Representation Theory of the Symmetric Group in Voting Theory and Game Theory
aa r X i v : . [ m a t h . R T ] A ug REPRESENTATION THEORY OF THE SYMMETRIC GROUPIN VOTING THEORY AND GAME THEORY
KARL-DIETER CRISMAN AND MICHAEL E. ORRISON
Abstract.
This paper is a survey of some of the ways in which the repre-sentation theory of the symmetric group has been used in voting theory andgame theory. In particular, we use permutation representations that arisefrom the action of the symmetric group on tabloids to describe, for example,a surprising relationship between the Borda count and Kemeny rule in vot-ing. We also explain a powerful representation-theoretic approach to workingwith linear symmetric solution concepts in cooperative game theory. Alongthe way, we discuss new research questions that arise within and because ofthe representation-theoretic framework we are using. Introduction
Symmetry arises in many ways in voting theory and game theory. In this paper,we highlight one such appearance by surveying some of the ways in which the representation theory of the symmetric group has been used in these fields. Ourprimary goal is to show how certain well-understood permutation representationsof the symmetric group can be used to make sense of foundational ideas in votingtheory and game theory, and how using these representations can in turn helpresearchers formulate novel questions and meaningful generalizations.Among other examples, we use the representation theory of the symmetric groupto describe a surprising relationship between the Borda count and the Kemeny rulein voting. We also use it to construct and make sense of certain infinite familiesof solution concepts in cooperative game theory. Along the way, we describe muchof the representation-theoretic framework used in voting theory and game theory,and we discuss several new research questions that arise therein.The permutation representations on which we focus are those associated to the usualaction of the symmetric group S n on simple combinatorial objects called tabloids.Representation theory experts will have no trouble seeing that we only scratch thesurface on what could be done with these representations in voting theory and gametheory. Another goal for this paper is therefore to encourage readers to contributeto these fields by going well beyond what we present here. Date : August 7, 2015.2010
Mathematics Subject Classification.
Primary 91B12, 91A12, 20C30.
As a brief introduction to the role that symmetry can play in voting, consider asituation in which voters are asked to choose their favorite candidate A , B , or C .If we end with a profile of (13 , ,
7) for
A, B, C respectively, then it is natural tosay this is essentially the same profile as (9 , , , ,
7) = (8 , ,
8) + (5 , − , −
1) and noting that the first vector on the right cap-tures how many voters there were, while the second vector captures the differentlevels of support the candidates received, which in the end is what really matters.Interestingly, such decompositions profiles arise by having the symmetric group S act on the candidates by permuting their labels, which in turn induces an action onthe set of profiles. Furthermore, by having the symmetric group act on the labelsof candidates in much more complicated voting situations, similar decompositionsof profiles arise, and we are able to use what we know about the resulting subspacesof voting data to say something worthwhile (as we will demonstrate in Section 3)about how profiles are used by different kinds of voting procedures.In the next section, we introduce most of the notation we will use for the rest of thepaper, and we motivate the use of tabloids for indexing the kind of data with whichwe will be working. Although we do not assume our readers will be familiar withthe ideas we will be presenting from voting theory and game theory, we will assumethat our readers have a basic working knowledge of the representation theory offinite groups (see, for example, [13, 38]).Finally, we encourage interested readers to view the ideas presented in this paperwithin the larger framework of harmonic analysis on finite groups [2, 3, 7, 42]. Inour opinion, doing so makes it much easier to see how the representation theory ofthe symmetric group (and other finite groups) has been applied outside of votingtheory and game theory, in fields, for example, such as statistics [1, 7, 8, 25] andmachine learning [11, 23, 24]. Doing so will also reveal that the approach we aretaking in this paper has already found success in a variety of other settings.2. Background
In this section, we define tabloids and their associated permutation representationsof the symmetric group. We also describe how tabloids can be used to index the kindof voting and game-theoretic data with which we will be working. Good referencesfor the material in this section are [12] and [36].Let n be a positive integer. A composition of n is a list λ = ( λ , . . . , λ m ) of positiveintegers whose sum is n . If it is also the case that λ ≥ · · · ≥ λ m , then we say λ is a partition of n . For example, (2 , , ,
3) is a composition of 9, and (4 , , , λ is a composition, then we will denote by λ the partitionobtained by reordering the numbers in λ so that they form a non-increasing list.For example, if λ = (2 , , , λ = (3 , , , λ is a composition of n , then the Young diagram of shape λ is the left-justifiedarray of boxes that has λ i boxes in its i th row (see Figure 1). If we fill these boxes EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 3 with the numbers 1 , . . . , n without repetition, then we create a
Young tableau ofshape λ . Two tableaux of shape λ = ( λ , . . . , λ m ) are then said to be row equivalent if they have the same set of λ numbers in the first row, the same set of λ numbersin the second row, and so on. An equivalence class of tableaux is then called a tabloid of shape λ . Figure 1.
The Young diagram of shape (2 , , , Figure 2.
Two row equivalent tableaux and their tabloid.Suppose λ is a composition of n . Let X λ denote the set of tabloids of shape λ ,and let M λ denote the vector space of real-valued functions defined on X λ . If x ∈ X λ , then f x will denote its associated indicator function with the property that f x ( x ) = 1 and f x ( y ) = 0 for all y = x . Note that the set of such indicator functionsforms an orthonormal basis for M λ with the respect to the usual inner product h· , ·i on M λ , which is defined by setting h f , g i = X x ∈ X λ f ( x ) g ( x )for all f , g ∈ M λ . We will call this basis of indicator functions the usual basis of M λ , and when necessary we will assume it has been ordered with respect to thelexicographic ordering of the associated tabloids in X λ .In this paper, we will be interested in a variety of real-valued functions defined on X λ for different choices of λ . We will also be interested in certain linear transfor-mations defined on M λ .For example, consider an election involving n candidates labeled 1 through n . Ifwe ask the voters to tell us their top favorite λ candidates, then their next topfavorite λ candidates, and so on, then we can interpret each of their responses asa tabloid x ∈ X λ . For example, if λ = (2 , , , KARL-DIETER CRISMAN AND MICHAEL E. ORRISON the other. The function f ∈ M λ in which we are then interested is simply the onefor which f ( x ) is the total number of voters who chose tabloid x ∈ X λ .We will mostly focus on three kinds of compositions in this paper. The first com-position is the all-ones partition (1 , . . . , X (1 ,..., naturally cor-respond to all of the possible permutations of the numbers 1 , . . . , n . Therefore, M (1 ,..., is helpful when describing voting situations in which voters are asked toprovide a full ranking of candidates. The second kind of composition has the form( k, n − k ) where 1 ≤ k ≤ n −
1. In this case, we will use the tabloids in X ( k,n − k ) toindex k -element subsets of { , . . . , n } , where each k -element subset corresponds tothe tabloid whose top row contains all of its elements. Finally, the third composi-tion is ( n ). The set X ( n ) consists of a single, one-rowed tabloid which we will useto index the entire set { , . . . , n } .As an example of how we will use M (1 ,..., and M ( k,n − k ) in this paper, in the nextsection we will consider the voting situation in which voters have been asked to fullyrank a set of n candidates. We will then use that information, which we will viewas an element of M (1 ,..., , to assign points to each of the individual candidates.The result may therefore be viewed as an element of M (1 ,n − . Furthermore, wewill do all of this using a linear transformation from M (1 ,..., to M (1 ,n − .Now it is natural in voting to insist that a voting method not depend on how welabel the candidates. Fortunately, we may easily express this property using a groupaction. More specifically, note that the symmetric group S n acts naturally on X λ where if σ ∈ S n and x ∈ X λ , then σ · x is the tabloid one gets by applying thepermutation σ to each entry of x .The action of S n on X λ extends to an action of S n on M λ where if σ ∈ S n , f ∈ M λ ,and x ∈ X λ , then ( σ · f )( x ) = f ( σ − · x ) . In other words, M λ may be viewed as a module over the group algebra R S n . Forexample, M ( n ) corresponds to the trivial R S n -module. This is because X ( n ) consistsof a single tabloid, and every permutation in S n fixes this tabloid.In the voting example above, we can view the action of S n on M (1 ,..., and M (1 ,n − as the result of changing the labels on the candidates. When the outcome of a votingmethod does not depend on the labels that have been assigned to the candidates,we say that the voting method is neutral (see, for example, [29, 41], among manyother references for general voting theory concepts). Insisting that a voting methodbased on a linear transformation T : M (1 ,..., → M (1 ,n − be neutral then simplybecomes the requirement that T must be an R S n -module homomorphism. We willfocus only on voting methods that are neutral in this paper.Fortunately, the representation theory of M λ is well-understood. The irreducible R S n -modules are parametrized by the partitions of n , and we denote the irreduciblemodule corresponding to the partition µ by S µ . These are the well-known Spechtmodules where, for example, S ( n ) corresponds to the one-dimensional trivial R S n -module. EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 5 If λ is a partition of n (and not just a composition), then M λ is isomorphic to adirect sum of Specht modules M λ ∼ = M µ κ µλ S µ where the κ µλ are Kostka numbers and are used here record the multiplicity of eachSpecht module in M λ (see, for example, Sections 2.3 and 2.11 of [36]).Note that if λ is a composition of n and not necessarily a partition, then themodule M λ is easily seen to be isomorphic to M λ by simply reordering the rows ofthe tabloids in X λ . We may therefore just as easily work with compositions as withpartitions of n when dealing with real-valued functions (e.g., voting data) definedon sets (e.g., rankings of candidates) that are indexed by tabloids.The modules M (1 ,..., , M ( k,n − k ) , and M ( n ) have particularly straightforward de-compositions in terms of Specht modules. For example, M (1 ,..., is isomorphic tothe regular R S n -module, and thus M (1 ,..., ∼ = M µ (dim S µ ) S µ . If 1 ≤ k ≤ n/ n − k, k ) is a partition of n ), then M ( k,n − k ) ∼ = M ( n − k,k ) ∼ = S ( n ) ⊕ S ( n − , ⊕ S ( n − , ⊕ · · · ⊕ S ( n − k,k ) . Lastly, because M ( n ) corresponds to the trivial R S n -module, we have that M ( n ) ∼ = S ( n ) (see Chapter 7 of [7] or Section 2.11 of [36]).Before leaving this section, we introduce some specialized notation and terminologythat we will use later in the paper.First, note that M (1 ,n − is a direct sum of two irreducible submodules. Thesesubmodules are easy to describe (see also p. 39 in [36]). One of the submodules,which we will denote by U , is isomorphic to S ( n ) and consists of all of the constantfunctions in M (1 ,n − : U = n f ∈ M (1 ,n − | f ( x ) = f ( y ) for all x, y ∈ X (1 ,n − o . The other submodule, which we will denote by U , is isomorphic to S ( n − , . It isthe orthogonal complement of U , and it consists of those functions whose valuessum to zero: U = f ∈ M (1 ,n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ X (1 ,n − f ( x ) = 0 . If f ∈ M (1 ,n − , then we will denote its projection into U by b f . Thus f = ( f − b f ) + b f where f − b f ∈ U and b f ∈ U . In fact, if we let ∈ M (1 ,n − denote the functionthat assigns the value 1 to every tabloid, and let f denote the projection of f onto , then f − b f = f .Next, note that the action of S n on X λ is transitive. In other words, given twotabloids x, y ∈ X λ , there exists at least one permutation σ ∈ S n such that σ · x = y .In this case, we have that σ · f x = f y . It follows that every module homomorphism KARL-DIETER CRISMAN AND MICHAEL E. ORRISON defined on M λ is determined by the image of any of the indicator functions f x . Forconvenience, when we construct such maps, we will focus on the image of f x , wherewe use x to denote the tabloid in X λ that appears first when the tabloids in X λ are listed lexicographically. More specifically, x ∈ X λ is the tabloid that containsthe tableau whose entries, when read from left to right and top to bottom, are thenumbers 1 , . . . , n in that order.Finally, suppose λ is a composition of n , and that T is a module homomorphismdefined on M λ . The module M λ can be written as the direct sum M λ = ker T ⊕ (ker T ) ⊥ of the kernel of T and its orthogonal complement. We will refer to the submodule(ker T ) ⊥ as the effective space of T , and we will denote it by E ( T ). Note that theeffective space E ( T ) is isomorphic to (and also determines) the image of T .3. Voting Theory
For algebraists interested in learning about the mathematics of voting, DonaldSaari’s papers [27, 30–32] and books [28, 29, 33, 34] are a fruitful place to start. Hiswork provides a friendly but mathematically sophisticated gateway to the subject,and although his approach is primarily geometric, linear algebra and symmetry areused to great effect.Some of Saari’s key results rely on decomposing vector spaces of voting data into ahandful of simple but meaningful subspaces. Despite the geometric flavor of many ofhis papers, algebraists will easily recognize many of these subspaces as submodulesof M (1 ,..., . This realization has been put to use in papers such as [5] and [6], andwhat follows in this section can be partly viewed as an introduction to those twopapers.We begin by considering an election in which there are n candidates labeled 1 , . . . , n .Suppose each voter has been asked to rank the candidates by choosing a tabloidin X (1 ,..., , where the candidate in the top row of the tabloid is their favorite, thecandidate in the second row is their second favorite, and so on.For example, if there are n = 3 candidates, then each voter is being asked to chooseone of the following tabloids from X (1 , , :123 , 132 , 213 , 231 , 312 , 321 .In this case, a voter who chooses the third tabloid above is saying that she preferscandidate 2 to candidate 1 to candidate 3.Let p ∈ M (1 ,..., be the function with the property that p ( x ) is the number ofvoters who chose the tabloid x . The function p is called a profile . Next, let w = [ w , . . . , w n ] t be a column vector in R n such that w ≥ · · · ≥ w n . The EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 7 vector w is called a weighting vector , and we will use this vector of “weights” tocreate a voting procedure as follows.For each tabloid x ∈ X (1 ,..., , if candidate i is in row j of x , then she will begiven p ( x ) w j points. We then sum over all of the tabloids in X (1 ,..., to find thetotal number of points received by candidate i . The candidates who receive themost points (there might be ties) are then declared to be winners. We refer to thisvoting procedure as the positional voting procedure associated to w . Note that theinequality condition above ensures that a voter’s “more favored” candidates receiveat least as many points from her as her “less favored” candidates.Note that w = [1 , , . . . , t corresponds to the plurality voting procedure, wherevoters are essentially being asked to vote for only one candidate. When w =[1 , . . . , , t , we get the anti-plurality voting procedure, where voters are essentiallybeing asked to vote against their least favorite candidate. The Borda count , wherea candidate receives ( i −
1) points for each time she is ranked in the i th positionby a voter, is given by the vector w = [ n − , n − , . . . , , , t .Let w ( i ) ∈ M (1 ,..., be defined by setting w ( i ) ( x ) = w j if candidate i is containedin the j th row of the tabloid x . We can use the function w ( i ) to find the totalnumber of points candidate i will receive, which is X x ∈ X (1 ,..., p ( x ) w ( i ) ( x ) = h p , w ( i ) i . Furthermore, we can use the tabloids in X (1 ,n − to index the candidates by associ-ating y ∈ X (1 ,n − with the single candidate in the top row of y . With that in mind,let c i ∈ M (1 ,n − denote the indicator function of the tabloid corresponding to can-didate i . We can then create the module homomorphism T w : M (1 ,..., → M (1 ,n − by setting T w ( f ) = h f , w (1) i c + · · · + h f , w ( n ) i c n . The coefficient in front of c i in T w ( p ) is therefore simply the number of points thatcandidate i received when we apply the positional voting procedure associated with w to the profile p .To see the motivation behind the notation defined above, consider encoding thecalculation of T w ( p ) as a matrix-vector multiplication. For example, let n = 3, andsuppose w = [1 , s, t for some s such that 1 ≥ s ≥
0. If the coordinate vector of p with respect to the usual basis of M (1 , , is [3 , , , , , t , then the coordinatevector of T w ( p ) with respect to the usual basis of M (1 , is given by s s s s s s = s s s . In this case, the functions w (1) , w (2) , w (3) correspond to the three rows of the 3-by-6 matrix, and we are simply dotting each of these rows with the coordinate vector KARL-DIETER CRISMAN AND MICHAEL E. ORRISON of p to find the coordinate vector [5 + 4 s, s, s ] t of points received by eachof the candidates. (Note that T w ( p ) = (5 + 4 s ) c + (6 + 6 s ) c + (3 + 4 s ) c .)There is, however, more to see here. Notice that each column of the 3-by-6 matrixcan be obtained by permuting the entries of the weighting vector w = [1 , s, t . Infact, if x ∈ X (1 , , is the tabloid with candidate i in row i , and σ ∈ S n , thenthe column of the matrix corresponding to the tabloid σ · x corresponds to thefunction (1) c σ (1) + ( s ) c σ (2) + (0) c σ (3) in M (1 , .From this point on we will slightly abuse our notation by identifying the weightingvector w with the function w c + · · · + w n c n ∈ M (1 ,n − . If x ∈ X (1 ,..., is thetabloid with candidate i in row i , and e f ∈ R S n is defined by setting e f ( σ ) = f ( σ · x ),we then have that T w ( f ) = X σ ∈ S n f ( σ · x )( w c σ (1) + · · · + w n c σ ( n ) )= X σ ∈ S n e f ( σ )( σ · w )= e f · w . In other words, we may interpret the function T w ( f ) as the result of the groupalgebra element e f ∈ R S n acting on the function w ∈ M (1 ,n − . Finding functions f ∈ M (1 ,..., so that T w ( f ) has certain desirable properties then becomes a questionof finding appropriate elements of the group algebra R S n to act on w ∈ M (1 ,n − .This was one of the key insights in [6], and it leads to theorems like the following,which is essentially a special case of Theorem 1 in that paper: Theorem 1.
Let n ≥ , and suppose w , . . . , w k ∈ U ⊂ M (1 ,n − form a linearlyindependent set of weighting vectors. If r , . . . , r k ∈ U , then there exist infinitelymany functions f ∈ M (1 ,..., such that T w i ( f ) = r i for all i such that ≤ i ≤ k . To begin to appreciate the connection between Theorem 1 and voting theory, itis first helpful to realize that if w = w + b w , where w ∈ U and b w ∈ U , then T w ( f ) = e f · w = e f · w + e f · b w . Furthermore, the ordinal ranking (i.e., who comesin first, who comes in second, and so on) provided by T w is going to depend onlyon b w . After all, e f · w is contained in U , and is therefore a constant function.Building on this insight, we will say that two weighting vectors w , w ′ ∈ M (1 ,n − are equivalent, and write w ∼ w ′ , if and only if there exist α, α ′ ∈ R such that α > w ′ = α w + α ′ . It should be clear that in this case, w ′ and w will yield thesame ordinal rankings. (Also, note that if α < w ′ would be the reverse of the ordinal rankings given by w .) The following theorem,which follows from Theorem 1 and Theorem 2.3.1 in [28], highlights the usefulnessof this equivalence relation on weighting vectors when dealing with positional votingprocedures: Theorem 2.
Let w , w ′ ∈ M (1 ,n − be weighting vectors. The ordinal rankingsof T w ( p ) and T w ′ ( p ) will be the same for all profiles p ∈ M (1 ,..., if and only if w ∼ w ′ . EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 9
In other words, if the weighting vectors w and w ′ are not equivalent, then theordinal rankings of the outcomes of elections that use w and w ′ might differ, evenif the same profile p ∈ M (1 ,..., is used in both elections. In fact, by Theorem 1,the results of such elections can be arbitrarily different from one another. Thisfollows from the fact that if w and w ′ are not equivalent, then their projectionsinto U will be linearly independent.Another way of appreciating the effect that Theorem 1 can have on our under-standing of positional voting procedures is to realize that the effective spaces of T w and T w ′ share very little in common unless w ∼ w ′ . This is because if w ∈ U is nonzero, then E ( T w ) ∼ = S ( n − , , and thus E ( T w ) is an irreducible submoduleof M (1 ,..., . This allows us to easily prove theorems like the following, which isTheorem 4 in [6]: Theorem 3.
Let w , w ′ ∈ U ⊂ M (1 ,n − be nonzero weighting vectors. Then E ( T w ) = E ( T w ′ ) if and only if w ∼ w ′ . Furthermore, if E ( T w ) = E ( T w ′ ) , then E ( T w ) ∩ E ( T w ′ ) = { } . In other words, if w and w ′ are not equivalent, then T w and T w ′ will use verydifferent subspaces of M (1 ,..., from which to pull the information necessary todetermine the outcome of an election. The projections of a profile p ∈ M (1 ,..., into those subspaces determine the cardinal rankings of the candidates. We shouldtherefore not expect any relationship between the associated ordinal rankings ofthe candidates.There are, of course, other voting methods besides positional voting procedures. Forexample, a simple ranking scoring function , or SRSF, takes a profile M (1 ,..., anduses it to assign points to full rankings instead of individual candidates (see [4],and the related paper [44] where these are posited as a subset of a much largerclass of generalized scoring functions). A winning ranking (as opposed to simplya winner, which is usually how one interprets normal positional rules) is a rankingthat receives at least as many points as all of the other rankings.As an example of an SRSF, let x ∈ M (1 ,..., be the tabloid (i.e., full ranking)that has candidate i in row i . Let z ∈ M (1 ,..., be a fixed function, and define T z : M (1 ,..., → M (1 ,..., by setting T z ( f ) = X σ ∈ S n f ( σ · x )( σ · z ) = e f · z for all f ∈ M (1 ,..., . Thus, if p ∈ M (1 ,..., is a profile, then a winning rankingin our election would correspond to a tabloid x ∈ M (1 ,..., with the property that T z ( p )( x ) ≥ T z ( p )( y ) for all y ∈ M (1 ,..., .How might we find a sensible function z ∈ M (1 ,..., to create an SRSF like theone above? One way is to use a metric defined on the tabloids in X (1 ,..., . Forexample, if d : X (1 ,..., × X (1 ,..., is a metric with maximum distance d max , thenwe can set z ( x ) = d max − d ( x, x ). A winning ranking with respect to T z would thencorrespond to a tabloid that is “closest” to the entire multiset of tabloids chosenby the voters (see [44] for a thorough discussion of “proximity rules”). One of the most popular metrics on X (1 ,..., is the Kendall tau distance , whichmeasures the number of pairwise disagreements between two rankings. To explain,let i, j ∈ { , . . . , n } where i = j . Let a ij ∈ M (1 ,..., be the function defined bysetting a ij ( x ) = 1 whenever candidate i is ranked above j in x (i.e., in the tabloid x , i is in a higher row than j ), and setting a ij ( x ) = 0 otherwise. Kendall’s taudistance is then given by d ( x, y ) = (cid:18) n (cid:19) − X i = j a ij ( x ) a ij ( y ) . The maximum distance between rankings in this case is (cid:0) n (cid:1) , and therefore wecould define the function z ∈ M (1 ,..., by setting z ( x ) = P i = j a ij ( x ) a ij ( x ). Theresulting voting procedure given by T z then becomes the well-known and well-studied Kemeny rule [17]. To distinguish this case, we will denote this particularinstance of T z by K : M (1 ,..., → M (1 ,..., .Perhaps not surprisingly, we can use the a ij to apply K to a profile p ∈ M (1 ,..., .More specifically, with a little bit of effort, and using the fact that the Kendall taudistance is invariant under the action of S n , one can show that K ( p ) = X i = j h p , a ij i a ij (see Proposition 1 in [40] for a similar expression). On the other hand, it mightnot be clear how a voting procedure like the Kemeny rule is related to positionalvoting procedures. We describe such a relationship next.First, it turns out that we can turn any positional voting procedure into an SRSFas follows. Let w ∈ M (1 ,n − be a weighting vector, and let b ∈ M (1 ,n − be theweighting vector for the Borda Count: b = ( n − c + ( n − c + · · · + (1) c n − + (0) c n . Note that although we will use b in the construction below, any weighting vectorwhose weights are strictly decreasing would also work. Next, recall that T w ( f ) = h f , w (1) i c + · · · + h f , w ( n ) i c n . If we compose T w with the adjoint T ∗ b : M (1 ,n − → M (1 ,..., , then we create themap T ∗ b ◦ T w : M (1 ,..., → M (1 ,..., where( T ∗ b ◦ T w )( f ) = h f , w (1) i b (1) + · · · + h f , w ( n ) i b ( n ) . It turns out that if we view T ∗ b ◦ T w as an SRSF, then it will return the rankingthat agrees with the ranking given by T w (see Proposition 1 in [4]).For such procedures, individual candidates may then be ranked based on the orderthey appear in a winning ranking, recovering the winning candidate(s) in the usualinterpretation of T w . However, for more general SRSFs (including the Kemenyrule), this may not always be meaningful. For example, suppose there are threecandidates A , B , and C , and we are given a profile where two voters prefer ABC ,two voters prefer
CAB , and only one voter prefers
BCA . The Kemeny rule thenproduces a tie between the rankings
ABC and
CAB . EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 11
Next, let c ij ∈ M (1 , ,n − be the indicator function corresponding to the tabloidthat has i in the top row and j in the second row. We will call the map P : M (1 ,..., → M (1 , ,n − given by P ( f ) = X i = j h f , a ij i c ij the pairs map defined on M (1 ,..., . Note that if p ∈ M (1 ,..., is a profile, then P ( p )simply catalogues the number of times each candidate was ranked over anothercandidate.For example, when n = 3, M (1 ,..., = M (1 , ,n − , and with respect to the usualbasis of M (1 , , , the pairs map P can be encoded as the matrix[ P ] = . Note, for example, that the first row of this matrix corresponds to a , the thirdrow corresponds to a , and the sum of these two rows corresponds to the constantall-ones function in M (1 , , .The pairs map P is not surjective. In particular, the codomain has the decompo-sition M (1 , ,n − ∼ = S ( n ) ⊕ S ( n − ⊕ S ( n − , ⊕ S ( n − , , but the effective space of P is isomorphic to S ( n ) ⊕ S ( n − , ⊕ S ( n − , , (see p. 682in [6]). Let W , W , and W denote the corresponding subspaces in M (1 ,..., , where W ∼ = S ( n ) , W ∼ = S ( n − , , W ∼ = S ( n − , , , and E ( P ) = W ⊕ W ⊕ W . Note that if P ∗ : M (1 , ,n − → M (1 ,..., is the adjoint of P , then( P ∗ ◦ P )( f ) = X i = j h f , a ij i a ij . In other words, K = P ∗ ◦ P . This should be compared with T ∗ b ◦ T w defined above,particularly in the case when w = b corresponds to the Borda count.If necessary, we can create the matrix encoding of K with respect to the usual basisof M (1 ,..., by simply taking the product of the matrix encodings of P ∗ and P . Forexample, when n = 3, the matrix encoding of K is[ K ] = which is the product [ P ] t [ P ]. Notice the columns of [ K ] are all permutations of thefirst column of [ K ], which corresponds to the function z ∈ M (1 , , that is based(as was described above) on the Kendall tau distance.Interestingly, it turns out that the effective space E ( T b ) of the Borda Count is W ⊕ W (see, for example, Section 7 of [6]). This is perhaps much more understandableonce we realize that b ( i ) = X j : i = j a ij . Furthermore, by Theorem 3, the only nontrivial positional voting procedures thatcontain W in their effective spaces are those whose weighting vectors are equivalentto b . This begins to explain theorems like the following, which is a combination ofparts of Theorem 3 and Theorem 4 in [35]: Theorem 4.
For n ≥ candidates, the Borda count always ranks the Kemenyrule top-ranked candidate strictly above the Kemeny rule bottom ranked-candidate.Conversely, the Kemeny rule ranks the Borda count top-ranked candidate strictlyabove the Borda count bottom-ranked candidate. For any positional voting methodother than the Borda count, however, there is no relationship between the Kemenyrule ranking and the positional ranking. Using the pairs map P , we can say more about the relationship between the Bordacount and the Kemeny rule. More specifically, because K = P ∗ ◦ P is self-adjoint,we know it is orthogonally diagonalizable. In fact, the eigenspaces of K are W , W , W , and ( W ⊕ W ⊕ W ) ⊥ with eigenvalues κ = n !2 (cid:0) n (cid:1) , κ = ( n +1)!6 , κ = n !6 ,and 0, respectively (see Theorem 3 in [40]).If we let T i : M (1 ,..., → W i denote the orthogonal projection onto W i , we thenhave that K = κ T + κ T + κ T . Furthermore, if β = ( n − n !2 (cid:0) n (cid:1) , β = n ( n +1)!12 , and β = 0 then it is possible toshow that T ∗ b ◦ T b = β T + β T + β T . The Borda count and Kemeny rule may therefore be viewed as members of thesame family of SRSFs whose maps all have the form K ( γ ,γ ,γ ) = γ T + γ T + γ T where γ , γ , γ ∈ R .When using K ( γ ,γ ,γ ) , the resulting ordering of the rankings in X (1 ,..., dependsonly on the ratio γ /γ . After all, γ T only contributes scalar multiples of theall-ones function to the outcome. This fact is exploited in [5] by setting γ = 0 andfocusing on a one-parameter family of procedures that is interpreted as being “be-tween” the Borda count and Kemeny rule. Such a family allows us, for example, tobetter understand when properties like being susceptible to the “no-show paradox”can arise in voting (see Proposition 5.16 in [5]).By describing everything up to this point in terms of M (1 ,..., and M (1 ,n − , wehope it is clear how one might extend all of these ideas to more general settings. EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 13
For example, suppose λ = ( λ , . . . , λ m ) is a partition of n . We could ask the votersto rank the candidates by choosing a tabloid in X λ , where the candidates in thetop row of their tabloid are their top favorite λ candidates, the candidates in thesecond row are their next favorite λ candidates, and so on.When voters are asked to provide rankings by choosing a tabloid from X (1 ,..., , wesay they are giving full rankings of the candidates. If they are choosing tabloidsfrom X λ where λ = (1 , . . . , partial rankings (ofshape λ ) of the candidates. We can therefore ask if any of the theorems abovecan be extended to partial rankings. For some of the theorems, the answer is yes.In fact, the paper by Daugherty et al. [6] focused primarily on such theorems forpositional voting.Notice also that the maps T w and T z were defined in essentially the same way.In both cases, we chose a vector v ∈ M µ for some partition µ , and then defined T v : M (1 ,..., → M µ by setting T v ( f ) = e f · v . We then used T v and a profile p ∈ M (1 ,..., to find the “best” tabloid in M µ based on “points” that T v ( p ) assignedto the standard basis vectors of M µ . What would happen if we were to extend thisconstruction to situations in which voters choose tabloids in X λ and points are thenassigned to tabloids in X µ ? Would we learn anything new? For readers interested inexploring this question, Section 3 in [6] might be a good place to start. It discussesmaps from M λ to M (1 ,n − and what happens to positional voting procedures whenvoters are asked to provide partial (instead of full) rankings of shape λ .Another possible direction one might take is to ask what would happen if we wereto replace the pairs map in the definition of the Kemeny rule with maps thatcatalogue information about triples of candidates, or quadruples of candidates, andso on. Doing so would then place the Kemeny rule in another potentially interestingfamily of voting procedures. We are not aware of any work in this direction, butanyone interested in pursuing such a project would almost certainly benefit fromthe discussions about inversions in [9] and [25].Finally, there are maximum likelihood estimator procedures which give the “mostlikely” ranking of the candidates, assuming that the voters all meant to choose thesame ranking but their votes were corrupted by a noise model. Young [43] showsthat the Borda count is an MLE when the desired outcome is a single winner, andthat the Kemeny rule is an MLE when the desired outcome is a full ranking. In [26],the question of which procedure is an MLE when the desired outcome is a top pair,top triple, and so forth is investigated. Given the connection between SRSFs andMLEs found in [4], we suspect that an algebraic approach to MLE procedures hastremendous promise. 4. Game Theory
For algebraists interested in learning about (transferable utility cooperative) gametheory, the pioneering work of Kleinberg and Weiss [18–22] will quickly make themfeel welcome. Consider, for example, the first few lines from the introduction of [18]:
Game theory and algebra become inextricably intertwined once onerecognizes that the notion of a permutation of players gives rise toa representation of the symmetric group in the space of automor-phisms of G , the vector space of games (in characteristic functionform).In this paper we use this observation to attempt to turn the usualapproach to game-theoretic problems on its head by analyzing thespace of games as an algebraist might. In this way, we can see ifthe mathematical structure of G has anything to say about whatconstructs are significant from a game-theoretic point of view.In this section, we focus on the role that tabloids play when navigating Kleinbergand Weiss’s work, as well as the recent work of Hern´andez-Lamoneda, Ju´arez, andS´anchez-S´anchez [10] and S´anchez-P´erez [37]. We then take advantage of our useof tabloids to suggest possible extensions of their work.We begin with some notation and terminology. Let n ≥
2, and suppose we have n players. We will label the players with the numbers 1 , . . . , n , and we will let N = { , . . . , n } be the set of all of the players.A cooperative game for N is a real-valued function defined on the set of all subsetsof N , with the convention that the empty set is always mapped to 0. The set of allsuch games G = { v : 2 N → R | v ( ∅ ) = 0 } models ways in which various subsets (“coalitions”) can be allotted a utility (“value”)based on the extent to which they are perceived to contribute to the entire set ofplayers (“the grand coalition”).One of the defining challenges in cooperative game theory is to find useful andmeaningful solution concepts , which are functions defined on G that are used todetermine a “payoff” for each individual player. More specifically, if we index theindividual players by tabloids in X (1 ,n − as we did in the last section on votingtheory, then we can think of a solution concept as a function ϕ : G → M (1 ,n − where if v ∈ G and i ∈ N , then the value that ϕ associates to player i is thecoefficient in front of c i in ϕ ( v ). If we denote this value by ϕ ( v ) i , then ϕ ( v ) = ϕ ( v ) c + · · · + ϕ ( v ) n c n . It is interesting to note that, algebraically, a solution concept looks similar to apositional voting procedure, in that the payoff that a solution concept assigns toa player is similar to the points that a positional voting procedure assigns to acandidate.The vector space G of games is of course much more than just a vector space. It isan R S n -module, where for all σ ∈ S n and v ∈ G ,( σ · v )( S ) = v ( σ − · S ) EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 15 for all subsets S ∈ N , and where σ − · S is the set you obtain by applying σ − toeach element of S . Furthermore, the irreducible submodules of G are straightfor-ward to describe.First, note that we can write G = G ⊕ G · · · ⊕ G n − ⊕ G n , where G k is the subspaceof all games that assign the value 0 to any subset of players that does not have size k . Next, if we index the k -element subsets of players using tabloids in X ( k,n − k ) ,then it becomes clear that G k ∼ = M ( k,n − k ) . When k ≤ n/
2, we know that M ( k,n − k ) ∼ = S ( n ) ⊕ S ( n − , ⊕ · · · ⊕ S ( n − k,k ) . We alsoknow that M ( k,n − k ) ∼ = M ( n − k,k ) . Thus, if we let U kj denote the submodule of G k that is isomorphic to the Specht module S ( n − j,j ) , we then have G = U ⊕ U G = U ⊕ U ⊕ U G = U ⊕ U ⊕ U ⊕ U ... G ⌊ n/ ⌋ = U ⌊ n/ ⌋ ⊕ U ⌊ n/ ⌋ ⊕ U ⌊ n/ ⌋ ⊕ · · · ⊕ U ⌊ n/ ⌋⌊ n/ ⌋ ... G n − = U n − ⊕ U n − ⊕ U n − G n − = U n − ⊕ U n − G n = U n . We may therefore write G as a direct sum of the irreducible U kj , where G = ⌊ n/ ⌋ M k =1 k M j =0 U kj M n M k> ⌊ n/ ⌋ n − k M j =0 U kj which is essentially equation (3) in [18].The above decomposition is helpful in cooperative game theory because the con-sensus since [39] is that linear and symmetric solution concepts are particularlyimportant, and these solution concepts coincide with the set of all R S n -modulehomomorphisms from G to M (1 ,n − . Moreover, because the G k have such sim-ple decompositions into irreducible submodules, these homomorphisms are easy todescribe.To explain, first recall that M (1 ,n − is the direct sum of U and U , where U ∼ = S ( n ) and U ∼ = S ( n − , . By Schur’s lemma, this means that any module homomorphismfrom G k ∼ = M ( k,n − k ) to M (1 ,n − must contain U kj in its kernel for all j ≥
2. Wemay therefore describe any homomorphism from G to M (1 ,n − by focusing only onhomomorphisms defined on the U k and U k as follows. Let T k : U k → U and T k : U k → U be any fixed isomorphisms. By a slightabuse of notation, we can view these isomorphisms as homomorphisms from G to M (1 ,n − by identifying them with ι ◦ T k ◦ p k and ι ◦ T k ◦ p k , respectively, where p k is the projection map onto U k , p k is the projection map onto U k , and ι and ι are the usual inclusion maps into M (1 ,n − .By Schur’s lemma, every module homomorphism from G to M (1 ,n − may be ex-pressed uniquely as a linear combination of the T k and T k . In other words, for everylinear symmetric solution concept ϕ : G → M (1 ,n − , there exist scalars c , . . . , c n and c , . . . , c n − such that ϕ = ( c T + · · · + c n T n ) + ( c T + · · · + c n − T n − ) . Working with and describing linear symmetric solution concepts therefore becomesa matter of manipulating and communicating the c j and c j . Moreover, there arechoices for the T k and T k that make certain properties of solution concepts easyto verify.For example, consider the following maps that appear in [21, 22]. First, let A ( v, k ) = (cid:18) nk (cid:19) − X S : | S | = k v ( S )denote the average value received by the coalitions of size k , and let γ ( k ) = (cid:18) n − k − (cid:19) . Then define T k by setting T k ( v ) i = k − A ( v, k ), and define T k by setting T k ( v ) i = γ ( k ) − X S : | S | = k and i ∈ S [ v ( S ) − A ( v, k )] . (A somewhat different approach is taken in [10], but the results are essentiallyscaled versions of the maps above.)As example of a property that can now be expressed in terms of the c j and c j ,consider the property of efficiency , where a solution concept ϕ is said to be efficient if ϕ ( v ) + · · · + ϕ ( v ) n = v ( N )for all v ∈ G . It turns out that a linear symmetric solution concept will be efficientif and only if c = · · · = c n − = 0 and c n = 1 (see Proposition 2 in [21]).As another example, a solution concept ϕ is said to be a marginal value if thereexist scalars m , . . . , m n such that ϕ ( v ) i = X S : i ∈ S m | S | ( v ( S ) − v ( S − i ))for all v ∈ G . In this case, it turns out we can recover ϕ by setting c k = k (cid:20) m k (cid:18) n − k − (cid:19) − m k +1 (cid:18) n − k (cid:19)(cid:21) and c k = γ ( k ) [ m k + m k +1 ] EPRESENTATION THEORY, VOTING THEORY, AND GAME THEORY 17 where we define m n +1 = 0 (see Proposition 1 in [22]).The above approach to describing linear symmetric solution concepts becomes evenmore powerful when we ask about solution concepts that have more than one suchproperty. For example, using the insights above, we may easily verify that theremust be precisely one linear symmetric solution concept that is also an efficientmarginal value. It is the well-studied Shapley value [39], which is obtained bysetting c = · · · = c n − = 0, c n = 1, and c = · · · = c n − = n − (see Section 3of [22]; see also Section 4.3 of [10]).Similarly, [10] uses this type of analysis to prove a simple and intuitive criterion fora solution concept to be a “self-dual” marginal value. To explain, they define the duality operator ∗ : G → G by setting ( ∗ v )( S ) = v ( N ) − v ( N − S ) for all v ∈ G .A linear symmetric solution ϕ is then said to be self-dual if ϕ ( ∗ v ) = ϕ ( v ) for all v ∈ G . It turns out that a linear symmetric marginal value will be self-dual if andonly if the m k defined above have the property that m j = m n − j − for all j < n (see Proposition 6 of [10]).By viewing a linear symmetric solution concept as a module homomorphism ϕ : G → M (1 ,n − , it is now easy to see how one might replace M (1 ,n − by M ( k,n − k ) toget linear symmetric solution concepts that assign payoffs to k -element subsets ofplayers instead of individual players. Such an idea was explored in [18], but there isroom for significant additional investigation, as the conjectures in that paper seemto imply. From a representation-theoretic point of view, we could easily begin byincluding in the discussion above module homomorphisms defined on the U kj when j ≥ useful structure. After all, this common kernel is the sum of all ofthe U kj where j ≥
2, and each of these submodules would begin to contributeto generalized solution concepts that involved payoffs for pairs, triples, and so on.Finding meaningful properties of the associated generalized solution concepts couldbe challenging, but doing so might ultimately prove to be illuminating, especiallyif they were to enhance our understanding of cherished properties (e.g., being amarginal value) that solution concepts might possess.The approach taken in [10] is extended in [37] to so-called games in partition func-tion form . In this setting, the value of a coalition S depends not just on thecoalition, but on how the players not in the coalition are themselves partitionedinto coalitions. In this way, an “embedded coalition” is a pair ( S, Q ) such that Q is a (set) partition of N and S ∈ Q . The dimension of the resulting game space ismuch larger, and its decomposition into irreducible submodules has only been donefor n = 3 and n = 4, which are (unsurprisingly) mostly U and U for those n . Cana decomposition of the general case be described? If so, can it be used to describeproperties (e.g., efficiency) of the associated generalized solution concepts? (See,for example, Corollary 4 in [37].) Finally, it is also possible to decompose non-cooperative games [15, 16]. In thesimplest version of this setting, players have strategies they individually decideupon, yielding varying payoffs. As is shown in [16], we can decompose such gamesinto “fully competitive” zero-sum components and “cooperative” components inwhich all of the players receive the same payoff. The main theorem in [15] takes thisfurther, while also giving a different interpretation to the zero-sum situation. As anexample of their results, they show that the space of 2 × Conclusion
As we have seen, the permutation representations arising from the action of thesymmetric group on tabloids provide a unifying framework for understanding andextending foundational ideas in both voting theory and game theory. There is,however, much more work that could be done, and we encourage interested read-ers to consider how they might use these and other ideas to contribute to ourunderstanding of voting theory, game theory, and other mathematical behavioralsciences.
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