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Oh... but the main point of the amulet contest was the actual AMULETS themselves.
As in, 36 physical pieces of metal that would be mailed out at the end. 6 of them from 99.99% pure gold, 12 in 99.99% pure silver, and 18 in copper. I.e., I'd buy some Canadian Maple Leaf coins and lost-wax-cast the metal into hand-carved occult amulets that are one-of-a-kind. At the end of the contest, there would be 36 winners.
Ah, I wasn't sure how important that part was to you, it makes it a lot harder to prevent collusion/other weird incentives. e.g. this is guaranteed to get an edge: Have x colluding accounts, if two accounts ever meet in an amulet game have the game thrown to the one with the highest probability of winning that amulet (usually the one with the higher score unless that player has a higher score on another amulet). Also if you're a couple of points away from the lead you can offer pretty good money to players to throw games to you (the anonymity in the game makes it harder, but there's these forums and there's the chatrooms [and anyone following these gets an edge because of the chance of offering or being offered money to throw games]). I'm pretty sure these edges can't be eliminated in this structure, except with hard-to-police rules against the behvaiour. So the question is what size edgte does this sort of thing give you? Supposing colluding accounts can't do anything to increase the probability of facing each other then one has the amulet then the cahnces that they get to face a colluder will be the number of colluder/number of players. 10% seems like a doable amount of colluders to get, which means you get a 10% chance of winning for nothing, which is a 5% edge for someone who normally wins 50% per game, or 3% edge if you're winning 70% of normal games. That 3% edge for the 70% player increases their chances of getting 5 wins in a row from 16.8% to 20.7%, or of winning 10 games in a row from 2.8% to 4.3%, so colluding isn't going to guarantee a win but does seem worth doing if you want an amulet, (plus don't forget colluders have a time advantage because they can win really quickly if the face a colluder). Or if we look at it per game between colluders: Let's suppose for simplicity and because I think it's true that each player is only likely to get their hands on a particular amulet once so it's all about getting winning streaks. Let's further suppose that the score needed to win this particular amulet is 8 points. Two colluding players are playing, A with a score of 2 and B with a score of 0 and both with 60% chance of winning a game against a random opponent. Then if A wins they only need to win 5 more games which they do with 7.8% chance while if B wins they only have a 2.8% of getting to 8. Without colluding A and B collectively have a chance 5.3% of get the amulet, whereas with colluding they have a 7.8% chance so one they meet colluding is 2.5% of the value of the winner's prize. If A has a higher score then they stand to gain even more by colluding (the maximum is ~49% if A is on 7 points and the minimum is ~1% if A is on 1 point). If those edges are within the rules they seem worth taking to me.
You really have to change the structure of the thing quite a bit to avoid this while keeping the amulets: e.g. every point you win gets you 100 coins to play with in a mini-tournament for the amulet at the end of the promotion, or you only play people with the same number of amulet points as you (there are few different ways of doing this e.g. Each amulet has rounds in which games a played: each round for each amulet two new players are chosen to play an amulet game, and if any two players are on the same score they play an amulet game with every game worth one amulet point, and whoever has the most point holding the amulet [so the first round 2 people play for 1 amulet point and the amulet, the second round 2 new people play for just 1 amulet point, in the 3rd round the winners of the 1st 2 have an amulet point each so play each and two new players are chosen to play a game to get to 1 point etc., not really the structure you're looking for but it does prevent game throwing]). Another thing you could do is hide who has the amulets from everyone including the amulet holders, and then limit the number of games someone can play against the same opponent - actually maybe not as bad as it sounds if you could work it into the them of the game somehow and maybe drop some hints occaissionally about what is happening with the amulets. Otherwise you have to live with the edges or just ban them even if you can't easily police them.
Wait, so you're suggesting that the prize FOR ME gets bigger the more points I score while holding the amulet?
I.e., if I score 10 before dropping it, and you score 8 before dropping it, if I win at the end, I get $1000 (whereas you would have just gotten $800).
Or are you suggesting that BOTH players would get prizes at the end? $1000 AND $800?
I'm really trying to limit the prize structure so that my total liability is capped and known up front. Like the Steal Real Money contest, I was never going to be liable for more than $3000 total.
My suggestion is both players receive prizes.
What I have in mind is a procedure like this:
1. Decide on number of amulets, points structure for amulet games, who gets to play in amulet games, and everything else about how the mechanics of the promotion will work (the points system for example might be 1 point to the winner of every amulet game, 0 points to the loser and the winner keeping the amulet, a simple system with some nice properties but no chance of really high stakes amulet games, or a new player winning just a couple of games and getting a decent score).
2. Based on past games data, and the system chosen, generate a model of how many amulets games are likely to be played and what the total sum of scores, T, is likely to be.
3. Take a number, n, such that T has ~5% chance of exceeding n.
4. Decide the maximum amount you are prepared to pay out in prizes $P.
5. Declare every point worth $P/n up to a maximum total prize pool of $P.
So your maximum liability is P which is built in to the rules of the promotion, but ~95% of the time the maximum wont be reached, so players can just worry about maximising their own score and it won't matter to them if that gives another player a chance to score well as well. The big advantage of this sort of system is you get a lot of control of player incentives so you can avoid even colluding players from wanting to throw games, slow-play, etc. It also seems sort of in the spirit of the game to have this sort of prize structure given that the real money nature of the game.
Snowballing. Eventually, one person would hold all 36 amulets, and that ball would pass from person to person.
Ah, I see. I was bound to be missing something obvious given noone had suggested it yet.
However, imagine an amulet holder who wants to bully other players into leaving through slow play. Let's say they want to drag it out for as many rounds as possible. If they never bet above the ante, that's a minimum of 14 rounds before they run out of coins. Even if their opponent bets as soon as possible and gets them to fold each round, they still gets 1 minute for their first pick, one minute to bet 0, and another minute to fold. So, 3 minutes per round... that's 42 minutes, guaranteed. If your opponent refuses to leave, you can tie them up for 42 minutes.
That's a good point. So ideally you want to make it so that players don't have an incentive to play slower or faster than normal. The simplest and most effective here would just have it so that the prize pool grows with the amulet points collected, with prizes proportional to points collected. Then the incentive is just to win points, if that takes time then you take it, if a quicker strategy is likely to get your more points you use that. To do this you'd have to have a pretty good idea of the range of possible total score there could be, and you'd still want to have a mximum prize pool, but you could make it so that was unlikely to be reached.
The other concerns you mention would be dealt with by revealing amulet games to both players, which you can do if amulet challengers are winners of regular games as suggested in my previous post. It also allows another possibility for dealing with slowplaying in amulet games (though it is a little more complex): Once an amulet game has gone x% longer than an average game either player can select an option which does the following:
1. The game is stopped at the end of the current round.
2. Two new amulet challengers are found.
3. Each new player is paired with one of the players of the stopped game. The new players start with 100 coins, the original players continue with the amount of coins they started with.
4. These games are played with the ante continuing from where it was in the stopped game.
5. If at least one of the new players wins, the winners of each game are paired against each other, keeping the same amount of coins they had when they won their game (or if their opponent left all their coins and all their opponents coins), antes continuing from the maximum ante reached in either game.
6. The winner of this game gets the amulet, and is recorded as having won two games (so if the amulet holder wins their score is added to as if they won their original game and one extra game, if someone else wins their score is what it would be if they were playing the original game won it and then won one more game).
7. If both the original players win they keep their coins and two new amulet challengers are found, each starting with 100 coins again and antes continuing from the maximum they reached in either game. This is repeated until one new player beats one of the original players, when they are paired with the winner of the other game for the amulet. The number of wins the eventual winner of the amulet is recorded as having is 1 + #pairs of new players found.
To prevent colluding players drawing a game out this process would kick in automatically after the game goes y% longer than average (y>x) as suggested in my previous post. (Natural values of x and y might be 20% and 100%, but I'm not sure).
Otherwise I would suggest continually losing points for holding an amulet, with a slower rate while playing a game, and completely stopped when waiting for a game. The problem with just having timers is that players are still incentivised to draw their time out to the length of the timer, which it makes it hard to avoid players slow-playing effectively while still giving players time to play normally. Losing points means that slow-playing is discouraged but if your slow strategy wins more points than you lose in that period it is still worth player. This makes it more analogous to the main game where if you're trying to make your money as fast as possible (as e.g. profressional poker players are).
In his "secret incantations revealed!" post, Jere suggests that "Sometimes you'll get bullied by someone who bets 20 coins on every opening pick. No worries. You only need to win 5% of rounds to beat them." How does that work exactly?
It's not quite accurate but if you imagine that there is ante of one each round, they bet 20 every round, and 5% of the time you know for sure you are going to win. Then you could fold the other 95% giving up 1*0.95=.95 chips a round and call when you know you're winning winning gaining 21*0.05=1.05, so that you lose 19/20 boards, but the other 1/20 you win more than enough to make up for your loss. Since you won't know for sure whether you will win on your first pick of course you'll have to call more often to make up for the fact that sometimes you will call and lose. But if they're not outpicking you and they're always betting, just calling will break even and if you fold your really bad first picks you'll be doing even better. If you want to do better and beat them as much as possible then it gets more tricky deciding exactly what you should call and what you should fold, but intuitively, calling average or better folding worse, should do pretty well depending how well you each play on later picks (you have to keep in mind they'll know much more about what you than you will about what they have when they're employing this strategy).
How about this for an idea if you don't mind having the amulet-holder wait a little. When an amulet holder starts a game there will be n games being played. The next amulet challenger will be the winner of the middle game to finish of these (ie number the games 1 to n, then the winner of game n/2 if n is even or (n+1)/2 if n is odd will be the chosen amulet challenger). The chosen amulet challenger may not be able to play another game, so thye have to accept the amulet game. If they don't the challenger will be the player they defeated, or if preferred the winner of the next game. (The reason for making it go to the opponent even though the opponent lost is to keep the games constant sum where only one of two players will accept a chance at an amulet game). There's no randomness but very hard to exploit with collusion or any other sort of non-standard play. How do you aim to be the middle game finished when you don't know how many games there currently are or when they finish? It also encourages everyone to play more as every win could be tour ticket to an amulet game. You could even have all the information about who has the amulet and which ones are currently waiting for a game revealed to everyone, eliminating a possible source of advantage for colluders in knowing where amulets are, and making it possible to have real-time amulet leader-boards wihtout worrying about what information people might be able to scrape from them. The downside is that you don't know how long someone with the amulet will have to wait for a game, but you could tweak it to be game n/3 or game 5 or whatever if the wait was too long. (With n/2 if you assume games are started at a constant rate and all take the same amount of time, then the amulet holder will have to wait 1/4 of this time. On the other hand if you assume that the length of a given game is an exponential random variable [more or less that the probability of finishing a game any given round that you play is constant] then the amulet holder can expect to wait half the length of an average game, so the true expected wait time will be somewhere between 1/2 and 1/4 of the length of an average game, if you went with n/3 it would be between 1/6 and 1/3.)
If they're betting 10x the ante every time simply calling when you get an average or better number folding if you have a worse number (though really it's about how likely they are to beat you at the end than what your number actually is [e.g. there may only be one number they could have higher than yours at this point, or maybe you can predict their future choices, then it doesn't really matter what you're initial number is]). If they're minimaxing you as well, this will be calling every time because you'll always start with a better than average pick. You might win or lose everything very quickly this way, but if the bully isn't out-picking you then you'll win more than you lose.
"The primary legal question with lotteries is: does an unskilled player have a chance of winning?"
I don't understand how this is consistent with the interpretation of radio contests as not lotteries but randomly assigning the amulet as a lottery. Does an unskilled player have a chance of winning a radio call-in contest? Absolutely I would have thought. Does an unskilled player have a chance of winning an amulet promotion with the amulet given randomly at the beginning? Not a significant one.
"I don't see this as being all that different from random pairings in a tournament. It sucks to be paired against the very best player first, but that doesn't make it an illegal lottery."
The difference I see is that being randomly paired with the best player is not a disadvantage intrinsic to the game state, you get the same benefits and costs from playing well or poorly relative to your opponent no matter who you're paired against. Having a chance at an amulet however is intrinsic to the game state and does changes (potentially drastically) the benefits you get from playing well relative to your opponent.
"Fixed time limit for holding the amulet (say, 1 hour max), to encourage players to play as much as possible while holding it (to rack wins before time runs out) and not delay."
I really don't like this. I've had a number of games go an hour or close to an hour, and a bunch more that have gone more than half an hour, so to get a more than 3 amulet wins (probably even to get 3), I would have to adopt a strategy with a much higher chance of getting faster decisive results.
What about my idea of just constantly losing points while you're holding the amulet (which you can give up at any time)? If you set the rate of losing points right you should be able to discourage stalling without distorting in game incentives too much. Though actually I'm inclined to argue against discouraging slow-playing in game at all. If 1 minute a round is decided upon as a reasonable time limit ideally players should not be penalised for taking that time to choose their moves, if that is seen as too slow then the time limit should be lower. If that is granted I don't think it is so bad to give players a strategic incentive to take their time on their moves, (and as I mentioned in the thread on time limits there already are incentives to take your time on your moves in the main game). That said, even if you grant me all that there still is a legitimate worry about slowplaying when two colluding accounts meet each other, they could drag a game out for about 70 rounds *8 minutes a round= 9 hours 20 minutes. There are other ways of dealing with this though: 1. After one hour in amulet games finish the game and declare the winner to be the player with the most chips (disadvantage, distorts the strategy when nearing the hour mark as you don't have to get all the coins to win. More complicated version which reduces distortion, after the hour each player keeps their coins and is paired with a new player who gets 100 coins, antes in each game continue from where they left off at the end of the hour and players have a reduced time limit, the winner of each game (the one who gets all the coins) is paired with the winner of the other game [unless both original players win, in which case they keep their are paired with new players again], the winner of that game gets the amulet and is counted as having won two games (+1 for every time both original players won, the number of wins it counts as will correspond to the number of coins the winner ends up with divided by 200) . 2. Have the prize pool increase with the amulet games played (ideally so that other people winning amulet games doesn't hurt me) then the incentive to stall is reduced as you can win more by playing more, but you can't lost so much from others winning games. (Of course the increase will have to stop or assymptote at some point so that the prize pool can't get out of control, but that could be at a point high enough that it's unlikely to be reached).
"--Amulets handed out initially based on how many games you played until the end in the last 24 hours (the same way we count amulet points, except you don't have to win a game to have it count).
--Amulets going back into the pool to be handed out again using the above criteria when an amulet-holder's time limit elapses."
A bot can get a lot of touches of an amulet with these rules. You could limit each player to only get one amulet assigned to them this way for the whole promotion, but then if you put in a large number of bots one of them is going to get each amulet every time one drops. I'm not sure whether I prefer this to my idea of assigning it to the winner of the first game to start after the amulet is dropped. My idea is open to collusion through timing drop and game starts, especially without random time delays. On the plus side, though, it encourages everyone to play more without feeling they can't sleep. With the 24 hour idea, I feel if I'm not a bot, and need to sleep I'm unlikely to play the most games in a 24 hour period and so I'm not incentivised to play more to get a chance at picking up a dropped amulet. Or even without bots, if I have to work then I'm not going to play the most games so why bother. (In general I think rewards you might get for playing a single game will encourage the average player to play more whereas rewards for playing the most games won't).
"--Two amulet holders face each other, the winner gets to pick which amulet to keep, the other amulet goes back in the pool. This prevents someone who holds copper from being blocked from gold, but doesn't assume they WANT gold (maybe they're very close to being top for that copper amulet)."
I don't mind this system but why not just have the winner take all the amulets in play with them all remaing active while they have them. That way an amulet holder gets the same benefit from being paired with another amulet holder as everyone else, and realistically it's unlikely anyone will hold multiple amulets for long. This would also be easier to incorporate with a points system like the one I suggested above to keep games constant sum.
One suggestion I have about this, especially if you want to reduce the standard average time per decision, is to give some extra time (relative to the other rounds) for the first decision, to give players some time to get a feel for the board. (Also a non-obtrusive countdown timer that you can see from the start, in the betting rounds I some times get in trouble with the 10 second timer because I have to press a lot of buttons to get the right bet size amount, and end up picking a different bet to what I wanted to avoid risk of leaving).
Yeah, Jere, the "luck" of being blessed with the football at the start is a problem. I wouldn't be legally allowed to do that randomly.
When you hold the amulet, you are paired with a totally random opponent (who is also currently looking to join or create a game).
If assigning the amulet at random puts you in legal difficulty (even though you only get something for it if you go on and win games of skill), wouldn't assigning players to amulet games at random put you in the same legal difficulty? In both cases you get a prize (amulet or amulet game) which is assigned at random though it is only worth something to you if you can go on and win games. All the other suggestions for preventing colluding with the amulet seem to have this issue as well.
If random amulet pairing does put you in legal difficulty maybe a completely different idea: For every X coins players pay in tribute they are given the opportunity enter special bonus games, which are play for free win for real events of some sort; could be amulet like, but need not be. For example the simplest would be that after being raked 20 coins you earn a ticket. Games can be created and joined for a ticket stake if you have a ticket, and the winner of a ticket game wins points, with a prize pool divided among points holders at the end of the promotion. But you could have it that you can continue playing bonus games while you keep winning them with benefits accruing exponentially, rewarding streaks like an amulet does (with ending streaks also increasing in value the longer they are). Shouldn't distort player behaviour too much in regular games, except some players might try to play really fast at low stakes to increase their coin tribute as quickly as possible. You could instead make it about time spent playing rather than amount actually tributed to avoid this. Also can't collude to get an advantage here, as to get into the bonus games you have to put in the game time (so long as the bonus games themselves are constant sum). This sort of thing has less of a competition feel to it and more of a poker sign-up bonus feel, but I'm not sure whether that's a feature or a bug. You could always run a tournament the following week or concurrently to satisfy the more competitive players though.
Another issue with the 5s after your opponent idea, is it reveals to your opponent exactly when you made your decision so it would encourage some players to wait so as not to reveal that information.
I don't think Jere is talking about psychoanalysis so much here, Apreche. To take a simple example, a lot of players like to choose the minimum maximum number to give to their opponent. I know this so I'm inclined to pick the collumn with the highest number in that row. If I think my opponent knows all of this (and a bot could easily know all of this) then I might expect them to give me a low number in that collumn, so I should change my collumn choice, but if I don't think they have figure this out I've got no reason to change. Spotting patterns in your opponents play and avoiding patterns in your own play that your opponent might spot is definitely part of the game. (Gasme theoretically optimal play doesn't require thinking about your actual opponent, but it still requires thinking about a theoretical opponent who knows exactly what strategy your using all the time, so doesn't ecatly make the decision simpler most of the time).
Also +1 to removing reveal stage and betting stages once a player is allin
Well I could easily have been one of these players you're annoyed about Apreche. I was worried I might annoy someone so sorry. It usually takes me that long to figure out what's going on, though. Unfortunately the game also encourages players to go close to the time limit at a couple of points even if their decision is easy: 1. When they have an easy decision but don't want the other player to know that it's easy. 2. With the simultaneous betting if you take up to the time limit to make your bet then if you have to wait you know your opponent didn't bet as much as you did at the first opportunity. I don't think there's a right answer for whether short or long time limits are better though, the game would just be different, and I'd also enjoy faster games. 1 minute per decision is much longer than you typically get per decision in online poker, say, which is what the game this gets compared to, (it's actually much longer than you typically even get per decision in online go). For this reason I'm sure this time limit is part of the experience Jason is trying to get with the game. That time limit means players can think more deeply about there moves and increases the investment (and so tension) in each board. It also makes it feel more like a strategy game and probably does less to encourage problem gambling.
In other strategy games like chess, go or poker the player creating the game usually gets to decide between a few options for time limits on thinking. I'd be for adding this option, though for tournaments/promotions you would probably need a fixed time limit depending on the structure.
Unique accounts stop someone from getting 100 bots colluding, but it doesn't stop a group of 10 friends doing so.
Also I really think that you need different amulets for different stakes rather than one that the amulet holder gets to choose the stakes for, as such an amulet would probably get stuck at some stake anyway (e.g. 1c to find weaker opposition or $20 to keep the player pool small).
Jason: Under my system you only get the points attached to the amulet once you drop it (or at the end of the promotion) so in your example you only have 675 points. The only colluding behaviour this is designed to deal with is throwing games, it does this by trying to make a pair of colluding players indifferent about the outcome of their game. So for example two colluding players are playing and the amulet has 600 points, if the amulet holder wins they end up with 900 point in the amulet (or 750 taking my proposed adjsutment), if they lose their opponent ends up with only 300 points, but the original holder has 600 points, etiher way they have 900 points between them (points on the amulet give more chance of winning more later though which is why I proposed the adjustment of the amulet holder only getting 25% added to the amulet).
Two players are still motivated to collude to get into an amulet game together in this system, but this has to be dealt with by other means as it is always good for players to be in amulet games. This was the reason I suggested having multiple amulets at different fixed stakes, but it would have to be combined with something like your random display time idea to avoid players controlling match ups. The reason I think it is still good to try to avoid incentivising game throwing is that there's going to be no way to stop players from working out they are playing each other (they could just send each screenshots of their board). The main downside I see to my proposal is that you basically have to have the prize pool be mostly cash (how else to make prizes proportional to points?)
I'm with Jere on revealing amulets during the game. I think it would be much cooler to know while you are playing and know that your opponent also knows. It does seem like it would make collusion harder though if it were hidden.
Here's an idea that I think might deal with some of the problems (well it's actually a bunch of more or less separable ideas). I'll just give the mechanics, not sure about the flavour. The idea here is to have amulets and amulet points won by winning games where one player has an amulet, prizes should be more or less proportional to points won to avoid providing incentives to collude (so you can have prizes for those with the most points but they should be small in dollar value compared to the total payout given to players proportional to their points). Have a 1c, 10c, $1 and 5$ amulet, the first game to start at each of these stakes is for the amulet of that stake. Each amulet starts with 100 points attached to it which is the minimum amulet stake. Every game at one of these stakes where one player has the corresponding amulet is for the amulet as well as an amulet stake of half the amulet points (rounded up) attached to the amulet (or 100 amulet points if there are less than 200 points attached to the amulet). if the amulet holder wins they keep the amulet, if they lose they first take all the points that were in the amulet and their opponent receives the amulet; in either case the amulet stake of the game is then added to the amulet. For example if the amulet has 300 points attached to it, then the next game the player with the amulet plays has an amulet stake of 150 points. This means if they win, they keep the amulet, the number of point attached to the amulet goes to 450 and the next game they play has an amulet stake of 225 points. If they lose they get the 300 points attached to the amulet, but the amulet goes to their opponent with the 150 point amulet stake attached and the next game they play will have an amulet stake of 100 points. At the end of the promotion of course the holder of the amulet gets all the points attached to the amulet at that time. To encourage the amulet holder to play, every time a coin is raked in a non-amulet game of the corresponding stake the amulet loses 1 point attached to it. (Probably only while the amulet is not playing though I'm a bit worried about a pair of people deliberately dragging out an amulet game in this case, especially towards the end of the promotion). Then to avoid this being too costly to someone who can't play, an amulet holder can voluntarily give it up at any time, in which case they get the amulet points attached to the amulet and the next game to start at the corresponding stake is for the amulet with the amulet stake reset to 100.
The idea of the above system is to avoid incentivising throwing games by making them constant sum. The above system isn't quite constant sum as the system doesn't take into account the extra value of having an amulet with more points attached to it (as it allows you to play games with higher amulet stakes). This just requires an adjustment of the points awarded for each result. Assuming players can keep finding amulet games without limit while they still have it, want to maximise their expected number of points and have 50% chance of winning each game, and don't have to worry about losing points for not playing then the simplest adjustment is to say that if the amulet holder wins instead of the points attached to the amulet increasing by 50% they increase instead by 25% keeping everything else constant. Of course when the amulet has less than 100 points attached to it the games are no longer constant sum, but the incentive among a pair of colluding players there is to pass the amulet on, so this is not much of a concern and can be dealt with in a number of ways. Also the amulet is worth more to stronger players so they could in theory increase their edge by paying people to lose to them, but with fixed stakes for amulet games and some measures to prevent players organising match-ups I think the realistic potential for this is minimal.
The rules for this system are a litle more complex than some other amulet systems, but the relevant things players need to know at particular points are fairly simple: If you win you'll get x number of points and keep the amulet, if you lose you'll get y points and lose the amulet, try to get as many points as possible. It doesn't have any random elements and rewards skill, but at the same time everybody has a chance to get some points, and even score well if they get a single win when there's a bunch of points stacked on the amulet. Also the fact that there are amulets at different stakes would make going for the lower stake ones seem less intimidating I think. I like the idea of trying to encourage some slightly higher stake games as well this way (that's why I suggested having a $5 amulet), but there is a risk that only a few people would compete for it and it wouldn't really work that well.
There are also lots of things you can tweak in there, or different elements you can add or remove. For example one thing that might be interesting to add is a bit of a push your luck element to holding the amulet, by either allowing/requiring the amulet holder to wager some amount of amulet points on each game they play (if they lose then they those points get subtracted from the points they take from the amulet, if they win point get added to the amulet). Or making it so that if someone loses the amulet they don't get the full amount of points attached to the amulet added to their score, whereas if they give it up voluntarily they do. These elements shouldn't discourage someone from playing as they can always give up the amulet voluntarily and then try to win it back again by playing more.
the amulet/football idea is interesting. Seems like a fun contest which meets a lot of the desiderata for the promotion. The main issue I have with it is that it seems to me with 5-10 colluding accounts you could get a *huge* advantage. My suspicion is that it is very hard to get long winning streaks at this game, so the winning streak number if this was done for a week would be in the 5-15 range (a statistical analysis on this would be useful and should be doable witht he data Conto Delirium has been collecting). I've been trying to think of ways around this which preserve the positive features but haven't come up with much. If you have a large pool of players that the amulet holder gets randomly paired that should bring the edge right down, but the pairings really have to be outside the players control. Or maybe games against players trying to get the amulet from you need to be interspersed with games against a different group of players who have more at stake in the games (so they can't have an incentive to throw them).
Context: you are forgetting about the picks for your opponent! Including picks for your opponent there are definitely 6!=720 ways of doing this.
How about this: entry fee is e.g $11, $1 is tribute leaving $10 to play with. In this case there are n rounds, in each round the aim is to double your money (no tribute is taken from tournament games as it is already taken with the entry fee). After a certain period of time those players who have not yet reached the target amount are required to play games where at least one of them is playing for the remainder of the money they have to play with, until all players have either $0 or at least the target score, those who have reached the target amount by this point would only be allowed to play with the excess. Repeat until there are only two players, then they play games until one has all the money. A certain portion of the prize pool in later rounds can be divided among the players proportionally to the amount of tournament money they currently have.
This structure still allows players to decide how much risk to take on each game, allows players to leave games when they think they're beat, and doesn't require match-making. The only way to get an advantage with colluding here is to have an account lose as much as another gains, this could be profitable if there is an inflated prize pool, but no more than just entering multiple times would be. Also your expected gain in prize money will be proportional to the tournament money you win so skill in this tournament should be basically the same as skill in the main game.
hmmm... tough one. To prevent motivating collusion or rewarding special tournament skills, you would really want to avoid non-zero sum interactions. Standard tournament prize structures tend to have a lot of non-zero sum interactions as whoever is closer to the the big prize money tends to have more at stake (poker tournaments are actually an exception here, where it actually the short stacks who have more at stake due to the potential to survive into the money). Off the top of my head I can ony think of two ways of avoiding this:
1. Have prizes proportional to winnings (unworkable as it requires negative prizes, it would just be everyone playing at higher stakes)
2. Have a knockout tournament (actually doesn't require match-making, but does require exactly 2^n players which is rather awkward).
My intuition is also that there aren't really significantly different ways of avoiding zero-sum interactions altogether. You need expected payoffs to be proportional to coins won each round, and it doesn't look like there's another way of doing that. There may be approximations around that avoid the issues with the above tournaments without opening the door too wide to collusion. For instance a Swiss system only requires an even number of players and approximates a knock-out, though you likely will have people with a shot at prize money playing people without such a shot in the final round (though a swiss requires match-making). Maybe a tournament where game entries double after a certain period of time might be another such approximation but I'd have to think about the details and their implications.