Some cards are described as swingy.
User rrenaud once compiled a sorted list, based on how much entropy was left in the win-when-in-kingdom rates once rating was factored out, and concluded that Goons, Ambassador and Tournament are high-skill cards, whereas Embassy is a high-randomness card.
What makes Embassy high-randomness is the fact that there is an Embassy-centered strategy (Embassy BM) which is fairly strong and easy to play correctly.
What makes Tournament high-skill, I think, is that it requires good judgment and game sense to know when to go for it, how best to go for it, and which prizes to pick when you connect it with a Province. Somehow, [joke]rrenaud's ironclad mathematical proof—proof, I tell you![/joke]—that Tournament is high-skill seems not to have changed the common perception of Tournament as swingy.
I think I have a different definition of swingy which classifies Tournament as swingy. It allows for a 2x2 matrix of swingy yes/no and high-skill yes/no; we could for example use rrenaud's measure to indicate whether a card is high-skill.
Whenever you make a play, you can compare you chances of winning (assuming optimal play) in the game state before and after making that play.
Weak cards are cards that reliably only improve your win chances by a small amount (or not at all, or even harm you). Strong cards are cards that usually do something better than weak cards for your win chances.
Swingy cards are cards that are usually ok-ish but sometimes have an unusually large impact on your win chance when played. That is, your win chance after is significantly better than your win chance before. Playing it causes a big swing.
I equivocated between "making a play" and "playing a card". I only used the latter phrase for convenience—Cultist can be highly swingy, in particular if exactly one player opens 5/2 (but also if it misses shuffles, or the first ruins hit vs. miss shuffles). In the case of opening 5/2, it's the dealing of the opening hands that's the swingy play. That play is made by RNGsus, so really the definition should read "the move greatly changes the win percentages of the players", without using 'you[r]'.
And considering this further, having Province in hand when you play Tournament (and being first to do so) is often largely attributable to shuffle luck. When the shuffle has been made in a way that favors you, actually making the swingy play is just playing optimally—so really, I think swingy is something like "some of the moves made by RNGsus have the capacity to greatly alter the win probabilities of the players".
The degree to which a card is swingy is then the capacity of RNGsus to impact win chances, averaged across all the kingdoms in which the cards occur. The inherent swinginess of Tournament is attributed a little bit to e.g. Smithy, because they can occur on the same kingdom, but Embassy's randomness is also attributed a little bit to Smithy by the same logic. I claim without evidence that each card's inherent swinginess will be attributed overwhelmingly to itself, and each card's measured swinginess will be its inherent swinginess, adjusted a little bit in the direction of the average inherent swinginess across all cards, but the ranking by measured swinginess and the ranking by inherent swinginess will be the same.
A small pedantic footnote: instead of win probabilities I should probably use "expected number of match points" (loss=0 and win,draw=1/#winners), such that the players'
win probabilities expected number of match points sum to 1.
I skipped over some rigor in a few places, but I think my definition roughly approximates what people mean by swingy. I would love to hear whether people agree or disagree, and especially if you think you can improve this definition. I think of it as a 'first draft'
Edit: today I relearned that the internet is a poor medium for carrying tone of voice, which is often crucial to indicate when you're joking. To address this, I have inserted [joke] tags in the appropriate places.