Do player odds increase with more players at the table in single deck blackjack? (Please show math to explain.)
Provided you aren't counting cards and you are just playing a consistent (lets say perfect)strategy, the answer is player odds are not dependent on the number of people playing.
They are also not dependent on the actions of other players at the table.
I'm not sure what mathematics you would like to see to prove this. If, however, you are playing a card counting strategy and vary your play (that is whether you hit, double or stand) dependent on this card counting, More players are better and the later you have to act the better.
Imagine a very large game (20 players) where you are the last person to act before the dealer. On the table, in full view ,are 50 of the 52 cards that were in the deck when the game began. By looking at these cards you can see that the remaining cards to be dealt are 2 kings. you have a 2 and a 5 in your hand (7 total), while the dealer holds a 9 and 6 (15 total). The obviously play here is to "double" because you are certain the dealer will bust when he draws the last king to hit 25.
Obviously this is why casinos don't have 20 person blackjack games, but a similar strategy "could" be applied to smaller games. However, varying your strategy (whether to stand, hit, double or split) dependent on cards already showing is very complicated, difficult to master and also increases your Expected Value by quite a small amount. This is why nobody does it.
If you are using the more common strategy of counting cards and varying your BET SIZE dependent on cards left in the deck, then LESS players improves your expected value. This is because you will have more opportunities to make big bets when the deck has alot of high cards remaining(this sways the odds in your favour)
Of course any casino that saw you playing one on one, single deck blackjack and varying your betting size wildly when the "COUNT" is favourable would kick you out very quickly and probably break your thumbs, so this whole conversation is fairly moot.
That's the common sense of it all anyway, if you need mathematical proofs etc, maybe someone else can pitch in.
The odds are not the same for the player and the dealer because in blackjack the player has the option to stay at whatever they want, where as, the dealer has to keep hitting until he gets between 17 to 21. Plus when it comes to a soft 17, the dealer is required to hit. So with more players at the table there are less cards available for the dealer to receive in order to get what he needs.