I’m currently working on a model that involves an Elo/Glicko/TrueSkill type component as part of its feature set.
The basic idea underlying this type of system is that if A usually beats B and B usually beats C then A will usually beat C. Well that isn’t the idea, but it is what will result from it.
While thinking about this I remembered Warren Buffets fascination with non transitive dice:
Which is basically you can construct a set of three sets of dice and if the other person picks first - you can always pick another set that gives you an advantage. In other words - A > B; B > C; but C > A
I’m curious if this occurs in more realistic situations
I can imagine games of skill where say a group of three boxers with certain mixes of skills, speed, strength where this would occur, but I know very little about sports.
I’m looking for any examples of a competition of skill where this is known to have occurred - it doesn’t have to be sports, and could be something like wars (but there are probably to many confounding variables if we are dealing with inter state conflicts) - or whatever, but it should be obvious that it isn’t a statistical fluke (as in it happened enough times)- and preferably it is known why the non transitive results occurred.
And no rock, paper, scissors doesn’t count.
It’s ok if luck is part of it - as long as it’s obvious that the non transitive nature wasn’t solely due to luck.
But the situation should be that people would expect that if A beat B and B beat C - A should beat C.
I put this in GQ as I think there is a factual answer to it, and I am not necessarily limiting it to games and sports.
A is good at attacking and likes to play risky gambits.
B is good at defence, but poor in endings.
C is good at endings, but poor at defending.
So A loses to B (risky gambit refuted.)
B loses to C (in the ending of a long game.)
C loses to A (the gambit works!)
P.S. The above would only apply to club strength players.
One season my club got grandmaster Morozevich to play top board. (He was rated in the World top 10.)
In his first game, he beat a grandmaster with a crushing attack. In the second game, he simplified down to an almost level ending - and won that too.
Amazing.
There are certain types of elections where voter preferences for multiple candidates can be non-transitive.
E.g., voting in some areas for city council seats where a voter can choose 2+ candidates can result in individuals being elected that aren’t generally the voters’ first choices.
Instant runoffs are another example. (These have a lot of flaws.)
(While not real life, my favorite example of a non-transitve game is Penney Ante.)
When I ran high school cross country, this happened once. We had three teams run together, but it was scored as 3 dual meets. In order to do that, you just removed the runners from the team not included and scored the other two teams to see who won. In cross country at that level, each team has 7 runners and the top 5 score. You add up the places of the top 5 and the team with the lower total wins.
At the end, A beat B by 1 point. B beat C by 1 point. C beat A by 1 point.
In early World War 2-era battles, anti-tank units > tanks > infantry > anti-tank units. Later on the line blurred when tanks started to get so armored that earlier anti tank rifles couldn’t hurt them, and infantry started to get bazooka-style weapons, and anti-tank units became more like tanks.
You can also get reversals at different skill levels. For instance, in medieval times and earlier, cavalry would always defeat low-quality infantry (low training, discipline, etc.), and high-quality infantry would always defeat cavalry, no matter what the cavalry’s skill level. So if everyone on the field is green, then the knights win, and if everyone on the field is a grizzled veteran, then the footmen win.
A conservative player who raises and calls only when he has very good cards will lose against a player who bluffs repeatedly, who will lose to a player who is likely to call with a moderately good hand, who will lose to the conservative player.
IIRC, Machiavelli noted that cavalry would obliterate lightly-armored shortsworders with little shields, but pit that cavalry against pikemen and they die – but pikemen against lightly-armored shortsworders with little shields? Despite the length and the weight, you could move that spearpoint around in time to level it at a horseman coming your way – but now you’re all facing quick parriers who’ll deftly slap your slow weapons aside and rush forward to play in-fighter.
In collectible card games like Magic: the Gathering or Hearthstone this happens often. A very general rule of thumb is that fast decks will generally beat slow decks, which beat mid-range decks, which beat fast decks. Fast decks will overwhelm slow decks before they can fully develop, but will be underpowered against decks that deploy just slightly slower than them such that the latter can weather the assault and win on the back of stronger overall cards. However the mid-range decks don’t have the staying power that the slow decks do, and at the same time aren’t fast enough to beat the slow decks before they set up. The exact details will vary from game to game, but I’m fairly sure that the basic premise holds to some extent. It won’t always be 100% RPS, but you generally will have to position your deck somewhere on the spectrum of winning quickly and having staying power, with your best results coming if your deck has slightly more staying power than the opponent or is much, much faster.
One of my senseis described three main types of martial art techniques: hard, soft, and hard/soft. Hard techniques are directed and focused, soft techniques are generally circular and manipulative, and hard/soft usually start as one and end as the other.
So, the example he gave us that would apply here is that soft beats hard, hard beats hard/soft, and hard/soft beats soft.
In Texas Hold’Em, assuming for simplicity that there are only two players and one is “all in” before the flop,
[ul][li] A player with a very small pair is favored against A-7 off-suit;[/li][li] A player with A-7 off-suit is favored against 9-8 suited;[/li][li] A player with 9-8 suited is favored against a very small pair.[/li][/ul]
Can we look for examples outside of games and sports? Oscillating reactions the Belousov-Zhabotinsky reaction depend on non-transitive chemistry.
Another example comes from the field of diachronic linguistics. Since highly inflected (fusional) languages tend to lose their inflections over time, a common question is: Why do fusional languages still exist? Languages all trace back tens of thousands of years; shouldn’t inflected forms have disappeared completely? The explanation depends on a non-transitive relationship: the isolational form when inflections are lost is inefficient semantically without affixes, so eventually an agglutinative language arises. And these are inefficient morphologically until they become fusional!
Some folks predicted this before Ali boxed Foreman, since Frazier had recently lost to Foreman with the tactic that had beaten Ali; Frazier’s style involved getting tagged by his opponent while moving in close to land a big left hook – which worked against Ali, since Frazier waded through light jabs to clock him with a knockdown blow.
But it was a horrible idea against a power puncher like Foreman, since giving that guy a couple of mid-range shots at your advancing head equals defeat.
But, the reasoning went, it’s not an A>B>C so A>C scenario, because Ali can land a long jab and then dance away from the slower Foreman; he won’t try to get in close, like Frazier did; and Foreman doesn’t try to get in close, like Frazier did; Ali will float like a butterfly and sting like a bee, and Foreman will fondly look back on the days when people obligingly ran into his punches chin-first.
It fell apart because Foreman knew it, and Ali knew that Foreman knew it, and as Foreman advanced, Ali didn’t dance away but covered up while leaning on springy ropes to tire him out. But if you ignore that slaying of a beautiful theory with an ugly fact – that folks aren’t locked into a style, and can change tactics – the idea is solid: if M has a speed advantage against G, he can win with little punches at long range; and if J can swiftly get in close against M, he’ll get hit with little punches before winning with a big punch; and if J tries that same style against G, he’ll get hit by big punches and go down before he can get in close, regardless of foot speed.
