Exploring the concept of degenerate strategies and degenerate game states in board games and TTRPGs. What does it mean and why does it matter? And how is it connected to the Black Lotus card in Magic?
I wonder if a good simple example is found in noughts and crosses (tic tac toe). It’s a common introductory game that just needs pencil and paper, but kids quickly understand that if you start in the centre position it is almost impossible to lose. Against experienced players it is always a draw, but you might find someone unfamiliar who gifts you the victory.
Soon people stop playing the game because the result is a foregone conclusion. Seems like a kind of degenerate state to me?
Incidentally I found a productive way of keeping playing a little longer was to start in a corner - people were less likely to be familiar with that, and were more likely to fall into traps!
So, as I understand it, in game theory, a degenerate game is one where it kind of doesn't matter what the one player does, if the first player does a certain thing. Game theory is usually expressed in payoff matrices and vectors, so one example is there being a column in the payoff matrix that the first player can choose, such that the row player gets the same value no matter what they do.
The game is "degenerate" because certain game theory concepts just don't work with degenerate games - in much the same way that it doesn't make sense to work with the area of a degenerate triangle - it's actually a line so the area is 0.
Imagine a game where the first player can hide a ball in their left hand or right hand, and the other player can choose a hand, they win 1 point if they choose the hand the ball is in but lose 1 point if they choose wrongly. The first player might not hide the ball in either hand, and instead eat it (You can argue whether that constitutes a violation of the rules, but assume it's not). That strategy means the second player cannot make a meaningful choice between left and right. They may as well not be playing. The game has degenerated into a state where nothing that player does matters any more.
That's what I think is key to the concept of a degenerate game or game state in our sense. It doesn't matter how awesome your combat rules are if by turtling you can always avoid it. You've reduced the game by choosing a strategy that eliminates that part of the game as if it never existed.
I don’t know about degenerate, but isn’t what you are describing weak strategic dominance? If the first player must put the ball in box 1, 2, or 3 and the second player must say whether the ball is in box 1 or box 2, then the first player strategy of always placing the ball in box 3 weakly dominates all her other strategies — it produces an outcome at least as good as any other strategy. (I am no mathematician. I just read this: <https://en.wikipedia.org/wiki/Strategic_dominance>.)
Actually eating the ball is slightly more interesting. How many can I eat before I am very ill? I know what to do to win, but actually doing it presents difficulties. Is this how a game becomes a sport? ;)
Of course — I say with more confidence than I feel — someone may discover a strategy for a game which no one knows how to beat even though that strategy is not even weakly dominant (over all other possible strategies).
I don't think so, even if your strategy is dominant, it would still matter what I choose in a non-degenerate game. The degenerate game isn't because you have a winning strategy, just that you have the ability to make my choices irrelevant.
“B strictly dominates (>) A: choosing B always gives a better outcome than choosing A, no matter what the other players do.”
It is true that even strict dominance allows that the second player’s action can have some effect. Consider a modified example: if the first player puts the ball in box 1 or box 2, she scores one point if the second player picks the other box (else zero); if the first player puts the ball in box 3, she scores two points if the second player picks box 1, but three points otherwise. Always putting the ball in box 3 strictly dominates putting it in box 1, putting it in box 2, and sometimes putting it in box 1, sometimes in box 2. The second player can affect how quickly the first player racks up points, but that doesn’t affect the first player’s choice of strategy. (If the first player had another strategy that was sometimes better in the case of some second player action, then “always put the ball in box 3” would not dominate it.)
Is the modified game “degenerate”? I don’t have a dog in that fight, but it is still boring for both players.
My wife and I found this when playing Sky Traders as a 2 player game.
Theoretically, it's a pretty standard pick-up-and-deliver game. Except you accumulate sludge, which you're supposed to go out of your way to deposit in a safe location. Dropping sludge on a city gets you all your points, saves you time... but tallies up a bounty on you. Other players can hunt you down for that bounty.
At 3+ players, it's always worth it to collect on a bounty. At 2 players, the bounties aren't high enough to offset the points you can get from taking your own turn. So in order to stay competitive, you ALSO need to dump sludge on cities. And the game becomes less about delivering goods, and more about competitively dropping sludge onto unsuspecting cities, and hoping a bounty hunter isn't randomly played.
This circumvents about half of the intended mechanics, and turns it into a game we call "Sludge Traders".
I might have some details wrong, it's been a few years since I've played this one (Star Wars: Outer Rim replaced it as our pick-and-deliver of choice)
as someone who loves both mathematics and language/the importance of words/definitions, this article was a lovely gift :D your articles never disappoint, but this one was as if made for me ^^ thank you for sharing <3
funny thing, i have never played Magic the Gathering, and know next to nothing about the game; though just the other day i heard Black Lotus being referenced on a ttrpg actual play ^^ finally looked it up when reading this article! (i played a ton of Yu-Gi-Oh growing up though, so i still was/am a big card game nerd x3)
now i want to find that counterexample of a degenerate game state that makes the game not worse – would love to find one that arguably makes it better, haha xD counterexamples are fun :) can't think of one off the top of my head though.
i voted for the "inherently changes the game" – going by the math degeneracy concept first and foremost :) [i'm personally most familiar with it from probability theory, see eg the wikipedia article about degenerate distributions! (as i specialise in probability/statistics)]
I voted for "follows the rules as written" in the poll because when I think of degeneracy, I think of ways of play that are technically 'correct' but result in problems. For example, think of the very common 'Free Parking' house rule in Monopoly. It makes the game play for way longer, and results in a generally worse play experience, but I don't consider it 'degenerate,' I consider it to be 'wrong' because it isn't the way the rules indicate how to play.
I'd have to think on it for a while, but it seems to me that some of the most fun I've had in games was exploring the possibility of 'breaking' them. Like finding that "One Stat" to rule them all in early console RPGs or some glitch or strategy that the designers apparently didn't catch during testing.
And then there are some games (like original Magic: The Gathering) that were seemingly designed to invite players to find and explore dueling degenerate strategies. Which of course led to the banning of some of those strategies (more specifically, the cards behind them) for tournament play.
It's fun (at least briefly) being on the winning end of a broken game; not so much fun being the one getting gamed by it.
Agree completely. That's why I think "makes the game less fun" isn't a particularly helpful criterion when identifying degenerate states in games. Someone might very much be having fun! ;)
Great article. I still shudder when I think about playing Dominion. Very hackable and therefore incredibly unfun once players been exploiting the game mechanics. Once someone exclaims, "It's not against the rules!" you know the rest of the game session (any game, not just Dominion) is going to quickly devolve into a mess of frustration (except for the winner of course).
Have you ever written an article about Fantasy Flight Games' X-Wing (1e)? I witnessed innumerable debates over the modal verbs printed on the game's cards. Wild.
As a child, I discovered Big Money independently, although I used the Mine card to upgrade my money using my Actions. Now I am really into the Chapel, trashing my deck until I have the most compact money-printing, province-buying machine possible.
In Magic the Gathering, Infinite Combos are often seen as degenerate. Winning the game out of nowhere unless your opponent can stop you at instant speed. However, in commander EDH format, it is seen as a part of the very fast, high-power metagame.
Would you consider writing an article about Player Skill vs Character Skill, or so called "Metagaming?" Would love to get your take!
You've got me thinking about how this intersects heavily with kinds of fun. Seeking dominant or degenerate strategies is incentivized for players who are principally seeking victory. I appreciated Alex White's tic-tac-toe comment; they noted that they find the game more fun when using a non-dominant strategy. I've found this sort of thing in some cases, too - if you enjoy playing a game and find a strategy that will always win, resulting in other players no longer wanting to play, it can incentivize actively avoiding that strategy for players who enjoy playing above winning.
Magic: the Gathering is actually a great example of this. There are something like 30,000 cards in the game's history; if you want to win, you can find a variety of card combinations to do that every time (Black Lotus being a prime example). This leads to all kinds of additional rules, stipulations, and formats for competitive and tournament play. It also leads to players themselves inventing new formats, almost as hacks or mods to the original game. One of my friends noted that he enjoys Magic because it's effectively become a game design space for him. He can use this catalog of thousands of cards and select a subset to build the game experience that he wants, crafting a "cube" of cards to play with. This allows him to play competitively but avoid cards that contribute to certain dominant or degenerate strategies. Other players will build "jank" decks with themes like "chair tribal" (making a legal deck where all the artwork contains chairs) - their primary goal is not really winning. On sites like EDHRec, having a "rule zero" discussion about expectations (and essentially kinds of fun) has become common advice because there are just so many possible ways to play the game.
I guess I find this interesting because it speaks to some of the consequences of degeneracy. Highly competitive play might see rules changes to fix the problem, while other players might find other creative solutions. In the case of Magic, the original game rules have arguably become less popular than the EDH/Commander format, a fan-created set of alternate rules. The methods by which players navigate around dominant strategies or degeneracy in alignment with the kinds of fun they're seeking arguably defines Magic as a game (or maybe constellation of games, given how significantly some formats shift the rules). Some people will walk away from the game when they discover degenerate game states, while those who enjoy the game sufficiently seem to find ways to continue playing that align with their desire for particular kinds of fun, and this may actually improve and enrich the game itself for many players. (Or maybe it's better to talk about this as the metagame? Still thinking about this.)
“Strategy B strictly dominates strategy A” — here A and B are choices for the *same* player, not opposing players. If B dominates all my alternatives, that does not mean it is a winning strategy for me.
In games with first player victory, there may be ways for the second player to lose more slowly. Right?
Avoiding a winning strategy reminds me of a cat playing with a mouse: creepily sadistic, but the mouse may escape. Instead, change the rules, play a game without victory conditions (perhaps with no scoring, at all), or join a string quartet? ;)
I wonder if a good simple example is found in noughts and crosses (tic tac toe). It’s a common introductory game that just needs pencil and paper, but kids quickly understand that if you start in the centre position it is almost impossible to lose. Against experienced players it is always a draw, but you might find someone unfamiliar who gifts you the victory.
Soon people stop playing the game because the result is a foregone conclusion. Seems like a kind of degenerate state to me?
Incidentally I found a productive way of keeping playing a little longer was to start in a corner - people were less likely to be familiar with that, and were more likely to fall into traps!
I had the same thought while reading the article, thus I guess thinking of that game might be indeed a solid example.
I think the platform is eating reader comments today. I swear I just read a good one and now it’s gone. 😐
So, as I understand it, in game theory, a degenerate game is one where it kind of doesn't matter what the one player does, if the first player does a certain thing. Game theory is usually expressed in payoff matrices and vectors, so one example is there being a column in the payoff matrix that the first player can choose, such that the row player gets the same value no matter what they do.
The game is "degenerate" because certain game theory concepts just don't work with degenerate games - in much the same way that it doesn't make sense to work with the area of a degenerate triangle - it's actually a line so the area is 0.
Imagine a game where the first player can hide a ball in their left hand or right hand, and the other player can choose a hand, they win 1 point if they choose the hand the ball is in but lose 1 point if they choose wrongly. The first player might not hide the ball in either hand, and instead eat it (You can argue whether that constitutes a violation of the rules, but assume it's not). That strategy means the second player cannot make a meaningful choice between left and right. They may as well not be playing. The game has degenerated into a state where nothing that player does matters any more.
That's what I think is key to the concept of a degenerate game or game state in our sense. It doesn't matter how awesome your combat rules are if by turtling you can always avoid it. You've reduced the game by choosing a strategy that eliminates that part of the game as if it never existed.
I don’t know about degenerate, but isn’t what you are describing weak strategic dominance? If the first player must put the ball in box 1, 2, or 3 and the second player must say whether the ball is in box 1 or box 2, then the first player strategy of always placing the ball in box 3 weakly dominates all her other strategies — it produces an outcome at least as good as any other strategy. (I am no mathematician. I just read this: <https://en.wikipedia.org/wiki/Strategic_dominance>.)
Actually eating the ball is slightly more interesting. How many can I eat before I am very ill? I know what to do to win, but actually doing it presents difficulties. Is this how a game becomes a sport? ;)
Of course — I say with more confidence than I feel — someone may discover a strategy for a game which no one knows how to beat even though that strategy is not even weakly dominant (over all other possible strategies).
I don't think so, even if your strategy is dominant, it would still matter what I choose in a non-degenerate game. The degenerate game isn't because you have a winning strategy, just that you have the ability to make my choices irrelevant.
“B strictly dominates (>) A: choosing B always gives a better outcome than choosing A, no matter what the other players do.”
It is true that even strict dominance allows that the second player’s action can have some effect. Consider a modified example: if the first player puts the ball in box 1 or box 2, she scores one point if the second player picks the other box (else zero); if the first player puts the ball in box 3, she scores two points if the second player picks box 1, but three points otherwise. Always putting the ball in box 3 strictly dominates putting it in box 1, putting it in box 2, and sometimes putting it in box 1, sometimes in box 2. The second player can affect how quickly the first player racks up points, but that doesn’t affect the first player’s choice of strategy. (If the first player had another strategy that was sometimes better in the case of some second player action, then “always put the ball in box 3” would not dominate it.)
Is the modified game “degenerate”? I don’t have a dog in that fight, but it is still boring for both players.
My wife and I found this when playing Sky Traders as a 2 player game.
Theoretically, it's a pretty standard pick-up-and-deliver game. Except you accumulate sludge, which you're supposed to go out of your way to deposit in a safe location. Dropping sludge on a city gets you all your points, saves you time... but tallies up a bounty on you. Other players can hunt you down for that bounty.
At 3+ players, it's always worth it to collect on a bounty. At 2 players, the bounties aren't high enough to offset the points you can get from taking your own turn. So in order to stay competitive, you ALSO need to dump sludge on cities. And the game becomes less about delivering goods, and more about competitively dropping sludge onto unsuspecting cities, and hoping a bounty hunter isn't randomly played.
This circumvents about half of the intended mechanics, and turns it into a game we call "Sludge Traders".
I might have some details wrong, it's been a few years since I've played this one (Star Wars: Outer Rim replaced it as our pick-and-deliver of choice)
Sludge Traders! Hah! Sounds like it might still be fun even if different?
1. Take Australia
2. Hold Australia
3. Wait for opportunity that never comes
4. Brainlessly drink beer
RISK's degenerate game state supported my degenerate life-state in college
There are worse ways to spend time. 😅
as someone who loves both mathematics and language/the importance of words/definitions, this article was a lovely gift :D your articles never disappoint, but this one was as if made for me ^^ thank you for sharing <3
funny thing, i have never played Magic the Gathering, and know next to nothing about the game; though just the other day i heard Black Lotus being referenced on a ttrpg actual play ^^ finally looked it up when reading this article! (i played a ton of Yu-Gi-Oh growing up though, so i still was/am a big card game nerd x3)
now i want to find that counterexample of a degenerate game state that makes the game not worse – would love to find one that arguably makes it better, haha xD counterexamples are fun :) can't think of one off the top of my head though.
i voted for the "inherently changes the game" – going by the math degeneracy concept first and foremost :) [i'm personally most familiar with it from probability theory, see eg the wikipedia article about degenerate distributions! (as i specialise in probability/statistics)]
I voted for "follows the rules as written" in the poll because when I think of degeneracy, I think of ways of play that are technically 'correct' but result in problems. For example, think of the very common 'Free Parking' house rule in Monopoly. It makes the game play for way longer, and results in a generally worse play experience, but I don't consider it 'degenerate,' I consider it to be 'wrong' because it isn't the way the rules indicate how to play.
I'd have to think on it for a while, but it seems to me that some of the most fun I've had in games was exploring the possibility of 'breaking' them. Like finding that "One Stat" to rule them all in early console RPGs or some glitch or strategy that the designers apparently didn't catch during testing.
And then there are some games (like original Magic: The Gathering) that were seemingly designed to invite players to find and explore dueling degenerate strategies. Which of course led to the banning of some of those strategies (more specifically, the cards behind them) for tournament play.
It's fun (at least briefly) being on the winning end of a broken game; not so much fun being the one getting gamed by it.
Agree completely. That's why I think "makes the game less fun" isn't a particularly helpful criterion when identifying degenerate states in games. Someone might very much be having fun! ;)
Great article. I still shudder when I think about playing Dominion. Very hackable and therefore incredibly unfun once players been exploiting the game mechanics. Once someone exclaims, "It's not against the rules!" you know the rest of the game session (any game, not just Dominion) is going to quickly devolve into a mess of frustration (except for the winner of course).
Have you ever written an article about Fantasy Flight Games' X-Wing (1e)? I witnessed innumerable debates over the modal verbs printed on the game's cards. Wild.
As a child, I discovered Big Money independently, although I used the Mine card to upgrade my money using my Actions. Now I am really into the Chapel, trashing my deck until I have the most compact money-printing, province-buying machine possible.
In Magic the Gathering, Infinite Combos are often seen as degenerate. Winning the game out of nowhere unless your opponent can stop you at instant speed. However, in commander EDH format, it is seen as a part of the very fast, high-power metagame.
Would you consider writing an article about Player Skill vs Character Skill, or so called "Metagaming?" Would love to get your take!
Particularly great article this week!
You've got me thinking about how this intersects heavily with kinds of fun. Seeking dominant or degenerate strategies is incentivized for players who are principally seeking victory. I appreciated Alex White's tic-tac-toe comment; they noted that they find the game more fun when using a non-dominant strategy. I've found this sort of thing in some cases, too - if you enjoy playing a game and find a strategy that will always win, resulting in other players no longer wanting to play, it can incentivize actively avoiding that strategy for players who enjoy playing above winning.
Magic: the Gathering is actually a great example of this. There are something like 30,000 cards in the game's history; if you want to win, you can find a variety of card combinations to do that every time (Black Lotus being a prime example). This leads to all kinds of additional rules, stipulations, and formats for competitive and tournament play. It also leads to players themselves inventing new formats, almost as hacks or mods to the original game. One of my friends noted that he enjoys Magic because it's effectively become a game design space for him. He can use this catalog of thousands of cards and select a subset to build the game experience that he wants, crafting a "cube" of cards to play with. This allows him to play competitively but avoid cards that contribute to certain dominant or degenerate strategies. Other players will build "jank" decks with themes like "chair tribal" (making a legal deck where all the artwork contains chairs) - their primary goal is not really winning. On sites like EDHRec, having a "rule zero" discussion about expectations (and essentially kinds of fun) has become common advice because there are just so many possible ways to play the game.
I guess I find this interesting because it speaks to some of the consequences of degeneracy. Highly competitive play might see rules changes to fix the problem, while other players might find other creative solutions. In the case of Magic, the original game rules have arguably become less popular than the EDH/Commander format, a fan-created set of alternate rules. The methods by which players navigate around dominant strategies or degeneracy in alignment with the kinds of fun they're seeking arguably defines Magic as a game (or maybe constellation of games, given how significantly some formats shift the rules). Some people will walk away from the game when they discover degenerate game states, while those who enjoy the game sufficiently seem to find ways to continue playing that align with their desire for particular kinds of fun, and this may actually improve and enrich the game itself for many players. (Or maybe it's better to talk about this as the metagame? Still thinking about this.)
“Strategy B strictly dominates strategy A” — here A and B are choices for the *same* player, not opposing players. If B dominates all my alternatives, that does not mean it is a winning strategy for me.
In games with first player victory, there may be ways for the second player to lose more slowly. Right?
Avoiding a winning strategy reminds me of a cat playing with a mouse: creepily sadistic, but the mouse may escape. Instead, change the rules, play a game without victory conditions (perhaps with no scoring, at all), or join a string quartet? ;)