Should We Sacrifice? The Basic Arithmetic
A sacrifice is worth it when your doubled penalty costs less than the opponents' game, and when their game was really going to make.
A sacrifice means bidding a contract you expect to go down in, because the penalty costs less than letting the opponents make their contract.
Compare the two numbers
Their game is worth:
about 420 (major) or 400 (3NT) not vulnerable
about 620 or 600 vulnerable.
Your doubled undertricks cost:
Not vulnerable: 1 down 100, 2 down 300, 3 down 500, 4 down 800.
Vulnerable: 1 down 200, 2 down 500, 3 down 800.
By vulnerability
Favourable (you not vulnerable, they vulnerable): you can go three down (500) against their 620. The best time to sacrifice.
Both not vulnerable: two down (300) against 420 gains; three down (500) loses.
Both vulnerable: one down (200) is fine; two down (500) against 620 still gains a little.
Unfavourable: one down (200) against 420 gains; two down (500) loses. Sacrifice rarely.
Example
The opponents, vulnerable, bid 4♠. You are not vulnerable with a ten-card heart fit and little defence. If 5♥ doubled goes two down, you lose 300 instead of 620: a big gain.
Two checks before you bid
Will their contract make? If you might beat it, defending is better (see "Avoiding Phantom Sacrifices").
Will they go on? They may bid one more and make it, or double you for more than you expected.
Key takeaways
Compare your doubled penalty with their game score.
Favourable vulnerability is the best time to sacrifice.
Only sacrifice when their game is likely to make.