tag:blogger.com,1999:blog-3891434218564545511.post1327785381905859034..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: Winners and losersAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger24125tag:blogger.com,1999:blog-3891434218564545511.post-68406616419173526062013-07-18T23:14:08.560-05:002013-07-18T23:14:08.560-05:00Well, it does seem like there are systems of bets ...Well, it does seem like there are systems of bets that are good and systems of bets that are bad, and suggestion 4 doesn't allow gauging that. (This is Elga's objection that imprecise probabilities get one to pass by good books.) But now that I think about it, this isn't so pressing a worry. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-48052091621439503472013-07-18T22:15:46.498-05:002013-07-18T22:15:46.498-05:00Correction to my first post, 2nd last paragraph: ”...Correction to my first post, 2nd last paragraph: ”...and 1/6 to all her partners...” should be 5/6.<br /><br />On logical relations: One could simply disallow cyclic pairings, perhaps as an extension of property 2. It would still be possible to pair all the players: First find all the first player’s partners. For each of these 6 players in turn, find the 5 remaining partners from previously unpaired players. For each of these 30 players in turn, repeat the process, etc. This procedure would pair everyone with no loops.<br /><br />Would you like to say more on why betting odds may be a problem for suggestion 4?IanShttps://www.blogger.com/profile/01896170184784056126noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-51113943179872747472013-07-18T00:12:02.708-05:002013-07-18T00:12:02.708-05:00I missed the point about there being such logical ...I missed the point about there being such logical relations. Nice point.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-37991758275291019662013-07-18T00:04:39.516-05:002013-07-18T00:04:39.516-05:00If I preferred more objective probabilities, I mig...If I preferred more objective probabilities, I might answer like this: Option 4, unknown rather than interval-valued probabilities, and change when I see a pair.<br /><br />I don’t know how the organizers do the pairing, so I can’t calculate probabilities that depend on the pairing. Suppose I see Jones and Smith paired on day one. The organisers may, for all I know, have a rule that they will pair Jones with Smith only on Jones’ winning day. This would make Jones a winner. Or maybe they will pair Jones with Smith only on Smith’s winning day. This would make Jones a loser. So I can say nothing about the probability that Jones is a winner (and interval-valued probabilities are ruled out).<br /><br />Some pairings lead to tricky logical relations. Suppose for example that Jones and Smith are paired on day 1, Smith and Robinson on day 2 and Robinson and Jones on day 3. Then Jones must have rolled either 1 or 3, so her partner on day 2 must have rolled 2 (and days 4, 5 and 6 are similar). So I have to say that the probabilities<i> that are not logically determined</i> are unknown.<br /><br />When to change? When I learn something that depends on the pairing. When I see Jones paired, I learn which player she is paired with. This is precisely what a placeholder like JonesPair1 does not tell me.IanShttps://www.blogger.com/profile/01896170184784056126noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-42258497843165294992013-07-17T23:58:45.990-05:002013-07-17T23:58:45.990-05:00If I liked indifference distributions, I might ans...If I liked indifference distributions, I might answer like this:<br /><br />Option 3 can be made to work, at least for any finite number of known pairs. The known pairs relate to a finite number (say N) of different players. For these players, list all 6^N combinations of dice rolls. Strike out all combinations that are inconsistent with the known pairs, and assign equal probabilities to the remaining ones. (All the other players are assigned 1/6 to each day, independently for each player.) The resulting distribution is as “uniform” as it can be, consistent with the known pairs.<br /><br />If, for example, my known pairs are all from day 1, I will assign each paired player 1/2 for day 1 and 1/10 for the other days. But if I know all Jones’ pairs and no other pairs, I will assign 1/6 to Jones for all days and 1/6 to all her partners for the day they are paired with her (work it out if you doubt it). This seems reasonable. This method, though tedious, can in principle be applied to any finite number of known pairs.<br /> <br />What about an infinite number? First, unlike the organizers (who have the super powers needed to form an infinite number of pairs in a finite time), I am an ordinary human. I can’t know an infinite number of pairs. Second, I already know that indifference does not work even in the simplest infinite case (the infinite lottery), so I don’t expect it to work here.IanShttps://www.blogger.com/profile/01896170184784056126noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-20293540821191668242013-07-16T18:42:01.424-05:002013-07-16T18:42:01.424-05:00Genesis 3 vs 1 does state that the serpent was the...Genesis 3 vs 1 does state that the serpent was the most cunning (subtle) of all the wild animals the Lord God had created.<br /><br />The thing about it is that that slight tilt doesn't look so bad at all. In fact, sometimes it will appear as something good. In fact, it need a lesser good in order to function.Dagmara Lizlovshttps://www.blogger.com/profile/14744785407281199347noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-41893820423504503802013-07-15T20:29:29.960-05:002013-07-15T20:29:29.960-05:00Indeed that slight tilt ruins lives. It's devi...Indeed that slight tilt ruins lives. It's devilishly clever.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-16753980242003312662013-07-15T20:02:06.153-05:002013-07-15T20:02:06.153-05:00Looks like here's a story that's very simi...Looks like <a href="http://arxiv.org/abs/1202.3376" rel="nofollow">here</a>'s a story that's very similar. (Thanks, LB, for pointing me to it.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-85311422097826122212013-07-15T18:47:09.279-05:002013-07-15T18:47:09.279-05:00"Actually, I think casinos have the system on..."Actually, I think casinos have the system only slightly tilted in their favor, ..."<br /><br />Here is a bit of a human face on that. Over 20 years ago I had a boyfriend whose father was a gambling addict. Many years ago, my sister had a friend whose husband was a gambling addict. I'm only too aware of the devastation that a "system only slightly tilted in their favor" can wreak on families and marriages.<br /><br />Back when I was working up in Trenton, New Jersey, I had a training class held in Atlantic City. It was one of those managing people one day motivational type seminars, and it was held in a room in one of the casinos. After class I tried my hand at the slots. I watch this one guy, obviously of lower income than myself, judging from his work uniform, constantly throwing in money into the slots and pulling the lever like there was no tomorrow, money he probably couldn't afford to loose. After loosing something like $40, I up and left. In the parking lot I met an older couple. They asked me if I won anything, I said no, I lost $40. They replied that that's nothing, they had lost several hundred dollars, and they sounded like they were trying to one up me even though they were far more substantially bigger losers.Dagmara Lizlovshttps://www.blogger.com/profile/14744785407281199347noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-66965907861341145362013-07-13T22:47:00.148-05:002013-07-13T22:47:00.148-05:00Actually, I think casinos have the system only sli...Actually, I think casinos have the system only slightly tilted in their favor, I've heard that rather less than lotteries, but still make lots of money thanks to the wonders of the law of large numbers.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-61435313911625906002013-07-13T20:24:54.411-05:002013-07-13T20:24:54.411-05:00"Actually, even in the case of a fair game, y..."Actually, even in the case of a fair game, you cannot do better than even just by deciding to walk away when you're ahead. The problem is that you might never end up ahead--you might just go down and never return up,..."<br /><br />Ah, but what will keep you there is the psychological impact of something called variable scheduled reward. You might get a little bit of a turn around every so often, and there is always a sense that you might just make it in the next go around. Here's how it works and I've thoroughly learned this technique from my Thoroughbred, Merlin.<br /><br />My problem was how to get Merlin to do things my way with the least amount of resistance on his part. My body had stopped being damage tolerant and rodeo riding is just not my cup of tea. I decided to go the route of variable scheduled rewards. When saddling up, I'd fill my saddle bags with a bunch of carrots broken up into pieces. Merlin knew they were there. Initially when I'd ask him to do something, I'd reward each success on his part. Then I'd change the reward schedule, sometimes I'd reward the first response to my cues, sometimes the second, sometimes the third etc. And I'd mix them up so that Merlin never knew which positive response to my cues would get him the reward. Then at some point, Merlin would hit the "jackpot". I'd hop of his back, make a big fuss over him and give him the rest of the carrots in the saddle bag. Only he never knew when he'd hit the "jackpot". You should have seen the gusto with which that horse would be responding to all my inputs!<br /><br />Casinos train us to keep playing the same way only they are a lot less benevolent, and the variable scheduled reward systems is very heavily tilted in their favor. One of the women I was boarding with was going to the casinos to gamble. I told her that the casino was doing nothing but training her to part with her hard earned cash the same way I trained my horse. She replied "I know what you've been doing to your horse, and it's not my money I'm gambling with."Dagmara Lizlovshttps://www.blogger.com/profile/14744785407281199347noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-37647437876051732342013-07-12T21:18:45.771-05:002013-07-12T21:18:45.771-05:00"I thought you could actually consistently ma..."I thought you could actually consistently maintain a modest per-hour win at blackjack if you knew what you were doing."<br /><br />I have heard that something like that myself. I remember my dad also saying some things about blackjack like that. I do recall my supervisor talking about the crap player. All this was a long time ago. I also remember about 20 years ago Dr. Robert Abernathy was giving a Montecarlo demonstration how you can definitely win in gambling when he came to our place to teach Weibull Analysis to our group. I cannot recall if the game was blackjack or not. I do remember that the super returns came after playing several thousand times, and if you have substantial cash to start with.<br /><br />For more on Dr. Abernathy:<br /><br />http://www.barringer1.com/drbob-bio.htm<br /><br />I have taken Weibull training from him and Wes Fulton a couple of times many, many years ago.Dagmara Lizlovshttps://www.blogger.com/profile/14744785407281199347noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-87364179635057510722013-07-12T20:05:56.616-05:002013-07-12T20:05:56.616-05:00Oh, as for what I conclude, I guess it's that ...Oh, as for what I conclude, I guess it's that our probabilistic methods of reasoning fall apart in such infinite contexts. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-1437369854033704772013-07-12T19:42:10.263-05:002013-07-12T19:42:10.263-05:00Actually, even in the case of a fair game, you can...Actually, even in the case of a fair game, you cannot do better than even just by deciding to walk away when you're ahead. The problem is that you might never end up ahead--you might just go down and never return up, in which case you'll have to stop playing some time (when you die or if you run out of money). The <a href="http://en.wikipedia.org/wiki/Optional_stopping_theorem" rel="nofollow">optional stopping theorem</a> tells you that there is no way of choosing a stopping time for a game (under some conditions) such that you do on average better than even.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-931623886283193822013-07-12T15:25:16.594-05:002013-07-12T15:25:16.594-05:00Hi Professor Pruss,
A couple things:
1) Yes, my ...Hi Professor Pruss,<br /><br />A couple things:<br /><br />1) Yes, my understanding is that Blackjack has the lowest house margin of any casino game, at roughly .01% Over the long run one will come close to even, but if one always walks away at a moderate upswing, I think a profit could be made.<br /><br />2) What is your conclusion from this paradox? That a real infinity is impossible?<br /><br />Thanks,<br />AustinAustinhttps://www.blogger.com/profile/12755810627119315845noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-81366487292048528402013-07-12T11:15:29.344-05:002013-07-12T11:15:29.344-05:00I thought you could actually consistently maintain...I thought you could actually consistently maintain a modest per-hour win at blackjack if you knew what you were doing, at least 20 years ago (I think there may have been some changes). When I was a student, I had a friend--very smart applied math guy--who practiced a lot of blackjack, doing computer simulations, before going to a computer graphics conference in Las Vegas. He said he was making about $10 per hour at blackjack.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-82766259325606539142013-07-08T18:26:55.041-05:002013-07-08T18:26:55.041-05:00About 20 years ago I worked for a boss who knew so...About 20 years ago I worked for a boss who knew some one who occasionally went to Atlantic City to gamble. This person was reasonable successful at gambling, and my boss gave me her secrets:<br /><br />1) She played craps. This was the game with the best probability of winning.<br /><br />2) Be satisfied with winning a modest sum.<br /><br />3) As soon as a modest sum is won - LEAVE THE TABLE. Leaving the table at this point is the crucial key.Dagmara Lizlovshttps://www.blogger.com/profile/14744785407281199347noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-59993752390227130222013-07-08T16:43:43.663-05:002013-07-08T16:43:43.663-05:00What matters is which sequence we picked from, sor...What matters is which sequence we picked from, sorry if that was unclear.<br /><br />I meant, if we first have arrangement (A) and then re-arrange to sequence (B).<br /><br />If we pick a number from (A), tag it somehow, then re-arrange to (B), the tagged number probably moves around in going into the (B) sequence, but it is still 1/2 that it was odd versus even when we picked it from (A).<br />Williamhttps://www.blogger.com/profile/12533263841520213358noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-11696101449859078592013-07-08T16:12:02.100-05:002013-07-08T16:12:02.100-05:00Why would the order matter for probability? Why would the order matter for probability? Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-67517704404655520102013-07-08T16:00:17.483-05:002013-07-08T16:00:17.483-05:00
Given a choice out of the infinite sequence
(A)...<br />Given a choice out of the infinite sequence<br /><br /> (A) 1,2,3,4,5,....<br /><br />the probability of picking an even counting number is 1/2.<br /><br />Given the sequence<br /><br /> (B) 1,2,4,6,8,10,3,12,14,16,18,5,20,22,24,26,...<br /><br />the probability of picking an even counting number is 4/5.<br /><br />The (B) sequence is an arrangement which incorporates a selection bias.<br /><br />Since we can rearrange (A) to be (B) and include all of A, it <br />makes a difference whether we re-arrange first or pick first!<br />Williamhttps://www.blogger.com/profile/12533263841520213358noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-19828682839736114652013-07-08T15:58:41.738-05:002013-07-08T15:58:41.738-05:00Suppose you're watching all of them?Suppose you're watching all of them? Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-37675912065840728932013-07-08T15:33:30.271-05:002013-07-08T15:33:30.271-05:00Did you pick Jones before or after the pairings we...Did you pick Jones before or after the pairings were completed?<br /><br />If before, 1/6, if after, 1/2.<br /><br />Selection bias in the pairings.<br />Williamhttps://www.blogger.com/profile/12533263841520213358noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-39534524176313043592013-07-08T14:12:22.581-05:002013-07-08T14:12:22.581-05:00No, all the winners are paired with all the losers...No, all the winners are paired with all the losers on each day.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-38244287513741406352013-07-08T13:42:11.957-05:002013-07-08T13:42:11.957-05:00If we choose 1/6 of an infinite sampling pool as w...If we choose 1/6 of an infinite sampling pool as winners on day 1 and match them with another 1/6 of the pool as losers, doesn't that mean that 4/6 of the infinite pool is infinitely deferred from being chosen for either role on that particular day?<br /><br />If so, the question is how we know Jones is going to be in the game at all?<br /><br />If we set up the game after Jones rolls, knowing Jones cannot be excluded, that is different (1/6) from the case when the game is fully set up, and then we choose first a pair, then a person we call Jones from the day 1 pairings (1/2).<br />Williamhttps://www.blogger.com/profile/12533263841520213358noreply@blogger.com