Talk:St. Petersburg paradox/Archive 3

Is there a mistake in the initial wealth and maximum cost statements?
Hi, I've tried to verify the below statements myself and I think they're mixed up: "For example, with log utility a millionaire should be willing to pay up to $10.94, a person with $1000 should pay up to $5.94, a person with $2 should pay up to $2, and a person with $0.60 should borrow $0.87 and pay up to $1.47."

According to my simulation of results in a spreadsheet, I calculate that a natural logarithm utility function player with an initial wealth of $1000 (not a million) would pay a maximum of $10.95, rounded to the nearest cent, to participate once in the game. I calculate different values for all the above listed costs. Here is a link to my spreadsheet: http://www.keithwoodward.com/random/wiki/StPetersburgParadoxBernoulliUtilitySolution.xlsx Apologies if I've made a silly error. Just wanted to check. Keithphw (talk) 21:45, 30 April 2016 (UTC) — Preceding unsigned comment added by Keithphw (talk • contribs) 21:44, 30 April 2016 (UTC)

I've found another person on MathExchange who did an analysis similar to mine and found the same result, so I believe that the current numbers on wikipedia are wrong so I will edit them. Here is the mathexchange article: http://math.stackexchange.com/questions/566414/maximum-amount-willing-to-gamble-given-utility-function-uw-lnw-and-w-100/569934#569934 Cheers, Keith Keithphw (talk) 07:30, 2 May 2016 (UTC)


 * I spot-checked this for the case w=1000, and the new number appears to be correct. Loraof (talk) 17:26, 4 May 2016 (UTC)

Hi Laraof, thanks for checking my fixed value. I also double-checked the result on the Mathematica forum and thanks to the kind help of Daniel Lichtblau and Sander Huisman (http://community.wolfram.com/groups/-/m/t/851650) we now have the algorithm to solve it in Mathematica: w = 1000000; util[c_?NumberQ] := NSum[1/2^n*(Log[w + 2^n - c] - Log[w]), {n, 1, Infinity}] FindRoot[util[c] == 0, {c, 2}] Best regards, KeithKeithphw (talk) 02:55, 7 May 2016 (UTC)

Nicolaus Bernoulli
In this article it is stated that Daniel Bernoulli came to the paradox through his cousin Nicolaus I Bernoulli, but in the French and English version of the article on Daniel Bernoulli it is stated that it was actually his brother Nicolaus II Bernoulli...! — Preceding unsigned comment added by 146.0.189.115 (talk) 21:15, 1 June 2018 (UTC)
 * The French article on Daniel Bernoulli says "...in which he articulates the Saint Petersburg Paradox - originating from discussions between him and his brother Nicolas". However, I can find no such claim in the English article on the same subject, and the entry on this paradox in French also claims Nicolas I Bernoulli (the cousin of Daniel) as its originator. So the claim in the French article on Daniel Bernoulli is likely to be mistaken. Citizen Canine (talk) 22:22, 1 June 2018 (UTC)

Description of Lottery is Vague
It's not clear in the beginning how this game is structured. It sounds like, and I'm inferring this from subsequent parts of the article, rather than the initial description, that players put up a sum of money as a fee to enter the game, and then they get unlimited plays to win as much money as they want until they get tired of playing. I hope I'm inferring this correctly. But whether or not this is the correct takeaway, this should not be merely implicit in the article. It should be spelled out explicitly. Readers should be able to understand how the game is structured from the initial description alone, so that in the subsequent sections, they are able to use their mental resources for understanding the implications of proposed solutions to the paradox and other discussions about the paradox, rather than having to use those mental resources to figure out if they've correctly deduced how the game is actually played. I understand that mathematicians discussing various curiosities in mathematical journals, letters to their friends, and in private conversations may skip some details because they know that their target audience is other mathematicians who've already studied many similar things and certain conventions may have arisen organically as a result. But since non-mathematicians will be reading the article, it should be structured so that they don't need to already have a phd just to follow along. Comiscuous (talk) 23:53, 22 February 2018 (UTC)


 * Maybe it has been edited since your visit, but the paradox description seems pretty clear now. Basically, you pay an entry fee to begin playing. A coin is tossed, and if it comes up heads the house adds $2 to a pot. If it comes up heads again they add 4, then 8, then 16, doubling every time, each time added to what's already in the pot. If it ever comes up tails, the game is over, and you take what's in the pot. You may then start over with a new entry fee if you'd like. The question is, what's a fair entry fee? Traditional "expected value" calculations are unhelpful. --Sam (talk) 13:58, 24 September 2018 (UTC)

Median value
- I don't have access to the journal article, but can you go into a bit more detail? As written, using the median value as a fair price seems obviously wrong, so I doubt that's what the paper was saying - I presume it's offering something a little more complex. (For example, a game with a 51% chance of returning $0 and a 49% chance of returning $100 would have a median value of 0 if you played it 10000 times and took the median of the resulting payouts, but that game is worth way more than nothing, it's far closer to the $49 expected value!) SnowFire (talk) 19:05, 4 August 2020 (UTC)
 * If you play your game only once then the median is 0. (After all, getting nothing is the most likely result of playing only once.) But already playing twice the median is $100, suggesting a price of $50 per game. Playing many times and the median per game will converge to $49. iNic (talk) 00:28, 5 August 2020 (UTC)
 * - If you have a better phrasing than my edit, please include it. I do think that some further clarification is required though per my original comment, though.  SnowFire (talk) 18:59, 9 August 2020 (UTC)
 * I do not see the need for a rephrasing at all. iNic (talk) 19:31, 9 August 2020 (UTC)
 * Sure it does. The median of [0, 0, 0, 100, 100] is 0.  Repeated single plays of the St. Petersbug paradox game would have a payout matrix that looks something like this (say over 16 plays to keep this nice & simple):
 * [2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 8, 8, 16, 32]
 * The median of this would be the average of 2 & 4, or 3 (or 2, or 4, depending on the median procedure for a set with an even number of entries). That is clearly an undervaluing of the game, for a similar reason as the toy game I described above.  Now, I appreciate your explanation, but it clearly requires assuming thousands of people play the game thousands of times each, not thousands of people play the game once.  Yet the "take the median of a bunch of single plays" is definitely the intuitive assumption based off the snippet you added to the article, so it requires further clarification in some manner.  SnowFire (talk) 19:42, 9 August 2020 (UTC)
 * You are calculating the median the wrong way. Playing your St P game 16 times has 106 as the median, which gives a value of 6,625 per game. iNic (talk) 20:30, 9 August 2020 (UTC)
 * I'm calculating it the way that I presume most people would assume based off the current text that simply says "calculate using the median" - that the median value of this game is obviously between 2 and 4. I agree it's "wrong" for what the paper said, so the text should be corrected to accurately reflect what is in the paper...  I don't see why this is an issue.  Like I said, I'm open to alternate phrasings, but don't you want to clearly rule out my "wrong" interpretation above based off single plays?  I agree with you that 16x plays of the above matrix results in 6.625, but 1x plays is 3, so that's why I added "over repeated plays" to the article.  (I think you may have misunderstood "16" - I had 16 entries in the matrix is all.  Those were each an event of playing the SPP game once, where half failed the first flip, 25% failed the second flip, etc.  I never intended to mean "play this matrix 16x times".)  SnowFire (talk) 21:37, 9 August 2020 (UTC)
 * What is meant by a median is already explained in the link median and should not require a special clarification here. Even if 100% of the readers here have no clue what a median is, this is still not right the place to provide that explanation. iNic (talk) 22:12, 9 August 2020 (UTC)
 * Math major here. I know what a median is, thanks.  You get a different answer if you take the median of single plays vs. repeated plays.  The problem isn't the word "median", it's the dataset the median is applied to.  SnowFire (talk) 23:43, 9 August 2020 (UTC)
 * The median isn't applied to a data set. Where did you get that from? Please check the definition of a median in probability theory in the median link. iNic (talk) 07:55, 10 August 2020 (UTC)

I genuinely do not understand why you don't acknowledge the alternate interpretation here, when you already explained what you meant above (which I appreciate!), so why not explain it in the article as well? Fine, I'm a big dumb-dumb who doesn't know what a median-in-probability is, I am so stupid, I can take the snide remarks, but again, don't you want to write text that makes the "correct" interpretation clearer? Here, I'll write it out again, but it's literally just what you already said. Let's assume our toy game has a 2/3 chance of returning 0 and a 1/3 chance of returning 100 for a payoff matrix of [0, 0, 100].
 * [0, 0, 100]. One play of the game.  Median is 0 (indisputably, you agreed with me already above that the median is 0 in such a case!)
 * [0, 0, 0, 0, 100, 100, 100, 100, 200]. Two plays of the game.  Median is 100 (value is 50).
 * [8x 0s, 12x 100s, 6x 200s, 1x 300]. Three plays of the game.  Median is still 100 (value is 33.33...).
 * [Some vastly huge amount of entries]. Limit of arbitrarily large numbers of plays.  Median approaches expected value (33.33 again in this particular case).

You're talking about the last median, the one with a vast number of repeated plays of the same game. However, a perfectly valid use of "median" is to read it as the first scenario, where the median is 0, which is not what this paper is talking about (since that interpretation leads to a vastly undervalued SPP game). Linking "median" won't help, because both these values are medians. What differs is the underlying list of data, as I said before, and you clearly understand this because you were talking about how my original toy game would go from 0->50->(...)->49. So... the underlying data should be made clear. Do we need to get a third opinion? SnowFire (talk) 14:13, 10 August 2020 (UTC)
 * If you are interested in this idea I recommend you to read the papers referred to. If you think the idea is stupid, no problem, tell these authors that, not me. But you don't want to read the papers and you don't even bother to read the definition of a median just one click away. I don't say that that is stupid, that's your interpretation. But a little lazy is it, I can admit that. You have invented half a dozen toy games and want me to explain to you what the median is for these games. If you are unsure on how to calculate the median for different games I recommend you take a course in probability theory. iNic (talk) 17:17, 10 August 2020 (UTC)

Third opinion

 * I was mock-mimicing your ragging on me, not the paper, since you kept including edit summaries like "please educate yourself what a median is". Like I already said, I don't have access to read the paper so I have no judgment on the paper itself, although I think the idea sounds cool.  This discussion is rapidly devolving into personal attacks, so I've made a third opinion request at Wikipedia talk:WikiProject Statistics.
 * I'm not mocking you. I genuinely think you should read the papers first before you explain on Wikipedia what the papers are about. iNic (talk) 19:43, 10 August 2020 (UTC)
 * For the record, you're talking to somebody who does actually comb through sources when they're not paywalled or made otherwise accessible to check them (see: Featured_article_candidates/1969_Curaçao_uprising/archive1 for one example ). If you want to email me a Google Drive link of the PDF, I'll read it.  However, I'm going to argue here that it's actually not super relevant since Wikipedia is written for a general audience.  If, hypothetically, 100% of the general audience "didn't understand what a median was" as you put it, that would absolutely be reason to change the phrasing.  You don't seem to believe me, but your first explanation here on the talk page "worked" - I get what the paper is doing, cool, thanks.  I'm asking you to believe me that what the text you added to the article doesn't adequately communicate that, and should be improved in some fashion that isn't "read the journal article" for future readers, who shouldn't be expected to have to comb through the sources to understand what is going on.  If you think my addition is inaccurate, great, propose something else, but I think your current version is also underspecified and misleading.  SnowFire (talk) 20:10, 10 August 2020 (UTC)
 * The current version is not underspecified or misleading at all. It precisely summarises the conclusions of both these papers. Of which just one is behind a pay wall, by the way. One of them is totally free to read but you haven't read it anyway... And if you are very interested in something you actually sometimes have to pay. To think that you know what the papers are saying while refusing to read them is silly. You do not understand what they are saying. But still you think you know what they are saying so strongly that you want to find a third "neutral" person to judge who is correct, you or me. This will be interesting. :-) iNic (talk) 20:38, 10 August 2020 (UTC)
 * For the 3O responder: see this edit by iNic adding some text about a recent journal article and then my edit here, which has been reverted by INic. Per above, I'm open to alternate phrasings, but think the current text unadorned does not properly explain what the journal article is really doing.  INic has graciously explained what it's doing here on the talk page (it's taking the median of repeated plays, not of a single play) but strongly rejects including this clarification in the article itself, feeling that this is self-evident.  I'm arguing that it's not self-evident and that a median calculated on single-plays of the relevant game is not what the journal article is referring to.  SnowFire (talk) 18:23, 10 August 2020 (UTC)
 * It is interesting that you have such a strong opinion about what an article is saying that you have never read. It is not saying what you think it is saying.iNic (talk) 20:03, 10 August 2020 (UTC)
 * This is the part that has been so insanely frustrating with discussing this with you: I understand what the journal article is saying, thanks. I do not think that the text of the Wikipedia article you added communicates what the journal article said.  An intelligent third-grader could calculate the median of this game and see it's 3 (off a single play).  I'm claiming that's what your current text is saying the journal article is saying, which we both agree is wrong.  I'm saying to fix the Wikipedia article text to accurately communicate what is actually happening.  SnowFire (talk) 20:24, 10 August 2020 (UTC)
 * Haha this is funny. How on earth can you know what two articles are saying that you haven't read? This is spooky. :-P iNic (talk) 20:49, 10 August 2020 (UTC)

Feller
In the subsection called 'Feller' there currently is the remark "when the games of infinite number of times are possible ...". This is imho a remark which should be rephrased or deleted alltogether, because a practical game or test should have an ending and therefor cannot consist of an infinite number of steps. Bob.v.R (talk) 11:03, 20 June 2020 (UTC)
 * This whole paragraph is very badly written so delete it and rewrite it from scratch. iNic (talk) 22:32, 23 October 2020 (UTC)

Samuelson argument, as presented, makes no sense
It's the equivalent of responding to an argument that the house has the edge in blackjack by saying, "well, no one would ever come to a casino if that were true." Clearly people make irrational bets! (Or rational - what if someone wanted to rid of all their money?)

It's possible that Samuelson's actual argument is deeper than this - but as described in the article, it's trivially wrong. The section should be edited or cut. — Preceding unsigned comment added by Chomskied (talk • contribs) 02:37, 3 December 2020 (UTC)

Solution from Ergodicity economics?
Hi all,

I would like to add a section under solutions covering the proposal by Ole Peters and others regarding the non-ergodic property of gambling. Given that the work has gained quite a lot of traction lately I think it warrants inclusion here, obviously with the necessary counterpoints and criticisms.

Cheers, --Fractalfalcon (talk) 13:13, 4 January 2021 (UTC)

Why Must Every Wikipedia Article Be Messed-Up by Some Busybody?
When the passage from Samuelson was first introduced into this article, discussion on this page made it plain that the resolution was reported by Samuelson but not one that he endorsed, and the entry was worded accordingly. Someone who didn't consult Samuelson's article and didn't read the discussion here made an inferential leap which was fallacious, and reworded the entry to make an incompetent claim about Samuelson.
 * Please don't use profanity, we can resolve things with rational language, Wikipedia is not your local pub. I changed to Messed-up.

--mcyp (talk) 04:10, 30 April 2021 (UTC)

By the way. No one has authority to rewrite my comment. If you don't like it, then delete it. Don't misrepresent my actual remarks. —172.58.19.124 (talk) 04:45, 3 January 2019 (UTC)
 * I feel you. However, it's also possible that the incorrect edit helps us have a better text in the long run. For instance, it sometimes happens that a sentence is correct but written in such a way that it's often misunderstood; when it gets "reworded" in a way which actually changes its meaning, we are allowed to realise that the original wording was misunderstood by some readers and eventually we get a version which succeeds better at conveying the intended meaning. Nemo 18:06, 16 December 2019 (UTC)

Kelly criterion
Could you help me understand why you reverted my addition about the kelly criterion? It's my understanding that maximizing expected log utility as described in the article is precisely the same as the kelly criterion, and it also happens to maximize the expected median bankroll. Is that not right? Stonkaments (talk) 00:05, 23 September 2021 (UTC)


 * The identity in itself might very well be correct. If it's not original research and you have a reference to back it up you are welcome to put it back in the article. However, it's not appropriate to put it under the heading "Rejection of mathematical expectation". The ideas in this section does not, for obvious reasons, rely on the concepts of expected value or expected utility.
 * I would suggest you put it somewhere in the section "Expected utility theory". As you placed it in now, it gave the false impression that the median solution of the paradox has something to do with utilities, which it of course hasn't. iNic (talk) 11:13, 23 September 2021 (UTC)