I've set out my math below (the probabilities are for "to hit", "to wound", then failed armour save) and then drawn conclusions below that. If you don't like math, you can jump ahead to the conclusions heading.
Berzerker w/ chainsword (WS3+, S5, 0 Rend):
vs T4, 2+ save = 2/3 * 2/3 * 1/6 = 2/27 wounds per attack
vs T5, 2+ save = 2/3 * 1/2 * 1/6 = 1/18 wounds per attack
vs T6, 2+ save = 2/3 * 1/3 * 1/6 = 1/27 wounds per attack
vs T7, 2+ save = 2/3 * 1/3 * 1/6 = 1/27 wounds per attack
vs T5, 3+ save = 2/3 * 1/2 * 1/3 = 2/18 wounds per attack
vs T6, 3+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack
vs T7, 3+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack
vs T4, 4+ save = 2/3 * 2/3 * 1/2 = 2/9 wounds per attack
vs T5, 4+ save = 2/3 * 1/2 * 1/2 = 1/6 wounds per attack
vs T6, 4+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack
vs T7, 4+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack
vs T4, 5+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T5, 5+ save = 2/3 * 1/2 * 2/3 = 2/9 wounds per attack
vs T6, 5+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack
vs T7, 5+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack
vs T4, 6+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T5, 6+ save = 2/3 * 1/2 * 5/6 = 5/18 wounds per attack
vs T6, 6+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack
vs T7, 6+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack
Berzerker w/ chainaxe (WS3+, S6, -1 Rend):
vs T4, 2+ save = 2/3 * 2/3 * 1/3 = 4/27 wounds per attack
vs T5, 2+ save = 2/3 * 2/3 * 1/3 = 4/27 wounds per attack
vs T6, 2+ save = 2/3 * 1/2 * 1/3 = 2/18 wounds per attack
vs T7, 2+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack
vs T4, 3+ save = 2/3 * 2/3 * 1/2 = 4/18 wounds per attack
vs T5, 3+ save = 2/3 * 2/3 * 1/2 = 4/18 wounds per attack
vs T6, 3+ save = 2/3 * 1/2 * 1/2 = 1/6 wounds per attack
vs T7, 3+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack
vs T4, 4+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T5, 4+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T6, 4+ save = 2/3 * 1/2 * 2/3 = 2/9 wounds per attack
vs T7, 4+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack
vs T4, 5+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T5, 5+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T6, 5+ save = 2/3 * 1/2 * 5/6 = 5/18 wounds per attack
vs T7, 5+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack
vs T4, 6+ save = 2/3 * 2/3 = 4/9 wounds per attack
vs T5, 6+ save = 2/3 * 2/3 = 4/9 wounds per attack
vs T6, 6+ save = 2/3 * 1/2 = 1/3 wounds per attack
vs T7, 6+ save = 2/3 * 1/3 = 2/9 wounds per attack
Now for the comparison in each fight activation, a berzerker will have 3 attacks with a chainsword and 2 with a chainaxe, so each model will do the following number of wounds:
vs T4, 2+ save = 6/27 (0.22) w/ CS and 8/27 (0.30) w/ CA - CHAINAXE WINS
vs T5, 2+ save = 3/18 (0.17) w/ CS and 8/27 (0.30) w/ CA - CHAINAXE WINS
vs T6, 2+ save = 3/27 (0.11) w/ CS and 4/18 (0.22) w/ CA - CHAINAXE WINS
vs T7, 2+ save = 3/27 (0.11) w/ CS and 4/27 (0.15) w/ CA - CHAINAXE WINS
vs T4, 3+ save = 12/27 (0.44) w/ CS and 8/18 (0.44) w/ CA - EVEN
vs T5, 3+ save = 6/18 (0.33) w/ CS and 8/18 (0.44) w/ CA - CHAINAXE WINS
vs T6, 3+ save = 6/27 (0.22) w/ CS and 2/6 (0.33) w/ CA - CHAINAXE WINS
vs T7, 3+ save = 6/27 (0.22) w/ CS and 2/9 (0.22) w/ CA - EVEN
vs T4, 4+ save = 6/9 (0.67) w/ CS and 16/27 (0.59) w/ CA - CHAINSWORD WINS
vs T5, 4+ save = 3/6 (0.5) w/ CS and 16/27 (0.59) w/ CA - CHAINAXE WINS
vs T6, 4+ save = 3/9 (0.33) w/ CS and 4/9 (0.44) w/ CA - CHAINAXE WINS
vs T7, 4+ save = 3/9 (0.33) w/ CS and 8/27 (0.30) w/ CA - CHAINSWORD WINS
vs T4, 5+ save = 24/27 (0.89) w/ CS and 20/27 (0.74) w/ CA - CHAINSWORD WINS
vs T5, 5+ save = 6/9 (0.67) w/ CS and 20/27 (0.74) w/ CA - CHAINAXE WINS
vs T6, 5+ save = 12/27 (0.44) w/ CS and 10/18 (0.56) w/ CA - CHAINAXE WINS
vs T7, 5+ save = 12/27 (0.44) w/ CS and 10/27 (0.37) w/ CA - CHAINSWORD WINS
vs T4, 6+ save = 30/27 (1.11) w/ CS and 8/9 (0.89) w/ CA - CHAINSWORD WINS
vs T5, 6+ save = 15/18 (0.83) w/ CS and 8/9 (0.89) w/ CA - CHAINAXE WINS
vs T6, 6+ save = 15/27 (0.56) w/ CS and 2/3 (0.67) w/ CA - CHAINAXE WINS
vs T7, 6+ save = 15/27 (0.56) w/ CS and 4/9 (0.44) w/ CA - CHAINSWORD WINS
CONCLUSIONS
Some interesting results to analyze. The first (and the most obvious IMO) is that the chainaxe is better against against 2+ armour. That is because each attack is now TWICE as likely to make it through the armour. The margin of betterness becomes even higher against T5/T6 because of the chainaxes other benefit (the +1S).
When looking at all of the other results, the chainsword tends to do better in the extremes where the chainsword's extra attack outweigh's the aggregate benefits from the chainaxe (being the extra point of strength (which is useless against T4 and T7) and the -1 rend (which is less useful when the enemy's armour is shitter)). After reviewing the math, it made sense to me that the chainaxe upgrade only costs 1 point - of the 16 scenarios where the opponent has 3+ armour or worse, the chainaxe won 8, the chainsword won 6, and they tied on two.
Interestingly enough, once you ignore 2+ armour saves, the decision appears to be more connected to toughness than armour. Look at T4 guys - the chainsword wins twice and they're tied the final time - the axe never wins. Same with T7. Compare that with T5/T6, where the chainaxe always wins.
Basically, if you think you're going to be facing mostly T5 and T6 (or 2+ armour), it's worth it to use your chainaxe, otherwise keep the chainsword.
Or you could do what I plan on doing for my tournament armies - which is to have a couple of chainaxes (maybe 33% of the unit) and then remove casualties depending on what's on the field opposite me. That way the list can be more balanced and I can adapt to the circumstances.
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