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a bit of algebra -- does this make sense?

Posted By: Chuck Bower
Date: Wednesday, 30 March 2011, at 8:13 p.m.

Suppose you face a situation where a 'reset' can occur. By 'reset', I mean after the next 1296 possible dicerolls (i.e. each player rolls and plays) the position is the same as it is now. First off, this situation is rare. But sometimes you face a position where an approximation occurs. (I'm thinking that Stick's problem #1 today is such a situation.)

Define p(r) as the probability that a reset occurs. Define E(r) as the equity of the position when the reset occurs. Define E(o) as the total equity for all the other rolls (i.e. when reset doesn't occur). Let 'E' be the current equity. Then:

E = p(r)*E(r) + [1 - p(r)]*E(o).

However, if the position resets to the current position then E(r) = E. So:

E = p(r)*E + [1 - p(r)]*E(o).

E - p(r)*E = [1 - p(r)]*E(o).

E*[1 - p(r)] = [1 - p(r)]*E(o).

And assuming p(r) can't be 1 (always true for backgammon) then:

E = E(o).

Does this makes sense (in particular, did I make any errors of concept or algebra)? If it does make sense, is it useful? (Presumably, it then SHOULD have been stated before. Kleinman?)

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