Thomas Ray's Reviews > The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy

The Compleat Strategyst
Being a Primer on the Theory of Games of Strategy

by John D. Williams

Free download pdf:
https://www.rand.org/pubs/commercial_...

pdf is 286 pages. 18 + book page = pdf page.
1954, Chapter 6 added 1966, RAND Corporation.
Library-of-Congress QA270
Dewey 519.3
ISBN 0486251012

RAND Corporation and the many collaborating universities were contracted by the US Air Force, intending to use game theory to win wars. Games are stated in terms of maximizing Blue bombers' chances of evading Red fighters, with various strategies on each side. pp. 48, 51-52.

The games are all, "Suppose Blue has a mission-critical bomber and an identical-looking support plane. Suppose, if the bomber leads, and is attacked, it has an 80% chance of surviving a Red fighter pass, with cover from the support plane's guns. Suppose the following plane, if attacked, has a 60% chance of surviving, thanks to the less-effective cover the leading plane can provide. The game is to get the mission-critical bomber past the Red fighter. If Blue /always/ leads with the bomber, and Red knows it, Blue wins 80%. But if Blue mixes it up, so Red doesn't know which to attack, and Blue leads with the bomber 2/3 of the time and follows with it 1/3 of the time, and Red attacks the leader 2/3 of the time and attacks the follower 1/3 of the time, then Blue wins (.8*4 + .6*1 + 1*4)/9 = 7.8/9 = .8667"

And similar.

The book is all about how to determine the optimal mix of strategies given the stated presumptions.

The stated presumptions have all been so immensely oversimplified as to have zero relation to the real world.

So it's a pure intellectual exercise.

One interesting idea is: since Blue doesn't want Red to know what Blue will do, it's important that Blue not decide until the last minute what to do, and then make that decision based on the random-number tables in the back of the book.

Cites /Theory of Games and Economic Behavior/, John von Neuman, 1944.

PROBABILITY
In ten Prussian army corps, in the 20 years 1875-1894, 122 soldiers were kicked to death by horses. The number of these deaths in a given corps, in a single year, year was

. . . . . . . . . . Corps-years . Corps-years
. . . . Corps-years . expected: .. expected:
Deaths .. Actual . .. Wolfram . .. Book p. 10
0 . . . . . 109 . . . 108.504 . .. 109 [should say 108]
1 . . . . .. 65 . . .. 66.520 . . . 66 [should say 67]
2 . . . . .. 22 . . .. 20.223 . . . 20
3 . . . . . . 3 . . . . 4.065 . . .. 4
4 . . . . . . 1 . . . .. .608 . . .. 1
5 . . . . . . 0 . . . .. .072 . . .. 0
6 . . . . . . 0 . . . .. .007 . . .. 0
7 . . . . . . 0 . . . .. .001 . . .. 0
8 . . . . . . 0 . . . .. .000 . . .. 0

The expected number of the 200 corps-years in which k of the 122 deaths occurred is

(122!/(k!*(122 - k)!))*200*(1 - 1/200)^(122 - k)*(1/200)^k

But the author doesn't tell us this. We can ask Wolfram Alpha to remind us how to compute binomial coefficients: (click "more details" for the formula)
https://www.wolframalpha.com/input/?i...

Here's the table:
Table[(122!/(k!*(122 - k)!))*200.*(1 - 1/200.)^(122. - k)*(1/200.)^k, {k, 0, 8}]
https://www.wolframalpha.com/input/?i...

Because the author has a sense of humor, he also tells us that in these years no horses were kicked to death by Prussian soldiers.