Author Archives: Larry Franz

What Should Sleeping Beauty Say, Logically Speaking? (Part 2)

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In yesterday’s post, I described the so-called “Sleeping Beauty Problem”. Mathematicians and philosophers have been debating this problem for the past 15 years (most recently in response to an article at the Quanta Magazine site). What should Sleeping Beauty say when she wakes up on Monday or Tuesday and is asked the following question about a coin flip that happened on Sunday: “What are the odds that the coin we flipped on Sunday came up heads?”.

To review: Sleeping Beauty doesn’t know what day it is when she wakes up or whether she’s already woken up during the experiment. But she is told that if the coin flip on Sunday came up heads, she’s only being awakened on Monday. If, however, the coin came up tails, she’s being awakened on both Monday and Tuesday. “Halfers” think the odds that the coin came up heads is 1/2. “Thirders” think the answer is 1/3.

Here’s the answer Pradeep Mutalik, the author of the Quanta article, gave in response to a whole lot of comments:

This is the crux of the idea: that there are two propositions in the Sleeping Beauty problem:

1) the probability of heads when the fair coin is tossed, which is obviously 1/2. This is the proposition modeled by halfers.
2) the probability of heads encountered later, which thanks to the experimental conditions that are asymmetric between heads and tails, is 1/3. This is the proposition modeled by thirders.

… the same problem statement can mean tossed if understood one way and encountered if interpreted another way.

Once you accept that there are two valid interpretations, the solution of the Sleeping Beauty problem can be expressed in one sentence:

A fair coin is tossed: The probability of it landing heads when tossed is ½; the probability of Sleeping Beauty encountering heads later, thanks to the asymmetric experimental conditions, is 1/3.

And that, in a nutshell, is all there is to it.

Mr. Mutalik also added this note in response to my own comment on his article, in which I came out in support of the 1/3 answer:

As in my reply to Rich, your answer is correct for the heads “encountered” probability. The heads “tossed” probability is, and remains ½ until SB learns the actual result of the coin toss.

The point of all this is to show that there is no incompatibility in the thirder and halfer arguments. The same person can simultaneously be both a halfer and a thirder.

As a new supporter of the thirder position, I’m not convinced by Mr. Mutalik’s response. As I wrote in the previous post, it’s clear that the odds of getting heads or tails when a coin is flipped are always 1/2 heads and 1/2 tails. That’s how coin flips work (putting aside the extremely rare occasions when they land on their edge and don’t fall either way). 

But the question presented to Sleeping Beauty is: “What are the odds that the coin flip came up heads?” Since she knows how the experiment is being conducted when she’s asked this question, shouldn’t she conclude that the odds are 1/3, not 1/2? After all, she knows that a coin that came up tails means this could be either Monday or Tuesday. If it came up heads, it can only be Monday. So the odds that the coin came up tails is twice as likely as the odds that it came up heads (i.e., 2/3 vs. 1/3).

It seems to me that asking Sleeping Beauty to ignore the information she’s been given about the experiment is asking her to give an answer she knows is incorrect. What a question means depends on its context. If the context in this case were simply: “Please explain how coin flips work. In particular, what are the odds that a fair, properly-executed coin flip will come up heads instead of tails, such as the coin flip we executed on Sunday?” In that context, the correct answer is obviously 1/2. Everybody should be a halfer when asked how coin flips work.

But that isn’t the context in which Sleeping Beauty is being asked to give an answer. She’s not being asked to explain how coin flips work in general. She understands that she’s involved in a bizarre, possibly unethical (the memory-destroying drug!) experiment and that the question she’s being asked is part of that experiment. She’s being asked what the odds are given her current situation. That’s why I think it’s more reasonable for Sleeping Beauty to be a thirder than a halfer. She should answer “1/3”. It’s twice as likely that the coin came up tails!

Whether she should be allowed to go back to sleep after giving the correct answer, so that she can later be awakened by a handsome young prince, or whether she should receive appropriate medical care and reacquire a normal sleeping pattern, is an ethical question, not a logical one, and beyond the scope of this post.

What Should Sleeping Beauty Say, Logically Speaking?

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Imagine that Sleeping Beauty agrees to take part in an experiment. First, she’s going to be given a special memory-loss drug that will make her forget what day it is. Then she’ll go to sleep. While she’s asleep, the people running the experiment will flip a coin. If the coin comes up heads, they’ll wake her up on Monday and ask her a question. If it comes up tails, they’ll wake her up on Monday, ask her the same question, give her the drug, let her fall asleep again, and then wake her up on Tuesday, asking the same question as before.

Because of the drug, however, she won’t know what day it is when she wakes up. Furthermore, she won’t know whether this is the first or second time she’s woken up. Each day will be a brand new experience for her. But each time Sleeping Beauty wakes up during the experiment, she’s told about the coin flip and the rule that says heads will mean she will wake up on Monday, and tails will mean she will wake up Monday and Tuesday. 

Now here’s the problem: Each time Sleeping Beauty wakes up and is told the rules of the experiment, she’s asked this question: “What are the odds that the coin flip came up heads?” That is, what are the odds that the coin came up heads and this is Monday (as opposed to tails and this is either Monday or Tuesday)? What should Sleeping Beauty say?

I came across this problem at the Quanta Magazine site yesterday. You may be surprised to learn that experts have been arguing about its solution for years:

The famous Sleeping Beauty problem has polarized communities of mathematicians — probability theorists, decision theorists and philosophers — for over 15 years…. This simple mathematical problem has generated an unusually heated debate. The entrenched arguments between those who answer “one-half” (the camp called “halfers”) and those who say “one-third” (the “thirders”) put political debates to shame… Halfers and thirders tend to remain firmly rooted in their view of the Sleeping Beauty problem. Both camps can certainly do the math, so what makes them butt heads in vain? Is the problem underspecified? Is it ambiguous?

So, should Sleeping Beauty say there is a 1/2 chance that the coin came up heads, since there’s always a 1/2 chance that coin flips come up heads and a 1/2 chance that they come up tails? Should Sleeping Beauty be a halfer?

Or should she be a thirder and say there is only a 1/3 chance it was heads? After all, if it’s Monday, she was awakened because the coin came up heads or tails. If it’s Tuesday, she’s awake because the coin came up tails. That means it’s twice as likely she’s awake because the coin came up tails. There’s a 2/3 chance the coin came up tails and a 1/3 chance it came up heads. It sounds like Sleeping Beauty should be a thirder.

But not so fast! There was only that one coin flip on Sunday and we all know that coins have a 1/2 chance of coming up heads! Maybe she should be a halfer?

The Quanta article is fairly long and delves into why halfers and thirders give the answers they do, as well as why they often resist changing their minds. The comments that follow the article are even longer and include mathematical formulas. I didn’t read all the comments, and I’ve never studied probability, but I left my own comment anyway:

I wasn’t a halfer or a thirder before reading the article. Now I’m a committed thirder.

If you were to simply ask Sleeping Beauty whether a fair coin toss came up heads, she should say the odds were 1/2. Without any other information, that’s the rational answer. But you’re asking what odds Sleeping Beauty should assign, given the additional information she’s been given about the experiment. Since today could be either Monday or Tuesday (as far as she knows), it’s more likely that the coin came up tails. The fully-informed, rational answer she should give is 1/3.

So I think Sleeping Beauty should say this each morning: “Given how coin flips work, the odds are 1/2 that heads came up. But given how coin flips work and given what you’ve told me about this peculiar experiment, the odds are only 1/3 that heads came up. I can easily flip back and forth between the halfer and thirder positions, but why should I ignore the additional information you’ve given me? Taking into account what I  know about the experiment, I must conclude that the odds are 1/3 that heads came up. If that makes me a thirder, so be it. Now where’s my prince?”

But she shouldn’t take my word for it. She should make up her own mind.

The author of the article, Pradeep Mutalik, responded to my comment and everyone else’s last night. I’ll post his response tomorrow in case you haven’t already rushed to the Quanta site to read it.

I’m Super, You’re Okay. Clinton, Sanders and Arithmetic.

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[Note: After I wrote this long post, I saw two articles at Salon that nicely capture what’s going on with the selection of delegates to the Democratic convention. You could go to the bottom of this post and read them, skipping everything I have to say. I mean, they were both written by professional journalists – the people we trust to report and explain the news.]

You may have heard that the Democratic Party has superdelegates. These are people who get to vote at the upcoming national convention because of who they are (they’re former Presidents, members of Congress, Democratic Party officials, and so on). There will be 712 superdelegates at this year’s convention. That’s roughly 15% of the total number of delegates (which is 4,765). It’s been reported that most of the superdelegates (469 of them so far) have said they’ll vote for Hillary Clinton at the convention, but all of them are free to vote for Bernie Sanders if they want. 

The other 85% of the Democratic delegates are selected as the result of primary elections and party caucuses. These contests began in Iowa on February 1st and will end in the District of Columbia on June 14th. How many of the Clinton or Sanders delegates are selected in these contests depends on how many votes Clinton or Sanders receives (as well as the rules of the local Democratic Party). Unlike the similar process taking place in the Republican Party, none of the Democratic contests are “winner take all”. Delegates are assigned roughly proportionately.

According to Wikipedia, Hillary Clinton has 1,310 or 54% of these pledged, non-super delegates so far. Bernie Sanders has 1,094 or 46%. Those percentages roughly correspond to the number of votes Clinton and Sanders have received. Some states don’t report vote totals, but for the states that do, the New York Times says Clinton has received 9.4 million votes (57%) and Sanders has received 7 million (43%).

Clearly, this process is totally rigged!  

That’s what people are saying anyway. From a news article in the Times:

Backers of Senator Bernie Sanders, bewildered at why he keeps winning states but cannot seem to cut into Hillary Clinton’s delegate count because of her overwhelming lead with “superdelegates,” have used Reddit and Twitter to start an aggressive pressure campaign to flip [superdelegate] votes [to Sanders].

From a comment (recommended by 612 readers) in response to that article:

Not only is the DNC primary process in conflict with Democracy, but it is borderline authoritarian. Party leaders picking a candidate before voters have yet to speak is the epitome of corruption. If Hillary Clinton should get the nomination in no small part due to super delegate power, the DNC is in for a very harsh reality.

From noted political scientist D. J. Trump:

“Think of this. So I watch Bernie, he wins. He wins. He keeps winning, winning. And then I see, he’s got no chance. They always say he’s got no chance. Why doesn’t he have a chance? Because the system is corrupt,” Trump argued. “This is a crooked system, folks.”

From the hosts of a morning talk show:

Co-hosts Joe Scarborough and Mika Brzezinski were exasperated by just how rigged the Democratic primary system must be for Sanders to have won eight of the last nine primaries and still fallen further back in the overall delegate count [note: Sanders actually earned 110 more delegates than Clinton in those nine contests, eight of which were caucuses, not primary elections].

“Bernie Sanders won Wyoming by 12 percent, but he might not even pick up a single delegate. Hillary Clinton was awarded 11 delegates, Bernie Sanders only seven,” Scarborough said. “Why does the Democratic Party even have voting booths? This system is so rigged” [note: Sanders and Clinton both received seven delegates to the state’s May 28th convention as a result of the Wyoming caucuses — see the update down below for more on this subject].

A wise person would ignore all the hot air and misinformation being spread about the shocking unfairness of the Democrats’ presidential nomination process. New York and Pennsylvania will be voting soon, after which it should become even more obvious which candidate has the most support among Democratic voters.

Nevertheless, a few points deserve mention (I’ve never claimed to be wise).

First, the number of states a candidate wins has nothing at all to do with how many delegates that candidate receives. That’s because delegates are apportioned to the states based on their populations and some states have much larger populations than others. Idaho, Hawaii, Alaska, Utah and Wyoming, all of which Sanders won last month, had 111 delegates up for grabs between them. New York, voting next week, has 291. Pennsylvania, the following week, has 210. It’s only in the anti-democratic U.S. Senate where every state, the huge and the tiny, gets the same number of votes.

Second, in terms of democracy, primary elections are more democratic than caucuses. Primary elections encourage relatively large numbers of voters to participate and everyone casts a secret ballot. Caucuses rely on a smaller number of voters who are willing and able to attend a meeting and then vote in public. That’s why primary elections have replaced caucuses in most states, especially the big states with the most delegates.

Of the 37 Democratic contests so far, Clinton has won 20 and Sanders has won 17. But 16 of Clinton’s 20 wins have been the result of primary elections. Only six of Sanders’s 17 wins have been. This is why Clinton has received so many more votes than Sanders. (By the way, states that have caucuses don’t apportion delegates based on the total statewide vote. Only states with primary elections do that. States that haven’t adopted the more modern primary election system generally have complicated rules that apportion delegates based on which caucuses a candidate wins. Caucuses are not elections. They’re glorified discussion groups.) 

Third, if there weren’t any superdelegates at all, Clinton would be well on her way to winning the nomination on the first ballot by virtue of her success in the primaries and caucuses. All she would need to do is to continue winning a majority of the delegates before the convention starts. The existence of the superdelegates, therefore, is what’s stopping Clinton from arriving at the convention with the nomination sewn up. The existence of the superdelegates means that a candidate needs to win roughly 60% of the non-superdelegates in order to get the nomination, instead of a simple majority. If a candidate were to win 60% of the non-superdelegates, he or she wouldn’t need any superdelegate votes at all. But Clinton has “only” won 54% of the delegates so far.

Since the superdelegates do exist (as they’ve existed in the Democratic Party for decades), there are only 2,404 delegates left to be selected before the convention. And that means Clinton would need to win 1,073 or 69% of those remaining 2,404 non-superdelegates (much more than her current 54%) in order to have the nomination in her grasp before the convention begins. Since Sanders would need to win even more of the remaining delegates (1,289 or 83% of those 2,404) to accomplish the same thing, it’s clear that neither candidate will have enough pledged delegates to win the nomination on the first ballot (unless one of them drops out before the convention, which isn’t going to happen, despite Clinton’s email issue). The arithmetic dictates that the superdelegates will get the last word on who becomes the Democratic nominee.

Now, since the rules say they get to vote, who should they vote for? Should the superdelegates vote for the candidate with the loudest supporters on the internet? Or the one perceived to have the most “momentum” this week? Or should they vote for the candidate who arrives at the convention with the most delegates? In purely democratic terms, given the numbers, the answer is clear: Hillary Clinton should receive a majority of the superdelegate votes and win the nomination.

If I were a superdelegate, however, I’d vote for whichever candidate I preferred, whether or not the voters in the primaries and caucuses agreed with me. Personally, I don’t see the Democratic Party’s decision to give senior members of the party a voice in selecting the party’s nominee as a terrible, un-democratic mistake. (If the Republicans had superdelegates, Trump would be less likely to be the Republican nominee.)

But considering what I’ve said above, all this overheated complaining from Sanders supporters and political commentators about unfairness and a rigged process is just silly (“He’s not getting the most votes or the most delegates and he won’t get the nomination. How unfair!”). Sanders doesn’t have a right to the Democratic nomination, no matter how much his supporters whine about unfairness and a lack of democracy. The fact is that the people are speaking and they’re saying that Hillary Clinton should be the Democratic nominee, the infamous superdelegates notwithstanding.

Let’s hope that everyone who is complaining so much and feeling so persecuted comes out in November and helps elect Democratic candidates, instead of staying home and moaning about how Bernie and they have been mistreated. We really do need to cast as many votes as possible for Democrats in November, because the Republican Party has completely gone off the rails. That’s the reality of our situation.

UPDATE:

These two articles were published this morning at Salon. Both refer to the results of the Wyoming caucuses. One article is called:

“This system is so rigged”: Outrage as undemocratic superdelegate system gives Clinton unfair edge over Sanders

The other is called:

Bernie is wrong about super-delegates: Why his tortured Dem Primary arguments try to have it both ways

One article dispenses a lot of heat. The other dispenses some light. Guess which one relies on facts instead of bluster.

The Metaphysics and Ethics of Relativism by Carol Rovane

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I would have to read this book at least one more time in order to feel confident about summarizing the conclusions the author reaches. However, here’s my impression after reading it once. As I understand her aims, Carol Rovane wants to clearly explain what relativism is with respect to science and ethics, and then determine whether we should endorse relativism with respect to either of those domains.

She begins by criticizing what she calls “the prevailing, consensus view” of relativism, which she says relies on the idea of disagreement. This is the idea that relativism arises “with a certain kind of disagreement that is said to be, first of all, ‘irresoluble’ [i.e. unsolvable], but also, second, ‘irresoluble’ for the specific reason that both parties are right” [15-16]. Rovane prefers defining relativism in terms of alternatives, which may or may not involve disagreement, and which are themselves explainable in terms of “normative insularity”.

According to Rovane, relativists believe that some alternative views in science or ethics are cut off from other scientific or ethical views. Logic neither “mandates, licenses or prohibits” inferences between them, so two people can hold alternative views about science or ethics and logic has nothing to say about the alternatives [94]. It’s as if, metaphorically speaking, people can occupy different scientific or ethical worlds. Non-relativists, on the other hand, believe that all truth bearers are logically related, either directly or indirectly. My scientific views aren’t insulated from your scientific views, and your ethical views aren’t insulated from mine. We all occupy the same scientific world and the same ethical world.

Rovane goes so far as to label the non-relativist and relativist positions in terms of how many “worlds” they mandate. What she calls “unimundialism” is the non-relativistic view that there is only one world (in which there is no “normative insularity” between propositions in science or ethics). “Multimundialism” is the relativistic view that there are many worlds (in which there is “normative insularity” between some scientific or ethical propositions). 

I think the conclusion she reaches is that scientific theories apply to a single world, so it’s best not to think of science in unimundial or non-relativistic terms. Reality is one, so alternative scientific theories can’t be equally correct. But unlike scientists, who all study the same world, people grow up and live their daily lives in various social conditions. These social conditions help determine which behavior is morally correct for them. Rovane thinks it’s fair to say, therefore, metaphorically speaking, that people inhabit different ethical worlds depending on their particular social conditions. Hence, multimundialism or relativism is an acceptable view with respect to ethics.

To help justify her relativistic conclusion regarding ethics, Rovane asks us to imagine two women. One woman was brought up in Europe or America and accepts the ethical importance of autonomy, i.e. every individual’s right to make their way in the world according to their own needs and desires while respecting the needs and desires of other people. The other woman was brought up in a village in India and sincerely believes she has an ethical obligation to obey her parents, even if it means giving up her right to pursue her own needs and desires.  Rovane argues that these two women live in very different ethical worlds. Their societies are so different when it comes to ethical issues that each woman is acting ethically, even though they are following very different paths and choosing to obey very different ethical principles.

Adventures in the Screen Trade: A Personal View of Hollywood and Screenwriting by William Goldman

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William Goldman is a novelist who became a successful screenwriter in the 1960s. His best-known screenplays include Harper (the Paul Newman detective movie), Butch Cassidy and the Sundance Kid, Marathon Man, All the President’s Men,  A Bridge Too Far and The Princess Bride. He wrote this book in 1982, partly as a memoir and partly as a guide to screenwriting. It’s a bit dated now, but it’s still a wonderful book for anyone who’s interested in how movies get made (and how many movies don’t). 

Goldman lists two key lessons for the novice screenwriter (the Roman numerals and capital letters are his):

I. NOBODY KNOWS ANYTHING (i.e. nobody in the movie business knows for sure what will work and what won’t)

II. SCREENWRITING IS STRUCTURE (i.e. there’s more to screenwriting than telling your story in the right sequence, but you’ll never write a good one if you don’t get the story’s structure right).

The biggest lesson I took away from the book, however, is that screenwriting is extremely frustrating. You can make a whole lot of money at it, if you’re very talented and/or very lucky, but you’ll spend most of your time writing scripts that never get made into movies, and when one of your scripts does get filmed, you won’t have any control over what the director, producers, actors, et al. do with it. Film making is a collaborative medium, but the screenwriter is rarely invited to collaborate after filming starts. It’s unlikely you’ll even be invited to a sneak preview.