What's the difference between a tempo and a tempi? At the start, how many tempo/tempi is White up? On average, how many tempo/tempi is one Pawn worth?

Question

1. What's the difference between a tempo and a tempi?

2. At the start, how many tempo/tempi is White up? Half, one, or two?

3. At the start of the game (before White has even made his first move), how many tempi is one Pawn worth?** (For players rated near ~2200 Elo.)

Answer 1

1.What's the difference between a tempo and a tempi?

As others said, tempi is a plural of tempo.

2.At the start, how many tempo/tempi is White up? Half, one, or two?

In order to explain this, we must define what tempo is.

Unfortunately, this definition is incomplete, or we can say imprecise, in my opinion.

Tempo is the speed with which you carry out a plan, in my opinion. There is an optimal number of moves required to carry out your plan and there is the number of moves you actually played to carry out your plan.

If you carried out your plan in less moves than optimal, then you gained tempo somewhere. If you carried out your plan with greater number of moves than optimal then you lost tempo somewhere. If optimal number of moves is equal to those you played to carry out your plan than you played normal game.

Just a small clarification:

[Title "White obtains extra tempo"]
[fen ""]

1.e4 d5 2.exd5!? Qxd5 3.Nc3!

In the opening, plan is to introduce all units into battle as soon as possible. The fastest way to do this is to introduce 1 unit per move. Therefore this is the unit of measure for tempo in the above example.

With his second move 2.exd5` White lost 1 tempo because he played twice with the same piece in the opening, delaying his plan by one move. This means that Black will carry out his plan first if he keeps introducing 1 piece per move ( if he keeps tempo ).

The reason this line is played at the top is the fact that White can gain tempo by attacking the queen, thus forcing her retreat. This means that Black will be forced to delay his plan by one move too, which would restore "status quo" from the beginning.

To answer your question: White is tempo up in the initial position because he moves first, thus he can carry out his plan faster.

He will be faster than Black exactly by one move and this can be well illustrated in symmetrical positions. This is the main reason why GMs never play symmetrical openings as Black -> they can only draw and that will be tough to achieve. Also, some openings where White exchanges early, but there is no symmetry present, have confirmed this fact. This is the reason why Black can not "copy" White but must make asymmetric position.

3.On average, how many tempo/tempi is one Pawn worth? (For players rated near ~2200 Elo.) (I would need a very accurate estimation for this one, so please try to find an estimation as precise as possible.)

This is entirely position dependent. In the old days ( ~1970's ) it was argued ( roughly though! ) that 1 tempo = 3rd of a pawn so I will leave the math to you...

Answer 2

What's the difference between a tempo and a tempi?

Tempi is the plural of tempo.

At the start, how many tempo/tempi is White up? Half, one, or two?

At the start, White is not up any tempo. There is no such thing as a half tempo, because tempo refers to a single move from one side.

On average, how many tempo/tempi is one Pawn worth?

This is hard to say. In open positions, often even 1 or 2 tempi might be enough compensation for a pawn and 3 to 4 tempi may make one's position better. In closed positions, tempi don't matter that much, so in those situations 3 to 4 or more tempi might compensate for the loss of a pawn.

I'll give an example-

Consider this opening -

     [Event "2 Tempi gained by White"]
     [FEN ""]

     1. e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Bf5 5. Bd3 Qxd4 6. Nf3 Qd8

If you compare the position after Black's 4th move and Black's 6th move, Black hasn't changed its position; white has developed two extra pieces (2 tempi) and it's White's turn to move. However, White lost a pawn in the process.

I analyzed this position with Houdini 1.5 at depth 20 and here's the assessment -

After Black's 4th move -

= (0.28): 5.Ng3 Bg6 6.Be2 e6 7.Nf3 Nd7 8.0-0 Bd6 9.c4 Ngf6...

After Black's 6th move -

= (0.06): 7.Qe2 Nf6 8.0-0 Bxe4 9.Bxe4 Qc7 10.Rd1 Nbd7...

So, as per Houdini's evaluation White's two tempi was not sufficient compensation for the lost pawn (because the score reduced from 0.28 to 0.06.

Now, suppose I give White an extra tempi, so that the following position is reached with White to play -

 [FEN "rn1qkbnr/pp2pppp/2p5/5b2/4N3/3B1N2/PPP2PPP/R1BQ1RK1 w kq - 0 7"]
 [Event "3 Tempi for White"]  

Here, Houdini's evaluation at depth 20 is

 +/= (0.54): 7.Qe2 Bxe4 8.Bxe4 Nd7 9.Bf4 e6 10.Rfd1 Ngf6 11.Ne5 Qb6...

So, Houdini thinks White is better with the 3 gained tempi after the loss of the pawn.

Of course, this is just computer evaluation. From a human perspective, we can say that 3 of White's pieces are already developed and White is castled. Black has only one developed piece. None of it's kingside pieces have moved, so Black's king will be in the center for a while. White's rooks will quickly occupy the open files, so White will develop enough pressure. Thus, White is slightly better.

In open positions, my judgment (subjective, of course) is that a pawn is worth somewhere between 2-3 tempi, whereas in closed positions, it may be worth between 3-4 tempi (if the side with the tempi uses the tempi to open up the position or gain a similar advantage). Thus on an "average", a pawn is worth 3 tempi.

Answer 3

The general rule taught to beginners is that one pawn is worth 3 tempi although in practice it depends on many things like position and which pawn. Pawn sacrifices are usually made to open up the position and given the pieces of one side freer / quicker development. b pawns and g pawns seem to be particularly poorly valued and in some openings appear to be jettisoned on a whim ;-) while d and e pawns, the center pawns, are valued much more highly.

None of this takes any account of the positional sacrifice where there is no gain in tempi rather, as in some IQP positions, lines are opened up or points of attack switched.