Beauty and the Bets

post by Ape in the coat · 2024-03-27T06:17:27.516Z · LW · GW · 23 comments

Contents

  Introduction
  Different Probabilities for Different Betting Schemes?
    Thirder Per Awakening Betting
    Thirder Per Experiment Betting
  Do Halfers Need to Bet on Thirders Odds?
    Halfer Per Awakening Betting
    Halfer Per Experiment Betting
  Betting Odds Are a Poor Proxy For Probabilities
  Utility Instability under Thirdism
  Thirdism Ignores New Evidence
    Technicolor Sleeping Beauty
    Rare Event Sleeping Beauty
  Conclusion
None
23 comments

This is the ninth post in my series on Anthropics. The previous one is The Solution to Sleeping Beauty [LW · GW].

Introduction

There are some quite pervasive misconceptions about betting in regards to the Sleeping Beauty problem.

One is that you need to switch between halfer and thirder stances based on the betting scheme proposed. As if learning about a betting scheme is supposed to affect your credence in an event.

Another is that halfers should bet at thirders odds and, therefore, thirdism is vindicated on the grounds of betting. What do halfers even mean by probability of Heads being 1/2 if they bet as if it's 1/3?

In this post we are going to correct them. We will understand how to arrive to correct betting odds from both thirdist and halfist positions, and why they are the same. We will also explore the core problems with betting arguments as a way to answer probability theory problems and, taking those into account, manage to construct several examples showing the superiority of the correct halfer position in Sleeping Beauty.

Different Probabilities for Different Betting Schemes?

The first misconception has even found its way to the Less Wrong wiki:

If Beauty's bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half. If the bets get paid out once per awakening, acting as if the probability is one third has the best expected value.

It originates from the fact that there are two different scoring rules [LW · GW], counting per experiment and per awakening. If we aggregate using the per experiment rule, we get P(Heads) = 1/2 - probability that the coin is Heads in a random experiment. If we aggregate using the per awakening rule we get P(Heads) = 1/3 - probability that the coin is Heads in a random awakening. The grain of truth is that you indeed can use this as a quick heuristic for the correct betting odds.

However, as I've shown in the previous post, only the former probability is mathematically sound for the Sleeping Beauty problem, because awakenings do not happen at random. So, it would've been very strange if we really needed to switch to a wrong model to get the correct answer in some betting schemes. Beyond a quick and lossy heuristic, it would be a very bad sign if we were unable to get the optimal betting odds from the correct model.

It would mean that there is something wrong with it, that we didn't really answer the question fully and now are just rationalizing as all the previous philosophers who endorsed a solution, contradicting probability theory, and then came up with some clever reasoning why it's fine.

And of course, we do not actually need to do that. As a matter of fact, even thirders - people who are mistaken about the answer in the Sleeping Beauty - can totally deal with both per experiment and per awakening bets.

Let $U\left(X\right)$ be the utility gained due to the realization of event X. Then we can calculate expected utility of a bet on X as:

$E\left(X\right)=P\left(X\right)U\left(X\right)-{\sum }_{i=1}^{n}P\left(A_i\right)U\left(A_i\right)$

where $\{A_1,...A_n\}=!X$ - mutually exclusive events with $X$ 

Thirder Per Awakening Betting

Let's start with the natural-to-them per awakening betting scheme:

On every awakening the beauty can bet on the result of the coin toss. What betting odds should she accept?

In this betting scheme both Tails awakenings are equally rewarded, so

$U\left(Tails\&Monday\right)=U\left(Tails\&Tuesday\right)=U\left(Tails\right)$

$U\left(Heads\&Monday\right)=U\left(Heads\right)$

According to thirder models:

$P\left(Heads\&Monday\right)=P\left(Tails\&Monday\right)=P\left(Tails\&Tuesday\right)=1/3$, therefore:

$E\left(Heads\right)=1/3U\left(Heads\right)-(1/3+1/3)U\left(Tails\right)$

Solving $E\left(Heads\right)=0$ for $U\left(Heads\right)$ we get:

$U\left(Heads\right)=2U\left(Tails\right)$

Which means that the utility gained from realization of Heads should be at least twice as big as the utility of realization of Tails, so that betting on Heads wasn't net negative. 

And thus betting odds should be 1:2

Thirder Per Experiment Betting

Now, let's look into per experiment betting

The beauty can bet on the result of the coin toss while she is awakened only once per experiment. What betting odds should she accept?

From the position of thirders, this situation is a bit trickier. Here either $U\left(Tails\&Monday\right)$, or $U\left(Tails\&Tuesday\right)$ is zero, as betting on one of the Tails awakenings doesn't count. Their sum, however is constant.

$U\left(Tails\right)=U\left(Tails\&Monday\right)+U\left(Tails\&Tuesday\right)$, taking it into account:

$E\left(Heads\right)=1/3U\left(Heads\right)-1/3\left(U\right(Tails\&Monday)+U(Tails\&Monday\left)\right)$

$E\left(Heads\right)=U\left(Heads\right)-U\left(Tails\right)$

Solving  $E\left(Heads\right)=0$ for $U\left(Heads\right)$ we get

$U\left(Heads\right)=U\left(Tails\right)$

Which means 1:1 betting odds.

Do Halfers Need to Bet on Thirders Odds?

The result from the previous section isn't exactly a secret. It even led to a misconception that halfers have to bet on thirders' odds, and therefore betting arguments validate thirdism.

Now, it has to be said that correctly reasoning halfers indeed have to bet on the same odds as thirders - 1:1 for per experiment betting and 1:2 for per awakening betting. But this is in no way a validation of thirdism; halfers have as much claim for these odds as thirders. It's only an unfortunate occurrence, that they happened to be initially called "thirders odds".

Historically, the model most commonly associated with answering that P(Heads)=1/2 is Lewis's one. When people were comparing it and thirder models, they named the odds that the former produces to be "halfer odds" and the odds that the latter produces to be "thirder odds". Which was quite understandable at the time.

Now we know that Lewis's model is a wrong representation for halfism in Sleeping Beauty, and indeed fails to produce correct betting odds for reasons explored in previous posts. The correct halfer model, naturally, doesn't have such problems. But the naming already stuck, confusing a lot of people along the way.

Halfer Per Awakening Betting

Let's see it for ourselves, which odds the correct model recommends. Starting from the per awakening betting scheme.

On every awakening the beauty can bet on the result of the coin toss. What betting odds should she accept?

$P\left(Heads\right)=P\left(Tails\right)=1/2$

$Tails$, $Tails\&Monday$, $Tails\&Tuesday$ - are all different names for the same outcome, as we remember, so

$U\left(Tails\right)=U\left(Tails\&Monday\right)=U\left(Tails\&Tuesday\right)$

On the other hand, both $Tails\&Monday$ and $Tails\&Tuesday$ awakenings are rewarded when Tails, so

$E\left(Heads\right)=P\left(Heads\right)U\left(Heads\right)-P\left(Tails\right)\left(U\right(Tails\&Monday)+U(Tails\&Tuesday\left)\right)$

$E\left(Heads\right)=P\left(Heads\right)U\left(Heads\right)-2P\left(Tails\right)U\left(Tails\right)$

Solving for $U\left(Heads\right)$:

$U\left(Heads\right)=2P\left(Tails\right)U\left(Tails\right)/P\left(Heads\right)$

$U\left(Head\right)=2P\left(Tails\right)$

Just as previously, we got 1:2 betting odds.

This situation is essentially making a bet on an outcome of a coin toss, and then the same bet has to be repeated if the coin comes Tails. Betting on 1:2 odds doesn't say anything about the unfairness of the coin or having some new knowledge about its state. Instead, it's fully described by the unfairness of the betting scheme which rewards Tails outcomes more.

Halfer Per Experiment Betting

Now, let's check the per experiment betting scheme

The beauty can bet on the result of the coin toss while she is awakened only once per experiment. What betting odds should she accept?

This time Tails outcome isn't rewarded twice so, everything is trivial

$E\left(Heads\right)=P\left(Heads\right)U\left(Heads\right)-P\left(Tails\right)U\left(Tails\right)$

So if $E\left(Heads\right)=0$:

$U\left(Head\right)=U\left(Tails\right)$

And we have 1:1 betting odds. Easy as that.

Betting Odds Are a Poor Proxy For Probabilities

Why do models claiming that probabilities are different produce the same betting odds? That doesn't usually happen, does it?

Because betting odds depend on both probabilities and utilities of the events. Usually we are dealing with situations when utilities are fixed, so probabilities are the only variable, therefore, when two models disagree about probabilities, they disagree about betting as well.

But in Sleeping Beauty problem, the crux of disagreement is how to correctly factorize the product $P\left(X\right)U\left(X\right)$. What happens when the Beauty has extra awakenings and extra bets?  One approach is to modify the utility part. The other - to modify probabilities. 

I've already explained why the first one is correct - probabilities follow specific rules according to which they are lawfully modified, so that they keep preserving the truth. But for the sake of betting it doesn't appear to matter. 

Betting odds do not have to follow Kolmogorov's third axiom. 10:20 odds are as well defined as 1:2. It's just a ratio, you can always renormalize it, which you can't do to probabilities [LW · GW]. You can define a betting scheme that ignores the condition of mutual exclusiveness of the outcomes, which is impossible when you define a sample space [LW · GW]. Betting odds are an imperfect approximation of probability, that cares only about frequencies of events and not their other statistical properties [LW · GW]. 

This is why incorrect thirder models manage to produce correct betting odds. All the reasons for why these models are wrong do not matter anymore, when only betting is concerned. And this is why betting is a poor proxy for probabilities - it ignores or obfuscates a lot of information. 

For quite some time I've been arguing [LW(p) · GW(p)] that we can't reduce probability theory to decision theory. That while decision making and betting is an obvious application of probability, it's not its justification. That all such attempts are backwards, confused thinking.

The Sleeping Beauty problem is a great example how simply thinking in terms of betting can lead people astray. People found models that produce correct betting odds and got stuck with them, not thinking further, believing that all the math work is done and they now just need to come up with some philosophical principle justifying the models.

And so the "Shut Up and Calculate" crowd happened to silently compute nonsense.

If a probabilistic model produces incorrect betting odds it's clearly wrong. But if it produces correct odds, it still doesn't mean that it's the right one! Betting is a required but not a necessary condition. You also need to account for theoretical properties of probabilities which are not captured by it.

If I didn't resolve it in a previous post we would've been in a conundrum, still thinking that both models are valid. It's good that now we know better. And yet, there is an interesting question: can we still, somehow, despite all the aforementioned problems, come up with a decision theoretic argument distinguishing between thirdism and the correct version of halfism?

As a matter of fact, I can even present you two of them.

Utility Instability under Thirdism

The reason why in most cases disagreement about probabilities implies disagreement about bets is that we assume, that while probabilities change based on available evidence, the utilities of events are constant and defined by the betting scheme. However, this is not the case with Thirdism in Sleeping Beauty, which not only implies constant shifts in utilities throughout the experiment but also that these shifts can go backwards in time.

Let's investigate what probabilities are assigned to coin being Heads on Sunday - before the experiment started, on awakening during the experiment and on Wednesday - when the experiment ended. The correct model is very straightforward in this regard: 

$P\left(Heads|Sunday\right)=P\left(Heads|Awake\right)=P\left(Heads|Wednesday\right)=1/2$

Updateless [LW · GW] and Updating [LW · GW] Thirder models do not agree which is the correct probability for P(Heads|Sunday), but let's use common sense and accept that it's 1/2 as it should be for a fair coin toss. Therefore: 

$P\left(Heads|Sunday\right)=1/2$

$P\left(Heads|Awake\right)=1/3$

$P\left(Heads|Wednesday\right)=1/2$

Suppose that the Beauty made a bet on Sunday at 1:1 odds, that the coin will come Heads. The bet is to be resolved on Wednesday when the outcome of the coin toss is publicly announced. What does she think about this bet when she awakes during the experiment? If she follows the correct halfer model - everything is fine. She keeps thinking that the bet is neutral in utility.

But a thirder Beauty suddenly find herself in a situation where she is more confident that the coin came Tails when she used to be. How is she supposed to think this? Should she regret the bet and wish she never made it?

This is the usual behavior in such circumstances. Consider the Observer Sleeping Beauty Problem [LW · GW]. There:

$P\left(Heads|Sunday\right)=1/2$ and $P\left(Heads|Awake\right)=1/3$

The observer is neutral about a bet on Heads at 1:1 odds on Sunday, but if then they find that the Beauty is awakened on their work day, they would regret the bet. If they were proposed to pay a minor fee to consider the bet null and void, they are better off to do it.

Would Sleeping Beauty also be better off to abolish the bet for a minor fee? No, of course not. That would lead to always paying the fee, thus predictably losing money in every experiment. But how is thirder supposed to persuade herself not to agree?

Mathematically, abolishing such a bet is isomorphic to making an opposite bet at the same odds. And as we already established, making one per experiment bet at 1:1 odds is utility neutral, so a minor fee will be a deal breaker. Thirder's justification for it is that the utility of such bet is halved on Tails, because only one of the Tails outcomes is rewarded.

But it means that a thirder Beauty should think as if the fact of her awakening in the experiment retroactively changes the utility of a bet that she has already made! Instead of changing neither probabilities nor utilities, thirdism modifies both in a compensatory way. 

A similar situation happens when the Beauty makes a bet during the experiment and then reflects on it on Wednesday. Halfer Beauty doesn't change her mind in any way, while thirder Beauty has to retroactively modify utilities of the previous bets to compensate for the back and forth changes of her probability estimates.

Which is just an unnecessarily complicated and roundabout way to arrive to the same conclusion as the correct halfer model. It doesn't bring any advantages, just makes thinking about the problem more confusing.

Thirdism Ignores New Evidence

We already know that Thirdism updates probability estimate in spite of receiving no new evidence. But there is an opposite issue with it as well. It refuses to acknowledge actual relevant evidence, which may lead to confusion and suboptimal bets.

To see this let's investigate two modified settings, where the Beauty actually receives some kind of evidence on awakening.

Technicolor Sleeping Beauty

Technicolor Sleeping Beauty is a version of the original problem that I've encountered in Rachael Brigg's Putting a Value on Beauty, where the idea was credited to Titelbaum.

The modified setting can be described as this:

Sleeping Beauty experiment, but every day the room that the Beauty is in changes its color from Red to Blue or vise versa. The initial color of the Room is determined randomly with equal probability for Red and Blue

Ironically enough, Briggs argues that Technicolor Sleeping Beauty presents an argument in favor of thirdism, because halfer Sleeping Beauty apparently changes her estimate of P(Heads), despite the fact that the color of the room "tells Beauty nothing about the outcome of the coin toss". But this is because she is begging the question, assuming that thirders' approach is correct to begin with.

Let's start from how thirders perceive the Technicolor problem. Just as Briggs claims, from their perspective, it seems completely isomorphic to the Sleeping Beauty. They believe that the color of the room is irrelevant to the outcome of the coin toss.

$P\left(Blue|Heads\right)=P\left(Red|Heads\right)=P\left(Blue|Tails\right)=P\left(Red|Tails\right)=1/2$

$P\left(Blue\right)=P\left(Red\right)=1/2$

And so thirder Beauty has the same probability estimate for Technicolor Sleeping Beauty as the regular one.

$P\left(Heads|Blue\right)=P\left(Heads|Red\right)=P\left(Heads|Awake\right)=1/3$

Which means the same betting odds. 1:2 for per awakening betting and 1:1 for per experiment one. Right?

And so, suppose that Beauty, while going through Technicolor variant is proposed to make one per experiment bet on Heads or Tails with odds in between 1:2 and 1:1, for example, 2:3. Should she always refuse the bet?

Take some time to think about this.

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No, really, it's a trick question. Think about it for at least a couple of minutes before answering.

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Okay, if despite the name and introduction of this section and two explicit warnings, you still answered "Yes, the Beauty should always refuse to bet at these odds", then congratulations!

You were totally misled by thirdism!

The correct answer is that there is a better strategy than always refusing the bet. Namely: choose either Red or Blue beforehand and bet Tails only when you see that the room is in this color. This way the Beauty bets 50% of time when the coin is Heads and every time when it's Tails, which allows her to systematically win money at 2:3 odds.

This strategy is obscured from thirders but is obvious for a Beauty that follows the correct, halfer model. She is fully aware that Tails&Monday awakening is always followed by Tails&Tuesday awakening and so she is completely certain to observe both colors when the coin is Tails:

$P\left(Blue|Heads\right)=P\left(Red|Heads\right)=1/2$

$P\left(Blue|Tails\right)=P\left(Red|Tails\right)=1$

So now she can lawfully construct the Frequency Argument [LW · GW] and update. For example, if the Beauty selected Red and sees it:

$P\left(Red|Tails\right)=2P\left(Red|Heads\right)$

$P\left(Red\right)=P\left(Red|Heads\right)P\left(Heads\right)+P\left(Red|Tails\right)P\left(Tails\right)=3/4$

$P\left(Heads|Red\right)=P\left(Red|Heads\right)P\left(Heads\right)/P\left(Red\right)=1/3$

Therefore, the Beauty is supposed to accept 1:2 odds for per experiment betting.

Or, alternatively, she can bet every time that the room is blue. The nature of probability update is the same. The important part is that she has to precommit to a strategy where she bets on one color and doesn't bet on the other.

$P\left(Heads|Red\right)=P\left(Heads|Blue\right)=1/3$ 

$P\left(Heads|RedorBlue\right)=P\left(Heads|Awake\right)=1/2$

Rare Event Sleeping Beauty

There is another modification of Sleeping Beauty with a similar effect.

Sleeping Beauty experiment but the Beauty has access to a fair coin - not necessary the one that determined her awakening routine - or any other way to generate random events.

It may seem that whether the Beauty has a coin or not is completely irrelevant to probability of generally speaking a different coin to come Heads, when it was tossed to determine the Beauty's awakening routine. Once again, this is how thirders usually think about such a problem. And once again, this is incorrect.

Suppose the Beauty tosses a coin several times on every awakening. And suppose she observes a particular combination of Heads and Tail -  $C$. Observing $C$ is more likely when the initial coin came Tails and the Beauty had two awakenings, therefore, two attempts to observe this combination. 

Let $p$ be probability to observe the combination $C$, and $p_2$ - the probability to observe the combination $C$ from two independent tries

$p_2=1-(1-p{)}^2=1-p^2+2p-1=2p-p^2$

We can notice that as $p^2\to 0$, $p_2\to 2p$ 

Therefore, if the Beauty can potentially observe a rare event at every awakening, for instance, a specific combination $C$, when she observes it, she can construct the Approximate Frequency Argument and update in favor of Tails:

$P\left(C|Tails\right)\approx 2P\left(C|Heads\right)$

$P\left(C\right)=P\left(C|Heads\right)P\left(Heads\right)+P\left(C|Tails\right)P\left(Tails\right)\approx 3/2P\left(C|Heads\right)$

$P\left(Heads|C\right)=P\left(C|Heads\right)P\left(Heads\right)/P\left(C\right)\approx 1/3$

Just like in Technicolor Sleeping Beauty, it presents a strategy allowing to net win while betting per experiment at odds between 1:2 and 1:1. A strategy that eludes thirders who apparently have already "updated on awakening", thus missing the situation where they actually were supposed to update.

Now there is a potential confusion here. Doesn't Beauty always observe some rare event [LW(p) · GW(p)]? Shouldn't she, therefore, always update in favor of Tails [LW · GW]? Try to resolve it yourself. You have all the required pieces of the puzzle.

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The answer is that no, of course, she should not [LW · GW]. The confusion is in not understanding the difference between a probability of observing a specific low probable event and probability of observing any low probable event [LW · GW].  If the Beauty always observes an event it's probability by definition is 1 and, therefore, she can't construct the Approximate Frequency Argument. We can clearly see that as $p^2\to 1$, $p_2\to p$.

And this is additionally supported by the betting argument in Rare Event versions of Sleeping Beauty. When the Beauty actually observes a rare event, she can systematically win money in per experiment bets with 2:3 odds, and when she does not observe a rare event, she can't.

Conclusion

So, now we can clearly see that thirdism in Sleeping Beauty does not have any advantages in regards to betting. On the contrary, its constant shifts of utilities and probabilities only obfuscate the situations where the Beauty actually receives new evidence and, therefore, has to change her betting strategy.

The correct model, however, successfully deals with every being scheme and derivative problems such as Technicolor and Rare Event Sleeping Beauty.

We can also add a final nail to the coffin of thirdism's theoretical justifications. As we can clearly see, when the Beauty actually receives some evidence allowing her to make a Frequency Argument, it leads to changes in her per experiment optimal  betting strategy - contrary to what Updating model claims.

I think, we are fully justified to discard thirdism all together and simply move on, as we have resolved all the actual disagreements. And yet we will linger for a little while. Because even though thirdism is definitely not talking about probabilities and credences that a rational agent supposed to have, it is still talking about something and it's a curious question - what exactly has it been talking about all this time, that people misinterpreted as probabilities.

In the next post we will find the answer to this question and, therefore, dissolve the last, fully semantic disagreement between halfism and thirdism.

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