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 B&T. • 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 Black’s 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 Black’s 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, McGarvey’s 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, McGarvey’s 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] • B+A: the amended bill (no federal aid to segregated schools] • Q: the status quo [no federal to education] Preference Profile: N DemsS DemsReps B+A B Q B Q B+A Q B+A 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 B+A, and B+A 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. B+A]. • Then a vote would be taken on the question of passing the bill (as amended or not) [implicitly B vs. Q or B+A vs. Q, depending on the outcome of the first vote]. • Sincere voting: the bill is amended (B+A beats B) and then the bill fails (Q beats B+A). • But remember that B or B+A 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 shouldn’t Northern Democrats (knowing everybody’s 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 can’t 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 Farquharson’s 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 we’ll 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 other’s preferences --- or at least the majority preference tournament). • This logic was later definitively characterized by McKelvey and Niemi. • Let’s 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, . . . , xmaccording 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 TC[W(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 TC[W(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 TC[W(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 Shepsle’s 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 beenleading 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 <==> 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 <==> 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 setUC(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 everyx and y such that y P x, we can form a cyclic triple, i.e., find somez such that x P z P y P x. • Pure allocation: three voters “divide the dollar” game x: x1x2x3Σx = 1 y: x1 - 2c x2 + c x3 + c Σy = 1 z: x1 - c x2 + 2c x3 - c Σz = 1 x: x1x2x3Σx = 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., TC[TC(X)] = TC(X). • But, as we have seen, the uncovered is not in general equal to its own closure i.e., UC2(X) = UC[UC(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 TC[UC(X)] = UC(X), • i.e., that UC(X) always contains a complete cycle. • A correct theorem says that TC[UCu(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.