Chapter 4 addition - Chap 4 prob 24 Game Theory The...

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Unformatted text preview: Chap. 4, prob. 24. Game Theory: The Minimeor The- orer‘n. We are given two players, A and 5', and A picks one of two numbers, 1 or 2. B then has to gue: what number A picked. If B guesses number s correctly [3' can be 1 or 2], then 5' gets t dollars from A. If E' is wrong, A gets it dollars from B. One strategyr is for B to randomize his guess, saying 1 with probability p and 2 with probability 1 — p. The problem will be about what strategy the two players should follow. We have already said 5' will try to randomize. [f A were to try to avoid losing too much to 3, you might think he should always pick 1, but then that gives 3 something to predict reliably, and he can count on winning 1 dollar from A every turn. So you see there is a problem of picking a strategy without your opponent being able to read exactly what it is and therefore being able to anticipate your move. That is also the reason why a random strategy might be useful. Set G equal to the random variable "B’s gain. So, with 5' using this random strategy, we get that stem chooses 13 = p - 1 — Eu — p}. and 3 EEG|A chooses?) = [l — p] - 2 — EFL First notice that E{G|Achooses 1] = EEG|A chooses 2) at p = %. Then min-[E[G|A chooses l],E{G|Achooses2]} ={ El—rJJ-fl—i-a :32 E. The maximum of this is attained at p = % and is equal to %. {You can see this by inspection from the graphs.) The “maximin” is %. Similarly, you can see that if we take the point of view of A trying to minimise her loses, we would look at G as her loss, and calculate first that q'1_g'll_‘1lr qEH,and max{E{G|Echooses 1], E(G|Bchooses2)} = Again, by inspection, the minimum of this is attained when :1 = %, and the minimum is 1%, which is the “minimsx”, and is equal, as predicted by Von Neumann, to the maximin. F113 63 3 1:1}; t=§ 2hr: - E Piss-ck e’“ : e “ maul fl: NIL-17H: re Pf mans; -r{ titre?) ~ Mei-=9 - rm rs =11) . ,_ are as iii—e"! -.___._-I-- 31' a: :' Dill-fir— Pg 175 #43. (illlii‘JEE-El2 + EilLE‘JAE-EII + [215 Pg 175 #41 [a] an: [b]: s 7 m .2 (E‘in‘u wig“. [fill E {Emu m“. [iii] .2 (Elsa —pr-* 1:5 1:5 1:4 1Where p = .17 for [a] and p = .3 for [b]. {c} 2 challenges. Pg 177 #53. {El The probability that an arbitrary couple were both bron on April ED is, assuming independence and an equal chance of having being born on any given date. [1,3365]? Hence, the number of such couples is approximately Poisson with mean Sflflflflflflfififi R: .5. Therefore1 the probability that at least one pair were both born on this date is approximately 1 — 34m. {bl The probability that an arbitrary eouple were born on the same day of the year is 13"335. Hence? the number of such couple is approximately Poisson with mean El].| flfl'flfflfiE = 219.13. Hence, the probability of at least one such pair is 1 — E_21g.1a s: 1. Pg 178 #El]. P{2| beneficialflftl FH | beneficial}3f£1 + P{2| not beneficial}1l.l’£1 -3393 E ii s.- + -.:' P{beneficial | 2} Pg 179 #74. [a] Pp: =n} = {$7 [blPiX:>2}=1_ 54+“ “+94 a 1D Pg 18H #77 P-{rejected} = 1 — [.Qj‘l' Pngl] #4 LetY=ex1then _ ‘3 yin ”(m—{Film} yen Pg 181 #13. Easiest to first take leg and then determine the 151 that maximizes leg P{X = k}. 1']!- legP-{X = k} = log (1:) +8: lug]: + [n — k} lug[1 — 15} By taking derimtives with respect ten 13, we knew that when p = kfn? ilegP-{X = k} = i — E = fl1mlugP{X = k} is going to be maxi— m1zed. Pg 181 #18. lugP{X = k} = —J‘L +8: lug}; — lugl[k!]l %lugP{X=k}=—l+’f=fl=>h=k Pg 181 #19 m '3'} Em = Ere-w=zr‘e-W i=|2| i=1 m . m ' = Zlfl_1e_"‘fiiflfii— 11!: Eu + 1]“‘154‘33Hffi i=1 i=0 DC“ . = 121.3" + 11:“—le"‘w.:i! = mm: + 11“] j=fl Hence E[X3] = mux+1fl = A ZIEE—Allfil'FflZiE—AA‘IIII+ZE—A1hifll i=0 1:” 1:” = Maxi] + 213m + 1} = m? +3.1 + 1) ...
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