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I am glad to find a new face in philosophy that likes to discuss Pascal’s wager and epistemology generally.   That she earned a doctorate from Notre Dame also makes me smile.  One of the papers that she published deals with an issue involving infinity and Pascal’s wager.  https://www.academia.edu/16612267/Salvaging_Pascals_Wager

She explains an issue by way of an analogy that is pretty helpful to understand it.  She says consider you are in a game show and if you choose door number 1 you have a 1% chance of getting an infinitely good reward.  If you choose door number 2 you have a 99% chance of getting the exact same infinitely good thing.  It would seem irrational to pick door number 1.  Since decision theory tends to favor the options that give us the highest chance of a good outcome.   

But infinities are crazy things.  And as it turns out the mathematicians can only really say that both options would be infinitely valuable because infinity multiplied by any positive number (even a small fraction) is infinity.   Well ok.  But we really should think about this a bit deeper to at least try to at understand where the mathematicians might be coming from and if this really makes sense.       And by “try to understand” I mean I am making absolutely no promises. 

First decision theory.  It is fairly straight forward.  For any given outcome for an option you multiply the potential gain by the chance of getting that gain for all the outcomes and then add them up.  This gives you the utility value of that option.  So if a ticket has a 30% chance of winning $100 dollars then we say the utility value of the ticket is $30. 

Ok now to infinity and beyond! But first infinity. 

Gregor Cantor has devised some proofs that suggest certain infinities that might seem bigger are the same size but also that some infinities seem to have “more” than others.    

 He had an ingenious proof that the shorter line segment has as many points as a longer line segment and indeed any line.  The trick is to simply bend the smaller line segment into a “c” and then for any sized line position it along the back of the c.  You can draw a line from any imagined point in the middle of the space of the c (which is just the shorter line bent) to the longer line.  That line will cross the “c” in a unique point for every unique point on the longer line. 

See the drawing I scribbled out below:

One of my undergrad philosophy papers actually showed how Galileo did something similar when he explained how a ball rolling down an inclined plane reached every speed of the ball falling straight down.  Anyway the concept is the same as Cantors.  Galileo just took the longer line and tilted it so that you could then draw a perpendicular line from the line showing the height of the ball with the inclined plane.   See the drawing I scribbled out above. 

This can also form various Zeno’s paradoxes.    How could the ball going straight down reach every speed of the ball going down the inclined plane?  If the ball is crossing more points along the longer inclined plane and at every point it is hitting a new speed wouldn’t this mean it must be hitting a more speeds?  And if it is hitting a new speed every increment of time and is rolling for longer wouldn’t it have hit more speeds than the ball that is falling for a shorter time?  Etc.  

The infinite is fun and frustrating at the same time.  I recommend AW Moore’s book “The Infinite” if you want to learn a bit more about how puzzling the infinite can be.   

Cantor also showed all counting numbers seem to be just as numerous as all even counting numbers.  How?  Well you can draw a correspondence to each counting number with an even number. The even number 2 corresponds with 1 and the even number 4 corresponds with 2 the even number 6 corresponds with 3 and on and on.  You can see there will always be even numbers to correspond with each counting number.  The same is true if we take numbers divisible by 100.  100 corresponds with 1 200 corresponds with 2 etc.    So it seems that half (or even one hundredth) of infinity is still infinity of the same amount as all the counting numbers! 

So the line proof shows that you can keep adding line segments which all have an infinite number of points but doing so will not actually increase the number of points. Comparing and drawing a correspondence to each even number with each counting number shows that halving or taking any other fraction of an infinite set of numbers will not actually decrease the infinity either. These concepts explain why multiplying infinity by any positive number does not actually yield a bigger number/infinity (and if the positive number is a fraction it won’t yield a smaller number/infinity). Thus as we see why the utility value of Homer Simpson’s God is no lower than the Christian God even if we admit it is less likely.

Not all infinities are equal though. He argued there are more real numbers than counting numbers. 

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Ok back to Pascal’s wager.  This notion that no matter how small the percentage chance of achieving the infinite is, it still yields infinite rewards seems to help Pascals Wager because it doesn’t matter how low you put the probability of God existing it will still be the winning choice.  But it also hurts because if there is even any chance that Homer Simpson’s God reigns then that small chance would also yield an infinite utility value.   And these utility values seem to be the same as per the above proof.    So if the “Homer Simpson God” that just gets more and more angry every time we go to Christian church because we are doing it wrong, has any positive chance of being the true state of affairs well that small chance multiplied by infinity equals infinity as well.

So even if we think the Homer Simpson God is less probable that doesn’t matter because the utility values end up the same.  Should this convince us that choosing the door that gives a 1 percent chance of eternal reward is just as rational as choosing the door that gives a 99 percent chance of the same reward?  I have a few concerns.  One is just how challenging any discussion of infinities can be for mere mortals.     So how sure are the mathematicians of every step here? I think I understand and agree with them on the math but still.

Studies show we react more to empathic suffering than to joy so I think it is worth asking a question of mathematicians who deal in this area.  If you choose option one you have a 1 percent chance of you and everyone you love suffering for an infinite amount of time but if you choose option 2 you have a 99 percent chance of you and everyone you love getting the same amount of infinite suffering.  How many do you think would really say its fine to flip a coin and choose either option?   

The infinite seems to be playing the same role when it hurts and it helps Pascal.  But I think there is a difference.  If for the sake of argument we assume Homer Simpson’s God is less probable than the Christian God it seems we need to take a few more steps of analysis to say it would be irrational to prefer one option over another.  I think those steps are much more controversial than the steps Pascal takes in saying a shot at infinite value will always exceed a shot a finite value. That is, the reasoning about the infinite that helps pascal seems much less controversial.  

First consider what helps Pascal. Why would we think an shot at infinite gain is always better than a shot at finite gain?  If you are better off suffering for a single day rather than two days and better two days then suffer for three etc. it seems infinite would be worse than any finite amount of time.  I mean on what day would I say ok keep the tooth ache going I am unwilling to pay any price to prevent the pain?  It would seem that we always want suffering to end and so the infinite suffering is worse.  We would all think yes we would prefer our suffering to end rather than continue and thus there is always some cost we would pay to end it for any given day.  Whatever finite cost we would pay to end it could thus be multiplied every day of eternity that we feel the pain and would add up to infinity.  How long are you going to pay for pills that help ease your pain?  As long as the pain lasts.  Therefore if it lasts infinitely long we would pay and infinite amount.  That is the analysis that works *for* Pascal’s argument and that seems to be consistent with everything we know about the world.  That seems the sort of intuition supported by math that I am comfortable betting on.

But the analysis that works against him is this notion that it doesn’t matter which door you would pick between the 1 percent or the 99 percent chance.     I don’t think it is irrational for me to say I think that is really a different case.    I’m not so sure anyone really has enough of a handle on the infinite to tell me that choosing either option or even flipping a coin is just as rational.   But let’s at least try to chart out why that might be.   

In a discussion with Dr. Jackson Cosmic Skeptic says it is like monty hall problem in that math shows our Intuitions are wrong.  

I think the Monty Hall problem can be enlightening here but I think it helps Dr. Jackson’s/Pascal’s case.    

The Monty Hall problem involves a scenario where someone is given 3 options/doors to choose from.  Behind one of the doors is a car and there is a goat behind each of the other 2 doors.  Now you want the car because it is more valuable than the goat.   You get to pick a door and let’s say you pick door number 3.  Now before that door is opened Monty Hall says “look I will open up door number 1” and he does and shows you it has a goat.  Now he asks if you want to change your choice.  You can now choose door number 2 instead of number 3.  Should you?  Yes.  It may seem counter-intuitive but you will have substantially better chances of getting the car if you choose door 2. 

How do we know this?  Well there are actually 2 ways.  The first is just to test it repeatedly through computer simulation or otherwise. 

The second way is to think it through further.  Consider that instead of three doors there are 1000 doors.  And you pick door number 58.  And then Monty Hall opens all the other doors except the door 58 (the one you originally chose) and door number 678.   Now are you going to change your vote?  Of course.  So we know not to trust our intuitions in the monty hall problem  due to testing and thinking it through more. This way of understanding the Monty Hall problem comes courtesy of Brian Blaise.

But perhaps most importantly I can understand how the testing and the conceptualizing are done to solve the monty hall problem.  If I just took someone’s word for it I might reasonably still have doubts.   

What about my certainty that picking option 1 where I have a 1 percent chance of getting an infinite reward is the same as option 2 where I have a 99 chance of getting the same infinite reward.   First can I conceptualize why my intuition to choose the 99% chance is equal to the 1 percent chance?  Not really.   In fact quite the opposite.    

It seems to me that there is a problem with how the “utility value” is being used here.  I understand that as soon as one person (call him “that guy”) who chooses the one percent option gets the infinite reward that whole column equals the 99% column.  After all even if 99 times that number get the same infinite reward it is just like adding line segments to the number of points on the longer line as compared with the shorter line.  Or it can be seen as taking every 100th number and matching it with the counting numbers.  I’m not disputing the math. 

But there still is this nagging concern about me having a much lower chance of being “that guy” that wins the infinite reward with the 1% chance and evens out the tables.  I don’t think the standard decision analysis deals with this concern to my satisfaction. 

See the thing is when we say the “utility value” of ticket that has a 30% chance of winning $100 is $30 that does not mean everyone gets $30.  On average about 70 people out of 100 will get nothing while the other 30 out of one hundred will get $100.   The same is true for this.  99% of the people will get nothing while one percent will get infinite rewards.  The utility value may end up equaling the other option where 99% get infinite rewards and 1 percent get nothing,  but I still want to be someone that gets the winnings.      

So as I try to conceptualize it I still think it is rational to want the 99% option.  I don’t think I am denying any math in saying so.    And I do think it would be irrational to choose the 1% route.  

I remember reading “the kluge.”  It is a fine book but I took issue with one thing the author said.  He said something like people would be irrational if they didn’t always follow this line of thinking:  If you could buy a lottery ticket for one dollar and that gave you Y% chance to win a lottery you should pay the same amount for a lottery ticket that gives you 1/50 Y%  chance to win 50 times that.  But I am not so sure I agree.  I think I would rationally prefer to pay 1 dollar for a lottery ticket that gave me 50 times the chance to win 1 billion dollars instead of 1 dollar for a ticket that gave me 1/50 the chance to win $50 billion.  I mean even if I could figure out what to do with the first $100 million I am not sure what I would do with the other $900 million for a billion dollar prize.  Let alone the other 49 billion.  How much lobster can I eat?   Now here my issue is that I value the first dollar more than the 1 billionth.  So it is not the same exactly.  But I do think it is similar.  I think utility value is a tool but the results can rationally be used differently by different people. 

Now what about testing this Dr. Jackson theory?  Perhaps we can!  Notice when we test the Monty Hall problem we don’t actually need to deliver goats and cars.  We run a computer simulation and just count how many would get the cars versus goats depending on their choice.  Since we don’t need the actual infinite prize perhaps this is as easy running the simulation.  And guess what we would find?   Those choosing the 99% chance have a much higher chance of winning the infinite reward than those choosing the one percent chance.  And I suspect that is pretty much all there is to it.  The fact that the prize for the population on the whole choosing the 1% option equals the prize on the whole for those choosing the 99% option doesn’t change the fact that only 1% will get the infinite prize in the 1% option and I like the 99% odds more.  

So imagine we are given Jackson’s choice.  And huge numbers of people choose option 2 and are happy with their infinite gift but of course 99% isn’t 100 percent so some don’t get the infinite reward.  But then people start to realize that it seems that more than 1 percent didn’t get the infinite reward!  I think most people would be like huh what do you mean?  Do you mean people chose option one with only the one percent chance?  Or people chose to flip a coin?  I suspect not many people who chose that option would raise their hand and say yep I chose option one. 

On the other hand if somehow I got this wrong and both options are the same somehow.  I have to admit those who chose option one would get infinity plus an envious amount of smugness. 

My own view as of now is that the aspect of infinity that helps Pascal (the notion that we would always pay a finite amount to end suffering or experience joy and that price would be infinite if we are dealing with an infinite suffering or joy) seems consistent with everything I know about the world.  But the view that choosing the 1 percent door or the 99 percent door are the same, seems contrary to what I know. 

This title had some mention of a bar and a cast of characters.  I talked about Galileo Cantor Pascal Monty Hall, Dr. Jackson and Cosmic Skeptic but what about the Seinfeld cast?  Well Ok when I was thinking about this last night I imagined the following scene. 

Monty Hall has this big game where you can win an infinite checking account!   Don’t worry both top republicans and democrats assured him everything would be fine and they could just keep printing the money.  So he decides to make a huge number of roulette wheels with 1000 numbered slots.  And people can choose any number between 1 and 1000.  And you have an option.   Option 1:  If the roulette ball lands on the number you pick you win but if it lands on any other number they lose (99.9% chance of losing) or they can choose option 2: if the roulette wheel lands on the number they pick they lose but if it lands on any other number they win (99.9% chance of winning).    

Everyone can play once and all the wheels are spun in the morning.  That night the bars are packed.  Huge numbers of people are celebrating their winnings!  But of course some people are going to lose so they are hitting the bar too.  But rumors start to spread that considerably more than 1 in one thousand people lost!  Hmm.  So yeah I am bussing tables because even though I picked option 2  (the 99.9% winning chance) I sometimes think I have one thousand times the bad luck of others so for once the roulette wheel landed on my number. 

But then I see George Costanza arguing with Seinfeld and Elaine.  I see Seinfeld looking at a very discouraged George and saying “you did what??” in disbelief.  And Elaine looking amazed at George with her mouth gaping.    Kramer walks in with a big smile and orders a round for the whole bar.  Costanza charges at him and yells “You!! You went with the 99.9% chance didn’t you!! You were the one who convinced me it didn’t matter!”  Kramer is initially taken aback but then says “you didn’t uh I mean didnt uh I mean did you uh…”  and George busts in and says “yes yes I went with option 1!”  The bar room falls silent.  Except George keeps on.  He yells “Some friends you are! you told me it didn’t matter!  *I* tried to say option two was clearly better but *you* guys just kept on saying it didn’t matter didn’t you?  Didn’t you!?”

Seinfeld Elaine and Kramer all look a bit sheepish but then Seinfeld says “yeah but we also said it was CRAZY. How could we know you would actually pick the crazy option?”  George then says “alright so you admit it was your fault!   So just buy me anything I want.”  After a pause “Come on you owe me that and you can certainly afford it.”  Seinfeld says “Well, you know, we signed an agreement not to just give money away since if everyone did that there would be no workers, you know, no one to make the cocktails.  I can’t break the agreement, they might take my infinity check book away.”  Kramer and Elaine seem to agree. 

Meanwhile I see Cosmic Skeptic bartending.  I was giving him a hard time because he chose to flip a coin and the flip landed on option one for him.   But for what he lacks in wisdom he tends to make up for in being quick witted.  So he sees Cantor getting sloshed in the corner with a nearly empty glass.  Of course, Cantor chose option two and won but his troubles aren’t always solved with money.  Cosmic Skeptic asks Cantor if he wants another drink and Cantor says yes.  So CS says “well you were the one who said even numbers are equal to the counting numbers.”  Cantor slurs “well actually I *proved* it.”   CS says “Yeah right, then you wouldn’t mind giving me a tip of ½ of your infinite checking account.   After all the rules say you can’t give the money away but this would be a tip.”  Cantor immediately smiles and agrees saying “sure just don’t tell anyone – you know someone has to make the cocktails.”  And sure enough CS ends up with just as much money as anyone else.