Solvability

Designing competitive strategy games is a constant fight against solvability. It's a struggle to make a system simple enough to understand, yet complex enough that players can't figure out the best way to play and then always play that same way.

 
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Pure Solution vs. Mixed Solution

It's a much different situation if a game has a pure solution rather than a mixed solution. To understand why, we'll first have to define those terms.

A pure strategy is a complete definition of how to play a game. It's a set of instructions describing the move the player should make for every situation they could face. If a certain pure strategy is the best way to play the game, we'll call that a pure solution. If you know a pure solution for a game, it's hardly a game anymore because there aren't any actual decisions left for you; you simply follow the instructions of the pure solution.

A mixed strategy is a set of pure strategies where you assign a probability to each one. So instead of your instructions being something like "If the opponent does X, I'll do Y," it's more like "If the opponent does X, I'll do Y 30% of the time and Z 70% of the time." If a certain mixed strategy is the optimal way to play, we'll call that a mixed solution.

If you knew the mixed solution for a game, it sounds like just as bad of a situation as if you knew a pure solution. You still wouldn't be making any decisions, just randomizing across a set of choices. But this is NOT correct; there's still a lot for you to do in the case of a mixed solution. To understand why we'll have to look more closely on what playing "optimally" really means.

Playing Optimally

We said that if you had several possible mixed strategies, the one that lets you play optimally we'll call a mixed solution (this is also a Nash Equilibrium). There's a lot of potential confusion there because the word optimal has two meanings: an ordinary English meaning and a specific mathematical definition. This article is always referring to the mathematics meaning, NOT the everyday usage of the word that means "the best way to play." The math meaning is that playing optimally is playing least exploitably.

Let's see what playing exploitably looks like. If you were playing rock, paper, scissors and you decided to play rock 100% of the time, that is extremely exploitable. Your opponent could pick up on that and shift to playing paper 100% of the time. Your opponent can exploit your strategy so fully that your win rate goes down to 0%. If instead you play rock only 80% of the time (and paper 10%; scissors 10%), that's still a bad idea but it's a bit less exploitable. Your opponent could still play paper 100% of the time, but at least you'll win 10% of the time, rather than 0%.

If you want to be the least exploitable possible, you'll have to play each option 33% of the time. If you do that, there's no strategy your opponent can use to do better than you. That's the optimal mixed strategy to simple RPS.

Optimal Is Not "Best"

Playing optimally sounds like the best you can do, but if your goal is to win a tournament, then playing optimally is very likely not to be the best idea. Imagine you entered a rock, paper, scissors tournament and face a player who is known to play rock 100% of the time and they do exactly that against you. If you play optimally, you'll play each option 33%, so each hand of RPS there's a 33% chance you'll lose. Meanwhile, another player in the tournament could choose to play 100% paper when facing the 100% rock player. Your so-called optimal strategy has a much higher chance of losing and getting you eliminated from the tournament than if you had played 100% paper, too.

By choosing to play optimally, you gave up a massive advantage that was right there for you to take. Your opponent was ridiculously exploitable, but you chose not to capitalize on it. That's poor play if your goal is to win the tournament. This is an extreme example but the concept is still true even if the opponent was playing 40% rock, or even 35%.

What if you do play 100% paper against the 100% rock player, but after several rounds of play they change their strategy? It's possible that they could exploit you because now you strayed from optimal play. Yes, that's correct, but it's still worth it to try. If you're worried about your opponent changing their strategy to exploit you, then you don't have to go all the way from 33% paper to 100% paper. If you went up to, say, 40% then you're more likely to win this match than someone who stuck to 33%, but you're still not all that exploitable. Also, how good is your opponent at a) recognizing that you strayed from optimal and b) correctly implementing a strategy against that? It's entirely possible that you are better at those things, in which case you should definitely exploit their strategy. As they slowly adjust to that, you adjust faster.

Donkeyspace

The term donkeyspace, coined by Frank Lantz, describes the space of suboptimal plays. As described in the previous section, a good player should intentionally enter donkeyspace (in other words: play in an exploitable way) in order to exploit opponents who are also playing in donkeyspace. If both players are good, they each might dance through different regions of donkeyspace, jockeying for advantages.

It's important to have some perspective here. You might be thinking that everyone is going to play optimally so there's no dance through donkeyspace in high level play. That's laughable if you think about actual competitive games though. First, even at a high level, it's very common for players to play far from optimal. Second, it's highly unlikely that any—much less ALL—opponents will be playing optimally or even close to it. In a good competitive game, it's incredibly difficult to know what optimal play even is. There can be rules of thumb, but to know exactly the right probabilities in which to play a mixed strategy of exactly the right moves in a specific game state that could have thousands of variables? Even in a popular, well-understood game like Poker, optimal play is not known perfectly and in practice players stray from it considerably. Knowing optimal play in Pandante or Yomi is way more hopeless than in Poker.

Remember that when other players are playing non-optimally, even if you did know how to play optimally and even if you could perfectly execute the mixed solution, you still need to closely monitor your opponents and react to their styles in order to maximize your win rate.

There are several other reasons why actually executing an optimal mixed strategy is extremely difficult, even if you did somehow know what it was:

  1. People are very bad at actually playing randomly, so it would be very difficult to choose some certain option 42.3% of the time, for example.
  2. When people fail to play randomly, they are probably falling back on tendencies they do not know they have, but that you can detect and exploit.
  3. People cannot help but let their personalities spill over into decisions about how conservative or risky they are.
  4. If it's a real-time game, then skills at timing and physically executing the right moves (such as a difficult combo in a fighting game or a precise shot in a shooter) mean no one is ever anywhere near playing mathematically optimally.

Item 2 on that list is especially interesting. In two studies by Lewicki, et al (1997 and 1998), they demonstrated that people learn patterns without knowing that they learned them and without being able to explain or express what they learned. Subjects were shown four quadrants of numbers and had to press one of four buttons corresponding to the quadrant containing a certain number. They did several trials of this, but weren't told that the location of the numbers across trials was not random. The locations followed a complex set of 10 rules. As subjects did more and more trials, they were able to perform more and more quickly, yet they weren't aware of there being any pattern and no subjects could explain a pattern even after they were informed one existed and even when they were offered $100. Furthermore, when the underlying pattern was secretly replaced with pure randomness, the subjects immediately did far worse. Hilariously, even the subjects who were fellow psychology professors in Lewicki's department who were aware of Lewicki's research were adamant in their belief that the trials containing a secret pattern were actually random. They learned to exploit the pattern, yet were convinced it didn't exist.

The point is that your unconscious mind will make you perform mixed strategies imperfectly, and you'll fall into patterns you won't know you're doing. And then your opponent will pick up on those patterns and be able to exploit them, even if your opponent isn't aware that's happening. Mixed strategy games and dances through donkeyspace involve interesting battles of unconscious minds vs. other unconscious minds in addition to the part where conscious minds might disagree on what optimal play even is.

Pure Solution Games Degenerate Faster Than Mixed Solution Games

So in a game with a mixed solution, you still must be highly sensitive to what your opponent is doing. You have be able to detect how far they are straying for optimal play and then you have to be able to correctly counter that strategy. These are very difficult things to do and they involve, among other things, your unconscious mind picking up subtle patterns.

In a game with a pure solution, you do not have to care what your opponent will do, ever. If you know that pure solution, it doesn't matter what the opponent tends to do or what you think is in their mind, etc. You should follow the optimal script and there's no gameplay left.

It's also very important to think about how a game with a mixed solution looks vs. one with a pure solution when the playerbase is on their way to knowing that solution but isn't fully there yet. They're learning more and more about each game over time, they're approximating what optimal play is more and more closely. For the game with the pure solution, that means pockets of the game here and there become entirely about memorization and not about what the opponent is doing. For example, solved endgames in Chess are this way (but not Chess 2, because the midline invasion rule prevents all those solved endgames from happening). Openings in Chess (but not Chess 2) are another good example of that. As more and more is known about Chess over the years, the more structured the opening books become (the set of known-good opening moves) and the more important memorizing them becomes so that you don't enter the midgame at too much disadvantage.

Meanwhile, when we get closer to an approximation of a mixed solution—in Poker, Pandante, or Yomi for example—these games do not start to collapse into memorization. They are still about being very responsive to what your opponent is doing. And while these approximations get closer to a complete mixed solution over time (which will not happen for Yomi in our lifetimes), remember that EVERYONE is in donkeyspace. Even when there are lots of good players, they aren't literally playing optimally at every single step. Everyone is in some sort of donkeyspace and they disagree on who is where. Playing the game is in some way a method of resolving that debate. "I think I should block 42% of the time in this particular situation," while another player believes blocking 60% is correct. They each try to exploit the other's incorrect approximation, if they can even detect it in the first place.

From a design standpoint, games with mixed solutions have an inherent advantage in fighting against solvability. It's much safer to design a game that is still very interesting even if solved than it is to design a game that necessarily degenerates closer and closer to pure memorization and no decisions as it gets closer to being solved.

Making Pure Solution Games vs. Mixed Solution Games

As I've explained, pure solution games are dangerous to make. On the one hand, if your game is deep enough then you could delay people finding the solutions for a very long time. Chess and Go have been around for many centuries without full solutions being known. On the other hand, you'd be hard-pressed to make a pure solution game that stands up anywhere near as long as those games. Checkers is already solved, for example. Furthermore, even if you have a game as deep as Chess, Chess shows that memorization becomes more and more of what a pure solution game is about as the playerbase gets better. That's an unfortunate fate for a competitive multiplayer game.

There's also some irony there. If I told you that a certain game had perfect information (you know the full state of the game at each moment you have to make a decision) and that it had no randomness, then you'd probably say it sounds very skill-based. If you like skill-based games, you'd say we're off to a good start. But actually, we just guaranteed that this game has a pure solution and that it will necessarily become LESS about skill (decisions in-the-moment) and more about memorization as the game develops. A similar game that had some unknown elements, hidden information, and/or randomness could actually be more skill-testing, not less skill-testing.

So in creating a design, I recommend looking for what those unknown elements will be. What that hidden information will be. What that random element will be. Randomness has a real stigma, but it's important to understand that it's a valid tool to keep your game out of the dangerous pure solution category.

Kongai

My game Kongai is an example of that. You make two decisions per turn and each of those decisions is double blind. That means you make the decision at the same time the opponent makes theirs, then you simultaneously reveal those decisions. This very much helps against solvability (it's no longer a perfect information game), but even then, the game would be dangerously solvable without some other unknown elements. I used randomness in hit rates (just like in the Pokemon game Kongai is based on) as well as randomness in proc rates (the chance that a move does a special thing on hit). This worked extremely well in fighting against solvability. Those hit and proc percentages make it very, very difficult to compute the possibility tree several moves ahead.

Some Kongai players intent on finding mixed solutions had to zero in on the most pared down, toy examples you can imagine. They tried to find out optimal play given a certain lineup of characters vs another certain lineup, with a certain set of items equipped, in the endgame only when each team was down to its last character so no switching or intercepting was possible, and hit points were down to the end. In this microscopic portion of the game, it took them a dozen pages of analysis to determine the right play. Doing this for the real full game is basically unthinkable.

Conclusion

Competitive multiplayer games have to strive to be as unsolvable as possible while at the same time being understandable to players. Games with pure solutions may seem skill-based, but over time will necessarily degenerate to becoming pure memorization. Meanwhile, mixed solution games remain strategically interesting far, far longer.

In order to make a game with a mixed solution, incorporate some sort of unknown elements, hidden information, or randomness. If your game is real-time rather than turn-based, you're even better off.

Designing Pandante

I didn’t intend for Pandante to be “Poker 2” but a lot of people, including semi-pro poker players, have told me they see it as exactly that. The original point of making Pandante was to design a game around the fun idea that if you claim something is true and no one challenges you, then it is!

It sure would be fun to do that in real life. You could claim that you’re an airline pilot, then if no one calls you out, you are. Or claim that you have diamonds in your pocket. If people believe you, then you do. The downside is that you if people do call you out, you’ll have to pay them off if you were lying, but that’s a small matter given the potential upside.

A gambling game is a natural foundation to explore this mechanic, so that’s why Pandante has a similar skeleton as Texas Holdem. In both games, you get two private cards and all players share a set of five community cards. You try to make a good hand out of some combination of your private cards and the public cards.

Player Elimination and Folding

Poker has player elimination, meaning that if you go down to $0, you’re eliminated from the game. That makes sense for a gambling game and Pandante does have a mode that lets you play that way, but it’s a really undesirable quality when it comes to fun factor. It’s a more social and fun experience if everyone is involved in the game.

Folding is when you give up on the current hand and sit the rest of it out, but you’re still in the game for the next hand. It makes sense for this option to exist, but if it’s smart play to fold a huge percentage of the time, like way over 50% of the time, then it’s problematic for the same reason as player elimination: there’s too much time over the course of a game where you’re not involved.

Removing player elimination and greatly reducing folding are good design goals in and of themselves. For Pandante, these design goals were much more important than usual though. Remember, the core concept is that you can claim things to be true and they become true unless someone calls you out. That concept doesn’t actually work if too many other players have already folded this hand or if too many players were eliminated from the game. We want everyone involved and still playing so that everyone can participate in calling out your lies, or in choosing to let you get away with them. That’s the fun!

Pandante doesn’t have player elimination unless you play the seriousface gambling mode. In the regular mode though, the game ends when any player ends a gambit (that’s what each round of play is called) with more than X gold. X depends on how many players are in the game. No matter how little gold you have, you always still have a chance of winning. If you win several gambits in a row, you’re denying the leading player from getting any more gold and you can always make a comeback. In words, Pandante has no lame duck situation: a situation where you're so far behind that you can't win the game, yet you're still stuck playing it. That's an important thing to avoid in game design.

Eliminating Player Elimination

If you go below 20 gold, then a magic fairy will refill you back to 20 next gambit. How fun! If you go below 0, the fairy will pay off all your debts and refill you back to 20 gold, but there’s a catch. If the fairy has to pay any of your debts (meaning you went below 0), then you have to spend the next gambit frolicing with the fairy. You don’t get to play that gambit at all, then the following gambit you return to play. This is a big enough disincentive that it prevents players from playing too recklessly where they throw all their money away on obviously stupid bets.

This rule has worked out really well, partly because it's fun and flavorful and people like to joke about frolicing with the fairy. It rarely actually happens though. It exists as a deterrent to keep your incentives aligned correctly, and to make sure that even players who are almost out of gold still have real decisions about whether they should lie or not. 

Reducing Folding

There’s a lot of things in Pandante that all work together to reduce the percentage of times you should fold. The first and simplest is that you literally aren’t even allowed to fold right away in a gambit.

Can't Fold Early

In Texas Holdem, you have a chance to bet before you see any community cards, and the only thing you have to go on is your two private cards. This is a bad time to even have bets happen, so so-called pre-flop bets don’t exist in Pandante. Whenever you bet, you have more to go on which is both better for skill and better for fun. It also means you can’t fold at that point.

When the first three community cards are revealed, it's called the splash. The fourth card is called the paws, and the fifth is called the tail. Players can bet or fold after the splash, after the paws, and after the tail. After the splash and and after the paws, they also have a chance to buy snacks.

Snacks

Another Pandante feature called snacks greatly reduces your need to fold and also increases the skill ceiling and fun factor of the game. After the splash and paws, you can buy snacks if you want. If you do, you get to draw a card, then discard a card. Because it’s possible to improve your hand this way, many more opening hands are reasonable to keep that you’d otherwise have to fold on if snacks didn’t exist.

Whether it’s worth it to buy snacks is often a tough call. During each of the three betting rounds (after the splash, paws, and tail), you put your bet on your board on the space corresponding to the hand you’re claiming to have. As the gambit goes on, you cannot claim a lower hand later; you must either maintain your claim or raise to a higher hand each time. The higher your claim, the cheaper your snacks will be. If you claim to have a better hand than anyone at the table (or tied for highest), you get snacks for free. This baits you into lying about your hand, which is a fun dynamic. Because you can’t actually get called out for lying until the end of the gambit, this feels a lot like buying things with a credit card. Lying about your hand now gives you a great benefit now: snacks (draw a card, then discard a card). You won’t have to “pay for it” until the end of the gambit when people might realize you didn’t actually make that hand. But if you DO make it, then you don’t have to pay that credit card bill after all!

If you aren’t the highest claimed hand, snacks costs 2 gold per space below the highest (see the board below). So the worse you claim your hand is, the more you’ll have to pay for snacks to improve it, but at least you’re safer from being called out at the end.

The Sleep ability is another factor that reduces the amount of folding, but lets cover that when we get to the rest of the abilities in Pandante.

All of these factors work together such that skilled play involves far less folding than in Poker.

The Possible Hands

Early in development, the possible hands you could get in Pandante were more similar to the hands in poker. There was a big opportunity for improvement here though. The problem with using a standard set of poker hands was that the set of “interesting hands” was way too small. Pandante has 10 spaces on the board to represent 10 possible hands, and too many of those were either way too easy to get or way too hard.

Having straight flush and five of a kind are fun things to have, and they should be there, but they are very very difficult to get. Most games you play, these hands won’t come up. So in practical terms, we’re down to 8 spaces now. On the opposite end of the board, we had “high card (meaning no pair) and two pair as spaces. You always have at least high card though. And having spaces for high card, pair, two pair, and three of kind was really a waste. ALL of those are so easy to get that they kind of aren’t even real choices.

What we need are more hands that are slightly hard to get, but that still come up frequently. Full house and 4 of a kind are both already in that good zone. The final version of Pandante adds two more hands special to Pandante that are also in that good, common probability zone: the floosh and the rainbow straight. A floosh is a miniature flush, only 4 cards of a single color rather than 5. A rainbow straight is a straight where each card is a different color (note that there are 6 colors in Pandante).

Having more hands in the “slightly difficult, but still possible” probability zone not only adds more fun, but it also reduces folding. Imagine if the only possible hands were either ridiculously easy to get or way below 1%. In that case, having an easy hand is useless because everyone has it. And the hard hands are so hard to get that you immediately see you don’t have them and won’t have them so you might as well give up now. Pandante gets a lot of fun factor out of having several realistically achievable hands, plus the ability to actually make those hands by getting more cards from snacks and more cards from abilities.

Another interesting property of the list of possible hands in Pandante is that improving your hand tends to let you jump ahead more than one space in the list. For example, three-of-a-kind is hand #2 on the board, and it's natural to upgrade that hand to a full house. But full house isn't hand #3; it's actually #4. This lets you skip past players who claimed a straight (hand #3) if you upgrade your hand. But then straight, #3 in the list, can sometimes be upgraded to rainbow straight, which is #6. Doing that lets the straight player leapfrog past full house and floosh players. People claiming floosh can leapfrog too though! Anyone with a floosh is in a position to possibly upgrade to a flush, and that would let them skip past players claiming rainbows straight or four-of-a-kind.

The careful ordering of hands creates interesting dynamics where even players with worse hands are usually able to represent that they can beat players with slightly better hands. 

Abilities

Special abilities are important to have because they raise the skill ceiling of the game. They give you even more room than you have in Human Poker to outplay worse players. They also add enough mischief to be fun, and  Each one does a fairly simple thing and they cover the various kinds of actions you really want to do.

The RAISE ability is scary because it can force people out of the gambit, but it doesn’t improve your hand.

The DRAW, NEW HAND, and ADD abilities are important because they’re the ones that change what hand you have. Players who use those abilities are likely trying to make a hand they claimed to have, but didn’t. That’s a tell, so keep an eye people who use those. NEW HAND in particular is a pretty desperate move!

The PEEK ability is a good finesse move. Often you really really need to know if the highest claimed hand is real or fake and this gives you a good clue about that.

The SLEEP ability is basically a "super fold." You get a head start on the next gambit, but you can only do SLEEP if you stayed in the current gambit until you reached the ability phase. The existence of SLEEP means you're more likely to stay in the gambit to see if things work out, since you have a safety net to fall back on if they don't.

That’s only the surface of how you can use these abilities. There’s a lot of nuance and tricks to it.

Golden Panda Coin

Some of the most exciting moments in poker are when you do a huge bluff and actually get away with it. You feel like a rock star. The problem is that not only is this moment not celebrated by the game system, it’s actually suppressed. In this amazing situation, your best move is to keep it secret that you bluffed. That’s the opposite of hype. It would be so much better if your strategically best move here was to throw it in everyone’s faces that you bluffed and got away with it, and for something amazing to happen. In Pandante, it does.

When you win a hand by lying, you get the Golden Panda Coin. The coin is a one-time use powerup that lets you draw 5 cards and discard 5 cards as an ability on a future gambit. It's fun and exciting to get and when you do, you better use it soon; if another player wins by lying, they steal the coin from you if you didn't use it yet. When you do the most fun thing in the game (lying and getting away with it), you get to make it public and have the most fun.

Joy

A lot of the details that went into Pandante’s design are mathy things that make sure the incentives of how you SHOULD play line up with what’s really fun to do. I also needed to make sure the game isn’t degenerate somehow, that it's rock-solid enough to play with real money, and that at high level it’s a strategically interesting game. It has a skill ceiling quite a bit higher than poker because there are more kinds of decisions you can make, so there’s more ways to show you’re skilled and separate yourself from bad players.

That said, Pandante had one of the most pleasant surprises I’ve encountered playing games. After all those geeky details were hammered out, it created an experience that non-experts REALLY enjoy. At least half the people are lying at least half the time, and everyone kind of knows it. That creates a light-hearted feel that’s very much unlike the psychological warfare of poker. Everyone is involved for most of the game, and players are usually laughing and smiling, far moreso than in any of my other games.

Pandante seems to bring joy to players in a way that I haven’t achieved in my other games. I usually focus on strategy that holds up during high level play, but I sort of stumbled upon that lining up with happiness in Pandante. So it really has two very different demographics: those who want a fun social, laughy game and those who want a serious, real-money gambling game that’s more fun and has a higher skill ceiling than poker.

You can get Pandante here. (The deluxe version is super nice and comes with professional grade, clay poker chips.)