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 Backgammon

Backgammon and variants.

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17. Iulie 2008, 00:09:23
playBunny 
Subiectul: Re: Doubles theory
wetware: No dice change here, Thad...as far as I know :-)

That's because you misunderstood Thad's post! He was supposing that in the existing game, with the existing dice, the receiver of the most doubles has the greater chance of winning. It's an intuitively appealing idea but he's asking whether there's any empirical data to confirm it. That's why he gave the post the title Doubles Theory rather than Dice Cheating.

16. Iunie 2008, 17:54:19
playBunny 
Subiectul: Re: Resigning and backgammons
nabla: The parallel is that in chess [...]

Yes, I see what you mean now. It's conceptually clean but I still wouldn't implement it because In chess it's game over, while gammons and backgammons occur within the context of a match. My priority for resignations of part of a match would be to score fairly rather than simplistically.

Who would like a sophisticated software to decide about the result of a game ?

Dailygammon already has this in the resignation code. The game is resigned as soon as the outcome is known, with exceptions, I believe, only accounting for the human aspect of playing.

16. Iunie 2008, 00:57:20
playBunny 
Subiectul: Re: Resigning and backgammons
nabla: In chess, if your opponent loses on time and you have one single pawn against an army, you still win the game (if you don't have the pawn, it's a draw

That is an interesting point. You tell me that In chess it's conditional, so why not backgammon? I wonder whether they'd change the rules if there were more than one point at stake? When resigning a chess game the player loses one point because that's the only option. That not the case in backgammon which is precisely why the question remains about when to award what.

15. Iunie 2008, 13:24:02
playBunny 
Subiectul: Re: Resigning and backgammons
nabla: I would discount Triple Gammon anyway as it is a flawed format given that timeouts and resignation have such an impact. Resignation of Triple Gammon matches simply should not be permitted. I don't know what the solution for timeouts is but they can royally screw it up as well. It would be better if there was some way to prevent them or score them fairly.

've no doubt that there are exploits in regular backgammon matches. But even if one were to occur in a tourney final, I still maintain that the excessive potential backgammons would far outweigh the missing actual backgammons.

I think that both ways of doing it are less than conceptual clean but the gammon one is fairer overall in terms of matching unresigned outcomes.

15. Iunie 2008, 00:52:59
playBunny 
Subiectul: Re: Resigning and backgammons
saeco: he resigned the game and lost a gammon. how is it not an example for the possibility to do just that?

If you had agreed with me then there would have been no query but you agreed with nabla who qualified what I said with the claim that backgammon gets lost if there's still contact. It thus seemed as if you were saying that you had an example where backgammon was awarded.

Frankly I'd be surprised if there a contact-derived backgammon . I don't think BrainKing does that and I don't agree with nabla that it should do that either. If you've escaped your opponent's home then a backgammon is very unlikley. It's by no means guaranteed even with 4 men on the bar. I think the number of generously awarded backgammons simply because there's contact would far outweight any lost points from players who might have won a backgammon but only get a gammon.

14. Iunie 2008, 17:37:58
playBunny 
Subiectul: Re: Resigning and backgammons
saeco: How is that an example?

13. Iunie 2008, 16:54:59
playBunny 
Subiectul: Re: Wow, I learn something new everyday!
saeco: i think that a resign means you lose a backgammon as long as you haven't moved out any piece

You mean moved a piece off the board? No, it's a backgammon only if there are pieces in the opponent's home or on the bar and none of your own already borne off.

iirc, you can't resign and lose a gammon

If you resign when you're clear of your opponent's home then you'll be asked to confirm losing a gammon.

12. Iunie 2008, 02:27:55
playBunny 
Subiectul: Re: Wow, I learn something new everyday!
coan.net: As a side note for cube games, I'm pretty sure you are given the points where the current pieces are

You are.

...so a early resignation will lose the person a gammon.

It'll be a backgammon until the backrunners are clear of the opponent's home and that's often way into the mid game and later.

21. Mai 2008, 23:16:01
playBunny 
Subiectul: Re:"Get over it"
alanback: Wow, that's rather uncharacteristically snappish! Chill out Karma Daddy-O!

14. Martie 2008, 15:11:55
playBunny 
Subiectul: Re: MATFixer
Gordon Shumway: Thanks, the Internet Explorer forum's back to normal!

You're welcome. I thought it was overlkill for posting here but if you'd like me to send you the full text of import.c that can be done.

13. Martie 2008, 19:09:54
playBunny 
Subiectul: Re: MATFixer
Gordon Shumway: Might I suggest that you edit your post and split those long lines in the [pre] block? Now that we understand what they're about, they don't need to be correctly formatted. As they are they force the page to be scrollably too wide.

13. Martie 2008, 00:40:02
playBunny 
Subiectul: Re: MATFixer
Gordon Shumway: The segment within brackets is parsed as a Snowie .txt file. The following is taken from import.c of the GnuBg source code.
/*
* Snowie .txt files
*
* The files are a single line with fields separated by
* semicolons. Fields are numeric, except for player names.
*
* Field no meaning
* 0 length of match (0 in money games)
* 1 1 if Jacoby enabled, 0 in match play or disabled
* 2 Don't know, all samples had 0. Maybe it's Nack gammon or
* some other variant?
* 3 Don't know. It was one in all money game samples, 0 in
* match samples
* 4 Player on roll 0 = 1st player
* 5,6 Player names
* 7 1 = Crawford game
* 8,9 Scores for player 0, 1
* 10 Cube value
* 11 Cube owner 1 = player on roll, 0 = centred, -1 opponent
* 12 Chequers on bar for player on roll
* 13..36 Points 1..24 from player on roll's point of view
* 0 = empty, positive nos. for player on roll, negative
* for opponent
* 37 Chequers on bar for opponent
* 38.39 Current roll (0,0 if not rolled)
*
*/

19. Februarie 2008, 05:55:56
playBunny 
Subiectul: Re: hit or run ?
alanback: But of course.


Rollout 1296 trials, 2-ply for 12 plies, 0-ply thereafter
# .. Move ......... Equity . (Diff) .... Win ... WinG .. WinBg ---- Lose .. LoseG . LoseBg
1 .. bar/19*/15 ... +0.976 ............ 98.8% .. 66.1% ... 4.2% --- 1.2% ... 0.0% ... 0.0%
2 .. bar/15 ....... +0.954 (-0.022) ... 97.7% .. 43.9% ... 1.1% --- 2.3% ... 0.0% ... 0.0%
Std Err ............ 0.001 ............. 0.1% ... 0.6% ... 0.4% --- 0.1% ... 0.0% ... 0.0%

And if the move is bar/15 followed by 42: 12/10*/6 the chances drop:

Rollout 1296 trials, 2-ply for 12 plies, 0-ply thereafter
# .. Move ......... Equity . (Diff) .... Win ... WinG .. WinBg ---- Lose .. LoseG . LoseBg
1 .. 12/10*/6 ..... -0.930 (-0.046) .... 3.5% .. 0.0% ... 0.0% --- 96.5% .. 27.1% ... 0.2%

19. Februarie 2008, 02:59:32
playBunny 
Subiectul: Re: hit or run ?
alanback: there's nothing to be gained by hitting.

Sure there is. Fun!

Your one goal is to get past his back men, and hitting makes that harder, not easier.

Is that true, though? It's not just getting past his back men, it's getting all his men off before the opponent does. Failing to hit gives 19 pips to the opponent. How many rolls would that save the opponent in equalising and getting ahead? That must be weighed against the extra chances of 4 men and a one-point table detaining the blot sufficient to win.

Actually, one possible continuation after not hitting is that the opponent rolls 2-4 and moves 12/10*/6 to make the 6-point. But even if it's a non-hitting 6, that's the 6-point made. Thus, after not hitting, 17 rolls will give an immediate two-point home table. That's four times the chances of dancing, and the splendid 6-6 off the bar is lost.

there's nothing to be gained by hitting.

Sure there is. Wins!

18. Februarie 2008, 15:11:28
playBunny 
Subiectul: Re: hit or run ?
Hrqls: Hit everything and fry up some gammon! except that gammons don't count, so just hit and have fun. It's Hypergammon with 1 man versus 4. He doesn't stand a chance.

16. Februarie 2008, 11:15:34
playBunny 
Czuch: Yep, for turn-based play it would be DailyGammon. For live play it's probably GamesGrid. I've never played there but I understand that several World Class players have an account there, Paul Magriel, for instance.

9. Februarie 2008, 00:09:18
playBunny 
Subiectul: Re: The rating chaos laboratory
Andersp: Hope you can accept my apology for not noticing what you've been up to...sorry .

Lol. Apology accepted if given. It's hard to tell with you when you're such the joker.

I wasnt thinking about you at all when i asked Alan about sophisticated, but all of the sudden you popped up.

You're the one who applied the "sophisticated" label to me. My contribution was about "High vs Low", specifically "players who prey exclusvely on the low rated players" and that could have been made by anyone. The rest was the usual you-and-me banter.

9. Februarie 2008, 00:00:37
playBunny 
Subiectul: Re: High and low
Andersp: So you study backgammon, but play Ludo?...was i right this time?

Yep, I just move the pretty coloured pieces around the track.

But I do have to admit that I've looked deeper into the game .. http://brainking.com/en/Board?bc=3160

8. Februarie 2008, 23:58:01
playBunny 
Subiectul: Re: The rating chaos laboratory
Modificat de playBunny (9. Februarie 2008, 00:04:17)
Andersp: I didnt notice that you are on strike until today,

I've said it here several times in the past. It makes sense that you'd continue to complain about me not playing (you've certainly delighted in noticing that much) if you've never heeded my reasons.

Has "the Man" changed the system or does he spoil your fun?

No, no, getting the Man to make changes is your task and for your comfort and joy (at least so you won't have to complain). It used to spoil my fun until I discovered DailyGammon and made it my home. I don't care what he does anymore. I have my Ludo matches to play ... for the next twenty years. There's complete fulfillment in that.

8. Februarie 2008, 23:51:42
playBunny 
Subiectul: Re: High and low
Andersp: but you agree that you are sophisticated? :)
Reply (box)


Certainly I have my share of expertise. Having studied the game at the level of obsession for three years I should do! Yet I have a huge amount yet to learn, especially cube and equity mathematics. When I understand those aspects then I will consider myself sophisticated. But it's a relative term. Compared to someone who plays but doesn't study then you could say that I'm already sophisticated.

8. Februarie 2008, 23:41:19
playBunny 
Subiectul: Re: The rating chaos laboratory
Modificat de playBunny (8. Februarie 2008, 23:43:22)
Andersp: Wrong..i just suggested that we should have one rating for us brave players and one for the sophisticated....

Have you so conveniently forgotten that you whinge at every opportunity about high raters not playing low rated ones and have been doing so for at least two years?

cant be so fun to know that you keep your rating because you dont play any player, is it?

LOL. It sure seems to spoiil your fun, Mr "I Don't Care About Ratings"!

My lack of playing is a protest against the crap rating system. It's as simple as that. If you're so keen to see me play then get the Man's fingers busy changing the rating code.

8. Februarie 2008, 23:34:53
playBunny 
Subiectul: Re: High and low
Andersp: just checked his profile...but he also consider himself being a top player in backgammon

You've got to learn to read accurately. The top players at Dailygammon represent considerable expertise beyond mine. My profile says if you think I'm good (because I'm top rated here) you should consider that I'm only about the 50 mark at DailyGammon. "Top in backgammon" and "Top in ToyTown" are very different things!

8. Februarie 2008, 23:29:45
playBunny 
Subiectul: Re: The rating chaos laboratory
gogul: The rating system applied to Ludo - that sucks more than with backgammon!!

8. Februarie 2008, 23:27:17
playBunny 
Subiectul: Re: High and low
Andersp: Apparently happy with the rating system yet somehow desperately complaining about the high rated players!

8. Februarie 2008, 23:11:27
playBunny 
Subiectul: Re: High and low
Andersp: No, it works fine having one site for you lot and another, sophisticated, site for us.

8. Februarie 2008, 22:41:38
playBunny 
Subiectul: Re: High and low
Andersp: i think the BKR system should be more fair if all players played low rated players too

If there's a proper rating system put in place then you'll find a couple of players who prey exclusvely on the low rated players. Will you complain then as well, at these rating manipulators?

23. Ianuarie 2008, 00:42:36
playBunny 
Subiectul: Re: brearing off
Hrqls: are there also positions in which its better to move only 1 piece into you home because it covers an empty spot (and leaving 1 outside your home)

Most definitely. It's more about the distribution than simply leaping over the bar. As nabla says, the 6-5-4 points are the most important. If you're stacked on 6 and 5 or 4 are empty then a roll that stacks the 6 higher is no good to you. You'll only waste rolls later unstacking it (unless you get 6-6). As you'll have no control over those later rolls but right now you have a known roll, why not place the checkers usefully.

Here's an example that I've just made up:
http://www.gellie.plus.com/DG_Extras/MakeBoard/ShowBoard.htm?MatchId=cIkmAAAAAAAA&PosnId=3HauAADc/CUBAA


With the 5-1 you could add two more men to the 6-point. Of the 20 moves shown in GnuBg, the 11/6 7/6 stacker is 17th and a Bad move (-0.48). It's much better to do 7/2 6/5. Putting one on a low point is greatly preferable to stacking. But the best move (it just edges it out) is 9/4 6/5. This builds on the 4-point, which is important because it's very weak in relation to the 6-point, and unstacks the 6-point a bit.

One additional point about the 4-point compared to the 3-point. If you roll a 3 and there's a hole on the 3-point then 6/3 will fill it. The next 3 will thus take a man off. If you have a hole on the 4-point and roll a 4 then you have to move a man deep from 6 or 5 and there's still a hole on the 4-point. The next 4 would thus waste another half a roll. The same applies to the 5-point but it's somewhat less troublesome. When the 6-point is cleared then a hole on the 5-point disappears automatically but a hole on the 4-point will still need the 5-point to be cleared before it disappears.

22. Ianuarie 2008, 16:38:22
playBunny 
Subiectul: Re: brearing off
Hrqls: I would definitely do 6/4/off.

If you do 6/1 2/off then a 2-1 on the next roll would leave an odd number of men and therefore be a wasted roll. That would be an unnecessary gift to your opponent.

If there had been one more man on the ace point (or one less) at the start of the 5-2 then it would make no difference which you did. The 2-1 would reduce your bearoff requirement by one roll whether one man came off or two.

22. Decembrie 2007, 21:31:22
playBunny 
Subiectul: Re: Woohoo, BK's got a bot!! :-D
nabla: I'd tend to agree that a bot should be clearly labelled as a bot. At Fibs it's not always clear but, as there's a clearly established bot ethos, it doesn't really matter too much, especially as there's a full range of abilities exhibited by them. "Bot" doesn't mean perfect by any means.

As for tournaments, well, you know my interest in getting DG "bot standardised", but here, where the rating system is *ahem*, there is no value in that. Even if there were, tournaments wouldn't be ideal anyway as the players are self-selected and so uniformity of the pool is absent. And so I'd agree that there's no need for it to play tournaments. I'd almost suggest that a bot should only play unrated except that people could have fun playing one-pointers against it and sucking points out of it! ;-)

22. Decembrie 2007, 21:01:44
playBunny 
Subiectul: Re: Woohoo, BK's got a bot!! :-D
puckish: I agree completely with wetware and alanback. Getting a bot's-eye view of a position only takes a few minutes, if that, so the longer someone takes the more likely they are to be having a personal ponder. I've sometimes visited and revisited a game several times over a few days (DailyGammon time controls permit this), each time deciding to defer the decision until it becomes clearer (or rather for me to become clearer, lol. It's amazing how one day's foolish "Yeah, I'll get the rolls" optimism becomes another day's "That would be madness!" rational thinking. )

22. Decembrie 2007, 04:12:21
playBunny 
Subiectul: Woohoo, BK's got a bot!! :-D
I think it's wonderful!

Not everyone has a bot at home. At VogClub there is a room just for playing bots (there are three, varying from walkover to mean, cruel bastard) and it's well attended. At Fibs there are anything between half a dozen and a dozen bots and they get a lot of attention. This place needs a bot!

A programmatic interface makes the most sense but if some human fancies typing in moves for the bot then that's great. Three cheers to Botman!

Mind you, a proper bot should accept all invitations from any player, rated or unrated, and for any match length or type.

17. Octombrie 2007, 18:49:50
playBunny 
Subiectul: Re:
coan.net: (But as a side note - since my post, no new points in the discussion has come up - just the same ones... wow, who could have predicted that.)

A discussion trailing off isn't necessarily vindication of a "why don't you shut up?" post, it's quite possibly a sign that you piped up redundantly. So who could have predicted it? Anybody.

I'm sorry that you think this is argument rather than discussion. This is the liveliest that this board has been for a long time. Perhaps a sleeping board is preferred?

17. Octombrie 2007, 05:08:45
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
Czuch: Well, all I have to say about that is insert what I don't have to say.

17. Octombrie 2007, 05:06:41
playBunny 
Subiectul: Re: Show me the dice!
alanback: Yes, it's a word game. You made a statement about postponement of gratification, ie. that it was applicable to this dice situation, and I asked a question about why it is applicable. You've done everything except answer that question, talking instead about ego and Spirit. That strikes me as playing word games.

17. Octombrie 2007, 04:58:35
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
Czuch: Only better because we would not have any wonder if the dice rolling is completely random and fair,

How does knowing what the dice are sooner than later inform you about their fairness or crookedness?

17. Octombrie 2007, 04:56:33
playBunny 
Subiectul: Re:
Modificat de playBunny (17. Octombrie 2007, 04:59:12)
coan.net: Is there anything else we need to say about this?

Speak for yourself. Do you need to say anything? If not then be silent. As for other's needs to say anything? You either have a need to know ASAP whether they do or not - or you are telling others that you have a need for them not to say what they have to say.

16. Octombrie 2007, 23:50:04
playBunny 
Subiectul: Re: Show me the dice!
alanback: You're just playing word games, Alan.

IMHO it's a silly thing to be concerned about, but if most people want it, it should be implemented. ...... Just more evidence of our inability to postpone gratification :-)

You introduced the ideas of gratification and postponement of gratification. It was your given!

16. Octombrie 2007, 15:00:11
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
Czuch: No, I tend to state a position and ask questions as accurately as possible and that way round is not what Alan was suggesting.

16. Octombrie 2007, 11:38:07
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
alanback: Hmmm, I wasn't querying whether there was a need for gratification - it's a given. My query is about its postponement in this instance, namely the benefits of postponement versus the harms of immediate gratification.

15. Octombrie 2007, 23:14:56
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
alanback: Just more evidence of our inability to postpone gratification :-)

Isn't postponement of gratification best done when there's reason to; some benefit from delaying? I'm not sure what that benefit would be in this instance, nor what harm comes from knowing what the dice are when you leave the table.

15. Octombrie 2007, 15:50:44
playBunny 
Subiectul: Re: Show me the money. Er, I mean show me the dice!
It's obvious to anyone who has played at GoldToken or Dailygammon and been glad that they can leave the board knowing the dice that have been rolled that it's a Very Good Thing.

8. Octombrie 2007, 23:22:59
playBunny 
Subiectul: Re:
Czuch Czuckers: Gosh, Czuch, I certainly do enjoy a good debate but please don't ever think that I agree with you about being rude to people simply because they have a different preference. It is almost as if you view other people as machines. I know you're incapable of understanding this but I seriously think that the attitude you show is self-centred and boorish!

But I dont understand how you can defend someone being offended or upset over someone else playing within the boundaries of the rules

There's an attitude which I truly and wholeheartedly despise and it's the traffic-warden mentality that "It says so in the rules therefore I don't have to think or exercise any judgement at all. I can act just like a robot and be correct according to what it says here". If you take that position in anything where human judgement could be employed you have thrown away any chance of my respect.

That's where I'm coming from.

Anders, on the other hand, is just along for the laugh.

8. Octombrie 2007, 21:41:17
playBunny 
Subiectul: Re: Anders, your point is...?
Andersp: but to me its still more rude than to resign a finished game :)

But that really is irrelevant. It's not about what you'd find rude, it's about what an opponent who wishes to continue play would think.

i (and most players here) know the real reason why you dont accept even a friendly invitation

Go on then, spell it out. Make sure you properly include the terms "in protest", "rating system" and "really sucks". And you can get bonus points if you add another sentence or two incorporating "DailyGammon" plus some choice ideas from my profile.

8. Octombrie 2007, 21:29:34
playBunny 
grenv: Yes, this is my way.. I resign in a lost position. I never said others had to do the same.

Nor did I say that you said that. But you have said that you do not accept their point of view. You've said that their definition of the end of the game is wrong, that their wanting to finish by removing the last piece is pointless, ridiculous, etc, and that you'd resign regardless of their wishes.

you are saying that others might be offended at my actions and request that I play out the game. Wouldn't this be a case of them asking me to do it their way, not the other way around?

If you know that they'd like to continue play and you resign in spite of that then it could be considered rude and, yes, they may well be offended.

If you say to them that it really pains you to have to play on then you are asking them to take your feelings into consideration. If they take you at your word and assume that it truly does "pain you", and they would only "quite like" to continue, then it would be inconsiderate to expect you to play (not that they can force it, of course). In that case the reasonable way forward is to accept your resignation.

It's very likely to be the case that they will let you have your way even if they believe that their preference is the stronger - because they also value being the "nice guy" about things. Probably most people are like that but I get the impression that you wouldn't be one of those.

Difficulties arise when both players feel adamant about getting their way and that's when you could call in the judge. In reality, however, the likely outcome is that the one with the power, ie. the resign button, simply exercises it, and rudeness be damned. Indeed, rudeness may well have already surfaced during the "negotiation".

It's about mutual respect. It's not about simply dismissing the other person because you think their position is ridiculous.

8. Octombrie 2007, 21:27:03
playBunny 
Subiectul: Re: Anders, your point is...?
Andersp: Oh im sorry i misunderstood...so you accept a friendly invitation for backgammon?

Lol. Yes, you have misunderstood and I dare say you know it, too.

It's very clear what my position is. It's completely certain that I will reject (by ignoring) a rude invitation (defined specifically as one with no text whatsoever). And it's very unlikely, in the case of Backgammon, that I will accept even friendly invitations. "Very unlikely" is, of course, not measured as 0%. This therefore means that some percentage of friendly invitations would be accepted.

think you should add that to your profile

It's all there in one short sentence. All you have to do is not misunderstand it!

8. Octombrie 2007, 20:54:08
playBunny 
Subiectul: Re: Anders, your point is...?
Andersp: To me its more rude to not accept a friendly invitation on a gamesite

Really? You feel obliged to play any game that someone sends an friendly invitation for, regardless of whether you want to play that kind of game or not??

8. Octombrie 2007, 18:41:18
playBunny 
Subiectul: Re: Etiquette
Czuch Czuckers: Answer this then.... Do you think ...

Nope, I won't answer that then or even now. It's a whole new topic. Repost it tomorrow.

8. Octombrie 2007, 18:39:15
playBunny 
Subiectul: Re: Etiquette
grenv: as in the law where the test is always "the reasonable person". If someone is unreasonably offended that is their problem not mine.

In Law we don't have the plaintiff acting as prosecutor, witnesses and judge as well. Whose definition of "reasonable" is the reasonable one here?

8. Octombrie 2007, 18:34:41
playBunny 
Subiectul: Re: Is this rude?
Andersp: "Even with a friendly greeting, I am very unlikely to accept Backgammon invitations."

That's from my profile, for those that don't know. You've quoted it out of context, Anders, which is quite inconsiderate of you.

The full quote is: I consider it rude if I don't know you and you invite me to a match yet say nothing in the invitation. I will ignore it until you take it away. ...... Even with a friendly greeting, I am very unlikely to accept Backgammon invitations.

And, people being people, some will take it personally and be offended, even though it expresses my general preference for not playing Backgammon.

Also from my profile: Check out the Play Bunny of the Month for May 1999. Isn't she a cutey? :-D

People being people, some will take personal offense to that.

Anders, your point is...?

8. Octombrie 2007, 18:25:25
playBunny 
Subiectul: Re: Would your opponent think it rude?
Andersp: do you think its a good idea that we ask Fencer if he can give us an option to add a "dont resign mark" to our nickname?

Asking Fencer for anything to do with backgammon is a long and hard struggle. In this instance we can ask as long as it "we" means Andersp!

....i've rsigned many games and no one has complained so why should i think that they are offended?

For you there's every reason to think that not one of them is even the slightest bit offended, ever. That even includes the ones who you know, for a fact, would prefer to play on (they being the ones that we've been discussing).

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