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Regret (decision theory)

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I am quite confused by the page Regret (decision theory), as it is categorized under both “Game theory” and “Statistics”, while a large part of the article discusses psychology. Should we create a separate Wikipedia page dedicated to regret in mathematics, without any links to regret as a “negative emotion”, as described in Regret (decision theory)? I believe there is sufficient material on mathematical regret to justify a standalone page. Adrien PREVOST (talk) 10:09, 31 December 2025 (UTC)[reply]

Regret (decision theory)'s lede and "Anticipated regret" sections certainly don't look mathematical, and an article on regret as a mathematical concept seems sensible. I've taken a cursory look at some sources, and regret is well-defined and studied in game-theory.
Decision theory looks mathematical in itself. Regret in game theory looks similar to regret in decision theory (at present, Regret (game theory) redirects to Regret (decision theory)), so it might be best, instead of creating a new page, to move the psychology sections to Regret or delete it entirely. The BooleanTalk 19:23, 1 January 2026 (UTC)[reply]
I agree that the psychological material should be moved out and the article refocused on the mathematical optimization aspects of the topic. See also competitive regret and swap regret; this appears to be a big topic in machine learning that our articles don't really cover well. —David Eppstein (talk) 19:48, 1 January 2026 (UTC)[reply]
I had a quick look at the article and I actually feel like the "psychological" part of the article is fairly light. If there are particular parts that we feel are too detailed, I wouldn't be against them getting shortened, but I would be opposed to completely removing the "psychological" aspects of regrets from that article. Are there particular paragraphs that are of concern?
This article is primary about regret in the concept of decision theory and game theory. These two fields are at the intersection of mathematics and psychology. They try to use mathematics to model psychological mechanisms. Focusing solely on the mathematical aspects, in my view, would remove the intuition behind the math, and make the article a lot less useful for most readers (especially the lead section, which is generally supposed to be less technical than the rest of the article). 7804j (talk) 11:56, 11 January 2026 (UTC)[reply]
I don't think that regret in game theory is at the intersection of maths and psychology. Regret is a very common quantity in game theory, especially in online games where it is the difference between the best achievable from an oracle that knows everything and what the player is doing. When regret appears in papers on online games there is no reference to psychology. Application of regret in those games is absolutely not linked to any psychological stuff.[1]
We can find such quantity in Reinforcement Learning[2], learning with experts[3] and especially in Bandit games[4] where even in papers on bandits but not on regret they even have to specify that it is not about regret.[5] There are probably more domains of online learning that use regret that I forget or don't know.
That's why I think it requires its own page with only maths stuff on it. Maybe we can separate the page between regret in decision theory and in game theory? Adrien PREVOST (talk) 00:08, 13 January 2026 (UTC)[reply]
It could be that regret in game theory and in decision require two distinct pages.
Regarding the psychological angle: there is for example this paper that's referenced in the current article that talks about the "feeling" of regret (i.e., in a psychological sense) in the context of auctions. But perhaps that's because auctions fall more within the realm of decision theory?
If the game theoretical definition of regret is almost universally understood, as you described, as the "difference between the best achievable from an oracle that knows everything and what the player is doing", I agree with you that it's not about psychology (and thus that it may require a different page than for decision theory). 7804j (talk) 20:59, 14 January 2026 (UTC)[reply]
I've looked more closely at the article, and it seems to be trying to say that regret in decision theory has applications in psychology. If that's the case, the psychological content belongs in the article similarly to how the article Calabi-Yau manifold discusses the manifold's importance in string theory, even though, in principle, the Calabi-Yau manifold is a purely mathematical object.
Regret has other applications outside psychology (particularly in machine learning), so the psychology section shouldn't be the article's exclusive focus, but it needn't be removed.
As for the difference between regret in decision theory and regret in game theory, I suspect that no difference exists and that both fields use the same definition. "Regret" appears to be a widespread mathematical term. However, I have been unable to find a source that explicitly references both regret and decision theory, which is what we'd need to provide the correct definition of regret in the context of decision theory. Based on this, I'd propose:
  • Cleaning up the article to clarify the relationship between regret and its applications
  • Moving the page to Regret (game theory) or Regret (mathematics).
  • Writing the lede so it explains regret, then explains its uses
The BooleanTalk 23:17, 14 January 2026 (UTC)[reply]

References

  1. ^ A Survey on Practical Applications of Multi-Armed and Contextual Bandits, 2019
    https://arxiv.org/abs/1904.10040
  2. ^ Achieving Tractable Minimax Optimal Regret in Average Reward MDPs, 2024
    https://arxiv.org/abs/1904.10040
  3. ^ Hedged learning: Regret-minimization with learning experts, 2005
  4. ^ Explore First, Exploit Next: The True Shape of Regret in Bandit Problems, 2018
    https://arxiv.org/abs/1602.07182
  5. ^ Optimal Best Arm Identification with Fixed Confidence, 2016
    https://arxiv.org/abs/1602.04589

Metric barnstar

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As the WikiProject Measurement is now inactive, I thought here was the best place to post this. Would it be all right if I added this barnstar I made to the list of topical barnstars here? - Wikipedia:Barnstars/Topical

This is the Metric Barnstar, which can be awarded to editors who have made significant contributions to articles related to the metric system. It features a round chart with metric base units (outer ring) and constants (inner ring).

Please comment below to let me know what you think. Thank you. Caravansera (talk) 07:19, 5 January 2026 (UTC)[reply]

I don't see why not. —David Eppstein (talk) 07:41, 5 January 2026 (UTC)[reply]
Seems fine to me. Stepwise Continuous Dysfunction (talk) 21:33, 5 January 2026 (UTC)[reply]
Ok, since it's been a week and two users have said it's okay and nobody has objected, I am going to add the metric barnstar to the list of topical barnstars.Caravansera (talk) 18:43, 12 January 2026 (UTC)[reply]

Redundant Definitions in Nemytskii operator

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The article on the Nemytskii operator currently defines the operator three times:

- The "Nemytskii superposition operator", defined on maps between arbitrary sets

- The "Nemytskii operator" in Euclidean space, induced by a function satisfying the Carathéodory conditions

- An operator between Lipschitz continuous maps, which isn't explicitly stated to be the Nemytskii operator.

From some quick research, I found a fourth definition of the Nemytskii operator in terms of vector spaces of functions between Banach spaces in "Concrete Functional Calculus", by Dudley and Norvaiša (p. 273), which also explicitly states the terms "superposition operator" and "Nemytskii operator" are used interchangeably.

It seems sensible to me to restructure the article around a single one of these definitions. I am asking here first because I do not feel I am familiar enough with Wikipedia etiquette to judge whether it is appropriate to set out on such a major restructuring on my own, and because I am interested in second opinions on which level of generality would be most appropriate. Yoneda-Emma (talk) 16:32, 8 January 2026 (UTC)[reply]

I do not know anything about this subject, but I have some general observations. If these are the same, then you should show that they are the same, begin with the most concrete and work up to the most general. If they are different, then would it be better to split the article rather than downgrading all but one of them. JRSpriggs (talk) 19:33, 8 January 2026 (UTC)[reply]

Good article reassessment for Newton's theorem of revolving orbits

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This is a physics subject, but it's mathematically involved enough that editors active here may want to chime in: Newton's theorem of revolving orbits has been nominated for a good article reassessment. Per the reassessment page, some problems have been resolved but there is still work to do. Stepwise Continuous Dysfunction (talk) 00:54, 14 January 2026 (UTC)[reply]

Sprague–Grundy non-theorem

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Our article on the Sprague–Grundy theorem must be wrong as written. It says "every impartial game under the normal play convention is equivalent to a one-heap game of nim". This is absurd. If there was only one heap, then either it is empty and the first player loses or it is non-empty and the first player can immediately win by taking the whole heap. So the complexity of real nim with multiple heaps would be eliminated. When there are multiple heaps you want to reduce the largest [some] heap to a value which makes the XOR of all the heaps' sizes in binary equal to zero. JRSpriggs (talk) 02:29, 15 January 2026 (UTC)[reply]

Oh! No, that's completely right actually. The complexity of nim is encoded in the size of the heap. You can reduce a standard, multiple heap game of nim to a single heap whose size is the binary XOR of all the heaps sizes. MEN KISSING (she/they) T - C - Email me! 02:54, 15 January 2026 (UTC)[reply]
I realize now, this needs one clarification (which I hope clears all this up): the only reason that much complexity can be removed at all, is because we're assuming both players are playing optimally. A more complicated game of nim may have more complexity than if it were just a single nim heap, but only if we were to assume the two players might make sub-optimal moves.
Perhaps this point isn't illustrated clearly enough in the article? MEN KISSING (she/they) T - C - Email me! 03:11, 15 January 2026 (UTC)[reply]
There's a key word in this sentence: equivalent.
If it meant merely that the game had the same outcome when played in alternation with perfect play, then every game would be equivalent either to a zero-coin nim game (in which the first player always loses) or a one-coin nim game (in which the first player always wins). But that is not what equivalent means here; see Sprague–Grundy theorem#Equivalence.
The equivalence of games X and Y means that for every impartial game Z, the sum of games X + Z has the same outcome (with optimal play) as the sum of games Y + Z. Here the sum is more or less what you do in nim when you have multiple piles: you can move in exactly one of them.
The issue of optimal play is also valid but is part of the definition of equivalence, which is the more subtle and more important point. —David Eppstein (talk) 04:41, 15 January 2026 (UTC)[reply]
Yes, yes. A nimber is required, it can't just be either 0 or Star. Otherwise, it wouldn't explain why a nim game of *2 + *1 is a win for the player going first but *1 + *1 is a win for the player going second.
As an aside: I'm surprised Conway's Winning Ways for Your Mathematical Plays isn't used as a source there. Is there a reason for that? If I recall correctly, that book makes a good case for the significance of the theorem in game theory, an area where the article seems to currently be lacking. I'm on something of a self imposed wikibreak, so I don't want to dig in to it right now, but maybe I could poke around there some more in a few days! I've been meaning to get more invested in this WikiProject; I kinda got distracted futzing around at RfD haha. MEN KISSING (she/they) T - C - Email me! 05:03, 15 January 2026 (UTC)[reply]

It is unreasonable to presume perfect play. If there are no heaps, then the first player loses. If there is one heap, then the first player can take the whole heap and win. If there are two unequal heaps, then the first player can reduce the larger heap to the same size as the smaller one and win. When there are three heaps, a≤b≤c, then reduce c to b-a [reduce one of them to XOR of the other two]. JRSpriggs (talk) 13:15, 15 January 2026 (UTC)[reply]

All of combinatorial game theory is presuming perfect play, haha.
Note that you can also think about a nim game of any number of heaps as being equivalent to two heaps, the second heap being the XOR of all the other heaps aside from the first heap. This checks out under the Sprague-Grundy theorem because XOR is commutative and associative. MEN KISSING (she/they) T - C - Email me! 16:36, 15 January 2026 (UTC)[reply]
It is completely reasonable to presume perfect play, in the definition of equivalence. If what you want is games that have the same game play in all situations regardless of perfection, then you are looking at a different and less interesting equivalence relation, isomorphism. —David Eppstein (talk) 18:41, 15 January 2026 (UTC)[reply]

I think that you-all are conflating nim with Grundy's game. Although both deal with heaps or piles of objects, they have different rules including different winning conditions. JRSpriggs (talk) 01:02, 16 January 2026 (UTC)[reply]

No, that's not what's happening here. Both games are heap games, and although they have different rules, they are both impartial games and the Sprague-Grundy theorem applies just the same.
Grundy's game is actually a great example of how impartial games can be reduced to nim sums! You can convert each Grundy's game heap into a nim heap according to the stack size, and it will remain equivalent under perfect play. The sequence OEIS: A002188 describes the conversion rule. Note that the Sprague-Grundy value is exactly equal to the nim heap stack size you need to convert to.
I'd like to recommend volume 1 of Conway's Winning Ways if you want all of this cleared up. Chapters 3 to 4 I think cover the topic and how it relates to other heap games best. Also, it's just a really fun book. It has my favorite diagram I've ever seen in any mathematical text. MEN KISSING (she/they) T - C - Email me! 01:18, 16 January 2026 (UTC)[reply]
That OEIS sequence is labeled "Sprague-Grundy value for Grundy's game when starting with n tokens.". It is describing Grundy's game, not nim. JRSpriggs (talk) 01:33, 16 January 2026 (UTC)[reply]
Why are you keeping on talking about Grundy's game? It is nim that the Sprague-Grundy theorem is formulated in terms of, and nim that the Sprague-Grundy theorem article talks about. Where are you seeing Grundy's game used in place of nim? —David Eppstein (talk) 01:38, 16 January 2026 (UTC)[reply]
Read reference 2 in the article on Grundy's theorem. Grundy says "The following game with piles of matches is an example:- The move consists of taking any pile and dividing it into two unequal parts. The game is one in which the last mover wins; ...". He is talking about his game, not nim. JRSpriggs (talk) 01:47, 16 January 2026 (UTC)[reply]
That's an example of a game whose positions are equivalent (in the specific sense described) to games of nim. It doesn't mean it's the same (isomorphic) game. —David Eppstein (talk) 02:12, 16 January 2026 (UTC)[reply]
Yes. MEN KISSING (she/they) T - C - Email me! 01:39, 16 January 2026 (UTC)[reply]
The theorem is about normal-play nim where the player to take the last coin wins. Nim is also played in a "misère" version where the player to take the last coin loses, but the Sprague-Grundy theorem applies only to normal-play games, not misère games. For nim both versions have almost the same game play but that is not true for other impartial games. —David Eppstein (talk) 01:20, 16 January 2026 (UTC)[reply]

Uncle and ouch. After reading a great deal more of our articles on games, I see that I was wrong, for the most part. But I still have two points: (1) The word equivalent is being used in a very technical sense and I think that the readers should be apprised of that fact where it is first used in the lead of some articles. (2) The theorem does not appear to allow for infinite games. I agree that every game should end in finite time, but that can be achieved by using ordinals which have no infinite descending sequences despite being infinite in the upward direction. Consider a single heap of 2ω=ω things. I do not think that this would be equivalent to *n for any natural number n. JRSpriggs (talk) 21:03, 16 January 2026 (UTC)[reply]

Aw, don't sweat it. Someone was wrong on the internet. It happens all the time! If it didn't happen, we wouldn't need to have Wikipedia at all haha.
The way I understand the Sprague-Grundy theorem, both finiteness and the definition of equivalence are pretty important to justifying the theorem. That would be a good point of improvement.
Winning Ways did also talk about the case of infinite games at length, including an infinite version of the Sprague-Grundy theorem! If you want to learn more about that, I think volume two of the book covers most of that topic. MEN KISSING (she/they) T - C - Email me! 21:40, 16 January 2026 (UTC)[reply]
Volume 1 (Games in General), Chapter 11 (Games Infinite and Indefinite), p. 313 (The Infinite Sprague-Grundy and Smith Theories). Every impartial game that is guaranteed to eventually end is equivalent to nim with a single ordinal number pile of coins (in the same sense of equivalence). —David Eppstein (talk) 00:04, 17 January 2026 (UTC)[reply]

Aside

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@David Eppstein and JRSpriggs: I feel the discussion here is getting a bit off-focus. JR suggested the Sprague-Grundy theorem article was factually inaccurate and needed to be reworked. An opinion to the contrary from a professor and from an undergraduate who happened to read a relevant book on the topic should be sufficient to refute this as far as Wikipedia is concerned.

If there are concerns about the understandability of the article, let's have that discussion instead. Questions about the Sprague-Grundy theorem would be more appropriate at WP:REFD MEN KISSING (she/they) T - C - Email me! 01:52, 16 January 2026 (UTC)[reply]

Well, if JR accepted the correction and backed off, that would be a reasonable thing to do but instead, after being corrected, they inserted their misunderstanding into both nimber [1] and Sprague–Grundy theorem [2]. As for credentials, they have them too; it's the material that should be the basis for understanding here, not the people making the arguments. (Beyond having a faculty position in not-mathematics, I do have one publication that actually involves nim-values, [3], but again I don't think we should be making arguments from authority.) —David Eppstein (talk) 02:17, 16 January 2026 (UTC)[reply]
I'm mostly concerned about WP:NOTAFORUM regarding the dispute here. I was unaware of those edits; they seem to have happened before JR's second reply here, so they may have just not read our objections before now. Ultimately, I just don't think it would be productive to make a big fuss out of all of this.
Good point about credentials, I'll keep that in mind. Still, two editors of any sort should suffice here. MEN KISSING (she/they) T - C - Email me! 02:33, 16 January 2026 (UTC)[reply]
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