Knowledge and Reality A

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Knowledge and Reality A

• Lecture Six Paradoxes I

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This Week

• Our look at formal epistemology.
• Our first look at some paradoxes.
• Our chance to advance the skills you are expected
  to acquire.

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Formal Epistemology

• Basically this is the way of formalising, using
  logic, probability theory and mathematics, how to
  make decisions.
• Its applied in numerous areas.
• Economics (microeconomics and macroeconomics)
• MAD scheme

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Formal Epistemology

• Some choices are easy.
• If you like Pepsi and hate Coke, what should you
  decide about which to drink?
• PEPSI!
• If you want to go to the pub with your mates but
  your lover will make your life hell because of
  it, what do you do?
• Depends how bad a hell!

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Formal Epistemology

• We do some really basic maths to show which
  option is better.

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Formal Epistemology

• Some choices are easy.
• If you like Pepsi and hate Coke, what should you
  decide about which to drink?
• PEPSI!
• If you want to go to the pub with your mates but
  your lover will make your life hell because of
  it, what do you do?
• Depends how bad a hell!

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Formal Epistemology

• We do some really basic maths to show which
  option is better.
• Fix a utility to picking Coke and Pepsi.
• If you hate Coke then youll lose out.
• Lets say -10 Utility Points.
• If you love Pepsi then youll gain.
• Lets say 10 Utility Points.
• The choice is clear one option gives you 10,
  the other option loses you 10.
• Nor does it matter how difficult it is to
  determine these Utility Points

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Formal Epistemology

• Some choices are easy.
• If you like Pepsi and hate Coke, what should you
  decide about which to drink?
• PEPSI!
• If you want to go to the pub with your mates but
  your lover will make your life hell because of
  it, what do you do?
• Depends how bad a hell!

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Formal Epistemology

• Going to the pub is worth 20 UP.
• Say that if your missus/bloke hating you isnt
  that bad (say you hate them too, or theyre too
  mild mannered to be really mean) then its -5 UP.
• Say that if your missus/bloke would do terrible
  things to you then its -25 UP.
• In the former you do it, in the latter you dont.
• And other things can be factored in!
• If youd feel guilty at doing it, maybe thats
  -20 UP and even in the first case you shouldnt
  do it.

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Formal Epistemology

• Things can get more complicated.
• Perhaps if your lover came home he/she would
  discover you gone and make your life hell to the
  tune of 25 UP.
• But you reckon theres only a 1 in 5 chance of
  them coming home.
• If you stay in theres a 100 chance of losing
  nothing and gaining nothing.
• If you go to the pub theres a 100 chance of you
  gaining 20 UP and a 20 chance of you losing 25.
• The calculation is simple your expected utility
  gain is
• 20

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Formal Epistemology

• Things can get more complicated.
• Perhaps if your lover came home he/she would
  discover you gone and make your life hell to the
  tune of 25 UP.
• But you reckon theres only a 1 in 5 chance of
  them coming home.
• If you stay in theres a 100 chance of losing
  nothing and gaining nothing.
• If you go to the pub theres a 100 chance of you
  gaining 20 UP and a 20 chance of you losing 25.
• The calculation is simple your expected utility
  gain is
• 20 (0.2 x 25 )

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Formal Epistemology

• Things can get more complicated.
• Perhaps if your lover came home he/she would
  discover you gone and make your life hell to the
  tune of 25 UP.
• But you reckon theres only a 1 in 5 chance of
  them coming home.
• If you stay in theres a 100 chance of losing
  nothing and gaining nothing.
• If you go to the pub theres a 100 chance of you
  gaining 20 UP and a 20 chance of you losing 25.
• The calculation is simple your expected utility
  gain is
• 20 (0.2 x 25 ) 20 5

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Formal Epistemology

• Things can get more complicated.
• Perhaps if your lover came home he/she would
  discover you gone and make your life hell to the
  tune of 25 UP.
• But you reckon theres only a 1 in 5 chance of
  them coming home.
• If you stay in theres a 100 chance of losing
  nothing and gaining nothing.
• If you go to the pub theres a 100 chance of you
  gaining 20 UP and a 20 chance of you losing 25.
• The calculation is simple your expected utility
  gain is
• 20 (0.2 x 25 ) 20 5 15

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Formal Epistemology

• Easy!
• But weird things can happen.
• MAD is one example.

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The Centipede Game

• Lets have another example.
• I want to play a game.
• The rules are easy!

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The Centipede Game

• Two people, each taking a turn.
• There are 8 coins. The aim of the game is to get
  the most coins.
• On your turn you can
• (i) Take a coin and then let it be the other
  players turn.
• (ii) Take two coins, and then the game ends and
  we tot up who has the most coins.
• The game ends when we run out of coins.

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Centipede Game

• So you have a principle to maximise utility.
• What you do is altered by what you think your
  opponent is going to do.
• With MAD it wasnt so bad, but here it fails
  miserably.
• It ends up that when the game starts I should
  just take two coins.
• Imagine we played it with 100 coins (you can play
  it with as many coins as you like!)

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Centipede Game

• When its my go, and there are two coins left,
  itd be irrational of me to take one coin and let
  it be your go.
• To make the most money (assume thats my aim, and
  I dont give a toss about you personally) I
  should take both coins and end the game.
• Id end up with 51 and you end up with 49,
  rather than us both having 50.
• Sound right?

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Centipede Game

• But youre a rational kinda person too.
• So when we get to three coins on the table, and
  its your go you know what Im going to do next
  turn.
• And itll mean you end with 49.
• So when there are three coins on the table, you
  should take two coins and end the game.
• You end up with 50 and I end up with 49.

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Centipede Game

• Now go back to a turn earlier, where there are
  four coins on the table and its my go.
• I know that, rationally, you will end the game
  the next turn.
• So at that stage I should take both coins and end
  the game.
• So Id have 50 and youd have 48.

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Centipede Game

• So now think about it again a turn earlier.
• There are five coins on the table, and its your
  go.
• You know that when it gets to four coins Ill end
  the game and youll only have 48.
• Better to end the game then, and you end up with
  49 (and I end up with 48).

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Centipede Game

• Do you see where this is going?
• We can do it again when there are six coins on
  the table and its my turn, I know youll end the
  game when there are five, so I better end it when
  there are six.
• When there are seven coins on the table and its
  your turn, you know thats what Ill do so youd
  better end it there and then.
• And so on and so forth.

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Centipede Game

• So we can keep going.
• Rationally, it seems, as soon as the game starts
  I should take two coins and end it.
• And that cant be right can it!
• I only end up with 2 and you end up with
  nothing.
• So, again, if we follow rational thinking through
  to its rational conclusions, we end up with
  apparently irrational activity.

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Principles of Formal Epistemology

• Where weve gone wrong is that we must be
  misapplying the principles we thought governed
  rational choice.
• For it is demonstrably irrational to take two
  coins to begin with!
• Lets spend the rest of the lecture looking at
  another example Newcombs Paradox.

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Principles of Formal Epistemology

• In this paradox the combination of two
  intuitively true principles lead us astray.
• The first is the principle of maximum expected
  utility.
• You should do what is likely to bring about the
  maximum amount of utility.
• Weve had examples of this already.

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Formal Epistemology

• Some choices are easy.
• If you like Pepsi and hate Coke, what should you
  decide about which to drink?
• PEPSI!
• If you want to go to the pub with your mates but
  your lover will make your life hell because of
  it, what do you do?
• Depends how bad a hell!

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Maximum Expected Utility

• Even in more complex cases, it seems good to go
  with it.
• If you have to gamble and betting it on Horse A
  is 80 likely to yield 30 and betting on Horse B
  is 20 likely to yield 200, what should you do?
• Horse A expected yield 0.8 x 30 24
• Horse B expected yield 0.2 x 200 40
• Ceteris paribus bet on Horse B
• And things might not be equal maybe the Mafia
  will break your legs unless you get them 30. In
  which case the utility isnt in line with
  monetary gain.
• But for demonstration purposes well stick with
  simple numbers and money.

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Newcombs Paradox

• Imagine there is a room with two boxes in it.
• One box is transparent it has 1,000.
• Another box is opaque but you know it either has
  nothing in it, or 1,000,000.
• You can take the contents of either one box, or
  both boxes.

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Newcombs Paradox

• Heres the twist.
• Theres some Derren Brown-esque character
  present.
• He has predicted, with 100 accuracy, which box
  you will pick.
• If you choose two boxes, there will be nothing in
  the opaque box.
• If you choose one box, there will be 1,000,000
  in the opaque box.
• So, how many boxes do you pick?

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Newcombs Paradox

• If you pick one box, and pick the one you can see
  into, you will definitely get 1,000.
• If you pick on box, and pick the one you cant
  see into, what are the chances of you getting
  1,000,000?
• Well, if Derren is a perfect predicator, the
  chance is 100.
• So youll definitely get 1,000,000.

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Newcombs Paradox

• If you pick two boxes, whats the chances of you
  getting what?
• As Derren is the perfect predictor, you have 100
  chance of having nothing in the second box.
• So you will definitely get 1000.

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Newcombs Paradox

• One boxer (transparent) 1000
• Two boxer 1000

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Newcombs Paradox

• One boxer (transparent) 1000
• Two boxer 1000
• One boxer (opaque) 1,000,000
• The Maximum Expected Utility principle says to
  pick just one box the one you cant see into.
• Sound good?

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Principles of Formal Epistemology

• The other principle is the Dominance Principle.
• If you have some choices, and one of them, choice
  X, has no downsides that the others dont have

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Principles of Formal Epistemology

• The other principle is the Dominance Principle.
• If you have some choices, and one of them, choice
  X, has no downsides that the others dont have
  and theres at least a chance that doing X will
  be good for you then you should do it.

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Principles of Formal Epistemology

• Example Stay at home, or go to the pub.
• Imagine that staying at home has no upsides or
  downsides.
• Imagine that going to the pub has no downsides (I
  get free beer, I have no problem with walking to
  the pub etc.) but a high chance Ill have fun.
• I should go to the pub!
• Imagine theres a small chance Ill have fun.
• I should still go to the pub!
• Imagine theres a one in a trillion chance Ill
  have fun.
• I should still go to the pub!
• At least, if theres no difference whatsoever
  between being in the pub and being at home.

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Principles of Formal Epistemology

• One choice dominates the others.
• It is clearly better, because it is as good as
  the other choices and has the benefit of
  (possibly, or even definitely) being better.
• How does this work in the Newcomb Paradox?

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Principles of Formal Epistemology

• One choice dominates the others.
• It is clearly better, because it is as good as
  the other choices and has the benefit of
  (possibly, or even definitely) being better.
• How does this work in the Newcomb Paradox?

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Newcombs Paradox

• Imagine your mother could see inside both boxes,
  and could advise you on how many to take.
• And lets imagine you trust your mother
  implicitly.
• If there were 1,000,000 in the opaque box and
  1,000 in the transparent box, how many boxes
  would she tell you to pick?
• Both of them! Youd come away with 1,001,000
  rather than 1,000,000.
• Clearly, then, youd be well advised to take the
  advice of your mother in that case.

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Newcombs Paradox

• What if there were only 1,000 in the transparent
  box and nothing in the opaque box?
• Well then your mother would still tell you to
  pick both boxes as youd walk away with 1,000
  rather than nothing (as if you picked one box,
  youd presumably pick the opaque, empty, box).
• So in that case you also should follow her advice
  and take both boxes!

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Newcombs Paradox

• Heres the rub.
• It doesnt matter whether your mother is there or
  not.
• You know that either way shed tell you to take
  both boxes.
• So you know that either way its in your
  interests to pick both boxes.
• That choice dominates the other choices.
• So, no matter what happens you should pick both
  boxes.

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Newcombs Paradox

• Think of it another way its too late for Derren
  to change the boxes now.
• Whatever you decide whats in the boxes is whats
  in the boxes.
• So, screw it, you may as well take as much as you
  can and take both boxes.

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Newcombs Paradox

• So NP is a paradox because one principle says to
  do one thing, and another principle says to do
  another.
• MEU Pick one box.
• DP Pick two boxes.
• And both principles look to be good, rational
  principles.
• Something has to go! Something has to be revised!
• A principle concerning how choices are made has
  to be altered!

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Broader Skills

• This topic builds into the idea of broadening
  your skills.
• This topic really is about your own responses to
  the material.
• In the others, Ive stressed that you could go
  off and read books on it.
• Not so easy with this one the books are hard!

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Broader Skills

• So much of philosophy is your own response, and
  your own tackling of the material.
• You need to go away and ruminate on these issues.

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Broader Skills

• If you do read the material, youll see that a
  lot of it is tricky.
• Lots of maths and tables and stuff.
• That can be okay. You need to master the art of
  learning what to read and what to ignore.
• Especially in a discipline like philosophy where
  you can be dipping in and out of many other
  disciplines (do applied ethics, read politics and
  economics do metaphysics, read physics journals
  etc.)

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Broader Skills

• You have to try and get out of an article what
  you can.
• The complex bits may be so complex you wont get
  anything out of them.
• Just skip them.
• You cant argue against them, so you may as well
  assume that it works.
• If the author says it, then you may as well
  believe it!
• Its called accepting for the purpose of
  argument

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Broader Skills

• Theres even a part of the article youve got for
  which youll need to do this
• Have fun!

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Next Lecture

• Induction.