CORDIAL MINUET ENSEMBLE

The trick is to only do it after they raise first. That way it forces the other player to fold away their chips, or call at a disadvantage. And if you do it enough, it pressures the other player to not raise early in the game, which is what you really want. Then as the frequency and amount of their raising lowers, you can incorporate strategies such as matching when you only know you probably have the better position, or even bluffing some of the time. It depends on what the other player is doing, and it's more of an art than a science. But the only potential risk is if you scare the other player into never raising, and in this case you can just leave the match without losing anything.

Incidentally, I think poker is a bad game to compare to Cordial Minuet in this particular case. In poker (at least Texas Hold 'em style poker), every round has the power to increase the uncertainty that you will win. For example, in a two person table, say you are dealt pocket aces, bet half your stake before the flop, and the other player calls you. At this point you know for certain that your position is as least as strong as the other player's. But what happens if the flop has a pair of twos? You know you've still probably won, but if the other players raises you all in, would you call? You know they are probably bluffing, but then why would they bluff in a situation where they know they'd be finished if you called. Do you think it's likely they had a two in the first place when they called such a high raise before the flop? Is it possible they even had pocket deuces? You went from a situation where you knew you were winning for certain, to a situation where you don't even know you have probably won anymore.

In Cordial Minuet, this would never happen. Every round in Cordial Minuet gives you more information about whether you have won than the previous round. But that applies to the other player as well. This means that the longer the game goes on, the worse it is to be the first person to bet. By betting, you are either telecasting the strength of your position to the other player, or if you're bluffing you're taking a huge risk that the other player could exploit. This is why in Cordial Minuet, I strongly believe the first person to bet is always at a disadvantage. The only problem is, if you never bet first, the other player can pressure you into folding too much. Going all in is the solution, because as long as the other player raises enough at times when they aren't sure they're in the lead, the will have to call you or you will be able to buy the pot too often for them to keep their lead. Their only choice to counteract what you are doing, is to never raise first (which is the strategy you are using, meaning no one ever raises), or to lower the amount they bet early in the game (which allows you to call them at significantly less risk). It's pretty much win win.

In my experience, going all in in Cordial Minuet has consistently worked better than slowly betting up the pot. When you try to slowly bet up the pot in a game where every round just tells you more about whether you've won, you're playing a dangerous game. Cordial Minuet just gives you so much information about whether you have won by the end of the second round. If you telegraph the strength of your position in any way, the other player will probably just fold. And being able to know when it's safe to bluff in the second round requires you to have a very good read of the board layout and what the most likely potential outcomes are. Miss one little detail that the other player saw, and it could be very costly. When two experienced players play Cordial Minuet, they tend to scare each other away from taking big risks, and they end up fighting over increasingly small margins. This is why I initially quit playing the game. It's not fun when the reward for beating a strong player is so much less than a weak player. But going all in so much solves the problem completely. Either a lot of money gets moved around, or no one gets to play. It made the game interesting for me again, and so far the strategy has paid off financially as well. I went from below $7 to above $11 in a very short time. I've since switched to $0.01 to refine my strategy further before completely committing to it, but so far I think it's a winner.

The premise that you are describing does not apply to this situation though. If you bet 100% of your entire bankroll every time, it is inevitable you would go bust eventually. And likewise, if you bet a fixed percent of your bankroll every time, you will also eventually go bust. You could easily make some simple modifications to my script to demonstrate this. But if you commit to betting the same amount every time, you can create a situation where going bust is statistically impossible. Allowing players to set the stake for a table allows them to exploit this. As long as you keep betting a fixed amount, over time the percentage of your bankroll that you're risking will decrease, lowering your odds of busting. And in practice can gradually increase the amount you bet, as long as the average increase percent is less than that of your statistical edge. Of course, the less you increase it the less risk you expose yourself it, but you will eventually be able to raise it a fair amount without ever putting yourself in serious danger of busting.

True, but if the other player never folds...

Anyway, the reason why going all in is such a good idea in situation where you know you have an edge and you know the other player will call is that it creates certainty. You have created a situation, where the longer you play, the more you will win. This would not necessarily be true if you didn't go all in, because uncertainty would give the other player room to manipulate your decision making by bluffing later in the game or by folding to limit their losses. Even if the edge you have is small, over the long term you can't lose. Don't believe me? Here is a simple ruby program that proves it:

You'll note that when you run the program with no statistical edge, the results are all over the place. You can win a large amount of money, or lose a large amount, or just about break even. But try changing the edge to 0.01 (1%) and see what happens. No matter how many times you run the program, you will always win. It's possible you may go negative at some points, but the longer you play, the less likely it is you will end up behind. After playing a million games, the chances of being behind are almost a statistical impossibility. It's true you might be able to make more money by playing with a more elaborate betting strategy, but honestly, why take the risk? Any other betting strategy you could use would make you more susceptible to manipulation, and if the other player's betting strategy was as good as yours it wouldn't give you an advantage either way.

Ah, yes, I see your point. But in my case, I already know the odds are in my favor, and getting another player to match my all in when I have a stronger position basically guarantees I will win more money over the long run than I lose. Even more so if I can exploit their picking strategy in the second/third rounds. It's the same trap set by casinos. Even if you are able to win some of the time, the longer you play the more you will lose. Calling my all ins so much when your position is weaker than mine is like giving me free money.

I think if players who were better at the game consistently played games that were higher stakes it might make things more interesting. One idea would be to create a ranking system that would be tied to the amount of money you were allowed to bet in a match. Things could be set up so a lower ranked player could only join or create matches in low stakes rooms, while a high ranked player would only be able to create and join high stakes matches. Of course, making it clear what the skill of the other player was before you entered their room would create an incentive to game the system, so I'm not sure how well this would work in practice.

In Cordial Minuet, the problem becomes that because the other player actually has fairly reliable information on how strong their position is compared to yours, it's extremely difficult to entrap a skilled player. Most of the money I lost from my original buy-in was because I spent some time studying how other players bluff, which of course required me to call in situations where I knew it wasn't the greatest idea. What I found is that really good players rarely bluff. So in general, unless you have a very good reason to believe the other player is bluffing, or you know your current position is one with good potential to win, you should always fold to a raise. On the other hand, if you know you have a strong position, you probably shouldn't raise anyway because another strong player would just fold unless they had reason to believe their position was even better. Worse still, if they then re-raise against you, you could be put in the position of either folding away all the money you just bet, or completely committing yourself to a round you aren't sure you will win. I'm not entirely sure the best betting strategy in Cordial Minuet isn't to just never be the first person to raise.

As far as the betting game, I am beginning to notice a trend. When the betting is mostly done with low stakes bets (which I believe is always the best strategy as it minimizes risk to yourself, and increases the risk to the other player if they bet more than you), you will actually tend to lose more money from the tribute than you will to the other player. For example, it's not uncommon at all to see games where one player is breaking even and the other player is losing money. This creates a game of chicken, because the lower the losing player's stack becomes, the more pressure they feel to take risks in order to avoid losing more money to the tribute. I cannot stress this enough, I think that taking tributes out of individual rounds is a bad decision in terms of the future of the game. It makes much more sense to just take a tribute from both players at the beginning of the match. Not only does this encourage more sensible betting, but it also discourages players from abandoning matches the second they gain a lead.

Edit: No wait I'm an idiot. ;P It should be taken at the end of a match depending on how much money has changed hands. If you take it at the beginning the other player could leave the match right away just to troll you.

I also feel that I should point out, I am noticing that variations on my original strategy of giving the other player their worst possible row are emerging as a dominant strategy. While you can't do it all the time or it would become predictable and take away your advantage, doing it as often as you can get away with it appears to be better strategy than any other that I have observed. Even if the picking game couldn't be broken to the point of achieving pure Nash Equilibrium, the existence of a dominant strategy like this takes so much out of the picking game that there is very little room for alternative strategies.

I was also wondering if you read my post on the Monty Hall problem and how it relates to your odds with winning in the final betting round. In my opinion, given the fact that the decision of which number you show the other player will be biased toward presenting your position as stronger (or weaker if you are certain you will win), the possible outcomes shown in the final round will always give a less reliable indication of your odds of winning than the possible outcomes shown before the numbers are revealed. The only way this could not be true is if you selected your number to show the other players randomly, but no one would ever do this because there is no advantage in doing so. This renders the final betting round pointless in many cases, because the information available in the final round is almost always less reliable than the previous round. The only exceptions are the cases where the information available in the final round tells you for certain you will win or lose. But in most cases you would have already known this before the numbers were revealed anyway, or at least known if one player had a much higher chance of winning than the other. So the whole showing a number thing just becomes reduced to a gimmick, and if both players are playing correctly, they should usually ignore the possible outcomes shown in the final round completely.

I know that you can't beat the other player using Nash Equilibrium, at least not in terms of picking. But there would probably always be enough human players around with bad betting strategies that you could find and exploit, and eliminating picking as a factor would make it much more simple to do so. Putting that aside, as I said earlier, I believe that in any case the best possible strategy for actually winning would be to use an equilibrium strategy as you default strategy, and then at random times deviate from it in order to pick columns that are "better" than the ones your equilibrium strategy tells you to pick. The ideal amount of deviation would be the absolute minimum required to increase your odds to the point where you could be profitable. I believe that it would be next to impossible for a human to effectively counter this tactic, so it would limit the ability of any human player to win if bots were involved.

Oh, and I forgot about the ladder strategy. I hadn't even heard of the superko strategy. The point that I was trying to make, is that I think there needs to be some aspect to Cordial Minuet that adds some additional asymmetry to the picking. Something it would not be trivial for a bot to deal with. However, I have no idea what that should be.

And I'm probably not going to be the one to write it. The fact that I know it exists is enough to discourage me for seriously perusing this game, and I'm not really that interesting in writing a bot to farm money from human players. You should also consider, that even if achieving Nash Equilibrium in computationally infeasible (and I don't think it is for various reasons), you could still probably write a program that can approximate it better than any human. Even coming up with a weighting algorithm that gets better results than the average human in the first round would be enough to make it difficult for an unassisted human to be successful at this game.

Betting strategies are complex, it's true. But part of what makes them complex is the game that you're betting on. For example, betting on a coin flip is technically gambling, but unless someone is using a rigged coin, no one is going to make any money in the long term, and there really is no betting strategy that is necessarily better than any other. In Cordial Minuet, by playing to achieve Nash Equilibrium you can actually deconstruct the game into a random number generator. Every round, you get a number that is either lower or higher than the average number, and your bet should be based on the value of that number. In practice it is more complicated, since to determine the true value of your position you must take into account not only what number you have, but what numbers you know the other player cannot have, and what choices you will have left in future rounds. Nevertheless, by correctly accounting for this information, it is possible to gain perfect statistical information on the value of your position. In other words, you have a number that tells you the odds that you will win. So in that sense, once you have successfully deconstructed the game to the point where you can achieve Nash Equilibrium, there is no game beyond the betting strategy.

But okay, so that's not really any different that Texas Hold'em, is it? In Texas Hold 'em, it's possible to calculate your odds of winning at every step of the game too. But the difference is (and this is key) in Cordial Minuet you often definitely know that the value if your position is better than the other players' (or at least probably is), while in Texas Hold 'em having that information is relatively rare. This greatly simplifies the betting strategy, and I've been narrowing in on something I believe is an ideal strategy for betting. I know at least, that if I can control the pace of the betting enough that the other player will not raise too often at times when there is a small gap between the value of our positions, I can probably come out ahead. But it's still very much a work in progress.

What I actually wasn't thinking of, and what I really should have realized sooner, is that any fundamentally fair game of chance results in a 50% chance of winning. It really sounds obvious when you say it that way, but still... A coin flip is a fundamentally fair game of chance, and the odds of winning are 50%. If you are playing Cordial Minuet in such a way that you have perfect information (or at least information that is not less accurate than the other players') about your chances of winning, and you play with a picking strategy designed to achieve Nash Equilibrium, the game become little more than a series of coin flips.

But what I was saying earlier about introducing a subtle bias is a point I should elaborate on. In Cordial Minuet there are choices that are objectively better than other choices. This is why you will actually tend to get lower values than the other player if you just pick randomly. Choices that give you a higher possible final score in relation to your opponent's are better than choices that give them a higher possible final score. By correctly weighing the relative values of your column choices, you can then pick a random column based on those weights, and achieve Nash Equilibrium. The second and third rounds are a bit more complex than the first, because they have additional asymmetric aspects you need to take into account, but it should still be possible to choose in a way that achieves equilibrium. Once you know how to generate weights that achieve equilibrium in each round, you can actually introduce a subtle bias in your choices that tends to favor columns that are better for you even more than their weight suggests you should. I believe that this is what you could call an optimal strategy for playing to win, because it would be incredibly difficult to figure out what you were doing. Another human player would probably never be able to make better choices without risking exposing an exploitable bias in their behavior. A machine playing for perfect Nash Equilibrium on the other hand, would on average beat you, but assuming that more players are not picking for equilibrium than are, you should still come out ahead.

Even if that were true, how would you address the possibility of bots playing to force a draw in order to farm money for the server owner?

I would just like to say, that personally, I am 100% in favor of tables with higher buy-ins and low bet limits. But then again, I'm trying to find ways to break the game, so that's probably not a good sign.

Well, my goal right now is to figure out a strategy that "breaks" the game. In other words, a strategy where the best counter strategy the other player could use would result in a 50% or less chance of winning. In that sense, I'm not really looking for big payouts, I'm looking for ways to exploit the game that will guarantee profit (if only a little bit). That's what testers are supposed to do right? It's not like I'm going to get rich playing $0.01 matches.

But at the moment, I'm thinking that my first idea was a failure, because it seems that when the other player does not pick the highest number in the column with the lowest high number consistently, losing the advantage of knowing which number they picked takes away too much of your advantage for the strategy to by viable. Now I'm trying something new, and I defy anyone to figure out what strategy I am actually using.