Mouse & Elephant

Some basic strategy

Seek out your pet foe!
Have a look at the following middle game position (not taken from actual play, but set up in order to illustrate the point), and ask yourself if it is really as balanced as it appears to be at first glance:

Neither side has an important space advantage, but on closer inspection you will have noticed that in positional terms Red has very much the upper hand, in that all of his pieces are "looking at" (= positioned directly opposite) their "pet foes", i.e. those enemy pieces they'd rather fight because they are ranked directly below them.
That this kind of consideration is fundamental in the case of positioning your mouse and westie goes without saying. But a lion, for example, should also always endeavour to confront the enemy wolf, while staying away from the enemy tiger.

Bodyguards
This point is a logical extension of the previous one. If we know, for instance, that our opponent's mouse will be staring hard at our elephant during the whole game, why don't we neutralize this menace by keeping our cat right next to or in front of the trunk-bearing one!? The same applies in the case of westie, polar bear and wolf.

Identical pieces with non-identical value
A westie is a westie is a westie, right? Wrong! If, for instance, we have managed to swap our tiger against the enemy polar bear, both players' westies will all of a sudden be of very uneven value: ours has lost its chief purpose in life, harassing the enemy polar bear, while our opponent's westie is still having a ball doing exactly this. No prizes for guessing which player should now try to force an exchange of westies at the earliest opportunity (again, the same logic applies - with a vengeance! - in the case of mice and elephants).

Material (dis)advantage
OK, you have blundered away material (sorry, I meant to say it "went missing due to a completely unforeseeable string of adverse rolls...") - should you now hang your head in sorrow and contemplate resignation? The hell you should - this is NOT chess, remember, and we can try to force our luck at any time! In a situation like this, there are basically two ways to go:
a) try to restore the material balance by means of some plucky tactical play (typically in the centre), and
b) outplay your opponent positionally, typically outflanking him. Look at the following position:

Blue has lost his polar bear, but has subsequently succeeded in creating superiority on one wing, while more or less abandoning the other to his opponent. The advanced position of his queen - already well on the way to the enemy baseline! - gives him excellent winning chances.
Inversely, if you're the player with the material advantage, watch out for similar manoeuvres by your opponent. If you "read" his intention early, you may be able to block the wing where he's trying to pull off his little stunt, or at least slow him down sufficiently, and waltz home your queen on the other wing, given up by him.
If, however, play takes place largely in the centre, you can try to adopt the ancient chess-strategy of swapping everything off until the one piece you're ahead is the only one left and gives you a winning advantage. But don't fall asleep in the process - your opponent is sure to have different ideas!

Bite the bullet!
In the endgame position below Red's lone queen, who has just rolled 4-3, is completely lost, isn't she?

Well, while Red's prospects are certainly none too bright, the moment has come for him to make up his mind whether he'd rather go out with a bang or a whimper! If he backs away into the corner (1), he is likely to survive another turn, but provided his opponent handles the situation competently (see the next paragraph: "Checkmating the Queen - or not!?"), his chances of escaping alive from there are virtually nil.
On the other hand, if the threatened queen "bites the bullet" now, steps out boldly to (2), sure enough she'll be gobbled up immediately with a chance of 35/36, but what about the remaining 1/36? If Beige's westie fails to capture the Red queen now, she stands e fair chance of eluding the dogged one with her following move, thus obliging the enemy queen to plunge for a (drawing) suicide attack, which she might not even succeed in, in which case the winner is.....?!?