well. Arrow's theorem [1] shows that (under mild and natural constraint 9 ) the only aggregation mechanisms that satisfy independence and consistency are the dictatorships.
The recent works of Kalai [24], Mossel[35], and Keller [25] proved similar results to the results we prove here.
Theorem ( [25]). There exists an absolute constant C such that the following holds: For any ǫ > 0 and k 3, if f is an aggregation mechanism for the preference agenda over k candidates that satisfies independence and C · ǫ/k 2 3 -consistency, , then there exists an aggregation mechanism G that satisfies independence and consistency such that d(F, G) < ǫ.

Connection to Property Testing
We think it is useful to phrase the question of approximate aggregation using terms of property testing. In this field we query a function at a small number of (random) points testing for a global property (In our case, the property is being a consistent independent aggregation mechanism). We discuss this connection further is section 5. For example a corollary of the results we present in this paper (in property testing terms): For any three binary functions f, g, h : (when the addition is in Z 2 and Z n 2 , respectively), then there exists three binary functions , and Pr[h(x) = h ′ (x)] are smaller than Cǫ for some constant C independent of n and Notice a special case of this result (for f = g = h) is the classic result of Blum, Luby, and Rubinfeld ( [4], [2]) for linear testing of boolean functions.

Techniques
We prove the main theorem by proving the specific case of independent aggregation mechanism for two basic agenda families: the conjunction agendas (agendas in which there is exactly one conclusion with is constrained to be the conjunction of the premises. 7.1) and the xor agendas (agendas in which there is exactly one conclusion with is constrained to be the xor of the premises. theorem 7.3). Later we show how to extend these theorems to the general theorem of relaxing both constraints (theorem 7.4).
We use two different techniques in the proofs. For the conjunction agendas we study influence measures 10 of voters on the issue-aggregating functions and for the xor agendas we use Fourier analysis of the issueaggregating functions.
An open question is whether one can find such bounds for any agenda or whether there exists an agenda for which the class of aggregation mechanisms that satisfy consistency and independence expands non trivially when we relax the consistency and independence constraints.
We proceed to describe the structure of the paper. In Section 2 we describe the formal model of aggregation mechanisms. In section 3 we give present the main agendas we deal with, truth-functional agendas, and specifically conjunction agendas and xor agendas. In section 4 we state the motivation to deal with approximate aggregation, and describe the known results for preference approximate aggregation by Kalai, Mossel, and Keller. In section 5 we describe the connection we find between Approximate Aggregation and the field of Property Testing. In sections 6 and 7 we describe our main theorems and outline the proof. Section 8 concludes.

The Model
We define the model similarly to [16] (which is Rubinstein and Fishburn's model [43] for the binary case) We consider a committee of n individuals that needs to decide on m issues. An opinion is a vector x = (x 1 , x 2 , . . . , x m ) ∈ {0, 1} m denoting an answer to each of the issues. An opinion profile is a matrix 10 Both the known influence (Banzhaf power index) and a new measure we define: The ignorability of a voter. X ∈ ({0, 1} m ) n denoting the opinions of the committee members so an entry X j i denotes the vote of the i th voter for the j th issue, the i th row of it X i states the votes of the i th individual on all issues, and the j th column of it X j states the votes of each of the individuals on the j th issue. In addition we assume that an agenda X ∈ {0, 1} m of the consistent opinions is given.
The basic notion in this field is an aggregation mechanism which is a function that returns an aggregated opinion (not necessarily consistent) for every profile (F : An aggregation mechanism satisfies Independence (and we say that the mechanism is independent) if for any two consistent profiles X and Y and an issue j, if X j = Y j (all individuals voted the same on the j th issue in both profiles) then (F (X)) j = (F (Y )) j (the aggregated opinion for the j th issue is the same for both profiles). This means that F satisfies independence if one can find m binary functions . . , f (X m ) for some issue aggregating function f , i.e., all issues are aggregated using the same function. We will use the notation f 1 , f 2 , . . . , f m for the independent aggregation mechanism that aggregates the j th issue using f j .
The main two measures we study in this paper are the inconsistency index IC X (F ) and the dependence index DI X (F ) of a given aggregation mechanism F and a given agenda X. These measures are relaxations of the consistency and independence criterion that are usually assumed in current works 13 . We define the measures by For an agenda X and an aggregation mechanism F for that agenda, the inconsistency index is defined to be the probability to get an inconsistent result.

Definition 2.2 (dependency index).
For an agenda X and an aggregation mechanism F for that agenda, the dependency vector DI j,X (F ) is defined as The definition can be seen as a test for independence of the j th issue as discussed in section 5 The dependency index DI X (F ) is defined by In contexts where the agenda is clear we omit the agenda superscript and notate these as IC(F ), DI j (F ), and DI(F ), respectively.
We define these two indexes using local tests. We prove that the more natural definition of distance to he class of aggregation mechanisms that satisfy consistence (or independence) is equivalent to the above (up to multiplication by δ by constant). Proposition 2.3. Let F be an aggregation mechanism for an agenda over m issues. Then F satisfies IC(F ) δ iff there exists a consistent aggregation mechanism H that satisfies d(F, H) δ. 11 We define the function for all profiles for simplicity but we are not interested in the aggregated opinion in cases one of the voters voted an inconsistent opinion. 12 Notice this property is a generalization of the IIA property for social welfare functions (aggregation mechanism for the preference agenda) so a social welfare function satisfies IIA iff it satisfies independence as defined here (when the issues are the pair-wise comparisons). 13 F satisfies consistency iff IC(F ) = 0 and independence iff DI(F ) = 0 Proposition 2.4. Let F be an aggregation mechanism and j an issue. If DI j (F ) ǫ, then there exists an aggregation mechanism H that satisfies DI j (H) = 0 and d(F, H) 2ǫ. If DI j (F ) ǫ, then every aggregation mechanism H that satisfies DI j (H) = 0, also satisfies d(F, H) 1 2 ǫ Proposition 2.5. Let F be an aggregation mechanism for an agenda over m issues that satisfies DI(F ) δ.
Then there exists an independent aggregation mechanism H that satisfies d(F, H) 2mδ.
These definitions include two major assumptions on the opinion profile distribution. First, we assume the voters pick their opinions independently and from the same distribution. Second, we assume a uniform distribution over the (consistent) opinions for each voter (Impartial Culture Assumption). The uniform distribution assumption, while certainly unrealistic, is the natural choice for proving 'lower bounds' on IC(F ). That is, proving results of the format 'Every aggregation mechanism of a given class has inconsistency index of at least γ(n)'. In particular, the lower bound, up to a factor δ, applies also to any distribution that gives each preference profile at least a δ fraction of the probability given by the uniform distribution. 14 Note that we cannot hope to get a reasonable bound result for every distribution. For instance, since for every aggregation mechanism we can take a distribution on profiles for which it returns a consistent opinion.

The Independence Property
The independence criterion is sometimes criticized as being unjustified normatively in most real-life scenarios 15 . The impossibility results of judgement aggregation can also be seen as 'empirical' argument against independence since they show that it contradicts the more natural assumption of consistency. While we accept this argument, we think our work quantifies the tradeoff between the two criteria. Moreover, in this section we claim that in the general case this criterion can be justified on several different grounds.
First, in a lot of cases it is justified to expect due to normative or legal reasons that changing an individual judgement on an issue should not change the collective judgement on another issue.
Secondly, there are works that defend this criterion by using manipulation-resistance arguments. In [15] Dietrich and List define the notion of manipulability of an aggregation mechanism 16 and proved that any aggregation mechanism that does not satisfy independence is manipulable. In this paper they further prove that this manipulability property is equivalent to a more game-theoretic property of strategy-proofness under some assumptions on players' preferences.
On the ground of simplicity of representation one can justify independence as a criterion that returns aggregation mechanisms that are easy to represent, calculate, or justify (For instance, justify a voting method to the public).
Other grounds of justification for such aggregation mechanisms are from the voter point of view. There might be a situations in which the decisions are made over time (different meetings) or place (different representatives of the same voting identity) so it is fair to assume that when voting on an issue or aggregating the votes it is unreasonable to depend on votes on other issues. Another argument might be than there are scenarios in which you need to define the aggregation method and only at a later stage choose from the 14 In successive works we relax this assumption and prove similar results for more general distributions. 15 Chapman( [8]) and Mongin([34]) attack this criterion and claim it removes the discipline of reason from social choice since it disregards the intra-issue dependencies which is the essence of the problem. According to this criterion the aggregation of 'complex' issues is done without regarding the reasons of the voters for their opinions and hence lacks the information for good aggregation. 16 An aggregation mechanism F is manipulable at the profile X by individual i(the manipulator) on issue j if X i j = (F (X)) j (The manipulator disagrees with the aggregated opinion on issue j), but X i j = (F (X ′ )) j for some profile X ′ that differs form X in i's vote only.(I.e. the manipulator can get his will on j by voting differently) set of issues the relevant one (For instance, the definition of Social Welfare Functions as returning a choice function so only at a later stage the society is faced with the menu of alternatives).

Binary Functions
Since this work deals with binary functions (for aggregating issues), we need to define several notions for this framework as well. To ease the presentation, throughout this paper we will identify True with 1 and False with 0 and use logical operators on bits and bit vectors (using entry-wise semantics). In addition we define a distance function over the binary functions. The distance between two functions f, g : {0, 1} n → {0, 1} is defined to be the probability of getting a different result (normalized Hamming distance). d(f, g) = Pr [f (x) = g(x)]. From this measure we will derive a distance from a function to a set of functions by d(f, G) = min g∈G d(f, g) One more notation we are using in this paper is x J for a binary vector x ∈ {0, 1} n and a coalition J ⊆ {1, 2, . . . n} for notating the entries of x that correspond to J. 17 An equivalent definition is: ∀x, y : f (x)+f (y) = f (x+y) when the addition is in Z2 and Z n 2 , respectively. 18 In the literature this criterion is sometimes referred to as Unanimity, e.g., in [32]. We choose to follow [17] and refer to it as Pareto to distinguish between it and the unanimity function which is the oligarchy of {1, 2, . . . , n}. 19 In the simple cooperative games regime, this is also called the Banzhaf power index of player i in the game f .

Agenda Examples
A lot of natural problems can be formulated in the framework of aggregation mechanisms. It is natural to divide the agendas to two major classes Truth-Functional Agendas and Non Truth-Functional Agendas.

Truth-Functional Agendas
In a truth-functional agenda the issues are divided into two types: k premises and (m − k) conclusions. The conclusion issues are binary functions over the k premises, Φ : An opinion is consistent if the answers to the conclusion issues are attained by applying the function Φ on the premise issues.
There are cases in which there might be more than one way to classify the issues to premises and conclusions. For instance, the 2-premises xor agenda described below X = {001, 010, 100, 111} can be defined both as A, B, A ⊕ B and as A, A ⊕ C, C . Since we choose to analyze the agenda as opinion sets (and not as a proposition set) we do not handle this point and notice that it is irrelevant for our results.
These agendas ,due their structure, seems to be a good point to start our work on approximate aggregation and in this paper we prove basic results for the following two families of truth-functional agendas. Later we derive results to a more general family of truth-functional agendas.

Conjunction Agendas
In the (m-premises) conjunction agenda A 1 , . . . , A m , m ∧ j=1 A j there are m + 1 issues to decide on and the consistency criterion is defined to be that the last issue is a conjunction of the other issues. For instance the Doctrinal Paradox agenda is the 2-premises conjunction agenda. A common description of the problem is of a group of judges or jurors that should decide whether a defendant is liable under a charge of breach of contract. Each of them should decide on three issues: whether the contract was valid (p), whether there was a breach (q) and whether the defendant is liable (r). In their decision making they are constrained by the legal doctrine that the defendant is only liable if the contract was valid and if there was indeed a breach (r ⇐⇒ (p ∧ q)).

Xor Agendas
Similarly, in the (m-premises) xor agenda A 1 , . . . , A m , m ⊕ j=1 A j there are m + 1 issues to decide on and the consistency criterion is defined to be that the third issue is True if the number of true-valued opinions for the first m is even. An equivalent way to define this agenda is constraining the number of True answers to be odd.

Non Truth-Functional Agendas
One can think on a lot of agendas that cannot be represented as a truth-functional agenda. Among such interesting natural agendas that were studied one can find the equivalence agenda [21], the membership agenda [42] [33], and the preference agenda described below.

Preference Aggregation
Aggregation of preferences is one of the oldest aggregation frameworks studied. In this framework there are s candidates and each individual holds a full strict order over them. We are interested in Social Welfare Functions which are functions that aggregate n such orders to an aggregated order. As seen in [36] and [12], this problem can be stated naturally in our framework by defining s 2 issues 20 .

Motivation & Known Results
We find the motivation for dealing with the field of approximate aggregation in three different disciplines.
• The consistent characterization are often regarded as 'impossibility results' in the sense that they 'permit' a very restrictive set of aggregation mechanisms. (e.g., Arrow's theorem tells us that there is no 'reasonable' way to aggregate preferences). Extending these theorems to approximate aggregation characterizations sheds light on these impossibility results by relaxing the constraints.
• , and a generalization of them to non-truth-functional agendas called sequential priority aggregation( [29], [14]). Another procedure in the literature is the distance-based aggregation( [40]) which is a well known for preference aggregation (E.g., Kemeny voting rule [26], Dodgson voting rule [3], and lately a more systematic analysis in [19]). Our work contribute to this discussion by pointing out where one should search for solutions while not leaving the consistency and independence constraints entirely.
• Connections to Property Testing as discussed in section 5 The first work studying approximate aggregation was done for the preference agenda over three candidate by Kalai [24] (although without stating the general framework of approximate aggregation). This work was generalized by Mossel[35], and Keller [25].
Theorem ([25] theorem 1.3). There exists an absolute constant C such that the following holds: For any ǫ > 0 and k 3, if f is an aggregation mechanism for the preference agenda over k candidates that satisfies independence and C · ǫ/k 2 3 -consistency, , then there exists an aggregation mechanism G that satisfies independence and consistency such that d(F, G) < ǫ.

Connection to Property Testing
In the words of [41], the field of property testing deals with the following: Given the ability to perform (local) queries concerning a particular object (e.g., a function or a graph), the task is to determine whether the object has a predetermined (global) property (e.g., linearity or bipartiteness), or is far from having the property. The task should be performed by inspecting only a small (possibly randomly selected) part of the whole object, where a small probability of failure is allowed. Property testing trades accuracy (the distance parameter) for efficiency (number of queries).
We think it is useful to view the Approximate Aggregation problem in the framework of Property Testing. Below we highlight the connection between Approximate Aggregation and a special case of Property testing 20 The issue i, j (for i < j) represents whether an individual prefers ci over cj .
In our case the global property we are trying to test is 'consistency and independence' of an aggregation mechanism. The class of satisfying aggregation mechanism is characterized by the current state of research. It is clear that each of the components of this property separately, consistency and independence of an issue, can be tested trivially. The consistency test consists of picking a (consistent) profile uniformly at random and checking whether the aggregated opinion is consistent. The test for independence of issue j consists of picking a (consistent) profile uniformly at random altering the opinion for each voter without changing the j th bit and check whether the aggregated opinion on the j th issue is changed due to the altering. For each of the two tests the probability to accept a non-satisfying mechanism is linear in the distance to the satisfying set (and equals IC(F ) and DI j (F ), respectively). The main question of this work (in property testing terms) can be stated as follows: What is the best test for being 'consistent and independent' one can assemble from running the (m+1) tests as black boxes (and therefore get information only on IC(F ) and DI j (F )). Similar question was asked lately in [9]. In [9] the authors query (among other similar questions) the conditions needed in order to deduce from testability of two properties the testability of the intersection of the two properties. Our work can be seen as studying this question for a specific domain in which the question seems to be natural and while adding the constraint that the test of the intersection property should be defined as a sequence of tests for the basic properties.
The main result of this paper is that for a class of mechanisms (corresponding to a natural class of agendas) one can assemble those tests to a test for the property 'consistent and independent'.
Similarly one can state questions dealing with sub-families of aggregation mechanisms. For example, as we stated in the introduction, the classic result of Blum, Luby, and Rubinfeld for linearity testing of boolean functions is a direct corollary of our result for the 2-premises xor agenda when considering systematic aggregation mechanisms.
Still, the target of the two fields is different. While Property Testing deals with finding the most efficient (query-wise) algorithm for testing a property (functions family), Approximate Aggregation deals with analyzing a specific family of tests.

Main Results
The main result of this paper is Theorem 6.1. For any ǫ > 0 and m, n 1, there exists δ IC , δ DI = n −2 ǫ m poly(m) , such that for every truth-functional agenda X over m issues, in which each non-premise issue is defined to be either conjunction of several premises or xor of several premises(up to negation of inputs or output) 21 , if F is an aggregation mechanism for X over n voters satisfying δ-independence and δ-consistency, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < ǫ Moreover, one can take δ IC = n −2 (1−βǫ)ǫ 5m m 2 +m−1 −β ǫ ǫ and δ DI = 1 2m β ǫ ǫ for any β ǫ ∈ 0, n −2 ǫ m m 2 +m−1 .
A direct corollary is the following impossibility result. X over m issues, in which each non-premise issue is defined to be either conjunction of several premises or xor of several premises(up to negation of inputs or output), no aggregation mechanism F for X over n voters satisfies the following three conditions: • δ-independence • δ-consistency • F is ǫ-far from any independent and consistent aggregation mechanism for X .
In the case of xor agenda (and its generalization, a truth-functional agenda in which all the conclusions are xor) we can get a much better result (e.g., no dependence on the number of voters) Theorem 6.3. Let m 3 and let the agenda be X = A 1 , . . . , A m−1 , A j . For any ǫ < 1 6 and any aggregation mechanism F : If F is an aggregation mechanism for X over n voters satisfying ǫ-independence and ǫ-consistency, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < m(2m + 3)ǫ Noticing that any affine agenda (i.e., an agenda that is an affine subspace) can be represented as a truth-functional agenda that uses xor conclusions only (lemma C.5) we can get the following corollary Corollary 6.4. For any ǫ > 0 and m, n 1, there exists δ = ǫ m(2m+3) , such that for every affine agenda X over m issues, if F is an aggregation mechanism for X over n voters satisfying δ-independence and δ-consistency, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < ǫ

Proof Sketch of the Main Theorem
In this section we sketch the techniques behind our proofs. The full proofs can be found in the appendices.
We prove the main theorem by proving three independent theorems. An approximation result for independent aggregation mechanisms for conjunction agendas (theorem 7.1). An approximation result for independent aggregation mechanisms for xor agendas (theorem 7.3). An agenda independent method of converting results for the independent case to the general case of relaxing both constraints (theorem 7.4). Using induction on the number of conclusions and noticing that negating (of the inputs and of the output) is renaming of opinions in our framework (and hence does not change the approximation results) we get theorem 6.1. For any ǫ > 0 and any independent aggregation mechanism F : If IC(F ) ǫ, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < 5m n 2 ǫ 1 m 2 +m−1 .

Conjunction Agenda
Technique: The main insight in the proof is that for any two different issue-aggregating functions, f and g, in F for two of the m premises, we can bound the product of the influence of a voter on f and the ignorability of the same voter for g using the inconsistency index of F by P i (f ) · I i (g) 4IC(F ).

Proof sketch.
Let F = f, g, h be an aggregation mechanism that satisfies IC(F ) ǫ. In case that f (or g) is close enough to the constant zero function, F is close to the consistent aggregation mechanism 0, g, 0 .
Otherwise, we define for a given function f : {0, 1} n → {0, 1} and a coalition J (the junta), the junta function f →J : {0, 1} n → {0, 1}. It is derived from f in the following way: I.e., for a given input, f →J reads only the votes of the junta members, iterates over all the possible votes for the members outside the junta, and returns the more frequent result (assuming uniform distribution over the votes of the voters outside J). We define f →J and g →J with regard to the junta of all the voters with small ignorability for either f or g.
We prove that f →J and g →J are close to f and g, respectively and that there exists an issue aggregation function h ⋆ such that f →J , g →J , h ⋆ is a consistent aggregation mechanism that is close to F .
There is a known characterization of the consistent independent aggregation mechanism for the conjunction agenda. (This characterization is a direct corollary from a series of works in the more general framework of aggregation, E.g., [37], [16]. We include a proof of it in the appendix) A corollary from this theorem and theorem 7.1 is a characterization of the approximate aggregation mechanisms for this agenda. Actually ,in the proof of theorem 7.1 we get a tighter characterization that distinguishes between the two cases of consistent independent aggregation mechanism. For any ǫ < 1 6 and any independent aggregation mechanism F : If IC(F ) ǫ, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) mǫ.

Proof sketch.
Given an independent aggregation mechanism F = f, g, h we analyze the expression E [f (x)g(y)h(xy)] when x and y are sampled uniformly and independently.
On one hand we show that E [f (x)g(y)h(xy)] = 1−2IC(F ). On the other hand we show that E [f (x)g(y)h(xy)] = χ∈Lin f (χ) g(χ) h(χ) when f (χ) equals 1−2 min (d (f, χ) , d (f, −χ)). Hence, when IC(F ) is small then this sum is close to one 22 The proof is similar to the analysis of the BLR (Blum-Luby-Rubinfeld) linearity test done in [2]. and hence there exists a linear function such that f ,g, and h are close to it (up to negation). Noticing that for any linear function χ, χ, χ, χ and the permutations of −χ, −χ, χ are consistent independent aggregation mechanism for this agenda gives us the result.

Extending to δ-independence Results
Theorem 7.4. If there exists a function δ(ǫ, n) s.t. for any ǫ > 0 and n 1, if F is an aggregation mechanism for X over n voters satisfying independence and IC(F ) δ(ǫ), then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < ǫ.
Then, for any ǫ > 0 and n 1, there exist δ IC , δ DI > 0, such that if F is an aggregation mechanism for X over n voters satisfying IC(F ) δ IC and DI(F ) δ DI , then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < ǫ. Moreover, one can take In order to extend the results for the δ-dependent case (DI(F ) = 0) we prove the following agendaindependent proposition.
Proposition (Proposition 2.5). Let F be an aggregation mechanism for an agenda over m issues that satisfies DI(F ) δ. Then there exists an independent aggregation mechanism H that satisfies d(F, H) 2mδ.
I.e., if F is δ-independent we can find a close consistent aggregation mechanism H and since it is close we can deduce bounds on the proximity of F to the consistent and independent aggregation mechanisms from bounds on this proximity of H. Similarly, since H is close to F , we can deduce that if F is δ-consistent than H is δ ′ -consistent for δ ′ close to δ. Combining these we get the theorem.

Summary and Future Work
In this paper we defined the issue of approximate aggregation which is a generalization of the study of aggregation mechanisms that satisfy consistency and independence. We defined measures for the relaxation of the consistency constraint (inconsistency index IC) and for the relaxation of the independence constraint (dependency index DI). To our knowledge, this is the first time this question is stated in its general form.
We proved that relaxing these constraints does not extend the set of satisfying aggregation mechanisms in a non-trivial way for any truth-functional agenda in which every conclusion is either conjunction or xor up to negation of inputs or output. We saw that every conclusion of two premises can be stated as such. We also saw that any affine agenda can be represented as truth-functional agenda with xor conclusions only and derived a better approximate aggregation characterization for this family. Particulary we calculated the dependency between the extension of this class (ǫ) and the inconsistency index (δ(ǫ)) (although probably not strictly) for two families of truth-functional agendas with one conclusion. The relation we proved includes dependency on the number of voters (n). In the works that preceded us for preference agendas ( [24], [35], [25]) the relation did not include such a dependency. An interesting question is whether such a dependency is inherent for conjunction agendas or whether it is possible to prove a relation that does not depend on n.
A major assumption in this paper is the uniform distribution over the inputs which is equivalent to assuming i.i.d uniform distribution over the premises. We think that our results can be extended for other distributions (still assuming voters' opinions are distributed i.i.d) over the space over premises' opinions which seem more realistic.
Immediate extensions for this work can be to extend our result to more complex truth-functional agendas and generalize our results to non-truth-functional agendas to get a result unifying our work and Kalai, Mossel, and Keller's works for the preference agenda.
A major open question is whether one can find an agenda for which relaxing the constraints of independence and consistency extends the class of satisfying aggregation mechanisms in a non-trivial way.

A.1 Propositions A.1,A.3
For a given pair of independent aggregation mechanisms, the following propositions connect between the pairwise distance between respective issue-aggregating functions (which we found easier to analyze in most cases) and both the distance between the mechanisms and the inconsistency indices of them.
Proof of Proposition . Direct use of the union-bound inequality.
Proof of Proposition .
As a corollary of the above we get
Let F be an aggregation mechanism and j an issue. If DI j (F ) ǫ, then there exists an aggregation mechanism H that satisfies DI j (H) = 0 and d(F, H) 2ǫ. If DI j (F ) ǫ, then every aggregation mechanism H that satisfies DI j (H) = 0, also satisfies d(F, H) 1 2 ǫ

Proof of Proposition .
With no loss of generality assume that j = 1.
• Let F be an aggregation mechanism. We define the functions G 1 , . . . , G m : X n → {0, 1} by: and an aggregation mechanism G(X) = G 1 (X), . . . , G m (X) . Clearly • Let F be an aggregation mechanism that is ǫ-close to satisfy DI 1 (F ) = 0. That is, we can find an aggregation mechanism G such that d(F, G) ǫ and DI 1 (G) = 0.

Proposition.
Let F be an aggregation mechanism for an agenda over m issues that satisfies DI(F ) δ. Then there exists an independent aggregation mechanism H that satisfies d(F, H) 2mδ.

A.4 Id Agenda
For completeness we add here an approximate aggregation theorem for the id agenda A, A Theorem A.4. For any ǫ > 0 and any independent aggregation mechanism F : If IC A,A ǫ, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) ǫ.
Proof. This theorem is trivial since Noticing that any aggregation mechanism of the form f, f is consistent for this agenda so we get the theorem. For any ǫ > 0 and any independent aggregation mechanism F : If IC(F ) ǫ, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) < 5m n 2 ǫ I.e., for a given input, f →J reads only the votes of the junta members, iterates over all the possible votes for the members outside the junta, and returns the more frequent result (assuming uniform distribution over the votes of the voters outside J). We prove the following lemma: . . , f m , h is ∆-close to F = 0, f 2 , . . . , f m , h and (ǫ + ∆)-close to F = 0, f 2 , . . . , f m , 0 which is a consistent mechanism.
We will use the following isoperimetric inequality on the boolean cube: Proposition (The Isoperimetric Inequality for The Boolean Cube [5]).
Assume that for issues j, f j is not the constant zero function. We will prove that f 1 = f 2 = .
. . = f m = h ∈ Olig by proving the following series of claims.
• For all issues j f j (1) = 1 With no loss of generality, assume for contraction that f 1 (1) = 0. Let x ∈ {0, 1} n . Then I.e. h ≡ 0. From that we can conclude that there exists an issue j s.t. f j ≡ 0 and get a contradiction.
• For all issues j f j = h We will prove that f 1 = h. The proof is similar for all j. For any ǫ < 1 6 and any independent aggregation mechanism F : If IC(F ) ǫ, then there exists an aggregation mechanism G that satisfies consistency and independence such that d(F, G) mǫ.

Proof.
The theorem is a corollary of the following lemma: Noticing that a j χ is a consistent mechanism for any linear function χ and signs a j s.t. f j (S). 23 Fourier transforms are widely used in mathematics, computer science, and engineering. The main idea is representing the function f over an orthonormal basis χ S when the inner product is define to be f, g = E[f (x)g(x)] and the basis vectors χ S are defined to be χ S (x) = i∈S xi for S ⊆ {1, . . . , n}. The coefficients of f according to the Fourier basis are notated f (S). I.e., f ≡ S f (S)χ S . For a good introduction to the subject see [38], [11].
In this proof we are using the following: • χ S (xy) = χ S (x)χ S (y)


Integrating place-specific livelihood and equity outcomes into global assessments of bioenergy deployment

Integrated assessment models suggest that the large-scale deployment of bioenergy could contribute to ambitious climate change mitigation efforts. However, such a shift would intensify the global competition for land, with possible consequences for 1.5 billion smallholder livelihoods that these models do not consider. Maintaining and enhancing robust livelihoods upon bioenergy deployment is an equally important sustainability goal that warrants greater attention. The social implications of biofuel production are complex, varied and place-specific, difficult to model, operationalize and quantify. However, a rapidly developing body of social science literature is advancing the understanding of these interactions. In this letter we link human geography research on the interaction between biofuel crops and livelihoods in developing countries to integrated assessments on biofuels. We review case-study research focused on first-generation biofuel crops to demonstrate that food, income, land and other assets such as health are key livelihood dimensions that can be impacted by such crops and we highlight how place-specific and global dynamics influence both aggregate and distributional outcomes across these livelihood dimensions. We argue that place-specific production models and land tenure regimes mediate livelihood outcomes, which are also in turn affected by global and regional markets and their resulting equilibrium dynamics. The place-specific perspective suggests that distributional consequences are a crucial complement to aggregate outcomes; this has not been given enough weight in comprehensive assessments to date. By narrowing the gap between place-specific case studies and global models, our discussion offers a route towards integrating livelihood and equity considerations into scenarios of future bioenergy deployment, thus contributing to a key challenge in sustainability sciences.

Introduction
Recent efforts to model climate change mitigation within possible future energy systems have suggested that bioenergy can play a major role. The special report on renewable energy sources (SRREN) (IPCC 2011) indicates a global potential deployment of biomass for bioenergy in the range of 100-300 EJ yr −1 in 2050, with possibly more than 100 EJ yr −1 coming from dedicated bioenergy cropscompared to about 11 EJ yr −1 in dedicated bioenergy crops in 2008 (Chum et al 2011). Similarly, the Global Energy Assessment suggests that the production of bioenergy crops may increase by a factor of 10 in 2050 (Coelho et al 2012). Such a high deployment of biomass for bioenergy would require a massive agricultural transformation with significant implications for both environmental and social change. The global warming effects of induced land-use change and carbon stock dynamics can negate mitigation from fuel substitution (Fargione et al 2008, Searchinger 2012, a result that has more recently been taken up in assessment models (Wise et al 2009, Creutzig et al 2012a. Models also consider potential widespread impacts that such an agricultural transformation would have on environmental resources, e.g. water, soil and biodiversity (Erb et al 2012, Gerbens-Leenes et al 2009, Popp et al 2011. But the global competition for land use (Lambin and Meyfroidt 2011) is intensified by higher demand for biofuels, and impacts not only ecological dimensions but also directly about 1.5 billion smallholders 7 and agro-pastoralists. Hence, the aforementioned studies and assessments remain deficient in two ways. First, they operationalize social impacts as economic efficiency, economic growth, and sometimes food prices, ignoring other important dimensions of human wellbeing such as change in socio-economic and health conditions. Second, the high level of spatial aggregation of the assessments makes them blind to place-specific drivers and distribution of impacts among different social groups and geographical areas within countries.
As an example of a global energy assessment, the SRREN mentions livelihoods as a relevant category (p 58 and p 90), indicating positive outcomes of bioenergy deployment for livelihoods (Chum et al 2011). However, it neither considers livelihood impacts as a constraint to deployment nor identifies factors shaping the interaction between bioenergy and livelihoods. The mandate of the SRREN focuses on cost-effective climate change mitigation and does not extend to evaluating goals such as poverty alleviation, other livelihood improvements, and ecological sustainability. But climate change mitigation and economic growth are ultimately not goals on their own but rather means to achieve human wellbeing. Place-specific livelihood impacts of bioenergy schemes are nuanced, differentiated and insufficiently reflected in current bioenergy assessments (German et al 2012, Creutzig et al 2012b). As a result, the livelihoods of workers, smallholder farmers and local populations, as well as distributional considerations are systematically underexplored (Corbera and Pascual 2012, 7 See FAO (2012). Creutzig et al 2012b). It is these gaps-livelihoods and equity-that we seek to narrow.
In this letter, we systematically identify routes by which bioenergy deployment can influence livelihoods, influenced by global market dynamics and place-specific production models. We analyze case studies that investigate livelihood effects of cultivating the most common feedstocks for first-generation biofuels, as these constitute the vast majority of bioenergy schemes. Among these schemes, we investigate specifically those in developing countries where agriculture often remains a predominant source of livelihoods en large. We focus on the cultivation of oil palm, jatropha, soy, cassava and sugarcane. Section 2 addresses both global and local drivers and effects of biofuel deployment on livelihoods. Section 3 summarizes the livelihood effects in a conceptual figure and an example. We then discuss how these insights could be better incorporated in integrated assessments of bioenergy deployment and provide an outlook on the potential of second-generation biofuels for improving livelihoods in section 4.

Place-specific and global processes shape how bioenergy affects livelihoods
Livelihoods include 'flows', like income and food availability, and 'stocks' such as land tenure, social and financial assets. Gaining an income through a variety of on-farm and off-farm activities constitutes a key pillar of most rural livelihoods, allowing households and individuals to participate in markets and to benefit from resource use (Ellis 2000, Tesfaye et al 2011. Food access is an obvious livelihood dimension, which is often closely related to income but can also be decoupled, particularly in non-market economies and subsistence farming. Access to land and natural resources is another crucial capability because it enables the production of food for subsistence or sale, and provides other goods and services including firewood and medicinal plants. Access to land can also translate into land rents and flows of income. Such access is mediated by tenure regimes, which encompass property rights, and both formal and informal social relations and systems of authority that influence who gets access to and exercises control over land resources (Ribot and Peluso 2003). Health constitutes a particularly important asset that we include in our analysis as it is influenced both by poverty levels and livelihood strategies, including agricultural practices. We discuss each dimension in turn, highlighting what has been covered in macro-economic studies and contrasting with insights from bottom-up studies of bioenergy feedstock cultivation.

Bioenergy and income
2.1.1. Macro-perspective. Biofuel deployment affects rural income in two ways: indirectly via market-based multiplier and equilibrium effects on the macro-scale, and directly through changes in farm income or plantation wage income at the micro-scale (Arndt et al 2011, Lima et al 2011. At the macro-scale, biofuel market expansion is mostly driven by government policies. The US and EU blending mandates are quantitatively most important, but China, India, Brazil and many other countries also have mandates, tax breaks or subsidies (Mitchell et al 2011). In models, the strongly increasing demand for bioenergy leads to an aggregate increase in generalized rural incomes (Golub et al 2012; see also section 4). If higher wages are paid in biofuel jobs they can put upward pressure on rural wages in general (Arndt et al 2011), which in turn may lead to increased expenditure and second-round market effects of a different kind. However, if agricultural wage levels increase, they can reduce the relative purchasing power of the non-farm poor (Golub et al 2012).

Place-specific perspective.
Specific income effects depend on the production model underlying feedstock cultivation. Common production modes include: plantations; contract farming or outgrower schemes (formal, informal, in association or not with an estate (nucleus) plantation, and cooperatives); independent smallholder farming (operating mainly through spot market transactions); and subsistence farming (mainly for own consumption). Sugarcane and soy tend to be grown in large-scale operations because mechanical harvesting substantially lowers labor costs-a production model of high capital and low labor intensity. Mechanization of production can increase wages and improve labor conditions for those in skilled positions, but also reduce the number of jobs in areas where plantations substitute smallholder farming (Clancy 2013). Unskilled work in plantations is mostly seasonal, low-paid, and conditions have been judged to be sometimes inhumane, fostering exclusion (Clancy (2013); see A1 in table 1). In some contexts, sugarcane contract farmers may not even benefit economically from cultivation due to the control that the sugarcane company exercises on the costs of tillage, seedcane and transportation costs (Waswa et al 2009). Local production of biofuels for local use is often cited as the most promising way for bioenergy to produce livelihood benefits (Achten et al 2010, Ejigu 2008, Milder et al 2008, but relatively few peer-reviewed studies of this production model exist. Unlike feedstocks that can also be marketed for food or other products, jatropha tends to be newly deployed for energy production, both in plantations and on smaller scales. The choice of feedstock affects growers' income because of the production cycle and labor requirements. Farmers who plant slow-growing feedstocks (e.g. jatropha, oil palm) must be able to defer income streams until plants produce a harvest-a situation that favors the participation of better-off farmers with other income sources or adequate savings (McCarthy 2010, Schoneveld et al 2011. The scale of production also shapes income gains and distribution. Smallholders face higher risk of crop failure or lower yields than large farmers due to smaller financial resources for upfront investment and management (see A2 in table 1). However, smallholder contract farming schemes can benefit from economies of scale in certain functions (e.g. standards conformity, input supply, and crop procurement) while enjoying the advantages of family farming regarding labor management and land tenure. The realization of these benefits and their distribution between the contractor and the contracted smallholders will depend in turn on several factors, especially adequate resources and good market access of the scheme operator, low barriers to scheme entry for smallholders, the distribution of costs in the production cycle, and the existence of alternative market outlets for the producers (Gibbon et al 2010, Key and Runsten 1999, Kirsten and Sartorius 2002, Waswa et al 2009. Local competition for produce can put upward pressure on the price paid to the outgrower but can increase the risk for the scheme operator, who depends on a given level of supply to recover operation costs and maintain market position.
Contractual arrangements are thus crucial to determine the distribution of income gains and losses in bioenergy production. Policies and power relations shape value chains, determining the distribution of value added, and the inclusion, expulsion, and repositioning of actors. Often 'downstream' actors such as importers and retailers set the terms of participation, seizing a substantial part of the value added (Bolwig et al 2010). Uneven benefit-sharing may also result from value capture at the local nodes of the chain, by local authorities and farmer cooperatives (Rist et

Bioenergy and food security
The cultivation of bioenergy feedstock interacts with food security by affecting individuals' physical or economic access to food, considering the nutritional, safety and preferential aspects of food consumption as well as the inter-temporality of access (Pinstrup-Andersen 2009). For instance, it can alter patterns of food production, household income and food affordability for farmers, workers, and the non-farm poor. These processes occur both at the macro-scale, e.g. via rising food prices, and at the micro-scale, e.g. via changes in crop cultivation patterns. Increased biofuel production (e.g. the shift from maize to ethanol production in the US) has been connected to increased food prices (in particular the maize price in Latin America) (Mitchell 2008), as well as political food crises, and even the Arab Spring (Werrell and Femia 2013). But increased biofuel production also reduces the prices of energy input needed for growing food plants, possibly also dampening food prices. Interactions between food and bioenergy markets are complex also on regional scale. One Table 1. Examples of the effects of first-generation biofuel crops on rural livelihoods.

Bioenergy and income A1: mechanization sugarcane and soy
In rural Brazil, mechanization trends have been linked to the continuation and even worsening of social exclusion, as better-off farmers in richer regions have benefited more from producing energy crops than poorer farmers in less-productive regions. Policies have played a significant role: financial incentives through the ProAlcool program created further advantages for large-scale sugar sugarcane producers. For biodiesel, large-scale soy is increasingly dominant despite policy measures that aimed to encourage castor production from small-scale farmers (Hall et al 2009) A2: capital requirement oil palm and Jatropha In Jambi Province, Indonesia, oil palm production favored resource-rich farmers and excluded those with fewer resources. Participants in an oil palm scheme for Javanese settlers received a stipend for one year, after which they relied on income from casual plantation labor while waiting for their oil palm plots to mature. These wages were so low that up to half of the settlers sold their plots and moved back to Java before their oil palm became productive (McCarthy 2010). High credit requirements were also found to create dependency among Jatropha growers in Tamil Nadu, India (Ariza-Montobbio et al 2010)

A3: contractual arrangements cassava
In Banteay Chhmar, Cambodia, cassava production (a driver of deforestation) produced low incomes but high risks for smallholders, whereas middlemen and traders had considerable profit rates. A collapse in cassava prices in 2009 required smallholders that had reduced their cropping diversity to sell their land to repay debts and acquire food (Hought et al 2012) Bioenergy and food security B1: food access oil palm In Papua, Indonesia, the establishment of 20 000 ha of oil palm plantations on former clan-controlled forest land led to a significant decline in food access for the indigenous population, mainly because of decreased opportunities for hunting, fishing, and collecting forest resources, and reduced access to farm land (Obidzinski et al 2012). Few people were able to compensate for this loss of direct food access through paid employment on the plantations, and individual oil palm growing was not possible. Hence 100% of the 'former landowners' and 'customary users' interviewed experienced reduced access to food, while 93% experienced an overall negative change in livelihoods. Among plantation workers, 56% experienced an increase in consumption. But 46% still reported an overall negative change in livelihood B2: food production Jatropha In Tamil Nadu, India, Jatropha growing was found to significantly reduce food production (edible oil, staples and vegetables) both for sale and household consumption, as Jatropha replaced food crops and occupied more than 50% of the landholding. It also reduced the availability of crop residues for animal fodder. Crop diversity was reduced as mature Jatropha was grown as a monoculture. Reduced food production was not offset by higher income; at observed yields, economic returns for growing Jatropha were consistently negative. Planting Jatropha decreased livelihood diversity and made the income of smallholder farmers more precarious (Ariza-Montobbio and Lele 2010) Bioenergy and land tenure C1: loss of customary land Jatropha In Ghana, a foreign biofuel company acquired large contiguous areas of land without regard to customary land rights. Sixty-nine households lost land without consent or compensation. Loss of access to farmland and forests, and in turn to food and various income sources, disproportionately impacted women and migrant farmers. In contrast, plantation employee households gained stability and security in income flow. Social conflict reportedly increased as plantation workers became less able to help with communal labor (Schoneveld et al 2011) C2: institutional shortcomings oil palm In Central Kalimantan Province, Indonesia, the establishment of oil palm plantations and palm oil mills since around 2000 has led to a significant deterioration of local water resources, with negative effects on fish stock, rice farming, access to drinking water, and health (skin disease) in three villages (Klocker-Larsen et al 2012). Environmental laws exist to protect water resources and to ensure that Environmental Impact Assessment are performed prior to obtaining concession permits, and a water law stipulates water use rights-including establishing recognition of customary rights. These environmental and social regulations were not implemented in practice; the losses of the villagers were not compensated and their grievances not addressed Burnt sugarcane harvesting in Brazil has been found to increase blood pressure among healthy male workers (Barbosa et al 2012). It has also been associated with higher incidence of respiratory problems and reduced lung function compared to non-harvesting periods, not only among plantation workers but also among residents of a nearby town (Prado et al 2012). In Sao Paolo province, the onset of biomass burning led to a 30% increase in asthma-related hospital admissions in a neighboring city (Arbex et al 2007). Hospitalizations due to asthma were also significantly higher during sugarcane field burning in Louisiana, United States (Boopathy et al 2002) D2: water pollution oil palm In three sites in Kalimantan and Papua, Indonesia, large-scale clearing of forest for oil palm plantations led to siltation of waterways and swamps used as sources of fresh water for domestic needs (Obidzinski et al 2012). In eastern Papua, river water could no longer be used for domestic use due risk of contamination from pesticides and herbicides. Air pollution from dust and smoke coming from the plantation and mill sites were also considered significant hazards in two sites. Respectively, 24%, 27% and 31% of respondents in the three sites reported an increase in human disease after the establishment of oil palm plantations modeling study suggests that biofuel expansion can reduce the area cultivated with traditional export cash crops (in Mozambique and Tanzania), but can lead either to a decrease (Mozambique) or an increase (Tanzania) in food production depending on local factors such as land availability and the labor intensity of traditional cash crop production (Arndt et al 2011). Appreciation of the real exchange rate caused by the rapid expansion of biofuel exports can explain this dynamic. In both cases, appreciation crowds out traditional export crops that experience a price reduction in domestic currency. In Tanzania some of this production is then substituted by food production; in contrast, biofuel production substitutes for food production in Mozambique. Other equilibrium effects would be an increase in food prices in Mozambique, and a decrease in Tanzania, and a reduction of income of traditional cash crop producers and workers.

Micro-perspective.
The relationship between feedstock cultivation and food security at the micro-scale is equally complex. Extended cultivation of cash crops has been identified as a key factor determining local food security (Bolwig 2012, Ericksen 2008, Fafchamps 1992, Von Braun 1995; this is likely to apply to bioenergy feedstocks as well. On one hand, it may increase food access by raising crop revenue, e.g. by providing better access to fertilizers, incentivizing an increase in volume, and reducing the unit cost of production. And it may lead to savings that become reinvested in food production, non-farm employment, or children's education for longer term benefits (Davis et al 2009). The income obtained from biofuel production can be used to buy additional food thereby increasing household nutrition. On the other hand, it may reduce own food production by diverting scarce factors of production away from food crop farming, reducing food access (Obidzinski et al (2012)-B1 in table 1). Also the timing of a feedstock's peak labor requirements can lead to decreased food production in situations of labor scarcity in a smallholder context (Mingorría and Gamboa 2010). These outcomes are shaped by place-specific gender dynamics, market design, and food price volatility which together render predictions of bioenergy deployment on food security uncertain (Alene et al 2008, Bolwig 2012, Raynolds 2002. Evidence suggests that the risks associated with relying solely on cash crop income for food access may render a partial self-sufficiency strategy more attractive (Fafchamps 1992). Partial cash cropping can provide direct benefits in terms of agricultural development to all farmers, including those who do not grow cash crops (Govereh and Jayne 2003), and also result in higher agro-biodiversity and in situ conservation (Bangwayo-Skeete et al 2012).

Bioenergy and land tenure
2.3.1. Macro-perspective. Biofuel deployment leads to a rise in land value and can reorganize tenure regimes. Policies, such as blending mandates, contribute to an appreciation of land value within the mandated region and world wide, depending on the degree of trade liberalization (Banse et al 2008). An increase in yield productivity can make land formerly considered 'marginal' commercially attractive, leading to further expansion of biofuel feedstocks.

Micro-perspective.
Forced or voluntary displacement from other land uses, such as subsistence agriculture, forest commons or livestock grazing, often leads to loss of customary land rights and land access conditions (Schoneveld et al (2011); see C1 in table 1), as well as to indirect land-use change (Lima et al 2011). At the local scale, households and communities without formal land titles can see their access to land compromised, or be excluded from production schemes and their associated benefits . In some circumstances, women loose informal access to land and resources more easily than men (Schoneveld et al 2011). Subsequently, women see their household and farm labor-time budgets disproportionately increase (Julia and White 2012, Mingorría andGamboa 2010, Skutsch et al 2011). Such distributional effects derived from the interplay of biofuel deployment and land tenure can be felt at distinct spatio-temporal scales, some being experienced even before biofuel deployment occurs, including displacement of 'landless' or informal tenure holders through investments or land grabs (Baka 2013).
The benefits, costs and distributional consequences of the interplay between bioenergy and land tenure can again be related to the production model. In Mozambique, Jatropha cultivation in outgrower schemes allowed smallholders to accrue land rents, whereas the land rents of sugarcane plantations were transferred to absentee investors (Arndt et al 2009). Where large-scale bioenergy production is organized through policy incentives and land-use planning from the top-down, the rural poor can become vulnerable to dispossession if they do not hold official land title (Cotula et al 2008, Vermeulen andCotula 2010). Traditional land owners and communal use of resources can experience restrictions as a result of both plantations and outgrower schemes, with households that previously relied on forest resources for income and food shifting to other sources of livelihood (Julia and White (2012), Obidzinski et al (2012); see also C2 in table 1). The transfer of land rights to large-scale operations reduces income and livelihood opportunities for local communities and leads regularly to conflict over land rights (Marti 2008, McCarthy 2010. In contrast, land tenure security may buffer against social conflict and facilitate biofuel and agricultural developments (Smalley and Corbera 2012). In situations where land is relatively plentiful and infringement on traditional land rights is scarce, biofuel deployment can lead to changes in land use without necessarily triggering social conflict. For example, Jatropha cultivation in Zambia did not seem to restrict local food crop production, although it still produced unequal economic outcomes (German et al 2011). In summary, biofuel deployment often but not always compromises land tenure of some groups, usually those with weaker position. The effect is distinctly shaped by production model, politics and power relations, and pre-existing land use.

Bioenergy and health
2.4.1. Macro-perspective. Biofuel plantations can indirectly improve public health if additional income streams enable financial access to drugs and health care. Food insecurity has direct impacts on public health (Yngve et al 2009). Hence, it is possible that health and food security could deteriorate in a vicious cycle.

Micro-perspective.
Health enables household members to engage in crop cultivation and other social and economic activities. Plantations can facilitate the transmission of malaria or dengue by allowing their vectors to survive throughout the year or find more suitable breeding and survival conditions (Tanga et al 2011). Exposure levels to herbicides and other agricultural chemicals and their composition are insufficiently monitored (Mink et al 2012, Williams et al 2012 and are often unknown to workers (Gupta 2012). Social relations can also be eroded by biofuel deployment. In Ghana, plantation workers had insufficient time to help with communal labor leading to social conflict (Schoneveld et al 2011) and in Guatemala, households employed by oil palm plantations dedicated less time to communal activities (Mingorría and Gamboa 2010). Biomass burning in sugarcane plantation leads to detrimental health outcomes (Goldemberg et al 2008) including respiratory diseases, both among workers and in neighboring towns (D1 in table 1).

Livelihood outcomes summarized
Section 2 has reviewed studies that illustrate actual or potential impacts of bioenergy feedstock deployment on rural livelihoods, considering global and local dynamics. Global and regional models can already capture potentially relevant income and food security effects, and can identify groups that profit or lose from bioenergy deployment. However, they fall short of scrutinizing changes in capabilities and assets, and remain restricted in their treatment of distributional equity due to crude resolution of distributional dynamics. At the micro-scale, research reveals that the production model plays a key role in determining income gains or losses, food security, land tenure conflicts and health effects for workers and communities. The smallholder production model benefits livelihoods more when it is labor intensive, when it enables smallholders to participate in value creation along the supply chain, and when it is pursued alongside food production. Large-scale production may involve higher land productivity, but also requires considerable financial and human capital, increasing the risk that income inequalities will result, sometimes mirrored in health impacts and social exclusion. While plantation workers usually experience increased income, reductions in food production and community activities due to time constraints must also be accounted for. Those who obtain or keep formal land rights tend to profit from bioenergy deployment by reaping a higher land rent, while those who lose formal or-more often-informal land rights and access to land and its ecosystem services see a reduction in their land-related livelihoods. This can be particularly acute if deployment takes place through plantations that do not offer new opportunities to the dispossessed. As those with formal land rights usually have more capital to start with, and those without rights have less access to institutions and money, distributional inequities are often increased by formalization and consolidation of land rights. These main effects are summarized in the conceptual diagram of figure 1.
Assessing how these effects are distributed is crucial. As an example, a recent study of oil palm production in Indonesia by independent growers and outgrowers contracted to nucleus estates reveals aggregate and distributional effects that generate winners and losers (Obidzinski et al 2012). Smallholders reported increased income from producing oil palm and non-participants benefited from spillover demand effects in other economic sectors. However, these economic benefits were unequally distributed: 'former land owners' and 'customary land users' lost while (migrant) landowners with formal title gained. Access to capital reportedly served as a barrier to participation and whilst most stakeholders reported improved access to food, oil palm replaced cash crops and some food crops on smallholder land. Land values also increased, benefiting those with plots, but conflicts over land occurred, both between communities and companies and between traditional landowners and migrants. The loss of customary rights to cropland, mentioned by over half (53%) of respondents in one site, was more likely to affect households without formal land titles. The expansion of oil palm also reduced access to forest products and services-including fallows for shifting agricultureaffecting women's livelihoods more than men's. Oil palm production reduced the quality and quantity of available fresh water, caused soil erosion and water logging, and increased flooding. If one looked only at aggregate outcomes, this case would suggest that mildly positive outcomes occurred in three of the four livelihood dimensions. But once distributional outcomes are also considered, the analysis would suggest that inequities were perpetuated or widened in all four livelihood dimensions. Particularly for land and income, gains (for some) have been offset by increased inequality in the distribution of outcomes (figure 2).

Reconsidering livelihoods in integrated assessments
In spite of this growing evidence that bioenergy deployment has effected or may effect income, food security, land tenure and other assets, global integrated assessment studies consider livelihoods only generically and are operationalized by focusing on global market effects. Golub et al (2012) investigate the livelihood impact of global land-based climate policies (not bioenergy deployment), focusing on income and food security. They find that restricting deforestation and agricultural expansion for food production leads to increased income and food availability in farm households due to price appreciation, but mostly a decrease in real income and food availability for the non-farm poor. As biofuel production puts similar pressure on arable land and food production, it is likely to produce similar livelihood effects on the agricultural food-sector and non-farming livelihoods. Golub et al's study is the first to try to quantify livelihood effects of climate policies systematically and globally. However, by modeling a representative farming household it assumes complete and equal value capture by rural households, and underestimates the distribution of benefits and costs at the micro-scale, e.g. by not representing livelihood assets.
Our analysis demonstrates that deployment schemes do not only affect aggregate income and food availability. Schemes also impact land tenure and the quality or availability of other assets, particularly finance and health. More importantly, the design of deployment schemes also has distributional consequences across the four outcome dimensions we considered (income, food, land, and health). We therefore argue that to be policy relevant, integrated assessments should comprehensively consider livelihood outcomes of bioenergy deployment, summarized in table 2. Our analysis suggests a likely tension between aggregate and equity impacts that needs to be made explicit in bioenergy assessments to better support decision-making.
A first attempt to bring livelihoods into integrated assessment models could include introducing distributional parameters. For example, distinct parameters could represent the fractions of households (percentage of affected  We are aware that such parameters are highly uncertain and would still be insufficient to cover all place-specific effects. But the current practice of ignoring the distributional outcome is also a strong normative choice that can lead to misguided policy. Integrated assessment would turn from optimization models towards multi-criteria assessments that rely on scenarios that take into account potential distributional consequences at household and community levels: the combined outcome in efficiency, GHG emissions, livelihoods, and equity could be the more complex but also the more comprehensive basis for policy decisions (Creutzig and Kammen 2009). One could still object that aggregate livelihood and equity issues are problems that can be analyzed and solved independently from each other (this is the main claim of the second welfare theorem), but our analysis clearly highlights that these issues remain entangled. The choice of production model, the underlying and evolving land tenure regimes, global biofuel markets and equilibrium effects on food markets, all affect both aggregate and distributional outcomes. Hence, equity should be included in a full set of criteria for bioenergy assessments. It is likely that this conclusion is also valid more generally for other climate and energy assessments, such as those focusing on other technological but also behavioral options. Our analysis remains limited in focusing on firstgeneration crop-based biofuels. Jatropha is however sometimes considered a second-generation biofuel, but is similarly prone to detrimental distributional issues (Findlater and Kandlikar 2011) that need to be explicitly considered in comprehensive assessments. More generally, we suspect that second-generation bioenergy plantations would on the one hand provide higher income and land rent with higher yields compared to first-generation biofuels but again would marginalize local populations that live from informal land tenure. The utilization of residues and the cascade utilization of biomass is however also seen as a promising route for providing energy (Haberl andGeissler 2000, WBGU 2009) and could improve existing livelihoods. Future research should explore this direction.
Finally, we recognize that our analysis does not tackle a key conundrum: how can livelihoods be assessed globally if each deployment scheme has effects that depend on various place-specific factors? A comprehensive assessment by human geographers and agricultural economists could elucidate which kind of distributional livelihood effects need to be accounted for, and which biofuel deployment schemes (crops, institutional arrangements, land tenure schemes) have more advantageous outcomes. The diversity of approaches, methods and analytical categories could be roughly mapped using integrated livelihood assessment figures resembling figure 2, and assessments could rank deployment scenarios in terms of their impact on livelihood dimensions. We see ample opportunity to soft-couple integrated assessment models with local livelihood analyses and CGE and partial equilibrium sector models. This could elucidate both risks and opportunities of bioenergy deployment for livelihoods. The ultimate art is to ensure that pathways and production models not only produce positive aggregate outcomes, but that they simultaneously respect and improve place-specific livelihoods. If specific deployment schemes are unlikely to achieve these goals, they should not be pursued.