Forum ST222 EXERCISE SHEET

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1. A guest user

   #1 11:08, Fri 24 Oct 2014

   Hey, I'm struggling with exercise 4. Is it as simple as setting up a single bet? So for an event A where P(A) = 1 and an individual believes P(A) > 1 we can offer them the bet where they pay $P(A) for A to happen with reward $1. Then they've payed us more than they're winning. Then P(A)<1 is similar? Thanks for any help. Could use this thread for talking about other questions too if people want.
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2. A guest user

   #2 22:10, Tue 28 Oct 2014

   Im unsure about your first part. An individual can only believe an event occurs with a probability in [0,1]!
   However, I am also unsure about what happens if we assign P(A) < 1, p say. 
   We know that in reality P(A) = 1, but have assigned P(A) = p < 1. 
   Could we then do the following?
   Bet £p on Omega, with a win of £p if Omega, £0 otherwise
   Bet £(1-p) on not Omega, with a win of £1 if not Omega, £0 if Omega
   
   Then
   Total Bet: £1
   Expected winnings: £p
   Net: £(p-1) < 0. 
   
   Im not sure this is right though. Taking a bet to win the same amount back seems silly. Do we assume that we would happily take this bet knowing the outcome is certain?
   
   
   
   

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3. A guest user

   #3 22:27, Tue 28 Oct 2014

   Discussion of Exercise Sheet 1

   Also, has anybody managed to tackle question 5? 
   
   So far I have basically followed the urn example from lectures for probability elicitation to get the following:
   We want the client's agent to take the urn bet rather than bet on events B and C. 
   We have 0 ≤ P(B) ≤ P(C) ≤ P(B U C) but P(B) + P(C) > P(B U C).
   
   Offer instead of a bet on the event B a bet on drawing a blue ball. 

   Offer instead of a bet on the event C a bet on drawing a green ball. 

   Note that P(Blue U Green) (the event that a blue or green ball is drawn) is the same as P(Blue) + P(Green). 
   
   Denote the number of blue balls by b, and the number of green by g. Increase b and g from 0 to n and let b*, g* be such that:-
   
   - The bets on B and C are preferred to the urn bets when b = b*, g = g*.
   - The urn bets are preferred to bets on B and C when b = b* + 1, g = g* + 1. 
   
   Hence, we must have
   P(B) ≥ P(Blue drawn from n, b* are blue) = b*/n
   P(B) ≤ P(Blue drawn from n, (b* + 1) are blue) = (b* + 1)/n 
   
   and also
   
   P(C) ≥ P(Green drawn from n, g* are green) = g*/n
   P(C) ≤ P(Green drawn from n, (g* + 1) are green) = (g* + 1)/n.

   but...how do we use this to construct a Dutch Book? 

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4. A guest user

   #4 18:11, Thu 6 Nov 2014

   Discussion of Exercise Sheet 1

   Not sure if you're still looking for an answer, interpreting as the client thinking B or C happening is higher than it actually is, I think this might work:

   Let P(BUC)=x and so according to client's belief let P(B)=y and P(C)=z st y+z>x

   So offer bet that costs £1 and pays £1/(y+z) if B or C happens

   According to client's belief E[money received] = (y+z)*(1/(y+z))=1 so they would take bet

   But in reality E[money received] = x*(1/(y+z) < 1

   So client is paying £1 for an actual expected return of <£1

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5. A guest user

   #5 18:16, Sun 9 Nov 2014

   Discussion of Exercise Sheet 1

   Thanks Callum, that looks like the idea!
   Was just struggling about how to do that in terms of the balls in the urn... but yeah I can figure that out now! :D 
   
   

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6. A guest user

   #6 17:24, Thu 27 Nov 2014

   Discussion of Exercise Sheet 1

   On exercise sheet 4 what did people get for the value of the game on the the final question? Did anyone else get zero? :/

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