Oct 23-27 - Gambling and Learning
 
1. Personal Probabilities (PIL Chapter 13)
 
• Our dispositions to accept certain gambles can be understood in terms of personal probabilities (‘PPs’) for various outcomes.
• Calibration of your PP that voter turn-out will be above 56% on the November 7th general election = A (it was 52% in 2002) – determining the value by your dispositions:
    • Pick a smallish prize for selecting the correct outcome – I would like an airport express so I can play music from my computer over my kitchen radio. What’s yours?
    • Would you rather ‘bet’ on A or …
      1. (i)… ¬A?
      2. (ii)… snow next Monday?
      3. (iii)… rain next Monday?
      4. (iv)… at least one H in 3 tosses of a fair coin?
      5. (v)… at least one H in 2 tosses?
      6. (vi) … H on the toss of a fair coin?
      7. (vii) … 1 or 2 on the roll of a fair die?
      8. (viii) … two Hs in a row?
      9. (ix) … 6 on the roll of a die?
      10. (x)… three Hs in a row?
 
2. Betting
 
• Suppose we now have a bet on whether or not A will be the outcome in the election: suppose you bet a certain amount (your stake) on A and I bet a certain amount (my stake) on ¬A.
    • (Your) stake for = what you give to the other player if you are wrong
    • (My) stake against = what you receive from the other player if you are right
    • Total stake = Stake for + Stake against
 
• Fill in the stakes in the following table:
Outcome
Winnings for A
Winnings for ¬A
Stake on A
Stake on ¬A
Total stake
Betting rate for A
Bet 1
            
A
$1
-$1
        
¬A
-$3
$3
        
Bet 2
            
A
$1
$1
        
¬A
$1
$1
        
Bet 3
            
A
$1
-$1
        
¬A
-¢33
¢33
        
Bet 4
            
A
$1
-$1
        
¬A
-¢14
¢14
        
• The Betting Rate on X is defined to be:
    q(X) = stake on X ÷ total stake
    What then is q(X) + q(¬X)?
• Fill in the betting rates for A.
 
3. Personal Probabilities and Betting
 
• If you are indifferent between the two sides of a bet, then the betting rate is fair:
    your fair betting rate for A = your personal probability for A
because, in the long run, you should expect to break even if you accept fair bets. Thus your fair betting rate is another calibration of your PP.
• Suppose you are offered odds of 13:2 on the Chicago Bears winning the Superbowl (those were the odds as of 10/22/06): i.e., for a stake of $2 you stand to receive $13 if they win. (N.B., gambling with overseas casinos – i.e., most internet gambling – was recently prohibited by law.)
    1. (a)What is the betting rate? What do you stand to win for a stake of $1?
    2. (b)Suppose that your PP for the Bears winning is 1/4. If you have to take one side of the bet, which one should you take? (What do you stand to win for a stake of $1?)
    3. (c)Suppose that your PP for the Bears winning is 1/10. If you have to take one side of the bet, which one should you take? (What do you stand to win for a stake of $1?)
    4. (d)Suppose that your PP for the Bears winning is 2/15. If you have to take one side of the bet, which one should you take? (What do you stand to win for a stake of $1?)
    5. (e)Do the bookmakers think that the chance of the Bears winning is more or less than 2/15? (I.e., what is their PP?)
• If your PP for X is p, then any bet for X at a rate q > p is a good one for you!
 
4. Gambling Everyday
 
• The framework is not only applicable to betting in the narrow sense – every single choice that we make involves assigning odds. The idea is that all our behavior can be understood in terms of personal probabilities. E.g.:
    • A slower bus comes first – the 145 rather than the 135. Should I take it or not? It depends on whether my PP for this 145 getting there first is greater or less than my PP for the next 135 getting there first.
    • Should I vote for Blagojevich or Topinka? It depends on my PPs for either achieving the goals I desire for the governor.
 
5. Learning From Experience (PIL Chapter 15)
 
• Coherence – your personal probabilities are not independent; they obey the laws of probability. Otherwise you should expect to lose fair bets, which is a contradiction.
 
• Because of coherence, we can model how we learn from accumulated evidence. E.g.,
    1. H = Butler murdered the Colonel
    2. E1 = Butler’s hair present at crime scene
    3. E2 = Butler’s brother MIA under Colonel’s command – a motive
    4. E3 = Butler’s gloves have Colonel’s blood on them
    5.  
Suppose:
Prior Probability
PP(H) = 1/10
PP(¬H) = 9/10
Likelihood in light of E1
PP(E1|H) = 7/10
PP(E1|H) = 5/10
Likelihood in light of E2
PP(E2|H) = 9/10
PP(E2|H) = 2/10
Likelihood in light of E3
PP(E3|H) = 8/10
PP(E3|H) = 1/10
  1. (a)What is PP(H|E1)?
  2. (b)What is PP(H|E1∧E2)?
  3. (c)What is PP(H|E1∧E2∧E3)?
 
• Bayesian updating: When you obtain evidence E, your personal probability for X should change from PP(X) to PP*(X) = PP(X|E) – i.e., in accord with Bayes’ Rule.
 
• Another example: according to the wave theory of light (‘WT’), a small bright spot will appear in the centre of the shadow cast by a round object (‘E’). This remarkable fact was highly influential in deciding the scientific community in favor of WT against the particle theory (‘PT’) when it was it was verified by Arago in 1818. For someone with these personal probabilities, the following calculation shows why.
Prior Probability
PP(WT) = 1/5
PP(PT) = 4/5
Likelihood in light of E
PP(E|WT) = 1
PP(E|PT) = 1/1000
What should PP*(WT) be after the spot is observed?