I used Katago to estimate the temperature of moves of 5 professional games. The median value increases from around 12 in the first move to almost 20 in the middlegame and only starts falling at the beginning of the endgame.

The value of having Sente is called temperature . You can roughly calculate the temperature by letting Katago analyze the current position, the temperature is the value of the position minus the value of the position after passing once. I chose 5 random professional games and manually went through them to estimate the temperature with Katago every 5 moves. The graph below contains the minimum, median and maximum temperature.

The value of the first move is roughly 12, not equal to Komi. Komi is roughly half the value of the first move.

The value of the next move changes throughout the game, but it does not fall monotonically. On average it increases from the opening to the middle game and only then falls as the endgame begins.

We need to define a few terms. Having the right to play the next move is usually called "having Sente". This right is valuable, as demonstrated by the fact that White needs Komi to make up for Black going first. The value of having Sente is called temperature . A Sente move is a move whose followup move is worth more than the move itself, like in the position displayed below: The endgame move is worth 3 points , but if White does not answer Black's play, the followup move is worth 6 points. Rational players would always answer a Sente move locally, because not doing so would give the opponent the option to play the higher-valued followup move.

You can roughly calculate the temperature by letting Katago analyze the current position, the temperature is the value of the position minus the value of the position after passing once.

I chose 5 random professional games and manually went through them to estimate the temperature with Katago every 5 moves. The graph below contains the minimum, median and maximum temperature at turn 0,5,10,15 and so on until move 200.

The value of the first move is roughly 12. You might expect the value to be equal to Komi, but that's a misunderstanding. Komi is roughly half the value of the first move: In the opening position, it's Black's turn, but White has Komi. If Black were to pass, White would be the first player to play a move, but White would still have Komi. Apart from the players switching roles, the only thing changed is that one player lost Komi and the other player gained Komi, and so the difference is twice Komi. But half of 12 is 6, so is the correct Komi 6? Katago thinks 7 Komi is correct and modern professional games also use 6.5 or 7.5. I'm not quite sure what's going on here, but it might just be a small inaccuracy in Katago's estimation.

By the way, the concept of a Sente move is harder to define or harder to make practical use of on the 9x9 board, because Sente requires a "local" answer. The 9x9 board is so small that essentially every move is a local answer and thus every opening move is kind of Sente. To really define Sente on 9x9, you would have to use the loss due to passing, but that is impractical. On 19x19 boards, playing tenuki is equivalent to passing, at least locally, which gives the concept of temperature practical value, because you can use it to figure out when it is time to ignore your opponent and take the initiative.

If there is overwhelming demand, I might make that graph again using a script, using more thinking time, for every move of 100s of games. The result should be a much smoother curve. I could even try to filter out obvious Sente moves by ignoring positions where the temperature increases by more than some threshold value. I think the shape of the curve wouldn't change much.

I recently found out that apparently many Go players are not aware of this: The value of the next move changes throughout the game, but it does not fall monotonically. On average it increases from the opening to the middle game and only then falls as the endgame begins. I wanted to demonstrate this with actual numbers.

Before we can start, we need to define a few terms. Having the right to play the next move is usually called "having Sente". This right is valuable, as demonstrated by the fact that White needs Komi to make up for Black going first. But Komi is only related to the value of the first move of the game (at least roughly - being precise would require a longer discussion) . In general, the value of having Sente is called temperature (or read here ) , a term from combinatorial game theory. Confusingly, a move that preserves the right to play is also called a Sente move (in chess terms: a move played with tempo) , but more accurately a Sente move is a move whose followup move is worth more than the move itself, like in the position displayed below: The endgame move is worth 3 points per played stone , but if White does not answer Black's play, the followup move is worth 6 points. That roughly means that it is correct for a rationally playing player to answer a Sente move locally, because not doing so would give the opponent the option to play a much bigger move. In other words: Rational and impossibly strong players (who don't make mistakes) would always play the move with the biggest value and playing a Sente move means that the biggest move is necessarily a local answer to the Sente move. If we try to be more precise, then of course we have to mention the option of answering a Sente move with another Sente move whose followup is even more valuable, leading to a potentially ever escalating series of moves that ignore the previous Sente move by making the game hotter. But at least in the endgame, this does not work forever. The board fills up, move values tend towards zero on average and consequently every Sente move eventually has to be answered, even if that answer is delayed by an intervening sequence of other Sente exchanges. In the opening, it's a different story. As the graph below shows, the temperature keeps increasing throughout the opening and early middle game.

You can roughly calculate the temperature by letting Katago analyze the current position, noting the estimated value in points (not the winning probability), then passing, analyzing the position again and then noting that value. The difference between these two values is roughly what you lose by passing. That works because Katago always has to take into account temperature when it makes its estimates, it does not just count territory. Without knowing that value, Katago wouldn't be able to choose between two different followup positions correctly. In terms of Chess, not knowing the temperature would be like avoiding all queen sacrifices, because you must move across a position where your queen has already been sacrificed, i.e. is missing. A short-sighted evaluation considers this a catastrophic loss and avoids it at all costs. But a deeper evaluation of course considers what happens after the sacrifice and thus figures out that sometimes even sacrificing a queen is justified by the compensation you can get. In Go terms, this short-sighted view would be completely ignoring the value of having the right to move and would estimate the value of the empty board as zero (because nobody has territory yet) , even though Black has the initiative and thus a big advantage. To use an even more drastic example: On a 5x5 board, the first move (on tengen) kills all potential White stones on the board, making 24 points of territory using Japanese counting (25 using Chinese counting) . The short-sighted view would state instead that it does not matter who moves first.

I chose 5 random professional games and manually went through them to estimate the temperature with Katago every 5 moves. The graph below contains the minimum, median and maximum temperature at turn 0,5,10,15 and so on until move 200. I rounded the values, so the values are integer numbers. There is also a small error because Katago isn't perfect and only had finite time for its estimates, so don't consider this graph the gospel truth yet.

As you can see, the value of the first move is roughly 12. You might expect the value to be equal to Komi, but that's a misunderstanding. Komi is roughly half the value of the first move as can be seen with a simple symmetry argument: In the opening position, it's Black's turn, but White has Komi. If Black were to pass, White would be the first player to play a move. If you realize that the choice of player color is arbitrary (you could just change the rules so that White goes first) and does not change the game in principle, the position after Black having passed is exactly like the initial position, except that the other player now has Komi. Passing thus changes the position from one where the second player has Komi to a position where the first player has Komi. Thus passing makes the first player lose Komi and the second player gain Komi and so the difference between these two positions is twice Komi. But half of 12 is 6, so is the correct Komi 6? Katago gives equal winning probability, i.e. a fair game, to a position with 7 Komi and modern professional games also use 6.5 or 7.5. I'm not quite sure what's going on here, but it might just be a small inaccuracy in Katago's estimation.

By the way, the concept of a Sente move is harder to define or harder to make practical use of on the 9x9 board, because the definition of a Sente move mentions a "local" answer. Because the 9x9 board is so small, it is actually not really possible to play a non-local answer - every move has the potential to influence the entire board. Using the usual definition, that makes every board on the 9x9 board a Sente move, at least in the opening and middle game. On 19x19 boards, the concept of Sente is only useful because the range of influence of most stones is limited and you can always play tenuki. But even on the 9x9 board, you could ask how much you lose when passing. That is a way of defining the temperature from a theoretical point of view, but it is not practical. On the 19x19 board, playing tenuki and passing is the same locally, which gives the concept of temperature practical value, because you can use it to figure out when it is time to ignore your opponent and take the initiative.

If there is overwhelming demand, I might make that graph again. I could write some code to estimate the temperature with Katago automatically and then use it to estimate it for every move, not just for every fifth move. I could also increase the analysis time per move. I could then use it on 100s of professional games and average that. The result should be a much smoother curve. I could even try to filter out obvious Sente moves by ignoring positions where the temperature increases by more than some threshold value. The point is that even when filtering out these moves, the temperature should still rise during the opening and early middlegame, showing that this is not just the side effect of having Sente moves in the data. You could say that during the opening and early middlegame, most moves are "kind of Sente", but you have far more options for replying, both locally and non-locally, than you would expect from your usual experience with Sente moves.