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I would like to present my chain of arguments for the use of Bonus Time (also known as Fischer time).
First, I will list some facts to be used in the following argumentation. Then, I will present some properties that a timing system for Go should have. Finally, I will show how Bonus Time achieves them, and how other popular time systems fare in these respects.
For 19x19 Go, a game that is ended by counting takes on the average about 260 to 280 moves, but games of 200 moves up to 320 moves are not rare. When games are resigned, they can be much shorter, naturally.
Most people are surprised when they discover that endgame usually starts around move 100, i.e. more than half of a game's moves are endgame, even after they already played Go for several years. This is an indication that they have no idea about how many moves have been played at a certain point. Add to this the fact that they do not know the total number of moves the game will have had at the end, and you see that they must have a high degree of uncertainty about the remaining number of moves.
If you use absolute time, players will either under- or overestimate the number of remaining moves, under- or overestimate their time needs for the remaining moves, under- or overestimate their current time use, or not think about it at all.
Players who "never" lose on time under absolute time use a big margin of safety, usually taking much less time for their game than they would have been allowed to. Thus, they put themselves at a disadvantage. Optimizing this disadvantage against the disadvantage of a higher probability of losing on time can be seen as a game in itself. Note that losing on time under absolute time is often seen as "bad luck" because of this.
Most moves in a casual, untimed game setting are made within a few seconds. The rest can take up to several minutes (the distribution might look a bit like Planck's law). The average time used per move does not depend on the stage (i.e. opening, middle game, endgame) of the game (see Timing Systems Redux).
This is obvious. We use clocks to ensure that the players do not take forever for their moves, so that we can keep schedules, e.g. in tournaments, or in a casual game where we want to go home before the significant other becomes angry.
This is something that follows from the players' aforementioned inability to manage their own time. A time system should enforce sensible time management. This frees the players' minds for playing Go.
It may seem paradox, but if unused time is "spilled", the game will last longer. A player who has decided on a move, but knows that he can use some time now that will be lost if he does not use it, will try to fit some "thinking" into this time, thus deliberately delaying his move.
As an example, if you use "japanese byoyomi" with 30 sec per move, the players will make their moves at a pace of a bit less than 30 sec, even though they would play most moves within a few seconds if there was no clock.
When a player looks at the clock during the game, he wants to get an indicator for how big time pressure is for him at that point. The clock should give it to him directly, preferably in a single number without the need for further calculation.
Both players start with a basic time. After each move, they get a bonus time added to their clock. If a player's clock drops below 0, he loses on time.
The time settings are usually noted as "XX/YY", where XX denotes the basic time per player in minutes and YY the bonus time per move in seconds.
If a game has n moves, it cannot take longer than n times the bonus time, plus the basic time of both players.
We can calculate an estimate for the maximum total time that a game can take by estimating a sensible maximum total number of moves.
Example: For 19x19, 300 moves is a good estimate for the maximum total number of moves, and convenient because 300 sec are exactly 5 min. The maximum total time for a setting of XX/YY can thus be calculated in minutes as MTT = 2 * XX + 5 * YY. For "Bonus Time 30/20", this is 2 * 30 + 5 * 20 = 160 (2:40 h).
The "sensible minimum" can thus be directly set.
The player knows that he will always have at least the bonus time for each remaining move. The time he reads on the clock is the current reserve that he has additionally. No further calculations or readings are needed.
(Please note that this differs from the actual name-giving byoyomi that is used in the japanese title matches)
Each player starts with a main time. When a player's clock drops below 0 for the first time, he is allotted a number p of byoyomi periods of n seconds. From then onwards, after each move, his clock is reset to n seconds. Whenever his clock drops below 0, his remaining number of byoyomi periods is reduced by 1 and the clock reset to n seconds. If his remaining number of byoyomi periods drops to 0, he loses on time.
The time settings are usually noted as "XX min & YY * ZZ sec" or similar, where XX denotes the main time, YY the number of periods, and ZZ the time per period.
A game of n moves cannot take longer than (n + twice the number of byoyomi periods - 2) times the byoyomi time, plus the main time of both players.
We can calculate an estimate for the maximum total time that a game can take by estimating
Example: As before, I will assume that a 19x19 game has a maximum total number of moves of 300 moves. I have observed that a sensible minimum number of moves for the main time is about 100 (50 per player), and that a sensible average for the time used in a period might be about two thirds of the length of a period. The maximum total time in minutes for a setting of XX min & YY * ZZ sec can thus be calculated as XX * 2 + (299 + YY - 100) * 2/3 * ZZ / 60. For "60 min & 5 * 30 sec japanese byoyomi", this gives 60 * 2 + (299 + 5 - 100) * 2/3 * 30 / 60 = 188 (3:08 h).
The sensible minimum can thus be directly set, although the actual time available is a little smaller than that, depending on the player's timing skills.
A player will therefore try to use the remaining time for other calculations. The extent of this will vary from player to player, of course.
The first value is the time, the second is the number of byoyomi periods remaining.
During a player's main time, the time displayed is the remaining main time. This is free time, the player has no obligation to play a certain number of moves during this time. The time pressure during this is dependent on how fast the byoyomi is, and on how much main time remains. When this time reaches 0, it just marks the start of byoyomi.
After that, the time displayed is the remaining time before a byoyomi period is deducted. The time pressure is now dependent on how much byoyomi time is left in this period, and on how many periods remain.
(Invented to overcome the need for a third person as time keeper for japanese byoyomi, when electronic clocks were not yet available)
Each player starts with a main time. When a player's clock drops below 0 for the first time, his clock is reset to a byoyomi time and he is allotted a certain number of byoyomi moves that he has to make during this time. Whenever he has made that many moves, his clock and byoyomi moves are reset again to the same number. If his clock drops below 0 before he has played his allotted number of moves, he loses on time.
The time settings are usually noted as "XX min & YY moves / ZZ min" or similar, where XX denotes the main time, and YY the number of moves he has to make in ZZ min afterwards.
A game of n moves cannot take longer than (n divided by the byoyomi moves) times the byoyomi time, plus the main time of each player.
We can calculate an estimate for the maximum total time that a game can take by estimating
Example: For 19x19, the first two are, as before, 300 moves total, and 100 moves during main time. A sensible average for the third might be about four fifths of the time for each overtime period. The maximum total time in minutes for a setting of XX min & YY moves in ZZ min can thus be calculated as XX * 2 + ((300 - 100) / YY) * 4/5 * ZZ. For "60 min & 15 moves/5 min canadian overtime", this gives 60 * 2 + (200/15) * 4/5 * 5 = 183.3 (3:03:20 h).
This is not as bad as in absolute time, since the number of remaining moves in that period is known. Thus the player still has to manage his time, but only in little chunks.
A player who has played all but one move for a running overtime period will therefore try to use the remaining time for other calculations. The extent of this will vary from player to player, of course.
The first value is the time, the second is the number of stones to be played in the current overtime period.
During a player's main time, the time displayed is the remaining main time. This is free time, the player has no obligation to play a certain number of moves during this time. The time pressure during this is dependent on how fast the overtime setting is, and on how much main time remains. When this time reaches 0, it just marks the start of overtime.
After that, the time displayed is the remaining time for playing the number of moves given as a second value. The time pressure now depends on the ratio of these two values. Some players calculate this ratio constantly, typically just after making a move.
Obviously, all time systems guarantee that a game is finished in finite time. You can also calculate an estimate for the maximum total time in each presented system.
However, while under Bonus Time, the only factor you need an estimate for is the maximum total number of moves, you need additional factors under the other time systems that are immanent in these time systems. Therefore, the calculation has a much higher degree of uncertainty under these systems.
Note that this calculation is very interesting for keeping schedules.
Bonus Time is a clear winner in this category, the other two seem to share their place.
All systems greatly reduce the amount of time management the players have to do.
During their respective main time, both Japanese Byoyomi and Canadian Overtime give the players much freedom, more than they would have with a comparable Bonus Time setting: they are allowed not to make a single move during all the main time, but under Bonus Time, they have this freedom only as long as the basic time holds. Basic time under Bonus Time is usually only about half of the main time under a comparable Japanese Byoyomi or Canadian Overtime setting, since the bonus time is added from the start. Of course, not using the main time, or using it only for a small number of moves will be both regarded as "stupid" and frowned upon by any organizer, whose time calculation is thrown off by this. Thus, this is not really a shortcoming of Bonus Time.
During byoyomi, however, the players are more restricted by Japanese Byoyomi or Canadian Overtime. Both dictate a certain pace. If a player is slower, he loses on time, but if he is faster, it will not benefit him -- unused time is always spilled. Japanese Byoyomi is a tighter corset in this regard: each move has the same forced pace, with a number of exceptions equal to the number of periods minus one. Canadian Overtime allows variation during a period, but the average over a period is prescribed. When a player's time runs low under Bonus Time, on the other hand, he also has a minimum pace, but if he moves faster, it benefits him, building the reserve again.
Another way to look at Bonus Time is this: each player magically gets an absolute time, the amount of which is dependent on the total number of moves. This is magic because the total number of moves is not known beforehand. However, the clock forces the players always to keep enough time to make the remaining moves. This is the whole restriction.
Japanese Byoyomi and Canadian Overtime share their place again in this category, Japanese Byoyomi having more restrictions but also freeing the player from time management to a bigger extent than Canadian Byoyomi. Bonus time is the clear winner, again, having less restrictions than Canadian Overtime while freeing the player from time management to a bigger extent than Japanese Byoyomi.
For this category, all has been said before. Japanese Byoyomi and Canadian Overtime both spill unused time, Bonus Time does not and is thus the winner.
Both Japanese Byoyomi and Canadian Byoyomi divide the whole game into two phases: main time and byoyomi. In the second phase, the time pressure is dependent on two values, but these values have a totally different meaning from the value during main time. It should be quite clear that Bonus Time, whose single value always has the same meaning, is a clear winner here, too.
The conclusion is quite simple. I do not even have to weigh different aspects to proclaim Bonus Time the clear winner overall in this little study. I welcome a lively discussion (press "Discuss page" at the top), and perhaps you found other desirable properties that should be added.