Some people think games are defined by their rules; Others define them by the behaviors of the players. This can lead to some misunderstandings, so I think it could be helpful foreach type of player to understand the other perspective.
Two Definitions
Here is the way that I would choose to define a game:
A game has one or more teams of players and some set of rules. Players choose actions according to the rules to try to achieve the goals of their team.
Often the goal is to win, but you can also have games that allow ties, multiple teams winning, degrees of victory/loss, etc. Let’s call players operating under this definition the mathematicians.
For a long time this seemed so obvious to me that I failed to recognize a second definition:
A game is some activity performed by some players. The players perform the activity while constrained by rules. Usually there is some way for players to win or lose.
Let’s call the players operating under this definition the sociologists.
The mathematicians have a stricter definition than the sociologists: the activity the mathematicians are performing is “plan for victory within these rules”, while for the sociologists, planning for victory may be only one component of the activity.
Examples
Alice and Carol are mathematicians, Bob and Dave are sociologists. I won’t give a complete description of the rules of each game, but hopefully you can mostly tell what is going on.
Candyland
In candyland there is a path with colored spaces. Players take turns drawing cards from a deck and advancing to the next space that matches the color of the card. The first player to the finish wins. There are some other rules that I’m forgetting, but the point is that there is no room for choice from the players.
Bob: …If I draw red then yellow I can win with only two cards.
-Bob draws a red-
Bob: Ha! Alice, I passed you!
Alice: This game is sooo boring! We may as well just use a random number generator to pick the winner and save ourselves some time.
Bob: Not every game is a strategy game! Can’t you just have some fun every once in a while? Finally we play a game I can beat you at sometimes and you just get mad!
Catan
In case you haven’t played: In settlers of catan, players build structures, which score points and produce resource cards. You can spend resource cards to build structures or trade resource cards with other players, but only on your turn. Each turn there is a 1/6 chance of activating the robber. When this happens, all players with more than 7 cards in hand must discard half of them.
-Carol rolls dice, Alice collects a bunch of cards-
Alice: Hey Carol, I have ten cards right now and I don’t want to get robbed. How about I give you four, then on my turn you keep a stone and give the rest back?
Carol: Cool! Robber insurance! Let’s do it! …I wonder what other financial instruments we could set up… the highly discrete resources are an issue…
Bob: Hey! That’s not allowed! Where in the rules does it say you can offer insurance?
Alice: Where does it say you can’t? I guess it says you can’t give cards for free, so I’ll give carol five cards for one, and then she’ll give me four for one.
Dave: Now you’re going against the spirit of the game. You’re not supposed to do these big complicated trades.
Spicy
This is a bluffing game where you play cards face down, claiming either a suit or a number. When you play a card, other players can choose to challenge it, and if it doesn’t have the suit or number you claimed then the challenge is successful, if it does the challenge fails.
Alice:…In most bluffing games, challenges are negative-sum, but here, the winner of the challenge scores points for the cards in the pile, while the loser only loses one point… so if the pile is large it’s very good to challenge.
Bob:…Carol sure looks suspicious over there… what are the chances she really has an eight?
-Alice Challenges a bunch of times, Bob challenges when he is suspicious, Alice wins-
Bob: Well. It only took you one try to break this game.
Go
-Alice recently reached 1 dan, and is playing online against a stranger-
Alice:…I’m pretty weak in joseki… Maybe if I mirror his moves I can make it safely into the middle game…
-Alice mirrors moves, opponent resigns and blocks-
rat-a-tat cat
At the beginning of the game each player looks at some of their cards, while the rest remain hidden. Throughout the game players can take actions which reveal or replace replace some of their cards. At any point in the game, any player can call for and end to the game, and the player with the best cards wins.
Alice:…I have better initial cards than average, and there’s no reason to expect I will do better than the others once we start playing… so I maximize my win probability by ending the game now. Especially since if Eve had even better cards, she would have thought the same thing and ended the game already.
-Alice ends the game-
Bob: Now you’ve broken this game too…
What’s going on?
When mathematicians play a game, they try to win. The rules of a game determine the constraints, and you plan within them. A fun game has room for players to make interesting plans, and leads to interesting, complicated strategies. If the rules of catan just describe how trade works but players naturally develop insurance, that’s super cool! It’s fun to experiment with such small-scale economics. Discovering new strategies is a big part of what makes playing games so fun. There is beauty in games like go that have extremely simple rules that nonetheless lead to complex behavior. When sociologists complain that mathematicians aren’t playing the game the way it was meant to be played, the mathematicians get frustrated. If the game was limited to what the designer was thinking of at the time, it wouldn’t be very interesting! The whole point is that the players can put a lot of thought into optimizing their play, so of course what they do will go far beyond what is directly explained in the rules. When sociologists want to play games that are decided almost entirely by luck, mathematicians will often get bored. To them, candyland and games like it are weird mockeries that take on the outside appearance of games with none of the substance. These games are broken from the start. There are also games that appear interesting at first, but turn out to be “broken” by a winning strategy that is overly simple or boring to execute.
When sociologists play a game, they hope to win, but the main goal is to make sure that everybody in the group is having fun. The rules of the game function to describe what you should do while playing the game, but what is really fundamental is the behaviors you execute while playing. The written rules sometimes just describe all of these behaviors directly, but sometimes they act as an efficient compression of the unwritten rules. The written rules of chess fit on a few pages, but when someone teaches you chess, they will also teach you about pins, forks, the value of different pieces, etc. These unwritten rules are the true rules. Games exist on a spectrum from social games to strategy games. Social games involve less planning and thinking - you generally just hang out and have a good time. Strategy games involve a lot more planning and thinking. Some games are in the middle. When a mathematician plays a game with a large social component but plays in a weird way and wins, that’s annoying! The winner is supposed to be whoever played the best (with some luck to keep it fair), but the mathematician played completely wrong and got an undeserved win. What’s even more infuriating is that they can point to three different places in the rules that imply that what they did is allowed, but it clearly wasn’t what was intended! Some party games have a clause that says “if it feels like cheating, it is”, and it helps sociologists nullify these strategies. In a strategy game, there will often be a lot of techniques that are part of the unwritten rules. mathematicians tend to win these games a lot, because they have extra techniques that are illegible (or secret). Sometimes a mathematician comes up with a legible technique. Sometimes the sociologists appreciate this and it is added to the unwritten rules. Sometimes they don’t, and declare that it breaks the game. I think it comes down to whether the new technique fundamentally changes the “feel” of the game, but I’m not enough of a sociologist to know for sure.
Peaceful Coexistence
What if a mathematician is winning too much?
A common misconception of sociologists is that mathematicians only care about winning (perhaps even at the expense of others), and don’t know how to have fun. This isn’t true. Mathematicians enjoy planning out their moves and searching for a way to win. For a mathematician, a hard-fought battle against a strong opponent is fun even if you lose! Mathematicians win a lot in practice against sociologists not because they are monomaniacally focused on winning at the expense of fun, but because the activity that is fun for them tends to result in a lot of winning. For some mathematicians, a generic group activity in a social situation can actually be kind of stressful, and trying to play a game while “going easy” and being judged for “overthinking” just makes it way worse.
So what do you do if a mathematician is winning all the time and it’s no fun for anybody else? Add a handicap! Give the mathematician a worse starting position until they win with the desired frequency. This will often be a lot more fun for a mathematician because the game is more interesting. Instead of easily gaining an advantage they have to fight their way back from a losing position, and everyone can win with approximately the same frequency. For some games like go, there is a really natural way to handicap a player. For other games you might have to get more creative. This is something a mathematician can help you with! For mathematicians, you might want be careful with suggesting a handicap. I have had people get offended at me for this, so you should probably find a tactful way of explaining that the point is not to humiliate other players by showing how much of a disadvantage you can win with, but rather to generate a more interesting, even game.
Broken games
For a mathematician, a broken game is one where the optimal strategy is uninteresting. For a sociologist, it is one where the written rules don’t accurately describe the intended play. If a mathematician is playing the wrong way, modify the rules so that they lead to the behavior you want. The mathematician might be fully on board if they think your intended behavior is more interesting, but they will feel a lot more comfortable with an explicit rule change. For example, with rat-a-tat cat, where the incentive is for a player with good cards to end the game immediately, mathematicians and sociologists are aligned in finding this behavior undesirable, but if a sociologist says “well just don’t end the game instantly”, the incentive is still there. A big part of the game becomes “how soon can I end the game without risking social disapproval”, which is stressful and not so interesting. Similar games that avoid this problem add a rule where if you end the game but don’t have the highest score, you lose points. This adds a reason for players to continue the game until they have gathered more information and fixes the problem for both types of players. It is really frustrating for a mathematician when a sociologist says “here are the rules, but I’ll be offended if you play this technically-legal winning strategy”, and a lot less frustrating to hear “I don’t like how that game turned out, let’s modify the rules to prevent [x]”.
Survival as a mathematician
What if you are a mathematician playing a frustrating game with a bunch of sociologists? If people don’t like you breaking a game, overthinking, or winning too much, maybe for them this isn’t the kind of game where you try to win. Sometimes this just isn’t an activity you enjoy, but sometimes if you stop thinking of what you are doing as playing a game and start thinking of it as playing a part it can be fun. Some games involve e.g. drawing or singing, and there are clever ways to win every time that will make everyone else dislike you. The point of these “games” is not the game - it’s just to get a bunch of people to start drawing or singing! It’s usually not the case that people resent you for being smart, just want to win without putting in effort, or want to force conformity for its own sake. They just don’t share your definitions.
For sociologists, you might actually help the mathematicians be a lot more comfortable if you don’t call something a game in the first place. Just say e.g. “let’s draw pictures and pass them to each other” or “we sing songs that match the clue on the card” as things that people in the group are going to do, not as rules in a game. If you bought a game, the printed rules might have a way of choosing a winner or assigning points. Sometimes that’s completely unnecessary! If you don’t want mathematicians to try to win, just don’t assign a score or a winner. For mathematicians, if you can tell that is what somebody meant you can pretend it’s what they said and act accordingly.
Some people think games are defined by their rules; Others define them by the behaviors of the players. This can lead to some misunderstandings, so I think it could be helpful foreach type of player to understand the other perspective.
Two Definitions
Here is the way that I would choose to define a game:
A game has one or more teams of players and some set of rules. Players choose actions according to the rules to try to achieve the goals of their team.
Often the goal is to win, but you can also have games that allow ties, multiple teams winning, degrees of victory/loss, etc. Let’s call players operating under this definition the mathematicians.
For a long time this seemed so obvious to me that I failed to recognize a second definition:
A game is some activity performed by some players. The players perform the activity while constrained by rules. Usually there is some way for players to win or lose.
Let’s call the players operating under this definition the sociologists.
The mathematicians have a stricter definition than the sociologists: the activity the mathematicians are performing is “plan for victory within these rules”, while for the sociologists, planning for victory may be only one component of the activity.
Examples
Alice and Carol are mathematicians, Bob and Dave are sociologists. I won’t give a complete description of the rules of each game, but hopefully you can mostly tell what is going on.
Candyland
In candyland there is a path with colored spaces. Players take turns drawing cards from a deck and advancing to the next space that matches the color of the card. The first player to the finish wins. There are some other rules that I’m forgetting, but the point is that there is no room for choice from the players.
Bob: …If I draw red then yellow I can win with only two cards.
-Bob draws a red-
Bob: Ha! Alice, I passed you!
Alice: This game is sooo boring! We may as well just use a random number generator to pick the winner and save ourselves some time.
Bob: Not every game is a strategy game! Can’t you just have some fun every once in a while? Finally we play a game I can beat you at sometimes and you just get mad!
Catan
In case you haven’t played: In settlers of catan, players build structures, which score points and produce resource cards. You can spend resource cards to build structures or trade resource cards with other players, but only on your turn. Each turn there is a 1/6 chance of activating the robber. When this happens, all players with more than 7 cards in hand must discard half of them.
-Carol rolls dice, Alice collects a bunch of cards-
Alice: Hey Carol, I have ten cards right now and I don’t want to get robbed. How about I give you four, then on my turn you keep a stone and give the rest back?
Carol: Cool! Robber insurance! Let’s do it! …I wonder what other financial instruments we could set up… the highly discrete resources are an issue…
Bob: Hey! That’s not allowed! Where in the rules does it say you can offer insurance?
Alice: Where does it say you can’t? I guess it says you can’t give cards for free, so I’ll give carol five cards for one, and then she’ll give me four for one.
Dave: Now you’re going against the spirit of the game. You’re not supposed to do these big complicated trades.
Spicy
This is a bluffing game where you play cards face down, claiming either a suit or a number. When you play a card, other players can choose to challenge it, and if it doesn’t have the suit or number you claimed then the challenge is successful, if it does the challenge fails.
Alice: …In most bluffing games, challenges are negative-sum, but here, the winner of the challenge scores points for the cards in the pile, while the loser only loses one point… so if the pile is large it’s very good to challenge.
Bob: …Carol sure looks suspicious over there… what are the chances she really has an eight?
-Alice Challenges a bunch of times, Bob challenges when he is suspicious, Alice wins-
Bob: Well. It only took you one try to break this game.
Go
-Alice recently reached 1 dan, and is playing online against a stranger-
Alice: …I’m pretty weak in joseki… Maybe if I mirror his moves I can make it safely into the middle game…
-Alice mirrors moves, opponent resigns and blocks-
rat-a-tat cat
At the beginning of the game each player looks at some of their cards, while the rest remain hidden. Throughout the game players can take actions which reveal or replace replace some of their cards. At any point in the game, any player can call for and end to the game, and the player with the best cards wins.
Alice: …I have better initial cards than average, and there’s no reason to expect I will do better than the others once we start playing… so I maximize my win probability by ending the game now. Especially since if Eve had even better cards, she would have thought the same thing and ended the game already.
-Alice ends the game-
Bob: Now you’ve broken this game too…
What’s going on?
When mathematicians play a game, they try to win. The rules of a game determine the constraints, and you plan within them. A fun game has room for players to make interesting plans, and leads to interesting, complicated strategies. If the rules of catan just describe how trade works but players naturally develop insurance, that’s super cool! It’s fun to experiment with such small-scale economics. Discovering new strategies is a big part of what makes playing games so fun. There is beauty in games like go that have extremely simple rules that nonetheless lead to complex behavior. When sociologists complain that mathematicians aren’t playing the game the way it was meant to be played, the mathematicians get frustrated. If the game was limited to what the designer was thinking of at the time, it wouldn’t be very interesting! The whole point is that the players can put a lot of thought into optimizing their play, so of course what they do will go far beyond what is directly explained in the rules. When sociologists want to play games that are decided almost entirely by luck, mathematicians will often get bored. To them, candyland and games like it are weird mockeries that take on the outside appearance of games with none of the substance. These games are broken from the start. There are also games that appear interesting at first, but turn out to be “broken” by a winning strategy that is overly simple or boring to execute.
When sociologists play a game, they hope to win, but the main goal is to make sure that everybody in the group is having fun. The rules of the game function to describe what you should do while playing the game, but what is really fundamental is the behaviors you execute while playing. The written rules sometimes just describe all of these behaviors directly, but sometimes they act as an efficient compression of the unwritten rules. The written rules of chess fit on a few pages, but when someone teaches you chess, they will also teach you about pins, forks, the value of different pieces, etc. These unwritten rules are the true rules. Games exist on a spectrum from social games to strategy games. Social games involve less planning and thinking - you generally just hang out and have a good time. Strategy games involve a lot more planning and thinking. Some games are in the middle. When a mathematician plays a game with a large social component but plays in a weird way and wins, that’s annoying! The winner is supposed to be whoever played the best (with some luck to keep it fair), but the mathematician played completely wrong and got an undeserved win. What’s even more infuriating is that they can point to three different places in the rules that imply that what they did is allowed, but it clearly wasn’t what was intended! Some party games have a clause that says “if it feels like cheating, it is”, and it helps sociologists nullify these strategies. In a strategy game, there will often be a lot of techniques that are part of the unwritten rules. mathematicians tend to win these games a lot, because they have extra techniques that are illegible (or secret). Sometimes a mathematician comes up with a legible technique. Sometimes the sociologists appreciate this and it is added to the unwritten rules. Sometimes they don’t, and declare that it breaks the game. I think it comes down to whether the new technique fundamentally changes the “feel” of the game, but I’m not enough of a sociologist to know for sure.
Peaceful Coexistence
What if a mathematician is winning too much?
A common misconception of sociologists is that mathematicians only care about winning (perhaps even at the expense of others), and don’t know how to have fun. This isn’t true. Mathematicians enjoy planning out their moves and searching for a way to win. For a mathematician, a hard-fought battle against a strong opponent is fun even if you lose! Mathematicians win a lot in practice against sociologists not because they are monomaniacally focused on winning at the expense of fun, but because the activity that is fun for them tends to result in a lot of winning. For some mathematicians, a generic group activity in a social situation can actually be kind of stressful, and trying to play a game while “going easy” and being judged for “overthinking” just makes it way worse.
So what do you do if a mathematician is winning all the time and it’s no fun for anybody else? Add a handicap! Give the mathematician a worse starting position until they win with the desired frequency. This will often be a lot more fun for a mathematician because the game is more interesting. Instead of easily gaining an advantage they have to fight their way back from a losing position, and everyone can win with approximately the same frequency. For some games like go, there is a really natural way to handicap a player. For other games you might have to get more creative. This is something a mathematician can help you with! For mathematicians, you might want be careful with suggesting a handicap. I have had people get offended at me for this, so you should probably find a tactful way of explaining that the point is not to humiliate other players by showing how much of a disadvantage you can win with, but rather to generate a more interesting, even game.
Broken games
For a mathematician, a broken game is one where the optimal strategy is uninteresting. For a sociologist, it is one where the written rules don’t accurately describe the intended play. If a mathematician is playing the wrong way, modify the rules so that they lead to the behavior you want. The mathematician might be fully on board if they think your intended behavior is more interesting, but they will feel a lot more comfortable with an explicit rule change. For example, with rat-a-tat cat, where the incentive is for a player with good cards to end the game immediately, mathematicians and sociologists are aligned in finding this behavior undesirable, but if a sociologist says “well just don’t end the game instantly”, the incentive is still there. A big part of the game becomes “how soon can I end the game without risking social disapproval”, which is stressful and not so interesting. Similar games that avoid this problem add a rule where if you end the game but don’t have the highest score, you lose points. This adds a reason for players to continue the game until they have gathered more information and fixes the problem for both types of players. It is really frustrating for a mathematician when a sociologist says “here are the rules, but I’ll be offended if you play this technically-legal winning strategy”, and a lot less frustrating to hear “I don’t like how that game turned out, let’s modify the rules to prevent [x]”.
Survival as a mathematician
What if you are a mathematician playing a frustrating game with a bunch of sociologists? If people don’t like you breaking a game, overthinking, or winning too much, maybe for them this isn’t the kind of game where you try to win. Sometimes this just isn’t an activity you enjoy, but sometimes if you stop thinking of what you are doing as playing a game and start thinking of it as playing a part it can be fun. Some games involve e.g. drawing or singing, and there are clever ways to win every time that will make everyone else dislike you. The point of these “games” is not the game - it’s just to get a bunch of people to start drawing or singing! It’s usually not the case that people resent you for being smart, just want to win without putting in effort, or want to force conformity for its own sake. They just don’t share your definitions.
For sociologists, you might actually help the mathematicians be a lot more comfortable if you don’t call something a game in the first place. Just say e.g. “let’s draw pictures and pass them to each other” or “we sing songs that match the clue on the card” as things that people in the group are going to do, not as rules in a game. If you bought a game, the printed rules might have a way of choosing a winner or assigning points. Sometimes that’s completely unnecessary! If you don’t want mathematicians to try to win, just don’t assign a score or a winner. For mathematicians, if you can tell that is what somebody meant you can pretend it’s what they said and act accordingly.