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I am working through chapter 3 of Rufus Isaacs's work on differential games which is devoted to discrete games. I am stuck trying to understand his section 3.3 Battles of Extinction game where there are 2 players E and P (E tries to maximize, P to minimize the payoff). I can not understand the moves allowed in the game as shown in the figure 3.3.1 below (in his language If y pertains to the maximizing player E, then the H will be —x on the y-half axis and y on the x-half axis but I cant make sense of this wording)

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and the payoffs derived and shown in the next figure 3.3.2. enter image description here

What are the rules and goals of the game?

Isaacs, Rufus, Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization., Mineola, NY: Dover Publications (ISBN 0-486-40682-2/pbk). xxii, 384 p. (1999). ZBL1233.91001.

user35202
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1 Answers1

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The moves are prescribed/given in figure 3.3.1. The goal is to hit the y=0 or x=0 axes the quickest. So if it is E's turn at the point (y=1,x=-1) and E tries the top allowed move E will end up at (y=3,x=0). If E tries the other allowed move E will end up at (y=0, x=1). E's payoff is then the max of 3 and 1 which is 3.

On the other end if it is P turn at the point (y=1,x=-1) and P does the 2 down 1 left move, P ends at (y=-1,x=0). If it tries the other move 1 right and 1 down, P ends at (y=0,x=-2). P's payoff is then the min of -1 and -2 so it is -2.

If the starting point is (y=1,x=-2) then E can not reach either of the axes. P can however reach the y=0 axis at x=-3 using one of the moves (the other move ends at y=-1,x=-1) so P's payoff is -3.

Proceeding this way we get figure 3.3.2 although there is a typo: P's payoff at (y=1,x=-5) is -6 not -5.

We can also fill other values (not in figure 3.3.2) by using the values in 3.3.2: For example if the starting point is (y=5,x=-1) then P moves lead it to either (y=3,x=0) or (y=4,x=-2). The payoff for E at those points is 3 and 6. So P's payoff is the min of 3 and 6 or 3.

Proceeding this way we can find the payoffs in the entire grid (figure 3.3.3 in the book)

user2175783
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