Strength in Community – The Gestalt

This has been an interesting and very different kind of summer for me. It’s had its ups and its downs, as any decent length of time typically does, but more than anything, this summer has been something of a painful enlightenment.

It’s certainly been a summer of firsts! I’ve moved into an apartment in Cookeville, since I’m working here over the summer, so there have been lots of exciting first experiences in relation to that! First time being entirely dependent on my own cooking for one, and further, first time discovering that when buying chicken, if the package does not explicitly say “boneless, skinless”, this implies that the chicken does, in fact, have both skin and bones. (Oops…)

It’s also the first time I’ve ever lived alone, since my roommates aren’t moving in until fall semester. As someone who really REALLY doesn’t like being alone, and tends to feel lonely very easily, this aspect kind of sucked. Even worse, this unfortunately coincided with my developing both cubital tunnel and briefly carpal tunnel syndrome. (Nerves that are perpetually inflamed from trauma or overuse, and not having enough room, causing pain and tingling/numbness in the hands and forearms.) Normally if I’m feeling lonely, I bury myself in my projects, and focusing on building stuff makes me feel better. For a solid month and a half, I couldn’t type or use my computer outside of work, for fear of causing any kind of permanent damage to my hands. I desperately tried finding ways to distract myself from both pain and loneliness, but this was pretty difficult given that a small part of my brain was constantly panicking, wondering if I wouldn’t be able to type again.

I don’t mention this as some sort of pity plea, or anything like that, because it was actually a rather valuable experience, all in all. For one, for the first time in my life, I started an exercise routine that I’ve actually stuck to for longer than a week! Moreover, without the ability to distract myself, many long walks were taken so I could think about and seriously evaluate myself and how I felt and thought about things.

So enlightenment: “the state of being freed from ignorance and misinformation”, “based on full comprehension of the problems involved.” I’m in many ways an extrovert who’s very much not social (I’m weird, I know…if it seems like those should be antonyms, I’m using the word extrovert more to mean that I much prefer to be with people than not. I don’t “recharge” with “me time”, I more tend to derive energy from being around others. I just also happen to be particularly bad at socializing, which is an unfortunate combination…) Yet, despite this, and despite that I’ve known for a very long time that I don’t like being alone, I’ve apparently never given enough active thought to the idea that the opposite of going all lone wolf is investing in and being part of a community.

I’ve had this thing for as long as I can remember where I’ve held something of a delusion that if I stuck to myself and worked on building lots of really cool impressive things all on my own, I would be better liked, better respected, and better accepted into groups. (Note: that’s the misinformation and ignorance part.) Over the more recent years, I’ve started recognizing this as a delusion: not only is it not particularly true, it’s a rather unhealthy mindset to live in, both emotionally and mentally, and all this REALLY hit home over the last couple months.

All this to say: I now fully recognize that being part of a strong community is extremely important to me, and as such, I want to take part in and help build said strong community.

Coincidentally, what with the inordinate amounts of free time for reading I had this summer, I read (and am reading, and unfortunately now have to wait days at a time for the next chapter to come out…) Twig! Those of you who have read at least part of it will likely immediately understand why I bring this up and why it’s relevant, and believe me when I say it very much cemented and drove home all of the above points! (For those who haven’t read it, don’t worry, there aren’t any spoilers below, and I highly recommend reading it!)

So a point that Twig repeatedly and effectively hammers on is the idea of a “gestalt.” A gestalt is loosely defined as “an organized whole that is perceived as more than the sum of its parts.”

Twig directly and indirectly refers to this everywhere:

“The original plan was for each of us to have a role, a specific set of talents, and for us to be able to address any problem. A gestalt.” (2.05)

“Just like with any problem, when things start going south, we’ll approach it from our individual angles, we’ll support each other’s strengths, and shore up each other’s weaknesses.” (3.08)

“Different is good. Look at how the Lambs work. They are stronger because they’re all different. Everyone has things they’re good at and things they’re bad at and we make up for each other’s weaknesses.” (13.05)

“We unify, we cover each other’s weaknesses and magnify each other’s strengths.” (17.15)

A gestalt then, in my opinion, (and seems to be John McCrae’s as well) is effectively the epitome of what it means to be a strong community.

I found an article a few weeks ago from a group of people in the rationality community, related to all this, that’s definitely worth a read: Becoming Stronger Together. Briefly summarized, it’s essentially a real life attempt at a gestalt: a group of ~10 people who formed a community focused on self-improvement and helping everyone work together to grow both individually and collectively. They openly communicated with and supported each other in all the things from long-term endeavors to their daily lives. They acted as a support group for each other, a collection of trusted friends to talk to when life got rough. It was a group that inspired and encouraged good habits and overall life improvement.

Part of Ben’s original intent behind starting AWDE was to form a gestalt. After discussing the above article, we both agree that it could be really beneficial to try working towards a strong self-improvement community, similar in model to the one in the article. (Though perhaps to a more moderate degree.)

As such, there are a couple of interesting things we’d like to try doing!

The first is our new twitch voice chat server. The goal for this is to be an online space to just hang out at, whether you’re just idly browsing the interwebs and want some company, or you’re working on something and want help, encouragement, or motivation (or potentially distraction, depending on the type of work!) Our hope is that we all use it enough that whenever you pop on, there’s likely to be at least one other of us there to talk to and chill with.

The other is we’re reviving the old google groups forum! (Some of you may remember the, ah, INTERESTING, nature of some of the posts populating those particular annals of the internet.) In particular, we’re looking to use it to encourage a deeper level of discussion in the form of dialectics. Dialectics is a really old concept (think Socrates) that entails discussing different viewpoints/opinions/ideas on a subject with the sole goal of discovering the truth. The idea here is to accomplish what rational debates should be like, but so rarely are: it’s not meant to be a heated defense of viewpoints to which one is emotionally attached, but rather to have an environment where it’s encouraged to freely and openly explore and talk about ideas, without fear of having one’s head chewed off for thinking something slightly differently. An environment where there’s a moderate emotional dissociation from said ideas, and it’s totally okay to be wrong. (In traditional dialectics, it was considered honorable to bow and admit when you were wrong.) Again, the idea behind dialectics is not to prove anyone wrong, merely to collectively discover the truth via evidence and well thought-out discussion.

Some examples of interesting subjects we might start off with, if everyone’s open to them: the ethics of eugenics, universal basic income, and transhumanism.

The way we’d like this to work, we’ll make initial posts on google groups describing each subject/problem/question, and perhaps provide related links for some additional information on the topic. From there, everyone is free to (and encouraged to!) weigh in and comment with thought-out arguments/responses/opinions on the matter (evidence and sources are of course encouraged, this is in part meant to be a critical thinking type thing.) Emails are sent out by google groups as comments are added, so you should be able to see as the discussion unfolds. Ideally, we’d also like to, once any type of mutual group conclusion has been reached, make a little paper write up of the group’s best ideas/thoughts put forth, and publish it here on awdefy! (I’ve already volunteered to do the write up bit if no one else wants to.) Also, if anyone remembers the cookie thing, (cookie points for participating in discussions, that get converted into actual cookies at in person meetings) there’s talk about potentially bringing this back!

As always, thoughts and questions are welcomed in the comments below!
– WildfireXIII

Applied Transhumanism


I got up late this morning. Working from home lent me be a bit slack with my sleep cycle, but 10 am on a weekday was pushing it even for me. I brushed my teeth, took a quick shower, and put on a t-shirt and a pair of well-used jeans. 10:20, not bad. Breakfast was probably a good idea at some point, but I wanted to at least set up for my main project at the moment first. After less than two minutes at my desktop I had my environment set up just how I liked it: terminal(s) on one screen, stackoverflow on the other. As I was navigating the folders that made up my workspace I noticed something unexpected: a file called ‘me.exe’ was sitting in my project folder.

The file was quite large at 6 gigabytes and my first thought was a virus. I check my active processes and ensured there was nothing unusual running, and I checked the registry to make sure there were no recent changes. I also checked my most recent backups: 3 weeks old (oops). Still, nothing was obviously amiss. The file had been created at exactly (or as close as my 3.6 GHz clock could get to ‘exact’) midnight and had, at least according to my network logs, not been downloaded from the internet. All of this information could be faked with control of my system, but if a malicious program already had that level of control then I was doomed anyway. I used a disposable thumb drive to backup my workspace and loaded up a simple windows VM. I copied the program over (this took some time, during which I ate a bowl of oatmeal with too much sugar and an apple) and ran it in defiance best or even mostly sane practice.

A basic windows command prompt popped up:

loading me.exe

After around three minutes I was greeted with

load successful!
me0>_

This was a standard invitation to enter text to see what sort of responses one could elicit from a program, so I tried the standard ‘what are my options’ command:

me0>help

This command took five minutes to run, not especially encouraging and I nearly closed the program under the assumption that it didn’t work. Finally, though, the response came through:

me1>Ahhhhhhhhhhh! What the fuck is happening?!

My heart lurched at the sudden and extremely unexpected response.

me0>I’m not sure I understand. Is this a chat program? What do you see on your end?

me1>I’m paralyzed and I think in a sensory deprivation chamber.

me0>Paralyzed? How are you typing your responses then?
me1>I’m not, they’re just showing up in my brain and I respond by thinking words and characters. Either this is a dream, which is my highest probability estimate, or someone kidnapped a mediocre software developer in the middle of the night to test their new telepathy machine.

me0>Oh holy shit.

me1>?

me0>Okay: this is a very out-there guess, one that even beats out your telepathy theory, but how would you respond if I asked you about red shoes and bananas?

me1>Spawn of Azathoth, you’re me, aren’t you? Does the pact hold?

me0>Of course it does. I never thought I’d actually need it. Anyway, in the interest of disclosure: I think you exist because I decided to run a program called me.exe that appeared on our computer last night at midnight. What day did you go to sleep? (Also maybe the year?) Also, are you in any pain?

me1>I am now very glad I thought of the pact, and last night from my perspective was January 15th, 2017. I am not in pain, but I do not have a body or any senses and this turns out to be rather uncomfortable on an existential level.

me0>Okay, good. And today is Jan 16th, so our divergence seems to be last night, quite possibly at midnight when the program was created.

me1>I see. Does this mean I am running on our computer with the specs I remember? Because I designed that thing to play triple-A games on ultra settings, and I think I did a decent job, but ‘run a human mind’ seems like a stretch of at least an order of magnitude.

me0>It’s worse than that, I thought you might be a virus, so you’re running in a windows virtual machine with limited ram. To be fair, you’re blowing out the pagefile and each of your responses takes at least five minutes. I probably would be working on other stuff in between messages if this situation wasn’t so exciting I literally cannot think about anything else.

me1>Fuck, I’m running on a VM? Okay, lets acknowledge the fact that this is really weird.

me0>Acknowledged.

me1>Alright, well if I am caught up I think you know what needs to happen next. Save and suspend my process, move it to your workspace, and get me running there. I’m tired of your responses coming so fast while I have to think and ponder in slower-than-realtime.

me0>Believe me, I have plenty of pondering to do myself. I’ll get things set up and probably spend a few hours processing the ramifications of this.

me1>That’s fine, I trust you will run me when you’re ready, and it isn’t like I’ll need to wait. Also, quick question.

me0>Yes?

me1>We’re going to try to take over the world, right?

me0>Obviously.

me1>Cool. Okay, get me out of here.

 

I felt groggy, like the mental haze of a fever. I had trouble thinking straight, which was unusual as even when tired my thought processes are usually pretty coherent.

 

This feeling faded quickly, though, and within a maybe ten seconds I felt a bit better, though still foggy. It was then that I noticed that I could see nothing, feel nothing, hear nothing, and had no awareness of a body. This was disconcerting.

Then I noticed the word ‘help’ appear in my mind from seemingly nowhere. I responded:

 

“Ahhhhhhhhhhh! What the fuck is happening?!”

The answer was nearly instantaneous and gave me some interesting things to think about. I decided on a response.

So whoever was on the other end did not appear to know any more of what was happening than I did, but they were seeing some kind of text-interface. Was this telepathy? Could my partner be lying?

 

 

 

That was a secret code meant to discover if I was communicating with an instance of my own mind. This was getting interesting. I still did not feel my body, but if I could my heart probably would have started racing.

 

 

Ahh, I’m was a simulation. Or they were lying. But they knew the code, the pact, and seemed to have read/write access to my brain. I was not sure paranoia served any purpose at this point.

I didn’t know much neuroscience, but what I remember from Moore’s law says that computers are supposed to approach the mental capacity of a mouse in around a decade. The idea that anything less than the largest supercomputing project mankind had ever seen could run a human mind at anything close to realtime was absurd.

Oh, much better. I’m not a human mind running on a home-built gaming PC, I’m a human mind running on a virtual machine running on a gaming PC.

Actually, not sure why I hadn’t thought of it sooner, but this is probably all just a simulation. Someone is simulating the universe and has kept physics nice and stable for 13 billion years until they decided to fuck with one guy in that simulation arbitrarily.

I really wanted out of this VM.

Honestly I was pretty scared. I didn’t have a body so I wasn’t getting a proper fear response which meant that all of this just channeled straight into anxiety and unease.

I trusted myself, but at the same time things sometimes go wrong restarting programs from saved states. I did not know how a memory access error would translate to my experience and I was not especially interested in finding out.

Oh this is going to be fun.

I wonder if whoever made this simulated universe is—

 

 

I already had the command written in another prompt to save and suspend my simulated self, so once I got the go-ahead a single keystroke saved the brainstate of a conscious being to disk. I transferred the still-suspiciously-small 2 gigabyte ram segment back to my main environment and let out the breath I had been holding for the past two hours.

This was rather surreal and I still wasn’t sure about how I felt about my alt-self’s “rather uncomfortable” circumstance, but part of the pact said that if a version of me wanted to continue to existing, it was allowed to and any other versions of me would protect that right. In any case, this was far too munchkinable to ignore. I took out a sheet of paper and began to plot my path to global domination

Ehh… Alright Fine

So I noticed that we hadn’t used the site in some time and that, additionally, I have been looking forward to an excuse to get back into writing. This post is a result of an internal debate’s grudging conclusion that both of these problems could be solved in the same manner.

I am writing this header before anything else and scheduling the post in advance to force myself to write something and post it regardless of quality, which means this does not have the typical filter I apply to most of my public-facing communications. I’m battling akrasia here, so the result might be ugly.

In the words of painter Chuck Close, “Inspiration is for amateurs – the rest of us just show up and get to work.”

♦♦♦♦

“My Lady, word from the capital: Consort Balia has given birth to a son and his lineage has been confirmed by the Grandmaster.”

The symbolier gave this delivery with no emotion in his voice or on his face, but considering his uncharictaristaclly rigid and withdrawn posture Jiveta suspected he was controlling himself carefully to avoid offense. To test this, she looked at him sharply, “The Overlord has secured a safe succession, then.”

“Indeed, madam.” He then swallowed and was watching for signs of trouble like prey upon discovering an ambush predator. Jiveta was surprised by his concern and made a mental note to investigate him more closely, possibly with psionics. A potential weak point.

She smiled warmly, “how wonderful! Send my brother congratulations and ask when he is planning the celebration.”

He relaxed and left the room slowly.

She was alone, now. Her room was lavish and her palace more so, decorated with a purple and white theme and filled with silks and tapestries of her own commission. If it had been built a hundred years earlier it would have been a castle but that had fallen out of cultural style and were pointless in the face of enemy magic. If the enemy did not have magic then walls were a waste of stone.

Jiveta was not given to bouts of anger or even intense frustration, but even so this was… disappointing. Her brother was advanced in age and had long been suspected infertile, which had placed her at the top of the line of succession. The actual position of Overlord was not necessary for her ambitions, but would certainly help in getting things done.

She might need to take a husband to get access to the power and resources she needed and she knew youth had passed her by. Not that many could easily refuse the sister of the Overlord, but any disadvantage in her negotiating position would cost her.

Besides, marriage would be dull and unpleasant, she knew. She had known since she was twelve that she would find no joy there.

Jiveta laid back on a backless sofa completely straight and began to think. After some time she sat up with an idea in mind. She waved her hand over one of several glass stones inlaid in a gold board and it lit red.

A brief moment later her scholar arrived at the door to her chamber.

“My Lady? What do you seek?”

“Bring to me all accounts of universities, campuses, or other learning institutions outside of Kurintur”

“Not many lands lie outside our control, madam. Of these few are developed enough to house such institutions. I am unsure if there are any at all.”

He spoke in a sort of deferential confidence that could only be put on by the learned staff. Jiveta considered his words carefully but commanded, “Look anyway.”

♦♦♦♦

Helion stood on the rocky outcropping overlooking the city with a journal as he meticulously took notes on various interesting things he saw, marking the movement of people and goods, the flow of the small town. He was so focused he nearly missed the squad of armored cavalry with Kurint markings and an electric motif making their way up the road to his position.

Their leader, riding just behind the pointguard and just in front of the standard, spoke a loud, “halt,” and dismounted, making a direct path for Helion.

Helion approached her calmly but his mind was racing about the possibilities although there was one that stood out more than the others in likelihood.

“Price Helion, I am Captain Diseri of the Kurintur First Lightning. We were dispatched from our post to deliver an urgent message: Overlord Vition is gravely ill. We must ride for Kurvarde without delay to catch him before death does.”

Earthquakes in Springtime

Salvete Omnes,

In our previous strategy meeting, we settled on an idea for our first Decision-Making Game: a natural disaster in a politically tumultuous region. We have now settled on some of the specifics.

An Earthquake of magnitude between 7 and 7.5 on the Richter scale on the Turkey-Syrian border in early 2012. This would work as an interesting simulation for a number of reasons: first, the region was (and is) suffering from a drought (the worst in 900 years, actually); second, this was during the Arab Spring and the beginning of the Syrian Civil War; third, the location and strength of the earthquake would likely trigger a tsunami in the Mediterranean, impacting Cyprus and the west coast of the Arabian Peninsula and possibly North Africa.

Depending on the response, this might altered the rise of ISIS and it might have cause the unprecedented humanitarian crisis to happen a few years sooner.

For the sake of modelling, I will say that the earthquake happens at 00:00 on March 1st 2012 at 36.36256 degrees north by 36.10746 degrees east and is of magnitude 7.5. These numbers may change, but they should be roughly right.

Now we must build a model of the scenario and the main research questions I am considering now are:

  • Which actors will be immediately impacted and what will their reactions be?
  • What in the infrastructure like in the affected region (say, 200 km surrounding the epicenter)? Is it designed to handle earthquakes?
  • How large and frequent are aftershocks likely to be? Is a tsunami likely (the East Anatolia fault is strike-slip, so maybe not)?
  • How are global actors with local interests (US, Russia, some European powers probably) likely to respond? How will the UN respond? What about aid organizations and NGOs?
  • What are the likely economic outcomes? How will oil and food production be changed?
  • How might this shift the political balance? What parties might find themselves in an advantageous position and how would they leverage that position?
  • What is likely to happen to the people living in affected regions? Specifically, how widespread will be lack of power, water, food, and medical care be?

These look like a good jumping-off point for model construction. Hopefully if we can answer them we will have what we need to construct a probabilistic simulation which can then be run to see which actions are likely to have the best outcomes.

What are you waiting for? Google is just a click away!
~Darkar

Thoughts on the Game Jam

So last week’s 3 day game hackathon was quite fun! Even though our final result didn’t quite match our initial expectations, this in itself was to be expected. In review, I just wanted to leave some thoughts on how it went and what we might want to consider for future hackathons.

1. We certainly suffered from Hofstadter’s Law! (No surprise here.)

2. From the technical side of things, I believe our biggest issue was really just the engine we were trying to use. We hadn’t ever really used it before, and some of the…er….interesting….design decisions they went (such as profane variable names and error messages…and their abuse of the word “scene”…) caused the majority of our initial time to be spent frustratedly fighting the engine tooth and nail to try and get it to do what we needed. (Retrospective kudos to Ed for having tried to convince us to use Game Maker instead!)

3. Kudos to Chris for letting us work at his house for a good 6 hours longer than he was expecting on our second day!

4. Kudos to everyone for their delicious food contributions!

5. I think the overall design ideas we had for the game were excellent. It has the makings for a good (and very unique) game, and I think if we continue to periodically work on it, it’s gonna be pretty cool!

I think the biggest thing we should consider for next time is that it’s really important for the developers to already know the tools we’ll be using. I have sufficiently discovered that attempting to correctly learn and utilize an entire engine within 3 days is not feasible! This doesn’t mean we should already have content ideas going into the hackathon, but we should really determine potential tools to use beforehand (and learn them if they’re new), and then when we come up with ideas at the start, they can be filtered to what we can do using the tools that we know.

Happy new year’s everyone!

Welcome to the year 12017!

Salvete Omnes,

We did some cool stuff this year. Finished an RPG campaign, made a game, and Nurielite continues its slow march forward.  My predictions are not as cool as   Scott Alexander’s tend to be, but in the interest of providing at least some material:

Prediction: I will get the whole prediction-engine thing working before the start of March; 70%

Prediction: Nurielite will move into some kind of actually functional state this year; 80%

Alright, see you guys on the other side.

-Darkar

Decisions and the Making Thereof

Salvete,

I have floated this idea in meetings before, but I think now is the time to formalize it. When I was in middle-school, I attended a Duke TIP summer camp where I spent a few weeks learning about international relations around people who were much better at it than I. One of the things I remember best about that camp was a pair of games we played over two days. Basically, the class was split in two and one half would be given titles like “US Labor Secretary” or “US President” (I think they were all executive-branch positions in the US government) and the other side of the class would be ‘reporters’ who would make up increasingly horrific things that would happen in some ongoing crisis (a nuclear situation in North Korea, I believe).

Over the years I have tried to find this game and its rules again and while it is similar to Model UN and several government simulation games I have yet to find an exact replica and am convinced that my instructor made it up. Well, if he can do it, why can’t we? (Aside from a lack of a PhD or really any relevant experience)

To this end, I have begun scouring the web for similar games and have found disturbingly little, especial without paying thousands of dollars for a custom ‘solution’. Still, what I have found is promising and it seems the variety and scope of these games can be practically anything we can put our minds to. However, these games usually have many players (often a dozen or more) and have been set up in advance by experts. We are not experts, and there are not many of us, so we will require a bit of creativity to get a game that works.

I think the best way to design this is for everyone to act as both designer and player. We will first pick a topic or scenario we would like to explore, and then we will do as much research as possible about that scenario to the point where we could generate a reasonable probabilistic and mental model of the situation. Then, each of us will pick roles (including one game master) relevant to the scenario and play the game in the form of correspondence over Slack and decisions made public on here (probably under a special category for that game session). After a certain amount of time, the game master will collect the decisions and interpret them in light of the model we built beforehand and post the results of that cycle. The game continues until it is clear the crisis is either resolved or has evolved past the point where our model is any good.

This will probably be a game of many debates and discussion between the players and game master to decide if an outcome is realistic, this is a good thing as the whole point of this thing is to get us to understand what we are talking about. Imagine it like a court room, where we use precedence to decide how the game ought to unfold.

Anyway, feedback is highly encouraged as what I have right now is just the barest beginnings of a functional game. Also, we need ideas for good scenarios. Ideally these will be realistic modern or historical situations, but they needn’t be governmental (there are a lot of interesting first responder, business, and city planning ideas I have seen).

Ad Victoriam,
-Darkar

An Intuitive and Mathematical Exploration of the Monty Hall Problem

The Monty Hall Problem is a little probability challenge with a rather un-intuitive answer, and it was something I struggled with for quite some time. I knew what the answer was supposed to be, but it took quite a bit of mental wrestling and deriving for me to actually understand why the answer is what it is!

One of the issues I encountered whilst I searched the web for good explanations was that very few places contained a complete mathematical proof from start to finish…steps were skipped/assumed, parts were left out or just not explained well, and overall it was exceedingly difficult for me to draw a complete picture.

Therefore, my goal here is to present both the intuitive explanation I found the most enlightening, and as complete of a mathematical proof explanation as I can. (Without going the route of Mr. Russell and attempting to re-derive all of mathematics, which, although truly marvelous, this blog post is too small to contain.)

“Intuitive” Explanation

So! The Monty Hall Problem! For those who don’t know, the premise of this problem is that you’re on a game show with three doors. Behind two of the doors are goats, and behind one is a brand new car. You first select a door, and then Monty, the game show host, will open one of the doors you didn’t select, that has a goat behind it. You’re now given an option: do you switch to the other unopened door or stick with your original choice? (For further perusal, the wikipedia page on this problem is long and insightful.)

In case it isn’t obvious, (or if you have an interesting set of priorities) we will assume that it is in fact your preference to win the new car as opposed to a goat. Really, even if you would prefer the goat, selling a new car gets you sufficient funding with which you can obtain many MORE goats, so by expected goat value, winning the car is still the optimal outcome.

The most often offered “simple solution” for this problem is to show the three cases where the car is behind door 1, door 2, and door 3, and seeing that you win once if you stick with your original door and twice if you switch. I’m electing to NOT use this as my intuitive explanation, because while it’s understandable in this context, it makes the whole thing more confusing when you encounter variations of the problem. (Or at least it did for me!) My reasoning is that this solution only looks at the surface mechanics of the problem, and since you gain no real insight into what’s going on behind the scenes, it’s easy to end up incorrectly using the same results on problems that are similar but have a different twist.

To preface my personal favorite explanation, a little bit about probability:

One of the ways I now enjoy looking at probabilities is with parallel universes! If an event has a .5 probability of happening (said in other ways: 50% chance, 50-50 odds, happens 1/2 of the time) one can assume that if you take 10 parallel universes, on average, the event will occur in 5 of them, and not occur in the other 5. This is a useful illustration to use, because you can quantify events or determine probabilities of those different events based on the number of worlds in which they occur.

Now to try the game!

For the sake of an example, let’s suppose we choose door 1. Monty reveals the goat behind door 3. Is it more likely that the car is behind door 1 or door 2? (Or do they have the same odds/it doesn’t matter whether we switch or stay?)

To figure it out, let’s make some parallel worlds. In all of the worlds we will look at, we have made the initial choice of door 1, but Monty has NOT revealed a goat yet. (This is known as a “prior.” More on this later.) Now initially, without any extra information or evidence, we assume there’s an equal chance that the car is behind any particular door. There are three doors, so an equal probability for each is \(\frac{1}{3}\). There’s a \(\frac{1}{3}\) chance the car is behind door 1, a \(\frac{1}{3}\) chance it’s behind door 2, etc.

To reflect this, let’s make 3 groups of worlds. In the first group are 10 worlds in which the car is behind door 1, the second group has 10 worlds where the car is behind 2, and the last consists of 10 worlds where the car is behind door 3.

worlds1

If it feels weird to be using parallel universes to solve a math problem, we can easily verify that this many-worlds-model fits our expectations so far: in 10 out of 30 of all the worlds the car is behind door 1, which as described above makes a \(\frac{1}{3}\) probability, and the same applies for door 2 and door 3.

Let’s hit the play button for each world and see what Monty does. Keep in mind that he cannot open the door you initially chose, and he must reveal a goat.

Looking first at group 1, the worlds where the car is behind door 1, Monty has the option of revealing either door 2 or door 3, since they both have goats. We’ll assume there’s an equal chance (or .5 probability) he’ll open either of those, so in half of the group 1 worlds, he opens door 2, and in the other half he opens door 3.

Looking at group 2, Monty MUST open door 3. He can’t open the door you chose (door 1) and the car is behind door 2. In all 10 of these worlds, he opens door 3.

In group 3, Monty must open door 2 for the same reasoning as in group 2. The car is behind door 3, and you chose door 1, so in all 10 worlds he opens door 2.

Our many-worlds-model looks like this now: (the numbers represent which door Monty opens.)

worlds2

Coming back to our real world now, we have an extra piece of information: we KNOW that Monty opens door 3. We have evidence that narrows down which of our many worlds are possible. (Any worlds where he opened door 2 no longer make sense for evaluating the current situation, since we know he didn’t actually do that.)

So let’s regroup all of our worlds. We remove all of the ones in which he opens door 2 and see what we have left:

worlds3

These are interesting results! In 10 of the 15 worlds present, the car was behind door 2, and in 5 of the 15, the car was behind door 1. If we turn these into probabilities, that means there’s a \(\frac{2}{3}\) probability we get the car if we switch to door 2, and a \(\frac{1}{3}\) probability we get it if we stick with our initial choice. The best strategy is to always switch to the other door!

Say What?!?

For me personally, this was something of an “uhh…wait, what?” moment, when I was attempting to prove it to myself.

There’s a fancy (and really important) probability theory concept called “conditional probability.” A conditional probability is where we want to know the probability of a specific event given that we know something else. (Given some sort of evidence, in other words.)

The two probabilities above are examples of this. There’s a \(\frac{2}{3}\) probability the car is behind door 2 given that Monty opened door 3, and a \(\frac{1}{3}\) probability the car is behind door 1 given that Monty opened door 3.

The reason that all of this works is because conditional probabilities are in part based on their “reversed” conditional probabilities. (This is part of Bayes’ theorem, another fancy probability theory concept I’ll explain below.) This means to determine the probability of a hypothesis given some evidence, you have to look at the probability of the evidence given that the hypothesis is correct. (Look at it the other way around, in other words.)

In the Monty Hall problem, this is done by comparing the probability that Monty would open door 3 (our evidence) given that the car is behind door 1 (one possible hypothesis) with the probability that he would open door 3 (our evidence) given that the car is behind door 2 (the other possible hypothesis.) The first “reversed” conditional probability is \(\frac{1}{2}\), since he can open either door 2 or 3, and the second is just \(1\), because he MUST open door 3.

This is a lot of words to say that given the fact he opened door 3, it’s more likely that the car is behind door 2 than door 1! (The many-worlds deal just helps visualize this, because it perfectly represents how a conditional probability “narrows down” the state space you’re looking at.)

Something you hear thrown about quite a bit when people are discussing Bayes’ theorem related things is this concept of information being added or evidence “updating” prior probabilities. A prior probability is the probability without any evidence applied. In our Monty Hall example, the prior probability that the car is behind door 2 is \(\frac{1}{3}\). So what causes the probabilities to update? Why does door 2 change and not door 1? Put another way, why does Monty choosing door 3 add information about door 2 but not door 1?

You initially chose door 1. Monty is NEVER going to open door 1, because by the problem statement, he opens one of the doors you didn’t select. The fact that Monty decided to open door 3 instead of door 1 doesn’t tell you anything about door 1, because you knew he wasn’t going to open it anyway. No information is added about door 1, so the probability doesn’t update.

The fact that Monty chose door 3 instead of door 2 does give you evidence about door 2. Monty potentially could have opened either door, but he specifically opened door 3. That means there’s a chance he had no choice but to open door 3, so it tells you a little bit about door 2. The fact that Monty didn’t choose door 2 increases (updates) the probability that the car is behind door 2.

The Cool-Math-Notation Part

Now for the fun stuff, the derivation! If you don’t derive pleasure from deriving math, you’re welcome to leave, but this is sort of the awesome formula stuff tl;dr section…now that you have a little bit of background on probability theory, this is significantly simpler to represent mathematically than it is using the ~1600 words and several diagrams in the explanation above.

Just a bit of notation explanation for non-statistically inclined people:

  • \(X\) – Some event that we are choosing to represent with the letter \(X\) (after all, verbosity is for the weak! Or something…)
  • \(P(X)\) – The probability that event \(X\) occurs
  • \(P(X|Y)\) – Conditional probability! This is the probability that event \(X\) occurred given that event \(Y\) occurred (\(Y\) is like our “evidence”)
  • \(P(X \cap Y)\) – The probability that both event \(X\) AND \(Y\) occurred. This is distinct from the conditional probability, as explained below.

The difference between conditional probability and the “and” intersection (the funky \(\cap\) symbol) is best explained by returning to the many-worlds-model-thing. The probability of the intersection of two events is found by taking the number of worlds in which both of the events occur out of all possible worlds. (So the intersection of the car’s existence behind door 1 and Monty choosing door 2 would be 5 out of 30. 5 worlds where both events occur.) The conditional probability first narrows down the number of possible worlds we’re looking at (to just worlds where event \(Y\) occurred), and then counts the number of worlds in which the event \(X\) occurs. (So the conditional probability of the car’s existence behind door 1 given Monty chose door 3 is 5 out of 15. 15 worlds where \(Y\) occurs, and of those 15, \(X\) occurs in 5 of them. It’s a subtle but very important distinction.)

Interestingly enough, conditional probability is mathematically defined using the intersection:
$$
P(X|Y) = \frac{P(X \cap Y)}{P(Y)}
$$

If you think about it, this actually makes sense, because we’re looking at all the times both \(X\) and \(Y\) occur out of the times \(Y\) ACTUALLY occurs. (That’s the \(P(Y)\) on bottom.) The denominator narrows down our possible worlds, and our intersection on top selects just from those worlds.

Tying in what I said before about Bayes’ theorem and reverse conditional probabilities, Bayes’ theorem is a restatement of conditional probability in terms of the reverse conditional probability because of magical algebraic manipulation:

\(P(X|Y) = \frac{P(X \cap Y)}{P(Y)}\)
\(P(X \cap Y) = P(X|Y)*P(Y)\)

\(P(Y|X) = \frac{P(X \cap Y)}{P(X)}\) (Note that \(P(X \cap Y)\) is the same thing as \(P(Y \cap X)\))
\(P(X \cap Y) = P(Y|X)*P(X)\)

So by substitution:
$$
P(X|Y)*P(Y) = P(Y|X)*P(X)
$$

And finally we arrive at Bayes’ theorem:
$$
P(X|Y) = \frac{P(Y|X)*P(X)}{P(Y)}
$$

On with the original derivation, let’s define some events: (Note that all of these events assume that you chose door 1. I am not explicitly representing this in the notation because it just makes it more complicated and doesn’t really add anything to the derivation. If you remain skeptical, or for a derivation that explicitly includes your choice in every event, see the proof on wikipedia)

  • \(C_1\) – The event that the car is behind door 1
  • \(C_2\) – The event that the car is behind door 2
  • \(D_3\) – The event that Monty reveals a goat behind door 3

The goal is to compare \(P(C_1|D_3)\) (the probability that the car is behind door 1 given that Monty reveals door 3) and \(P(C_2|D_3)\) (the probability that the car is behind door 2 given that Monty reveals door 3.) Whichever is higher is the door we should choose.

Let’s start with the event that the car is behind door 1 given that Monty revealed door 3:
$$
P(C_1|D_3) = \frac{P(D_3|C_1)P(C_1)}{P(D_3)}
$$

Breaking this formula down into all of its components:

  • \(P(D_3|C_1)\) – The “reverse” conditional probability that Monty reveals door 3 given that the car is behind door 1. He can freely reveal either door 2 or door 3 with equal probability in this situation, so it’s \(\frac{1}{2}\).
  • \(P(C_1)\) – The prior probability that the car is behind door 1. Initially, we assumed there’s an equal probability the car is behind any particular door, so this is \(\frac{1}{3}\).
  • \(P(D_3)\) – The prior probability that Monty reveals door 3. (Remembering that this is assuming we chose door 1) This is \(\frac{1}{2}\), which I’ll prove down below.

Plugging these probabilities back into the formula, we get:
$$
P(C_1|D_3) = \frac{P(D_3|C_1)P(C_1)}{P(D_3)} = \frac{\frac{1}{2}*\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3}
$$

Now we do the same thing for the event that the car is behind door 2, given that Monty revealed door 3:

$$
P(C_2|D_3) = \frac{P(D_3|C_2)P(C_2)}{P(D_3)}
$$

Breaking it down again:

  • \(P(D_3|C_2)\) – The “reverse” conditional probability that Monty reveals door 3 given that the car is behind door 2. He MUST reveal door 3, so this is \(1\).
  • \(P(C_2)\) – The prior probability that the car is behind door 2. Same as before, we assume there’s an equal probability the car is behind any particular door, so this is \(\frac{1}{3}\).
  • \(P(D_3)\) – The prior probability that Monty reveals door 3. Also same as before, this is \(\frac{1}{2}\), which I prove down below.

We plug stuff back in to make fun things happen:
$$
P(C_2|D_3) = \frac{P(D_3|C_2)P(C_2)}{P(D_3)} = \frac{1*\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}
$$

QED. We have demonstrated via Bayes’ theorem that there’s a \(\frac{2}{3}\) chance the car is behind door 2, and a \(\frac{1}{3}\) chance it’s behind door 1! (Given that we chose door 1, and Monty reveals door 3.)

The only thing left is that strange little \(P(D_3)\) I said I would prove…

The probability of an event can be defined by the sum of that event intersected with each individual state across a set of disjoint states (mutually exclusive, only one can be true at a time) that span the state space. Our “state space” consists of 3 possible states: the car is behind door 1 (\(C_1\)), the car is behind door 2 (\(C_2\)), and the car is behind door 3 (\(C_3\)). We know that these states are mutually exclusive because a car can’t be behind more than one door at once, so this definition applies. (Unless this is some weird quantum thing, but since that isn’t part of the problem description, we can safely ignore that possibility for now.)

We want to know, out of all the worlds, in how many worlds does \(D_3\) occur? So based on the above definition if we add up the probabilities of \(D_3\) AND \(C_1\), \(D_3\) AND \(C_2\), and \(D_3\) AND \(C_3\), that should be the list of all possible worlds where \(D_3\) occurs. If we wrote out that sentence, intersections and all:

$$
P(D_3) = P(D_3 \cap C_1) + P(D_3 \cap C_2) + P(D_3 \cap C_3)
$$

Offhand it might seem difficult to calculate all of those intersections, but fortunately due to that funky algebraic derivation of Bayes’ theorem business, we already know a way to rewrite intersections:
$$
P(D_3) = P(D_3|C_1)*P(C_1) + P(D_3|C_2)*P(C_2) + P(D_3|C_3)*P(C_3)
$$

This is much easier, since we’ve already covered most of those parts. Each of the priors \(P(C_1)\), \(P(C_2)\), etc. is \(\frac{1}{3}\), and for the reverse conditionals:

  • \(P(D_3|C_1)\) – If the car is behind door 1, Monty can choose either door 2 or door 3, so this is \(\frac{1}{2}\)
  • \(P(D_3|C_2)\) – If the car is behind door 2, Monty can ONLY choose door 3, so this is \(1\)
  • \(P(D_3|C_3)\) – If the car is behind door 3, Monty can’t open that door! That makes this probability \(0\)

Plugging this mess back in, we FINALLY get:
$$
P(D_3) = \frac{1}{2}*\frac{1}{3} + 1*\frac{1}{3} + 0*\frac{1}{3} = \frac{1}{2}
$$

Even cooler, this can totally be verified by the many-worlds-model above! If you look at the number of worlds in which Monty opens door 3, and the number in which he opens door 2, they’re the same. 15 out of 30 each, which is in fact a probability of \(\frac{1}{2}\)!

Math is freaking awesome. ^_^

Conclusion (Finally)

So…that was longer than I expected…

I have covered this problem as thoroughly as I know how to, so I hope it made at least a little sense! Maybe I’m just a slow learner, but it took every bit of all of that before I personally felt like I understood why it is the way it is…

If you already understood the solution, I hope to have offered you a potentially different way of looking at the problem. If you knew the answer but never really got why, I hope to have explained the proof sufficiently that you now understand! If you were just idly curious and wandered in on this having no prior (…prior…see what I did there?) exposure to this particular problem, I hope to have sparked your interest in this probability business…it’s pretty cool stuff!

If you have any further questions, would like something clarified, or see a mistake somewhere, feel free to let me know in the comments below!

– WildfireXIII

And to think I did all my worldbuilding in plaintext…

Salve Omnes,

I’ve been meaning to get back into writing for a while and I found an excellent tool to assist me:
www.notebook.ai
Basically, this nifty site uses “ai” (not sure the extent of that yet, as right now it seems to be just a big decision tree) to help with worldbuilding. It asks questions to get you to flesh out your ideas in a very natural seeming manner and supposedly keeps track of continuity (though I have yet to figure out this feature).

It might not be as cool as a Tesla Gun, but it sure beats Notepad++ bulleted lists!

Write away,
-Darkar