In their final game of the group phase at the 2006 Olympic ice-hockey tournament, a surprisingly lethargic Swedish team lost to Slovakia. The result meant they finished third in their group, when a win would have guaranteed at worst a second placed finish. As the top four teams in each of the two groups qualified for the quarter-finals, Sweden remained in the tournament after this abject performance, but with a lower seeding for the playoffs. However, everything turned out well in the end as they crushed both their quarter- and semi-final opponents ( against Switzerland and against the Czech Republic respectively), before lifting the gold after a narrow win over Finland in the final.
The Slovakia match has gained notoriety because of persistent rumours that Sweden threw the game in order to avoid ending up in the same half of the playoff draw as Canada and Russia, the two traditional giants of ice-hockey. Indeed, in an interview in 2011, Peter Forsberg, one of Sweden’s top stars, seemed to admit as much111See www.expressen.se/sport/hockey/tre-kronor/forsberg-slovakien-var-en-laggmatch, though controversy remains about the proper interpretation of his words. Whatever the truth in this regard, it certainly seems as though Sweden were better off having lost the game.
Instances like this in high-profile sports tournaments, where a competitor is accused of deliberately losing a game, are rare and tend to attract a lot of attention when they occur. This could be considered surprising given that deliberate underperformance in sport is nothing unusual. For example, quite often a team will decide to rest their best players or give less than effort when faced with an ostensibly weaker opponent, having calculated that the risk in so doing is outweighed by future potential benefits. Note that this could occur even in a single-elimination knockout tournament, with a team deciding to trade an elevated risk of an early exit for higher probability of success later on. Of course, in such a tournament it can never be in a team’s interest to actually lose. However, many tournaments, including Olympic ice-hockey, are based on the template of two phases, the first being a round-robin event (everyone meets everyone) which serves to rank the teams, and thereby provide a seeding for the second, knockout phase222In 2006, the Olympic ice hockey tournament employed a minor modification of this template. There were teams. In the first phase, they were divided into two groups of six, each group playing round-robin. The top four teams in each group qualified for the knockout phase. The latter employed standard seeding (c.f. Figure 1), but with the extra condition that teams from the same group could not meet in the quarter-finals. This kind of modification of the basic two-phase template, where the teams are first divided into smaller groups, is very common since it greatly reduces the total number of matches that need to be played. Teams are incentivized to perform well in the first phase by often, only higher ranking teams qualify for the second phase, and standard seeding (c.f. Figure 1) aims to place high ranking teams far apart in the game tree, with higher ranking teams closer to lower ranking ones, meaning that a high rank generally gives you an easier starting position.
The example of Sweden in 2006 illustrates the following phenomenon of two-phase tournaments. Since a weaker team always has a non-zero probability of beating a stronger one in a single match, a motivation to throw a game in the first phase can arise when it seems like the ranking of one’s potential knockout-phase opponents does not reflect their actual relative strengths. Sweden’s loss to Slovakia meant they faced Switzerland instead of Canada in the quarter-final and most observers would probably have agreed that this was an easier matchup, despite Switzerland having finished second and Canada third in their group (Switzerland also beat Canada in their group match).
The above phenomenon is easy to understand and begs the fascinating question of why instances of game-throwing seem to be relatively rare. We don’t explore that (at least partly psycho-social) question further in this paper. However, even if game-throwing is rare, it is still certainly a weakness of this tournament format that situations can arise where a team is given the choice between either pretending to be worse than they are, or playing honestly at the cost of possibly decreasing their chances of winning the tournament.
In a 2000 paper , Allan Schwenk studied the question of how to best seed a knockout tournament from a mathematical point of view. One, perhaps counter-intuitive, observation made in that paper is that standard seeding does not necessarily benefit a higher-ranking players, even when the ranking of its potential opponents accurately reflects their relative strengths. Consider a matchplay tournament with competitors, or “players” as we shall henceforth call them, even though the competitors may be teams. In Schwenk’s mathematical model, the players are numbered through and there are fixed probabilities such that, whenever players and meet, the probability that wins is . Draws are not allowed, thus . Suppose we impose the conditions
(i) whenever ,
(ii) whenever and .
Thus, for any , wins against with probability at least , and for any other player , has at least as high a probability of beating as does. It then seems unconstestable to assert that player is at least as good as player whenever . Indeed, if we imposed strict inequalities in (i) and (ii) we would have an unambiguous ranking of the players: is better than if and only if . This is a very natural model to work with. It is summarized by a so-called doubly monotonic matrix , whose entries equal along the main diagonal, are non-decreasing from left to right along each row, non-increasing from top to bottom along each column and satisfy for all . We shall refer to the model as the doubly monotonic model (DMM) of tournaments. It is the model employed throughout the rest of the paper.
In , Schwenk gave an example of an doubly monotonic matrix such that, if the standard seeding method (illustrated in Figure 1) were employed for a single-elimination tournament, then player would have a higher probability of winning than player .
As an evident corollary, assuming the same mathematical model one can concoct situations in two-phase tournaments of the kind considered above in which it is a player’s interest to lose a game in the first phase even when, say, in every other match played to that point, the better team has won.
Many tournaments consist of only a single phase, either round-robin333or, more commonly, a league format, where each pair meet twice. or single-elimination. As opposed to the aforementioned two-phase format, here it is not hard to see that it can never be in a team’s interest to lose a game. Indeed, this is clear for the single-elimination format, as one loss means you’re out of the tournament. In the round-robin format, losing one game, all else being equal, only decreases your own total score while increasing the score of some other team. As Schwenk showed, the single-elimination option, with standard seeding, may still not be fair, in the sense of always giving a higher winning probability to a better player. The obvious way around this is to randomize the draw. Schwenk proposed a method called Cohort Randomized Seeding444It is easy to see that the standard method cannot result in a player from a lower cohort, as that term is defined by Schwenk, having a higher probability of winning the tournament than one in a higher cohort., which seeks to respect the economic incentives behind the standard method555The standard format ensures the romance of “David vs. Goliath” matchups in the early rounds, plus the likelihood of the later rounds featuring contests between the top stars, when public interest is at its highest. Schenk used the term delayed confrontation for the desire to keep the top ranked players apart in the early rounds. while introducing just enough randomization to ensure that this basic criterion for fairness is satisfied. According to Schwenk himself, in email correspondence with us, no major sports competition has yet adopted his proposal666On the other hand, uniformly random draws are commonly employed. An example is the English FA Cup, from the round-of- onwards..
Even tournaments where it is never beneficial to lose a match often include another source
of unfairness, in that players may face quite different schedules, for
reasons of geography, tradition and so on. For example, qualifying for the
soccer World Cup is organized by continent, an arrangement that effectively
punishes European teams. The host nation automatically qualifies
for the finals
and is given a top seeding in the group phase, thus giving it an unfair
advantage over everyone else. In the spirit of fair competition, one would
ideally wish for a tournament not to give certain players any special
treatment from the outset, and only break this symmetry after seeing how the
teams perform within the confines of the tournament. Note that a
single-elimination tournament with standard seeding is an example of such
“asymmetric scheduling”, unless the previous performances upon which the
seeding is founded are considered part of the
The above considerations lead us to the question on which this paper is based. Suppose the rules of a tournament ensure both
- honesty, meaning it is impossible for a situation to arise where it is in a player’s interest to lose a game, and
- symmetry, meaning that the rules treat all the players equally.
In particular, the rules should not depend on the identity of the players, or the order in which they entered the tournament.
Must it follow that the tournament is fair, in the sense that a better player always has at least as high a probability of winning the tournament as a worse one? Having defined our terms precisely we will show below that the answer, perhaps surprisingly, is no. Already for three players, we will provide simple examples of tournaments which are symmetric and honest, but not fair. The question of “how unfair” a symmetric and honest tournament can be seems to be non-trivial for any number of players. For we solve this problem exactly, and for we formulate a general conjecture. The rest of the paper is organized as follows:
Section 2 provides rigorous definitions. We will define what we mean by a (matchplay) tournament and what it means for a tournament to be either symmetric, honest or fair. The DMM is assumed throughout.
Sections 3 and 4 are the heart of the paper. In the former, we consider -player tournaments and describe what appear to be the simplest possible examples of tournaments which are symmetric and honest, but not fair. Theorem 3.6 gives a precise characterization of those probability vectors which can arise as the vectors of win-probabilities for the players in a symmetric and honest tournament777These results may remind some readers of the notion of a truel and of the known fact that, in a truel, being a better shot does not guarantee a higher probability of winning (that is, of surviving). See https://en.wikipedia.org/wiki/Truel. Despite the analogy, we’re not aware of any deeper connection between our results and those for truels, nor between their respective generalizations to more than three “players”.; here fairness would mean .
In Section 4 we extend these ideas to a general method for constructing symmetric, honest and unfair -player tournaments. We introduce a family of -vertex digraphs and an associated convex polytope of probability vectors in and show that every interior point of this polytope arises as the vector of win-probabilities of some symmetric and honest -player tournament. The polytope includes all probability vectors satisfying , but is shown to have a total of corners, thus yielding a plethora of examples of symmetric and honest, but unfair tournaments. Indeed, we conjecture (Conjecture 4.2) that the vector of win-probabilities of any symmetric and honest -player tournament lies in .
Section 5 considers the notion of a frugal tournament, namely one which always begins by picking one player uniformly at random to take no further part in it (though he may still win). The tournaments constructed in Sections 3 and 4 have this property, and the main result of Section 5 is, in essence, that frugal tournaments provide no counterexamples to Conjecture 4.2.
Section 6 introduces the notion of a tournament map, which is a natural way to view tournaments as continuous functions. We describe its relation to the regular tournament concept. Using this, we show (Corollary 6.4) that any symmetric and honest tournament can be approximated arbitrarily well by one of the form described in the section. We further provide three applications.
- The first is to strictly honest tournaments, which means, informally, that a player should always be strictly better off in winning a match than in losing it. We show that any symmetric and honest tournament can be approximated arbitrarily well by a strictly honest one.
- The second application is to tournaments with rounds. For simplicity, we assume in the rest of the article that matches in a tournament are played one-at-a-time, something which is often not true in reality. Extending the notion of honesty to tournaments with rounds provides some technical challenges, which are discussed here.
- The final application is to prove that the possible vectors of win-probablities for symmetric and honest -player tournaments form a finite union of convex polytopes in , minus some boundary points. This provides, in particular, some further evidence in support of Conjecture 4.2.
In Section 7, we consider the concept of a futile tournament, one in which a player’s probability of finally winning is never affected by whether they win or lose a given match. We prove that, in a symmetric and futile -player tournament, everyone has probability of winning. This is exactly as one would expect, but it doesn’t seem to be a completely trivial task to prove it.
Finally, Section 8 casts a critical eye on the various concepts introduced in the paper, and mentions some further possibilities for future work.
2. Formal Definitions
The word tournament has many different meanings. In graph theory, it refers to a directed graph where, for every pair of vertices and , there is an arc going either from to or from to . In more common language, a matchplay refers to a competition between a (usually relatively large) number of competitors/players/teams in which a winner is determined depending on the outcome of a number of individual matches, each match involving exactly two competitors. We concern ourselves exclusively with matchplay tournaments888Athletics, golf, cycling, skiing etc. are examples of sports in which competitions traditionally take a different form, basically “all-against-all”.. Even with this restriction, the word “tournament” itself can be used to refer to: a reoccurring competition with a fixed name and fixed format, such as the Wimbledon Lawn Tennis Championships, a specific instance of a (potentially reoccurring) competition, such as the 2014 Fifa World Cup, or a specific set of rules by which such a competition is structured, such as “single-elimination knock-out with randomized seeding”, “single round-robin with randomized scheduling”, etc. We will here use tournament in this last sense.
More precisely, we consider an -player tournament as a set of rules for how to arrange matches between players, represented by numbers from to . The decision on which players should meet each other in the next match may depend on the results from earlier matches as well as additional randomness (coin flips etc.). Eventually, the tournament should announce one of the players as the winner. We assume that:
A match is played between an (unordered) pair of players . The outcome of said match can either be won, or won. In particular, no draws are allowed, and no more information is given back to the tournament regarding e.g. how close the match was, number of goals scored etc.
Matches are played sequentially one-at-a-time. In practice, many tournaments consist of “rounds” of simultaneous matches. We’ll make some further remarks on this restriction in Subsection 6.2.
There is a bound on the number of matches that can be played in a specific tournament. So, for example, for three players we would not allow “iteration of round-robin until someone beats the other two”. Instead, we’d require the tournament to break a potential three-way tie at some point, e.g. by randomly selecting a winner.
Formally, we may think of a tournament as a randomized algorithm which is given access to a function PlayMatch that takes as input an unordered pair of numbers between and and returns one of the numbers.
In order to analyze our tournaments, we will need a way to model the outcomes of individual matches. As mentioned in the introduction, we will here employ the same simple model as Schwenk . For each pair of players and , we assume that there is some unchanging probability that wins in a match between them. Thus, by (1) above. We set and denote the set of all possible matrices by
where denotes the all ones matrix. We say that is doubly monotonic if is decreasing in and increasing in . We denote
We will refer to a pair consisting of an -player tournament and a matrix as a specialization of . Note that any such specialization defines a random process where alternatingly two players are chosen according to to play a match, and the winner of the match is chosen according to . For a given specialization of a tournament, we let denote the probability for player to win the tournament, and define the win vector . For a fixed tournament it will sometimes be useful to consider these probabilities as functions of the matrix , and we will hence write and to denote the corresponding probabilities in the specialization
We are now ready to formally define the notions
of symmetry, honesty and fairness.
Symmetry: Let be an -player tournament. For any permutation and any , we define by for all . That is, is the matrix one obtains from after renaming each player . We say that is symmetric if, for any , and any , we have
This definition is meant to capture the intuition that the rules
“are the same for everyone”. Note that any tournament can be turned
into a symmetric one by first randomizing the order of the players.
Honesty: Suppose that a tournament is in a state where matches have already been played, and it just announced a pair of players to meet in match . Let denote the probability that wins the tournament conditioned on the current state and on being the winner of match , assuming the outcome of any subsequent match is decided according to . Similarly, let denote the probability that wins the tournament given that is the loser of match . We say that is honest if, for any possible such state of and any , we have .
The tournament is said to be strictly honest if in addition, for all , the above inequality is strict, and all pairs of players have a positive probability to meet at least once during the tournament. Here denotes the set of matrices such that . It makes sense to exclude these boundary elements since, if for every , then player cannot affect his destiny at all. For instance, it seems natural to consider a single-elimination tournament as strictly honest, but in order for winning to be strictly better than losing, each player must retain a positive probability of winning the tournament whenever he wins a match.
To summarize, in an honest tournament a player can never be put in a strictly better-off position by throwing a game. In a strictly honest tournament, a player who throws a game is always put in a strictly worse-off position.
We note that the “state of a tournament” may contain more information than what the players can deduce from the matches played so far. For instance, the two-player tournament that plays one match and chooses the winner with probability and the loser with probability is honest if the decision of whether to choose the winner or loser is made after the match. However, if the decision is made beforehand, then with probability we would have and . Hence, in this case the tournament is not honest.
Fairness: Let be an -player tournament. We say that is fair if for all .
The main purpose of the next two sections is to show that there exist symmetric and honest tournaments which are nevertheless unfair.
3. Three-player tournaments
It is easy, though non-trivial, to show that every -player symmetric
and honest tournament is fair - see Proposition 3.4 below.
Already for three players, this breaks down however. Let and
consider the following two tournaments:
Tournament : The rules are as follows:
Step 1: Choose one of the three players uniformly at random. Let denote the chosen player and denote the remaining players.
Step 2: Let and play matches.
- If one of them, let’s say , wins at least matches, then the winner of the tournament is chosen by tossing a fair coin between and .
- Otherwise, the winner of the tournament is chosen by tossing a
fair coin between and .
Tournament : The rules are as follows:
Step 1: Choose one of the three players uniformly at random. Let denote the chosen player and denote the remaining players.
Step 2: Let and play matches.
- If one of them wins at least matches, then he is declared the winner of the tournament.
- Otherwise, is declared the winner of the tournament.
It is easy to see that both and are symmetric and honest (though not strictly honest), for any . Now let and , so that the matrix is doubly monotonic, and let’s analyze the corresponding specializations , of each tournament as .
Case 1: Player
is chosen in Step 1. In Step 2, by the law of large numbers, neithernor will win at least matches, asymptotically almost surely (a.a.s.). Hence, each of and wins with probability tending to , while a.a.s. wins .
Case 2: Player is chosen in Step 1. In Step 2, player will win all matches. Hence, each of and wins with probability , while wins .
Case 3: Player is chosen in Step 1. In Step 2, neither nor will win at least matches, a.a.s.. Hence, each of and wins with probability tending to , while a.a.s. wins .
Hence, as , we find that
Indeed, we get unfair specializations already for , in which case the dichotomy in Step 2 is simply whether or not a player wins both matches. One may check that, for ,
We can think of as trying to give an advantage to player over player , and trying to give an advantage to player over player . It is natural to ask if it is possible to improve the tournaments in this regard. Indeed the difference in winning probabilities for players and in is only , and similarly the winning probabilities for players and in only differ by . In particular, is it possible to modify such that goes below or such that goes above ? Is it possible to modify such that goes above ? The answer to both of these questions turns out to be “no”, as we will show below. In fact, these two tournaments are, in a sense, the two unique maximally unfair symmetric and honest -player tournaments.
We begin with two lemmas central to the study of symmetric and honest tournaments for an arbitrary number of players.
Let be a symmetric -player tournament. If for all , then .
Follows immediately from the definition of symmetry by taking to be the permutation that swaps and .
Let be an honest -player tournament and let . Then, for any , is increasing in .
As the proof of this lemma is a bit technical, we will delay this until the end of the section.
In applying Lemma 3.2, it is useful to introduce some terminology. We will use the terms buff and nerf to refer to the act of increasing, respectively decreasing, one player’s match-winning probabilities while leaving the probabilities between any other pair of players constant999These terms will be familiar to computer gamers..
Let and let be a symmetric and honest -player tournament. For any and any we have .
Every symmetric and honest -player tournament is fair. Moreover, for any , there is a specialization of an honest and symmetric -player tournament where and .
By Proposition 3.3, any doubly monotonic specialization of such tournament satisfies and thereby . On the other hand, for any , if and the tournament consists of a single match, then .
Let be a symmetric and honest -player tournament. Then, for any , , and .
The second inequality was already shown in Proposition 3.3.
Let us consider the bound for player . Given we construct a matrix by nerfing player such that he becomes identical to player . That is, we let and . This reduces the winning probability of player , i.e. , and by symmetry . We now claim that this common probability for players and is at least . To see this, suppose we construct from by buffing player to become identical to players and , i.e. for all . On the one hand, this increases the winning probability of player , i.e. , but on the other hand, by symmetry we now have . Hence, and hence , as desired.
The bound for player can be shown analogously. We first buff player to make him identical to player , and then nerf to become identical to the other two players.
For each , let denote the convex polytope of -dimensional probability vectors, i.e.:
Let be the closed, convex subset
We call the -dimensional fair set. A vector will be said to be achievable if there is a matrix and a symmetric, honest -player tournament such that . We denote by the closure of the set of achievable vectors in . Note that Proposition 3.4 says that , whereas we already know from (3.1) that .
The following result summarizes our findings for symmetric and honest -player tournaments. This is illustrated in Figure 2.
Denote the above set by . By Proposition 3.5, we know that , so it only remains to prove that . We start with two observations:
is a convex polygon with five vertices:
Suppose , are specializations of symmetric and honest -player tournaments , respectively, and with the same matrix . For we let denote the tournament: “With probability play and with probability play ”. Clearly, is also symmetric and honest for any and, if is its specialization for the matrix , then .
It follows from these observations that, in order to prove that , it suffices to construct, for each , a sequence of symmetric and honest tournaments such that as , where is the specialization of by the unique matrix satisfying , .
Indeed, we’ve already constructed appropriate sequences for
, by (3.1), so it remains to take care of
Tournament : Play iterations of round-robin. Choose the winner uniformly at random from among the players with the maximum number of wins.
It is clear that is symmetric and honest and
Tournament : Play iterations of round-robin. Choose a player uniformly at random from among those with the minimum number of wins. Flip a coin to determine the winner among the two remaining players.
It is clear that is symmetric and honest and that
Tournament : Just choose the winner uniformly at random. Obviously and the tournament is symmetric and honest.
To conclude this section, we finally give the proof of Lemma 3.2.
Proof of Lemma 3.2..
Fix and . Consider two matrices such that , and whenever
. The proof will involve interpolating between the specializationsand by a sequence of what we’ll call “tournaments-on-steroids”.
For a given we imagine that we play the tournament where, in the first matches, winning probabilities are determined by , and after that according to . The idea is that, at the beginning of the tournament, we give player a performance enhancing drug that only works against , and only lasts for the duration of matches (regardless of whether he plays in those matches or not). With some slight abuse of terminology, we will consider these as specializations of , and denote them by , and the corresponding winning probability of a player by . Clearly , and taking equal to the maximum number of matches played in , it follows that . Hence, it suffices to show that is increasing in .
Suppose we run the specializations and until either chooses a pair of players to meet each other in match
, or a winner is determined before this happens. As both specializations evolve according to the same probability distribution up until this point, we may assume that both specializations have behaved identically so far. The only way the winning probability for playercan differ in the two specializations from this point onwards is if match is between players and . Assuming this is the case, let denote the probability that wins the tournament conditioned on him winning the current match and assuming all future matches are determined according to , that is, according to the specialization . Similarly denotes the probability that he wins conditioned on him losing the match. This means that the winning probability for is in and in . But by honesty, , from which it is easy to check that the winning probability is at least as high in as in . We see that, for any possibility until match is played, the probability for to win in is at least as high as in . Hence , as desired.
(i) The above proof still works without assuming a bound on the number of matches in . The only difference will be that is now the limit of as .
(ii) If is strictly honest, one can see that for any and any such that there is a positive probability that match is between players and . Hence, is strictly increasing in in this case.
4. -Player Tournaments
Already for , it appears to be a hard problem to determine which win vectors are achievable. The aim of this section is to present partial results in this direction. As we saw in the previous section, can be completely characterized by the minimum and maximum win probability each player can attain. Thus, a natural starting point to analyze for is to try to generalize this. For each , let
In other words, (resp. ) is the least upper bound (resp. greatest lower bound) for the win probability for player , taken over all doubly monotonic specializations of all symmetric and honest -player tournaments.
It is not too hard to construct a sequence of doubly monotonic specializations of symmetric and honest tournaments such that . Thus we have and for all . Moreover, by Proposition 3.3, for all . We can extract a little more information by using the the technique of “buffing and nerfing a player” which was used in Propositions 3.3 and 3.5.
(i) For every , is a decreasing function of .
(i) Suppose, on the contrary, that , for some and . Then there must exist some symmetric and honest -player tournament and some matrix such that . Now buff player until he is indistinguishable from (according to the same kind of procedure as in the proof of Proposition 3.3). Let be the resulting matrix. By symmetry and honesty we then have , a contradiction.
(ii) Let be any symmetric and honest -player tournament and let . Perform the following three modifications of the specialization:
Step 1: Buff player until he is indistinguishable from .
Step 2: Nerf player until he is indistinguishable from and .
The latter equality implies, in particular, that
A third application yields
Putting all this together, we have
(iii) As before, let be any symmetric and honest -player tournament and let . We must show that . Perform the following two modifications of the specialization:
Step 1: Nerf player until he is indistinguishable from .
We next present a way to construct many symmetric and honest but unfair tournaments. For each , let denote the family of labelled digraphs (loops and multiple arcs allowed) on the vertex set whose set of arcs satisfies the following conditions:
Rule 1: There are exactly two arcs going out from each vertex.
Rule 2: Every arc satisfies .
Rule 3: If and are the two outgoing
arcs from , then or . In other
words, if the two arcs have the same destination, then either they are both
loops or the destination is vertex .
To each digraph we associate a vector according to the rule
Note that since, by Rule 1, each vertex has outdegree , we can also write this formula as
In what follows, each vector will be interpreted as the win vector of a certain symmetric and honest tournament. According to (4.8), the arcs of
instruct us how to “redistribute” win probabilities amongst the players, starting from the uniform distribution, where each arc “carries with it”of probability.
Let denote the convex hull of all vectors , . It is easy to see that is the single point - the only digraph in consists of the single vertex with two loops. For , the number of digraphs in is since, for each , the possibilities for the two outgoing arcs from vertex are:
- send both to ( possibility),
- send both to ( possibility),
- send them to distinct
The number of corners in the convex polytope is, however, much less than this. For a digraph to correspond to a corner of , there must exist some vector such that is the unique maximizer, in , of the sum . We can assume that the coefficients are distinct numbers. For a given vector , a digraph which maximizes the sum is determined by the following procedure: List the components of in decreasing order, say . Now draw as many arcs as possible first to , then to and so on, all the while respecting Rules 1,2,3 above.
We see that the resulting digraph depends only on the ordering of the components of , not on their exact values. In other words, there is a well-defined map from permutations of to corners of , , where, for , the digraph is given by the procedure:
“Draw as many arcs as possible first to vertex , then to and so on, all the while respecting Rules 1, 2, 3”.
Table 1 shows how this works for and . The map is not injective for any and the exact number of corners in is computed in Proposition 4.5 below. For the time being, the crucial takeaway from Table 1 is that and . Recall also that .
We are ready to formulate
, for every .
Our main result in this section is
, for every .
We’ve already observed that for
. We divide the remainder of the proof into two cases.
Case I: . Since we can form a “convex combination of tournaments” - see the proof of Theorem 3.6 - it suffices to find, for any fixed and for each , a sequence of symmetric and honest tournaments such that as .
Let be any doubly monotonic matrix such that unless either or . The matrix is henceforth fixed. Let
In other words, is half the minimum difference between two distinct numbers appearing in the matrix .