For those not familiar with it, the Doey Don't works as follows:
On a come-out roll, make both a pass and don't pass bet, for the same amount.
If a point is rolled, bet the odds on it.
The thinking is the don't pass bet is a hedge against a 7-out on the come-out roll. Players who play the doey don't would probably say the pass and don't offset each other, letting the player enjoy the odds bet with no house advantage.
The flaw is if a 12 is rolled on the come-out roll. The pass will lose and the don't pass will push, resulting in a loss of one unit. The probability of a 12 is 1/36, making the expected loss with this strategy 1/36 = 2.78% of one unit. Meanwhile, the expected loss making just a pass bet and taking the odds 7/495 = 1.41% of one unit.
However, the Doey Don't has less volatility. Assuming 3-4-5x odds, here is the standard deviation both ways:
Pass + full odds: 4.915632
Doey Don't: 4.085789
The bottom line is I don't recommend the Doey Don't, because the expected loss is 1.36% of a unit more.