Rankings for Bipartite Tournaments via Chain Editing
RRankings for Bipartite Tournaments via Chain Editing
Joseph Singleton
Cardiff [email protected]
Richard Booth
Cardiff [email protected]
ABSTRACT
Ranking the participants of a tournament has applications in voting,paired comparisons analysis, sports and other domains. In this pa-per we introduce bipartite tournaments , which model situations inwhich two different kinds of entity compete indirectly via matchesagainst players of the opposite kind; examples include education(students/exam questions) and solo sports (golfers/courses). In par-ticular, we look to find rankings via chain graphs , which correspondto bipartite tournaments in which the sets of adversaries defeatedby the players on one side are nested with respect to set inclusion.Tournaments of this form have a natural and appealing ranking as-sociated with them. We apply chain editing โ finding the minimumnumber of edge changes required to form a chain graph โ as a newmechanism for tournament ranking. The properties of these rank-ings are investigated in a probabilistic setting, where they arise asmaximum likelihood estimators, and through the axiomatic methodof social choice theory. Despite some nice properties, two problemsremain: an important anonymity axiom is violated, and chain edit-ing is NP -hard. We address both issues by relaxing the minimisationconstraint in chain editing, and characterise the resulting rankingmethods via a greedy approximation algorithm. A tournament consists of a finite set of players equipped with a beating relation describing pairwise comparisons between each pairof players. Determining a ranking of the players in a tournament hasapplications in voting in social choice [5] (where players representalternatives and ๐ฅ beats ๐ฆ if a majority of voters prefer ๐ฅ over ๐ฆ ),paired comparisons analysis [14] (where players may representproducts and the beating relation the preferences of a user), searchengines [23], sports tournaments [3] and other domains.In this paper we introduce bipartite tournaments , which consistof two disjoint sets of players ๐ด and ๐ต such that comparisons onlytake place between players from opposite sets. We consider rankingmethods which produce two rankings for each tournament โ one foreach side of the bipartition. Such tournaments model situations inwhich two different kinds of entity compete indirectly via matchesagainst entities of the opposite kind. The notion of competitionmay be abstract, which allows the model to be applied in a varietyof settings. An important example is education [15], where ๐ด rep-resents students, ๐ต exam questions, and student ๐ โbeatsโ question ๐ by answering it correctly. Here the ranking of students reflectstheir performance in the exam, and the ranking of questions reflectstheir difficulty . The simultaneous ranking of both sides allows oneranking to influence the other; e.g. so that students are rewardedfor correctly answering difficult questions. This may prove partic-ularly useful in the context of crowdsourced questions provided by students themselves, which may vary in their difficulty (see forexample the PeerWise system [8]).A related example is truth discovery [16, 21]: the task of findingtrue information on a number of topics when faced with conflictingreports from sources of varying (but unknown) reliability. Manytruth discovery algorithms operate iteratively, alternately estimat-ing the reliability of sources based on current estimates of the trueinformation, and obtaining new estimates of the truth based onsource reliability levels. The former is an instance of a bipartitetournament; similar to the education example, ๐ด represents datasources, ๐ต topics of interest, and ๐ defeats ๐ by providing true infor-mation on topic ๐ (according to the current estimates of the truth).Applying a bipartite tournament ranking method at this step maytherefore facilitate development of difficulty-aware truth discoveryalgorithms, which reward sources for providing accurate informa-tion on difficult topics [12]. Other application domains include theevaluation of generative models in machine learning [19] (where ๐ด represents generators and ๐ต discriminators) and solo sports contests(e.g. where ๐ด represents golfers and ๐ต golf courses).In principle, bipartite tournaments are a special case of gener-alised tournaments [7, 14, 22], which allow intensities of victoriesand losses beyond a binary win or loss (thus permitting draws ormultiple comparisons), and drop the requirement that every playeris compared to all others. However, many existing ranking methodsin the literature do not apply to bipartite tournaments due to theviolation of an irreducibility requirement, which requires that thetournament graph be strongly connected. In any case, bipartitetournament ranking presents a unique problem โ since we aim torank players with only indirect information available โ which webelieve is worthy of study in its own right.In this work we focus particularly on ranking via chain graphs and chain editing . A chain graph is a bipartite graph in which theneighbourhoods of vertices on one side form a chain with respectto set inclusion. A (bipartite) tournament of this form representsan โidealโ situation in which the capabilities of the players areperfectly nested: weaker players defeat a subset of the opponentsthat stronger players defeat. In this case a natural ranking can beformed according to the set of opponents defeated by each player.These rankings respect the tournament results in an intuitive sense:if a player ๐ defeats ๐ and ๐ โฒ ranks worse than ๐ , then ๐ must defeat ๐ โฒ also. Unfortunately, this perfect nesting may not hold in reality:a weak player may win a difficult match by coincidence, and astrong player may lose a match by accident. With this in mind,Jiao et al. [15] suggested an appealing ranking method for bipartitetournaments: apply chain editing to the input tournament โ i.e. findthe minimum number of edge changes required to form a chaingraph โ and output the corresponding rankings. Whilst their workfocused on algorithms for chain editing and its variants, we look to a r X i v : . [ c s . M A ] J a n tudy the properties of the ranking method itself through the lensof computational social choice. Contribution.
Our primary contribution is the introduction of aclass of ranking mechanisms for bipartite tournaments defined bychain editing. We also provide a new probabilistic characterisationof chain editing via maximum likelihood estimation. To our knowl-edge this is the first in-depth study of chain editing as a rankingmechanism. Secondly, we introduce a new class of โchain-definableโmechanisms by relaxing the minimisation constraint of chain edit-ing in order to obtain tractable algorithms and to resolve the failureof an important anonymity axiom.
Paper outline.
In Section 2 we define the framework for bipartitetournaments and introduce chain graphs. Section 3 outlines howone may use chain editing to rank a tournament, and characterisesthe resulting mechanisms in a probabilistic setting. Axiomatic prop-erties are considered in Section 4. Section 5 defines a concretescheme for producing chain-editing-based rankings. Section 6 in-troduces new ranking methods by relaxing the chain editing re-quirement. Related work is discussed in Section 7, and we concludein Section 8. Note that some proofs are omitted in the body of thepaper and can be found in the appendix.
In this section we define our framework for bipartite tournaments,introduce chain graphs and discuss the link between them.
Following the literature on generalised tournaments [7, 14, 22],we represent a tournament as a matrix, whose entries representthe results of matches between participants. In what follows, [ ๐ ] denotes the set { , . . . , ๐ } whenever ๐ โ N . Definition 2.1. A bipartite tournament โ hereafter simply a tour-nament โ is a triple ( ๐ด, ๐ต, ๐พ ) , where ๐ด = [ ๐ ] and ๐ต = [ ๐ ] forsome ๐, ๐ โ N , and ๐พ is an ๐ ร ๐ matrix with ๐พ ๐๐ โ { , } for all ( ๐, ๐ ) โ ๐ด ร ๐ต . The set of all tournaments will be denoted by K .Here ๐ด and ๐ต represent the two sets of players in the tourna-ment. An entry ๐พ ๐๐ gives the result of the match between ๐ โ ๐ด and ๐ โ ๐ต : it is 1 if ๐ defeats ๐ and 0 otherwise. Note that wedo not allow for the possibility of draws, and every ๐ โ ๐ด facesevery ๐ โ ๐ต . When there is no ambiguity we denote a tourna-ment simply by ๐พ , with the understanding that ๐ด = [ rows ( ๐พ )] and ๐ต = [ columns ( ๐พ )] .The neighbourhood of a player ๐ โ ๐ด in ๐พ is the set ๐พ ( ๐ ) = { ๐ โ ๐ต | ๐พ ๐๐ = } โ ๐ต , i.e. the set of players which ๐ defeats. Theneighbourhood of ๐ โ ๐ต is the set ๐พ โ ( ๐ ) = { ๐ โ ๐ด | ๐พ ๐๐ = } โ ๐ด ,i.e. the set of players defeating ๐ .Given a tournament ๐พ , our goal is to place a ranking on each of ๐ด and ๐ต . We define a ranking operator for this purpose. Definition 2.2. An operator ๐ assigns each tournament ๐พ a pair ๐ ( ๐พ ) = (โชฏ ๐๐พ , โ ๐๐พ ) of total preorders on ๐ด and ๐ต respectively. Note that ๐ด and ๐ต are not disjoint as sets: is always contained in both ๐ด and ๐ต , forinstance. This poses no real problem, however, since we view the number merely a label for a player. It will always be clear from context whether a given integer shouldbe taken as a label for a player on the ๐ด side or the ๐ต side. ๐ข ๐ข ๐ข ๐ฃ ๐ฃ ๐ฃ ๐ฃ Figure 1: An example of a chain graph
For ๐, ๐ โฒ โ ๐ด , we interpret ๐ โชฏ ๐๐พ ๐ โฒ to mean that ๐ โฒ is ranked atleast as strong as ๐ in the tournament ๐พ , according to the operator ๐ (similarly, ๐ โ ๐๐พ ๐ โฒ means ๐ โฒ is ranked at least as strong as ๐ ). Thestrict and symmetric parts of โชฏ ๐๐พ are denoted by โบ ๐๐พ and โ ๐๐พ ,As a simple example, consider ๐ count , where ๐ โชฏ ๐ count ๐พ ๐ โฒ iff | ๐พ ( ๐ )| โค | ๐พ ( ๐ โฒ )| and ๐ โ ๐ count ๐พ ๐ โฒ iff | ๐พ โ ( ๐ )| โฅ | ๐พ โ ( ๐ โฒ )| . Thisoperator simply ranks players by number of victories. It is a bipartiteversion of the points system introduced by Rubinstein [20], andgeneralises Copelandโs rule [5].
Each bipartite tournament ๐พ naturally corresponds to a bipartitegraph ๐บ ๐พ , with vertices ๐ด โ ๐ต and an edge between ๐ and ๐ whenever ๐พ ๐๐ = The task of ranking a tournament admits a particularlysimple solution if this graph happens to be a chain graph . Definition 2.3 ([25]).
A bipartite graph ๐บ = ( ๐ , ๐ , ๐ธ ) is a chaingraph if there is an ordering ๐ = { ๐ข , . . . , ๐ข ๐ } of ๐ such that ๐ ( ๐ข ) โ ยท ยท ยท โ ๐ ( ๐ข ๐ ) , where ๐ ( ๐ข ๐ ) = { ๐ฃ โ ๐ | ( ๐ข ๐ , ๐ฃ ) โ ๐ธ } isthe neighbourhood of ๐ข ๐ in ๐บ .In other words, a chain graph is a bipartite graph where theneighbourhoods of the vertices on one side can be ordered so as toform a chain with respect to set inclusion. It is easily seen that thisnesting property holds for ๐ if and only if it holds for ๐ . Figure 1shows an example of a chain graph.Now, as our terminology might suggest, the neighbourhood ๐พ ( ๐ ) of some player ๐ โ ๐ด in a tournament ๐พ coincides withthe neighbourhood of the corresponding vertex in ๐บ ๐พ . If ๐บ ๐พ is achain graph we can therefore enumerate ๐ด as { ๐ , . . . , ๐ ๐ } suchthat ๐พ ( ๐ ๐ ) โ ๐พ ( ๐ ๐ + ) for each 1 โค ๐ < ๐ . This indicates that each ๐ ๐ + has performed at least as well as ๐ ๐ in a strong sense: everyopponent which ๐ ๐ defeated was also defeated by ๐ ๐ + , and ๐ ๐ + may have additionally defeated opponents which ๐ ๐ did not. Itseems only natural in this case that one should rank ๐ ๐ (weakly)below ๐ ๐ + . Appealing to transitivity and the fact that each ๐ โ ๐ด appears as some ๐ ๐ , we see that any tournament ๐พ where ๐บ ๐พ is achain graph comes pre-equipped with a natural total preorder on ๐ด ,where ๐ โฒ ranks higher than than ๐ if and only if ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) . Theduality of the neighbourhood-nesting property for chain graphsimplies that ๐ต can also be totally preordered, with ๐ โฒ ranked higherthan ๐ if and only if ๐พ โ ( ๐ ) โ ๐พ โ ( ๐ โฒ ) . Moreover, these total A total preorder is a transitive and complete binary relation. ๐ด โ ๐ต is the disjoint union of ๐ด and ๐ต , which we define as {( ๐, A) | ๐ โ ๐ด } โช{( ๐, B) | ๐ โ ๐ต } , where A and B are constant symbols. Note that this is a more robust notion of performance than comparing the neighbour-hoods of ๐ ๐ and ๐ ๐ + by cardinality , which may fail to account for differences in thestrength of opponents when counting wins and losses. reorders relate to the tournament results in an important sense: if ๐ defeats ๐ and ๐ โฒ ranks worse than ๐ , then ๐ must defeat ๐ โฒ also.That is, the neighbourhood of each ๐ โ ๐ด is downwards closed w.r.tthe ranking of ๐ต , and the neighbourhood of each ๐ โ ๐ต is upwardsclosed in ๐ด .Tournaments corresponding to chain graphs will be said to sat-isfy the chain property , and will accordingly be called chain tourna-ments . We give a simpler (but equivalent) definition which does notrefer to the underlying graph ๐บ ๐พ . First, define relations โฉฝ A ๐พ , โฉฝ B ๐พ on ๐ด and ๐ต respectively by ๐ โฉฝ A ๐พ ๐ โฒ iff ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) and ๐ โฉฝ B ๐พ ๐ โฒ iff ๐พ โ ( ๐ ) โ ๐พ โ ( ๐ โฒ ) , for any tournament ๐พ . Definition 2.4.
A tournament ๐พ has the chain property if โฉฝ A ๐พ isa total preorder.According to the duality principle mentioned already, the chainproperty implies that โฉฝ B ๐พ is also a total preorder. Note that therelations โฉฝ A ๐พ and โฉฝ B ๐พ are analogues of the covering relation fornon-bipartite tournaments [5]. Example 2.5.
Consider ๐พ = (cid:104) (cid:105) . Then ๐พ ( ) โ ๐พ ( ) โ ๐พ ( ) , so ๐พ has the chain property. In fact, ๐พ is the tournamentcorresponding to the chain graph ๐บ from Figure 1. We have seen that chain tournaments come equipped with naturalrankings of ๐ด and ๐ต . Such tournaments represent an โidealโ situation,wherein the abilities of the players on both sides of the tournamentare perfectly nested. Of course this may not be so in reality: thenesting may be broken by some ๐ โ ๐ด winning a match it oughtnot to by chance, or by losing a match by accident.One idea for recovering a ranking in this case, originally sug-gested by Jiao et al. [15], is to apply chain editing : find the minimumnumber of edge changes required to convert the graph ๐บ ๐พ into achain graph. This process can be seen as correcting the โnoiseโ in anobserved tournament ๐พ to obtain an ideal ranking. In this sectionwe introduce the class of operators producing rankings in this way. To define chain-editing in our framework we once again present anequivalent definition which does not refer to the underlying graph ๐บ ๐พ : the number of edge changes between graphs can be replacedby the Hamming distance between tournament matrices.
Definition 3.1.
For ๐, ๐ โ N , let C ๐,๐ denote the set of all ๐ ร ๐ chain tournaments. For an ๐ ร ๐ tournament ๐พ , write M ( ๐พ ) = arg min ๐พ โฒ โC ๐,๐ ๐ ( ๐พ, ๐พ โฒ ) โ K for the set of chain tournamentsclosest to ๐พ w.r.t the Hamming distance ๐ ( ๐พ, ๐พ โฒ ) = |{( ๐, ๐ ) โ ๐ด ร ๐ต | ๐พ ๐๐ โ ๐พ โฒ ๐๐ }| . Let ๐ ( ๐พ ) denote this minimum distance.Note that chain editing, which is NP -hard in general [15], amountsto finding a single element of M ( ๐พ ) . We comment further on thecomputational complexity of chain editing in Section 7. The follow-ing property characterises chain editing-based operators ๐ . Note that the ordering of the ๐ต s is reversed compared to the ๐ด s, since the larger ๐พ โ ( ๐ ) the worse ๐ has performed. The decision problem associated with chain editing โ which in tournament terms isthe question of whether ๐ ( ๐พ ) โค ๐ for a given integer ๐ โ is NP -complete [9]. (chain-min) For every tournament ๐พ there is ๐พ โฒ โ M ( ๐พ ) suchthat ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) . That is, the ranking of ๐พ is obtained by choosing theneighbourhood-subset rankings for some closest chain tournament ๐พ โฒ . Operators satisfying chain-min will be called chain-minimal . Example 3.2.
Consider ๐พ = (cid:104) (cid:105) . ๐พ does not have the chainproperty, since neither ๐พ ( ) โ ๐พ ( ) nor ๐พ ( ) โ ๐พ ( ) . The set M ( ๐พ ) consists of four tournaments a distance of 2 from ๐พ : M ( ๐พ ) = (cid:110)(cid:104) (cid:105) , (cid:104)
01 1 0 0 (cid:105) , (cid:104) (cid:105) , (cid:104) (cid:105)(cid:111) The corresponding rankings are ( , { } ) , ( , { }) , ( , { }) and ( , { } ) . Example 3.2 shows that there is no unique chain-minimal oper-ator, since for a given tournament ๐พ there may be several closestchain tournaments to choose from. In Section 5 we introduce a prin-cipled way to single out a unique chain tournament and therebyconstruct a well-defined chain-minimal operator. So far we have motivated chain-min as a way to fix errors in atournament and recover the ideal or true ranking. In this sectionwe make this notion precise by defining a probabilistic model inwhich chain-minimal rankings arise as maximum likelihood es-timates. The maximum likelihood approach has been applied for(non-bipartite) tournaments (e.g. the Bradley-Terry model [4, 14]),voting in social choice theory [10], truth discovery [24], belief merg-ing [11] and other related problems.In this approach we take an epistemic view of tournament rank-ing: it is assumed there exists a true โstate of the worldโ whichdetermines the tournament results along with objective rankings of ๐ด and ๐ต . A given tournament ๐พ is then seen as a noisy observation derived from the true state, and a maximum likelihood estimate is astate for which the probability of observing ๐พ is maximal.More specifically, a state of the world is represented as a vectorof skill levels for the players in ๐ด and ๐ต . Definition 3.3.
For a fixed size ๐ ร ๐ , a state of the world is atuple ๐ = โจ ๐ , ๐ โฉ , where ๐ โ R ๐ and ๐ โ R ๐ satisfies the followingproperties: โ ๐, ๐ โฒ โ ๐ด ( ๐ฅ ๐ < ๐ฅ ๐ โฒ = โ โ ๐ โ ๐ต : ๐ฅ ๐ < ๐ฆ ๐ โค ๐ฅ ๐ โฒ ) (1) โ ๐, ๐ โฒ โ ๐ต ( ๐ฆ ๐ < ๐ฆ ๐ โฒ = โ โ ๐ โ ๐ด : ๐ฆ ๐ โค ๐ฅ ๐ < ๐ฆ ๐ โฒ ) (2)where ๐ด = [ ๐ ] , ๐ต = [ ๐ ] . Write ฮ ๐,๐ for the set of all ๐ ร ๐ states.For ๐ โ ๐ด , ๐ฅ ๐ is the skill level of ๐ in state ๐ (and similarly for ๐ฆ ๐ ).These skill levels represent the true capabilities of the players in ๐ด and ๐ต in state ๐ : ๐ is capable of defeating ๐ if and only if ๐ฅ ๐ โฅ ๐ฆ ๐ .Note that (1) suggests a simple form of explainability : ๐ โฒ can only bestrictly more skilful than ๐ if there is some ๐ โ ๐ต which explains thisfact, i.e. some ๐ which ๐ โฒ can defeat but ๐ cannot ((2) is analogous forthe ๐ต s). These conditions are intuitive if we assume that skill levelsare relative to the sets ๐ด and ๐ต currently under consideration (i.e. Here ๐ ๐ ๐ is shorthand for the ranking ๐ โบ ๐ โบ ๐ of ๐ด , and similar for ๐ต .Elements in brackets are ranked equally. For simplicity we use numerical skill levels here, although it would suffice to have apartial preorder on ๐ด โ ๐ต such that each ๐ โ ๐ด is comparable with every ๐ โ ๐ต . hey do not reflect the abilities of players in future matches againstnew contenders outside of ๐ด or ๐ต ). Finally note that our states of theworld are richer than the output of an operator, in contrast to otherwork in the literature [4, 10, 14]. Specifically, a state ๐ containsextra information in the form of comparisons between ๐ด and ๐ต .Noise is introduced in the observed tournament ๐พ via false pos-itives (where ๐ โ ๐ด defeats a more skilled ๐ โ ๐ต by accident) and false negatives (where ๐ โ ๐ด is defeated by an inferior ๐ โ ๐ต bymistake). The noise model is therefore parametrised by the falsepositive and false negative rates ๐ถ = โจ ๐ผ + , ๐ผ โ โฉ โ [ , ] , which weassume are the same for all ๐ โ ๐ด . We also assume that noiseoccurs independently across all matches.
Definition 3.4.
Let ๐ถ = โจ ๐ผ + , ๐ผ โ โฉ โ [ , ] . For each ๐, ๐ โ N and ๐ = โจ ๐ , ๐ โฉ โ ฮ ๐,๐ , consider independent binary random variables ๐ ๐๐ representing the outcome of a match between ๐ โ [ ๐ ] and ๐ โ [ ๐ ] , where ๐ ๐ถ ( ๐ ๐๐ = | ๐ ) = (cid:40) ๐ผ + , ๐ฅ ๐ < ๐ฆ ๐ โ ๐ผ โ , ๐ฅ ๐ โฅ ๐ฆ ๐ (3) ๐ ๐ถ ( ๐ ๐๐ = | ๐ ) = (cid:40) โ ๐ผ + , ๐ฅ ๐ < ๐ฆ ๐ ๐ผ โ , ๐ฅ ๐ โฅ ๐ฆ ๐ (4)This defines a probability distribution ๐ ๐ถ (ยท | ๐ ) over ๐ ร ๐ tournaments by ๐ ๐ถ ( ๐พ | ๐ ) = (cid:214) ( ๐,๐ ) โ[ ๐ ]ร[ ๐ ] ๐ ๐ถ ( ๐ ๐๐ = ๐พ ๐๐ | ๐ ) Here ๐ ๐ถ ( ๐พ | ๐ ) is the probability of observing the tournamentresults ๐พ when the false positive and negative rates are given by ๐ถ and the true state of the world is ๐ . Note that the four cases in (3)and (4) correspond to a false positive, true positive, true negativeand false negative respectively. We can now define a maximumlikelihood operator. Definition 3.5.
Let ๐ถ โ [ , ] and ๐, ๐ โ N . Then ๐ โ ฮ ๐,๐ is a maximum likelihood estimate (MLE) for an ๐ ร ๐ tournament ๐พ w.r.t ๐ถ if ๐ โ arg max ๐ โฒ โ ฮ ๐,๐ ๐ ๐ถ ( ๐พ | ๐ โฒ ) . An operator ๐ is a maximum likelihood operator w.r.t ๐ถ if for any ๐, ๐ โ N and any ๐ ร ๐ tournament ๐พ there is an MLE ๐ = โจ ๐ , ๐ โฉ โ ฮ ๐,๐ for ๐พ suchthat ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ฅ ๐ โค ๐ฅ ๐ โฒ and ๐ โ ๐๐พ ๐ โฒ iff ๐ฆ ๐ โค ๐ฆ ๐ โฒ .Now, consider the tournament ๐พ ๐ associated with each state ๐ = โจ ๐ , ๐ โฉ , given by [ ๐พ ๐ ] ๐๐ = ๐ฅ ๐ โฅ ๐ฆ ๐ and [ ๐พ ๐ ] ๐๐ = ๐พ ๐ is the unique tournament with non-zero probabilitywhen there are no false positive or false negatives. Expressed interms of ๐พ ๐ , the MLEs take a particularly simple form if ๐ผ + = ๐ผ โ ,i.e. if false positives and false negatives occur at the same rate.Lemma 3.6. Let ๐ถ = โจ ๐ฝ, ๐ฝ โฉ for some ๐ฝ < . Then ๐ is an MLE for ๐พ if and only if ๐ โ arg min ๐ โฒ โ ฮ ๐,๐ ๐ ( ๐พ, ๐พ ๐ โฒ ) . Note that a false positive for ๐ is a false negative for ๐ and vice versa. This is a strong assumption, and it may be more realistic to model the false posi-tive/negative rates as a function of ๐ฅ ๐ . We leave this to future work. Proof (sketch). Let ๐พ be an ๐ ร ๐ tournament. It can be shown(and we do so in the appendix) that for any ๐ โ ฮ ๐,๐ ๐ ๐ถ ( ๐พ | ๐ ) = (cid:16) (cid:214) ๐ โ ๐ด ๐ผ | ๐พ ( ๐ )\ ๐พ ๐ ( ๐ ) |+ ( โ ๐ผ โ ) | ๐พ ( ๐ )โฉ ๐พ ๐ ( ๐ ) | ( โ ๐ผ + ) | ๐ต \( ๐พ ( ๐ )โช ๐พ ๐ ( ๐ )) | ๐ผ | ๐พ ๐ ( ๐ )\ ๐พ ( ๐ ) |โ (cid:17) Plugging in ๐ผ + = ๐ผ โ = ๐ฝ and simplifying, one can obtain ๐ ๐ถ ( ๐พ | ๐ ) = ๐ (cid:214) ๐ โ ๐ด (cid:18) ๐ฝ โ ๐ฝ (cid:19) | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | where ๐ โณ ๐ = ( ๐ \ ๐ ) โช ( ๐ \ ๐ ) is the symmetric difference oftwo sets ๐ and ๐ , and ๐ = ( โ ๐ฝ ) | ๐ด |ยท| ๐ต | is a positive constant thatdoes not depend on ๐ . Now, ๐ ๐ถ ( ๐พ | ๐ ) is positive, and is maximalwhen its logarithm is. We havelog ๐ ๐ถ ( ๐พ | ๐ ) = log ๐ + log (cid:18) ๐ฝ โ ๐ฝ (cid:19) โ๏ธ ๐ โ ๐ด | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ )| = log ๐ + log (cid:18) ๐ฝ โ ๐ฝ (cid:19) ๐ ( ๐พ, ๐พ ๐ ) Since log ๐ is constant and ๐ฝ < / (cid:16) ๐ฝ โ ๐ฝ (cid:17) <
0, it fol-lows that log ๐ ๐ถ ( ๐พ | ๐ ) is maximised exactly when ๐ ( ๐พ, ๐พ ๐ ) isminimised, which proves the result. โก This result characterises the MLE states for ๐พ as those for which ๐พ ๐ is the closest to ๐พ . As it turns out, the tournaments ๐พ ๐ that arisein this way are exactly those with the chain property.Lemma 3.7. An ๐ ร ๐ tournament ๐พ has the chain property if andonly if ๐พ = ๐พ ๐ for some ๐ โ ฮ ๐,๐ . The proof of Lemma 3.7 relies crucially on (1) and (2) in thedefinition of a state. Combining all the results so far we obtain ourfirst main result: the maximum likelihood operators for ๐ถ = โจ ๐ฝ, ๐ฝ โฉ are exactly the chain-minimal operators.Theorem 3.8. Let ๐ถ = โจ ๐ฝ, ๐ฝ โฉ for some ๐ฝ < . Then ๐ is a maxi-mum likelihood operator w.r.t ๐ถ if and only if ๐ satisfies chain-min . Proof (sketch). First note that by Lemma 3.6, a state ๐ is anMLE for an ๐ ร ๐ tournament ๐พ iff ๐พ ๐ is closest to ๐พ amongstall other tournaments { ๐พ ๐ โฒ | ๐ โฒ โ ฮ ๐,๐ } . But by Lemma 3.7, thisset is exactly the ๐ ร ๐ tournaments with the chain property. Itfollows from the definition of M ( ๐พ ) that ๐ is an MLE if and onlyif ๐พ ๐ โ M ( ๐พ ) . Consequently, ๐พ โฒ โ M ( ๐พ ) if and only if ๐พ โฒ = ๐พ ๐ for some MLE ๐ for ๐พ . We see that chain-min can be equivalentlystated as follows: for all ๐พ there exists an MLE ๐ such that ๐ ( ๐พ ) = ( โฉฝ A ๐พ ๐ , โฉฝ B ๐พ ๐ ) . Using properties (1) and (2) in Definition 3.3 for ๐ it isstraightforward to show that ๐ โฉฝ A ๐พ ๐ ๐ โฒ iff ๐ฅ ๐ โค ๐ฅ ๐ โฒ and ๐ โฉฝ B ๐พ ๐ ๐ โฒ iff ๐ฆ ๐ โค ๐ฆ ๐ โฒ for all ๐, ๐ โฒ โ ๐ด , ๐, ๐ โฒ โ ๐ต (where ๐ = โจ ๐ , ๐ โฉ ). This meansthat the above reformulation of chain-min coincides with thedefinition of a maximum likelihood operator, and we are done. โก Similar results can be obtained for other limiting values of ๐ถ . If ๐ผ + = ๐ผ โ โ ( , ) then the MLE operators correspond to chaincompletion : finding the minimum number of edge additions requiredto make ๐บ ๐พ a chain graph. This models situations where falsepositives never occur, although false negatives may (e.g. numericalntry questions in the case where ๐ด represents students and ๐ต exam questions [15]). Similarly, the case ๐ผ โ = ๐ผ + โ ( , ) corresponds to chain deletion , where edge additions are not allowed. Chain-minimal operators have theoretical backing in a probabilisticsense due to the results of Section 3.2, but are they appropriate rank-ing methods in practise? To address this question we consider the normative properties of chain-minimal operators via the axiomaticmethod of social choice theory. We formulate several axioms forbipartite tournament ranking and assess whether they are compati-ble with chain-min . It will be seen that an important anonymity axiom fails for all chain-minimal operators; later in Section 5 wedescribe a scenario in which this is acceptable and define a classof concrete operators for this case, and in Section 6 we relax the chain-min requirement in order to gain anonymity.
We will consider five axioms โ mainly adaptations of standard socialchoice properties to the bipartite tournament setting.
Symmetry Properties.
We consider two symmetry properties.The first is a classic anonymity axiom, which says that an operator ๐ should not be sensitive to the โlabelsโ used to identify participantsin a tournament. Axioms of this form are standard in social choicetheory; a tournament version goes at least as far back as [20].We need some notation: for a tournament ๐พ and permutations ๐ : ๐ด โ ๐ด , ๐ : ๐ต โ ๐ต , let ๐ ( ๐พ ) and ๐ ( ๐พ ) denote the tournamentobtained by permuting the rows and columns of ๐พ by ๐ and ๐ respectively, i.e. [ ๐ ( ๐พ )] ๐๐ = ๐พ ๐ โ ( ๐ ) ,๐ and [ ๐ ( ๐พ )] ๐๐ = ๐พ ๐,๐ โ ( ๐ ) .Note that in the statement of the axioms we omit universal quan-tification over ๐พ , ๐, ๐ โฒ โ ๐ด and ๐, ๐ โฒ โ ๐ต for brevity. (anon) Let ๐ : ๐ด โ ๐ด and ๐ : ๐ต โ ๐ต be permutations. Then ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ ( ๐ ) โชฏ ๐๐ ( ๐ ( ๐พ )) ๐ ( ๐ โฒ ) . Our second axiom is specific to bipartite tournaments, and ex-presses a duality between the two sides ๐ด and ๐ต : given the twosets of conceptually disjoint entities participating in a bipartitetournament, it should not matter which one we label ๐ด and whichone we label ๐ต . We need the notion of a dual tournament . Definition 4.1.
The dual tournament of ๐พ is ๐พ = โ ๐พ โค , where denotes the matrix consisting entirely of 1s. ๐พ is essentially the same tournament as ๐พ , but with the roles of ๐ด and ๐ต swapped. In particular, ๐ด ๐พ = ๐ต ๐พ , ๐ต ๐พ = ๐ด ๐พ and ๐พ ๐๐ = ๐พ ๐๐ =
0. Also note that ๐พ = ๐พ . The duality axiom states that theranking of the ๐ต s in ๐พ is the same as the ๐ด s in ๐พ . (dual) ๐ โ ๐๐พ ๐ โฒ iff ๐ โชฏ ๐๐พ ๐ โฒ . Whilst dual is not necessarily a universally desirable propertyโ one can imagine situations where ๐ด and ๐ต are not fully abstractand should not be treated symmetrically โ it is important to con-sider in any study of bipartite tournaments. Note that dual implies ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ โ ๐๐พ ๐ โฒ , so that a dual -operator can be defined bygiving the ranking for one of ๐ด or ๐ต only, and defining the other byduality. This explains our choice to define anon (and subsequent axioms) solely in terms of the ๐ด ranking: the analogous anonymityconstraint for the ๐ต ranking follows from anon together with dual . An Independence Property.
Independence axioms play a crucialrole in social choice. We present a bipartite adaptation of a clas-sic axiom introduced in [20], which has subsequently been called
Independence of Irrelevant Matches [14]. (IIM) If ๐พ , ๐พ are tournaments of the same size with identical ๐ -thand ๐ โฒ -th rows, then ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ โชฏ ๐๐พ ๐ โฒ . IIM is a strong property, which says the relative ranking of ๐ and ๐ โฒ does not depend on the results of any match not involving ๐ or ๐ โฒ .This axiom has been questioned for generalised tournaments [14],and a similar argument can be made against it here: although eachplayer in ๐ด faces the same opponents, we may wish to take the strength of opponents into account, e.g. by rewarding victoriesagainst highly-ranked players in ๐ต . Consequently we do not view IIM as an essential requirement, but rather introduce it to facilitatecomparison with our work and the existing tournament literature.
Monotonicity Properties.
Our final axioms are monotonicityproperties, which express the idea that more victories are better .The first axiom follows our original intuition for constructing thenatural ranking associated with a chain graph; namely that ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) indicates ๐ โฒ has performed at least as well as ๐ . (mon) If ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) then ๐ โชฏ ๐๐พ ๐ โฒ . Note that mon simply says โชฏ ๐๐พ extends the (in general, partial)preorder โฉฝ A ๐พ . Yet another standard axiom is positive responsiveness . (pos-resp) If ๐ โชฏ ๐๐พ ๐ โฒ and ๐พ ๐ โฒ ,๐ = for some ๐ โ ๐ต , then ๐ โบ ๐๐พ + ๐ โฒ ,๐ ๐ โฒ , where ๐ โฒ ,๐ is the matrix with 1 in position ( ๐ โฒ , ๐ ) and zeros elsewhere. That is, adding an extra victory for ๐ should only improve itsranking, with ties now broken in its favour. This version of positiveresponsiveness was again introduced in [20], where together with anon and IIM it characterises the points system ranking method forround-robin tournaments, which simply ranks players according tothe number of victories. The analogous operator in our frameworkis ๐ count , and it can be shown that ๐ count is uniquely characterisedby anon , IIM , pos-resp and dual . Finally, note that pos-resp alsoacts as a kind of strategyproofness : ๐ cannot improve its ranking bydeliberately losing a match. Specifically, if ๐พ ๐๐ = ๐ โชฏ ๐๐พ ๐ โฒ ,then pos-resp implies ๐ โบ ๐๐พ โ ๐๐ ๐ โฒ . We come to analysing the compatibility of chain-min with theaxioms. First, the negative results.Theorem 4.2.
There is no operator satisfying chain-min andany of anon , IIM or pos-resp . The counterexample for anon is particularly simple: take ๐พ = (cid:2) (cid:3) . Swapping the rows and columns brings us back to ๐พ , so anon implies 1 , โ ๐ด rank equally. However, it is easily seen forevery ๐พ โฒ โ M ( ๐พ ) , either ๐พ ( ) โ ๐พ ( ) or ๐พ ( ) โ ๐พ ( ) , i.e nochain-minimal operator can rank 1 and 2 equally.The MLE results of Section 3.2 provides informal explanation forthis result. For ๐พ above to arise in the noise model of Definition 3.4here must have been two โmistakesโ (false positives or false neg-atives). This is less likely than a single mistake from just one of1 , โ ๐ด , but the likelihood maximisation forces us to choose oneor the other. A similar argument explains the pos-resp failure.It is also worth noting that anon only fails at the last step ofchain editing, where a single element of M ( ๐พ ) is chosen. Indeed,the set M ( ๐พ ) itself does exhibit the kind of symmetry one mightexpect: we have M ( ๐ ( ๐ ( ๐พ ))) = { ๐ ( ๐ ( ๐พ โฒ )) | ๐พ โฒ โ M ( ๐พ )} . Thismeans that an operator which aggregates the rankings from all ๐พ โฒ โ M ( ๐พ ) โ e.g. any anonymous social welfare function โ wouldsatisfy anon . The other axioms are compatible with chain-min .Theorem 4.3. For each of dual and mon , there exists an operatorsatisfying chain-min and the stated property.
Despite the simplicity of mon , Theorem 4.3 is deceptively dif-ficult to prove. We describe operators satisfying chain-def and dual or mon non-constructively by first taking an arbitrary chain-minimal operator ๐ , and using properties of the set M ( ๐พ ) to pro-duce ๐ โฒ satisfying dual or mon . Note also that we have not yetconstructed an operator satisfying dual , mon and chain-min si-multaneously, although we conjecture that such operators do exist. The counterexample for chain-min and anon suggests that chain-minimal operators require some form of tie-breaking mechanismwhen the tournaments in
M ( ๐พ ) cannot be distinguished whilerespecting anonymity. While this limits the use of chain-minimaloperators as general purpose ranking methods, it is not such a prob-lem if additional information is available to guide the tie-breaking.In this section we introduce a new class of operators for this case.The core idea is to single out a unique chain tournament close to ๐พ by paying attention to not only the number of entries in ๐พ thatneed to be changed to produce a chain tournament, which entries.Specifically, we assume the availability of a total order on the set ofmatrix indices N ร N (the matches ) which indicates our willingnessto change an entry in ๐พ : the higher up ( ๐, ๐ ) is in the ranking, themore acceptable it is to change ๐พ ๐๐ during chain editing.This total order โ called the match-preference relation โ is fixed forall tournaments ๐พ ; this means we are dealing with extra informationabout how tournaments are constructed in matrix form , not extrainformation about any specific tournament ๐พ .One possible motivation for such a ranking comes from caseswhere matches occur at distinct points in time. In this case thematches occurring more recently are (presumably) more represen-tative of the playersโ current abilities, and we should therefore preferto modify the outcome of old matches where possible.For the formal definition we need notation for the vectorisation ofa tournament ๐พ : for a total order โด on N ร N and an ๐ ร ๐ tournament ๐พ , we write vec โด ( ๐พ ) for the vector in { , } ๐๐ obtained by collect-ing the entries of ๐พ in the order given by โด โพ ( ๐ด ร ๐ต ) , startingwith the minimal entry. That is, vec โด ( ๐พ ) = ( ๐พ ๐ ,๐ , . . . , ๐พ ๐ ๐๐ ,๐ ๐๐ ) ,where ( ๐ , ๐ ) , . . . , ( ๐ ๐๐ , ๐ ๐๐ ) is the unique enumeration of ๐ด ร ๐ต such that ( ๐ ๐ , ๐ ๐ ) โด ( ๐ ๐ + , ๐ ๐ + ) for each ๐ .The operator corresponding to โด is defined using the notion ofa choice function : a function ๐ผ which maps any tournament ๐พ to This denotes the restriction of โด to ๐ด ร ๐ต , i.e. โด โฉ (( ๐ด ร ๐ต ) ร ( ๐ด ร ๐ต )) . an element of M ( ๐พ ) . Any such function defines a chain-minimaloperator ๐ by setting ๐ ( ๐พ ) = ( โฉฝ A ๐ผ ( ๐พ ) , โฉฝ B ๐ผ ( ๐พ ) ) . Definition 5.1.
Let โด be a total order on N ร N . Define an operator ๐ โด according to the choice function ๐ผ โด ( ๐พ ) = arg min ๐พ โฒ โM ( ๐พ ) vec โด ( ๐พ โ ๐พ โฒ ) (5)where [ ๐พ โ ๐พ โฒ ] ๐๐ = | ๐พ ๐๐ โ ๐พ โฒ ๐๐ | , and the minimum is taken w.r.tthe lexicographic ordering on { , } | ๐ด |ยท| ๐ต | . Operators generatedin this way will be called match-preference operators . Example 5.2.
Let โด be the lexicographic order on N ร N so thatvec โด ( ๐พ โ ๐พ โฒ ) is obtained by collecting the entries of ๐พ โ ๐พ โฒ row-by-row, from top to bottom and left to right. Take ๐พ from Example 3.2.Writing ๐พ , . . . , ๐พ for the elements of M ( ๐พ ) in the order that theyappear in Example 3.2 and setting ๐ฃ ๐ = vec โด ( ๐พ โ ๐พ ๐ ) , we have ๐ฃ = (
00 0000 ) ; ๐ฃ = ( ) ๐ฃ = ( ) ; ๐ฃ = ( ) The lexicographic minimum is the one with the 1 entries as farright as possible, which in this case is ๐ฃ . Consequently ๐ โด ranks ๐พ according to ๐พ , i.e. 1 โบ ๐ โด ๐พ โบ ๐ โด ๐พ โ ๐ โด ๐พ โ ๐ โด ๐พ โ ๐ โด ๐พ ๐ผ โด ( ๐พ ) as the unique closest chain tour-nament to ๐พ w.r.t a weighted Hamming distance, and thereby avoidthe need to enumerate
M ( ๐พ ) in full as per eq. (5).Theorem 5.3. Let โด be a total order on N ร N . Then for any ๐, ๐ โ N there exists a function ๐ค : [ ๐ ] ร [ ๐ ] โ R โฅ such that forall ๐ ร ๐ tournaments ๐พ : arg min ๐พ โฒ โC ๐,๐ ๐ ๐ค ( ๐พ, ๐พ โฒ ) = { ๐ผ โด ( ๐พ )} (6) where ๐ ๐ค ( ๐พ, ๐พ โฒ ) = (cid:205) ( ๐,๐ ) โ[ ๐ ]ร[ ๐ ] ๐ค ( ๐, ๐ ) ยท | ๐พ ๐๐ โ ๐พ โฒ ๐๐ | . For example, the weights corresponding to โด from Example 5.2and ๐ = ๐ = ๐ค = (cid:2) . .
25 1 . . . . (cid:3) . Having studied chain-minimal operators in some detail, we turn totwo remaining problems: chain-min is incompatible with anon ,and computing a chain-minimal operator is NP -hard. In this sectionwe obtain both anonymity and tractability by relaxing the chain-min requirement to a property we call chain-definability . We goon to characterise the class of operators with this weaker propertyvia a greedy approximation algorithm, single out a particularlyintuitive instance, and revisit the axioms of Section 4. The source of the difficulties with chain-min lies in the minimi-sation aspect of chain editing. A natural way to retain the spiritof chain-min without the complications is to require that ๐ ( ๐พ ) corresponds to some chain tournament, not necessarily one closestto ๐พ . We call this property chain-definability . Note that ๐พ โ ๐พ โฒ is 1 in exactly the entries where ๐พ and ๐พ โฒ differ. That is, ( ๐,๐ ) โด ( ๐ โฒ ,๐ โฒ ) iff ๐ < ๐ โฒ or ( ๐ = ๐ โฒ and ๐ โค ๐ โฒ ). chain-def) For every ๐ ร ๐ tournament ๐พ there is ๐พ โฒ โ C ๐,๐ suchthat ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) . Clearly chain-min implies chain-def . โChain-definableโ opera-tors can also be cast in the MLE framework of Section 3.2 as thosewhose rankings correspond to some (not necessarily MLE) state ๐ .At first glance it may seem difficult to determine whether a givenpair of rankings correspond to a chain tournament, since the num-ber of such tournaments grows rapidly with ๐ and ๐ . Fortunately, chain-def can be characterised without reference to chain tourna-ments by considering the number of ranks of โชฏ ๐๐พ and โ ๐๐พ . In whatfollows ranks (โชฏ) denotes the number of ranks of a total preorder โชฏ , i.e. the number of equivalence classes of its symmetric part.Theorem 6.1. ๐ satisfies chain-def if and only if | ranks (โชฏ ๐๐พ ) โ ranks (โ ๐๐พ )| โค for every tournament ๐พ . According to Theorem 6.1, to construct a chain-definable operatorit is enough to ensure that the number of ranks of โชฏ ๐๐พ and โ ๐๐พ differby at most one. A simple way to achieve this is to iteratively selectand remove the top-ranked players of ๐ด and ๐ต simultaneously, untilone of ๐ด or ๐ต is exhausted. We call such operators interleavingoperators . Closely related ranking methods have been previouslyintroduced for non-bipartite tournaments by Bouyssou [2].Formally, our procedure is defined by two functions ๐ and ๐ which select the next top ranks given a tournament ๐พ and subsets ๐ด โฒ โ ๐ด , ๐ต โฒ โ ๐ต of the remaining players. Definition 6.2. An A - selection function is a mapping ๐ : K ร N ร N โ N such that for any tournament ๐พ , ๐ด โฒ โ ๐ด and ๐ต โฒ โ ๐ต :(i) ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) โ ๐ด โฒ ; (ii) If ๐ด โฒ โ โ then ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) โ โ ; (iii) ๐ ( ๐พ, ๐ด โฒ , โ ) = ๐ด โฒ .Similarly, a B - selection function is a mapping ๐ : K ร N ร N โ N such that (i) ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) โ ๐ต โฒ ; (ii) If ๐ต โฒ โ โ then ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) โ โ ; (iii) ๐ ( ๐พ, โ , ๐ต โฒ ) = ๐ต โฒ .The corresponding interleaving operator ranks players accordingto how soon they are selected in this way; the earlier the better. Definition 6.3.
Let ๐ and ๐ be selection functions and ๐พ a tour-nament. Write ๐ด = ๐ด , ๐ต = ๐ต , and for ๐ โฅ ๐ด ๐ + = ๐ด ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) ; ๐ต ๐ + = ๐ต ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) For ๐ โ ๐ด and ๐ โ ๐ต , write ๐ ( ๐ ) = max { ๐ | ๐ โ ๐ด ๐ } and ๐ ( ๐ ) = max { ๐ | ๐ โ ๐ต ๐ } . We define the corresponding interleaving op-erator ๐ = ๐ int ๐ ,๐ by ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ ( ๐ ) โฅ ๐ ( ๐ โฒ ) and ๐ โ ๐๐พ ๐ โฒ iff ๐ ( ๐ ) โฅ ๐ ( ๐ โฒ ) .Note that ๐ด ๐ and ๐ต ๐ are the players left remaining after ๐ applica-tions of ๐ and ๐ , i.e. after removing the top ๐ ranks from both sides.Before giving a concrete example, we note that interleaving is notjust one way to satisfying chain-def , it is the only way.Theorem 6.4. An operator ๐ satisfies chain-def if and only if ๐ = ๐ int ๐ ,๐ for some selection functions ( ๐ , ๐ ) . We show in the appendix that the recursive procedure eventually terminates with ๐ด ๐ and ๐ต ๐ becoming empty (and remaining so) after finitely many iterations, so ๐ and ๐ are well-defined. Table 1: Iteration of the interleaving algorithm for ๐ CI ๐ ๐พ ๐ด ๐ ๐ต ๐ ๐ ๐ ๐พ โฒ ๐ (cid:20) (cid:21) { , , , } { , , , , } { } { } (cid:20) (cid:21) (cid:20) (cid:21) { , , } { , , , } { } { , } (cid:20) (cid:21) (cid:20) (cid:21) { , } { , } { } { } - (cid:20) (cid:21) { } { } { } { } - - โ โ โ โ - Theorem 6.4 justifies our study of interleaving operators, andprovides a different perspective on chain-definability via the selec-tion functions ๐ and ๐ . We come to an important example. Example 6.5.
Define the cardinality-based interleaving operator ๐ CI = ๐ int ๐ ,๐ where ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = arg max ๐ โ ๐ด โฒ | ๐พ ( ๐ ) โฉ ๐ต โฒ | and ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = arg min ๐ โ ๐ต โฒ | ๐พ โ ( ๐ ) โฉ ๐ด โฒ | , so that the โwinnersโ ateach iteration are the ๐ด s with the most wins, and the ๐ต s withthe least losses, when restricting to ๐ด โฒ and ๐ต โฒ only. We take thearg min/arg max to be the emptyset whenever ๐ด โฒ or ๐ต โฒ is empty.Table 1 shows the iteration of the algorithm for a 4 ร ๐พ . In each row ๐ we show ๐พ with the rows and columns of ๐ด \ ๐ด ๐ and ๐ต \ ๐ต ๐ greyed out, so as to make it more clear how the ๐ and ๐ values are calculated. For brevity we also write ๐ and ๐ in placeof ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) and ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) respectively.The ๐ and ๐ values can be read off as 0, 2, 1, 3 for ๐ด and 0, 3, 1, 1,2 for ๐ต , giving the ranking on ๐ด as 4 โบ โบ โบ
1, and the rankingon ๐ต as 2 โ โ โ โ
1. Note also that each ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) is arank of โชฏ ๐๐พ (and similar for ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) ), so the rankings can in factbe read off by looking at the ๐ and ๐ columns of Table 1.The interleaving algorithm can also be seen as a greedy algorithmfor converting ๐พ into a chain graph directly. Indeed, by settingthe neighbourhood of each ๐ โ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) to ๐ต ๐ , and removingeach ๐ โ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) from the neighbourhoods of all ๐ โ ๐ด ๐ + , weeventually obtain a chain graph. We show this process in the ๐พ โฒ ๐ column of Table 1, where only three entries need to be changed. The selection functions ๐ and ๐ can therefore be seen as heuristics with the goal of finding a chain graph โcloseโ to ๐พ .The operator ๐ CI from Example 6.5 uses simple cardinality-basedheuristics, and can be seen as a chain-definable version of ๐ count (which is not chain-definable). It is also the bipartite counterpart torepeated applications of Copelandโs rule [2]. Note that ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) and ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) can be computed in ๐ ( ๐ ) time at each iteration ๐ ,where ๐ = | ๐ด | + | ๐ต | . Since there cannot be more than ๐ iterations,it follows that the rankings of ๐ CI can be computed in ๐ ( ๐ ) time. We now revisit the axioms of Section 4 in relation to chain-definableoperators in general and ๐ CI specifically. Firstly, the weakening of chain-min pays off: chain-def is compatible with all our axioms. Note that while ๐ and ๐ for ๐ CI are independent of the greyed out entries, we donot require this property for selection functions in general. In this example
M ( ๐พ ) contains a single tournament a distance of 2 from ๐พ , so ๐ CI makes one more change than necessary. heorem 6.6. For each of anon , dual , IIM , mon and pos-resp ,there exists an operator satisfying chain-def and the stated property. Unfortunately, these cannot all hold at the same time. Indeed,taking ๐พ = (cid:2) (cid:3) โค and assuming anon and pos-resp , the rank-ing on ๐ด is fully determined as 1 โบ โ โบ
4, and ranks (โชฏ ๐๐พ ) = anon with dual implies the ranking of ๐ต is flat, i.e. ranks (โ ๐๐พ ) =
1. This contradicts chain-def by Theorem 6.1, yield-ing the following impossibility result.Theorem 6.7.
There is no operator satisfying chain-def , anon , dual and pos-resp . For the specific operator ๐ CI we have the following.Theorem 6.8. ๐ CI satisfies chain-def , anon , dual and mon , anddoes not satisfy IIM or pos-resp . Note that anon is satisfied. This makes ๐ CI an important exam-ple of a well-motivated, tractable, chain-definable and anonymousoperator, meeting the criteria outlined at the start of this section. On chain graphs.
Chain graphs were originally introduced byYannakakis [25], who proved that chain completion โ finding theminimum number of edges that when added to a bipartite graphform a chain graph โ is NP -complete. Hardness results have sub-sequently been obtained for chain deletion [18] (where only edgedeletions are allowed) and chain editing [9] (where both additionsand deletions are allowed). We refer the reader to the work of Jiaoet al. [15] and Drange et al. [9] for a more detailed account of thisliterature. Outside of complexity theory, chain graphs have beenstudied for their spectral properties in [1, 13], and the more generalnotion of a nested colouring was introduced in [6]. On tournaments in social choice.
Tournaments have importantapplications in the design of voting rules, where an alternative ๐ฅ beats ๐ฆ in a pairwise comparison if a majority of voters prefer ๐ฅ to ๐ฆ . Various tournament solutions have been proposed, which select aset of โwinnersโ from a given tournament. Of particular relevanceto our work are the
Slater set and
Kemeneyโs rule [5], which findminimal sets of edges to invert in the tournament graph such thatthe beating relation becomes a total order. These methods areintuitively similar to chain editing: both involve making minimalchanges to the tournament until some property is satisfied. A roughanalogue to the Slater set in our framework is the union of thetop-ranked players from each ๐พ โฒ โ M ( ๐พ ) . Solutions based on thecovering relation โ such as the uncovered and Banks set [5] โ alsobear similarity to chain editing.Finally, note that directed versions of chain graphs (obtained byorienting edges from ๐ด to ๐ต and adding missing edges from ๐ต to ๐ด ) correspond to acyclic tournaments , and a topological sort of ๐ด becomes a linearisation of the chain ranking โฉฝ A ๐พ . This suggests aconnection between chain deletion and the standard feedback arcset problem for removing cycles and obtaining a ranking. Note that a ranking, such as we consider in this paper, induces a set of winners bytaking the maximally ranked players. Note that like chain editing, Kemenyโs rule also admits a maximum likelihoodcharacterisation [10].
On generalised tournaments. A generalised tournament [14] isa pair ( ๐,๐ ) , where ๐ = [ ๐ก ] for some ๐ก โ N and ๐ โ R ๐ก ร ๐ก โฅ is a non-negative ๐ก ร ๐ก matrix with ๐ ๐๐ = ๐ โ ๐ . In this formalism eachencounter between a pair of players ๐ and ๐ is represented by two numbers: ๐ ๐ ๐ and ๐ ๐๐ . This allows one to model both intensities ofvictories and losses (including draws) via the difference ๐ ๐ ๐ โ ๐ ๐๐ , andthe case where a comparison is not available (where ๐ ๐ ๐ = ๐ ๐๐ = ๐ ร ๐ bipartite tournament ๐พ has a natural generalised tour-nament representation via the ( ๐ + ๐ ) ร ( ๐ + ๐ ) anti-diagonal blockmatrix ๐ = (cid:104) ๐พ๐พ (cid:105) , where the top-left and bottom-right blocks arethe ๐ ร ๐ and ๐ ร ๐ zero matrices respectively. However, suchanti-diagonal block matrices are often excluded in the generalisedtournament literature due to an assumption of irreducibility , whichrequires that the directed graph corresponding to ๐ is strongly con-nected. This is not the case in general for ๐ constructed as above,which means not all existing tournament operators (and tourna-ment axioms) are well-defined for bipartite inputs. Consequently,bipartite tournaments are a special case of generalised tournaments in principle , but not in practise.
Summary.
In this paper we studied chain editing, an interestingproblem from computational complexity theory, as a ranking mech-anism for bipartite tournaments. We analysed such mechanismsfrom a probabilistic viewpoint via the MLE characterisation, andin axiomatic terms. To resolve both the failure of an importantanonymity axiom and NP -hardness, we weakened the chain edit-ing requirement to one of chain definability , and characterised theresulting class of operators by the intuitive interleaving algorithm. Limitations and future work.
The hardness of chain editingremains a limitation of our approach. A possible remedy is to lookto one of the numerous variant problems that are polynomial-timesolvable [15]; determining their applicability to ranking is an in-teresting topic for future work. One could develop approximationalgorithms for chain editing, possibly based on existing approxi-mations of chain completion [17]. The interleaving operators ofSection 6.2 go in this direction, but we did not yet obtain any theo-retical or experimental bounds on the approximation ratio.A second limitation of our work lies in the assumptions of theprobabilistic model; namely that the true state of the world can bereduced to vectors of numerical skill levels which totally describethe tournament participants. This assumption may be violated whenthe competitive element of a tournament is multi-faceted , since asingle number cannot represent multiple orthogonal componentsof a playerโs capabilities. Nevertheless, if skill levels are taken as aggregations of these components, chain editing may prove to be auseful, albeit simplified, model.Finally, there is room for more detailed axiomatic investigation.In this paper we have stuck with fairly standard social choice axiomsand performed preliminary analysis. However, the indirect natureof the comparisons in a bipartite tournament presents unique chal-lenges; new axioms may need to be formulated to properly evaluatebipartite ranking methods in a normative sense. We note that Slutzki and Volij [22] side-step the reducibility issue by decomposing ๐ into irreducible components and ranking each separately, although their methodsmay give only partial orders. CKNOWLEDGMENTS
We thank the anonymous AAMAS reviewers for their helpful com-ments.
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A PROOFS
This appendix contains proofs that were omitted (or only sketched)in the main paper.
A.1 Proof of Lemma 3.6
The proof of Lemma 3.6 requires a lemma of its own.Lemma A.1.
Let ๐พ be an ๐ ร ๐ tournament, ๐ถ โ [ , ] and ๐ โ ฮ ๐,๐ . Then ๐ ๐ถ ( ๐พ | ๐ ) = (cid:214) ๐ โ ๐ด ๐ผ | ๐พ ( ๐ )\ ๐พ ๐ ( ๐ ) |+ ( โ ๐ผ โ ) | ๐พ ( ๐ )โฉ ๐พ ๐ ( ๐ ) | ( โ ๐ผ + ) | ๐ต \( ๐พ ( ๐ )โช ๐พ ๐ ( ๐ )) | ๐ผ | ๐พ ๐ ( ๐ )\ ๐พ ( ๐ ) |โ Proof. Write ๐ ๐๐,๐พ for ๐ ๐ถ ( ๐ ๐๐ = ๐พ ๐๐ | ๐ ) . Expanding theproduct in definition 3.4, we have ๐ ๐ถ ( ๐พ | ๐ ) = (cid:214) ๐ โ ๐ด (cid:214) ๐ โ ๐ต ๐ ๐๐,๐พ Let ๐ โ ๐ด . Note that ๐ต can be written as the disjoint union ๐ต = ๐ต โช ๐ต โช ๐ต โช ๐ต , where ๐ต = ๐พ ( ๐ ) \ ๐พ ๐ ( ๐ ) ๐ต = ๐พ ( ๐ ) โฉ ๐พ ๐ ( ๐ ) ๐ต = ๐ต \ ( ๐พ ( ๐ ) โช ๐พ ๐ ( ๐ )) ๐ต = ๐พ ๐ ( ๐ ) \ ๐พ ( ๐ ) Recall that ๐ โ ๐พ ๐ ( ๐ ) iff ๐ฅ ๐ โฅ ๐ฆ ๐ (where ๐ = โจ ๐ , ๐ โฉ ). It follows that โข ๐ โ ๐ต iff ๐พ ๐๐ = ๐ฅ ๐ < ๐ฆ ๐ โข ๐ โ ๐ต iff ๐พ ๐๐ = ๐ฅ ๐ โฅ ๐ฆ ๐ โข ๐ โ ๐ต iff ๐พ ๐๐ = ๐ฅ ๐ < ๐ฆ ๐ โข ๐ โ ๐ต iff ๐พ ๐๐ = ๐ฅ ๐ โฅ ๐ฆ ๐ Note that this correspond exactly to the four cases in (3) and (4)which define ๐ ๐๐,๐พ ; we have ๐ ๐๐,๐พ = ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃด๏ฃด๏ฃณ ๐ผ + , ๐ โ ๐ต โ ๐ผ โ , ๐ โ ๐ต โ ๐ผ + , ๐ โ ๐ต ๐ผ โ , ๐ โ ๐ต Consequently (cid:214) ๐ โ ๐ต ๐ ๐๐,๐พ = (cid:169)(cid:173)(cid:171) (cid:214) ๐ โ ๐ต ๐ผ + (cid:170)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) (cid:214) ๐ โ ๐ต ( โ ๐ผ โ ) (cid:170)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) (cid:214) ๐ โ ๐ต ( โ ๐ผ + ) (cid:170)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) (cid:214) ๐ โ ๐ต ๐ผ โ (cid:170)(cid:174)(cid:172) = ๐ผ | ๐ต |+ ( โ ๐ผ โ ) | ๐ต | ( โ ๐ผ + ) | ๐ต | ๐ผ | ๐ต |โ = ๐ผ | ๐พ ( ๐ )\ ๐พ ๐ ( ๐ ) |+ ( โ ๐ผ โ ) | ๐พ ( ๐ )โฉ ๐พ ๐ ( ๐ ) | ( โ ๐ผ + ) | ๐ต \( ๐พ ( ๐ )โช ๐พ ๐ ( ๐ )) | ๐ผ | ๐พ ๐ ( ๐ )\ ๐พ ( ๐ ) |โ Taking the product over all ๐ โ ๐ด we reach the desired expressionfor ๐ ๐ถ ( ๐พ | ๐ ) . โก roof of Lemma 3.6. Let ๐ โ ฮ ๐,๐ . From lemma A.1 we get ๐ ๐ถ ( ๐พ | ๐ ) = (cid:214) ๐ โ ๐ด ๐ฝ | ๐พ ( ๐ )\ ๐พ ๐ ( ๐ ) |+| ๐พ ๐ ( ๐ )\ ๐พ ( ๐ ) | ( โ ๐ฝ ) | ๐พ ( ๐ )โฉ ๐พ ๐ ( ๐ ) |+| ๐ต \( ๐พ ( ๐ )โช ๐พ ๐ ( ๐ )) | Note that | ๐พ ( ๐ ) \ ๐พ ๐ ( ๐ )| + | ๐พ ๐ ( ๐ ) \ ๐พ ( ๐ )| = | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ )|| ๐พ ( ๐ ) โฉ ๐พ ๐ ( ๐ )| + | ๐ต \ ( ๐พ ( ๐ ) โช ๐พ ๐ ( ๐ ))| = | ๐ต | โ | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ )| and so ๐ ๐ถ ( ๐พ | ๐ ) = (cid:214) ๐ โ ๐ด ๐ฝ | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | ( โ ๐ฝ ) | ๐ต |โ| ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | = (cid:214) ๐ โ ๐ด (cid:18) ๐ฝ โ ๐ฝ (cid:19) | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | ( โ ๐ฝ ) | ๐ต | = ( โ ๐ฝ ) | ๐ด |ยท| ๐ต | (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = ๐ (cid:214) ๐ โ ๐ด (cid:18) ๐ฝ โ ๐ฝ (cid:19) | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | = ๐ (cid:214) ๐ โ ๐ด (cid:18) ๐ฝ โ ๐ฝ (cid:19) | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ ) | where ๐ is a positive constant that does not depend on ๐ . Now, ๐ ๐ถ ( ๐พ | ๐ ) is positive, and is maximal when its logarithm is. Wehave log ๐ ๐ถ ( ๐พ | ๐ ) = log ๐ + โ๏ธ ๐ โ ๐ด | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ )| log (cid:18) ๐ฝ โ ๐ฝ (cid:19) = log ๐ + log (cid:18) ๐ฝ โ ๐ฝ (cid:19) โ๏ธ ๐ โ ๐ด | ๐พ ( ๐ ) โณ ๐พ ๐ ( ๐ )| = log ๐ + log (cid:18) ๐ฝ โ ๐ฝ (cid:19) ๐ ( ๐พ, ๐พ ๐ ) Noting that ๐ฝ < / (cid:16) ๐ฝ โ ๐ฝ (cid:17) <
0, it follows that for any ๐, ๐ โฒ โ ฮ ๐,๐ : ๐ ๐ถ ( ๐พ | ๐ ) โฅ ๐ ๐ถ ( ๐พ | ๐ โฒ ) โโ log ๐ ๐ถ ( ๐พ | ๐ ) โ log ๐ ๐ถ ( ๐พ | ๐ โฒ ) โฅ โโ log (cid:18) ๐ฝ โ ๐ฝ (cid:19)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) < [ ๐ ( ๐พ, ๐พ ๐ ) โ ๐ ( ๐พ, ๐พ ๐ โฒ )] โฅ โโ ๐ ( ๐พ, ๐พ ๐ ) โค ๐ ( ๐พ, ๐พ ๐ โฒ ) which proves the result. โก A.2 Proof of Lemma 3.7
We need a preliminary result.Lemma A.2.
Let ๐ = โจ ๐ , ๐ โฉ โ ฮ ๐,๐ . Then for all ๐, ๐ โฒ โ ๐ด and ๐, ๐ โฒ โ ๐ต :(1) ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ๐ โฒ ) iff ๐ฅ ๐ โค ๐ฅ ๐ โฒ (2) ๐พ โ ๐ ( ๐ ) โ ๐พ โ ๐ ( ๐ โฒ ) iff ๐ฆ ๐ โค ๐ฆ ๐ โฒ . Proof. We prove (1); (2) is shown similarly. Let ๐, ๐ โฒ โ ๐ด . Firstsuppose ๐ฅ ๐ โค ๐ฅ ๐ โฒ . Let ๐ โ ๐พ ๐ ( ๐ ) . Then ๐ฆ ๐ โค ๐ฅ ๐ โค ๐ฅ ๐ โฒ , so ๐ โ ๐พ ๐ ( ๐ โฒ ) also. This shows ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ๐ โฒ ) .Now suppose ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ๐ โฒ ) . For the sake of contradiction,suppose ๐ฅ ๐ > ๐ฅ ๐ โฒ . By (1) in the definition of a state (definition 3.3),there is ๐ โ ๐ต such that ๐ฅ ๐ โฒ < ๐ฆ ๐ โค ๐ฅ ๐ . But this means ๐ โ ๐พ ๐ ( ๐ ) \ ๐พ ๐ ( ๐ โฒ ) , which contradicts ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ๐ โฒ ) . Thus (1) isproved. โก Proof of Lemma 3.7. The โifโ direction follows from lemma A.2part (1): if ๐ = โจ ๐ , ๐ โฉ and ๐, ๐ โฒ โ ๐ด then either ๐ฅ ๐ โค ๐ฅ ๐ โฒ โ in whichcase ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ๐ โฒ ) โ or ๐ฅ ๐ โฒ < ๐ฅ ๐ โ in which case ๐พ ๐ ( ๐ โฒ ) โ ๐พ ๐ ( ๐ ) .Therefore ๐พ ๐ has the chain property.For the โonly ifโ direction, suppose ๐พ has the chain property.Define ๐ = โจ ๐ , ๐ โฉ by ๐ฅ ๐ = |{ ๐ โฒ โ ๐ด | ๐พ ( ๐ โฒ ) โ ๐พ ( ๐ )}| ๐ฆ ๐ = (cid:40) min { ๐ฅ ๐ | ๐ โ ๐พ โ ( ๐ )} , ๐พ โ ( ๐ ) โ โ + | ๐ด | , ๐พ โ ( ๐ ) = โ It is easily that since the neighbourhood-subset relation โฉฝ A ๐พ isa total preorder, we have ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) if and only if ๐ฅ ๐ โค ๐ฅ ๐ โฒ .First we show that ๐พ ๐ = ๐พ by showing that ๐พ ๐๐ = [ ๐พ ๐ ] ๐๐ =
1. Suppose ๐พ ๐๐ =
1. Then ๐ โ ๐พ โ ( ๐ ) , so ๐ฆ ๐ = min { ๐ฅ ๐ โฒ | ๐ โฒ โ ๐พ โ ( ๐ )} โค ๐ฅ ๐ and consequently [ ๐พ ๐ ] ๐๐ = [ ๐พ ๐ ] ๐๐ =
1. Then ๐ฅ ๐ โฅ ๐ฆ ๐ . We must have ๐พ โ ( ๐ ) โ โ ; otherwise ๐ฆ ๐ = + | ๐ด | > | ๐ด | โฅ ๐ฅ ๐ . We can thereforetake ห ๐ โ arg min ๐ โฒ โ ๐พ โ ( ๐ ) ๐ฅ ๐ โฒ . By definition of ๐ฆ ๐ , ๐ฅ ห ๐ = ๐ฆ ๐ โค ๐ฅ ๐ .But ๐ฅ ห ๐ โค ๐ฅ ๐ implies ๐พ ( ห ๐ ) โ ๐พ ( ๐ ) ; since ห ๐ โ ๐พ โ ( ๐ ) this gives ๐ โ ๐พ ( ห ๐ ) and ๐ โ ๐พ ( ๐ ) , i.e. ๐พ ๐๐ =
1. This completes the claim that ๐พ = ๐พ ๐ .It only remains to show that ๐ satisfies conditions (1) and (2) ofdefinition 3.3. For (1), suppose ๐ฅ ๐ < ๐ฅ ๐ โฒ . Then ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) , i.ethere is ๐ โ ๐พ ( ๐ โฒ ) \ ๐พ ( ๐ ) = ๐พ ๐ ( ๐ โฒ ) \ ๐พ ๐ ( ๐ ) . But ๐ โ ๐พ ๐ ( ๐ โฒ ) gives ๐ฆ ๐ โค ๐ฅ ๐ โฒ , and ๐ โ ๐พ ๐ ( ๐ ) gives ๐ฅ ๐ < ๐ฆ ๐ ; this shows that (1) holds.For (2), suppose ๐ฆ ๐ < ๐ฆ ๐ โฒ . Clearly ๐พ โ ( ๐ ) โ โ (otherwise ๐ฆ ๐ = + | ๐ด | is maximal). Thus there is ๐ โ ๐พ โ ( ๐ ) such that ๐ฆ ๐ = ๐ฅ ๐ .This of course means ๐ฅ ๐ < ๐ฆ ๐ โฒ ; in particular we have ๐ฆ ๐ โค ๐ฅ ๐ < ๐ฆ ๐ โฒ as required for (2).We have shown that ๐พ = ๐พ ๐ and that ๐ โ ฮ ๐,๐ , and the proof iscomplete. โก A.3 Proof of Theorem 3.8
Proof. First we show that for any ๐, ๐ โ N and any ๐ ร ๐ tournament ๐พ it holds that ๐ is an MLE state for ๐พ if and only if ๐พ ๐ โ M ( ๐พ ) .Indeed, fix some ๐, ๐ and ๐พ . Write K ฮ ๐,๐ = { ๐พ ๐ | ๐ โ ฮ ๐,๐ } .By lemma 3.6, ๐ is an MLE if and only if ๐ ( ๐พ, ๐พ ๐ ) โค ๐ ( ๐พ, ๐พ ๐ โฒ ) for all ๐ โฒ โ ฮ ๐,๐ , i.e. ๐พ ๐ โ arg min ๐พ โฒ โK ฮ ๐,๐ ๐ ( ๐พ, ๐พ โฒ ) . But bylemma 3.7, K ฮ ๐,๐ is just C ๐,๐ , the set of all ๐ ร ๐ tournamentswith the chain property. We see that arg min ๐พ โฒ โK ฮ ๐,๐ ๐ ( ๐พ, ๐พ โฒ ) = arg min ๐พ โฒ โC ๐,๐ ๐ ( ๐พ, ๐พ โฒ ) = M ( ๐พ ) by definition of M ( ๐พ ) . Thisshows that ๐ is an MLE iff ๐พ ๐ โ M ( ๐พ ) .Now, by definition, ๐ satisfies chain-min iff for every tourna-ment ๐พ there is ๐พ โฒ โ M ( ๐พ ) such that ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) . Usinglemma 3.7 and the above result, ๐พ โฒ โ M ( ๐พ ) if and only if ๐พ โฒ = ๐พ ๐ for some MLE ๐ for ๐พ . We see that chain-min can be equivalentlystated as follows: for all tournament ๐พ there exists an MLE ๐ suchthat ๐ ( ๐พ ) = ( โฉฝ A ๐พ ๐ , โฉฝ B ๐พ ๐ ) . But by lemma A.2 we have ๐ โฉฝ A ๐พ ๐ ๐ โฒ iff ๐ฅ ๐ โค ๐ฅ ๐ โฒ and ๐ โฉฝ B ๐พ ๐ ๐ โฒ iff ๐ฆ ๐ โค ๐ฆ ๐ โฒ (where ๐ = โจ ๐ , ๐ โฉ ). The abovereformulation of chain-min now coincides with the definition ofa maximum likelihood operator, and we are done. โก .4 Proof of Theorem 4.2 Proof. We take each axiom in turn. Let ๐ be any operator satis-fying chain-min . anon: Consider ๐พ = (cid:2) (cid:3) , and define permutations ๐ = ๐ = ( ) , i.e. the permutations which simply swap 1 and 2. It is easilyseen that ๐ ( ๐ ( ๐พ )) = ๐พ . Supposing ๐ satisfied anon , we would get1 โชฏ ๐๐พ ๐ ( ) โชฏ ๐๐ ( ๐ ( ๐พ )) ๐ ( ) iff 2 โชฏ ๐๐พ
1, which implies 1 โ ๐๐พ M ( ๐พ ) = (cid:8)(cid:2) (cid:3) , (cid:2) (cid:3) , (cid:2) (cid:3) , (cid:2) (cid:3)(cid:9) Since ๐ satisfies chain-min and 1 , โ ๐ด rank equally in โชฏ ๐๐พ , theremust be ๐พ โฒ โ M ( ๐พ ) such that 1 and 2 rank equally in โฉฝ A ๐พ โฒ , i.e. ๐พ โฒ ( ) = ๐พ โฒ ( ) . But clearly there is no such ๐พ โฒ ; all tournaments in M ( ๐พ ) have distinct first and second rows. Hence ๐ cannot satisfy anon . IIM:
Suppose ๐ satisfies chain-min and IIM . Write ๐พ = (cid:104) (cid:105) , ๐พ = (cid:104) (cid:105) Note that the first and second rows of ๐พ and ๐พ are identical, so by IIM we have 1 โชฏ ๐๐พ โชฏ ๐๐พ
2. Both tournaments have a uniqueclosest chain tournament requiring changes to only a single entry:
M ( ๐พ ) = (cid:110)(cid:104) (cid:105)(cid:111) , M ( ๐พ ) = (cid:110)(cid:104) (cid:105)(cid:111) Write ๐พ โฒ and ๐พ โฒ for these nearest chain tournaments respectively.By chain-min , we must have ๐ ( ๐พ ๐ ) = ( โฉฝ A ๐พ ๐ โฒ , โฉฝ B ๐พ ๐ โฒ ) . In particular,1 โบ ๐๐พ โบ ๐๐พ
1. But this contradicts
IIM , and we are done. pos-resp:
Suppose ๐ satisfies chain-min and pos-resp , andconsider ๐พ = (cid:20) (cid:21) ๐พ has a unique closest chain tournament ๐พ โฒ : M ( ๐พ ) = { ๐พ โฒ } = (cid:26)(cid:20) (cid:21)(cid:27) chain-min therefore implies ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) . Note that ๐พ โฒ ( ) = ๐พ โฒ ( ) , so we have 1 โ ๐๐พ
2. In particular, 1 โชฏ ๐๐พ
2. Since ๐พ =
0, wemay apply pos-resp to get 1 โบ ๐๐พ +
2. But ๐พ + is just ๐พ โฒ . Sincethe chain property already holds for ๐พ โฒ , we have M ( ๐พ โฒ ) = { ๐พ โฒ } and consequently ๐ ( ๐พ + ) = ๐ ( ๐พ โฒ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) = ๐ ( ๐พ ) so in fact 1 โ ๐๐พ +
2, contradicting pos-resp . โก A.5 Proof of Theorem 4.3
For ease of presentation we establish the compatibility of chain-min with dual and mon separately.Proposition A.3.
There exists an operator ๐ satisfying chain-min and dual . Proposition A.4.
There exists an operator ๐ satisfying chain-min and mon . It is clear that these two propositions will together prove Theo-rem 4.3. For Proposition A.3 we use the following result.Lemma A.5.
Let ๐พ be a tournament. Then(1) โฉฝ B ๐พ = โฉฝ A ๐พ (2) ๐พ โฒ โ M ( ๐พ ) if and only if ๐พ โฒ โ M (cid:0) ๐พ (cid:1) Proof. Fix an ๐ ร ๐ tournament ๐พ .(1) Note that for any ๐ โ ๐ต , we have ๐พ โ ( ๐ ) = ๐ด \ ๐พ ( ๐ ) . Indeed,for any ๐ โ ๐ด = ๐ด ๐พ = ๐ต ๐พ , ๐ โ ๐พ โ ( ๐ ) โโ ๐พ ๐๐ = โโ โ ๐พ ๐๐ = โโ ๐พ ๐๐ = โโ ๐ โ ๐พ ( ๐ ) This means that for any ๐, ๐ โฒ โ ๐ต , ๐ โฉฝ B ๐พ ๐ โฒ โโ ๐พ โ ( ๐ ) โ ๐พ โ ( ๐ โฒ )โโ ๐ด \ ๐พ ( ๐ ) โ ๐ด \ ๐พ ( ๐ โฒ )โโ ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ )โโ ๐ โฉฝ A ๐พ ๐ โฒ so โฉฝ B ๐พ = โฉฝ A ๐พ .(2) ( โ ) Suppose ๐พ โฒ โ M ( ๐พ ) . First we show that ๐พ โฒ has thechain property. It is sufficient to show that โฉฝ B ๐พ โฒ is a total preorder, since part (1) then implies โฉฝ A ๐พ โฒ is a total preorder and ๐พ โฒ has thechain property by definition.Since โฉฝ B ๐พ โฒ always has reflexivity and transitivity, we only needto show the totality property. Let ๐, ๐ โฒ โ ๐ต and suppose ๐ ฬธ โฉฝ B ๐พ โฒ ๐ โฒ .We must show ๐ โฒ โฉฝ B ๐พ โฒ ๐ , i.e. ( ๐พ โฒ ) โ ( ๐ โฒ ) โ ( ๐พ โฒ ) โ ( ๐ ) . To that end,let ๐ โ ( ๐พ โฒ ) โ ( ๐ ) .Since ( ๐พ โฒ ) โ ( ๐ ) โ ( ๐พ โฒ ) โ ( ๐ โฒ ) , there is some ห ๐ โ ( ๐พ โฒ ) โ ( ๐ โฒ ) with ห ๐ โ ( ๐พ โฒ ) โ ( ๐ ) . That is, ๐ โฒ โ ๐พ โฒ ( ห ๐ ) but ๐ โ ๐พ โฒ ( ห ๐ ) . Since ๐ โ ๐พ โฒ ( ๐ ) , we have ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ห ๐ ) . By the chain property for ๐พ โฒ ,we get ๐พ โฒ ( ห ๐ ) โ ๐พ โฒ ( ๐ ) . Finally, this means ๐ โฒ โ ๐พ โฒ ( ห ๐ ) โ ๐พ โฒ ( ๐ ) , i.e ๐ โ ( ๐พ โฒ ) โ ( ๐ โฒ ) . This shows ๐ โฒ โฉฝ B ๐พ โฒ ๐ as required.It remains to show that ๐ ( ๐พ, ๐พ โฒ ) is minimal. Since every tour-nament is the dual of its dual, any ๐ ร ๐ chain tournament is ofthe form ๐พ โฒโฒ for an ๐ ร ๐ tournament ๐พ โฒโฒ . The above argumentshows that the chain property is preserved by taking the dual, sothat ๐พ โฒโฒ has the chain property also. Since ๐พ โฒ โ M ( ๐พ ) , we have ๐ ( ๐พ, ๐พ โฒโฒ ) โฅ ๐ ( ๐พ, ๐พ โฒ ) . It is easily verified that the Hamming distanceis also preserved under duals, so ๐ ( ๐พ, ๐พ โฒ ) = ๐ ( ๐พ, ๐พ โฒ ) โค ๐ ( ๐พ, ๐พ โฒโฒ ) = ๐ ( ๐พ, ๐พ โฒโฒ ) We have shown that ๐พ โฒ is as close to ๐พ as any other ๐ ร ๐ tour-nament with the chain property, which shows ๐พ โฒ โ M (cid:0) ๐พ (cid:1) asrequired.( โ ) Suppose ๐พ โฒ โ M (cid:0) ๐พ (cid:1) . By the โonly ifโ statement above, wehave ๐พ โฒ โ M (cid:16) ๐พ (cid:17) . But ๐พ = ๐พ and ๐พ โฒ = ๐พ โฒ , so ๐พ โฒ โ M ( ๐พ ) asrequired. โก Note that we claim this holds for any ๐พ โฒ with the chain property in the body of thepaper, but this has not yet been proven. roof of Proposition A.3. Let ๐ be an arbitrary operator sat-isfying chain-min . Then there is a function ๐ผ : K โ K suchthat ๐ ( ๐พ ) = ( โฉฝ A ๐ผ ( ๐พ ) , โฉฝ B ๐ผ ( ๐พ ) ) and ๐ผ ( ๐พ ) โ M ( ๐พ ) for all tourna-ments ๐พ . We will construct a new function ๐ผ โฒ , based on ๐ผ , suchthat ๐ผ โฒ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) .Let โช be a total order on the set of all tournaments K . Write ๐ = { ๐พ โ K | ๐พ โช ๐พ } Note that since ๐พ โ ๐พ for all ๐พ , exactly one of ๐พ and ๐พ lies in ๐ .Informally, we view the tournaments in ๐ as somehow โcanonicalโ,and those in K \ ๐ as the dual of a canonical tournament. We usethis notion to define ๐ผ โฒ : ๐ผ โฒ ( ๐พ ) = (cid:40) ๐ผ ( ๐พ ) , ๐พ โ ๐๐ผ ( ๐พ ) , ๐พ โ ๐ First we claim ๐ผ โฒ ( ๐พ ) โ M ( ๐พ ) for all ๐พ . Indeed, if ๐พ โ ๐ then ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) โ M ( ๐พ ) by the assumption on ๐ผ . Otherwise, ๐ผ ( ๐พ ) โ M (cid:0) ๐พ (cid:1) , so Lemma A.5 part (2) implies ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) โM (cid:16) ๐พ (cid:17) = M ( ๐พ ) .Next we show ๐ผ โฒ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) . First suppose ๐พ โ ๐ . Then ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) and ๐พ โ ๐ , so ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) = ๐ผ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) as required. Similarly, if ๐พ โ ๐ then ๐พ โ ๐ , so ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) ,and ๐ผ โฒ ( ๐พ ) = ๐ผ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) . Taking the dual of both sides, we get ๐ผ โฒ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) .Finally, define a new operator ๐ โฒ by ๐ โฒ ( ๐พ ) = ( โฉฝ A ๐ผ โฒ ( ๐พ ) , โฉฝ B ๐ผ โฒ ( ๐พ ) ) .Since ๐ผ โฒ ( ๐พ ) โ M ( ๐พ ) for all ๐พ , ๐ โฒ satisfies chain-min . Moreover,using Lemma A.5 part (1) and the fact that ๐ผ โฒ ( ๐พ ) = ๐ผ โฒ ( ๐พ ) , for anytournament ๐พ and ๐, ๐ โฒ โ ๐ต we have ๐ โ ๐ โฒ ๐พ ๐ โฒ โโ ๐ โฉฝ B ๐ผ โฒ ( ๐พ ) ๐ โฒ โโ ๐ โฉฝ A ๐ผ โฒ ( ๐พ ) ๐ โฒ โโ ๐ โฉฝ A ๐ผ โฒ ( ๐พ ) ๐ โฒ โโ ๐ โ ๐ โฒ ๐พ ๐ โฒ which shows ๐ โฒ also satisfies dual . โก Next we prove Proposition A.4. We will proceed in three stages.First, Lemma A.7 shows that if ๐พ ( ๐ ) โ ๐พ ( ๐ ) and ๐พ โฒ โ M ( ๐พ ) issome closest chain tournament with the reverse inclusion ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ ) , then swapping ๐ and ๐ in ๐พ โฒ yields obtain another clos-est chain tournament ๐พ โฒโฒ โ M ( ๐พ ) . Next, we show in Lemma A.8that by performing successive swaps in this way, we can find ๐พ โฒ โ M ( ๐พ ) such that ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ ) whenever ๐พ ( ๐ ) โ ๐พ ( ๐ ) (note the strict inclusion). Finally, we modify this ๐พ โฒ in Lemma A.9to additionally satisfy ๐พ โฒ ( ๐ ) = ๐พ โฒ ( ๐ ) whenever ๐พ ( ๐ ) = ๐พ ( ๐ ) .This shows that there always exist an element of M ( ๐พ ) extend-ing the neighbourhood-subset relation โฉฝ A ๐พ , and consequently it ispossible to satisfy chain-min and mon simultaneously. Definition A.6.
Let ๐พ be a tournament and ๐ , ๐ โ ๐ด . We denoteby swap ( ๐พ ; ๐ , ๐ ) the tournament obtained by swapping the ๐ Note that K is countable, so such an order can be easily constructed. Alternatively,one could use the axiom of choice and appeal to the well-ordering theorem to obtain โช . and ๐ -th rows of ๐พ , i.e. [ swap ( ๐พ ; ๐ , ๐ )] ๐๐ = ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃณ ๐พ ๐ ,๐ , ๐ = ๐ ๐พ ๐ ,๐ , ๐ = ๐ ๐พ ๐,๐ , ๐ โ { ๐ , ๐ } Lemma A.7.
Suppose ๐พ ( ๐ ) โ ๐พ ( ๐ ) and ๐พ โฒ โ M ( ๐พ ) is suchthat ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ ) . Then swap ( ๐พ โฒ ; ๐ , ๐ ) โ M ( ๐พ ) . Proof. Write ๐พ โฒโฒ = swap ( ๐พ โฒ ; ๐ , ๐ ) . It is clear that ๐พ โฒโฒ hasthe chain property since ๐พ โฒ does. Since ๐พ โฒ โ M ( ๐พ ) , we have ๐ ( ๐พ, ๐พ โฒโฒ ) โฅ ๐ ( ๐พ, ๐พ โฒ ) . We will show that ๐ ( ๐พ, ๐พ โฒโฒ ) โค ๐ ( ๐พ, ๐พ โฒ ) also,which implies ๐ ( ๐พ, ๐พ โฒโฒ ) = ๐ ( ๐พ, ๐พ โฒ ) = ๐ ( ๐พ ) and thus ๐พ โฒโฒ โ M ( ๐พ ) .To that end, observe that for any tournament ห ๐พ , ๐ ( ๐พ, ห ๐พ ) = โ๏ธ ๐ โ ๐ด | ๐พ ( ๐ ) โณ ห ๐พ ( ๐ )| Noting that ๐พ โฒ ( ๐ ) = ๐พ โฒโฒ ( ๐ ) for ๐ โ { ๐ , ๐ } and ๐พ โฒโฒ ( ๐ ) = ๐พ โฒ ( ๐ ) , ๐พ โฒโฒ ( ๐ ) = ๐พ โฒ ( ๐ ) , we have ๐ ( ๐พ, ๐พ โฒ ) โ ๐ ( ๐พ, ๐พ โฒโฒ ) = โ๏ธ ๐ โ{ , } (cid:0) | ๐พ ( ๐ ๐ ) โณ ๐พ โฒ ( ๐ ๐ )| โ | ๐พ ( ๐ ๐ ) โณ ๐พ โฒโฒ ( ๐ ๐ )| (cid:1) = | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| โ | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )|+ | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| โ | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| To simplify notation, write ๐ = ๐พ ( ๐ ) , ๐ โฒ = ๐พ โฒ ( ๐ ) , ๐ = ๐พ ( ๐ ) \ ๐พ ( ๐ ) and ๐ โฒ = ๐พ โฒ ( ๐ ) \ ๐พ โฒ ( ๐ ) so that ๐พ ( ๐ ) = ๐ ; ๐พ ( ๐ ) = ๐ โช ๐๐พ โฒ ( ๐ ) = ๐ โฒ โช ๐ โฒ ; ๐พ โฒ ( ๐ ) = ๐ โฒ and ๐ โฉ ๐ = ๐ โฒ โฉ ๐ โฒ = โ . Rewriting the above we have ๐ ( ๐พ, ๐พ โฒ ) โ ๐ ( ๐พ, ๐พ โฒโฒ ) = | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| + | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )|โ | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| โ | ๐พ ( ๐ ) โณ ๐พ โฒ ( ๐ )| = | ๐ โณ ( ๐ โฒ โช ๐ โฒ )| + |( ๐ โช ๐ ) โณ ๐ โฒ |โ | ๐ โณ ๐ โฒ | โ |( ๐ โช ๐ ) โณ ( ๐ โฒ โช ๐ โฒ )| (7)Each of the symmetric differences in eq. (7) are depicted in Fig-ure 2. Note that each of these sets can be expressed as a union of the8 disjoint subsets of ๐ โช ๐ โช ๐ โฒ โช ๐ โฒ shown in the figure. Expandingthe symmetric differences in eq. (7) and consulting Figure 2, it canbe seen that most terms cancel out, and in fact we are left with ๐ ( ๐พ, ๐พ โฒ ) โ ๐ ( ๐พ, ๐พ โฒโฒ ) = | ๐ โฉ ๐ โฒ | โฅ ๐ ( ๐พ, ๐พ โฒโฒ ) โค ๐ ( ๐พ, ๐พ โฒ ) , and the proof is complete. โก Notation.
For a relation ๐ on a set ๐ and ๐ฅ โ ๐ , write ๐ ( ๐ฅ, ๐ ) = { ๐ฆ โ ๐ | ๐ฅ ๐ ๐ฆ } ๐ฟ ( ๐ฅ, ๐ ) = { ๐ฆ โ ๐ | ๐ฆ ๐ ๐ฅ } for the upper- and lower-sets of ๐ฅ respectively.Lemma A.8. For any tournament ๐พ there is ๐พ โฒ โ M ( ๐พ ) such thatfor all ๐ โ ๐ด : ๐ ( ๐, < A ๐พ ) โ ๐ ( ๐, โฉฝ A ๐พ โฒ ) That is, ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) implies ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ โฒ ) for all ๐, ๐ โฒ โ ๐ด . ๐๐ โฒ ๐ โฒ (a) ๐ โณ ( ๐ โฒ โช ๐ โฒ ) ๐ ๐๐ โฒ ๐ โฒ (b) ( ๐ โช ๐ ) โณ ๐ โฒ ๐ ๐๐ โฒ ๐ โฒ (c) ๐ โณ ๐ โฒ ๐ ๐๐ โฒ ๐ โฒ (d) ( ๐ โช ๐ ) โณ ( ๐ โฒ โช ๐ โฒ ) Figure 2: Depictions of the sets in Equation (7)
Proof. Write ๐ด = { ๐ , . . . , ๐ ๐ } , ordered such that | ๐ฟ ( ๐ , โฉฝ A ๐พ )| โค ยท ยท ยท โค | ๐ฟ ( ๐ ๐ , โฉฝ A ๐พ )| . We will show by induc-tion that for each 0 โค ๐ โค ๐ there is ๐พ ๐ โ M ( ๐พ ) suchthat: 1 โค ๐ โค ๐ = โ ๐ ( ๐ ๐ , < A ๐พ ) โ ๐ ( ๐ ๐ , โฉฝ A ๐พ ๐ ) ( โ )The result follows by taking ๐พ โฒ = ๐พ ๐ .The case ๐ = ๐พ to bean arbitrary member of M ( ๐พ ) . For the inductive step, suppose( โ ) holds for some 0 โค ๐ < ๐ . If ๐ ( ๐ ๐ + , < A ๐พ ) = โ then we mayset ๐พ ๐ + = ๐พ ๐ , so assume that ๐ ( ๐ ๐ + , < A ๐พ ) is non-empty. Takesome ห ๐ โ min ( ๐ ( ๐ ๐ + , < A ๐พ ) , โฉฝ A ๐พ ๐ ) . Then ห ๐ has (one of) the smallestneighbourhoods in ๐พ ๐ amongst those in ๐ด with a strictly largerneighbourhood than ๐ ๐ + in ๐พ .If ๐พ ๐ ( ๐ ๐ + ) โ ๐พ ๐ ( ห ๐ ) then we claim ( โ ) holds with ๐พ ๐ + = ๐พ ๐ .Indeed, for ๐ < ๐ + โ ) holds since it does for ๐พ ๐ .For ๐ = ๐ +
1, let ๐ โ ๐ ( ๐ ๐ + , < A ๐พ ) . The definition of ห ๐ implies ๐พ ๐ ( ๐ ) โ ๐พ ๐ ( ห ๐ ) ; since ๐พ ๐ has the chain property this means ๐พ ๐ ( ห ๐ ) โ ๐พ ๐ ( ๐ ) .Consequently ๐พ ๐ ( ๐ ๐ + ) โ ๐พ ๐ ( ห ๐ ) โ ๐พ ๐ ( ๐ ) , i.e. ๐ โ ๐ ( ๐ ๐ + , โฉฝ A ๐พ ๐ ) = ๐ ( ๐ ๐ + , โฉฝ A ๐พ ๐ + ) as required.For the remainder of the proof we therefore suppose ๐พ ๐ ( ๐ ๐ + ) โ ๐พ ๐ ( ห ๐ ) . The chain property for ๐พ ๐ gives ๐พ ๐ ( ห ๐ ) โ ๐พ ๐ ( ๐ ๐ + ) . Since ๐พ ๐ โ M ( ๐พ ) and ๐พ ( ๐ ๐ + ) โ ๐พ ( ห ๐ ) , we may apply Lemma A.7. Set ๐พ ๐ + = swap ( ๐พ ๐ ; ๐ ๐ + , ห ๐ ) โ M ( ๐พ ) . The inclusion in ( โ ) is easyto show for ๐ = ๐ +
1: if ๐ โ ๐ ( ๐ ๐ + , < A ๐พ ) then either ๐ = ห ๐ โin which case ๐พ ๐ + ( ๐ ๐ + ) โ ๐พ ๐ + ( ๐ ) by construction โ or ๐ โ ห ๐ and ๐พ ๐ + ( ๐ ๐ + ) = ๐พ ๐ ( ห ๐ ) โ ๐พ ๐ ( ๐ ) = ๐พ ๐ + ( ๐ ) . In either case ๐ โ ๐ ( ๐ ๐ + , โฉฝ A ๐พ ๐ + ) as required.Now suppose 1 โค ๐ < ๐ +
1. First note that due to our assump-tion on the ordering of { ๐ , . . . , ๐ ๐ } , we have ๐ ๐ โ ห ๐ (indeed, if ๐ ๐ = ห ๐ then ๐พ ( ๐ ๐ + ) โ ๐พ ( ๐ ๐ ) and | ๐ฟ ( ๐ ๐ , < A ๐พ )| > | ๐ฟ ( ๐ ๐ + , < A ๐พ )| ).Since ๐ ๐ โ ๐ ๐ + also, ๐ ๐ was not involved in the swapping in theconstruction of ๐พ ๐ + , and consequently ๐พ ๐ + ( ๐ ๐ ) = ๐พ ๐ ( ๐ ๐ ) . Let ๐ โ ๐ ( ๐ ๐ , < A ๐พ ) . We must show that ๐พ ๐ + ( ๐ ๐ ) โ ๐พ ๐ + ( ๐ ) . We con-sider cases. Case 1: ๐ = ห ๐ . Using the fact that ( โ ) holds for ๐พ ๐ we have ๐พ ๐ + ( ๐ ๐ ) = ๐พ ๐ ( ๐ ๐ ) โ ๐พ ๐ ( ห ๐ ) โ ๐พ ๐ ( ๐ ๐ + ) = ๐พ ๐ + ( ห ๐ ) Case 2: ๐ = ๐ ๐ + . Here ๐พ ( ๐ ๐ ) โ ๐พ ( ๐ ๐ + ) โ ๐พ ( ห ๐ ) , i.e. ห ๐ โ ๐ ( ๐ ๐ , < A ๐พ ) . Applying the inductive hypothesis again we have ๐พ ๐ + ( ๐ ๐ ) = ๐พ ๐ ( ๐ ๐ ) โ ๐พ ๐ ( ห ๐ ) = ๐พ ๐ + ( ๐ ๐ + ) Case 3: ๐ โ { ห ๐, ๐ ๐ + } . Here neither ๐ ๐ nor ๐ were involved in theswap, so ๐พ ๐ + ( ๐ ๐ ) = ๐พ ๐ ( ๐ ๐ ) โ ๐พ ๐ ( ๐ ) = ๐พ ๐ + ( ๐ ) .By induction, the proof is complete. โก Lemma A.9.
Let ๐พ be a tournament and suppose ๐พ โฒ โ M ( ๐พ ) is such that ๐ ( ๐, < A ๐พ ) โ ๐ ( ๐, โฉฝ A ๐พ โฒ ) for all ๐ โ ๐ด . Then there is ๐พ โฒโฒ โ M ( ๐พ ) such that โฉฝ A ๐พ โ โฉฝ A ๐พ โฒโฒ . Proof. Let ๐ด , . . . , ๐ด ๐ก โ ๐ด be the equivalence classes of โ A ๐พ ,the symmetric part of โฉฝ A ๐พ . Note that ๐ โ A ๐พ ๐ โฒ iff ๐พ ( ๐ ) = ๐พ ( ๐ โฒ ) , sowe can associate each ๐ด ๐ with a neighbourhood ๐ต ๐ โ ๐ต such that ๐พ ( ๐ ) = ๐ต ๐ whenever ๐ โ ๐ด ๐ .Our aim is to select a single element from each equivalence class ๐ด ๐ , which we denote by ๐ ( ๐ด ๐ ) , and modify ๐พ โฒ to set the neighbour-hood of each ๐ โ ๐ด ๐ to ๐พ โฒ ( ๐ ( ๐ด ๐ )) . To that end, construct a function ๐ : { ๐ด , . . . , ๐ด ๐ก } โ ๐ด such that ๐ ( ๐ด ๐ ) โ arg min ๐ โ ๐ด ๐ | ๐ต ๐ โณ ๐พ โฒ ( ๐ )| โ ๐ด ๐ Define ๐พ โฒโฒ by ๐พ โฒโฒ ๐๐ = ๐พ โฒ ๐ ( [ ๐ ]) ,๐ , where [ ๐ ] denotes the equivalenceclass of ๐ . Then ๐พ โฒโฒ ( ๐ ) = ๐พ โฒ ( ๐ ([ ๐ ])) for all ๐ .Next we show that ๐พ โฒโฒ โ M ( ๐พ ) . Note that ๐พ โฒโฒ has thechain property, since ๐ โฉฝ A ๐พ โฒโฒ ๐ iff ๐ ([ ๐ ]) โฉฝ A ๐พ โฒ ๐ ([ ๐ ]) , and ๐ ([ ๐ ]) , ๐ ([ ๐ ]) are guaranteed to be comparable with respect to โฉฝ A ๐พ โฒ since ๐พ โฒ has the chain property. To show ๐ ( ๐พ, ๐พ โฒโฒ ) is minimal,observe that ๐ ( ๐พ, ๐พ โฒโฒ ) = โ๏ธ ๐ โ ๐ด | ๐พ ( ๐ ) โณ ๐พ โฒโฒ ( ๐ )| = ๐ก โ๏ธ ๐ = โ๏ธ ๐ โ ๐ด ๐ | ๐ต ๐ โณ ๐พ โฒ ( ๐ ( ๐ด ๐ ))| By definition of ๐ , we have | ๐ต ๐ โณ ๐พ โฒ ( ๐ ( ๐ด ๐ ))| โค | ๐ต ๐ โณ ๐พ โฒ ( ๐ )| for all ๐ โ ๐ด ๐ . Consequently ๐ ( ๐พ, ๐พ โฒโฒ ) โค ๐ก โ๏ธ ๐ = โ๏ธ ๐ โ ๐ด ๐ | ๐ต ๐ โณ ๐พ โฒ ( ๐ )| = ๐ ( ๐พ, ๐พ โฒ ) = ๐ ( ๐พ ) which implies ๐พ โฒโฒ โ M ( ๐พ ) .e are now ready to prove the result. Suppose ๐ โฉฝ A ๐พ ๐ โฒ i.e. ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) . If ๐พ ( ๐ ) = ๐พ ( ๐ โฒ ) then [ ๐ ] = [ ๐ โฒ ] , so ๐พ โฒโฒ ( ๐ ) = ๐พ โฒ ( ๐ ([ ๐ ])) = ๐พ โฒ ( ๐ ([ ๐ โฒ ])) = ๐พ โฒโฒ ( ๐ โฒ ) and in particular ๐พ โฒโฒ ( ๐ ) โ ๐พ โฒโฒ ( ๐ โฒ ) . If instead ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) , then ๐พ ( ๐ ([ ๐ ])) = ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) = ๐พ ( ๐ ([ ๐ โฒ ])) , i.e. ๐ ([ ๐ ]) < A ๐พ ๐ ([ ๐ โฒ ]) .By the assumption on ๐พ โฒ in the statement of the lemma, this means ๐ ([ ๐ ]) โฉฝ A ๐พ โฒ ๐ ([ ๐ โฒ ]) , and so ๐พ โฒโฒ ( ๐ ) = ๐พ โฒ ( ๐ ([ ๐ ])) โ ๐พ โฒ ( ๐ ([ ๐ โฒ ])) = ๐พ โฒโฒ ( ๐ โฒ ) In either case ๐พ โฒโฒ ( ๐ ) โ ๐พ โฒโฒ ( ๐ โฒ ) , i.e. ๐ โฉฝ A ๐พ โฒโฒ ๐ โฒ . Since ๐, ๐ โฒ werearbitrary, this shows that โฉฝ A ๐พ โ โฉฝ A ๐พ โฒโฒ as required. โก The pieces are now in place to prove Proposition A.4Proof of Proposition A.4. For any tournament ๐พ , write M mon ( ๐พ ) = { ๐พ โฒ โ M ( ๐พ ) | โฉฝ A ๐พ โ โฉฝ A ๐พ โฒ } By Lemma A.8 and Lemma A.9, M mon ( ๐พ ) is non-empty. Let โช beany total order on the set K of all tournaments. Define a function ๐ผ : K โ K by ๐ผ ( ๐พ ) = min (M mon ( ๐พ ) , โช) โ M mon ( ๐พ ) Note that the minimum is unique since โช is a total order. Definingan operator ๐ by ๐ ( ๐พ ) = ( โฉฝ A ๐ผ ( ๐พ ) , โฉฝ B ๐ผ ( ๐พ ) ) , we see that ๐ satisfies chain-min and mon , as required. โก A.6 Proof of Theorem 5.3
The following preliminary result is required.Lemma A.10.
Let ๐ and ๐ be integers with โค ๐ โค ๐ . Then ๐ โ๏ธ ๐ = ๐ โ ๐ < โ( ๐ โ ) Proof. This follows from the formula for the sum of a finitegeometric series: ๐ โ โ๏ธ ๐ = ๐ ๐ = โ ๐ ๐ โ ๐ which holds for all ๐ โ
1. In this case we have ๐ โ๏ธ ๐ = ๐ โ ๐ = ๐ โ๏ธ ๐ = โ ๐ โ ๐ โ โ๏ธ ๐ = โ ๐ = ๐ โ๏ธ ๐ = (cid:18) (cid:19) ๐ โ ๐ โ โ๏ธ ๐ = (cid:18) (cid:19) ๐ = โ (cid:16) (cid:17) ๐ + โ (cid:16) (cid:17) โ โ (cid:16) (cid:17) ๐ โ (cid:16) (cid:17) = (cid:16) โ ๐ โ โ( ๐ + ) (cid:17) = โ( ๐ โ ) โ โ ๐ (cid:124)(cid:123)(cid:122)(cid:125) > < โ( ๐ โ ) as required. โก Proof of Theorem 5.3. Let โด be a total order on N ร N and let ๐, ๐ โ N . For ๐ โ [ ๐ ] and ๐ โ [ ๐ ] , write ๐ ( ๐, ๐ ) = + |{( ๐ โฒ , ๐ โฒ ) โ [ ๐ ] ร [ ๐ ] : ( ๐ โฒ , ๐ โฒ ) โ ( ๐, ๐ )}| for the โpositionโ of ( ๐, ๐ ) in โด โพ ([ ๐ ] ร [ ๐ ]) (where 1 correspondsto the minimal pair). Define ๐ค by ๐ค ( ๐, ๐ ) = + โ ๐ ( ๐,๐ ) If we abuse notation slightly and view ๐ค as an ๐ ร ๐ matrix, wehave, by construction, vec โด ( ๐ค ) = ( + โ , . . . , + โ ๐๐ ) . Notingthat | ๐พ ๐๐ โ ๐พ โฒ ๐๐ | = [ ๐พ โ ๐พ โฒ ] ๐๐ for any tournaments ๐พ, ๐พ โฒ , andletting โข denote the dot product, it is easy to see that ๐ ๐ค ( ๐พ, ๐พ โฒ ) = vec โด ( ๐ค ) โข vec โด ( ๐พ โ ๐พ โฒ ) = ( + โ , . . . , + โ ๐๐ ) โข vec โด ( ๐พ โ ๐พ โฒ ) = ๐ ( ๐พ, ๐พ โฒ ) + ๐ โข vec โด ( ๐พ โ ๐พ โฒ ) where ๐ = ( โ , . . . , โ ๐๐ ) and ๐ ( ๐พ, ๐พ โฒ ) is the unweighted Ham-ming distance. In particular, since ๐ and vec โด ( ๐พ โ ๐พ โฒ ) are non-negative, we have ๐ ๐ค ( ๐พ, ๐พ โฒ ) โฅ ๐ ( ๐พ, ๐พ โฒ ) .Now, we will show that for any ๐ ร ๐ tournament ๐พ and ๐พ โฒ โC ๐,๐ with ๐พ โฒ โ ๐ผ โด ( ๐พ ) we have ๐ ๐ค ( ๐พ, ๐ผ โด ( ๐พ )) < ๐ ๐ค ( ๐พ, ๐พ โฒ ) . Since ๐ผ โด ( ๐พ ) โ M ( ๐พ ) โ C ๐,๐ by definition, this will show that ๐ผ โด ( ๐พ ) is the unique minimum in Equation (6), as required.So, let ๐พ be an ๐ ร ๐ tournament and ๐พ โฒ โ C ๐,๐ . To ease notation,write ๐ฃ = vec โด ( ๐พ โ ๐ผ โด ( ๐พ )) and ๐ฃ โฒ = vec โด ( ๐พ โ ๐พ โฒ ) . There are twocases. Case 1: ๐พ โฒ โ M ( ๐พ ) . In this case we have ๐ ( ๐พ, ๐พ โฒ ) โฅ ๐ ( ๐พ ) + ๐ ๐ค ( ๐พ, ๐ผ โด ( ๐พ )) = ๐ ( ๐พ, ๐ผ โด ( ๐พ )) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = ๐ ( ๐พ ) + ๐ โข ๐ฃ = ๐ ( ๐พ ) + ๐๐ โ๏ธ ๐ = โ ๐ ยท ๐ฃ ๐ (cid:124)(cid:123)(cid:122)(cid:125) โค โค ๐ ( ๐พ ) + ๐๐ โ๏ธ ๐ = โ ๐ (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) < โ = < ๐ ( ๐พ ) + โค ๐ ( ๐พ, ๐พ โฒ )โค ๐ ๐ค ( ๐พ, ๐พ โฒ ) where Lemma A.10 was applied in the 4th step. This shows ๐ ๐ค ( ๐พ, ๐ผ โด ( ๐พ )) < ๐ ๐ค ( ๐พ, ๐พ โฒ ) , as required. Case 2: ๐พ โ M ( ๐พ ) . In this case we have ๐ ( ๐พ, ๐ผ โด ( ๐พ )) โ ๐ ( ๐พ, ๐พ โฒ ) = ( ๐ ( ๐พ ) + ๐ โข ๐ฃ ) โ ( ๐ ( ๐พ ) + ๐ โข ๐ฃ โฒ ) = ๐ โข ( ๐ฃ โ ๐ฃ โฒ ) Now, since ๐พ โฒ โ M ( ๐พ ) , ๐ฃ โฒ appears as one of the vectors overwhich the arg min is taken in Equation (5). By definition of ๐ผ โด we therefore know that ๐ฃ strictly precedes ๐ฃ โฒ with respect to thelexicographic order on { , } ๐๐ . Consequently there is ๐ โฅ ๐ฃ ๐ = ๐ฃ โฒ ๐ for ๐ < ๐ and ๐ฃ ๐ < ๐ฃ โฒ ๐ . That is, ๐ฃ ๐ = ๐ฃ โฒ ๐ =
1. Thismeans ๐ ( ๐พ, ๐ผ โด ( ๐พ )) โ ๐ ( ๐พ, ๐พ โฒ ) = ๐ โข ( ๐ฃ โ ๐ฃ โฒ ) = ๐๐ โ๏ธ ๐ = โ ๐ ( ๐ฃ ๐ โ ๐ฃ โฒ ๐ ) = ๐ โ โ๏ธ ๐ = โ ๐ ( ๐ฃ ๐ โ ๐ฃ โฒ ๐ ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = + ๐๐ โ๏ธ ๐ = ๐ โ ๐ ( ๐ฃ ๐ โ ๐ฃ โฒ ๐ ) = โ ๐ ( ๐ฃ ๐ โ ๐ฃ โฒ ๐ ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = โ + ๐๐ โ๏ธ ๐ = ๐ + โ ๐ ( ๐ฃ ๐ โ ๐ฃ โฒ ๐ ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) โค โค โ โ ๐ + ๐๐ โ๏ธ ๐ = ๐ + โ ๐ < โ โ ๐ + โ ๐ = ๐ ๐ค ( ๐พ, ๐ผ โด ( ๐พ )) < ๐ ๐ค ( ๐พ, ๐พ โฒ ) , and the proof is complete. โก A.7 Proof of Theorem 6.1
Proof. First we set up some notation. For a total preorder โชฏ ona set ๐ and ๐ง โ ๐ , write [ ๐ง ] โชฏ for the rank of โชฏ containing ๐ง , i.e. theequivalence class of ๐ง in the symmetric closure of โชฏ : [ ๐ง ] โชฏ = { ๐ง โฒ โ ๐ | ๐ง โชฏ ๐ง โฒ and ๐ง โฒ โชฏ ๐ง } Also note that โชฏ can be extended to a total order on the ranks bysetting [ ๐ง ] โชฏ โค [ ๐ง โฒ ] โชฏ iff ๐ง โชฏ ๐ง โฒ .( โ ) We start with the โonly ifโ statement of the theorem. Suppose ๐ satisfies chain-def , and let ๐พ be a tournament. We need to showthat | ranks (โชฏ ๐๐พ ) โ ranks (โ ๐๐พ )| โค ๐พ โฒ with the chain property suchthat ๐ โชฏ ๐๐พ ๐ โฒ iff ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ โฒ ) and ๐ โ ๐๐พ ๐ โฒ iff ( ๐พ โฒ ) โ ( ๐ ) โ( ๐พ โฒ ) โ ( ๐ โฒ ) . Write X = {[ ๐ ] โชฏ ๐๐พ | ๐ โ ๐ด, ๐พ โฒ ( ๐ ) โ โ }Y = {[ ๐ ] โ ๐๐พ | ๐ โ ๐ต, ( ๐พ โฒ ) โ ( ๐ ) โ โ } for the set of ranks in each of the two orders, excluding those whohave empty neighbourhoods in ๐พ โฒ . Note that [ ๐ ] โชฏ ๐๐พ = [ ๐ โฒ ] โชฏ ๐๐พ ifand only if ๐พ โฒ ( ๐ ) = ๐พ โฒ ( ๐ โฒ ) (and similar for ๐ต ).We will show that |X| = |Y| . Enumerate X = { ๐ , . . . , ๐ ๐ } and Y = { ๐ , . . . , ๐ ๐ก } , ordered such that ๐ < ยท ยท ยท < ๐ ๐ and ๐ < ยท ยท ยท < ๐ ๐ก . First we show |X| โค |Y| .For each 1 โค ๐ โค ๐ , the ๐ ๐ be an arbitrary element of ๐ ๐ . Then ๐ โบ ๐๐พ ยท ยท ยท โบ ๐๐พ ๐ ๐ , so โ โ ๐พ โฒ ( ๐ ) โ ยท ยท ยท โ ๐พ โฒ ( ๐ ๐ ) . Since theseinclusions are strict, we can choose ๐ , . . . , ๐ ๐ โ ๐ต such that ๐ โ ๐พ โฒ ( ๐ ) and ๐ ๐ + โ ๐พ โฒ ( ๐ ๐ + ) \ ๐พ โฒ ( ๐ ๐ ) for 1 โค ๐ < ๐ .It follows that ๐ ๐ โ ( ๐พ โฒ ) โ ( ๐ ๐ ) \ ( ๐พ โฒ ) โ ( ๐ ๐ + ) , and thus ( ๐พ โฒ ) โ ( ๐ ๐ ) โ ( ๐พ โฒ ) โ ( ๐ ๐ + ) . Since ๐พ โฒ has the chain property, thismeans ( ๐พ โฒ ) โ ( ๐ ๐ + ) โ ( ๐พ โฒ ) โ ( ๐ ๐ ) , i.e. ๐ ๐ โ ๐๐พ ๐ ๐ + . We now have ๐ โ ๐๐พ ยท ยท ยท โ ๐๐พ ๐ ๐ ; a chain of ๐ strict inequalities in โ ๐๐พ . The corresponding ranks [ ๐ ] , . . . , [ ๐ ๐ ] are all distinct and lieinside Y . But now we have found ๐ = |X| distinct elements of Y ,so |X| โค |Y| as promised.Repeating this argument with the roles of X and Y interchanged,we find that |Y| โค |X| also, and therefore |X| = |Y| .To conclude, note that ranks (โชฏ ๐๐พ ) โ {|X| , |X| + } , since therecan exist at most one rank which was excluded from X (namely,those ๐ โ ๐ด with ๐พ โฒ ( ๐ ) = โ ). For identical reasons, ranks (โ ๐๐พ ) โ{|Y| , |Y| + } . Since |X| = |Y| , it is clear that ranks (โชฏ ๐๐พ ) and ranks (โ ๐๐พ ) can differ by at most one, as required.( โ ) Now we prove the โifโ statement. Let ๐พ be a tournament. Wehave | ranks (โชฏ ๐๐พ ) โ ranks (โ ๐๐พ )| โค
1, and must show there is tour-nament ๐พ โฒ with the chain property such that ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) .Let ๐ < ยท ยท ยท < ๐ ๐ and ๐ < ยท ยท ยท < ๐ ๐ก be the ranks of โชฏ ๐๐พ and โ ๐๐พ respectively. By hypothesis | ๐ โ ๐ก | โค
1. Define ๐ : { , . . . , ๐ } โ{ , . . . , ๐ก } by ๐ ( ๐ ) = (cid:40) ๐, ๐ โ { ๐ก โ , ๐ก } ๐ โ , ๐ = ๐ก + | ๐ โ ๐ก | โค ๐ โ [ ๐ ] , write ๐ ๐ = (cid:216) โค ๐ โค ๐ ( ๐ ) ๐ ๐ where ๐ : = โ . Note that ๐ ( ๐ + ) = ๐ ( ๐ ) +
1, and consequently ๐ ๐ + = (cid:216) ๐ โค ๐ ( ๐ )+ ๐ ๐ = ๐ ๐ โช ๐ ๐ ( ๐ )+ = ๐ ๐ โช ๐ ๐ ( ๐ + ) Since ๐ ( ๐ + ) > ๐ ๐ ( ๐ + ) โ โ , and thus ๐ ๐ + โ ๐ ๐ for all ๐ < ๐ .Now, for any ๐ โ ๐ด , let ๐ ( ๐ ) โ [ ๐ ] be the unique integer such that ๐ โ ๐ ๐ ( ๐ ) ; such ๐ ( ๐ ) always exists since { ๐ , . . . , ๐ ๐ } is a partitionof ๐ด . Note that due to the assumption on the ordering of the ๐ ๐ , wehave ๐ โชฏ ๐๐พ ๐ โฒ if and only if ๐ ( ๐ ) โค ๐ ( ๐ โฒ ) .Let ๐พ โฒ be the unique tournament such that ๐พ โฒ ( ๐ ) = ๐ ๐ ( ๐ ) foreach ๐ โ ๐ด . Since ๐ โ ยท ยท ยท โ ๐ ๐ , we have ๐ โชฏ ๐๐พ ๐ โฒ โโ ๐ ( ๐ ) โค ๐ ( ๐ โฒ )โโ ๐ ๐ ( ๐ ) โ ๐ ๐ ( ๐ โฒ ) โโ ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ โฒ )โโ ๐ โฉฝ A ๐พ โฒ ๐ โฒ (8)i.e. โชฏ ๐๐พ = โฉฝ A ๐พ โฒ . Since โชฏ ๐๐พ is a total preorder, this shows that ๐พ โฒ hasthe chain property.It only remains to show that โ ๐๐พ = โฉฝ B ๐พ โฒ . First note that if ๐ โ ๐ ๐ and ๐ โ ๐ ๐ , the fact that { ๐ , . . . , ๐ ๐ก } are disjoint implies ๐ โ ( ๐พ โฒ ) โ ( ๐ ) โโ ๐ โ ๐พ โฒ ( ๐ ) = ๐ ๐ = (cid:216) โค ๐ โค ๐ ( ๐ ) ๐ ๐ โโ ๐ โค ๐ ( ๐ ) Hence ( ๐พ โฒ ) โ ( ๐ ) only depends on ๐ : every ๐ โ ๐ ๐ shares the sameneighbourhood ๐ ๐ , given by ๐ ๐ = (cid:216) ๐ โ[ ๐ ] : ๐ ( ๐ ) โฅ ๐ ๐ ๐ ote that if 1 โค ๐ < ๐ก , ๐ ๐ = (cid:216) ๐ โ[ ๐ ] : ๐ ( ๐ ) โฅ ๐ ๐ ๐ = (cid:169)(cid:173)(cid:171) (cid:216) ๐ โ[ ๐ ] : ๐ ( ๐ ) โฅ ๐ + ๐ ๐ (cid:170)(cid:174)(cid:172) โช (cid:169)(cid:173)(cid:171) (cid:216) ๐ โ ๐ โ ( ๐ ) ๐ ๐ (cid:170)(cid:174)(cid:172) = ๐ ๐ + โช (cid:216) ๐ โ ๐ โ ( ๐ ) ๐ ๐ Since 1 โค ๐ < ๐ก we have ๐ โ ( ๐ ) = (cid:40) { ๐ } , ๐ โ { ๐ก โ , ๐ก }{ ๐ + } , ๐ = ๐ก + ๐ โ ( ๐ ) โ โ , which means (cid:208) ๐ โ ๐ โ ( ๐ ) ๐ ๐ โ โ and thus ๐ ๐ โ ๐ ๐ + for all 1 โค ๐ < ๐ก .Finally, since ( ๐พ โฒ ) โ ( ๐ ) = ๐ ๐ for ๐ โ ๐ ๐ and ๐ โ ยท ยท ยท โ ๐ ๐ก , anargument almost identical to (8) shows that โ ๐๐พ = โฉฝ B ๐พ โฒ .We have shown that ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) and that ๐พ โฒ has thechain property, and the proof is therefore complete. โก A.8 Proof that the interleaving procedureeventually terminates
Proposition A.11.
Let ( ๐ , ๐ ) be selection functions. Fix a tourna-ment ๐พ and let ๐ด ๐ , ๐ต ๐ ( ๐ โฅ ) be as in Definition 6.3. Then there are ๐, ๐ โฒ โฅ such that ๐ด ๐ = โ and ๐ต ๐ โฒ = โ . Moreover, there is ๐ก โฅ suchthat both ๐ด ๐ก = ๐ต ๐ก = โ . Proof. Suppose ๐ โฅ ๐ด ๐ โ โ . Then properties (i) and(ii) for ๐ in Definition 6.2 imply that โ โ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) โ ๐ด ๐ , andconsequently ๐ด ๐ + = ๐ด ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) โ ๐ด ๐ .Supposing that ๐ด ๐ โ โ for all ๐ โฅ
0, we would have ๐ด โ ๐ด โ ๐ด โ ยท ยท ยท which clearly cannot be the case since each ๐ด ๐ lies inside ๐ด which is a finite set. Hence there is ๐ โฅ ๐ด ๐ = โ .Moreover, since ๐ด ๐ โ ๐ด ๐ + โ ๐ด ๐ + โ ยท ยท ยท , we have ๐ด ๐ = โ for all ๐ โฅ ๐ .An identical argument with ๐ shows that there is ๐ โฒ โฅ ๐ต ๐ โฒ = โ and ๐ต ๐ = โ for all ๐ โฅ ๐ โฒ .Taking ๐ก = max { ๐, ๐ โฒ } , we have ๐ด ๐ก = ๐ต ๐ก = โ as required. โก A.9 Proof of Theorem 6.4
Proof. Throughout the proof we will refer to a pair of totalpreorders (โชฏ , โ) as โchain-definableโ if there is a chain tournament ๐พ such that โชฏ = โฉฝ A ๐พ and โ = โฉฝ B ๐พ .( โ ) First we prove the โifโ direction. Let ๐ = ๐ int ๐ ,๐ be an interleav-ing operator with selection functions ( ๐ , ๐ ) , and fix a tournament ๐พ . We will show that ๐ ( ๐พ ) is chain-definable.As per Proposition A.11, let ๐, ๐ โฒ โฅ ๐ด ๐ = โ and ๐ต ๐ โฒ = โ . Then we have ๐ด โ ยท ยท ยท โ ๐ด ๐ โ โ ๐ด ๐ = โ and ๐ต โ ยท ยท ยท โ ๐ต ๐ โฒ โ โ ๐ต ๐ โฒ = โ .Recall that, for ๐ โ ๐ด , we have by definition ๐ ( ๐ ) = max { ๐ | ๐ โ ๐ด ๐ } , which is the unique integer such that ๐ โ ๐ด ๐ ( ๐ ) \ ๐ด ๐ ( ๐ )+ .Since ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ ( ๐ ) โฅ ๐ ( ๐ โฒ ) , it follows that the non-empty sets ๐ด \ ๐ด , . . . , ๐ด ๐ โ \ ๐ด ๐ form the ranks of the total preorder โชฏ ๐๐พ (thatis, the equivalence classes of the symmetric closure โ ๐๐พ ). Thus, โชฏ ๐๐พ has ๐ ranks. An identical argument shows that โ ๐๐พ has ๐ โฒ ranks. It follows from Theorem 6.1 that ๐ ( ๐พ ) is chain-definable if andonly if | ๐ โ ๐ โฒ | โค
1. If ๐ = ๐ โฒ this is clear. Suppose ๐ < ๐ โฒ . Then ๐ด ๐ = โ and ๐ต ๐ โ โ . By property (iii) for ๐ in Definition 6.2, wehave ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ( ๐พ, โ , ๐ต ๐ ) = ๐ต ๐ . But this means ๐ต ๐ + = ๐ต ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ต ๐ \ ๐ต ๐ = โ . Consequently ๐ โฒ = ๐ +
1, and | ๐ โ ๐ โฒ | = | โ | = ๐ > ๐ โฒ , then a similar argument using property (iii) for ๐ in Definition 6.2 shows that ๐ = ๐ โฒ +
1, and we have | ๐ โ ๐ โฒ | = | | = | ๐ โ ๐ โฒ | โค ๐ ( ๐พ ) is chain-definable asrequired.( โ ) Now for the โonly ifโ direction. Suppose ๐ satisfies chain-def . We will define ๐ , ๐ such that ๐ = ๐ int ๐ ,๐ . The idea behind theconstruction is straightforward: since ๐ and ๐ pick off the next-top-ranked ๐ด s and ๐ต s at each iteration, simply define ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) asthe maximal elements of ๐ด ๐ with respect to the existing ordering โชฏ ๐๐พ ( ๐ will be defined similarly). The interleaving algorithm willthen select the ranks of โชฏ ๐๐พ and โ ๐๐พ one-by-one; the fact that ๐ ( ๐พ ) is chain-definable ensures that we select all the ranks before theiterative procedure ends. The formal details follow.Fix a tournament ๐พ . By Theorem 6.1, | ranks (โชฏ ๐๐พ )โ ranks (โ ๐๐พ )| โค
1. Taking ๐ก = max { ranks (โชฏ ๐๐พ ) , ranks (โ ๐๐พ )} , we can write ๐ , . . . , ๐ ๐ก โ ๐ด and ๐ , . . . , ๐ ๐ก โ ๐ต for the ranks of โชฏ ๐๐พ and โ ๐๐พ respectively, possibly with ๐ = โ if ranks (โ ๐๐พ ) = + ranks (โชฏ ๐๐พ ) or ๐ = โ if ranks (โชฏ ๐๐พ ) = + ranks (โ ๐๐พ ) . Note that ๐ ๐ , ๐ ๐ โ โ for ๐ >
1. Assume these sets are ordered such that ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ โค ๐ whenever ๐ โ ๐ ๐ and ๐ โฒ โ ๐ ๐ (and similar for the ๐ ๐ ). Also notethat the ๐ ๐ โฉ ๐ ๐ = โ for ๐ โ ๐ (and similar for the ๐ ๐ ).Now set ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = (cid:40) max ( ๐ด โฒ , โชฏ ๐๐พ ) , ๐ต โฒ โ โ ๐ด โฒ , ๐ต โฒ = โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = (cid:40) max ( ๐ต โฒ , โ ๐๐พ ) , ๐ด โฒ โ โ ๐ต โฒ , ๐ด โฒ = โ It is not difficult to see that ๐ and ๐ satisfy the conditions ofDefinition 6.2 for selection functions. We claim that for with ๐ด ๐ , ๐ต ๐ denoting the interleaving sets for ๐พ and ( ๐ , ๐ ) , for all 0 โค ๐ โค ๐ก wehave ๐ด ๐ = ๐ก โ ๐ (cid:216) ๐ = ๐ ๐ , ๐ต ๐ = ๐ก โ ๐ (cid:216) ๐ = ๐ ๐ (9)For ๐ = ๐ , . . . , ๐ ๐ก contains all ranks of โชฏ ๐๐พ wehave (cid:208) ๐ก โ ๐ = = ๐ โช ยท ยท ยท โช ๐ ๐ก = ๐ด = ๐ด (and similar for ๐ต ).Now suppose (9) holds for some 0 โค ๐ < ๐ก . We will show that ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ๐ก โ ๐ by considering three possible cases, at least oneof which must hold. Case 1: ( ๐ด ๐ โ โ , ๐ต ๐ โ โ ). Here we have ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = max ( ๐ด ๐ , โชฏ ๐๐พ ) = max ( ๐ โช ยท ยท ยท โช ๐ ๐ก โ ๐ , โชฏ ๐๐พ ) = ๐ ๐ก โ ๐ since the ๐ ๐ form (disjoint) ranks of โชฏ ๐๐พ with ๐ ๐ โบ ๐ ๐ for ๐ < ๐ . Here max ( ๐, โชฏ) = { ๐ง โ ๐ | (cid:154) ๐ง โฒ โ ๐ : ๐ง โบ ๐ง โฒ } , for any set ๐ and a total preorder โชฏ on ๐ (with strict part โบ ). ase 2: ( ๐ต ๐ = โ ). Here we have (cid:208) ๐ก โ ๐๐ = ๐ ๐ = โ . Since ๐ก โ ๐ โฅ ๐ ๐ โ โ for ๐ >
1, it must be the case that ๐ก โ ๐ = ๐ต ๐ = ๐ = โ . Consequently by the induction hypothesis we have ๐ด ๐ = (cid:208) ๐ = ๐ ๐ = ๐ , and thus ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ( ๐พ, ๐ด ๐ , โ ) = ๐ด ๐ = ๐ = ๐ ๐ก โ ๐ Case 3: ( ๐ด ๐ = โ ). By a similar argument as in case 2, we musthave ๐ก โ ๐ = ๐ด ๐ = ๐ = โ . Using the fact that ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) โ ๐ด ๐ we get ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ( ๐พ, โ , ๐ต ๐ ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) โโ = โ = ๐ = ๐ ๐ก โ ๐ We have now covered all cases, and have shown that ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ๐ก โ ๐ must hold. Consequently, using again the fact that the ๐ ๐ aredisjoint, ๐ด ๐ + = ๐ด ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = (cid:169)(cid:173)(cid:171) ๐ก โ ๐ (cid:216) ๐ = ๐ ๐ (cid:170)(cid:174)(cid:172) \ ๐ ๐ก โ ๐ = ๐ก โ( ๐ + ) (cid:216) ๐ = ๐ ๐ as required. By almost identical arguments we can show that ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ ) = ๐ ๐ก โ ๐ , and thus ๐ต ๐ + = (cid:208) ๐ก โ( ๐ + ) ๐ = ๐ ๐ also. By induc-tion, (9) holds for all 0 โค ๐ โค ๐ก .It remains to show that ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ โชฏ ๐ int ๐ ,๐ ๐พ ๐ โฒ and that ๐ โ ๐๐พ ๐ โฒ iff ๐ โ ๐ int ๐ ,๐ ๐พ ๐ โฒ .For ๐ โ ๐ด , let ๐ ( ๐ ) be the unique integer such that ๐ โ ๐ ๐ ( ๐ ) , i.e. ๐ ( ๐ ) is the index of the rank of ๐ in the ordering โชฏ ๐๐พ . Note that wehave ๐ โ ๐ด ๐ = ๐ โช ยท ยท ยท โช ๐ ๐ก โ ๐ โโ ๐ก โ ๐ โฅ ๐ ( ๐ ) and therefore ๐ ( ๐ ) = max { ๐ | ๐ โ ๐ด ๐ } = max { ๐ | ๐ก โ ๐ โฅ ๐ ( ๐ )} = ๐ก โ ๐ ( ๐ ) Using the fact that ๐ ๐ โบ ๐ ๐ for ๐ < ๐ , we get ๐ โชฏ ๐ int ๐ ,๐ ๐พ ๐ โฒ โโ ๐ ( ๐ ) โฅ ๐ ( ๐ โฒ )โโ ๐ก โ ๐ ( ๐ ) โฅ ๐ก โ ๐ ( ๐ โฒ )โโ ๐ ( ๐ ) โค ๐ ( ๐ โฒ )โโ ๐ โชฏ ๐๐พ ๐ โฒ A similar argument shows that ๐ โ ๐๐พ ๐ โฒ iff ๐ โ ๐ int ๐ ,๐ ๐พ ๐ โฒ for any ๐, ๐ โฒ โ ๐ต . Since ๐พ was arbitrary, we have shown that ๐ = ๐ int ๐ ,๐ asrequired. โก A.10 Proof of Theorem 6.6
Proof. Since chain-min implies chain-def , Theorem 4.3 im-plies the existence of an operator with chain-def and dual , and anoperator with chain-def and mon . Moreover, the trivial operatorwhich ranks all ๐ด s and ๐ต s equally satisfies anon and IIM . It onlyremains to show that there is an operator satisfying both chain-def and pos-resp .To that end, for any tournament ๐พ , define ๐พ โฒ by ๐พ โฒ ๐๐ = (cid:40) , ๐ โค | ๐พ ( ๐ )| , ๐ > | ๐พ ( ๐ )| Note that ๐พ โฒ ( ๐ ) = { , . . . , | ๐พ ( ๐ )|} for | ๐พ ( ๐ )| >
0. Consequently ๐พ โฒ ( ๐ ) โ ๐พ โฒ ( ๐ โฒ ) iff | ๐พ ( ๐ )| โค | ๐พ ( ๐ )| . We see that ๐พ โฒ has the chainproperty, and the operator ๐ defined by ๐ ( ๐พ ) = ( โฉฝ A ๐พ โฒ , โฉฝ B ๐พ โฒ ) satisfies chain-def . In particular, ๐ โชฏ ๐๐พ ๐ โฒ iff | ๐พ ( ๐ )| โค | ๐พ ( ๐ โฒ )| .To show pos-resp , suppose ๐ โชฏ ๐๐พ ๐ โฒ and ๐พ ๐ โฒ ,๐ = ๐, ๐ โฒ โ ๐ด and ๐ โ ๐ต . Write ห ๐พ = ๐พ + ๐ โฒ ,๐ .Since ๐ โชฏ ๐๐พ ๐ โฒ implies | ๐พ ( ๐ )| โค | ๐พ ( ๐ โฒ )| , we have | ห ๐พ ( ๐ โฒ )| = + | ๐พ ( ๐ โฒ )| > | ๐พ ( ๐ )| = | ห ๐พ ( ๐ )| , and therefore ๐ โบ ๐ ห ๐พ ๐ โฒ as requiredfor pos-resp . โก A.11 Proof of Theorem 6.7
Proof. For contradiction, suppose there is an operator ๐ satis-fying the stated axioms. Consider ๐พ = (cid:20) (cid:21) and two tournaments obtained by removing a single 1 entry: ๐พ = (cid:20) (cid:21) , ๐พ = (cid:20) (cid:21) Now, anon in ๐พ gives 1 โ ๐๐พ ๐ = ( ) , ๐ = id ๐ต ). Inparticular, 1 โชฏ ๐๐พ
2, so pos-resp implies 1 โบ ๐๐พ
2. A similar argumentwith ๐พ shows that 3 โ ๐๐พ โบ ๐๐พ anon to ๐พ directly with ๐ = ( ) and ๐ = ( ) , we see that 2 โ ๐๐พ
3. The ranking of ๐ด is thus fullydetermined as 1 โบ โ โบ
4. In particular, ranks (โชฏ ๐๐พ ) = ๐พ = (cid:2) (cid:3) andapplying permutations ๐ = ( ) and ๐ = ( ) , we obtain 1 โ ๐๐พ anon , i.e. the ๐ด ranking in ๐พ is flat. By dual this implies the ๐ต ranking in ๐พ is flat, i.e. ranks (โ ๐๐พ ) =
1. We see that ranks (โชฏ ๐๐พ ) and ranks (โ ๐๐พ ) differ by 2, contradicting chain-def according toTheorem 6.1. โก A.12 Proof of Theorem 6.8
We require a preliminary result providing sufficient conditions foran interleaving operator ๐ int ๐ ,๐ to satisfy various axioms.Lemma A.12. Let ๐ = ๐ int ๐ ,๐ be an interleaving operator.(1) If for any tournament ๐พ , ๐ด โฒ โ ๐ด , ๐ต โฒ โ ๐ต and for any pair ofpermutations ๐ : ๐ด โ ๐ด and ๐ : ๐ต โ ๐ต we have ๐ ( ๐ ( ๐ ( ๐พ )) , ๐ ( ๐ด โฒ ) , ๐ ( ๐ต โฒ )) = ๐ ( ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ )) ๐ ( ๐ ( ๐ ( ๐พ )) , ๐ ( ๐ด โฒ ) , ๐ ( ๐ต โฒ )) = ๐ ( ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ )) hen ๐ satisfies anon .(2) If for any tournament ๐พ and ๐ด โฒ โ ๐ด , ๐ต โ ๐ต we have ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = ๐ ( ๐พ, ๐ต โฒ , ๐ด โฒ ) then ๐ satisfies dual .(3) If for any tournament ๐พ , ๐ด โฒ โ ๐ด , ๐ต โฒ โ ๐ต and ๐, ๐ โฒ โ ๐ด โฒ wehave ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) = โ ๐ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) or ๐ โฒ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) then ๐ satisfies mon . Proof. We take each statement in turn.(1) Let ๐พ be a tournament. For brevity, write ๐พ โฒ = ๐ ( ๐ ( ๐พ )) .Let us write ๐ด ๐ , ๐ต ๐ and ๐ด โฒ ๐ , ๐ต โฒ ๐ ( ๐ โฅ ) for the sets defined inDefinition 6.3 for ๐พ and ๐พ โฒ respectively. We claim that forall ๐ โฅ ๐ด โฒ ๐ = ๐ ( ๐ด ๐ ) , ๐ต โฒ ๐ = ๐ ( ๐ต ๐ ) (10)For ๐ = ๐ด โฒ = ๐ด = ๐ ( ๐ด ) = ๐ ( ๐ด ) since ๐ is a bijection. The fact that ๐ต โฒ = ๐ ( ๐ต ) is shown similarly.Suppose that (10) holds for some ๐ โฅ
0. Then applying ourassumption on ๐ : ๐ด โฒ ๐ + = ๐ด โฒ ๐ \ ๐ ( ๐พ โฒ , ๐ด โฒ ๐ , ๐ต โฒ ๐ ) = ๐ ( ๐ด ๐ ) \ ๐ ( ๐พ โฒ , ๐ ( ๐ด ๐ ) , ๐ ( ๐ต ๐ )) = ๐ ( ๐ด ๐ ) \ ๐ ( ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ )) = ๐ ( ๐ด ๐ \ ๐ ( ๐พ, ๐ด ๐ , ๐ต ๐ )) = ๐ ( ๐ด ๐ + ) (note that ๐ ( ๐ ) \ ๐ ( ๐ ) = ๐ ( ๐ \ ๐ ) holds for any sets ๐, ๐ due to injectivity of ๐ ). Using the assumption on ๐ we canshow that ๐ต โฒ ๐ + = ๐ ( ๐ต ๐ + ) in a similar manner. Therefore, byinduction, (10) holds for all ๐ โฅ
0. This means that for any ๐ โ ๐ด we have ๐ ( ๐ ) โ ๐ด โฒ ๐ โโ ๐ ( ๐ ) โ ๐ ( ๐ด ๐ ) โโ ๐ โ ๐ด ๐ and therefore, with ๐ ๐พ and ๐ ๐พ โฒ denoting the functions ๐ด โ N defined in Definition 6.3 for ๐พ and ๐พ โฒ respectively, ๐ ๐พ โฒ ( ๐ ( ๐ )) = max { ๐ | ๐ ( ๐ ) โ ๐ด โฒ ๐ } = max { ๐ | ๐ โ ๐ด ๐ } = ๐ ๐พ ( ๐ ) From this it easily follows that ๐ โชฏ ๐๐พ ๐ โฒ iff ๐ ( ๐ ) โชฏ ๐๐พ โฒ ๐ ( ๐ โฒ ) ,i.e. ๐ satisfies anon .(2) Once again, fix a tournament ๐พ and let ๐ด ๐ , ๐ต ๐ and ๐ด โฒ ๐ , ๐ต โฒ ๐ denote the sets from Definition 6.3 for ๐พ and ๐พ respectively.It is easy to show by induction that the assumption on ๐ and ๐ implies ๐ด โฒ ๐ = ๐ต ๐ and ๐ต โฒ ๐ = ๐ด ๐ for all ๐ โฅ ๐ โ ๐ต ๐พ : ๐ ๐พ ( ๐ ) = max { ๐ | ๐ โ ๐ต ๐ } = max { ๐ | ๐ โ ๐ด โฒ ๐ } = ๐ ๐พ ( ๐ ) which implies ๐ โ ๐๐พ ๐ โฒ iff ๐ โชฏ ๐๐พ ๐ โฒ , as required for dual .(3) Let ๐พ be a tournament and ๐, ๐ โฒ โ ๐ด such that ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) .We must show that ๐ โชฏ ๐๐พ ๐ โฒ .Suppose otherwise, i.e. ๐ โฒ โบ ๐๐พ ๐ . Then ๐ ( ๐ โฒ ) > ๐ ( ๐ ) . Notethat by definition of ๐ , we have ๐ โ ๐ด ๐ ( ๐ ) \ ๐ด ๐ ( ๐ )+ = ๐ ( ๐พ, ๐ด ๐ ( ๐ ) , ๐ต ๐ ( ๐ ) ) . Since ๐ ( ๐ โฒ ) โฅ ๐ ( ๐ ) + ๐ด ๐ ( ๐ ) โ ๐ด ๐ ( ๐ )+ โ ๐ด ๐ ( ๐ )+ โ ยท ยท ยท , we get ๐ โฒ โ ๐ด ๐ ( ๐ )+ โ ๐ด ๐ ( ๐ ) .In particular, ๐ โฒ โ ๐ ( ๐พ, ๐ด ๐ ( ๐ ) , ๐ต ๐ ( ๐ ) ) .Piecing this all together, we have ๐, ๐ โฒ โ ๐ด ๐ ( ๐ ) , ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) , ๐ โ ๐ ( ๐พ, ๐ด ๐ ( ๐ ) , ๐ต ๐ ( ๐ ) ) and ๐ โฒ โ ๐ ( ๐พ, ๐ด ๐ ( ๐ ) , ๐ต ๐ ( ๐ ) ) .But this directly contradicts our assumption on ๐ , so we aredone. โก Proof of Theorem 6.8. We take each axiom in turn. Let ๐ and ๐ be the selection functions corresponding to ๐ CI from Example 6.5. chain-def. Since ๐ CI is an interleaving operator, chain-def fol-lows from Theorem 6.4. anon. Let ๐พ be a tournament and let ๐ : ๐ด โ ๐ด and ๐ : ๐ต โ ๐ต be bijective mappings. Write ๐พ โฒ = ๐ ( ๐ ( ๐พ )) . We will show that theconditions on ๐ and ๐ in Lemma A.12 part (1) are satisfied.Let ๐ด โฒ โ ๐ด and ๐ต โฒ โ ๐ต . We have ๐ ( ๐พ โฒ , ๐ ( ๐ด โฒ ) , ๐ ( ๐ต โฒ )) = arg max ห ๐ โ ๐ ( ๐ด โฒ ) | ๐พ โฒ ( ห ๐ ) โฉ ๐ ( ๐ต โฒ )| = ๐ ( arg max ๐ โ ๐ด โฒ | ๐พ โฒ ( ๐ ( ๐ )) โฉ ๐ ( ๐ต โฒ )|) where we make the โsubstitutionโ ๐ = ๐ โ ( ห ๐ ) . Using the defintionof ๐พ โฒ = ๐ ( ๐ ( ๐พ )) it is easily seen that ๐พ โฒ ( ๐ ( ๐ )) = ๐ ( ๐พ ( ๐ )) . Also,since ๐ is a bijection we have ๐ ( ๐ ) โฉ ๐ ( ๐ ) = ๐ ( ๐ โฉ ๐ ) for any sets ๐ and ๐ , and | ๐ ( ๐ )| = | ๐ | . Thus ๐ ( ๐พ โฒ , ๐ ( ๐ด โฒ ) , ๐ ( ๐ต โฒ )) = ๐ ( arg max ๐ โ ๐ด โฒ | ๐พ โฒ ( ๐ ( ๐ )) โฉ ๐ ( ๐ต โฒ )|) = ๐ ( arg max ๐ โ ๐ด โฒ | ๐ ( ๐พ ( ๐ )) โฉ ๐ ( ๐ต โฒ )|) = ๐ ( arg max ๐ โ ๐ด โฒ | ๐ ( ๐พ ( ๐ ) โฉ ๐ต โฒ )|) = ๐ ( arg max ๐ โ ๐ด โฒ | ๐พ ( ๐ ) โฉ ๐ต โฒ |) = ๐ ( ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ )) as required. The result for ๐ follows by a near-identical argument.Thus ๐ CI satisfies anon by Lemma A.12 part (1). dual. Fix a tournament ๐พ and let ๐ด โฒ โ ๐ด , ๐ต โฒ โ ๐ต . Note that for ๐ โ ๐ต โฒ we have | ๐พ โ ( ๐ ) โฉ ๐ด โฒ | = |( ๐ด \ ๐พ ( ๐ )) โฉ ๐ด โฒ | = | ๐ด โฒ \ ๐พ ( ๐ )| = | ๐ด โฒ | โ | ๐พ ( ๐ ) โฉ ๐ด โฒ | Consequently ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) = arg min ๐ โ ๐ต โฒ | ๐พ โ ( ๐ ) โฉ ๐ด โฒ | = arg min ๐ โ ๐ต โฒ (cid:0) | ๐ด โฒ | โ | ๐พ ( ๐ ) โฉ ๐ด โฒ | (cid:1) = arg max ๐ โ ๐ต โฒ | ๐พ ( ๐ ) โฉ ๐ด โฒ | = ๐ ( ๐พ, ๐ต โฒ , ๐ด โฒ ) and, by Lemma A.12 part (2), ๐ CI satisfies dual . mon. Once again, we use Lemma A.12. Let ๐พ be a tournamentand ๐ด โฒ โ ๐ด , ๐ต โฒ โ ๐ต . Suppose ๐, ๐ โฒ โ ๐ด โฒ with ๐พ ( ๐ ) โ ๐พ ( ๐ โฒ ) . Weneed to show that either ๐ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) or ๐ โฒ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) Suppose ๐ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) . Then ๐ โ arg max ห ๐ โ ๐ด โฒ | ๐พ ( ห ๐ ) โฉ ๐ต โฒ | , so | ๐พ ( ๐ )โฉ ๐ต โฒ | โฅ | ๐พ ( ๐ โฒ )โฉ ๐ต โฒ | . On the other hand ๐พ ( ๐ )โฉ ๐ต โฒ โ ๐พ ( ๐ โฒ )โฉ ๐ต โฒ ,o | ๐พ ( ๐ ) โฉ ๐ต โฒ | โค | ๐พ ( ๐ โฒ ) โฉ ๐ต โฒ | . Consequently | ๐พ ( ๐ ) โฉ ๐ต โฒ | = | ๐พ ( ๐ โฒ ) โฉ ๐ต โฒ | , and so ๐ โฒ โ ๐ ( ๐พ, ๐ด โฒ , ๐ต โฒ ) . This shows the property required byLemma A.12 part (3) is satisfied, and thus ๐ CI satisfies mon . pos-resp. We have show that ๐ CI satisfies chain-def , anon and dual ; due to impossibility result of Theorem 6.7, ๐ CI cannot satisfy pos-resp . IIM.
Write ๐พ = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป , ๐พ = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป Note that the first and second rows of each tournament are identical,so
IIM would imply 1 โชฏ ๐ CI ๐พ โชฏ ๐ CI ๐พ
2. However, it is easilyverified that 1 โบ ๐ CI ๐พ โบ ๐ CI ๐พ
1. Therefore ๐ CI does notsatisfy IIM ..