THE BIOGRAPHY OF THE UNCOVERED SET

1
THE BIOGRAPHY OF THE UNCOVERED SET

• Nicholas R. Miller
• April 2007
• Creighton University Talks

2
Brief Personal Biography

• Undergraduate government major at Harvard.
• Likes of Gary King, Ken Shepsle, etc,
  unimaginable in the Harvard Government Department
  of the time.
• Two exceptions
• In my sophomore year, I discovered a brand new
  book by a Harvard economist Thomas Schelling
  Strategy of Conflict, which I found absolutely
  fascinating and quite unlike anything I had read
  before.
• Edward Banfield taught an upper-level/grad course
  on Political Economizing with Arrow, Black, and
  other on the syllabus.
• I sat in on class for first couple of days but
  did not take.
• Also no math beyond first-year calculus and
  analytical geometry, plus some statistics in
  latter part of graduate career.

3
Brief Personal Biography (cont.)

• Graduate study at Berkeley not Rochester
• There was no formal theory (or methodology)
  subfield at the time, and no formal theory course
  until 1969.
• Nevetheless, I became aware of Arrow, Riker,
  Downs, and Buchanan and Tullock within first few
  years, and wrote a seminar paper on Downs and
  BT.
• I hand in mind to write a theoretical
  dissertation further developing a theory of party
  competition.
• After surviving prelims, I undertook a extended
  (self-supervised) reading course in formal
  political theory
• I found the setup of Duncan Blacks Theory of
  Committees and Elections especially appealing.
• I became very interested in the cyclical majority
  phenomenon.

4
Brief Personal Biography (cont.)

• In 1968-69, I took the first offering of a course
  on Formal Models in Politics, taught by Robert
  Axelrod and Michael Leiserson
• We read a book by Otomar Bartos (a mathematical
  sociologist) on Simple Models of Group Behavior.
• It included a chapter on dominance structures
  in societies and employed adjacency matrices and
  directed graphs and tournaments in particular
  to represent and analyze such structures.
• A tournament is a complete asymmetric directed
  graph.

5
Majority Preference Tournaments

• It occurred to me that the eight three-person
  dominance patterns also represent the eight
  possible majority preference patterns over three
  alternatives (e.g., candidates) A, B, and C
• By (my) notational convention x is majority
  preferred to y (x P y) is indicated by an arrow
  from x to y (sometimes the reverse convention is
  used).
• And by (my) verbal language, x beats y (under
  majority rule).
• Note that, apart from the labelling of
  alternatives, the eight dominance patterns reduce
  to just two dominance structures
• transitive and cyclical.
• More generally given an odd number of voters with
  strong preferences,
• a tournament could be used to represent the
  majority preference relation, and
• some of the that deductions Bartos presented
  concerning social dominance structures (many of
  which were standard graph-theoretic theorems) had
  analogs for majority voting.
• In my seminar paper (and prospective dissertation
  chapter), I set out to use the tournament
  technology to systematize and extend Duncan
  Blacks work on committee voting.

6
Majority Preference Tournaments (cont.)

• I discovered with some disappointment that
  Michael Taylor had already proposed the idea of
  applying graph theory to social choice problems.
• Nevertheless, the tournament technology allowed
  me to make some modest advances in the Black
  program.
• For example, Black provided an example with three
  voters and four alternatives in a cycle of
  majority preference showing that, for each
  alternative, there was some voting order under
  ordinary committee procedure (what we now call
  amendment procedure) that would lead to its
  winning under sincere voting.
• Using the tournament technology, I was able to
  extend this result to any odd number of voters
  (indeed, the tournament device made it
  unnecessary to specify a particular number of
  voters) and to any number of alternatives in a
  cycle.
• This became Proposition 4 in Miller, 1977.
• Michael Taylor, Graph-Theoretical Approaches to
  the Theory of Social Choice. Public Choice,
  1968.
• Nicholas R. Miller, Nicholas R.
  Graph-Theoretical Approaches to the Theory of
  Voting. American Journal of Political Science,
  1977.

7
Majority Preference Tournaments

• Clearly, if we have an odd number of voters with
  strong preferences over any odd number m of
  alternatives, majority preference can be
  represented by a tournament with m vertices.
• The adjacent figure shows a tournament with eight
  vertices.
• Moreover, McGarveys Theorem tells us that every
  tournament we might construct represents majority
  preference for some profile of transitive
  preference orderings, one for each of some finite
  number of voters.
• David McGarvey, "A Theorem on the Construction
  of Voting Paradoxes," Econometrica, 1953.
• Thus any tournament we might construct represents
  a logically possible majority voting situation.

8
Majority Voting Tournaments (cont.)

• Since tournament may contain cycles, McGarveys
  Theorem also tells us that majority preference
  may be cyclical.
• In the adjacent figure, it is evident that there
  is a cycle of majority preference encompassing
  all eight alternatives.
• Tournament diagrams drove home a point that I had
  not before adequately appreciated
• if you arrange four or more points around a
  circle and draw arrows around the perimeter so
  as to create a cycle encompassing all the points,
  this cycle also has an internal structure,
• that is, arrows must also extend across the
  interior of the circle.

9
Majority Voting Tournaments (cont.)

• Moreover, given five points or more, cycles of
  given length may have different internal
  structures.
• Clearly there are a great many other eight-cycles
  (with the alternatives remaining in the same
  order in the cycles) that have different
  interior structures.
• These interior structures have important
  implications -- in particular, for the covering
  relation.

10
Condorcet Winners

• An alternative is a Condorcet Winner if and only
  if it beats every other alternative.
• The previous tournaments, with all-encompassing
  cycle, had no Condorcet winners.
• But the presence of a Condorcet winner does not
  preclude a (less than all-encompassing) cycle in
  majority preference.

11
Top Cycle Sets

• A top cycle set TC(X) is a minimal set of points
  such that every point in TC(X) beats every point
  outside TC(X).
• If TC(X) includes more than one point (if there
  is no CW), TC(X) must contain at least three
  points and a complete cycle (hence the name).
• Every tournament has a unique top cycle set.
• At one extreme, a CW (if it exists) is the TC
  set.
• At the other extreme, the TC may be
  all-encompassing.

12
All Possible Tournament Structures with Four
Alternatives.

• Given four alternatives (and apart from their
  labelling), there are just four possible
  tourna-ment structures
• fully transitive
• a Condorcet winner plus a three-element bottom
  cycle
• a three element top cycle plus a Condorcet
  loser and
• an all-encompassing four-element cycle.
• Increasing the number of alternatives beyond four
  greatly increases the number of distinct
  structural possibilities.

13
Amendment Procedure

• Amendment Procedure
• vote on two alternatives, then
• pair the winner with a third alternative, and
• so forth until all alternatives have entered the
  voting.
• The alternative that survives the final vote is
  the winner.
• Duncan Black called this ordinary committee
  procedure.
• Though it is not immediately apparent, this type
  of voting resembles the kind of amendment
  procedure used in Anglo-American legislatures.
• See Nicholas R. Miller, Committees, Agendas, and
  Voting, 1995, Chapter 2

14
Sincere Voting under Amendment Procedure

• Consider this four-element cyclical tournament.
• Apart from the labelling of alternatives, there
  is just one such tournament
• If the alternatives are voted on under amendment
  procedure in the order zyxv, v wins if voting is
  sincere.
• THEOREM. No point outside of TC(X) can win under
  any voting order.
• THEOREM. For every point x in TC(X) set, there
  is some voting order that makes x the sincere
  winner.
• PROOF IDEA. Arrange the TC alternatives in a
  cyclical fashion, and let the voting order follow
  the cycle upstream.
• COROLLARY. An alternative wins under every
  voting order if and only if it is the Condorcet
  Winner.

15
The Powell Amendment (1950s)

• Alternatives
• B the unamended bill federal aid to education
• BA the amended bill (no federal aid to
  segregated schools
• Q the status quo no federal to education
• Preference Profile
• N Dems S Dems Reps
• BA B Q
• B Q BA
• Q BA B
• Three bloc of voters, none a majority.
• Latin Square (non-value restricted) Matrix each
  alternative appears once in each rank.
• Cyclical majority preference B beats Q, Q beats
  BA, and BA beats B.

16
Sincere Voting on the Powell Amendment

• Under standard procedure, the first vote would be
  on the question of amending the bill implicitly
  B vs. BA.
• Then a vote would be taken on the question of
  passing the bill (as amended or not) implicitly
  B vs. Q or BA vs. Q, depending on the outcome of
  the first vote.
• Sincere voting the bill is amended (BA beats B)
  and then the bill fails (Q beats BA).
• But remember that B or BA could be passed under
  other voting orders (that might arise under other
  parliamentary situations, e.g., if the content of
  the Powell Amendment was incorporated in the
  original bill and/or if the status quo were
  different).

17
Strategic Voting on the Powell Amendment

• But shouldnt Northern Democrats (knowing
  everybodys prefer-ences or at least the M-P
  tournament) realize that the Powell Amendment is
  a killer amendment that will defeat the bill if
  it is so amended, and therefore vote
  strategically (contrary to their sincere
  preferences) against the amendment on the initial
  vote.
• Indeed, if they do so and nothing else changes, B
  wins.
• Are strategic countermoves available to either
  Southern Democrats or Republicans to defeat this
  Northern Democratic ploy?
• It is evident, that on the final (here second)
  vote, no one has a strategic reason to vote
  otherwise than sincerely, so
• strategic issues arise with respect to the first
  vote only.
• Southern Democrats are now getting their first
  preference, so they have no reason to change
  their first vote.
• Republicans are now being outvoted by Democrats
  on the first vote, so changing their first vote
  cant change the outcome.
• So strategic (as opposed to sincere) voting
  apparently changes the outcome from defeat of the
  bill (Q) to passage of the unamended bill (B).
• As under sincere voting, changing the voting
  order would probably change the strategic voting
  outcome.

18
Strategic Voting (cont.)

• The prior strategic analysis of the Powell
  Amendment voting was rather ad hoc.
• In a last-minute addendum to my seminar paper, I
  made a stab at deriving a Black-style results
  under strategic voting.
• Using a three-dimensional normal (or strategic or
  matrix) form for a voting game, I was able to
  show that the same anything in the Top Cycle can
  win result held under strategic voting with
  three blocs of voters and three alternatives.

19
Matrix Form of a Three-Voter, Three-Alternative
Voting Game (Successive Procedure)
20
Strategic Voting (cont.)

• I could also show that the voting orders that led
  to victory by a given alternative were different
  under sincere and strategic voting and
• in particular, under sincere voting the last
  alternative (e.g., Q) in a three-alternative
  cycle to enter the voting wins, but under
  strategic voting the first such alternative
  (e.g., B) wins.
• But I had no good idea as to how to proceed
  beyond three voters and three alternatives in the
  strategic voting case.

21
Strategic Voting Farquharson

• By the winter of 1970, Robin Farquharsons Theory
  of Voting had just been published, and I studied
  it eagerly.
• His Table of Results in Appendix I confirmed my
  results for both sincere and strategic (or
  sophisticated) voting in the three-voter
  three-alternative case.
• But I was completely mystified by his statement
  in the Preface that these results . . . can be
  readily extended to cover any desired number of
  voters and/or alternatives,
• I had been using essentially the same kind of
  strategic form matrix analysis that Farquharson
  was manifestly using in the body of his text, and
• I had found it to be impossibly burdensome to
  extend to a larger number of voters and/or
  alternatives.

22
Farquharson (cont.)

• But Farquharson also introduced a device he
  called an outcome tree device (a highly
  compressed extensive form that well call an
  agenda tree) to concisely represent binary
  voting games.

23
Farquharson (cont.)

• At some point that winter, while staring at such
  an agenda tree, I suddenly saw how backwards
  induction readily solved binary voting games
  when voters are strategic (and know each others
  preferences --- or at least the majority
  preference tournament).
• This logic was later definitively characterized
  by McKelvey and Niemi.
• Lets see how an agenda tree and backwards
  induction can quickly solve the Powell Amendment
  strategic voting game.
• Richard D. McKelvey and Richard G. Niemi. A
  Multistage Game Represen-tation of Sophisticated
  Voting for Binary Procedures. Journal of
  Economic Theory, 1978.

24
Powell Amendment Agenda Treewith Sophisticates
Equivalents
25
An Agenda Tree, M-P Tournament, and Backwards
Induction
26
Strategic (Sophisticated) Voting

• I now could readily determine the strategic
  voting outcome for any number of voters and
  relatively many alternatives under any voting
  order.
• Moreover, it was not even necessary to draw a new
  voting tree each time.
• I devised an algorithm by which I could quickly
  determine the strategic voting outcome for any
  M-P tournament and voting order.

27
Win Sets

• Note the term win set was introduced in the
  early 1980s by Shepsle and Weingast and has
  become very standard.

28
A Sophisticated Voting Algorithm forAmendment
Procedure

• Label the alternatives x1, x2, . . . , xm
  according to the order of voting.
• Examine W(xm) if W(xm) is empty, xm is the
  winning outcome otherwise, the sophisticated
  outcome belongs to W(xm).
• Examine TCW(xm) if this top cycle is a
  one-element set or has a top element, say xg,
  xg is the winning outcome otherwise, the
  sophisticated outcome belongs to TCW(xm) and
  depends on which alternative in this set comes
  last in the voting order designate this
  alternative xk.
• Alternative xk cannot be the outcome examine the
  intersection W(xk) with TCW(xm) and, more
  particularly, the top cycle of this intersection
  if this top cycle is a one-element set, say xh,
  xh is the outcome otherwise, the outcome belongs
  to this top cycle and depends on which
  alternative in it comes last in the voting order.
• And so forth. (Since A is a finite set, and
  since the set of possible decisions is reduced at
  each stage, the method must terminate at some
  stage.)
• In many cases, the strategic voting outcome is
  identified after applying only the two or three
  steps.

29
An Anomaly under Strategic Voting

• With these tools in hand, I enthusiastically
  began to derive Black-style propositions for
  strategic voting.
• An apparent anomaly quickly turned up
• when I extended the number of top cycle
  alternatives in a cycle from three to four, I
  discovered that there was always one alternative
  in the cycle that could not win under any voting
  order.

30
The Anomaly
31
The Algorithm Applied
32
The Strategic Voting Anomaly

• This was surprising and somewhat disconcerting,
  since I expected sincere and strategic voting
  results to run in parallel.
• Something was preventing one alternative from
  winning, regardless of the voting order.
• Evidently it had to do with the fact that, when
  internal structure is considered, the
  alternatives in a four-element cycle (unlike
  those in a three-element cycle) occupy distinct
  positions in the tournament structure.
• But this structural asymmetry did not preclude a
  degree of symmetry with respect to sincere voting
  outcomes, so the discrepancy remained puzzling.
• I further discovered that, given a cycle of five
  (or more) alternatives, different internal
  structures are possible, some of which made it
  impossible for certain alternatives to win while
  others did not.
• In my subsequent dissertation chapter, I simply
  presented the following proposition
• Under ordinary amendment procedure, there may
  be a motion alternative in the Condorcet top
  cycle set that is not the sophisticated voting
  decision under any voting order.

33
The Origins of the Uncovered Set

• In 1976, I had a revise and resubmit decision on
  a paper largely derived from this dissertation
  chapter.
• My main task was to revise the presentation of
  the material (on the basis of wise editorial
  guidance provided by AJPS editor Phillips
  Shively) rather than its substance.
• But in revisiting the analytical issues, I
  noticed two additional points concerning the
  strategic voting anomaly.
• An alternative y that could not win under any
  voting order (in the relatively small cyclic
  tournaments I was examining) was dominated in a
  particularly strong way by some other alternative
  x --- namely,
• not only did x beat y but also x beat everything
  that y beats.

34
The Origins of the Uncovered Set (cont.)

• The second point I noticed was that whenever x is
  unanimously preferred to y, then x dominates y in
  this strong sense.
• I incorporated the latter point into the revised
  paper.
• An alternative x Pareto-inferior is some other
  alternative y is unanimously preferred to x.
• THEOREM. Under amendment procedure, no
  Pareto-inferior alternative can win under
  strategic voting
• Note that this proposition is not trivial a
  Pareto-inferior alternative can win under sincere
  voting.
• Nicholas R. Miller, Graph-Theoretical Approaches
  to the Theory of Voting, American Journal of
  Political Science, 1977.

35
The Top Cycle Set May IncludePareto-Inferior
Alternatives
36
Covering

• I then turned quickly to explore this strong
  dominance more thoroughly and discovered that it
  had many interesting properties.
• Given an arrangement of x, y, and the
  alternatives beaten by y into a three-level
  structure with x at the top and with
  downward-pointing arrows representing majority
  preference, in my own mind it seemed natural to
  say that x covers y
• For better or worse, the terminology stuck.

37
The Origins of the Uncovered Set (cont.)

• This exploration led to a paper that I presented
  at the 1978 APSA meeting.
• It drew almost no attention (probably because it
  was one of six papers squeezed into a two-hour
  panel, one of which was Kenneth Shepsles early
  statement of his structure-induced equilibrium
  setup).
• Kenneth A. Shepsle, "Institutional Arrangements
  and Equilibrium in Multidimensional Voting
  Models," American Journal of Political Science,
  February, 1979
• It did attract attention once it was published.
• Nicholas R. Miller, A New Solution Set for
  Tournaments and Majority Voting, American
  Journal of Political Science, 1980.
• The joint appearance of these two papers at the
  APSA panel previewed two theoretical responses to
  majority rule choas, both of which have been
  put to extensive theoretical use.

38
Two Responses to Chaos

• The cyclical and seemingly chaotic nature of
  majority rule revealed by the theoretical work on
  voting and social choice of Plott, McKelvey,
  Schofield and others suggested that political
  processes rarely achieve equilibrium and might
  wander all over the place.
• But this theoretical conclusion was anomalous
  because actual political choice processes appear
  to be considerably more stable than the theory
  suggested.
• In the face of this anomaly, formal political
  theorists have pursued two different, though not
  mutually exclusive, lines of inquiry.
• The first, exemplified most notably by Shepsle,
  recognizes that political choice is always
  embedded in some kind of institutional structure,
  which may constrain processes so as to create
  (perhaps rather arbitrary) equilibria that would
  not otherwise exist.
• The second, in contrast, focuses directly on pure
  majority rule and seeks to find some deeper
  interior structure and coherence within the
  system of majority preference that may constrain
  or guide political choice processes, even in the
  face of apparent chaos and independently of
  particular institutional arrangements. The
  uncovered set (and the Banks set) have been
  leading contributions of the latter line of
  theorizing.

39
Covering and the Uncovered Set in the Finite
Alternative Case

• Except for one conjecture, Miller (1980) deals
  only with the finite as a opposed to spatial
  alternative set.
• Moreover, it assumes an odd number of voters with
  strong preferences, i.e., a majority preference
  tournament.
• In this context, covering can be defined simply
• x C y ltgt W(x) is a subset of W(y)
• which implies x P y so the inclusion must be
  strict.
• If majority rule is fully transitive, x C y ltgt
  x P y.

40
Covering and the Uncovered Set in the Finite
Alternative Case (cont.)

• x C y is a transitive subrelation of x P y.
• Accordingly, the covering relation has maximal
  elements, i.e., the uncovered set UC(X) x y
  C x for all y.
• If x is uncovered, then for all y either x P y or
  there is some z such that x P z P y the
  two-step or strategic principle.
• If there were no such z, y would cover x.
• UC(X) is a subset (perhaps proper) of TC(X), so
• if there is a Condorcet winner x, UC(X) x.
• Define x UP y x is unanimously preferred to y.
• PO(X) x y UP x for all y Pareto (or
  Pareto-optimal) set
• UC(X) is a subset of PO(X).

41
Covering and the Uncovered Set in the Finite
Alternative Case (cont.)

• In the absence of a Condorcet Winner, UC(X)
  includes at least three alternatives in a cycle.
• However, UC(X) may be a proper subset of TC(X),
  so to that extent UC has some cycle-busting
  (anti-chaos) power.
• TC(X) always contains a complete cycle but (if
  the number alternatives in TC(X) exceeds four)
  its degree of cyclicity can vary.
• The size of UC(X), relative to TC(X), depends on
  the degree of cyclicity TC(X) which depends on
  the interior structure of its complete cycle.

42
Degree of Cyclicity

• Consider a tournament with m alternatives in a
  complete cycle, so X TC(X).
• Its degree of cyclicity can be measured by the
  proportion of all triples of alternatives that
  are cyclic.
• A minimally cyclic tournament has m-2 cyclic
  triple,
• in which case, UC(X) includes just three
  alternatives.
• In a maximally cyclic tournament, every pair of
  alternatives belongs to a cyclic triple,
• so no covering exists and UC(X) TC(X).

43
A Minimally Cyclic Top Cycle
44
Another Minimally Cyclic Top Cycle
45
Maximally Cyclic Majority Preference
Distributive Politics not in Miller, 1980

• For every x and y such that y P x, we can form a
  cyclic triple, i.e., find some z such that x P z
  P y P x.
• Pure allocation three voters divide the dollar
  game
• x x1 x2 x3
  S x 1
• y x1 - 2c x2 c x3 c S y
  1
• z x1 - c x2 2c x3 - c S
  z 1
• x x1 x2 x3
  S x 1
• In this case X TC(X) UC(X), where X is the
  set of efficient allocations.
• Benjamin Ward, "Majority Rule and Allocation,"
  Journal of Conflict Resolution, 1961.
• David Epstein, "Uncovering some Subtleties of the
  Uncovered Set Social Choice Theory and
  Distributive Politics" Social Choice and
  Welfare, January, 1998.

46
The Uncovered Set and Voting Processesfrom
Miller (1980)

• Given strategic voting under an amendment agenda,
  every possible voting outcome belongs to the
  uncovered set UC(X).
• CONJECTURE every alternative in UC(X) is a
  possible strategic voting outcome.
• If voters bargain among themselves before voting,
  they will agree to enact some alternative that
  belongs to UC(X).
• Given competition between two victory-seeking
  political parties/candidates, both propose a
  uncovered platform, so the resulting government
  policy belongs to UC(X) and indeed to UCu(X).

47
Covering Depends on Environment

• The covering relation depends on irrelevant
  alternatives.
• Suppose m 2 and x P y. Then x C y.
• Now suppose a third alternative z is added to the
  tournament.
• If x P z, it remains true that x C y.
• But if y P z and z P y, then x no longer covers
  y.
• Conversely, even if x fails to cover y in a
  tournament T, x may cover y in a subtournament of
  T.

48
The Ultimately Uncovered Set

• The top cycle set is equal to its own closure,
• i.e., TCTC(X) TC(X).
• But, as we have seen, the uncovered is not in
  general equal to its own closure i.e., UC2(X)
  UCUC(X) may be a proper subset of UC(X).
• Likewise, UC3(X) may be a proper subset of
  UC2(X), and so forth.
• Given that the number of alternatives is finite,
  we must reach some u such that UCu(X) UCu-1(X),
• i.e., the ultimately uncovered set.
• Moreover (in the absence of a Condorcet Winner),
  at three least alternatives belong to UCu(X).

49
The Ultimately Uncovered Set
50
The Ultimately Uncovered Set (cont.)

• There was an incorrect theorem in Miller (1980)
• it claimed that TCUC(X) UC(X),
• i.e., that UC(X) always contains a complete
  cycle.
• A correct theorem says that TCUCu(X) UCu(X).
• Nicholas R. Miller, The Covering Relation in
  Tournaments Two Corrections, American Journal
  of Political Science, May 1983.
• The following example was provided in the
  correction.

51
The Ultimately Uncovered Set (cont.)
52
Precursors of the Uncovered Set

• LANDAUS THEOREM. In a tournament, a point with
  maximum score a king chicken beats every
  other point in one or two steps, i.e., is
  uncovered.
• H.G. Landau, On Dominance Relations and the
  Structure of Animal Societies, Bulletin of
  Mathematical Biophysics, 1953.
• Also
• Richard D. McKelvey, and Peter C. Ordeshook,
  Symmetric Spatial Games Without Majority Rule
  Equilibria, American Political Science Review,
  1976
• Peter C. Fishburn, Condorcet Social Choice
  Functions, SIAM Journal of Applied Mathematics,
  1977.
• Ronald A. Heiner, Length and Cycle
  Equalization, Journal of Economic Theory, 1981.

53
The Banks Set

• Banks was first show that the set of possible
  strategic voting outcomes under amendment
  procedure could be a proper subset of UC(X).
• Making use of a slightly different version of the
  sophisticated voting algorithm for amendment
  procedure due to Shepsle and Weingast, Banks
  showed that
• an alternative x is a possible voting outcome if
  and only if x is the maximal element in an
  externally stable chain.
• Kenneth A. Shepsle and Barry Weingast,
  "Uncovered Sets and Sophisticated Voting Outcomes
  with Implications for Agenda Institutions,"
  American Journal of Political Science, 1984.
• Jeffrey S. Banks, "Sophisticated Voting Outcomes
  and Agenda Control," Social Choice and Welfare,
  1985.
• Nicholas R. Miller, Bernard Grofman, Scott L.
  Feld, "The Structure of the Banks Set," Public
  Choice, 1990.

54
The Banks Set (cont.)

• Here is an intuitive statement of what this
  means
• We first pick some alternative x1 from the set of
  alternatives.
• We next pick a second alternative x2 such that x2
  P x1 and put x2 on top of x2 . .
• We next pick a third alternative x3 such that x3
  P x2 and x3 P x1 and put x3 on top of both x2
  and x1.
• In this manner, we are creating a chain or a
  cycle-avoiding trajectory, i.e., a sequence of
  alternatives such that each higher alternative
  in the sequence beats all the alternatives
  below it.
• We continue until we have built the chain with a
  top element xk such that we can expand the chain
  no further upward.
• We have now created an externally stable chain,
• i.e., every alternative outside the chain is
  beaten by some alternative in the chain.
• If this were not so, we could expand further
  upward.

55
The Banks Set (cont.)

• The top alternative xk of an externally stable
  chain
• is a Banks alternative and
• is the strategic voting outcome given by the
  voting order reflected in the chain, i.e., x1 is
  voted on last, x2 second to last, etc.
• Other inocuous may be inserted into the voting
  order.
• The Banks set is the set of all Banks
  alternatives.

56
The Uncovered Set in a Spatial Context

• Miller (1980) conjectured that
• in a spatial context, the uncovered set would be
  a relatively small subset of the Pareto set,
  centrally located in the distribution of ideal
  points, and that it would shrink in size as the
  number and diversity of ideal points increase.
• I also suggested that, given sophisticated voting
  under amendment procedure, the covering relation
  put important qualifications on McKelveys famous
  concluding comments about the implications of his
  global cycling theorem for agenda control.
• If the Chairman has complete control over the
  agenda, he can construct an agenda which will
  arrive at any point in the space.
• Richard D. McKelvey, Intransitivities in
  Multidimensional Voting Models and Some
  Implications for Agenda Control, Journal of
  Economic Theory, 1976.

57
The Uncovered Set in a Spatial Context
58
McKelveys demonic Chairman must construct a
(more or less) minimally cyclic agenda, and he
fails to get his way if he must announce his
agenda in advance and voting is sophisticated.
59
The Uncovered Set in a Spatial Context

• At the time I did not have the analytical tools
  at hand to pursue covering in a spatial context
  effectively.
• I sent my uncovered set paper to Richard
  McKelvey and invited him to apply his expertise
  to the problem.
• I like to think that this helped lead to McKelvey
  (1986).
• Richard D. McKelvey, Covering, Dominance, and
  Institution Free Properties of Social Choice,
  American Journal of Political Science, 1986.

60
Review The Covering Relation

• Alternative x covers alternative y iff
• x beats y, and
• x beats every alternative that y beats, so
• W(x) is a proper subset of W(y).
• The covering relation is transitive so
• maximal (uncovered) alternatives exist under
  relevant circumstances.
• Strategic Property an uncovered point beats
  every other alternative in no more than two steps.

61
Review The Covering Relation (cont.)

• Given finite alternatives and a majority
  preference tournament, UC(X)
• coincides with the Condorcet winner (if it
  exists)
• is a subset of the top cycle set and
• is a subset of the Pareto set.
• The size of UC(X) depends on the degree of
  intransitivity in the tournament.

62
Review The Covering Relation (cont.)

• In a two-dimensional spatial context (with
  Euclidean preferences), the same three properties
  hold.
• However, the second loses all of its punch, since
  (in the absence of Plott symmetry and a
  Condorcet winner) the top cycle encompasses the
  entire space.
• Moreover, while majority rule in a
  two-dimensional space is almost always cyclical,
• its degree of cyclicity varies with the nature of
  the ideal point (preference) configuration, and
• it is never maximally cyclic.
• Cyclicity increases with the dimensionality of
  the space, and
• is maximal when the number of dimensions equals
  the number of voters (as in the distributive
  case).
• However, in a spatial context one additional
  bound on the uncovered set was established by
  McKelvey.

63
In a (two-dimensional Euclidean) spatial context

• In the unlikely event that a Condorcet winner
  exists, the uncovered set coincides with it (as
  in non-spatial context).
• The uncovered set lies within the Pareto set (as
  in non-spatial context).
• The uncovered set lies within a circle centered
  on the yolk with a radius four times that of the
  yolk (McKelvey, 1986).

64
The Case of Three Voters

• Hartley and Kilgour (1987) established precise
  boundaries on the uncovered set for
  configurations of three voters with Euclidean
  preferences in a two-dimensional space.
• In the event ideal points form the vertices of an
  equilateral triangle, the uncovered set coincides
  with the Pareto set.
• Otherwise, the uncovered set excludes portions of
  the Pareto triangle in the vicinity of the one
  (if the Pareto triangle is acute) or two (if it
  is obtuse) relatively extreme ideal points.
• An implication of their analysis was that, at
  least in the three-voter case, even the
  conjunction of the Pareto bound and McKelveys 4r
  bound is overgenerous.
• Richard Hartley and D. Marc Kilgour, The
  Geometry of the Uncovered Set, Mathematical
  Social Sciences, 1987.
• Scott L. Feld, Bernard Grofman, Richard Hartley,
  Marc Kilgour, Nicholas R. Miller, and Nicolas
  Noviello, The Uncovered Set in Spatial Voting
  Games, Theory and Decision, 1987.

65
Uncovered Set with n 3
66
Almost Collinear n 3
67
The Size of the Yolk

• When the concept was first propounded, there was
  a widespread intuition that the yolk
• is centrally located relative to the
  configuration of ideal points, and
• tends to shrink in size as the number and
  diversity of voters increases.
• However, it was difficult to confirm this
  intuition or even to state it in a theoretically
  precise fashion.
• Feld et al. (1988) took a few very modest first
  steps.
• Tovey (1990) took a considerably larger step by
  showing that, if ideal point configurations are
  random samples drawn from a centered continuous
  distribution, the expected yolk radius approaches
  zero as the number of ideal points increases
  without limit.
• Scott L. Feld, Bernard Grofman, and Nicholas R.
  Miller, Centripetal Forces in Spatial Voting
  Games On the Size of the Yolk, Public Choice,
  1988.
• Craig A. Tovey, The Almost Surely Shrinking
  Yolk, Naval Postgraduate School, Monterey,
  California, October 1990.

68
In Search of the Uncovered Set

• Until recently one major problem pertaining to
  the uncovered set in the spatial context
  remained.
• In the context of spatial voting games of two or
  more dimensions, voting theorists have had only
  incomplete or rough knowledge concerning the
  location, size, and shape of the uncovered set.
• This problem motivated Bianco, Jeliazkov, and
  Sened BJS to employ a grid-search computational
  algorithm to generate pictures of uncovered sets
  in a variety of spatial voting scenarios.
• William T. Bianco, Ivan Jeliazkov, and Itai
  Sened, The Uncovered Set and the Limits of
  Legislative Action, Political Analysis, 2004.

69
BJS Figure 1 Computing the Uncovered Set
70
BJS Figure 5 Computing the Uncovered Set
(cont.)
71
BJS Figure 2 Computing the Uncovered Set
(cont.)
72
BJSs Theoretical Claims

• Based on the computational results displayed in
  their figures, BJS made three theoretical claims
  concerning the location and size of the uncovered
  set.
• The uncovered set can be much larger than our
  expectations based on conventional wisdom and
  previous work, as all their figures seem to
  illustrate.
• The uncovered set is not necessarily centrally
  located. If ideal points are polarized (as in
  the contemporary House), the uncovered set does
  not lie in the center of the distribution of
  legislators ideal points but is skewed toward
  the majority caucus, as illustrated by BJS
  Figure 5.
• The size, shape, and location of the uncovered
  set are very sensitive to the distribution of
  ideal points.
• With respect to size, this sensitivity is quite
  dramatically illustrated by their Figure 2 and is
  less dramatically illustrated by comparing panels
  in Figure 5.
• With respect to location, such sensitivity is
  illustrated by the first panel of their Figure 4
  and by a comparison of the last two panels of
  their Figure 5.

73
Observations on BJS Figure 2

• BJS Figure 2 is distinctive in that the uncovered
  set appears to have straight line boundaries that
  coincide with certain median lines.
• Furthermore, in several of the panels the
  uncovered set appears to be similar to the
  Hartley-Kilgour construction for the three-voter
  case in some way, the two additional ideal
  points (to the left and right) have no effect on
  the size and location of the uncovered set.

74
BJS and NRM

• Bianco and I interacted fairly extensively as BJS
  put their paper together comments and I was a
  referee for several journals.
• When it was accepted by Political Analysis, its
  editor (Bob Erikson) asked whether I would be
  interested in writing a commentary on it.

75
NRM and Joseph Godfrey

• Shortly thereafter, I met up with Joseph Godfrey,
  who made (an early version of) his CyberSenate
  software available to me.
• Godfrey modified CyberSenate to incoporate a
  procedure (broadly similar to BJSs) to make
  uncovered set calculations.
• But CyberSenate also has a much wider range of
  capabilities.
• My commentary on BJS ended up as a substantial
  paper that also served to showcase CyberSenate.
• Nicholas R. Miller, In Search of the Uncovered
  Set, Political Analysis, 2007.

76
CYBERSENATE

• CyberSenate was developed by Joseph Godfrey of
  the WinSet Group, LLC.
• It can create configurations of ideal points
• by point and click methods,
• generate them by Monte Carlo methods, or
• derive them from empirical data.
• Indifference curves, median lines, Pareto sets,
  win sets, yolks, cardioid bounds on win sets,
  uncovered set approximations, and other
  constructions can be generated on screen.
• CyberSenate produced some of the preceding and
  many of the following figures.

77
CyberSenate for me is a dream come true --
compare the figures below with those that follow.
78
Reminder The One-Dimensional Case

• In the one-dimensional cases, there is no chaos
  to be straightened out by covering.
• In a one-dimensional context with (standard
  assumptions about preferences),
• (strict) majority preference is fully transitive
• if x beats y, W(x) is always a subset of W(y) so
• covering is identical to (strict) majority
  preference.
• If the number of voters is odd, a Condorcet
  Winner always exists.
• It corresponds to the median ideal point in the
  one-dimensional space.
• This is Duncan Blacks Median Voter Theorem.
• Duncan Black, On the Rationale of Group
  Decision-Making, Journal of Political Economy,
  1948.
• Duncan Black, The Theory of Committees and
  Elections, Cambridge University Press, 1958.

79
Theory of Two-Dimensional Spatial Voting

• In the spatial context, we refer to alternatives
  as points.
• The set X of all alternatives is the set of all
  points in the space.
• There is a finite odd number n gt 3 of voters with
  Euclidean preferences.
• Each voter i
• has an ideal point xi in the space, and
• prefers any point closer to his ideal point to
  one that is more distant.
• This implies that the set of points Pi(x) that i
  prefers to x is the set of points bounded by a
  circle that is centered on xi and passing through
  x.
• Indifference curves are concentric circles around
  ideal points).
• The Pareto set is the convex hull of voter ideal
  points.

80
The Pareto Set
81
Theory (cont.)

• If some majority of m (n1)/2 voters prefers x
  to y, I say x beats y.
• The win set W(x) is the set of all points in X
  that beat x.
• The set of points that a particular majority of
  voters prefers to x is the intersection of all
  sets Pi(x) such that i belongs to that majority.
  
• W(x) is the union all such majority preference
  sets.
• Thus the boundary of a win set is everywhere
  demarcated by segments of individual voter
  indifference curves (segments of circles in the
  Euclidean context).
• In a spatial context, x beats essentially all
  points not in W(x).
• There are some majority preference ties but, in
  order to simplify exposition, I overlook
  technical issues pertaining to points that lie on
  the boundaries of sets.

82
Indifference Curves and the Win Set of a Point
Inside the Pareto Set
83
Indifference Curves and the Win Set of a Point
Outside the Pareto Set
84
Ditto with Five Ideal Points
85
Collinearity

• A configuration of ideal points diverse if no two
  ideal points precisely coincide.
• A key feature of a spatial voting game is whether
  the configuration of ideal points exhibits
  collinearities
• that is, whether three or more ideal points lie
  precisely on the same straight line.
• Collinearity always exists when ideal points
  coincide but clearly may be found in diverse
  configurations as well.
• Non-diversity and collinearity may both be deemed
  exceptional in the sense that, if hypothetical
  ideal points were randomly thrown into a policy
  space, non-diversity and collinearity would
  almost never occur.
• Of course, we can (and will) deliberately
  contrive non-diverse and collinear configurations
  e.g., BJS Figure 2.
• In empirical work, where ideal point locations
  estimated from interest group rating scales or
  similar data are typically expressed in whole
  numbers, it is likely that several legislators
  have identical scores on a given dimension,
  producing non-diversity and other collinearities.
• Collinearity produces a variety of peculiarities
  in particular, the invisible voter phenomena
  discussed in the next section.

86
Median Lines

• A any straight line L partitions the set of voter
  ideal points into three subsets
• those that lie on one side of L,
• those that lie on the other side of L, and
• those that lie on L itself.
• If it partitions the ideal points so that no more
  than half of the ideal points lie on either side,
  L is a median line, which we henceforth label M.
  
• Every ideal point lies on some median line and,
• if n is odd,
• every median line M passes through some ideal
  point,
• fewer than half of the ideal points lie on either
  side of M, and
• no other median line is parallel to M.

87
A (Non-Limiting) Median Line
88
Limiting Median Lines

• If n is odd, a typical median passes through just
  one ideal point.
• A limiting median line passes through two or more
  ideal points.
• Typically pairs of limiting median lines pass
  through a given ideal point, with non-limiting
  median lines sandwiched them.
• A median line that passes through the three or
  more (necessarily collinear ideal points) is a
  stand-alone limiting median line in which the
  sandwich of non-limiting median lines is
  reduced to zero thickness.

89
A Limiting Median Line
90
Limiting and Non-Limiting Median Lines
91
A Stand-Alone Limiting Median Line
92
Induced Ideal Points

• Each voter i has an induced ideal point, i.e., a
  most preferred point, on any line L.
• Given Euclidean preferences, voter is induced
  ideal point is the point on L closest to xi,
  i.e., the intersection of L with the line through
  xi perpendicular to L.
• The n induced ideal points appear on L in some
  (possibly weak) order and (since n is odd) we can
  identify the median induced ideal point(s) on L.
• Voter is induced preferences over L have the
  standard Euclidean property.

93
Induced Ideal Points
94
The Median Line Perpendicular to L

• The perpendicular line through the median induced
  point on L is itself the unique median line
  perpendicular to L.
• By standard Euclidean median voter logic, a point
  x on L is beaten by another point y on L if and
  only if y lies in the interval between x and its
  reflection point x' such that x and x' are
  equidistant from the median induced ideal point
  on L.
• In the event that x coincides with the median
  induced ideal point, x beats every other point on
  L.

95
The Median Induced Ideal Point on L
96
A Condorcet Winner in Two-Dimensional Space

• This last consideration implies that, if a point
  x lies off any median line M, x is beaten by
  points on M.
• It follows that a point x in the space is
  unbeaten (and a Condorcet winner) if and only if
  it lies on every median line,
• which is possible if and only if all median lines
  intersect at the single point x (which itself
  must be an ideal point).
• This in turn can hold only in the presence of a
  sufficient (and unlikely) degree of Plott
  symmetry in the configuration of ideal points.
• Such a Condorcet winner is structurally
  unstable.
• Charles R. Plott, A Notion of Equilibrium and
  Its Possibility Under Majority Rule, American
  Economic Review, 1967.
• James M. Enelow and Melvin J. Hinich, On Plott's
  Pairwise Symmetry Condition for Majority Rule
  Equilibrium, Public Choice,1983.

97
Full Plott Symmetry (Showing Limiting Median
Lines)
98
Sufficient Plott Symmetry(with Non-diverse Ideal
Points)
99
The Yolk

• The yolk is the set of points bounded by the
  smallest circle that intersects every median
  line.
• The location of the yolk is given by its center
  c, which indicates the generalized center (in the
  sense the median) of the configuration of ideal
  points.
• The size of the yolk is given by its radius r,
  which indicates the extent to which the
  configuration of ideal points departs from one
  exhibiting a degree of Plott symmetry sufficient
  for the existence of a Condorcet winner.
• The yolk circle is inscribed within the yolk
  triangle formed by three median lines to which
  the circle is tangent.
• Typically, but not always due to Tovey
  anomalies, these are limiting median lines.

100
A Yolk with Zero Radius
101
A Yolk with Small Radius
102
A Yolk with Large Radius
103
Win Sets in Two-Dimensional Space

• To get a preliminary sense of the size, shape,
  and location of a win set W(x) in the spatial
  context, consider the special case in which there
  is only one voter i.
• In this event, the center of the yolk is xi,
• the yolk radius is zero,
• and W(x) coincides with Pi(x),
• which (given Euclidean preferences) is the circle
  with a center at c and a radius of d, where d is
  the distance from x to c.
• Whenever the yolk has zero radius, W(X) Pi(x),
  where i is the central voter.

104
A Win Set with a Single Voter
105
Plott Symmetry and a Circular Win Set
106
Bounds on Win Sets

• In the general case of multiple voters with
  diverse ideal points and for a point x outside
  the yolk, the boundary of W(x) is approximated by
  the same circle centered on c with a radius of
  d.
• The accuracy of this approximation depends on the
  size of the yolk, as given by its radius r,
  according to this 2r Rule
• point x beats all points more than d  2r from
  the center of the yolk, and x is beaten by all
  points closer than d - 2r to the center of the
  yolk
• put otherwise, the boundary of W(x) everywhere
  falls between two circles centered on the yolk
  with radii of d 2r and d - 2r respectively (the
  inner constraint disappears if d lt r and the two
  circles coincide if r 0)
• Tighter bounds on W(x), especially in the
  vicinity of x itself, are provided by the outer
  and inner cardioids, originally described in
  Ferejohn et al. (1984).
• John A. Ferejohn, Richard D. McKelvey, and Edward
  W. Packel, Limiting Distributions for
  Continuous State Markov Voting Models, Social
  Choice and Welfare, 1984.

107
The Circular Bound on a Win Set
108
Circular Bound on a Win Set (d lt 2r)
109
Cardioid Bound on a Win Set
110
Global Cycling

• For any point x, construct the circle with its
  center at c that passes through x.
• The previous figures (and the next ones)
  illustrate the following proposition (Miller et
  al., 1989.
• THEOREM. In the absence of Plott symmetry, for
  any point x there is some other point y that both
• beats x, and
• is further from c than x is.
• Indeed, the boundary of a win set intersects the
  cardioid bound at three points.
• Nicholas R. Miller, Bernard Grofman, and Scott
  L. Feld, The Geometry of Majority Rule,
  Journal of Theoretical Politics, 1989.

111
W(x) Intersects the Cardioid at Three Points
112
In the Absence of Plott Symmetry, All Win Sets
Are At Least Slightly Chaotic
113
Even Slightly Chaotic Win Sets Produces Global
Cycling

• However, the majority preference path from a
  centrally located point to an extreme one may
  take many steps.
• The distance outwards take at each step cannot
  exceed 2r (by the 2r Rule).
• The number of steps required therefore depends on
  the size yolk, i.e., the yolk radius r.
• Put otherwise, it depends on the extent to which
  the configuration of ideal points deviates from
  Plott symmetry.

114
Orderly Win Sets

• In two dimensions, a win set W(x) is orderly if
  it is a subset of some open half space about x.
• This implies that there is some voter i such
  that, within the vicinity of x, W(x) is a subset
  of Pi(x), and likewise for the win sets of other
  points in the vicinity of x.
• This guarantees that majority preference, while
  (almost always) globally cyclical, is transitive
  (being consistent with is preferences) in the
  vicinity of x.
• This also implies that local covering operates
  in the vicinity of x.
• That is, if y and z are both in the vicinity of
  x, and y P z, then
• W(y) U N(x) is a subset of W(z) U N(x), where
  N(x) is a small area surrounding x (a
  neighborhood of x).
• However, outside of N(x), there may be points
  that z beats but beat y,
• so y does not (globally) cover z.

115
An Orderly Win Set with Locally Transitive
Majority Preference
116
Local Covering vs. (Global) Covering
117
Disorderly Win Sets

• A win set W(x) is disorderly if it is not a
  subset of any half space about x, but rather has
  multiple small petals that point in all
  directions from x.