How would two moves for the black after the first white move affect the odds?

In team sports when choosing teams there is a common practice that the second captain gets to choose two players instead of one. After that the process continues with one for each every turn until all the players are selected.

Quote from Wikipedia:

"In chess, there is a general consensus among players and theorists that the player who makes the first move (White) has an inherent advantage. Since 1851, compiled statistics support this view; White consistently wins slightly more often than Black, usually scoring between 52 and 56 percent."

I'm a novice chess player and my question is how would it affect the winning odds if black could make two moves after the first white move?

I'm going to answer this as if black moves first but then white moves twice. It's the same question but it's just easier to answer that way.

I think it would simply favor white because for example 1.d4 Nf6 is a top tier opening but 1.e4 Nf6 is not. Black also can't start with Nc6 or e5... and now that I think about it, I'm not sure what a good first move is... yeah I think it would favor the person who moves twice too much.

If you want to make the starting position close to equal then probably a better rule would be something like "white's first move has to be a pawn, and it can't move two squares forwards."

So white would typically choose between 1.b3, 1.d3, 1.e3, or 1.g3

That does make sense. But it also brings more questions into my mind.

1) How would that rule you mentioned affect the probabilities exactly?

2) With superhuman computers and great computing power, why can't we design a more equal start for a chess game? AlphaZero needed 4 hours to become a beast. Why couldn't we use it to find the best possible opening rules for a chess game as well while still maintaining the classic rules of the game?

3) Do you think any rule changes would ruin the history and culture of chess?

I have no way of answering that question.

We've already found equal positions in chess960.

In any case, "equal" and "best" are two different things. For example some equal positions are boring and others aren't.

Well, in one sense, the "history of chess" is the history of a game whose rules have changed over time to create a game rich in strategy and tactics (meaning both long term and short term considerations). I think this is a major reason it has maintained its popularity. For example the game "go" (weiqi, paduk) is rich in strategy. Shogi has tactics. Chess is a mix of both.

So changing rules is not necessarily bad.

However since white's advantage is so slight, it's not so important. More important would be a rule change that got rid of draws somehow. In other words every game would end as a win for one of the players.

It would just make white black and black white
E.g. 1 d4 d5 2 (void) c5

You didn't think about this very much did you?

1.e4 d5 2.(void) dxe

This is the easiest example. There are many openings that are eliminated.

#8
1 e4 d5 2 (void) dxe4 = 1 d4 e5 2 dxe5

The Englund Gambit isn't the Ruy Lopez.

So like I said... there are many openings that are eliminated... the player who moves twice has the advantage.... well... I guess after a move like 1.e3 or 1.g3 maybe it's not much of an advantage, but I think white may be restricted to openings like this.

#10
No not at all. White just has to avoid 1 e4. Nimzovich called 1 e4 premature in standard chess and it is more premature in this variant. If white just avoids 1 e4, white is black is OK.
1 d4 d5 2 (void) c5 = 1 d4 d5 2 c4
1 Nf3 e5 2 (void) e4 = 1 e4 Nf6 2 e5
1 c4 e5 2 (void) Nf6 = 1 e4 c5 2 Nf3

A far greater swing to black than currently enjoyed by white. Armaggedon is much better. White is worth 0.2 centipawns and 40% of GM games are drawn so give black the draw and a 20% time forfeit. The sytem has been active for 20 years and is dead level for a 5 minute decider. The concept could be divvied up to the same effect but who wants a County Championship round robin on neutral grounds? You take the rough with the smooth and a final Armageddon solution if necessary.

#12
Indeed white is like playing black but with fewer options.

1 e4 Nf6 2 (void) Nxe4
1 d4 c5 2 (void) cxd4
1 Nf3 g5 2 (void) g4

Are not appealing for white, however
1 c3 e5 2 (void) d5 = 1 e4 c6 2 d4 = Caro-Kann = OK
1 e3 e5 2 (void) d5 = 1 e4 e6 2 d4 = French = OK

Thus the black advantage in this variant is larger than the white advantage it is supposed to cure. Anyway, chess has a draw problem, not a white advantage problem.

Another variant could be: white makes 1 move, black makes 2 moves, white makes 2 moves and so on. This might be balanced.
  

An edge of even just 2-4% is huge in any field. Just ask hedge funds, casinos or poker players. If a non-skill-related advantage in any game can be removed with a reasonable solution, I think it should be done. Again, I think powerful engines could be used to explore different solutions.

I leave the draw problem for an another thread. But I just say that maybe chess could learn something from football (soccer) in that regard.