I've tried to figure out the EV of accepting doubles. You have two options; decline the offer and lose a point (your opponent gains is equivalent to you losing if you look at it from the perspective of a single score scale), or accept the offer and see what happens. I came up with the following equation in order to solve the dilemma of accepting doubles:

x = % of winning
(1-x) = the opposite of winning, hence losing

-1 = 2x -2(1-x)
-1 = 2x -2 + 2x
1 = 4x
1/4 = x

According to my equation above, you must accept a double offer if you are capable of winning at least 25% of the time. Is this correct, because intuitively this seems incredibly wrong.

Also if you want me to explain the math better I can go into more detail. If you have an insight on a mathematical solution to this problem please let me know.

You are correct.

This one is beautiful, and regrettably not mine:

We lose 1 extra point by taking when we lose the game, but we gain 3 points by taking when we win the game (drop = -1, win = 2). Winning : losing = 3 : 1 = 75 : 25

The 25% figure you calculated has a name. It is called the dead-cube take point. In unlimited games (i.e., money play), whenever the game will be decided on the next roll, it gives the minimum percentage of games the taker must win in order to accept a double. So long as he can win 1 game in 4, he has a take.

Most doubles, however, occur with a live cube. That just means that after taking, the game will continue for some time, and the taker may have an opportunity to redouble if things turn around. Because most of the taker’s losses will occur on a 2-cube, while many of his wins will occur on a 4-cube (after a redouble), his live-cube take point is a few percentage points lower than his dead-cube take point. In a smoothly changing, non-volatile game, it can be demonstrated that the theorectical live-cube take point is 20%. In practice, it is hard to time your redoubles well enough to reach that ideal figure. Most experts, therefore, use 21.5% as their live-cube take point in unlimited games.

But even that does not tell the whole story. Your take point must be adjusted further in order to take into account the effects of gammons and backgammons. The numbers we have been discussing thus far assume that neither side will win or lose any. When you factor them in, the result is called the gammon-adjusted take point, but I’ll leave that calculation for another day!

If you are interested, you can search for any of the terms above, and in addition, others, such as recube vig, gammon price, and doubling window. You may also enjoy reading this old thread where some of the concepts appear.

Hope this helps.