Danger Sense

In DnD 4th edition there are several feats that increase your initiative.  The way it works is everyone rolls a d20 and adds their initiative bonus and then takes their turns in order, highest going first.  In the case of a tie the person with the highest initiative bonus wins the tie.  The two feats in we care about are:

Improved Initiative:  +4 to initiative.
Danger Sense:  When you roll initiative you can roll twice and take the best roll.

So which is better?  Initially I figured out that Danger Sense increases the average of your final initiative value by ~3.5 and so I used that value but it turns out the formula is actually much more complicated.  The reason this is so is that if one character has a drastically different value than the other then rolling several times is often irrelevant while adding +4 can make victory guaranteed (or make success possible).  The other thing that is interesting is that tiebreakers are huge.  If the +4 gets you above the other guy's bonus rather than below you effectively gain +5 instead so the breakpoints for how good Danger Sense is become pretty interesting.

You can see the little chart below which tells you how much + to initiative Danger Sense is worth depending how much you are ahead or behind your opponent in current bonus to initiative.  The bottom axis is borked though because it should read from -10 to +10.  The dip in the middle is because Danger Sense doesn't get you over the tiebreaker hump like actual bonuses to initiative do.  Essentially this tells us that if your initiative bonus is low you want Improve Initiative but if you initiative bonus is high (higher than the enemies the majority of the time) you want Danger Sense.

I like math and I just can't lie.