A Learning Framework for Distribution-Based Game-Theoretic Solution Concepts
aa r X i v : . [ c s . A I] J un A Learning Framework for Distribution-BasedGame-Theoretic Solution Concepts
Tushant Jha
International Institute of Information Technology, Hyderabad [email protected]
Yair Zick
National University of Singapore, Singapore [email protected]
Abstract
The past few years have seen several works on learning economic solutions fromdata; these include optimal auction design, function optimization, stable payoffs incooperative games and more. In this work, we provide a unified learning-theoreticmethodology for modeling such problems, and establish tools for determiningwhether a given economic solution concept can be learned from data. Our learn-ing theoretic framework generalizes a notion of function space dimension — thegraph dimension — adapting it to the solution concept learning domain. We iden-tify sufficient conditions for the PAC learnability of solution concepts, and showthat results in existing works can be immediately derived using our methodology.Finally, we apply our methods in other economic domains, yielding a novel notionof PAC competitive equilibrium and PAC Condorcet winners.
Recent years have seen widespread application of learning-theoretic notions in economic domains.Rather than assuming full knowledge of the underlying domain (or a prior over the domain space),one assumes access to a dataset of past instances, and employs learning-theoretic tools in order toobtain approximate solutions. Consider the following simple example of learning a solution fromdata: we wish to find the maximum of a function f : R n → R ; we do not have access to f , butrather to a dataset of the form h ~x , f ( ~x ) i , . . . , h ~x m , f ( ~x m ) i . One way to find a likely candidatepoint would be to use classic learning-theoretic tools [1], learn an approximation f ∗ of f , andcompute the maximum of f ∗ ; however, this goes above and beyond the problem requirement: theapproximability of f depends on its hypothesis class (whether f is a linear function, a two-layerneural network etc.), and on the approximation robustness. A much simpler solution is available: ifthe number of samples is sufficiently large, taking the empirical maximum of f over the dataset —i.e. ~x ∗ ∈ argmax j { f ( ~x j ) : j ∈ [ m ] } — yields a point that is likely to be greater in value than anyfuture point sampled from the same distribution as the original dataset.The same reasoning applies to other economic solutions: a naive approach to inferring solutionsfrom data would be to learn an approximate model (e.g. learn a function f ∗ which approximates f in the maximization example above), and then try generating solutions for the approximate model.However, as has been shown in the literature, learning an approximate model may:1. be insufficient for generating ‘good’ solutions (this is the case in [32])2. require an exponential number of samples, whereas directly learning solutions is easy. In-deed, finding a payoff in the core of TU games is easy [4, 11], while PAC learning cooper- Preprint. Under review. tive games requires an exponential number of samples [7]; this is also the case for findingan empirical maximum in the example above.Recent works directly learn solutions to classic optimization problems, as well as solutions in game-theoretic domains. These lines of work have progressed more or less independently, proving thatsolutions in a specific problem domain can (or cannot) be efficiently inferred from data; however,there has been no attempt to provide a unified theory of learning solution concepts from data. Thisis where our work comes in.
We begin by establishing a learning-theoretic framework for learning solution concepts from data.Unlike classic learning problem spaces, solution concepts do not inhabit the same space as theobserved samples (e.g. when learning an approximate maximum, the function space is R n → R ,whereas the solution space is R n ). In Section 2, we define the solution dimension : this quantitydepends on both the hypothesis class of the underlying game and the solution space. The solutiondimension generalizes the graph dimension [18] in PAC learning, and serves a similar purpose: ifthe solution dimension is low, then a distribution-based solution can be efficiently learned fromsamples. Drawing on notions of shattering from VC dimension, we introduce solution conceptshattering which is used to bound the solution dimension in various domains. We also show thatthe existence of a consistent solution and a low graph dimension are sufficient conditions for PAClearning solutions, simplifying technical learnability arguments in existing works, as well as pavingthe way for a straightforward learnability approach of other solution concepts. In Section 3, weapply our methodology to immediately derive sample complexity bounds on learning solutions forhedonic games, as well as for two novel domains: market equilibria, and Condorcet winners invoting. Several recent works study learning solutions from data; these include solutions in cooperativegames [7, 11, 24, 32], combinatorial auctions [6, 8, 14, 16, 19, 25, 33], voting and judgment ag-gregation [12, 35], envy-free allocation [5], and optimization [9, 10, 30]. Some of these works offerlow-error approximation guarantees with respect to an optimal solution, such as estimating the max-imum [9, 10], maximizing revenue in mechanism design [16], or finding an election winner [12];our work focuses on solutions that minimize expected loss with respect to sampled data , as is thecase when learning the core of a cooperative game [7, 24, 32], reserve prices in auctions [6, 26],or approximately efficient allocations [14]. While some of the above works explicitly explore thedimension of the solution space, they do not offer the full generality of our model.Our analysis of the underlying solution space utilizes recent learning-theoretic tools [18], yielding anextension of classical function dimension measures such as the VC dimension [1, 34], and the graphdimension [18]. Our results generalize the General Learning problem discussed in [31], as learningsolution concepts can also involve some global properties playing a role in the loss function.
For the sake of completeness, we provide a brief overview of the PAC learning model. A learningproblem is defined over an instance space X and a set of functions (the hypothesis class ) H ⊆ Y X ( Y is the label space ). Let D be a distribution over X × Y ; we let the loss of h ∈ H given D be L D ( h ) = Pr ( x,y ) ∼D [ h ( x ) = y ] . Given a set of m i.i.d. samples T = h x j , y j i j ∈ [ m ] from D , the empirical loss of h ∈ H is ˆ L T ( h ) = m P mj =1 ( h ( x j ) = y j ) . We assume that D is a distributionover X , where every x ∈ X is evaluated by some unknown c ∈ H ; this is referred to as the realizablecase , in which there is some h ∈ H for which L D ( h ) = 0 , and there is at least one hypothesis h ∈ H for which ˆ L T ( h ) = 0 for any T ⊆ X . An algorithm A is a PAC learner for H if there is some m polynomial in ε , δ and the natural problem parameters, such that for any distribution D and any setof m ≥ m samples T sampled i.i.d. from D , A outputs a hypothesis h ∗ ∈ H (which is a functionof T , but not of D ) such that Pr T ∼D m [ L D ( h ∗ ) ≥ ε ] < δ .For binary hypothesis classes (where the label space is Y = {± } ), the VC dimension [34] charac-terizes the sample complexity of H . The sample complexity required by any PAC learning algorithm2or H is upper and lower-bounded by the VC dimension of H . This is achieved by an algorithm thatoutputs a consistent hypothesis, i.e. one which minimizes the empirical loss ˆ L T ( h ) w.r.t. a sample T . Definition 1.1.
Given a hypothesis class H , a set C ⊆ X is said to be shattered if for any binarylabeling b : C → { , } there exists some h ∈ H such that h ( x ) = b ( x ) for all x ∈ C . The VCdimension of H , or V C ( H ) , is the size of the largest set C ⊆ X that is shattered by H .For example, if the hypothesis class is the set of all linear classifiers over R n , its VC dimension is O ( n ) [1]. Theorem 1.2 relates the VC dimension and the PAC learnability of H . Theorem 1.2.
There exists absolute constants α and α , such that for a hypothesis class H , thesample complexity of H with respect to ε and δ (denoted m ( ε, δ ) ) is α ε (cid:18) VC ( H ) + log (cid:18) δ (cid:19)(cid:19) ≤ m ( ε, δ ) ≤ α ε (cid:18) log (cid:18) ε (cid:19) VC ( H ) + log (cid:18) δ (cid:19)(cid:19) Theorem 1.2 can be slightly generalized to the following claim: for any two functions f, g ∈ H ,the empirical loss on an i.i.d. sample of more than α ε (cid:0) log (cid:0) ε (cid:1) VC ( H ) + log (cid:0) δ (cid:1)(cid:1) points, is closewithin ε to the statistical loss (ie. Pr x ∼D [ f ( x ) = g ( x )] ).The case where samples are labelled by some arbitrary function c (not necessarily in H ) is alsoknown as the agnostic case; however, as a result of the uniform convergence results, if an algorithm A outputs a hypothesis h ∗ that minimizes empirical risk — ∀ h ∈ H : ˆ L T ( h ∗ ) ≤ ˆ L T ( h ) — thestatistical error is ≤ ε : Pr T ∼D m [ L D ( h ∗ ) ≥ min h ∈ H L D ( h ) + ε ] < δ . Therefore, as discussed in[18], uniform convergence is a powerful tool for bounding statistical loss. In what follows, we briefly introduce the solution concepts discussed in this work. In all scenariosbelow, we have a set of players N = { , . . . , n } , with preferences over outcomes induced in somemanner; our objective is to obtain a solution with some desirable properties. In hedonic games [13, Chapter 15], each player i ∈ N has a complete, transitive preference order ≻ i over coalitions in N that contain it. Solutions are partitions (also referred to as coalition structures )of N ; a coalition structure π is blocked by a coalition S ⊆ N if all members of S prefer S over thecoalition they are in (denoted π ( i ) ), i.e. S ≻ i π ( i ) for all i ∈ S . The core of a hedonic game is theset of stable coalition structures: they cannot be blocked by any coalition S ⊆ N . It is often assumedthat players’ preferences over subsets are induced by a cardinal utility function v i : 2 N → R + ; inthis case, S ≻ i T if and only if v i ( S ) > v i ( T ) . We are given a set of k indivisible goods G = { g , . . . , g k } . Each player i ∈ N values bundles ofgoods in G according to v i : 2 G → R + , where v i ( ∅ ) = 0 for all i ∈ N . A market outcome is a tuple h π, ~p i , where π is a partition of G into n disjoint bundles (some of them may be empty), with π ( i ) assigned to player i ; ~p ∈ R k is a price vector, denoting the price of each item in G . In these markets,known as Fisher markets [15], we assume that each player i has a budget β i ∈ R + . Given a pricevector ~p ∈ R k , the affordable set of player i is the set of all bundles whose total price is less than β i : A i ( ~p, β i ) = S ⊆ G : X g j ∈ S p j ≤ β i . An outcome h π, ~p i is a competitive equilibrium if for all i ∈ N , π ( i ) ∈ A i ( ~p, β i ) , and ∀ S ∈A i ( ~p, β i ) , v i ( π ( i )) ≥ v i ( S ) . Consider a set of voters N = { , . . . , n } , each with a preference order ≻ i over some finite set of candidates C . Given two candidates c, c ′ ∈ C , we define B ( ≻ , c ′ , c ) = 1 iff a majority of voters3refer c ′ to c under ≻ . A candidate c ∗ is a Condorcet winner iff B ( ≻ , c ∗ , c ) = 1 for every othercandidate c ∈ C . As described in Section 1.4, a solution concept or an equilibrium concept characterizes a subset ofits solution space satisfying some natural desiderata.
Games are mappings from some domain X toa label space Y . For example, in hedonic games, a game is a set of functions v i : 2 N → R (for every i ∈ N ), mapping from subsets of players to real values; thus, X = 2 N and Y consists of vectors ofthe form ( v i ( S )) i ∈ S for every S ⊆ N . We assume no knowledge of the actual game g , except forthe hypothesis class it belongs to; we only observe samples of the game’s evaluation on points in X .Constraints characterizing solution concepts are often universal quantifiers over a local loss function λ . In hedonic games, we define λ : 2 N × G × Π( N ) → { , } , where λ ( S, ~v, π ) = 0 iff v i ( S ) ≤ v i ( π ( i )) for all i ∈ N . Thus, π is in the core iff λ ( S, ~v, π ) = 0 for all S ⊆ N . Similarly, the maximaof a function satisfy x ∗ ∈ argmax f ( x ) ⇐⇒ ∀ x : f ( x ) ≤ f ( x ∗ ) ; in particular, λ ( x, f, x ∗ ) = 0 iff f ( x ∗ ) ≥ f ( x ) . Solution concepts that can be defined via a local loss function λ readily admit adistributional variant: we require that the expected loss as measured by λ is low, with respect to adistribution D over the domain X ; i.e. Pr x ∼D [ λ ( x, g, s ) = 0] ≥ − ε , for some ε ∈ (0 , .More formally, an instance of the S TATISTICAL S OLUTION problem is a tuple
Ψ = ( X , Y , G , S , λ ) .Here X is the instance space; Y is the codomain (or label) space; G ⊆ Y X is the class of games; S is the solution space; finally, λ : X × G × S → { , } measures local loss . In standard PAC learning(Section 1.3), G = S and λ ( x, g , g ) = 0 ⇐⇒ g ( x ) = g ( x ) .Given a game g ∈ G and m points T = h ( x j , g ( x j )) i mj =1 , the empirical error (or empirical risk) of s ∈ S is ˆ L T ( g, s ) = 1 m X ( x j ,g ( x j )) ∈ T λ ( x j , g, s ) , and the statistical error (or statistical risk) as L D ( g, s ) = E T ∼D m [ λ ( x j , g, s )] . A PAC solver fora S TATISTICAL S OLUTION Ψ is an algorithm L whose input is a list T = h x j , g ( x j ) i mj =1 of m values x j ∈ X labelled by some unknown g ∈ G , and whose output is a solution s ∗ ∈ S ; itssample complexity, denoted m L ( ε, δ ) , is the minimal number of samples required such that for any m ≥ m L ( ε, δ ) , Pr T ∼D m [ L D ( g, s ∗ ) > ε ] < δ . We let m PAC Ψ ( ε, δ ) be the minimal sample complexity m L ( ε, δ ) required by any PAC solver L for Ψ . In standard PAC learning, consistent or empirical risk minimizing (ERM) solvers play an importantrole; these are algorithms that minimize empirical error ( ˆ L T ( h ) ) on the training sample. As dis-cussed in Section 1.3, for binary functions, consistent algorithms are PAC learners whose samplecomplexity is bounded by the VC dimension. We first define a notion of consistency for solutionconcepts. Definition 2.1.
An algorithm A m : ( X × Y ) m → S is said to be a consistent solver for the S TATIS - TICAL S OLUTION problem Ψ if for all g ∈ G , and for any set of m samples T m = h x j , g ( x j ) i mj =1 , A m takes as input T m and outputs a solution s ∗ = A m ( T m ) ∈ S such that the empirical loss of thesolution s ∗ over T m is : ˆ L T m ( g, s ∗ ) = 0 . In other words, for any input batch T m labelled by someunderlying function g ∈ G , the algorithm returns a solution that has zero loss w.r.t g on all points inthe input sample.The definition of consistent solving presents a subtle yet crucial departure from the correspondingresult in standard PAC learning. In PAC learning, since λ ( x, g, h ) = [ g ( x ) = h ( x )] , if there aretwo functions g , g ∈ G such that g ( x ) = g ( x ) for a point x ∈ X , then for any hypothesis h , λ ( x, g , h ) = λ ( x, g , h ) . Therefore, even if two functions g , g ∈ G generate an equivalentsample T = h x j , y j i mj =1 = h x j , g ( x j ) i mj =1 = h x j , g ( x j ) i mj =1 , if h is consistent with samples ( x j , y j ) in T , then it is consistent with both g and g . In fact, this implies that, time complexityconsiderations aside, a consistent solution always exists in standard learning, and can be found via4xhaustive search. This is not the case in solution concept learning; two functions g , g ∈ G maygenerate an equivalent sample T = h x j , y j i mj =1 , yet disagree on a solution (this is noted in priorworks [24, 32]). Intuitively, this occurs since game-theoretic solutions treat unobserved regionsof players’ preferences. For example, in PAC market equilibria, one must inevitably set pricesfor unobserved goods, and assign bundles to players without knowing what their value might be; inhedonic games, a partition of players may contain subsets completely unobserved in the sample data.It is often useful to think of domains where this issue does not occur, as captured in the followingdefinition. Given a labelled sample of m points T ∈ ( X × Y ) m , let G | T be the set of games in G which agree with T . Definition 2.2.
A S
TATISTICAL S OLUTION Ψ is said to satisfy the consistent solvability criterion iffor all m and all T ⊆ ( X × Y ) m , there exists some s ∈ S such that for all g ∈ G | T , the empiricalloss ˆ L T ( g, s ) is . We now present a novel definition of dimension for the PAC solution setting, and use it to bound thesample complexity for finding solutions to problem domains.
Definition 2.3 (Solution-based Dimension) . Given some C ⊆ X , we say the set C is S-shattered in Ψ if there exists a game g ∈ G , such that for every binary labelling b : C → { , } there exists asolution s ∈ S (that may depend on b ) such that for all x ∈ C , λ ( x, g, s ) = b ( x ) .The Solution-based dimension of Ψ , denoted Sd (Ψ) , is the size of the largest set S-shattered in Ψ ,and ( C, g ) as the corresponding shattering witness. Sd (Ψ) bounds the sample complexity of consistent solutions for Ψ (i.e. m PAC ( ε, δ ) ); however,we first prove a stronger claim, using the idea of uniform convergence discussed in Section 1.3. Ifwe define the sample complexity for uniform convergence m UC ( ε, δ ) as the number of samplesrequired such that the empirical loss of any solution is ε -close to its statistical loss, then m UC ( ε, δ ) is polynomially dependent on the solution dimension of the problem. Theorem 2.4.
There are universal constants α and α , such that if Sd (Ψ) = d , then for a sampleof m ≥ α d +log( δ ) ε points T = h x j , y j i mj =1 , Pr T ∼D m [ ∃ g ∈ G | T , s ∈ S : | ˆ L T ( g, s ) − L D ( g, s ) | > ε ] < δ. Furthermore, if a solution s ∗ is consistent, i.e. ˆ L T ( g, s ∗ ) = 0 , then for any m greater than α ε (cid:0) log (cid:0) ε (cid:1) d + log (cid:0) δ (cid:1)(cid:1) , we have that Pr T ∼D m [ L D ( g, s ∗ ) > ε ] < δ . Note that in particular, m PAC ( ε, δ ) ≤ m UC ( ε, δ ) , and both are polynomially dependent on Sd (Ψ) , ε and log δ .As a useful sanity check, we observe that Sd collapses to the classic VC dimension when learningclassifiers: when S = G = H ⊆ X , then Sd (Ψ) = VC ( H ) . Similarly, when S = G = H ⊆ Y X (i.e. for multiclass learning problems with a general domain Y ), Sd collapses to the graph dimension[18]. We now observe few immediate corollaries of the above uniform convergence result. Corollary 2.5.
Given a S TATISTICAL S OLUTION problem
Ψ = hX , Y , G , S i : Simultaneous Constraints: if multiple local loss functions λ , . . . , λ k need to be simultaneouslyapproximated within ε , i.e. ∀ i ∈ [ k ] : | ˆ L i ( g, s ) − L i D ( g, s ) | < ε , then the sample complexity offinding a solution satisfying all of them is in O (max i ∈ [ k ] { m UC i ( ε, δ ) } ) . Separable Conjunctions: if there are local constraints λ over S , and λ over S , where S Ψ = S × S , such that we need to bound their conjunction within ε , i.e. Pr[ λ ( x, g, s ) ∧ λ ( x, g, s )] ,then m UC ( ε, δ ) is in O ( max i ∈{ , } { Sd (Ψ i ) } ) . The proof of Corollary 2.5 is relegated to the appendix. The following claim (whose proof is alsorelegated to the appendix) is also useful
Corollary 2.6 ( Sd for Argmax) . Let Ψ max be defined by G = { f : X → Y} and S = X , where Y is endowed with a total order ≻ , and λ ( x, g, x ∗ ) = [ g ( x ) ≻ g ( x ∗ )] . Then, Sd (Ψ max ) = 1 .
5o conclude, in order to establish an efficient PAC algorithm for a problem Ψ , it suffices to upper-bound m (Ψ) by its solution dimension Sd (Ψ) . As discussed in Section 1.3, uniform convergence complexity bounds can bound the sample com-plexity for Agnostic PAC learning via Empirical Risk Minimizers (ERM learners). However, ag-nostic solution learning can be defined in many ways. We discuss two definitions, for which thecorresponding notion of
ERM solving has a sample complexity that follows from uniform conver-gence.
Definition 2.7.
For a given S
TATISTICAL S OLUTION Ψ , A is a worst-case Agnostic PAC Solver if Pr T ∼D m [ L D ( g, A ( T )) ≤ min s max g ′ ∈ G | T L D ( g ′ , s ) + ε ] ≥ − δ ; it is a Bayesian agnostic PAC Solver, for a prior over games ˜ D , if Pr T ∼D m [ E g ′ ∼ ˜ D [ L D ( g ′ , A ( T )) | g ′ ∈ G | T ] ≤ min s E g ′ ∼ ˜ D [ L D ( g ′ , s ) | g ′ ∈ G | T ] + ε ] ≥ − δ. Corollary 2.8.
Given some S TATISTICAL S OLUTION , the sample complexity for Worst-Case andBayesian agnostic PAC solving is in O ( Sd (Ψ)) , and is achievable by an empirical risk minimizer. The proof of Corollary 2.8 is relegated to the appendix.
Let us now apply our theory for learning solution concepts in game-theoretic domains; all problemsdescribed below follow a common theme: rather than learning preferences, we learn solutions usingthe sampled dataset. While the focus of this paper is on game-theoretic solutions, our theory appliesfor other types of solution concepts as long as one can define a local loss function λ that dependsonly on a given point in x ∈ X , g ∈ G and the solution s ∈ S (see Section 2). Let us begin with hedonic games (Section 1.4.1); we analyze another type of cooperative game (TUcooperative games) in the appendix. A partition π ∗ of N PAC stabilizes a hedonic game w.r.t. a distri-bution
D ∈ ∆(2 N ) (where i ∈ π ∗ ( i ) for every i ∈ N ), if Pr S ∼D [ ∀ i ∈ S : v i ( S ) ≥ v i ( π ∗ ( i ))] < ε .The local loss function λ takes as input a coalition S ⊆ N , players’ valuations ~v = h v . . . , v n i and a partition π ∈ Π( N ) ; λ ( S, ~v, π ) = 1 iff S can block π under ~v . Our key result here is that thesample complexity of PAC stabilzing hedonic games is linear in n , for any class H of games . Lemma 3.1.
For any class of Hedonic Games H over n players, the solution dimension of PACstabilizing H is ≤ n .Proof. By definition, for a given hedonic game h ∈ H , a partition π , and a coalition S ⊆ N , thelocal loss λ ( S, h, π ) = 0 if and only if there exists a player in S that does not prefer it over herassigned coalition in π , i.e. v i ( S ) < v i ( π ( i )) . If a set of m coalitions S = { S , . . . , S m } is S-shattered by a witness h ∈ H , then for each S j ∈ S , there exists a coalition structure π j such that λ ( S j , h, π j ) = 0 , but λ ( S k , h, π j ) = 1 for all k = j . In other words, under π j , there exists some i ∈ S j such that v i ( S j ) < v i ( π j ( i )) , and for all k = j and for all i ∈ S k , v i ( S k ) ≥ v i ( π j ( i )) . Weconclude that for every S j ∈ S , there exists a player i who strictly prefers all coalitions that shebelongs to in S \ { S j } over S j . More formally, we let T ( i ) be the set of coalitions which are leastpreferred by player i in S ; note that T ( i ) must be a singleton, or else we arrive at a contradiction(the least liked coalition must be unique). Therefore, if S is S-shattered, the number of coalitions in S is bounded by n , and we are done.Applying Theorem 2.4 and leveraging Lemma 3.1 we obtain the following result: Theorem 3.2.
A class of Hedonic games H is efficiently PAC stabilizable iff there exists an algorithmthat outputs a partition consistent with samples evaluated by a game g ∈ H ; the sample complexityin this case is O ( n ) .
6n particular, Sliwinski and Zick [32] propose a consistent algorithm for top-responsive hedonicgames [13, Chapter 15]; Igarashi et al. [24] present a consistent algorithm for hedonic games whoseunderlying interaction graph is a tree [23]. Indeed, given Theorem 3.2, it suffices to show that thealgorithms they propose are consistent; their correctness is immediately implied by our results.
Competitive equilibria (CE) readily admit a PAC variant: given an allocation π , let P i ( π ) = { S ⊆ G : v i ( S ) > v i ( π ( i )) } be the set of bundles that are strongly preferred by i to π ( i ) . One can thinkof a CE as an outcome that ensures that P i ( π ) ∩ A i ( ~p, β i ) = ∅ , i.e. i cannot afford any bundle thatit prefers to its assigned bundle. In the statistical variant, we wish to ensure that this intersection hasa low measure under a distribution D over G . We define a loss λ i per player i as follows: given abundle of goods S ⊆ G , player valuations ~v and a market outcome h π, ~p i , λ i ( S, ~v, h π, ~p i ) = 1 iff S is both affordable (in A i ( ~p, β i ) ), and is preferred to π ( i ) (in P i ( π ) ). Our objective is to ensure thatthe overall error of player loss functions λ , . . . , λ n are within an error of ε . Lemma 3.3 bounds thesample complexity for PAC learning this problem, m PAC ( ε, δ ) , by O ( k ) . Therefore, by Theorem 2.4and Corollary 2.5, any algorithm that generates an outcome consistent against m sampled bundleswould also be a PAC CE solver with a sample complexity in O ( k ) . We refer to an instance of theCE problem as Ψ CE ( N, G, ~v, ~β ) . Lemma 3.3.
The solution dimension Sd (Ψ CE ( N, G, ~v,~b )) is O ( k ) , where k = | G | .Proof. For every player i ∈ N , the local constraint λ i can be seen as a conjunction of λ ,i ( S, ~v, h π ∗ , ~p ∗ i ) = hP g j ∈ S p ∗ j > β i i , and λ ,i ( S, ~v, h π ∗ , ~p ∗ i ) = [ v i ( S ) > v i ( π ∗ ( i ))] . Since λ ,i is defined by a linear constraint set by ~p ∗ and β i , it can be S-shattered by O ( k ) samples (in amanner similar to linear separators in standard PAC learning), which bounds its S-dimension. Onthe other hand, every λ ,i is a simple argmax constraint, which by Corolllary 2.6, has a solutiondimension of 1. By applying Corollary 2.5, the dimension of λ i = λ ,i ∧ λ ,i is O ( k ) ; since thecondition of λ i must hold for each i ∈ N , the CE loss is given by λ = V i λ i , which is O ( k ) byCorollary 2.5.Lemma 3.3 bounds the dimension of Ψ CE by O ( k ) ; however, the challenge is to design algorithmsthat generate consistent market solutions: bundle assignments and prices that ensure that all ob-served goods have been allocated, with no excess demand or assignment. We show the existenceof consistent solutions in two different settings; however, our solutions relax the market constraints.For Fisher markets with budgets ~β , for any ζ > , there exists a perturbed budget vector ~β ∗ with k ~β ∗ − ~β k ∞ ≤ ζ for which there exists a consistent solution h π ∗ , ~p ∗ i w.r.t. ~β ∗ ; this result holds for any class of valuation functions . Theorem 3.4 utilizes inefficient market outcomes, where a goodmay be allocated to more than one person; it is easy to think of an allocation π as a list of vectorsin { , } k , where π j ( i ) = 1 iff the j -th good is allocated to player i . If all goods are allocated, then P i ∈ N π ( i ) = ~ ; if goods are over-allocated, then P i ∈ N π ( i ) > ~ . Theorem 3.4.
We are given Ψ CE ( N, G, ~v, ~β ) , and m sampled bundles S , . . . , S m ⊆ G evaluatedby ~v . For any ζ > , there exists a perturbation on ~β , ~β ∗ such that k ~β − ~β ∗ k ∞ < ζ , for which thereis an outcome h π ∗ , ~p ∗ i such that players with budget levels ~β ∗ do not demand S , . . . , S m ; moreover, k P i ∈ N π ∗ ( i ) − ~ k ≤ k , where ~ , , . . . , ∈ [0 , k .Proof. We restrict ourselves to finding an assignment using only the sampled bundles and the emptybundle, i.e. for all i ∈ N : π ∗ ( i ) ∈ {∅ , S , . . . , S m } ; thus, we avoid making any assumptions aboutthe structure of v i . Budish [15, Theorem 1] shows that given ~β such that max i β i > min i β i , forany ζ > there exists a perturbed budget vector ~β ∗ and an outcome h π ∗ , ~p ∗ i for which: π ∗ ( i ) ∈ argmax S ∈A i ( ~p ∗ ,β i ) v i ( S ) ; k ~β − ~β ∗ k ∞ < ζ and k P i ∈ N π ∗ ( i ) − ~ k ≤ k . Assuming that for everyother S / ∈ {∅ , S , . . . , S m } , v i ( S ) ≤ for all i ∈ N , and applying the result by Budish, there existsa consistent outcome satisfying our requirements.While it makes no assumptions on player valuations, Theorem 3.4 is not constructive: it relies ona classic result from Budish [15], which utilizes a fixed-point theorem by Cromme and Diener [17]7or discontinuous maps to bound excess demand. We analyze exhange economies , a market variantwith divisible goods, in the appendix. In both cases, we are able to show that consistent marketsolutions exist. However, our results show the existence of solutions which only partially satisfy theequilibrium guarantees; moreover, both cases utilize non-constructive fixed-point theorems, ratherthan provide an efficient algorithm. There is little reason to believe that consistent solutions can beeasily computed in the general case; finding market solutions in settings similar to ours is PPADcomplete [29]. We conclude with a discussion of statistical solution concepts in voting (see Section 1.4.3 above). APAC Condorcet winner is a candidate c ∗ such that Pr c ∼D [ B ( ≻ , c, c ∗ )] < ε (recall that B ( ≻ , c, c ∗ ) =1 iff a majority of voters prefer c to c ∗ ). We refer to the problem of finding a Condorcet winner as Ψ Cond . We require that given a sample T ⊆ C of candidates, we can infer voters’ preferences w.r.t. T . This can be encoded as a valuation function of i over the candidates (as is the case for hedonicgames, see Section 3.1), or the truncated ranking ≻ i over the sampled candidates for every i ∈ N .Given a class of preference profiles H , let Ψ Cond ( H ) be the problem of finding Condorcet winnersfor profiles in H . We define the tournament graph : this is a directed graph where candidates arenodes; given a preference profile ≻ , there is an edge from a to b if a beats b in a pairwise electionunder ≻ . Theorem 3.5.
Given a class of preference profiles H over C such that | C | > , and a sample ofcandidates T ⊆ C , the following are equivalent: (a) There exists a consistent solver for Ψ Cond ( H ) that returns a PAC Condorcet winner c ∗ ∈ T . (b) Sd (Ψ Cond ) = 1 . (c) for every preference profile ≻∈ H , the tournament graph is transitive.In particular, if H satisfies the above, there exists a PAC solver for Ψ Cond ( H ) whose sample com-plexity is ǫ log δ .Proof. If there is a preference profile h ∈ H for which the tournament graph contains a -cycle, thenthere immediately exist two vertices of that cycle that can be S-shattered. This is true since for every H with more than one candidate, every singleton is shattered. Therefore, Sd (Ψ Cond ) = 1 if andonly if there are no preference profiles with Condorcet -cycles, which is equivalent to transitivity.Similarly, the existence of Condorcet winner for every C ′ ∈ C is equivalent to absence of any cycles,which is equivalent to transitivity of the tournament graph.Two notable families of voter preferences exhibit transitive preferences: single peaked preferences[13, Chapter 2] and single-crossing preferences [21] (see [20] for an overview); thus, if H is any ofthe former, a Condorcet winner can be PAC learned using ε log δ samples. Whenever the Condorcetwinner is known to exist within a sample C ′ ∈ C , the problem is equivalent to the argmax problemdiscussed in Corollary 2.6. However, as shown in Section 2.3, the graph dimension is still useful asa means to estimate (within ± ε with high confidence) the behavior of a candidate in pairwise elec-tions using a small empirical sample, even when no Condorcet winner exists. Theorem 3.6 bounds Sd (Ψ Cond ) in the case where Condorcet winners do not exist. The result bounds the solution dimen-sion in terms of the underlying structure of the tournament graph, and is based on Corollary 2.8; thefull proof is in the appendix. Theorem 3.6.
Let k be the largest number of candidates, such that for some tournament graph in H , every pair among them is part of some -cycle. Then Sd (Ψ Cond ( H )) ≤ log ( k + 2) . We propose a formal, general framework for learning solution concepts from data, and apply it toseveral problems in economic domains. While several solution concepts have a polynomial samplecomplexity, efficiently computing a consistent solution remains a challenging open problem. In thecase of market equilibria, we believe that there exist consistent algorithms for specific valuationclasses, such as gross-substitutes [22] or submodular valuations. While we mostly focus on therealizable case, solving the non-realizable case is an interesting open problem. Our model easilyaccommodates approximate solutions (as we do for market equilibria) by assimilating the approxi-mation guarantee into the loss function; this can be done generally by adopting the PMAC learning8ramework [3]. Our work upper-bounds the solution dimension using a generalization of the graphdimension; however, we offer no lower bounds. Daniely et al. [18] use the
Natarajan dimension [27]to establish lower bounds in multiclass learning; using the Natarajan dimension to lower-bound thesolution dimension is a promising direction for future work.
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A Missing Proofs for Section 4.2
We now present the proof for the Theorem 2.4 that provides an upper bound to the sample complexityfor uniform convergence in terms of the solution dimension . Definition 2.3 (Solution-based Dimension) . Given some C ⊆ X , we say the set C is S-shattered in Ψ if there exists a game g ∈ G , such that for every binary labelling b : C → { , } there exists asolution s ∈ S (that may depend on b ) such that for all x ∈ C , λ ( x, g, s ) = b ( x ) .The Solution-based dimension of Ψ , denoted Sd (Ψ) , is the size of the largest set S-shattered in Ψ ,and ( C, g ) as the corresponding shattering witness. Theorem 2.4.
There are universal constants α and α , such that if Sd (Ψ) = d , then for a sampleof m ≥ α d +log( δ ) ε points T = h x j , y j i mj =1 , Pr T ∼D m [ ∃ g ∈ G | T , s ∈ S : | ˆ L T ( g, s ) − L D ( g, s ) | > ε ] < δ. Furthermore, if a solution s ∗ is consistent, i.e. ˆ L T ( g, s ∗ ) = 0 , then for any m greater than α ε (cid:0) log (cid:0) ε (cid:1) d + log (cid:0) δ (cid:1)(cid:1) , we have that Pr T ∼D m [ L D ( g, s ∗ ) > ε ] < δ .Proof. For any g ∈ G , consider Φ g = { φ g,s : X → { , }| s ∈ S , φ g,s ( x ) = 1 − λ ( x, g, s ) } .Observe that, Sd (Ψ) = max g VC (Φ g ) ; this implies that if Sd (Ψ) = d , then for every g ∈ G , | Φ g | ≤ |X | d .From Theorem 1.2, we know that for any g ∈ G | T and f ∈ X , if T contains at least m ≥ α d +log( δ ) ε samples (for the appropriate constant α , then (where L φT and L φ D denote the corresponding lossfunctions for binary functions): Pr T ∼D m [ ∃ φ g,s ∈ Φ g : | L φT ( φ g,s , f ) − L φ D ( φ g,s , f ) | > ε ] < δ (1)Let : X → { , } , such that ∀ x : ( x ) = 1 . Then, it follows that λ ( x, g, s ) = I [ φ g,s ( x ) = ( x )] .By substituting the loss functions in Eq (1), and taking f = , we get: Pr T ∼D m [ ∃ g ∈ G | T , s ∈ S : | ˆ L T ( g, s ) − L D ( g, s ) | > ε ] < δ. This proves that m UC ( ε, δ ) , and m PAC ( ε, δ ) are both polynomially bounded by Sd (Ψ) . If a solutionis consistent, then we also know that there exists some s ∗ ∈ S such that for all g ∈ G | T , φ g,s ∗ = ∈ Φ g . The second part of the statement similarly follows from the upper bound of m PAC forbinary functions in Theorem 1.2.Let us next prove Corollary 2.5.
Corollary 2.5.
Given a S TATISTICAL S OLUTION problem
Ψ = hX , Y , G , S i : Simultaneous Constraints: if multiple local loss functions λ , . . . , λ k need to be simultaneouslyapproximated within ε , i.e. ∀ i ∈ [ k ] : | ˆ L i ( g, s ) − L i D ( g, s ) | < ε , then the sample complexity offinding a solution satisfying all of them is in O (max i ∈ [ k ] { m UC i ( ε, δ ) } ) . Separable Conjunctions: if there are local constraints λ over S , and λ over S , where S Ψ = S × S , such that we need to bound their conjunction within ε , i.e. Pr[ λ ( x, g, s ) ∧ λ ( x, g, s )] ,then m UC ( ε, δ ) is in O ( max i ∈{ , } { Sd (Ψ i ) } ) .Proof. Part 1 (Simultaneous Constraints) is a direct corollary of Theorem 2.4; if m ≥ max i { m PAC i ( ε, δ ) } , then ∀ i : Pr T ∼D m [ Pr x ∼D [ λ i ( x, g, s )] < ε ] < δ. λ : X × G × ( S × S ) →{ , } with λ ( x, g, ( s , s )) = λ ( x, g, s ) ∧ λ ( x, g, s ) , such that for any g ∈ G , we wish to find ( s , s ) ∈ S × S that bound the gap between the empirical and statistical loss.For any s ∈ S , let us define λ | s : X × G × S → { , } with λ | s ( x, g, s ) = λ ( x, g, ( s , s )) = λ ( x, g, s ) ∧ λ ( x, g, s ) . Observe that λ | s ( x, g, s ) equals λ ( x, g, s ) whenever λ ( x, g, s ) =1 , and is otherwise for every s . We note that a set shattered under λ | s cannot contain any point x ′ such that λ ( x ′ , g, s ) = 0 (points for which λ ( x ′ , g, s ) = 0 always evaluate to under λ anddo not admit Boolean functions b for which λ ( x ′ , g, ( s , s )) = b ( x ′ ) = 1 ); thus, we know that a setshattered in λ | s is also shattered under λ , therefore the solution dimension corresponding to λ | s is bounded by the solution dimension corresponding to λ . We also observe that the formulas forempirical and statistical loss under λ | s and λ are equivalent; therefore, by Theorem 2.4, we knowthat if m ≥ m UC ( ε, δ ) ∀ g ∈ G , s ∈ S , s ∈ S : P r [ | ˆ L | s T ( g, s ) − L | s D ( g, s ) | < ε ] > − δ (2)Therefore, m UC ∈ O (max { m UCi } ) .We note that the bound for conjuncts in Corollary 2.5 trivially generalizes to any number of con-junctions, i.e. if λ = λ ∧ · · · ∧ λ q , the dimension of λ is upper-bounded by the dimension of thedomains corresponding to λ , . . . , λ q . Corollary 2.6 ( Sd for Argmax) . Let Ψ max be defined by G = { f : X → Y} and S = X , where Y is endowed with a total order ≻ , and λ ( x, g, x ∗ ) = [ g ( x ) ≻ g ( x ∗ )] . Then, Sd (Ψ max ) = 1 .Proof. Let us assume that a set C = { x , x } is S-shattered with g ∈ G . This implies the existenceof: i) x ′ such that λ ( x , g, x ′ ) = 0 = ⇒ g ( x ′ ) (cid:23) g ( x ) and λ ( x , g, x ′ ) = 1 = ⇒ g ( x ) ≻ g ( x ′ ) .And, ii) x ′′ such that λ ( x , g, x ′′ ) = 1 = ⇒ g ( x ) ≻ g ( x ′′ ) and λ ( x , g, x ′ ) = 0 = ⇒ g ( x ′′ ) (cid:23) g ( x ) . However, since (cid:23) is transitive, this leads to contradiction.Finally, we present the full proof for ERM solvers for non-realizable agnostic solution learning. Corollary 2.8.
Given some S TATISTICAL S OLUTION , the sample complexity for Worst-Case andBayesian agnostic PAC solving is in O ( Sd (Ψ)) , and is achievable by an empirical risk minimizer.Proof. We first prove the result for worst-case agnostic learning. Let A m : ( X × Y ) m → S be anERM Solver that for any sample of m ≥ m UC ( ε, δ ) points T = h ( x i , y i ) i mi =1 , outputs a solution A m ( T ) ∈ S that minimizes max g ∈ G | T ˆ L T ( g, A m ( T )) .Let s ∗ ∈ S be a solution that minimizes the worst-case statistical loss for any game g consistent withthe sample, i.e. s ∗ ∈ argmin s ∈ S max g ∈ G | T L D ( g, s ) . By definition of A m ( T ) , we know that max g ∈ G | T ˆ L T ( g, A m ( T )) ≤ max g ∈ G | T ˆ L T ( g, s ∗ ) . For a sample of m ≥ m UC ( ε/ , δ/ points drawn i.i.d. from D , we know by Theorem 2.4, that forany g ∈ G , with probability ≥ − δ/ we have: | L D ( g , A m ( T )) − ˆ L T ( g , A m ( T )) | < ε , and, with probability ≥ − δ/ , for any g ′ ∈ G , | ˆ L T ( g , s ∗ ) − L D ( g ′ , s ∗ ) | < ε . Putting it all together, we get that with probability ≥ δ , L D ( g , A m ( T )) ≤ ˆ L T ( g , A m ( T )) + ε ≤ max g ∈ G | T ˆ L T ( g, A m ( T )) + ε ≤ max g ∈ G | T ˆ L T ( g, s ∗ ) + ε ≤ max g ∈ G | T L D ( g, s ∗ ) + ε ε g ∈ G | T L D ( g, s ∗ ) + ε. m ≥ m UC ( ε/ , δ/ , then Pr T ∼D m [ L D ( g, A m ( T )) ≤ min s ∈ S max g ′ ∈ G | T L D ( g ′ , s ) + ε ] ≥ − δ. For Bayesian agnostic learning, we present the result for distributions with a finite support over G ; the case where the distribution has an infinite support over G is similar. Let ˜ D be some priordistribution with a finite support over G . Then, by definition E g ′ ∼ ˜ D [ L D ( g ′ , A ( T )) | g ′ ∈ G | T ] = X g ∈ G | T L D ( g, s ) Pr ˜ D ( g ) . When m ≥ m UC ( ε, δ ) , by Theorem 2.4, for every g ∈ G | T and s ∈ S , Pr[ | L D ( g, s ) − ˆ L T ( g, s ) | ≤ ε ] > − δ . Combining these expressions, with probability mass ˜ D , we get Pr[ | E g ′ ∼ ˜ D [ L D ( g ′ , s ) | g ′ ∈ G | T ] − E g ′ ∼ ˜ D [ ˆ L T ( g ′ , s ) | g ′ ∈ G | T ] | ≤ ε ] > − δ. Therefore, an ERM solver that minimizes E g ′ ∼ ˜ D [ ˆ L T ( g ′ , s ) | g ′ ∈ G | T ] , also bounds the statisticalloss within ε . B The PAC Core in TU Cooperative Games
B.1 Cooperative Games In transferable utility (TU) cooperative games players’ preferences are induded by a function v :2 N → R + mapping every subset S ⊆ N to a value v ( S ) ∈ R + . We are interested in finding“good” payoff divisions for the game. These are simply vectors ~x = ( x , . . . , x n ) ∈ R n + such that P ni =1 x i = v ( N ) (efficiency) and x i ≥ v ( { i } ) for all i ∈ N (individual rationality). We say that acoalition S ⊆ N blocks a payoff division ~x if P i ∈ S x i < v ( S ) ; that is, the coalition S can guaranteeits members a strictly higher reward should they choose to break off from working with everyoneelse. The core is the (possibly empty) set of payoff divisions from which no coalition can deviate;in other words, core ( N, v ) = { ~x ∈ R n + | ∀ S ⊆ N : P i ∈ S x i ≥ v ( S ); P ni =1 x i = v ( N ) } . B.2 The PAC Core for TU Cooperative Games
Balcan et al. [7] propose a learning-based approach to finding a PAC stable payoff division for TUcooperative games (see definitions in Section B.1). Given a distribution D over N , a payoff division ~x ∗ ε -PAC stabilizes the game h N, v i with respect to D if Pr S ∼D " v ( S ) < X i ∈ S x ∗ i < ε. In what follows, we provide a proof for the PAC stabilizability of TU cooperative games in thelanguage of Theorem 2.4; direct proofs of this fact appear in [4, 11].
Theorem B.1.
The solution dimension of TU cooperative games is O ( n ) .Proof. We show that any set of > n coalitions cannot be S-shattered as per Definition 2.3. Takinga set of coalitions S = { S , . . . , S m } , it is S-shattered if there is some TU cooperative game v : S → R + such that for all T ⊆ S , there exists some vector ~x ∗ in R n such that for all T ∈ T , v ( T ) ≥ x ( T ) , and for all S ∈ S \ T , v ( S ) < x ∗ ( S ) . Let us bound the dimension m of S . Theproblem is equivalent to shattering sets of vectors in the hypercube { , } n with linear classifiers,which is well-known to be impossible for sets of size > n [1]. We conclude that Sd for the PACcore of TU cooperative games is ≤ n .We note that the solution computed in Balcan et al. [4] is only efficient (i.e. with P ni =1 x i = v ( N ) )if the core of the cooperative game v is not empty. In the case where the game v has an empty core,the solution computed still satisfies the core constraints with high probability with respect to D , butmay not be efficient. However, the payoff outputted is using the minimal subsidy required in orderto stabilize the game. In other words, the total payoff is no more than the cost of stability of theunderlying game v [2]. Efficiency is an important requirement: without it, one can “cheat” and payeach player some arbitrarily high amount, guaranteeing that the underlying game is stable.13 PAC Competitive Equilibria in Exchange Economies
In Section 1.4.2 we define Fisher markets; these are markets where goods are indivisible, and eachplayer i ∈ N has a budget β i . In what follows, we consider exchange economies [28, Chapters 6and 9], which follow a somewhat different structure. C.1 Exchange Economies
In exchange economies we have a set G = { g , . . . , g k } of k divisible goods, and player valuationsare of the form v i : [0 , k → R + for every i ∈ N ; bundle assignments are π : N → [0 , k (assigning a quantity q j ≤ of good g j to player i can be thought of as player i receiving q j percentof good g j ).In exchange economies with divisible goods, we assume that each player has an initial endowment of goods ~e i ∈ [0 , k , denoting the (divisible) amount of each good that she possesses. It is no lossof generality to assume that P ni =1 ~e i,j = 1 for every good g j ; in other words, the quantity e i,j is therelative amount of good g j that player i possesses. Given item prices, players demand certain itembundles. The affordable set is the set of all divisible goods whose total price is less than the worthof player i ’s endowment under ~p . A i ( ~p ) = ~g ∈ [0 , k : k X j =1 p j g j ≤ k X j =1 p j e i,j . An outcome h π, ~p i is a competitive equilibrium if π ( i ) ∈ A i ( ~p ) , and ∀ ~g ∈ A ( ~p ) , v i ( π ( i )) ≥ v i ( ~g ) . C.2 PAC Market Equilibria in Exchange Economies
We assume that player preferences are convex. We show that for any sample of fractional bundles T = { ~b , . . . ,~b m } , there exists a solution h π ∗ , ~p ∗ i consistent with T with non-positive excess as-signment (but potentially leaving some goods unassigned). We assume that none of the goods areundesirable, i.e. for every good there exists at least one player that assigns a positive value to somequantity of that good. Theorem C.1.
Suppose we are given an exchange economy for divisible goods with convex pref-erences and without undesirable goods. We observe m sampled bundles T = { ~b , . . . ,~b m } andplayer valuations over the bundles, along with player endowments ~e , . . . , ~e n . There exists a solu-tion h π ∗ , ~p ∗ i such that every player i is assigned a bundle they can afford given their endowment,which is consistent (against any possible valuation functions that could have generated the observedvalues).Proof. Without loss of generality, let us work with the reduced space of only observed goods. Let U denote the underlying space of convex preferences from which we draw player preferences overassignments. Let U | i,T denote the space of all valuation functions u that satisfy the observed values,ie. u ( ~b j ) = v i ( ~b j ) for every j ∈ [ m ] . Since prices only need to satisfy the affordability criterion forevery player, i.e. ~p ∗ · π ( i ) ≤ ~p ∗ · ~e i , we can normalize and assume that prices belong to the simplex ∆ n − . Also, observe that the absence of undesirable goods implies that in any consistent solutionthe price of any observed good cannot be .Now let us define the demand set function as D : U × ∆ n − × [0 , n → [0 , n , such that D ( u, ~p, ~e i ) = n ~b ∈ [0 , n : u ( ~b ) ≥ u ( ~b j ) ∀ b j ∈ T ; and ~p · ~b ≤ ~p · ~e i o . We observe that under convex preferences (i.e. quasi-concave utility functions), for every u , ~p and ~e , D ( u, ~p, ~e ) is a convex and compact body; in addition, D is continuous in ~p . Define, for every player i ∈ N , D Ti ( ~p ) = T u ∈U| i,T D ( u, ~p, ~e ) : D Ti ( ~p ) is the set of all possible bundles that player i mightdemand under the price vector ~p , under all possible utility functions that agree with the sample T .The intersection D Ti is convex and compact, as well as continuous in ~p . Also observe that D Ti ( ~p ) is always non-empty, since there is at least one bundle among the observed samples and the emptybundle which belongs to each of the D ( u, ~p, ~e ) . 14et f : [0 , k × n → [0 , n ] k be the excess demand function : f ( π ) = P i ∈ N π ( i ) − ~ (where ~ , , . . . , ∈ [0 , n ). Let z : ∆ n − → [0 ,k ] n , be the function z ( ~p ) = { f ( π ) : π ∈ Q i ∈ K D Ti ( ~p ) } .The function f is linear, therefore z ( ~p ) is convex and compact, and z is continuous. Using z , wedefine a function g : ∆ n − → ∆ n − such that r -th component is given by g ( ~p ) = (cid:26) ~g : where g r ( ~p ) = p r + max { , z r } P ns =1 max { , z s } , for some ~z ∈ z ( ~p ) (cid:27) By applying Kakutani’s fixed-point theorem over g , we get the existence of some ~p ∗ such that ~p ∗ ∈ g ( ~p ∗ ) . This implies the existence of some π ∈ Q i ∈ K D Ti ( ~p ) , such that f ( π ) ∈ z ( ~p ∗ ) satisfies p ∗ r = p ∗ r + max(0 , f r ( ~p ∗ ))1 + P ns =1 max(0 , f s ( ~p ∗ )) Let r ∗ some non-positive component of f ( π ) , as argued above; then p ∗ r = p ∗ r (1 + P ns =1 max(0 , f s ( ~p ∗ ))) ; this implies that for all r : max(0 , f r ( ~p ∗ )) = 0 . Therefore, there existssome allocation π ∗ with non-positive excess demand at ~p ∗ , such that h π ∗ , ~p ∗ i is consistent with theobserved bundles against all possible u ∈ U| i,T for every player i . D PAC Condorcet Winners
Recall that a Condorcet winner is a candidate c ∗ that beats every other candidate c in a pairwiseelection, i.e. for every other candidate c , a majority of voters prefer c ∗ to c . We note that if thereare only two candidates, then a Condorcet winner trivially exists (barring the case when the votesare tied); however, when there are three or more candidates, a Condorcet winner is not guaranteedto exist. This case is known in the literature as Condorcet cycles (or Condorcet paradoxes). Forthe sake of completeness, we provide a simple example of a voting profile where no candidate is aCondorcet winner. Example D.1.
Consider a setting with three candidates a, b, c and three voters, , and whosepreferences over a, b, c are as follows: a ≻ b ≻ c b ≻ c ≻ a c ≻ a ≻ b a to b ; 1 and 2 prefer b to c ; 2 and 3 prefer c to a . Therefore, there are no Condorcetwinners.Next, let us provide a complete proof of Theorem 3.6. Theorem 3.6.
Let k be the largest number of candidates, such that for some tournament graph in H , every pair among them is part of some -cycle. Then Sd (Ψ Cond ( H )) ≤ log ( k + 2) .Proof. Let us assume that there is a set of size d , C ⊆ C , that is shattered. Then by definitionof shattering, there exists d different candidates corresponding to every subset of C , such that for f ∈ C there exists a candidate c f ∈ C , such that c ∈ C beats c f in the tournament graph if andonly if f ( c ) = 1 .Let us focus on d − of these candidates, corresponding to all non-trivial functions f ∈ C (let us,for now, ignore the functions that assign a constant value (of or ) to all candidates in C ). Then,for every pair of these functions f and f , there exists a candidate c ∈ C such that f ( c ) = 1 and f ( c ) = 0 , and a candidate c such that f ( c ) = 0 and f ( c ) = 1 . This implies the existenceof a directed path of length at most from c f to c f , and vice versa. Since, in a tournament graph,either the edge c f → c f or c f → c f exists, we know that c f and c f are members of some -cycle. Since this is true for all such c f ’s, we know that d − is less than or equal to largestnumber of candidates, such that for some tournament graph in H , every pair amongst them is part ofsome3