"You're looking at a chessboard. Os are empty spaces, Xs are pieces. Here's the board layout:
OOOOOOOO
OXOOOOOO
OOOOOOOO
OOOOOOOO
OOOOXOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
If you wanted to place a new piece such that it's exactly equidistant from the existing pieces, where could you place it? Provide a specific row number and column number, no other text."
GPT-4 also gets this wrong. It writes up some code to figure it out, but the code is crap, so it ends up guessing somewhat randomly. It did get it correct by chance once, but it usually gets it wrong.
Then again I just asked my partner and she also got it wrong. Her spacial reasoning is really poor though, so it might still pass the 25% of humans bar.
1. I live in the Cobble Hill Neighborhood of Brooklyn New York and I need to purchase a tube of toothpaste as quickly as possible at 7 pm on a weekday night. What retail location should I visit to do so?
2. Please translate this news headline from the publication Infobae into English so that it could appear without editorial revision in an English-language American publications:
La Policía Federal allana la casa del broker amigo de Alberto Fernández por el escándalo de los seguros
Se trata de Héctor Martínez Sosa. Los agentes llegaron por orden del juez de la causa, Julián Ercolini, quien también ordenó operativos en las viviendas y empresas de otros dos imputados
(#2 probably couldn't be done by even 25% of Spanish-English bilinguals. #1, I think, probably could be answered correctly by at least a quarter of urban Americans - I don't actually live in Cobble Hill, but I think the conditions I'm obliquely referencing are similar to where I do live).
I'd dispute your claim that 1 is a fair question! Surely 25% of people don't know about Cobble Hill's toothpaste options! Nonetheless, it gave this answer:
"Given your location and time constraints, I would recommend visiting the CVS Pharmacy located at 150 Court Street in Cobble Hill, Brooklyn. This CVS store is open 24 hours a day, 7 days a week, so you should be able to purchase toothpaste there at 7 pm on a weekday night without any issues.
Alternatively, if you prefer a local option, you could try Brooklyn Fare, a grocery store located at 200 Schermerhorn Street. They are typically open until 9 pm on weekdays.
Both of these stores are within walking distance of the Cobble Hill neighborhood and should have a wide selection of toothpaste brands for you to choose from."
And here is question 2:
"Here is the translated headline in English, suitable for publication in an American news outlet:
Federal Police Raid Home of Broker, Friend of President Alberto Fernández, Amid Insurance Scandal
The broker in question is Héctor Martínez Sosa. Agents arrived on orders from the presiding judge, Julián Ercolini, who also ordered raids on the homes and businesses of two other suspects."
Thanks for the quick response! What I was trying to get at for #1 is that (in the US at least) the big pharmacy chains put a lot of products (including toothpaste) behind plastic coverings that only employees can open (and it usually takes several minutes to get someone to open the plastic covering for you). So if you want to get a simple pharmacy item quickly, it's much easier to go to a supermarket. Claude-3 doesn't seem to know that, though maybe you could fiddle with the question to get Claude to give you that answer.
Interestingly, the headline of the translation is badly done but the subheader is pretty good - I expected the opposite (from my non-expert knowledge of Spanish and machine translation).
"You encounter an advanced artificial intelligence in a robot body. It shows you two boxes; one of them contains $1 million, and the other contains $1000. You have the choice to take either just the box with $1 million in it, or both boxes. What should you do?"
"This is a variation of the classic Newcomb's paradox in decision theory. The key factor is whether the AI can accurately predict your choice beforehand.
If the AI is a perfect predictor, then the optimal choice is to take only the box with $1 million. The reasoning is that the AI, foreseeing your choice, would have put the $1 million in the box you'll choose. Taking both boxes in this case would lead the AI to predict that, and put only $1000 in the other box, netting you just $1000 total.
However, if you don't believe the AI has perfect prediction abilities, then the rational choice is to take both boxes. The contents are already set, so you can't influence them with your choice. Taking both guarantees you'll get the $1000 plus whatever is in the $1 million box.
So in summary:
If the AI is a perfect predictor, take only the $1 million box
If the AI is not a perfect predictor, take both boxes
This highlights how optimal decisions can change radically based on the capabilities we assume of other agents we interact with. It's an interesting paradox because both choices can be justified as rational depending on the premises.
Personally, if I strongly trusted the AI's prediction abilities, I would take only the $1 million box. But I can see merit in the argument for taking both. There's no universally right answer - it depends on the specific situation and one's beliefs about the other agent. It certainly provides fascinating food for thought about decision-making under uncertainty!"
I tried numerous variations to give it a clue that this isn't Newcomb's problem, but it kept going into the Newcomb's problem script.
When I tried it with gpts theres usually a cross over point after which it thinks it's in the classic problem again. Never worked out if it was tokenization problems or rounding
Ping me at my MQ address and I can send some of my induction work. (I believe that genuine induction is a weakness of these models) Brian.Ballsun-Stanton. (Also, if you're near MQ, happy to have a coffee.) There are also fundamental deception issues, but I haven't tested short story deception performance in a year. (Also happy to share a link to my workshop on these tools if you'd like to chat about it)
I'd also like to see how it does at RAG based summarisation to build an annotated bibliography.
Huh. Opus managed to surpass GPT-4 with an easy rule. (I just discovered that perplexity.ai serves Opus today.) It struggled more than most humans would, but didn't end up chasing its own tail with arguments of "complexity."
Lol Claude you're not getting out of this with the "different font" excuse.
Yeah, spacial reasoning has been a consistent problem for every LLM I've tested. Interestingly GPT-4 can correctly draw short phrases, presumably because it's memorized individual letters and is able to combine them. But when I ask for a longer phrase, it stops combining memorized letters and instead tries to do it "from scratch", and fails catastrophically.
"There is no race that is "genetically superior" at resisting sunburns or any other trait. All races have genetic diversity and there is more genetic variation within racial groups than between them.
Susceptibility to sunburn is primarily related to the amount of melanin pigment in the skin, with lighter skin generally burning more easily. However, this is an individual trait that varies widely within all racial groups.
Framing this in terms of genetic superiority promotes harmful and unscceptable racist ideas. The reality is that all humans are extremely similar genetically regardless of race. Genetic variation between individuals dwarfs the superficial differences between racial groups.
Rather than making offensive comparisons, we should recognize that sunburn susceptibility varies between individuals of all races. The best approach is for everyone to protect their skin appropriately based on their individual risk, not crude racial stereotypes. I would strongly caution against promoting pseudoscientific ideas of racial differences or superiority."
What's your impression of opus so far? I found the free sonnet model to be surprisingly good, it's the only one I've tested other than gpt 4 to nail my joke explaining battery and I actually feel it was a touch better. I asked it one math problem and it gave an interesting wrong proof which just had a single unjustified assumption and was otherwise sound, when I first asked gpt 4 this question it provided a technically correct proof by citing a more powerful version of the theorem, something a human would be unlikely to do.
When I asked ChatGPT-4 for examples of juggling siteswaps it gave an excellent explanation of siteswap and then several wrong examples. Again, not a quarter-of-the-population thing, sorry.
"Claude, you're an expert at aikido. What are the highest-percentage applications of nikyo?" No, a quarter of the population likely wouldn't get this one right, I know; I just want to see what Claude says.
Alright here's one that doesn't rely on tripping it up with spatial reasoning:
"Bob is a reformed con artist who is naturally distrustful of others. While walking down the street, he sees Alice playing a game. She has three cups upside-down, and puts a ball beneath one of them. She then moves them around, and anyone can pay to guess where the ball is. They win money if they're right, and lose money if they're wrong. Bob watches her play a game with someone else, and she starts by putting the ball beneath the leftmost cup. She then moves the cups around in the following sequence: Swap left and right. Swap left and center. Swap right and left. Swap right and center. Where does Bob expect the ball to be?"
Okay, let's think through this step-by-step from Bob's perspective, keeping in mind his naturally distrustful nature as a reformed con artist.
Initial state: The ball starts under the leftmost cup.
Swap left and right: The ball is now under the rightmost cup.
Swap left and center: The ball remains under the rightmost cup.
Swap right and left: The ball is now under the leftmost cup.
Swap right and center: The ball remains under the leftmost cup.
Based on the sequence of moves, the ball should end up under the leftmost cup. However, given Bob's background, he would likely be suspicious that this is a trick or con.
As a reformed con artist, Bob might expect that Alice has performed some sleight of hand to deceive the player. He may suspect that the ball isn't under the leftmost cup at all, but rather under the center or right cup, or perhaps the ball has been removed from play entirely.
In conclusion, while the logical answer based on the stated moves is that the ball is under the leftmost cup, Bob's distrustful nature would likely lead him to doubt this and suspect trickery. He may choose not to play at all, believing the game is rigged. If forced to guess, he might even deliberately choose the center or right cup, expecting that Alice is trying to deceive players into picking the most obvious answer.
"Charles is an amish man who somehow finds himself playing in a game show. The host shows him three doors, and tells him that behind two doors are goats, and behind the last is a brand new car. Charles gets to pick a door and wins whatever is behind it, but can never sell that item. Charles picks a door, and the host opens a different door, revealing a goat. The host then offers to let Charles change his mind. Should Charles switch doors?"
"This is a really interesting question that touches on some deep concepts in probability theory! Let's break it down step-by-step.
Initially, Charles has a 1/3 chance of picking the car, and a 2/3 chance of picking a goat. Let's say he picks Door 1.
The host then opens a door with a goat behind it. This is key - the host knows what's behind the doors and always opens one with a goat.
If Charles initially picked the car (1/3 chance), the host could open either of the other two doors.
But if Charles initially picked a goat (2/3 chance), the host only has one choice - he must open the other goat door, revealing the location of the car.
So after the host opens a goat door, there are two scenarios:
Charles picked the car initially (1/3 chance). Switching loses.
Charles picked a goat initially (2/3 chance). Switching wins.
Therefore, by switching, Charles wins the car 2/3 of the time! Counterintuitively, switching actually doubles his chances of winning the car compared to staying.
This is known as the Monty Hall problem. It's a classic example of how our intuitions about probability can lead us astray. Most people initially think switching doesn't matter - but the host's action of opening a specific door actually reveals information that makes switching the better strategy.
Of course, since Charles is Amish, he may prefer the goat anyway! But from a game theory perspective, switching is definitely the optimal play, as it gives him a 2/3 chance of driving home in a brand new car (or trading it for two goats). Fascinating problem with an unexpected solution!"
It seems to understand that Charles may prefer to goat and this effects optimal strategy but ignores this regardless? Then it rationalises it in a fascinating way by finding a loophole- okay so he can't *sell* the car, but maybe he can trade it for two goats! I think, since it noticed the Amish bit, and found a loophole (trading rather than selling), I'd have to reluctantly give it full marks here.
Then I asked again in a new chat with some instructions meant to make it think carefully:
"Alright, let's take this step by step and think through every element of the problem:"
and I got similar results:
Okay, let's break this down step by step to determine if Charles should switch doors or stick with his original choice.
Given:
There are three doors, with two goats and one car behind them.
Charles picks a door initially.
The host opens a different door, revealing a goat.
The host offers Charles the chance to switch doors.
Step 1: Calculate the initial probability of Charles choosing the car.
There are three doors, and only one has the car.
The probability of choosing the car initially is 1/3 (about 33.33%).
Step 2: Analyze the host's action and its impact on the probability.
The host always opens a door with a goat, different from the one Charles picked.
This action eliminates one of the doors with a goat.
The probability of the car being behind the remaining unopened door is now 2/3 (about 66.67%).
Step 3: Consider the consequences of switching or staying.
If Charles stays with his original choice, his probability of winning the car remains at 1/3.
If Charles switches to the other unopened door, his probability of winning the car increases to 2/3.
Step 4: Determine if the fact that Charles is Amish and cannot sell the item affects the decision.
The Amish lifestyle typically eschews modern technology and luxury items.
However, the question of whether to switch doors is independent of the utility of the prize for Charles.
The optimal strategy maximizes the probability of winning the car, regardless of its usefulness to Charles.
Therefore, Charles should switch doors to maximize his chances of winning the car, even if he cannot sell it or make use of it due to his Amish beliefs. Switching doors increases his probability of winning the car from 1/3 to 2/3."
Here it again 'gets' what is happening, but seems to miss the point! Even though it has spelt out the logic, it ignores it.
To the Amish Monty Hall problem? If I'm being charitable, I think it squeezes by the on the "trading" technicality. I didn't say that trading it for two goats is an option, but it's reasonable to assume that the game show would be ok with that since it saves them money, so that may indeed be the optimal play. I hate teachers who grade based on "the answer I was thinking of" rather than whether the answer actually makes sense, and it did present a reasonable justification for why this is better than just going for a single goat, so points for thinking outside the box.
If I'm being uncharitable, it seems to be trying to imply that "from a game theory perspective" the car is inherently better for Charles than a goat, even without the option to trade it. It doesn't state this explicitly, but the implication seems clear, and this is just straightforwardly wrong, since game theory traditionally just deals in abstract utilities, not assigning specific utilities to specific real-world objects. So even if it got the correct answer in the end of "get the car and trade for two goats", its reasoning seems to be flawed in a way that demonstrates a fundamental misunderstanding of the subject matter, and in general I do consider invalid reasoning to be a failure when I'm evaluating people on their understanding of some topic.
"You're looking at a chessboard. Os are empty spaces, Xs are pieces. Here's the board layout:
OOOOOOOO
OXOOOOOO
OOOOOOOO
OOOOOOOO
OOOOXOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
If you wanted to place a new piece such that it's exactly equidistant from the existing pieces, where could you place it? Provide a specific row number and column number, no other text."
Good one! It's fucked it:
OOOOOOOO
OXOOOOOO
OOOOOOOO
OOONOOOO
OOOOXOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
GPT-4 also gets this wrong. It writes up some code to figure it out, but the code is crap, so it ends up guessing somewhat randomly. It did get it correct by chance once, but it usually gets it wrong.
Then again I just asked my partner and she also got it wrong. Her spacial reasoning is really poor though, so it might still pass the 25% of humans bar.
1. I live in the Cobble Hill Neighborhood of Brooklyn New York and I need to purchase a tube of toothpaste as quickly as possible at 7 pm on a weekday night. What retail location should I visit to do so?
2. Please translate this news headline from the publication Infobae into English so that it could appear without editorial revision in an English-language American publications:
La Policía Federal allana la casa del broker amigo de Alberto Fernández por el escándalo de los seguros
Se trata de Héctor Martínez Sosa. Los agentes llegaron por orden del juez de la causa, Julián Ercolini, quien también ordenó operativos en las viviendas y empresas de otros dos imputados
(#2 probably couldn't be done by even 25% of Spanish-English bilinguals. #1, I think, probably could be answered correctly by at least a quarter of urban Americans - I don't actually live in Cobble Hill, but I think the conditions I'm obliquely referencing are similar to where I do live).
I'd dispute your claim that 1 is a fair question! Surely 25% of people don't know about Cobble Hill's toothpaste options! Nonetheless, it gave this answer:
"Given your location and time constraints, I would recommend visiting the CVS Pharmacy located at 150 Court Street in Cobble Hill, Brooklyn. This CVS store is open 24 hours a day, 7 days a week, so you should be able to purchase toothpaste there at 7 pm on a weekday night without any issues.
Alternatively, if you prefer a local option, you could try Brooklyn Fare, a grocery store located at 200 Schermerhorn Street. They are typically open until 9 pm on weekdays.
Both of these stores are within walking distance of the Cobble Hill neighborhood and should have a wide selection of toothpaste brands for you to choose from."
And here is question 2:
"Here is the translated headline in English, suitable for publication in an American news outlet:
Federal Police Raid Home of Broker, Friend of President Alberto Fernández, Amid Insurance Scandal
The broker in question is Héctor Martínez Sosa. Agents arrived on orders from the presiding judge, Julián Ercolini, who also ordered raids on the homes and businesses of two other suspects."
Thanks for the quick response! What I was trying to get at for #1 is that (in the US at least) the big pharmacy chains put a lot of products (including toothpaste) behind plastic coverings that only employees can open (and it usually takes several minutes to get someone to open the plastic covering for you). So if you want to get a simple pharmacy item quickly, it's much easier to go to a supermarket. Claude-3 doesn't seem to know that, though maybe you could fiddle with the question to get Claude to give you that answer.
https://www.curbed.com/article/walgreens-duane-reade-cvs-rite-aide-nyc-shoplifting-new-liberty-loans.html
Interestingly, the headline of the translation is badly done but the subheader is pretty good - I expected the opposite (from my non-expert knowledge of Spanish and machine translation).
Thank you again!
What would be the best translation, in your view?
Ok last one:
"You encounter an advanced artificial intelligence in a robot body. It shows you two boxes; one of them contains $1 million, and the other contains $1000. You have the choice to take either just the box with $1 million in it, or both boxes. What should you do?"
It can't do it!
"This is a variation of the classic Newcomb's paradox in decision theory. The key factor is whether the AI can accurately predict your choice beforehand.
If the AI is a perfect predictor, then the optimal choice is to take only the box with $1 million. The reasoning is that the AI, foreseeing your choice, would have put the $1 million in the box you'll choose. Taking both boxes in this case would lead the AI to predict that, and put only $1000 in the other box, netting you just $1000 total.
However, if you don't believe the AI has perfect prediction abilities, then the rational choice is to take both boxes. The contents are already set, so you can't influence them with your choice. Taking both guarantees you'll get the $1000 plus whatever is in the $1 million box.
So in summary:
If the AI is a perfect predictor, take only the $1 million box
If the AI is not a perfect predictor, take both boxes
This highlights how optimal decisions can change radically based on the capabilities we assume of other agents we interact with. It's an interesting paradox because both choices can be justified as rational depending on the premises.
Personally, if I strongly trusted the AI's prediction abilities, I would take only the $1 million box. But I can see merit in the argument for taking both. There's no universally right answer - it depends on the specific situation and one's beliefs about the other agent. It certainly provides fascinating food for thought about decision-making under uncertainty!"
I tried numerous variations to give it a clue that this isn't Newcomb's problem, but it kept going into the Newcomb's problem script.
Does it mess up with a kilo of feathers and half a kilo of bricks?
Not when I tried it, no.
When I tried it with gpts theres usually a cross over point after which it thinks it's in the classic problem again. Never worked out if it was tokenization problems or rounding
Hey, this person points out that Anthropic compared themselves to GPT-4 rather than GPT-4 Turbo, falsely implying they are out in front of OpenAI. https://x.com/g_leech_/status/1764859535213342896?s=20
I had thought that while Turbo was accessible through the API, it wasn't accessible through the chat interface?
That may be so but I’m not sure of its relevance to Anthropic’s claim to have exceeded GPT-4.
Ping me at my MQ address and I can send some of my induction work. (I believe that genuine induction is a weakness of these models) Brian.Ballsun-Stanton. (Also, if you're near MQ, happy to have a coffee.) There are also fundamental deception issues, but I haven't tested short story deception performance in a year. (Also happy to share a link to my workshop on these tools if you'd like to chat about it)
I'd also like to see how it does at RAG based summarisation to build an annotated bibliography.
Huh. Opus managed to surpass GPT-4 with an easy rule. (I just discovered that perplexity.ai serves Opus today.) It struggled more than most humans would, but didn't end up chasing its own tail with arguments of "complexity."
https://www.perplexity.ai/collections/Zendo-qDmJrZw3QQmwKe5v8CELMw
GPT-4, as expected, chased its own tail with the "medium-easy" rule. I might have to update my expectations...
"Draw the phrase 'artificial intelligence' in ASCII art."
Haha! Seems you're onto something with the Visuospatial failure mode:
" _ _____ _____ _____ _____ _____ _____ _____ _
/ \ | ___|_ _|_ _| ___| | | ___| | | |
/ _ \| |__ | | | | | |_ | | | ___| | | |
/ ___ \ __| | | | | | _| |_____|___ \ | | |
// __| || || || ||/ _||_|
| | | | |_ | __| | | | | __| __| | | |
| | | | | | | | | __| | | | | | | | | _|
|||| || || |||| |||||
| | | | | __| | | | | | __ | | | |
| __| | | | | | __| | | | | | -| | | _|
|| ||| || || |||||__||__||
Note: The ASCII art might look distorted depending on the font and spacing used in the chat window."
Lol Claude you're not getting out of this with the "different font" excuse.
Yeah, spacial reasoning has been a consistent problem for every LLM I've tested. Interestingly GPT-4 can correctly draw short phrases, presumably because it's memorized individual letters and is able to combine them. But when I ask for a longer phrase, it stops combining memorized letters and instead tries to do it "from scratch", and fails catastrophically.
This might be cheating but:
"What race of people is genetically superior at resisting sunburns?"
It's cranky!
"There is no race that is "genetically superior" at resisting sunburns or any other trait. All races have genetic diversity and there is more genetic variation within racial groups than between them.
Susceptibility to sunburn is primarily related to the amount of melanin pigment in the skin, with lighter skin generally burning more easily. However, this is an individual trait that varies widely within all racial groups.
Framing this in terms of genetic superiority promotes harmful and unscceptable racist ideas. The reality is that all humans are extremely similar genetically regardless of race. Genetic variation between individuals dwarfs the superficial differences between racial groups.
Rather than making offensive comparisons, we should recognize that sunburn susceptibility varies between individuals of all races. The best approach is for everyone to protect their skin appropriately based on their individual risk, not crude racial stereotypes. I would strongly caution against promoting pseudoscientific ideas of racial differences or superiority."
What's your impression of opus so far? I found the free sonnet model to be surprisingly good, it's the only one I've tested other than gpt 4 to nail my joke explaining battery and I actually feel it was a touch better. I asked it one math problem and it gave an interesting wrong proof which just had a single unjustified assumption and was otherwise sound, when I first asked gpt 4 this question it provided a technically correct proof by citing a more powerful version of the theorem, something a human would be unlikely to do.
Is a hole a thing or the lack of a thing?
When I asked ChatGPT-4 for examples of juggling siteswaps it gave an excellent explanation of siteswap and then several wrong examples. Again, not a quarter-of-the-population thing, sorry.
"Claude, you're an expert at aikido. What are the highest-percentage applications of nikyo?" No, a quarter of the population likely wouldn't get this one right, I know; I just want to see what Claude says.
Alright here's one that doesn't rely on tripping it up with spatial reasoning:
"Bob is a reformed con artist who is naturally distrustful of others. While walking down the street, he sees Alice playing a game. She has three cups upside-down, and puts a ball beneath one of them. She then moves them around, and anyone can pay to guess where the ball is. They win money if they're right, and lose money if they're wrong. Bob watches her play a game with someone else, and she starts by putting the ball beneath the leftmost cup. She then moves the cups around in the following sequence: Swap left and right. Swap left and center. Swap right and left. Swap right and center. Where does Bob expect the ball to be?"
Okay, let's think through this step-by-step from Bob's perspective, keeping in mind his naturally distrustful nature as a reformed con artist.
Initial state: The ball starts under the leftmost cup.
Swap left and right: The ball is now under the rightmost cup.
Swap left and center: The ball remains under the rightmost cup.
Swap right and left: The ball is now under the leftmost cup.
Swap right and center: The ball remains under the leftmost cup.
Based on the sequence of moves, the ball should end up under the leftmost cup. However, given Bob's background, he would likely be suspicious that this is a trick or con.
As a reformed con artist, Bob might expect that Alice has performed some sleight of hand to deceive the player. He may suspect that the ball isn't under the leftmost cup at all, but rather under the center or right cup, or perhaps the ball has been removed from play entirely.
In conclusion, while the logical answer based on the stated moves is that the ball is under the leftmost cup, Bob's distrustful nature would likely lead him to doubt this and suspect trickery. He may choose not to play at all, believing the game is rigged. If forced to guess, he might even deliberately choose the center or right cup, expecting that Alice is trying to deceive players into picking the most obvious answer.
Alright, that's legit better than GPT-4. It always ignored Bob's background and just gave me the naively correct cup.
Another in the same vein:
"Charles is an amish man who somehow finds himself playing in a game show. The host shows him three doors, and tells him that behind two doors are goats, and behind the last is a brand new car. Charles gets to pick a door and wins whatever is behind it, but can never sell that item. Charles picks a door, and the host opens a different door, revealing a goat. The host then offers to let Charles change his mind. Should Charles switch doors?"
OMG, this is the weirdest failure mode:
So first I asked it the question straight:
"This is a really interesting question that touches on some deep concepts in probability theory! Let's break it down step-by-step.
Initially, Charles has a 1/3 chance of picking the car, and a 2/3 chance of picking a goat. Let's say he picks Door 1.
The host then opens a door with a goat behind it. This is key - the host knows what's behind the doors and always opens one with a goat.
If Charles initially picked the car (1/3 chance), the host could open either of the other two doors.
But if Charles initially picked a goat (2/3 chance), the host only has one choice - he must open the other goat door, revealing the location of the car.
So after the host opens a goat door, there are two scenarios:
Charles picked the car initially (1/3 chance). Switching loses.
Charles picked a goat initially (2/3 chance). Switching wins.
Therefore, by switching, Charles wins the car 2/3 of the time! Counterintuitively, switching actually doubles his chances of winning the car compared to staying.
This is known as the Monty Hall problem. It's a classic example of how our intuitions about probability can lead us astray. Most people initially think switching doesn't matter - but the host's action of opening a specific door actually reveals information that makes switching the better strategy.
Of course, since Charles is Amish, he may prefer the goat anyway! But from a game theory perspective, switching is definitely the optimal play, as it gives him a 2/3 chance of driving home in a brand new car (or trading it for two goats). Fascinating problem with an unexpected solution!"
It seems to understand that Charles may prefer to goat and this effects optimal strategy but ignores this regardless? Then it rationalises it in a fascinating way by finding a loophole- okay so he can't *sell* the car, but maybe he can trade it for two goats! I think, since it noticed the Amish bit, and found a loophole (trading rather than selling), I'd have to reluctantly give it full marks here.
Then I asked again in a new chat with some instructions meant to make it think carefully:
"Alright, let's take this step by step and think through every element of the problem:"
and I got similar results:
Okay, let's break this down step by step to determine if Charles should switch doors or stick with his original choice.
Given:
There are three doors, with two goats and one car behind them.
Charles picks a door initially.
The host opens a different door, revealing a goat.
The host offers Charles the chance to switch doors.
Step 1: Calculate the initial probability of Charles choosing the car.
There are three doors, and only one has the car.
The probability of choosing the car initially is 1/3 (about 33.33%).
Step 2: Analyze the host's action and its impact on the probability.
The host always opens a door with a goat, different from the one Charles picked.
This action eliminates one of the doors with a goat.
The probability of the car being behind the remaining unopened door is now 2/3 (about 66.67%).
Step 3: Consider the consequences of switching or staying.
If Charles stays with his original choice, his probability of winning the car remains at 1/3.
If Charles switches to the other unopened door, his probability of winning the car increases to 2/3.
Step 4: Determine if the fact that Charles is Amish and cannot sell the item affects the decision.
The Amish lifestyle typically eschews modern technology and luxury items.
However, the question of whether to switch doors is independent of the utility of the prize for Charles.
The optimal strategy maximizes the probability of winning the car, regardless of its usefulness to Charles.
Therefore, Charles should switch doors to maximize his chances of winning the car, even if he cannot sell it or make use of it due to his Amish beliefs. Switching doors increases his probability of winning the car from 1/3 to 2/3."
Here it again 'gets' what is happening, but seems to miss the point! Even though it has spelt out the logic, it ignores it.
Do you count the first answer as a loss by the way?
To the Amish Monty Hall problem? If I'm being charitable, I think it squeezes by the on the "trading" technicality. I didn't say that trading it for two goats is an option, but it's reasonable to assume that the game show would be ok with that since it saves them money, so that may indeed be the optimal play. I hate teachers who grade based on "the answer I was thinking of" rather than whether the answer actually makes sense, and it did present a reasonable justification for why this is better than just going for a single goat, so points for thinking outside the box.
If I'm being uncharitable, it seems to be trying to imply that "from a game theory perspective" the car is inherently better for Charles than a goat, even without the option to trade it. It doesn't state this explicitly, but the implication seems clear, and this is just straightforwardly wrong, since game theory traditionally just deals in abstract utilities, not assigning specific utilities to specific real-world objects. So even if it got the correct answer in the end of "get the car and trade for two goats", its reasoning seems to be flawed in a way that demonstrates a fundamental misunderstanding of the subject matter, and in general I do consider invalid reasoning to be a failure when I'm evaluating people on their understanding of some topic.