Let's take a simple one-sided hypothesis test (say the two sample t-test). The confidence level we want (probability of rejecting the null when it is true) is defined as $\alpha$ - usually set to 5%. Now, I want to compare the two sample t-test with a different hypothesis test (say I use the normal distribution instead of the t-distribution in the test) and see how much better it is. One way to decide which test is better is to get the $\beta$ (defined as the probability of failing to reject the null hypothesis when it is false - meaning the alternate hypothesis is true). Then, $1-\beta$ becomes the power of the test corresponding to the $\alpha$ we choose. The test with higher power is better.
To calculate $\beta$, we need to basically compute the CDF of a distribution under the alternate hypothesis. The problem is, the alternate hypothesis is generally defined as the treatment group have a "larger" mean (in the context of the two sample t-test) than the control group. This would become a whole family of distributions (how much larger).
Now, which member of this family do we choose to calculate $\beta$?
One option is to define an effect size we want to capture (say we're interested in a difference of mean: 10), get the distribution under this difference assumption and get the $\beta$ using it.
Another option is to integrate over all positive effect sizes ($0$ to $\infty$). This is obviously going to be significantly harder than the first approach.
Is there any other approach we can use to compare the powers? And between the two approaches mentioned above, will the second one be substantially better than the first one (for which we have to choose an "arbitrary" effect size)?