The Flames of Rebirth: War in Fire Emblem Three Houses

During the summer of 2019 I bought myself a Nintendo switch, and a copy of Fire Emblem: Three Houses purely on a whim. I don’t normally buy games at launch, but I needed a first title for my new game system and had never played a fire emblem game before, so I decided to take the plunge. I finished my first play through of the game that fall, and by early 2020 I was working on my first maddening run, the hardest difficulty. Today I’ve completed the game front to back five times. It is easily my favourite single player game in a very long time. Partly because of its interesting, but flawed, game-play, but mostly because it has something interesting to say about conflict that has been in my mind since first playing the game.

(Spoiler warning!!! I’m going to be talking about the ENTIRE three houses story line in detail. If you have any interest in the game stop reading now and play this masterpiece for yourself.)

A game about war.

¡Bias warning!

The main quality that sets Three Houses apart from other war games is the subtle ways it communicates both the horrors and complexities of war. A major issue when designing war games is the limitations of the medium itself. Players want to win, and this simple fact reinforces a very black and white view on warfare that is difficult to get around; there are enemies to defeat, and allies to save. The plot may mess around with the morality of one side or the other, muddy the waters between allies and enemies, by having sympathetic villains, making characters switch sides, questioning the motivations of the hero, or just having the player take side with the villain, but ultimately the final level must have an enemy to overcome and an objective to clear; war is a game with two sides.

At a gameplay level Three Houses is no different. Similar to how most strategy campaigns allow players to pick opposing factions, players in three houses can find themselves on any of four different routes. However, unlike the typical campaign approach, Three Houses hides the branching of the story from the players until absolutely necessary. Players do not need to play through all campaigns in order to get a complete game experience. During the first half of the game all four routes follow an almost identical trajectory. The game only significantly branches the story in the second half. As well, the writers were careful enough to only branch when absolutely necessary. Certain events are shared across routes making it possible to experience the same battles from opposing perspectives. Regardless of which faction the player sides with, they still get to be the hero of their own story, but the interaction between these parallel routes allows us to see biases and complexities that none of the routes individually can express. What we get is a game, fundamentally about war, which is able to express the reality of conflict as more than just two competing sides vying for power.

The titular “Three Houses” refers to the three main student groups at the Officer’s academy. Each house, Blue Lions, Golden Deer, and Black Eagles, represent one of the main political factions on the continent of Fódlan and are each led by one of the games three main protagonists: Dimitri, Claude, and Edelgard. The player character, Byleth a silent protagonist1, is a mercenary recruited by the church of Seiros to teach at their officer’s academy. Early in the game Byleth is given a choice to lead one of these three houses. The player needs to make this choice before any main plot is revealed, and the consequences of this decision isn’t fully appreciated by the player until much later in the game. Each house is made of of eight students and only the mechanical stats and abilities of each ‘unit’ along with a single introductory sentence revealing their personality is revealed to the player before choosing them. The player is encouraged to choose based on little more than mechanical game elements and artistic preference making the decision feel like a character creation screen. However, this choice turns out to be the most consequential decision in the game.

The game is broken into two main parts. The academy phase, or “white clouds”, is a single narrative viewed through the lens of whichever house you choose. Each month, or chapter, the church sends you and your students on a mission that is common across all routes. As the missions progress your class tries prevent an evil ‘Flame Emperor’ and and the loosely associated band of cultists ‘Those who slither in the dark’ from interfering in the affairs of the church. The second half chronicles a war that breaks out over the entire land of Fódlan. There are four possible routes that can be taken depending on the choices made in the first phase; each telling a related but different story2

Mechanics

On no! The creepy librarian was actually evil the whole time.

The game’s mechanics are a big reason why the story of Three Houses works so well. An important part of the fire emblem series as a whole is a mechanic known as permanent death. Unlike other strategy games that might protect playable characters with resurrection items or by returning fallen characters, possibly through an injury mechanic, after the mission is done, dead characters in Three Houses stay dead. Most no longer appear in cinematics, they can no longer be selected for missions, and their story is no longer progressed. Some plot important characters might ‘retreat’ from combat allowing them to fulfill their plot relevant role; however, something terrible inevitably happens to them off screen once the plot no longer needs them. Notoriously, the game can soft lock if the player loses too many of their units, especially on harder difficulties, as they will no longer have the resources to clear later levels. Mechanically, death is this universe has meaning and the plot uses this to great effect.

During the Academy phase the player spends a great deal of time and effort building relationships with the students. Between each monthly mission you can explore the academy campus and talk to nearly every character in the game, who reacts uniquely to the events of the month. You can offer gifts to students, return lost items, share meals, and even invite them to tea parties. All of these activities build support between you and the students. Support offers a small amount of in game benefit by offering bonuses to units with high support when they fight together, but it is mostly used as a mechanic to further the plot. At certain support thresholds the game unlocks support conversations where, in a cut scene, students and faculty members share a little about their life and personal struggles. Importantly, you can interact with students outside your house in this way as well. External students can appear as guest characters in some missions, they can offer up quests for Byleth to complete, and, under certain conditions, can be recruited into your house. During the war phase, all students are forced to pick sides in the oncoming war. All character you recruit side with you, but other character can find themselves on opposite sides of the conflict. This allows the later portions of the game to serve up these characters as enemies instead of other faceless minions. Just like how your characters die permanently, characters you face in combat also die when defeated. The act is designed to be emotionally unpleasant, and it is not uncommon to hear of players recruiting as many characters as possible simply so they won’t be forced to kill them later in the game.

To the level designer who put Bernadetta here: I hate you.

Even background characters are treated like this. Rarely do missions require you to go up against nameless bandits, instead enemies are frequently relatives of one of the playable characters. Chapter three has the player put down a rebellion against the church led by Lord Lonato, the adopted father of Ashe a student of the blue lions house. Chapter five has the player fight off bandits, led by Miklan the older brother of Sylvain another member of the blue lions house, who have stolen a hero’s relic. When playing as any other house, these characters are disposable villains, but to both Ashe and Sylvain these are important life changing events that colour the story for the remainder of the game. Likewise, the web of noble houses introduced throughout the game means that generals defeated in the war phase are rarely just the monster of the week, but are instead someone’s relative who might have gotten more screen time if the player had sided with a different faction.

All of this works together to create a world where death matters, which adds necessary emotional weight to the war phase of the game. Edelgard reveals herself to be the Flame Emperor, becomes emperor of Adrestia, and declares war on the church of Seiros at the climax of White Clouds. It is here that the the game loses its silly and naive video game veneer and transforms into something extremely brutal. Few characters escape death, and death itself becomes the primary driver of the story. In this way the game escapes portraying the war as just heroic, but also as the tragedy that it really is.

Edelgard’s War: A war to end war.

Edelgard von Hresvelg born the ninth child of Emperor Ionius IX of the Adrestian Empire. In many ways her story is the story of the Adrestian Empire itself. At a young age she was taken to the Kingdom of Faerghus during a conflict between the emperor and the ruling nobles resulting in a transfer of power away from the Emperor. Upon returning to Adrestia, Edelgard, along with her siblings, found themselves as test subjects in experiments conducted by the cult, Those Who Slither in the Dark, who had at this point deeply embedded themselves in the Adrestian military. They were trying to artificially embed crests, a magical brand that granted the wielder great power, into these children. The emperor, in his much diminished capacity, disproved of these actions, but could do nothing to stop them. Only Edelgard herself survived, making her the de facto heir to the Adrestian throne.

Edelgard stated goals in the war is to overthrow a toxic world order. Her upbringing soured her permanently on the idea of crests which she saw as a physically manifested caste system. Those who have crests wield power, both physical and political. Crest bearers can wield ancient and powerful weapons, known as hero’s relics, while those without crests are driven to madness and transform into giant abominations by those same weapons. Normally crests are inherited genetically, and Fódlan’s noble families are defined by them. The nobles engage in aggressive breading practices in order to create children who bear crests so that they can go on and lead the family into the next generation. Those without crests, even those from the nobility, form a permanent underclass. Noble children without crests often find themselves viewed as lesser to their siblings in the best of cases and frequently outcasts in their own family. Everyone else are commoners and peasants.

Edelgard sets her sights on the church of Seiros because it is the source of stability to the crest system. All crests find their origin in the founding myth of the church Seiros as each crest is linked to the family line of those warriors who aided Seiros in defeating the King of Liberation long ago. It is the church that perpetuates and maintains the entire system. The church sits at the centre of the continent, acts as an intermediary in political disputes between the three ruling powers, and most importantly supports the noble families and their hero relics. This is made clear in chapter five as Miklan, a crestless son of house Gautier, steals the families’ relic. The church’s response is to kill him, retrieve the relic, and return it to house Gautier. Before the system can change, the church must be removed.

Edelgard wishes to see the world free of crests; a world where where those with and without crests can interact as equals. As well she also envisions a world free of the church of Seiros; a world where humans determine their own fate free from the interference of a God. She vowed to use the power that she was unwillingly given to bring about this future; at any cost. However, Edelgard’s war is as much a civil war as it is a foreign war. Destroying the crest system also involves unseating the Adrestian elite just as much as destroying the church, and as one would expect, Edelgard’s coronation also corresponded with the assassination and removal of many of these ‘corrupt’ nobleman. However, this purge has one very notable exception; Lord Arundel.

Lord Arundel is Edelgard’s uncle and the second most powerful man in Adrestia. During Crimson Flower, Edelgard’s route, he represents the Adrestian wing of Those Who Slither in the Dark. Edelgard openly dislikes the cult and many times throughout the game refuse to be associated with them. During chapter eight, after Byleth stops the cult from experimenting on a village full of civilians, Edelgard, disguised as the flame emperor, tries to distance herself from the actions of the cult.

And yet, she never does. During all routes, even her own, the cult are present at nearly every major battle. The death knight, Edelgard’s vassal, is present during most of the cults experiments in the early chapters of the game, they are present in the chapter twelve attack on Garreg Mach in all routes except Crimson Flower, and most notably they are present in Edelgard’s final stand in the Azure Moon route3. In crimson flower, the cult mostly disappears, but Arundel takes their place, and Edelgard seems unwilling or unable to check his power. He is seen as a necessary evil, and while Byleth never works with them directly, they continue to operate in the background unhindered.

“Their power is essential to us at present.”

The most egregious example of the cults relationship with Edelgard happens after the the battle of Arianrhod chapter sixteen. In this chapter Edelgard attacks and executes Cornelia, a kingdom mage, who was involved with forbidden crest magic. Notably, in the Azure moon route, Cornelia betrays the kingdom in favour of the Adrestian Empire. In Crimson Flower, Lord Arundel condemns Edelgard for executing the mage because if, “that were the case, would it not have been better to keep her as an ally?” Implying that she was associated with the cult. Lord Arundel then warns Edelgard to avoid such mistakes in the future. Moments later, the entire fortress and both armies inside are destroyed in heavenly flame, a weapon that in other routes is tied to the cult. Edelgard suspects the attack came from Arundel, but warns both Byleth and Hubert to keep this secret to themselves. In the next chapter she protects Arundel by blaming the destruction of Arianrhod on the church and uses it as further justification to attack the Kingdom capital directly.

“I will be praying. Praying that the Empire will not become another Arianrhod.”

The Crimson Flower route is contentious among the fan-base because it is the shortest route with the least number of missions. It, in many cases, feels lacking and doesn’t do a good job of letting Edelgard tell her own side of the story. Notably the game ends after Seiros, who in a rage transforms into a dragon and sets the kingdom capital and everyone in it on fire, is defeated. The fate of those who slither in the dark, to many disappointed fans, is not resolved. While I believe the story could have been better fleshed out, I do believe the omission of the resolution with the cult is on purpose. Many of the epilogues imply that even though the war is over, Edelgard and Byleth continue to fight an underground war against the cult.

Sadly, this is not a screenshot. Had to retrieve this ending from a fan site here.

It is hard to believe that Arundel would have gone quietly, which implies that this underground war is just a gentle way to label a much larger civil war. But, the bigger question is what a “world where people can rise and fall by their own merits” actually looks like. Edelgard made it clear that her own feelings on the subject were secondary to her actions. Neither her friends, her enemies, nor her own doubts could convince her to part from her chosen path, and anyone who got in her way wound up dead. Her actions tell a story much stronger than her stated goals. Merit, in Edelgard’s world, is just a pseudonym for useful. The cult aren’t allowed to exist because they deserve it, they are ignored because they are useful. It’s doubtful Edelgard could just suddenly turn off such a fundamental part of her personality, even in peacetime. The above epilogue (There are several depending on how the player ships various characters) implies a happy ending, but it is left unsaid how they go about fixing the issues with society beyond just saying that they did. Yet, this is just to prevent reality from spoiling Edelgard getting to be the hero of her own story. She has already set the precedent that her way to a better world is through the corpses of those in her way, and that she is willing to cooperate with evil so long as it’s useful. Why would she ever let anybody undo that victory? So she does create a better world, one where people who agree with Edelgard can prosper, and everybody else likely met the same fate Dimitri did.

Playing to Win: TicTacToe

While attending university, I spent a couple of summers working as a counsellor at various overnight children’s camps. One year, we played a game where each senior counsellor would set up an activity. Each cabin, led by an activity leader, would compete to see who could complete the most activities in a set period of time. The activity I put together required one volunteer from each cabin to challenge me to a game, anything they could come up with. If they beat me, they would get my point. Most of the challenges were things I was doomed to fail at from the start: a cartwheel contest, staring contest, a race to see who could count to ten the fastest. However, the one that stuck with me the most is the one that I shouldn’t have lost at all. A child foolishly challenged me to a game of TicTacToe. After trading ties for a few rounds, something strange happened: I lost.

TicTacToe is the frictionless surface of the game theory world; it’s less a game, and more a theoretical demonstration of what it means for a game to be unwinnable. If two players enter the game, and play perfectly, it is impossible for either player to win; this fact is common knowledge. I knew it, and the child across from me knew it too. Yet, she still won, I still signed her cabin’s activity sheet proving that she won, and I went home knowing that I lost the last game of TicTacToe I would ever play for real stakes.

Discussions about TicTacToe are common in the world of computer programmers. It is the perfect first project for anyone trying to teach themselves game theory and computer AI. A computer sees a game like TicTacToe as little more than an optimization problem: given any particular board state, return the optimal next move. Thus, all discussions around the game are discussions about what constitutes the most optimal move. The most important position, and thus the most discussed, is the opening position. The most common argument goes as follows.

  1. The best move is one that is most likely to result in a win.
  2. Assuming we are playing against a player who plays randomly, the best move is the one that creates the most losing opportunities for the opposing player.
  3. <math>… 7 > 4 …</math>
  4. Therefore, playing in the corner is best.

This discussion might include some scientifically collected statistics of game results pitting various levels of computer AI against each other1, or an admission that the analysis applies only to perfect play and more analysis is necessary to account for imperfect play2. However, the results are always the same. The reader learns some interesting facts about what move is best in certain situations, but is still just as likely to lose to a child when challenged to real stakes: as I did.

The problem is that such discussions don’t really talk about winning at TicTacToe, they are about beating dumber computer programs, which highlights a fundamental difference between how humans and computer approach games and decision making in general: randomness. Human intelligence is utterly incapable of randomness, while artificial intelligence depends almost completely on it3. In TicTacToe, randomness represents the baseline player; it is the dumbest possible computer program we are capable of generating4. However, it does not represent the dumbest possible human strategy. In fact it is not a possible human strategy at all, nor is it a reasonable approximation of one.

To demonstrate, take a look at the following position.

Now, try as hard as you can to place yourself in the body of a ten year old who knows nothing about the game. It’s your turn to play. Where do you move?

A random player is just as likely to play in any open position, but did you pick randomly? If not, why did you place the piece where you did? Did if feel strongest? Do you know it’s strongest? Did it fulfill some sort of pattern in your brain? If you did pick randomly, how did you choose which one was your random choice? Did you simply pick the one that feels the most random?

How humans feel out what position to play in is not a fundamentally part of game theory, but it is of absolute importance when talking about winning at real games. Whatever your answer, hold onto it; we’ll get back to it soon.

Some AI Basics

First let’s go over a simple but important concept quickly: minimax tree search.

In the above position, it’s black’s turn to play. Who wins?

Black wins. Playing in the top left hand corner ends the game with a win. They could, if they want, play somewhere else, but that would allow red to win or force a draw and is a mistake. For the time being we will assume players don’t make those. A game is said to be ‘winning for black’ if it’s black’s turn and they have at least one move that wins.

We have now travelled one turn back in time. It’s red’s turn. Who wins?

Black still wins. Black has two winning moves available to them and red can only block one of them on their turn. Thus this position is still ‘winning for black’ because all of red’s moves transition the game to a position where black has at least one winning move.

Through computer analysis, we can analyze every state of the game starting from winning positions and working backwards to decide who wins in every stage. If the current player has any winning move available to them then the position as a whole is winning for that player. If all of the moves available are losing for them then the position as a whole is losing as well. Anything else is a tie. The reason TicTacToe as a whole is considered an unwinnable game is because none of the moves available at the start of the game are winning.

The following widget will allow you to play TicTacToe against yourself or a computer player (by clicking on the AI button). It includes an option to “show hints”. If selected, each empty positions will gain a highlight: green indicates that the move is winning for the current player, red means losing, and yellow is a tie. Before moving on, I recommend playing around with the widget until you understand this concept thoroughly5.

The Opening Positions

Let’s consider the available opening moves.

The corner is the simplest opener. Either the opponent plays in the centre, or they lose.

The centre opening is a simple 50-50. For computers, half of the available squares are losses. For humans things simplify around corners and edges. All of the corners lead to one outcome, while all of the edges lead to another. The probability of winning is then just the probability that the player you are up against is the type of person who doesn’t play in corners or edges.

The edge opening is the hardest to understand completely. Like centre, the computer player is looking at a solid 50-50. However, there are no easy rules of thumb a human player can use to memorize all safe positions. Instead of just deciding between edges and corners the opponent now has to also consider distance; It’s less obvious which corners and which edges are safe.

If we considering only winning and losing positions, the corner opener certainly seems like the best option. There is only one response, and this simple fact is commonly why it’s considered the best; all other openers simply offer less losing moves. The edge opener feels like the worst, as it is mathematically inferior to the corner opener. Not only is the center square safe, but others are as well. However, corner opener has one profound weakness. The singular correct response for red is also the one square simply begging to have a token placed on it. Earlier, did your inner ten year old play in the centre? I won’t say they did, but I will confidently bet that much more than one in eight of you did. I say this for one main reason; it is the most symmetrical. It has the most triples running through it and therefore it just feels more powerful to anyone aware of the goal of the game.

Consciously or unconsciously humans always have a reason for everything we do, even if that reason is simply to create an subjectively aesthetic pattern. Positions don’t exist in a vacuum, they are always in relationship with each other. In any situation where we don’t know the correct answer, the action we take will still fulfill some internal criteria: maybe we placed our token next to the black tile, maybe across from it6, or maybe we took the answer from some unrelated part of our environment (decision anchoring). Either way, simply counting how many safe squares are available to the opponent is not a good indicator of how good that opener is. We must also take into account how likely it is for a human to know the correct answer, and also how much the wrong answers feel like right answers.

Corner play loses to both of these. Our internal pattern matching system tells us that there are three, not eight, possible replies to opening corner: corners, edges and the centre7. Two of these end the game immediately, meaning a new player, learning the game, will only have to play a maximum three games before completely exhausting the outcomes of each of these replies, assuming they don’t play center immediately just because it feels right. Likewise, it’s easy to remember once learned: centre is strong. Remember that and you will never lose to a corner opener again.

Now this doesn’t mean probabilistic modelling is useless. While we can’t model any particular individual as a random number generator, we can model communities as a whole. Imagine a million children getting asked to play as red after we play corner. Someone is going to play in every tile, but some tiles will get played more than others. What comes out is a probability distribution; the probability that any particular player will be the type of player who plays in certain squares. This is refereed to as a games ‘meta’, a generalization of what a community believes is powerful at any given point in time. Metas are not static, they change and grow as the community changes and grows. They are a weird mix a human intuition and learned behaviour that can change radically depending on the geography, size, common experiences, unwritten rules, and theoretical knowledge of the community as a whole.

Understanding a local meta is vital when picking openers. If we can assume that a community thinks that playing corner is powerful, then they are more likely to know that centre is the right response. In such situations playing centre could be better. Yes there is a higher probability in random play that they will guess right, but that is still better than them just knowing what the right response is, or worse feeling what the right response is.

Complexity

TicTacToe is a game of mistakes, and if we are to find a winning strategy it needs to be one that allows our opposition as many opportunist to make a mistake as possible. Even if someone knows the correct response to an opener, that doesn’t mean they know the correct response for later positions. It’s much hard to memorize the correct responses to a sequence of moves, then the response to a single hard position.

One way to get an idea of how complex an opener is, is to visualize how complicated the resulting game tree becomes. There are thousands of possible games of TicTacToe, so in order to come up with a useful set of positions to visualize, we need to simplify the game by making a few assumptions on what constitutes a reasonable game.

Firstly, let’s assume mostly perfect play.

  1. If a winning move is presented to a player they will always take it.
  2. Players always block a simple win (see below graphic).
  3. Either player will sometimes, at very low probability, pick a losing move. We are primarily interested in states where this can happen.
  4. Once a win is no longer possible, given the above three points, ignore all further variations.
  5. We ignore positions that are just rotations or mirrors images of positions we have already considered8.
No reasonable red player will ever play anything but top edge here.

In the below visualization each node is a key/value pairing. Each key represents an action a player could make; the key “x_1_1” means that the X player places a token at the row one column one square (counting starts at zero). The value associated with the key represents the result of that decision. An integer value means that the game is over: -1 means X has lost, 1 means X has won, and 0 represents a tie. If the game continues a button appears allowing us to reveal the key/value pairs for the next stage of the game.

The centre opener is the least complicated and produces only one real line of play. After the O player responds in any corner, X has only two moves that don’t immediately end the game in a tie. Playing next to their opening only succeeds at creating a route to victory for O and should be avoided. Playing in the opposite corner challenges the O player to play in one more corner before the game ends in a tie. Memorizing this sequence is trivial, even for a small child.

Starting in the corner is a bit better.

After the O player responds in the centre, the X player is given a choice between two lines; both can lead to a win. Playing in the opposite corner mixes things up by forcing the O player to play on an edge before ending the game in a tie. Playing on an opposite edge is a bit more interesting. If the O player is aware of this unusual position they can play in the opposite corner and force the X player to dodge a single bad tile before ending the game in a tie. Everywhere else is a minefield that the O player needs to wade through, at least one position creating a second such minefield. Either way, ties are a lot harder to stumble into than the centre start.

Now the last one; the edge start.

I can’t summarize this position in one paragraph. The most important line, where the O player responds in the centre, alone is about as complicated as the other two openers combined. In fact, it actually contains some of the corner openers more complicated lines. As well, a lot more variations here end at six moves, meaning that the O player is frequently required to play at least three times before ending the game. A relatively deep understanding of the game is necessary to navigate this opener safely. If you know more about the game than your opponent, opening edge is a really good way to offer your opponent plenty of opportunities to screw up.

Conclusion

If we focus only on the perspective of game theory or AI, we get this warped perspective of what it means to play a game. AIs are optimizers, programs who find the optimal action given some rules and contexts. However, few games actually have truly optimal actions or strategies. Much more common are games that seem like they have best strategies, but acting on those strategies can result in a worse performance: the prisoners’ dilemma for example. To play such games well we need more than just theoretical insight into the game itself; we also need a model of our opponent. Focusing on what’s optimal, or by relying on strategies that are optimal given a specific meta, makes us predictable and easy to manipulate. It’s like finding a bug in computer software; once found it can be exploited indefinitely until the code is changed. What is far more powerful is first understanding what moves a player is likely to make and then searching the game for positions that punish that action. If someone is known to play in corners, then any position where a corner is a losing move is optimal. Even better would be to coax them into playing corners more often by priming them to think of corners as being good by opening with a relatively safe move: like the centre.

This is the fate of the corner opener. The fact that it is viewed so favourably means that any somewhat competent opponent is likely to already have studied it, and know the proper responses. Edge play, conversely, can punish people with some knowledge of the game, as such a player is more likely to ignore their gut instinct and ‘shake things up’ by playing in a square they may erroneously think is safe. Likewise, edge play is the only opener that forces a player to consider not just the differences between edges and corners, but also the differences between different corners. However, playing corner isn’t a bad move; it’s a test. When my opponent plays in the corner they are testing my knowledge of the game. They do so knowing that they are not likely to win. However, maybe winning right away isn’t their plan. Maybe I’m being set up for something bigger, an attempt to put my brain into autopilot, an attempt to convince me that TicTacToe is a simple game. Because, after I start believing that, my brain shuts off, they play edge, put me in a position I was unprepared for, and win.

Corner opener might be the best way for a new player to beat another new player, but opening edge is the best opener at higher levels as it is the only opener that forces both players to prove that their knowledge of the game transcends the Dunning-Kruger effect9. Whoever was lying will eventually lose. If neither was lying then, and only then, do we reach perfect play and the game becomes unwinnable.

A Theory Of Games

My family had a play structure in our back yard that my brothers and I would use to defend against real and imagined invaders. The structure had two floors. The ground floor was a converted sandbox with with four walls and old carpet for flooring. The upper floor had walls on two sides, a ladder on the third, and a slide on the fourth. I spent a lot of time thinking about how one might defend this structure. The ground floor was a deathtrap; while one of the outer walls was chest high and could be used as cover while lobbing water balloons at invaders, it only had one exit which was easy for larger kids to block and force surrender out of smaller kids through liberal use of the garden hose. The top floor; however, was different. The slide was difficult to get up, especially when wet, but easy to go down, and the ladder required whoever was trying to traverse it to drop their weapons momentarily in order to climb. Even better both the ladder and slide were easy escape routes and even the walls could be vaulted over in case of emergency. Thus it was perfectly defensible.

I have a memory of a game we played once using this structure. My brothers and I were defending the fort against aliens, zombies, or possibly something in between, who were attacking us on all sides. One of us covered the slide while another lobbed invisible explosives indiscriminately over the wall into the yard below. I was responsible for protecting the ladder. Now ammunition, even pretend ammunition, is a limited resource, and if the invaders were going to break into our stronghold it was definitely going to be at the ladder. So, just as the mindless slaughtering of unidentifiable alien zombies was about to get boring, something grabbed my leg. I tried as hard as I could to shake it off, but my Super Soaker was out of both real and pretend ammunition and I eventually succumbed to my injuries. The brother on the slide tried to help, but that only gave the zombie aliens an opportunity to scale the slide and take him out as well. My final brother made a valiant last stand before he too succumbed and declared the game over. There was much fun to be had, but also loss. The joy in an activity like this comes from the interaction with others, and thus that joy ends when your older brother goes inside to clean up. That’s how one losses a game of Calvinball. 1

I am a gamer. That means that I choose to dedicate a large portion of my time to both playing and thinking about games. Games to me are a pastime, a means of taking in story, and also a lens through which I can see and understand the world around me. Likewise, my relationship with games has grown and changed as I myself have grown and changed. As a child, games were an avenue of wonder; a way to experience things I couldn’t normally experience. As a teen, games were a convenient distraction; a way to establish limited control over my otherwise uncontrollable life. As a young adult games were a way of measuring personal growth; a lens through which I could see my skills develop and improve. Today, games are just a part of who I am and an important lens through which I understand the world around me.

It’s not easy to talk about what a game is because the word means different things to different people. Games are a thing that children play and adults are supposed to grow out of. Ludwig Wittgenstein uses the term “language-game”2 as a way of characterizing how we use language. Countries frequently engage in “war games” to train and ready their troops even as other commentaries on the subject explicitly exclude war itself from being a game. 3 I operate primarily in the world of technology and under that umbrella games are primarily a business; they are things, products, nouns, something that one entity designs for other entities to consume. There is a lot of literature around what exactly a game is, especially in the world of commercial video games; however, there is no easy consensus to point to as to what games actually are. If we consider the perspective of a game designer, or games as object, we could come up with a list of attributes that distinguishes a game from some other consumer object like a movie. Jesper Juul in his work “The Game, the Player, the World: Looking for a Heart of Gameness”4 summarizes some of this rhetoric by compiling a general list of qualities that appear in most game definitions: rules, variable outcomes, player effort, players attachment to the outcome, negotiable consequences of outcome, etc. However, I believe this approach fails to convey any real insight into what a game truly is, and worse tends to draw the discussion into pointless debates about what is and isn’t a game. As an example, if games must be voluntary and unproductive as Roger Caillois5 asserts then Warfare must not be a game. However true this statement might be, it uselessly offers no insight into mathematical game theories fascination with warfare, the game industries obsession with warfare, or the simple fact that we simulate and analysis warfare as if it were a game. Likewise, Juul refers to simulation games like Sim City as being “borderline cases” because they contain no predefined objectives; the game never unambiguously declares the player a winner. However, these simulations are still sold, unambiguously, as computer games in computer game markets and are reviewed as if they were games.

The following definition of a game was given by Bernard Suits in his book “The Grasshopper”:

“…to play a game is to engage in activity directed towards bringing about a specific state of affairs, using only means permitted by rules, where the rules prohibit more efficient in favour of less efficient means, and where such rules are accepted just because they make possible such activity.”

This definition defines two primary components along with one observation. A game, in Suits eyes, requires two things: rules and a desired state of affairs. A player desires a certain outcome, and the rules limit how that player may bring about that outcome. It is important to suits that the rules prohibit the most efficient means of bringing about this outcome. The fastest way to get to the top of a mountain would be to take a helicopter but the sport of mountain climbing prohibits such an act in favour of the less efficient method of climbing via ones own power. A mountain climber engages in the game of mountain climbing if they wish to arrive at the top of the mountain without using a helicopter. The observation Suits makes is that it is this restriction of the most efficient means that makes the game possible. If one wanted to climb a mountain using only ones own power then using a helicopter would not fulfill that desire, so the game of mountain climbing is invented in order to create a structure that encourages one to engage in the activity they wanted to engage in. As another example, the most efficient means of getting a ball into a hole would be to use ones hands to put the ball into the hole, but this is not what the game of golf is all about. Instead we choose to use a stick to hit a ball from a fixed distance away into the hole. Thus we have voluntarily chosen to use less efficient means, the stick, to bring about a desired state of affairs, balls in a hole. In reality, golf is not about getting balls into holes, it is about getting certain balls into certain holes starting from a fixed distance and using only regulation sticks. Thus, we can’t play golf unless we follow the rules of golf and therefore ‘playing golf’ is only made possible by adherence to its rules.

To Suits, “efficient means” implies that a player can use any means available to bring about their desired state of affairs. If two players simply wanted to overpower each other using any and all means at their disposal, then their struggle wouldn’t be a game. However, Suits explores such a scenario and discovers that the simple act of agreeing on a start time qualifies as an agreement to inefficient means and therefore makes it a game. Under this definition suits would have a hard time disqualifying any human activity from being a game because of the pervasiveness of unwritten social norms acting as a limitation of efficient means.

One way to think of the structure of a game is as being an alternate physics. Chess is a good example of this. Chess pieces are only able to move about the board in set ways. An invisible force, the rules of the game, prevent the bishop from moving in any direction except diagonally in much the same way that gravity prevents us from walking anywhere except along the surface of a large object. The thing that separates the physics of chess from the physics of the real world is that we created the rules of chess, we have to enforce them, and we can change them if we so desire; we did not create the laws of gravity and have no say in its enforcement. Assuming inefficient means, all other games contain some form of alternate physics. Softer game systems, like mountain climbing, are subject to both enforceable rules, like the prohibition of flying, and physical rules, like gravity. The allowed ‘moves’ in a game of mountain climbing are governed primarily by the physical world with some restrictions on technologies that we impose on ourselves. The outcomes of the game are a mix of physical results, “Did the player reach the top of the mountain?”, as well as results requiring a human judge, “Did they use only legal means to do so?” Any particular instance of a game is also impossible to reproduce exactly but some formal record of the event, like a recording or an entry in log book, may stick around. The softest game structures are those of make belief and includes the game of Calvinball I detailed in the introduction. These games exist primarily within the human mind and contain no normalized rule systems whatsoever. One might argue that such games have no structure as the rules, dictated by the mind of a child, can change suddenly and without warning. However, a child always knows when an adult has made an illegal move and thus, at least to their own subjective experience, the structure exists even if it can’t be communicated. The outcome of the game is entirely up to the humans, and, much the dismay of my inner child, an instance of a game is impossible to repeat in any form. Once fun has been had once, it can never happen exactly the same way again.

In all these cases the rules of a game act as a sort of simulation running on some sort of medium: a chessboard, physics, or the mind of a child. Video games fit into this model quite well. They are a simulation somewhere between the chess board and the mountain. Video games are simulations that run on the medium of computer hardware. Computer software is a mathematical structure and thus chess can be represented as a video game. However, most computer simulations are complex enough that unintended side effects are common. In chess it is impossible to make an illegal move; however, bugs and exploits that allow the player to act in unintended ways are nearly unavoidable in all sufficiently complex video games or real life simulations. Thus, like the mountain climber, in competitive video games we generally defer to the computer simulation to determine what actions are allowed, and step in sometimes with human judges when the need arises.

The structure of a game exists to moderate our interaction with the game and with each other through the game. However, a structure alone does not make a game. It is possible to follow all of the rules of golf and still not be playing golf. They rules of golf explicitly state that whoever completes every hole with the least number of strokes is the winner, it has no way of enforcing this goal if the player has no interest in winning. There is nothing in the rules that forbid a player from purposefully hitting a ball away from the goal. Worse, the game has no way of forcing a player to even progress through the game short of skipping the remainder of a hole after a set amount of strokes.

Juul attempts to get around this by by claiming that “Player attachment to the outcome” is a necessary part of the game. Suits definition requires a player to want to “bring about a specific state of affairs”. While superficially similar, these two requirements are not the same. Juul is coming at it from the perspective of a game developer. He wants his players to be attached to the outcomes as dictated by the rules of the game, and by extension the game developer. If a game has a celebratory ending sequence, then the player needs to be attracted by the possibility of experiencing it. However, in Suits definition the player themselves dictates the desired state of affairs. The mountain climber wants to reach the top of the mountain, but they might not care if they get there first. So even though the rules state that the winner is the one who gets there first the player may only be attached to the physical act of making it to the top and care substantially less about their placement.

Suits uses the example of ping pong to explore this point. If two professional ping pong players square off against each other in a competitive match, but mutually decide that winning isn’t important they could instead choose to attempt a long rally and hit the ball back and forth indefinitely. If their purpose is to generate the longest rally possible, we might still call what they are doing a game, just not the game the spectators expected them to play. However, it is important to note that this new game, the ping pong rally, exists within the exact same structure as the ping pong match the spectators expected. There is even a judge trying his best to enforce the rules of a game not being played. The thing that makes it a new game is the players expectations, not the structure.

Juul’s definition requires that, “As a player you agree to be happy if you win the game, unhappy if you loose the game.” The ping pong rally could exist under Juul’s definition, but only if we switched the referee out for one who is actively measuring the length of the rally. Juul assumes that a game can only be a game if there is agreement between the game designers and the game player. This is why under his definition simulation games like Sim City are not fully games because the game designer doesn’t prescribe an outcome for the player to valorize.

How a player approaches a game structure dramatically changes how they experience the game. A player could be “playing to win” meaning that they only take actions that purposefully maximizing the probability of achieving an outcome prescribed by the game. They could be “playing for fun” or “playing casually” meaning that they seek to achieve some sort of experience facilitated by, but separate to, the rules of the game itself. I am reminded of a friend in high school who proudly showed me their Elder Scrolls Morrowind save file in which they were in the process of murdering every NPC in the game; a state of affairs certainly allowed by the rules of the game, but not necessarily intended by the developers.

In my view, the separation between Chess which has a goal enshrined in the rules and Sim City which doesn’t isn’t philosophically important. Both are systems of rules, or alternate physics, through which humans can generate a wide variety of experiences. Commercial games, in whatever medium they appear in, are just game structures, and even though these structures may or may not include some prescribed end state, it is only when a player approaches these game structures with purpose do they actually become games. Indeed if we only look at a game in the way rules intend it to be played we frequently miss out on most of what the game becomes as a social phenomenon. Does the rules of chess say anything about chess grand-masters? About chess tournaments? Or chess clocks? Or the social phenomenon of cheating? No. If one wishes to know anything about chess, pure knowledge of its mathematical structure is only partial knowledge of the game itself. If we start by assuming that a game can only be played by people who’s purpose is to play the game as it is designed, then the only knowledge we will ever achieve is of the machines we designed to play them.

So then, what is a game? Well as mentioned above, no single definition will ever suffice. I fully admit that Juul’s and Suits’s definitions are useful when talking about games in their respective fields but fail as a general definition. However, I have nothing definitive to add. The definition I find most useful for my purposes is that games are a metaphor. Games are an attempt to section off some small portion of our lived experience into a much more understandable reality. We limit means because we have no other way of shrinking the enormity of the real world into something we can understand. We humans create games and define the boundary between what is and is not our game, and we do this to fulfill some purpose that is both uniquely personal and uniquely human.

In short games are small worlds; they are miniature universes that humans create and inhabit whenever the real world becomes to large and complex to understand.

Re: Barry Bonds Without a Bat

So, first a disclaimer: I know very little about actual baseball.

I do, however, love games, numbers, strategy, and game theory. So when Chart Party (a recurring feature on the sports YouTube channel SB Nation hosted by Jon Bois) ran the numbers on what would happen if Barry Bonds, one of the greatest baseball players of all time, played without a bat, I was intrigued. The following is my response to the question posed at the end of the video. I suggest watching it first before continuing.

To answer the first question: yes, I agree Bois’ methodology is correct, and the result is a little bit puzzling. How can the performance of a great batter not be affected by the removal of his bat? To answer that, we will need to abstract a little bit away from baseball as a holistic game and just talk about the interaction between the batter and the pitcher.

In baseball, the pitch qualifies as a sub-game, described in the following payout matrix:

Swing No swing
Strike ???? -1/3
Ball -1/3 1/4

There are two players, the batter and the pitcher, and each has two actions that they could perform. The pitcher acts first and attempts to throw either a ball or a strike; the batter must react to this decision. If a ball is thrown and the batter does not swing, then the batter scores a “ball”, and doing so four times results in a free walk to first base. If the pitcher throws a strike, the batter must swing, otherwise the batter scores a strike; three of these results in a strike-out. Swinging at a ball also results in a strike. The remaining situation involves the entire rest of the team and is hard to assign a value to so we will ignore it for the time being.

Now, in the Chart Party experiment, Bois modeled the pitcher as a random number generator. This may seem unfair, because common sense says that professional players shouldn’t be throwing balls randomly; however, this is actually a good way of modeling high-level play. A professional pitcher pitching to a low-level player, like me, would quickly adapt to my inability to hit the ball and strike me out every time. Likewise, a professional batter would just as quickly adapt to my inability to throw a ball and would either launch one out of the park or take the free walk. However, things change when two professional players play each other. In this case, both players would notice and adapt to any pattern exhibited by the other; therefore, the best strategy is to not exhibit any patterns at all, which is the definition of random.

(Note: since the batter reacts to the pitcher, the batter doesn’t need to swing randomly, only avoid indicating to the pitcher what the batter plans on swinging at.)

Finally, we are only interested in ‘On Base Percentage’ (OBP) or the amount of times the batter leaves home base successfully. Runs don’t matter and for this simulation getting to first base is just as good as hitting a home run. So, with all these assumptions and simplifications in place, the pitching sub-game becomes extremely easy to model mathematically. If the batter never swings, we are left with a simple binomial distribution: six pitches at a probability of throwing a ball at 58.7% (as reported in the video). We are interested in the probability of a game resulting in at least 4 balls. So, if we take a quick calculator break….

https://stattrek.com/online-calculator/binomial.aspx

we arrive at a probability of 51.7%. In terms of baseball, that would be an OBP of 0.517.

Now, this number doesn’t include intentional walks and hit by pitches, which the video sadly lumps together with normal walks, so I cannot accurately calculate their effect. However, the video does report Bonds’ total walk rate to be 0.381 and if even a quarter of those are intentional walks (not a difficult assumption, given the background in the video) that would easily push his OBP to the reported value of 0.608.

The above graph demonstrates the relationship between a random pitcher, their probability of throwing a ball, and the probability of being walked, assuming the batter doesn’t swing. In a meta where pitchers threw balls less than 30% of the time, the effect on the batter’s OBP is minimal. Not swinging would result in getting to base only slightly more than 5% of the time. However, the numbers start changing quickly in metas with higher probabilities. A single 10% jump from 30% to 40% triples the expected OBP of the batter, while each of the next two 10% jumps both double it again. Suffice it to say, tiny changes in the pitching meta can have a massive impact on the expected OBP of the batter.

So where does the batter’s skill come into this?

Allowing the batter to start swinging would likely have a negative effect on their OBP. The simple fact that a batter can swing at balls will always negatively impact their score. Obviously, the better the batter is, the less this effect will be, but unless they play perfectly, inclusion of this option will always drop their OBP by at least a small amount. As noted, swinging at strikes has several possible outcomes. The batter can swing and get a strike, swing and not make it to base, or swing and make it to base. So the outcome of the rest of the entire game will have a variable effect depending on the skill of the team.

For the case of Bonds, we know the outcome. Swinging a bat had no effect on his overall OBP, which means he successfully swung at enough strikes to counteract the negative effect of swinging at balls. Seeing as he is one of the greatest batters of all time, I would assume that this is the upper limit of batting performance and lesser batters would perform a lot worse. In terms of this experiment, I hypothesize–although cannot prove– that for most regular batters, taking away their bat would actually improve their OBP.

Is this bad for baseball? No. In the real world, if Bonds showed up without a bat the pitcher would adapt quickly and strike him out; it’s a dumb strategy. What this is though is a good indicator that OBP is a terrible statistic, and likely shouldn’t be used as a proxy for a batter’s skill.