the SECOND largest number?

We always know that the goal in these circumstances is to maximize the SECOND largest number, i.e.

what will be left to attempt the recapture after the opponent captures your largest die and rerolls.

On the other hand, if he will have to make TWO attacks, maybe the THIRD largest number matters.

(though in this case, maybe that isn't true, since R. only needs to capture the 10 and 12 sided dice in order to win.

Oh, but wait -- does that mean R. should ignore the 6 sided die? That's not true in ALL circumstances, is it?)

In any case, the question isn't what R. should do, but what *I* should do. If the goal is to maximize the expected value of my SECOND largest die, I might be better off rerolling the 12 sided die. But doesn't rerolling the 10 sided die make more sense?

Doesn't it?

Button Men Game #679774 [some values tweaked to make the question more interesting]
Tournament Legal challenge, copying communication from game 679239

Skills in this game: Focus, V Swing, X Swing, Speed

Player: ElihuRoot *Dead Dude* *Fanatic*
Your Button Man: Ken (8 10 z12 20 V) Score: 38 (-0.6 sides) Rounds Won/Lost/Tied: 0 / 0 / 0 (Out of 3 wins)
Your Captured Dice: 10-sided die, 10-sided die, X Swing 4-sided die

10-sided die	Speed 12-sided die	V Swing (with 6 sides)		8-sided die
2	3	5		1 Captured last turn
Focus 6-sided die	16-sided die
2	16

Opponent: Russian *2^12th game cookie* *Ngozi's Disciple*
Button Man: Tsusuko (f6 10 10 16 X) Score: 39 (0.6 sides) Rounds Won/Lost/Tied: 0 / 0 / 0
Captured Dice: 20-sided die, 8-sided die