Problem_Session_1_-_Thumbtack_Derivation

MS&E 352 Decision Analysis II Problem Session 1:

MS&E 352 Decision Analysis II Problem Session 1

Slide 2:

Thumbtack Toss

The Setup:

The Setup Designate Heads and Tails Toss thumbtack twice If you can bet on Same or Different, which is a better deal?

Most Simple Setup:

Most Simple Setup What we want to calculate What we know {“Same” | &} {“Head” | &} = p ASSUME: {Two “Heads” | &} = ({“Head” | &}) 2 = p 2 {“Same” | &} = {Two “Heads” | &} + {Two “Tails” | &} = ( p ) 2 + (1- p ) 2 If you can bet on Same or Different, which is a better deal?

Example:

Example What we want to calculate You believe {“Same” | &} {“Head” | &} = p = 0.7 ASSUME: {Two “Heads” | &} = ({“Head” | &}) 2 = p 2 {“Same” | &} = {Two “Heads” | &} + {Two “Tails” | &} = ( p ) 2 + (1- p ) 2 = 0.7 2 +0.3 2 = 0.49 + 0.09 = 0.58 If you believe p ≠ 0.5 (either H or T), you are strictly better off betting on {“Same”|&}

Discrete Distributions:

Discrete Distributions What if you are not really sure which way the thumbtack is biased, but you believe that there is a chance p that it could be biased towards heads. What does this mean? Why might this be useful? We need to find {“Two Heads”|bias,&} and {“Two Tails”|bias,&} We could consider the value of tests for refining our beliefs about the distribution of p . {“Same” | &}

Example:

Example What I want to calculate What I believe {“Same” | &} p = 0.2 OR p = 0.8 In words : “I know the thumbtack is biased (with 4:1 odds), but I’m not sure which way…” I think heads or tails bias is equally likely. {“Same” | &}

Continuous Distributions:

Continuous Distributions What if, rather than having discrete possibilities for p , we wanted to express a continuous distribution for p ? Hint: Think of the continuous distribution as summing infinitely many possibilities of p . It identical to the previous example, except that the distribution for p is no longer discrete . {“Same” | &} Discrete (e.g. 2-degrees) Continuous

Distribution on result of thumbtack toss:

Distribution on result of thumbtack toss What if you are not really sure, but you have a distribution on how likely the thumbtack is to come up heads, and know the mean of the distribution (like the weight-of-the-chair distribution from DA 1)? φ If you toss the thumbtack many times, φ is the (long-range) fraction of heads that you will see. You are uncertain about φ and hence come up with a distribution for it. Between someone who believes p = 0.7, and someone who has a distribution like the one above, with a mean of 0.7, who is better off?

Case 1: DA 1 Certificate (Single Toss):

Case 1: DA 1 Certificate (Single Toss) Neither one is better off! Person 1 {Head|&} = 0.7 φ 0.7 φ 0.7 Person 2 {Head|&} Notation: < φ |&> is the first moment (or the mean) of the distribution on φ < φ 2 |&> is the second moment of the distribution on φ

Case 2: DA 2 Certificate (Two Tosses):

Case 2: DA 2 Certificate (Two Tosses) Person 1 {Same|&} = { φ |&} 2 +{1- φ |&} 2 = 0.58 φ 0.7

Case 2: DA 2 Certificate (Two Tosses):

Case 2: DA 2 Certificate (Two Tosses) What can you say about this expression? φ 0.7 Person 2 {Same|&}

Case 2: DA 2 Certificate (Two Tosses):

Case 2: DA 2 Certificate (Two Tosses) What can you say about this expression? We know that: Var( φ ) = < φ 2 |&> - < φ |&> 2 < φ 2 |&> = < φ |&> 2 + Var( φ ) Compare with Person 1 {Same|&} = { φ |&} 2 +{1- φ |&} 2

Case 2: DA 2 Certificate (Two Tosses):

Case 2: DA 2 Certificate (Two Tosses) We know that: Var( φ ) = < φ 2 |&> - < φ |&> 2 < φ 2 |&> = < φ |&> 2 + Var( φ ) Compare with Person 1 {Same|&} = { φ |&} 2 +{1- φ |&} 2 Identical with Person 1 More value added through uncertainty on φ You value deal more if you are uncertain on φ