GSOE9210 Engineering Decisions

Transcription

1 GSOE9 Engineering Decisions Problem Set 5. Consider the river roblem described in lectures: f f V B A B + (a) For =, what is the sloe of the Bayes indifference line through A? (b) Draw the Bayes indifference curves for = and through A and B. (c) Draw the Bayes indifference curve for which an agent would be indifferent between A and B, resectively. What is the sloe of the line? (d) For which robability (i.e., value of ) would an agent be indifferent between A and B under the Bayes decision rule? (e) What is the Bayes value associated with the indifference curve through A and B? (f) For which values of would an agent refer A to B? (a) The indifference curves are given by the oints (v, v ) which, for fixed u R, satisfy: v + ( )v = u (b) In gradient-intercet form, v = u v, where m = ; e.g., for =, m = / =. f = B A = f (c)

2 f = B A 5 The line AB laces A and B on the same indifference curve. The sloe of the line is given by: m AB = = (d) We saw above that m AB = ; i.e., =. Hence = ; i.e., =. Therefore =. Alternatively, = y x+ y = + =. f m m = = Alternatively, where m is the gradient of the line, = =. (e) Because the indifference line AB goes through A (and B), we can associate with it the Bayes value of A; i.e., u A = V B (A) = = =. (f) From the grah, for values >, the sloe is steeer (m < ) than that of line AB, and hence B is below the indifference line through A; i.e., A would be referred to B. Alternatively, analytically: V B (A) > V B (B) iff > + iff > iff >. Reeat the above exercises for regret. What can you infer about the Bayes decision rule when alied to the original values versus regrets? The regrets in regret sace are shown in the grah below. Since we want to minimise regret under the minimax Regret rule, lower-left (regret) indifference lines are referred (i.e., corresond to lower more referred Bayes regret values).

3 r = = A B = r The Bayes regret value for a strategy A is given by the Bayes value of A written V BR (A) with A situated in regret sace. Bayes regrets are calculated in the same way, using regrets instead of the original values. Indifference lines for given are obtained by fixing the Bayes regret value: r + ( )r = u A is at (, ) in regret sace. The Bayes value along the indifference line through A for = is given by setting r =, r = in the exression for V BR (A) above: u A = ( ) = ( ) = B is at (, ), so for =, the Bayes value of the indifference line through B is given by u B = =. AB has sloe m =, hence it corresonds to =. Moreover, V BR(B) = u B = = =. When considering regret, strategy A is referred to B when its Bayes regret value is lesser, which is the case for robabilities that roduce lines steeer than gradient (m < ); i.e., V BR (A) < V BR (B) iff m < ; i.e., > iff >. Note that as comarison of Bayes values and Bayes regret values, V B (A) and V BR (B), deend, in both cases, only on the sloe of their indifference curves. It follows that the Bayes decision rule is invariant under original values and regrets; i.e., V B (A) > V B (B) iff V BR (A) < V BR (B). That is, A is referred to B under the Bayes decision rule for the original values if an only if it is also referred under the Bayes decision rule for regrets.. Consider the generic two-strategy roblem below: s s A a a B b b Assume neither strategy dominates the other.

4 (a) Prove that an agent will be indifferent between A and B under Bayes when: y = x + y where (b) Prove that: where m = y x Cartesian lane. y = a b x = a b = m m is the sloe of the line joining A and B in the (a) If neither strategy is dominated then (b a )(b a ) < ; i.e., b a < iff b a >. Setting V B (A) = V B (B): (b) From lectures: V B (A) = a + ( )a V B (B) = b + ( )b a + ( )a = b + ( )b (a a ) + a = (b b ) + b (a b + b a ) = b a b a = (a b ) + (a b ) y = x + y = m = m m m = (m ) = m m. Consider the decision table below, with P (s ) = : A s5 s AB 5 CB C 5

5 (a) For which value of would the agent be indifferent between A and C? (b) Plot the Bayes values for the strategies as varies from to. (c) For which values of are A, B, and C referred, resectively, under the Bayes decision rule? s 5 C M = A = = 5 B 5 s (a) Sloe of AC: m = 5 5 =. Hence: = = 5 = = 5 Hence for < 5, C is referred. For > 5, A is referred. Note that B is (strongly) dominated, hence is not admissible, and therefore is never referred. (b) Consider the lot of the Bayes values of the strategies against : V B 5 C B A M 5 5

6 (c) From the grah it is clear that for < < 5, C is referred. For < <, A is referred Each day, a drinks vendor must urchase stock of several tyes of drink to sell in her sho. The tyes of drink which may be stocked are: a) hot chocolate; b) iced tea; c) lemonade; d) orange juice. She knows, from ast exerience, that on warm (w) days she ll make sales totalling $ on hot chocolate, $ on iced tea, $ on lemonade, and $ on orange juice. On cool (c) days, however, her sales total is $ on hot chocolate, $ on iced tea, $ on lemonade, and $ on orange juice. Assume days are either warm or cool, but she will not know which before she must order her stock. (a) Produce a decision table for this roblem. (b) What roortion of drinks should she stock to maximise her guaranteed (i.e., minimum) sales total regardless of the temerature? (c) Find the Bayes strategies for =,,,,. (d) What is the least favourable robability distribution on warm and cool (not warm) days? (e) Reeat the above analysis for the minimax Regret rule. (f) Define the admissibility frontier for this roblem. (a) Consider the decision table below, with P (s ) =. Values are exressed in tens of dollars. The associated grah is also shown. w c HC IT Le OJ c HC = M Le = = where: w warm day c cold day OJ IT w (b) She would maximise her guaranteed sales by having the mixture of stock which maximises the minimum sales irresective of whether the day is warm or cold. It is clear from the grah that the otimal mixture should comrise hot chocolate and lemonade only. Let m w be the average sales of the relevant mixture of drinks on a warm day and m c the mixture s average sales on a cool day. If µ is the desired roortion of hot chocolate in the mixture, then M = (m w, m c ) = (, ) + µ[(, ) (, )]; i.e., m w = + ( )µ = µ m c = + ( )µ = + µ 6

7 Setting m w = m c to find the Maximin mixed strategy: µ = + µ = µ µ = That is, she should have a mixture consisting of one third of the units on sale being hot chocolate and the other two thirds lemonade. That is, a ratio of two units of lemonade er unit of hot chocolate. (c) Consider the lot of the Bayes values of the strategies against : V B HC Le OJ M IT From the grah: Bayes strategy HC HC Le & OJ OJ IT & OJ For robabilities for which multile ure strategies are Bayes strategies, mixtures of those strategies involved would also be Bayes strategies; e.g., for =, any mixture of Le and OJ would also be a Bayes strategy. (d) The least favourable robability distribution is the one that minimises the value of the Bayes strategies, and corresonds to the robability associated with the indifference curve on which the Maximin strategy lies. This is obtained from the sloe of the segment on which M lies; i.e., the segment joining HC and Le. Since this sloe is m =, the robability is = + =. This is verified by insection of the above grah of the Bayes values against. (e) The maximum regret indifference curves are shown on the grah below (right). Since minimax Regret seeks to minimise the maximum regret, reference is for curves to the lower left (instead of uer right, which would corresond to reference under Maximin). w c HC IT Le OJ 7 where: w warm day c cold day

8 = = r c IT OJ Le M HCr w = Notice that the minimax Regret mixed strategy is the ure strategy Le, and that this does not agree with the Maximin strategy which is a mixture of HC and Le. Consider the lot of the Bayes regret values of the strategies against : V B IT HC Le OJ Notice that this grah resembles the other one but is inverted, and the values at = have been shifted by. Because of the similarity, the grahs of the lines for the strategies relative to each other are reserved, and hence the Bayes strategies remain unaffected for every value of ; i.e., Bayes strategies are invariant under regret. This can also be seen from the grah in regret sace; the strategies are rotated (double reflection) in the same relative ositions relative to each other, so the sloes (i.e., robabilities) will still roduce the same strategies under the Bayes decision rule when minimising Bayes regret rather than maximising the original Bayes values. (f) Consider: c IT OJ Le HC w Notice that iced tea (IT) is weakly dominated by OJ, and hence is not on the admissible frontier; in fact, the entire set of non-degenerate 8

9 mixtures of IT with OJ (the segment joining IT and OJ, excluding OJ itself) are inadmissible. When minimising regret, the admissibility frontier has the same shae, but is inverted (rotated). 6. Show that a strategy is admissible iff it is a Bayes strategy for some robability distribution. Consider an arbitrary inadmissible strategy A; i.e., there exists some strategy B such that for each of A s ayoffs, a i, for the corresonding ayoff b i under B, we have b i > a i. For an arbitrary robability distribution, let i be the robability of ayoffs a i and b i. It follows that: b i > a i iff i b i > i a i iff i b i > i i i a i iff V B (B) > V B (A) Therefore, B will be referred over A under the Bayes decision rule for any robability distribution, and hence A will not be a Bayes strategy. Conversely, suose A is admissible, then for any other strategy B, for some i, a i b i. So for any robability distribution such that i = (i.e., j = for all j i), V B (A) = k ka k = i a i i b i = k kb k = V B (B). It follows that for some robability distribution, A is a Bayes strategy. The two aragrahs above conclude the roof. 7. Show that a Maximin strategy is always a Bayes strategy for some robability distribution. A roof sketch is outlined for the case of two states. Let M = (m, m ) be a Maximin strategy. (Does there always exist a Maximin strategy?) There are two cases to consider:a) M is a ure strategy; or b) M is a mixture. If M is a ure strategy then there must be some state s i in which m i a i for any other strategy A. In this case M is admissible, and hence, by the result above, a Bayes strategy for some robability distribution. If M is a mixture then we saw that for the least favourable robability distribution P, M will receive a Bayes value no less than any admissible mixture. So M will be a Bayes strategy for P. In both cases M is a Bayes strategy, which comletes the roof. 8. Prove that for any two actions A and B, if A weakly dominates B, and all state robabilities are non-zero, then the Bayes decision rule will strictly refer A over B. Suose A weakly dominates B; i.e., for all i, a i b i and for some j, 9

10 a j > b j. Since for all i, i >, then it follows that for all i, i a i i b i and j a j > j b j. But then V B (A) = i ia i = i j ia i + j a j > i j ib i + j b j = V B (B).