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However this only concerns one part of the problem: dropping unclaimed amulets. And that's actually only a small problem. Even during the launch contest, most games will be amulet games whenever there are less than 75 active players because of the 2N+3 players rule (number of amulet equal to about half the number of players active in last two minutes).
Agreed. That was the reason for my suggestion for selecting amulet challengers in the same way. I guess that suggestion may be unappealing because it is a larger change. For what it's worth I think the positives of my suggestion are worth the size of the changes; I doubt we're going to find smaller changes which don't distort the main game more. But hey I'm not the one who has to code them! So here are some smaller changes which I think deal with the issue to some extent I prefer them less:
1. Amulets should only be dropped into the player pool when there are at least 4N players, maybe even 6N or 8N, the original idea after all was that these would be rare. A lower density of amulets makes it harder for colluders and creates smoother player experience (because it is easier to find games).
2. Go back to the idea of having the random delay be a counter on games joined which pulls a player joining or having their game joined (call either of these players a "joiner") once the counter reaches a randomly determined N.
[To determine N I'd suggest is something like the following: take the number of games in progress, G, and the number of amulets A, then let M=G/max(4,min(A,20)) and choose N uniformly from the numbers 1 to M. The idea is assuming that games take on average 20 minutes, there'll be G/20 games being joined per minute, so M=G/4 gives an expected wait time of 2.5 minutes, while M=G/20 gives an expected wait time of 30 seconds. The reason for having the expected wait time vary with the number of amulets is that it is easier to collude with less players and so there is more reason to extend wait times to avoid it, in fact you may well want to extend it beyond 2.5 minutes.]
3.Have each player only *considered* for an amulet game once every 20 minutes. More precisely each player has a cooldown which starts at 0. When a game-joining causes an amulet game counter (ie a counter as above) reaches N, if both joiners have cooldowns of 0 one is selected at random to be pulled into the amulet game, if only one has a cooldown of 0 that player is pulled and if neither, the process is repeated on the next game-joining. Whenever a game-joining causes an amulet game counter to increment, once it is determined if either joiner is to be pulled into an amulet game, any joiner whose counter is at 0 is reset to 20. Finally every player's cooldown reduces by 1 every minute.
This really hurts the colluders spamming games against each other because now they don't get any benefit over players playing one game every every 20 minutes. What about an amulet holder trying get into a game with an alt, though? They can still do it reliably so long as the have 2M alt accounts. So for example if there are 1000 active players G=500, M=500/20=25, so they'd need 50 accounts. If there are 100 active players, G=50, A=floor(50/4)=12, M=4, so only 8 accounts. They will aslo need an extra account for every extra game they want to play in a 20 minute period, but that wouldn't be a significant concern for them.
Ok so time delay is better for preventing someone trying to get into a game with an alt, if you replaced the game counters with random time delays of up to 1 minute with large numbers of amulets in play and up to 5 minutes when there's only a few then a player with the numbers of alts I mentioned above I believe would only achieve a 66% chance of getting paired with one of their alts. You could still implement my idea, with a time based delay: a player's cooldown just gets reset every time they join a game while an amulet holder is waiting for an amulet game. 66% still sounds like a pretty high chance of getting paired with an alt to me though, and I don't know how to reduce this further with this sort of system.
My previous suggestion did much better than this at foiling colluders trying to get into amulet games with each other. When challengers are chosen from game winners, you basically need 66% of non-amulet holders to be alts for you to get a 66% chance of being paired of getting into an amulet game with one (you can do a little better than this, depending on how much flexibility is given to amulet challenger on when they offer their games, and some other details) but (I claim) you can't do significantly better. A second benefit of my original suggestion over this current one (apart from benefits mentioned previously) is that it doesn't have an arbitrary cut-off at the 20 minute mark. The arbitrary cut-off will create some weird player incentives, where there is much more motivation to finish a game quickly after the 20 minute mark, and if a game takes 15 minutes you've got some reason to wait 5 minutes before joining another game.
With Storeroom's suggestion, amulet games could be set at a more meaningful stake, say one or two dollars. Those games would probably be hard-fought, using more conventional strategies, without insane, arbitrary all-ins.
I also like having the fixed stake be $1. Having stakes which are in some sense meaningful is more in the spirit of the game as you say, and I think there should be some encouragement to play at at least slightly higher stakes than the typical 1-10c stakes we've mostly seen pre-launch. I would expect some protest though, and I'm a bit worried about it being intimidating to some players; Getting a chance to play for an amulet should feel like a reward for winning your game, not a threat to money. My response would be that the maximum you can lose trying for an amulet this way is $1 per opportunity, and I doubt there will be many people playing who cannot afford to risk $1 every now and again.
Finally, there's the issue that the core game might change too much with this "Amulet Challenger" idea. This will be most people's first experience with the game, and Jason has said he doesn't want the game to look that much different during the contest. The whole amulet thing should just be "floating beneath the surface," as I recall.
Actually I think this is another advantage of my proposal. During the test amulet contests the game was quite different for everyone. There was strong motivation not to play for more than $3 stakes, strong motivation to play fast, often a good chance that it was a game for a particular amulet which gave different incentives to different players, etc. I think my system comes much closer to Jason's original vision of something that lies under the surface of the regular game. In my system non-amulet players can play for any stakes they wish and are rewarded for playing, but not for playing fast. The distortions to non-amulet player incentives involved are pretty minor compared to those that existed in previous amulet contests.
Well, it sorta seems like any game where you can reliably force a draw is inherently broken for the purpose of this discussion. So, let's use Hex as the example here.
Why is a forced draw broken but a forced win not broken? What if we calleda draw in chess a "small win for black" instead of a draw? Does the presence or absence of a forced draw somehow affect the skill cap?
If you are playing the optimal strategy as player 1, you can do no better than to play that strategy. There's nothing about a weak opponent that is worth exploiting. A win is a win.
That's true but I think that's peculiar to hex rather than indicative of perfect information games in general. For example consider just being the second player in hex, knowing the equilibrium play won't help you because EVERY strategy is in equilibrium for the second player, however there is still room for the second player to do better than just playing random moves (if they are not playing against an ideal opponent.
Well, I guess it's the same as Checkers, then, were the opt-vs-opt states is a draw, and the only way to win is to play sub-optimally and draw your opponent out into sub-optimal play. In Checkers, you'd be hoping that they would be trying to exploit you for better than a draw, and thus play sub-optimally, and thereby open themselves up to exploitation.
It's not quite the same. Call a strategy co-optimal if no equilibrium strategy "beats" it. In a game with mixed equilibria any pure strategy which is mixed between in an equilibrium strategy is co-optimal. These are the sub-optimal strategies you'd want to play to entice an equilibrium player to dance in donkeyspace, as if they don't they don't get any benefit from your "mistakes". In a game like checkers the co-optimal (but non-equilibrium) strategies still exist but look very different, they are strategies which only go wrong once the opponent has already gone wrong. As a result they can't be used to draw an opponent out into donkeyspace, because unless they go out there voluntarily they'll never know you're playing co-opimtally rather than optimally, and once they do make such a mistake you are best off playing optimally from then on.
Since we're talking about exploits again here's an exploit of the current system that I think would be very likely pick up an amulet even with the suggested modifications here (with Caravan Disturber's suggestion one of the accounts would no doubt pick up the penalty amulet though). Have 2 accounts spam games with each other, each time they check if they got into a game with each other, if so leave immediately. If they are not against each other whichever one is in a game has a decent chance of being in an amulet game, so play some strategy which has some decent chance of winning, and is fairly quick (I'd guess 30% should be fine and achievable with a bot, I would try the following: approximate a Nash equilibrium for the game where you always give your opponent the column with the lowest maximum number and then always go allin in the first betting round [if I've done my maths right this only has 1800 information sets per player so it's easy to play close to equilibrium for a computer]), if it's two human accounts the humans can just play normally when they're not against each other, but bots would work better because bots don't need to sleep. With just two account you're best off dropping amulets upon receiving them and continuing to spam games with each other, but with three accounts you're best off holding the amulet and spamming games with the other two accounts so you have a very good chance of getting into a game with one of them.
I think the above shows the fundamental problem here is not about dropping amulets. The fundamental problem is that the parts of the games rewarded with amulets are the beginning and the end, not the middle; play 20 games in the time another plays one and you've got 20 times the chance of picking up an amulet. Since it is trivially easy for colluding accounts to play a ridiculously large number of games against each other, it is very easy for colluders to get a large advantage. My original suggestion for handing out dropped amulets did not have this issue, so I'll suggest it again here. When an amulet is dropped record which games are in progress and count how many there are (call it n), start a counter at 0 and every time one of *these* games (i.e only games started BEFORE the amulet is dropped) is finished increment the counter by 1. If upon doing so the counter reaches floor(n/2) then award the last one standing in that game the amulet. If you're happy to add some randomness to it though, it is simpler just to assign the amulet to a random game in progress at the time of the drop (by which I mean "dropped into player pool" of course rather than "dropped by a player" of course). There is a slight incentive in this system to drawing out a games you're winning, and to leaving games when you're behind, but they are slight (I'm not sure it would be worth paying the leaving penalty and forfeiting the chance to win an amulet if the game has already been assigned an amulet for example) and don't provide advantages to colluders.
In fact even if you did just implement this suggestion for dropped amulets, my strategy above still looks pretty good, as my colluding accounts are going to get to play much more games against people holding their amulets (and as noted before they can get a very good chance of getting in games with each other once they get an amulet with 3 accounts). My suggestion would be that each time an amulet game starts, pick a non-amulet game in progress at random and have the last one standing in that game become an amulet challenger. When amulet challengers go to the new game page, they have an option for amulet games like amulet holders, if they select it they get paired with an amulet holder who has also selected amulet game (or they wait if there are none). If an amulet challenger does not select amulet game for 15 minutes they lose the status and a new random game is selected for an amulet challenger to be chosen from. When an amulet is dropped into the player pool, it is given to a waiting amulet challenger, and 2 games in progress are selected to take an amulet challenger from. Amulet games would all be at some fixed stake and non-amulet games at *any* stake could be chosen to select the amulet challenger from.
The 2 * N + 3 number comes from realising that you need 2 * N + 1 players to provide match ups for N amulet games. If you have 2 * N + 2, you can't hand out an amulet, because then you'll have N+1 amulets, and will need 2 *(N+1) + 1 players, but that's 2 * N + 3, and you only have 2*N + 2.
Thus, if you do have 2 * N + 3 and you hand out another amulet, you're still good.
I'd recommend (at least) 4N rather than 2N+3 if amulet holders can't play each other. Otherwise you still have a very high density of amulet games which makes things easier for colluders, means longer wait times for amulet holders, less chance to have games played at a variety of stakes. Also, this system doesn't get rid of the fundamental problem of having amulet holders sitting around for 2 hours unable to get a game, as that is the mechanism to reduce the number of amulets in circulation as I understand it.
There's not a good way to deal with the "other side of the world" issue.... recruit more people from your side of the world!
Well much simpler and more profitable for me will be to create two alt accounts which (when and only when I'm waiting for an amulet game) spam 1c games and leave any game they find themselves in immediately. This won't give a noticable edge with a thousand players playing at the same time, but it would give a huge one with <80 players, and I think still a noticeable one with 100. I'd go from being at a significant disadvantage to having a huge advantage, shading into the territory of "I won the contest with one simple trick" territory I think. This sort thing seem unavoidably to give an edge without completely changing the whole thing. But allowing amulet holders to play each other would certainly reduce it significantly especially if you can count on having 36 players playing at any one time. Putting limits on number of times you can play any particular opponent will help reduce the advantage from this as well, but it will also make things even worse for non-colluding players playing at "unpopular" (less than 80(!) active players) times.
Oh I should add if you don't want to implement any of my three simpler suggestions, a compromise would be to have it so players couldn't have a negative score. (Ideally you'd have it so they couldn't lose points by picking up an amulet, but you can't easily do this without giving players a motive to stall if they have a high score on an amulet and are lucky enough to pick it up again; mind you I'm not sure how worried one should be about this, and am still inclined to think amulet holder should not be losing so many points while playing a game).
I'm not sure what sort of feedback will be useful for Jason for this contest, but I thought it would be useful to have a separate thread for it, and I might as well give my thoughts from playing in it so far:
1. Definitely felt increased tension playing amulet games, and even games that might be amulet games, so I think it achieves that goal fairly well.
2. The current system feels quite unfair to those who pick up a dropped amulet. Not only have they won an amulet game without getting points for it but if they then go on to win a second game they'll have significantly *less* points than somebody who just wins one game against an amulet holder. Even worse if they win the game to pick up the amulet and then lose their next game they'll end up on *negative* points. Plus you'll have lost the chance to play other amulet holders, there will definitely be cases where picking up an amulet will be bad in expectation for the player that does it at the moment which is not what you want.
3. The system is currently feels incredibly punishing to those who play at unpopular times (e.g. those of us on the other side of the world!) As well as having to play for longer to get the same game volume because of the lower number of players you can be sitting waiting for games, uncontrollably losing points for long periods of time. This doesn't feel very fair and is not very fun. Mind you I suppose players playing at unpopular times are compensated by the higher density of amulet games. In this practice, probably not sufficient as the density is always going to be high, and I've been waiting about an hour losing points on an amulet and I'm not even sure this is an unpopular time. Numbers like that are going to make it very difficult to rack up points (and it's no good getting amulets if you've got no chance of getting very high scores on them; you need to get the *highest* number of points and there are going to be players who get the amulet during the popular times with much lower wait times). Maybe the extra players from the launch will mitigate this issue but that brings me to my next point:
4. One thing I don't think anyone's mentioned about the current system, and which I hadn't thought about until now is that it requires a *lot* of players to have everyone playing games or even to avoid long waits. Just not allowing amulet holders to play each other, and assuming amulet holders only want to play amulet games, means you need twice as many players as amulets (so in the real competition this will be 72 players playing). Further, currently for an amulet holder to get a game there need to be two non-amulet holders, one who proposes a game and another who joins. Add in that the random delay will mean multiple games will have to start on average before the amulet holder gets a game and we're talking about 80 players active at all times to avoid amulet holders having large wait times. These sort of numbers make me think that collusion will be more of a problem than I had thought. If you've got a decent chance of sitting there waiting for an amulet for awhile you can just start an amulet game and then get your alts to keep starting games with each other. After my experience today I'd probably make alt accounts just to make sure I don't end up uncontrollably losing points.
Suggested fixes (roughly from what seem simplest to most complex):
1. Whenever someone wins a game to win an amulet, give them 200 points (even if the amulet they won was dropped).
2. Don't have amulet holders lose points while waiting for an amulet game.
3. Give amulet holders the option of just dropping an amulet.
I'd would argue for all 3 together, though any one or two on their own would mitigate the issues above. Also none of them get the fundamental issue about the requirement for large numbers of players to be active, and playing at times when there are not those sort of numbers not being very rewarding (unless colluding when it will be amazing). For that I think you would need bigger changes e.g.:
1. Reduce the number of amulets. Halve the number of amulets and you almost halve the number of active players you need to avoid long wait times.
2. Allow amulet holders to play each other. In fact have that as default behaviour have an amulet holder waiting for a game to be about as likely to be pulled as a player just joining a game, and definitely be pulled rather than having both amulet holders waiting there for a few minutes (You haven't said how the current delay on amulet games works so I don't know the details about how you would do this). This would basically deal with the problem I think (for instance at the moment I wouldn't be surprised that there are 3 or 4 players all waiting for a game, and with this change we'd be playing each other).
When will the contest end?
Another minor tweak. Just noticed this:
After a full two hours of no amulet match completion, they drop the amulet into the player pool.
I'll record my vote for having this changed to "two hours with no amulet match beginning", or "two full hours of inactivity (no playing or offering amulet games)". Or if that two hours would then be too long, reduce it to one hour. I don't think you want amulet holders needing to finish a game by a particular time at all costs (that was on of the major advantages of having amulet holders lose a point per minute as I saw it).
Not sure if this counts as a new exploit or not but it seems worse than Josh suggests. In the current system without a random delay in when the random stake amulet game shows up colluders are going to have a very good shot at getting in a game together 50% of the time just from choosing a random stake that pops up when the amulet holder starts the game. Josh's system would also require random delays to avoid a similar or larger advantage to colluders of course. But I guess I'll focus on just exploiting things in the practice contest and see how that goes more than making criticisms/proposals here from now on?
Regarding tweaks to the current system this requires a little more work than just changing a parameter but the main one I'd advocate is getting rid of the leaving penalty in amulet games, or (better) just allow the amulet holder to drop the amulet. The amulet system is already punishing enough for those who might have to leave the game at some point while holding the amulet.
It's interesting to think about how there's no skill cap in CM as compared to perfect information games.
In those game, a perfect strategy exists, and after you master that, you can get no better. Hypothetically, of course, because for most games beyond tic-tac-toe, the perfect strategy is unknown.
Here, even if you discover and play the Nash equilibrium strategy (and one does exist), you still are not maximizing the amount you win against a non-equilibrium player, so you can STILL get better. It seems like the dance in donkeyspace can recurse into infinite layers of complexity, as you develop a strategy to exploit the strategy that is supposed to exploit the strategy that your opponent thinks you are using, etc.
This picture isn't qutie accurate. Even in a perfect information game a player can still sometimes improve on their equilibrium strategy against a poor player. For example, you pick an equilibrium strategy for checkers. A poor player memorises the main line of your strategy and can get a draw every time, until you change your strategy (maybe even to a non-equilibrium one) where they won't know what they are doing and lose. I'd argue that the reason that poker players for example, focus much more on explouiting weak play than chess, checkers or go players (and these players do look at playing sub-optimally to exploit weak opponents, Emmanuel Lasker was famous for making "psychological moves", and there are opening traps etc.), is the increased granularity of the result. Even in go results are scored generally as win-loss, and there's no benefit to winning by more points, whereas in poker you are given a monetary reward which can be much smaller or larger. As a result a chess player has no need to play differently against a weak player as a win is still a win, might as well assume they're going to see that 10-move winning combination, even though they usually can't see two moves ahead. In poker if you assume your opponent is going to play well then you won't win as much money from them. So in perfect information games there is still room for getting better at exploiting weaker players by playing off equilibrium strategies, just like imperfect information games.
i think the difference between perfect and imperfect information games in this ballpark is that in imperfect information games equilibrium strategies are typically mixed strategies. Games with mixed equilibria do have interesting properties which arguably adds a level of skilll not found in other games. For instance, there's actually no particular strategy against which mixing has any benefit at all (every pure strategy has a calculable EV against any given strategy, so against that strategy just pick any of the pure strategies which maximise your EV against that strategy, no need to mix them). When you mix then it must be because you think your opponent will guess your strategy and counter it, but of course if you can predict what strategy she will guess you have then you can just counter her counter etc. Mixing only makes sense if you think they can model your thought process [including your model of their thought process!] better than you can model them. So picking the equilibrium strategy is sort of acknowledging you can't win the modelling war (in poker they call it a "levelling" war). This modelling war is an element of skill not captured by the nash equilibrium, and cannot be present in perfect information games where both players know an equilibrium strategy.
I hear the suggestion about different tiers for different amulets, but that puts gold amulets more out of reach.
Oh if you *like* the arbitrariness of whether you pick up a gold, silver or bronze amulet, you could preserve this in a tiered system by having the same distribution of amulets available in each tier. It does encourage people to play in whatever stakes they expect to be less popular to get more chance of picking up an amulet, but I think this is 1. Less distorting than other proposals and 2. Good for the game because having a spread of stakes is good for the game.
Oh, crap, this would make the amulet games slowly rise in stakes, system-wide, over time. The following numbers ignore the tribute for simplicity. If you play a 62-cent game (half of your $1.24 max) and lose, then your opponent has now WON 62-cents on that amulet. Which means their limit will START at 62-cents and go up from there. Say they win three games, averaging stakes of 31, 46, and 70 (because the stake cap rises after each win). Their stake cap is now $2.09, and their next game will have average stakes of $1.04. If their opponent wins, they will START with a stake cap of $1.04.
This is completely opposite to the worry I have about this. First why worry about max stakes rising? You can still get an amulet by playing a 1c game and if you do, your next game you are only up for 1c. Even if the max stakes are rising low stakes players are not put at any disadvantage (quite the opposite actually). Indeed I would argue you probably need the max stakes to rise in expectation over time for this sort of thing to work. However, does the expected max stake of the amulet in fact rise as the game progresses in the suggested system? No, I don't think so. (What follows is a bit of maths, which may be hard to follow, explaining it would take a bit of work but I'm very confident in it and the upshot is that max stakes of amulet games would not be expected to increase and they would in fact likely hang around only a few cents) Call the max stake for game n of the amulet X(n), the actual stake of game n, Y(n) and the probability that the amulet holder wins a game p (assume for simplicity that p doesn't depend on n). Then the expected value of X(n+1) given the value of X(n) is p(X(n)+E(Y(n))+1-p(E(Y(n)))=pX(n)+E(Y(n)). If we assume all players are equally skilled so p=1/2 and that all stakes are equally like so E(Y(n))=1/2 then E(X(n+1)|X(n))=E(X(n), that is the expected max stake of a game is exactly the max stake of the previous game. In reality we can expect amulet holders to be more skilled on average than non-amulet holders so that p>1/2 but we can also expect most games available to be low stakes so E(Y(n))<<X(n)/2. As a result I wouldn't expect the max stakes to rise over time. In fact I expect there to be downward pressure on the max stake which will keep at a few cents (e.g <10c for most of the promotion).
Even if the system is changed it so that max stake did rise over time in expectation you would still have the issue that everyone who wants an amulet is encouraged to play 1c games as then they're sure to be in with a shot at the amulet. If you increase the min stakes as well, then you get a weird guessing game about what stakes the amulets are currently at, or put the low stakes players out of the running for it. I'm pretty pessimistic about finding a way to get it all to work, as cool as having rising stakes for amulet games would be.
In terms of the new player experience, well, super-conservative new player play isn't totally what I had in mind. Yeah, you CAN put in only $2, but you should probably put in more. So if you want a chance at an amulet, put in at least $5.
Just my 2 cents on this. I think the current intro page anchors people a lot on that $2 figure, it's the only realistic possibility put on the screen and a bunch is said about it. I'd suggest giving an example with $20-$50 with 10c-$1 games. This would be the price a lot of people would pay for a videogame (with no chance of winning money!) and is not going to break anyone's budget. I'm not saying you want to reduce people's respect for money like a casino will try to do, but ultimately it will be good for the game if you get people p[laying more of a spread of stakes.
By the way that's another advantage of the tiered amulet system it encourages a people to think about trying a few different stake levels and decide what's best for them which has got to be good.
But once that problem is solved, it becomes frustrating enough to collude, and colluding gives you no guarantee of winning. I need to balance a cool, simple, exciting contest against preventing any kind of collusion edge.
HOLY CRAP I'VE GOT THE AMULET.
I need to preserve that moment.
Well my other suggestion of having the points represent starting amounts for a tournament to played at the end of the promotion does preserve that moment, and keeps the feel of the promotion generally intact I think. I remember you saying you didn't want a tournament because you didn't want to have a structure that would just go to one of the top players without the other gettting a look in. Well now you've got 36 amulets, there's no way Jere is going to get his hands on every one in the promotion week, and even if he did he's bound to lose *some* games and so could find himself in a tournament having only won 1 point, so really having his work cut out for him to win a tournament with players start with 5-10x what he has, where losing one or two games could see him out. You could also limit everyone to only playing in one tournament each, then Jere could only play in one tournament anyway. This would create some non-zero sum interaction again, but there would be much less potential for collusion here than in the current system.
I'm totally with Cullman on the tiered amulets thing. It lets players choose their stakes without creating possibilities for collusion. Plus you can have more valuable amulets at higher stakes which makes which one you get seem less arbitrary.
1. It doesn't solve the fundamental problem of collusion, because the $5 tier would be much more sparsely populated than the 50-cent tier. So far, 90% of players have deposited less than $10. The vast majority of people are playing for very low stakes, and $5 games tend to sit there for a while. If you let the amulet holder control the stakes, even tiered stakes, you let them control the stakes. And colluders also control the timing.
The system I envisage would have each amulet locked to a particular stake so the amulet holder would not be able to choose their stake, non-amulet holders would be able to.
2. It funnels the behavior of non-amulet holders in a way I don't like. This is your first time playing the game, and there's all this extra stuff on the screen. All these tier buttons.
The current stake picking screen is clean and wide open. I want non-amulet holders to have that experience (whoa, I can play for 1 cent or 2 cents or 11 cents...). Ideally, for non-amulet holders, the game should look and behave identically to how it does now.
If this is important you can have each amulet locked to a range of stakes, e.g. 1-6c, 10c-30c, 66c-99c,$1-$1.49 $1.50-$2, $5-$6.66. You could creat ranges to fill in the gaps, expand those ranges or leave the gaps there depending on how much variance you can live with for the amulet holder, whether you want opt out stakes (as Cullman suggests) and how much you're concerned about funnelling stake selection. Then when an amulet holder decides to join a game they get assigned to a random player joining or having their game joined as suggested by Josh (there'd have to a random time delay here of course). You wouldn't have to add anything to the create game screen, non-amulet holders would get to control their cash stake precisely, everyone would know their cash stakes and amulet holders who are only playing penny games couldn't suddenly find themselves in a $5 game.
I still don't like that non-amulet holders don't know whether they are in an amulet game though personally. So I'll suggest a modified version of my idea in the other thread: When an amulet holder proposes a game, a random game in progress or waiting to be joined gets assigned as a game for the right to challenge for an amulet. There are a number of benefits to this, it rewards length of time playing the game, but it doesn't reward any particular part of the process over others, and so doesn't reward any particular behaviour given that you are playing. For example the current system rewards just the waiting for a game or joining part of the game, ss a result it rewards players who are in that part of the process at the right time. So if you get a tip-off about when an amulet game is to be proposed you have an advantage over other currently active players, and you are better off playing short games to get to the joining/proposing part of the game more frequently (this is balanced a little by not letting people know whether they are playing amulet games, but 1. It still encourages higher variance play if it doesn't reduce your winning chances too much. 2. If two colluders always tried to join each other's game they would often be successful if you don't fiddle with the mechanics for non-amulet games much, and they would always know if they weren't successful). It also means more players get the excitement of amulet games, you get all the amulet challengers knowing they are playing for an amulet, PLUS you get all the people who are playing for the chance to play in an amulet game. It also automatically reduces collusion edges as to even get an amulet you now have to win 2 games against 2 different opponents.
The major disadvantage of course is that it means that amulet holder might be waiting around a bit for a game, but you could reduce this a couple of ways, the idea I currently like is this: at the start of the promotion not only assign amulets but also amulet challengers (one for each amulet, though a challenger can play actually play for any amulet, unless you have a tiered stake amulet system when they can play for any amulet in the right tier). Then each time a new amulet game is started you select one game to be for the right to challenge for the amulet, then the average wait time for amulet holders or amulet challengers will be less than the average difference in length of time between two different games. (Challengers would of course lose the right to challenge if they don't accept an amulet game within a particular period).
This system has practically 0 collusion potential if you combine it with one of the zero-sum structures I proposed in the other thread (there's still a small edge that a colluding amulet holder can avoid starting an amulet game while there non-amulet holding partner is briefly afk, giving them a slightly smaller penalty for taking a break from the game, not a significant worry). With the current structure of amulet points there is a bit of potential for amulet challengers to try to hang around to make sure they get paired with a colluding account that has a few points. But this could be reduced by making the right to challenge for amulet games only last for a short period of time, and upping the penalty for sitting around with an amulet without proposing or playing a game. Also it requires that the two accounts get into quite a specific situation already which they can't collude to get into.
You're not missing anything. You could wait until your timer was nearly up before betting, when if you have to wait you're almost certain they have had to think about your raise. In general though if they've put in the same amount of chips as you then the chances are they saw your bet and called it rather than picked the same amount.
OWEVER, compare the cases for player A where AA does and does not exist. This isn't the friend case, but the case of whether or not to create an alt account. If A has scored 4 points but has no chance of scoring 5, having a second account AA with 4 points would give A a 10% chance of getting to 5 points, where without the alt account, A will have a 0% chance of getting to 5 points.
In this example having the extra account play is increasing everyone's chances of getting a higher score, AA is sort of just one of 9 players getting an advantage. In a tournament setting , with an entry fee A will be paying into a prize pool which will then be shared out by the other players including AA (if 5 points is enough to get a share of the prize pool), so if you're AA you get something for having A in the tournament but not as much as A loses for being there in the first place (because most of that goes to the other players in the tournament), so there's no reason to make the alt account. BUT If there is no entry fee then of course there's no reason not to make the alt account, and it can be beneficial to do so.
There's a bit more to your example though. It showed me this rather unintuitive fact: that with no entry fees you can have a motive to create accounts which has no chance of winning the event and which doesn't collude with you (if your alt loses to everyone it can't really be considered colluding if it loses to you as well and can't control who it's paired with), which you can't ensure you're paired with and which doesn't change the size of the prize pool. That is you can benefit from having a bad player in a tournament even if that doesn't increase the prize pool and they're no more likely to lose to you as anyone else: Suppose a knockout structure where the winnings are split between 1st and 2nd (this structure does not incentivise any sort of collusion). Now suppose there are 3 other players, all of equal skill but much better than me, so I have a no chance of beating any of the other 3. If I add 4 accounts that are guaranteed to lose to any other account (except themselves), then I still have no chance of winning the tournament (I'm always going to have to play at least 1 genuine account), but now I have a chance of playing fake accounts in the first two rounds and so getting into 2nd. In fact I will benefit from those fake accounts so long as my chances of beating a genuine opponent are less than 50%. Surprising but true! What is going on here is that the fake accounts essentially add a random element to a skill-game in which I was outclassed, and adding random elements benefits the player with less skill, so by the same token if I have more than 50% chance of winnning against a genuine opponent, having a bad player enter the tournament is actually bad for me if they don't increase the prize pool, even if they have 0 chance of winning.
Question: Are there situations where a *skilled* player (>50% chance of winning against a random opponent) benefits from adding an account that plays no better than them, and plays no worse against them than anyone else? What do these situations look like, and what bounds, if any, can we put on the incentives here? (the effect above is not so worrying as benefit comes to a player who isn't getting much of the prize pool either way, but if you had something similar with a skilled player that could be a concern).
Note that these effects don't go against what I said before as they don't involve collusion just manipulating the composition of the player pool.
Just issuing a new board each round gives the advantage to both players in a way that approaches a perfect 50/50 split in the limit.
Yes but so long as you don't pick out some players for a special advantage some way this will be true of any random element.
Isn't chess a forced win for white?
We're not likely to know any time soon. We don't even know that black doesn't have a forced win, though most players assume that they don't (see this short piece by a friend of mine giving some thoughts on it if you're interested: http://www.csse.monash.edu.au/~dld/Publ … Q_p158.jpg, http://www.csse.monash.edu.au/~dld/Publ … _p159.jpg). My money is is on chess being a draw though, white scores about 54-56% at the highest level which is much closer than to 50% than 100%, the vast majority of theorists think that and it generally feels much more like mistakes were made when black loses then when there's a draw. It's very rare to see a game where black lost but it seems they couldn't have improved their play but this is fairly common with drawn games. This Wikipedia page has lots of interesting stuff on this http://en.wikipedia.org/wiki/First-move … e_in_chess. The chess example is still good though. White seems to have an inherent advantage but chess would surely be a paradigm example of a pure skill game. In tournaments though there are always some systems in place to mitigate this effect and it generally involves very little randomness.
A simple solution:
If a player leaves the game before they commit to any decisions then both players get their coin back.
In theory players should just keep refusing until they get what they're sure is a favourable board in this case. In practice personally I'd be refusing probably at least 50% of boards.
The solution is to simply present each board twice in a row, rotated 90 degrees the second time. Thus, whatever slight advantage you had on the board on turn 1, your opponent will have on turn 2. Turn 3 you will get a completely fresh board.
This will also add another avenue for expert play (players have long been asking for a chance to see the board from their opponent's perspective).
Sorry to go way back in this thread, but I really like this idea. I do think the initial board layout does affect the outcome and can give one player a strategic advantage (both given particular player styles or given optimal play). I also like the idea of being able to compare my play with my opponents on the same board.
As Anohito mentions though you can still get an advantage depending on who sees which board first, because the stack sizes will be different, you could even have situations where the board is advantageous to player one where the effective stack is 50 times the ante but gives an advantage to the other player if the effective stack size is less than 10 antes. Then a player could get an advantage from both sides of the same board. Plus of course there's always a chance that a player loses all their coins the first time a board is seen, so whoever had an disadvantage on that board is uncompensated. So just having each board shown twice won't eliminate randomness.
Two ideas for eliminating it these factors, both are less friendly to new players though:
1. Every two rounds take the smaller stack and divide it in 2 (throw away remainder if small stack is an odd number), this is the maximum bet size for the following 2 rounds, ante, ante is the same for both rounds.
2. Display both boards simultaneously and have players makes their picks and bets for each board at the same time. You could only allow bet sizes up to half the small stack before. This would probably be more of a fun variant than a possible adjustment to the main game though, trying to play two boards at once would be too intimidating for new players and probably quite a few more experienced players too (there's plenty enough to think about on just one board) but I think it would be fun.
No, I meant getting an edge somehow through the chance of two colluding accounts playing each other and thereby giving one account a free win.
Ok sounds like we have a genuine substantive disagreement here. You can make a structure such that if you have two colluding accounts of equal skill they have no edge over two non-colluding accounts of the same skill. All the structures I suggested had this property or were close to having this property.
One of your suggestions here (amulet players only matched with other amulet players who have the same points so far for that amulet), having two accounts playing through for that same amulet still helps you, because there's still a chance those two accounts, that have both built up points on the amulet, will play each other later.
Yes two colluding accounts might meet each other, and one could throw the game to the other but that only benefits one account at the expense of the other, and crucially the benefit the winner gets is exactly equal to the cost the loser pays. For colluders to get an advantage by throwing games they have to prefer that one account wins over the other, but in this structure both accounts have the same position in the tournament, so it won't matter to the colluders which account wins.
I think it will help to think about about friends with existing accounts colluding here rather than people creating multiple accounts. So suppose my friend and I are playing in the promotion with a structure like the one just mentioned where people only play other people with the same number of points for a given amulet. My friend and I realise that we have the same score, n, and are going to play each other next round, we've agreed to split any winnings from the promotion. What can we do to increase our combined chances of winning the tournament now that we're playing each other? Nothing, whatever we do we'll end up we'll end up with one of us on n point and one on n+1 points. (If one of us was significantly more skilled than the other it is better for us if they win the game, but if we play honestly then the significantly more skilled player is very likely to win anyway). As a result we have no incentive to play any differently to non-colluding players, and if we don't play any differently to non-colluding players we can't have an advantage over them. So colluding players get no advantage in this structure over non-colluding players. QED
This is different from the structure currently proposed for the promotion. If me and my friend meet while they're on 4 points and I'm on 0 points, it is much better for us if my friend wins, because it is much more likely that someone on 5 points with the amulet will win than that *either* of us will win if one has 4 points but no amulet and one has 1 point and the amulet. So I will throw the game to my friend and they'll get to 5 points with the amulet. Two non-colluding players in the same situation would only half of the time end up with one player with 5 points and an amulet, the other half of the time they will end up with one player with 1 point and the amulet and one with 4 points but no amulet. Making the assumptions I did in my previous post me and my colluding friend have a 60% cubed = 21.6% chance of one of us winning the amulet from this position, two non-colluding players in the same situation would only have 50*60%^3+50%*60%^7 = 11.6% chance.
If you have two horses in the race, and both are playing well, they are likely to meet each other near the end, where one can throw the game. Better meet your accomplice near the end then some stranger.
Well of course if you've got two horses in a race of course you have more chances to win but the possibility of one of your horses deliberately losing to the other doesn't necessarily help. For example suppose that it's a knockout structure, and both your horses are in the semi-finals and racing well (so they each win against the other horses 75% time), then if they get paired against each other you have 75% chance of one winning (guaranteed to get to the final and 75% chance of winning that). This is actually UNLUCKY for you, if they weren't paired against each other the chances of winning would be ~84%: in that case you have 3/4*3/4= 9/16 chance that both make it to the grand final (when one is sure to win). You also have 2*3/4*1/4=3/8 chance that only one makes it to the final when you have 75% chance of winning, giving you 3/8*3/4=9/32 chance of winning. 9/32+9/16=27/32 which is about 84% chance of winning. So you prefer it if your horses DON'T meet each other (for the simple reason that it's better to be paired with weaker opponents).
Essentially, in any structure where WINNING counts, having two accounts in the game will give you some edge.
Yes it will double your chances of winning, but the accounts don't need to collude, and by the same token your entry fees have also doubled; entering two tournaments also doubles your chances of winning one of them in a similar fashion. In the current working rule set for this promotion your chances more than double if the accounts collude, but in the structures I suggested they don't (and this is mathematically provable). As I say in this promotion your likely giving away more in prizes than the entry fees (card processing fees and tribute in the games) so doubling the entry fee and doubling your share of the prizes would still be worth it, but that really is unavoidable without changing the sign up process, if you reward account creation you motivate someone to create multiple accounts!
An edge through collusion is impossible to avoid in any scenario where we don't verify the identity of each participant. Well, except in the main game, where we're lucky that collusion doesn't help you win more money. I'm talking about tournaments of any kind, etc. Someone with 2 accounts always has an edge over someone with 1 account, because there will always be some chance that those two accounts will play each other.
Not so! All my suggestions either provably avoided any number of colluding accounts from getting any edge by playing each other or left the potential small (e.g. with amulets losing points in game colluding players have an incentive to finish the game quickly).
But maybe we mean something slightly different by getting an edge from collusion? In all my suggestions 2 colluding accounts have more chance than 1 account of winning, but no more chance than 2 non-colluding accounts, and they do not increase that chance by playing each other. As a result in nearly all my suggestions two, ten or a hundred friends have no incentive to team up, and in the normal course of events someone doesn't get any benefit to having multiple accounts, even when tournaments are run (in the sense that their rate of return for the tournament doesn't change, they double their expected share of teh prize pool by doubling their entry cost). A promotion rewards a player having multiple accounts because it rewards every account upon creation with free entry into a contest with real prizes, under my suggestions for the most part that was the *only* reward someone got from creating multiple accounts. The collusion I mention above increases the expected payout *per account* and this is preventable.
For each full two hours that a player holds an amulet, they lose 1 point for that amulet.
I would advocate instead an amulet game being worth 60 points and players losing 1 point for every 2 minutes, otherwise your incentives start getting really weird around the 2 hour mark. This also has the benefit of making ties unlikely. If you didn't mind adding a little complexity I'd also suggest having it be 1 point for every 2 minutes (maybe even more minutes/point) in game, 2 points for every minute out of game (maybe even more points/minute) and no points lost between hitting start amulet game and the game starting. Also I think there needs to be an option of dropping an amulet voluntarily if you lose points while holding it.
Definition: Amulet dropped into the player pool.
An amulet passes to the next player, system-wide, who is the last player standing in a non-amulet match and who does not currently hold an amulet.
With 3 colluding accounts you can pass an amulet here fairly reliably by having the two non-amulet holding accounts start a game, then drop the amulet and have one player leave. You could keep feeding amulets to a main account this way. Limiting each account to only receiving one dropped amulet for the entire promotion would stop this, but it wouldn't stop someone who just needs a a couple more points colluding with others to get the relevant amulet. Giving to the next player who has never played a game for that amulet would fix that, though it wouldn't stop buying and selling of amulets. Buying and selling might not be such a concern though since you still have to earn points for them to be worth anything. But to avoid this altogether I'd recommend replacing next with last standing player in game n/2 to finish after the drop where n is the number of games in progress when the amulet is dropped.
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