Playing Game Theory Optimal 40K: A New Way of Thinking About Warhammer

We’re going to have fun today. Well, the nerds among you are going to have fun today. The people who aren’t nerds will be really bored by this article. But let me let you in on a secret: You came here to read long-form content about 40K today. You are a nerd, whether you know it or not. I think you’ll have fun here.

In this article, the goal is not to talk about specific tactics (although there did end up being a lot of tactical discussion). Rather, we’re going to talk about how to fundamentally view a game of Warhammer. We’re going to give you a framework to gain a deeper understanding of what is happening during a game of 40K, and over the long run, that will pay dividends by improving your play. If you’re a nerd like me, it’s also just plain fun.

Without further ado, let’s dive into today’s topic at Warphammer: Playing Game Theory Optimal 40K.

What Does Game Theory Optimal Mean?

If I’m going to write an article about playing Game Theory Optimal 40K, I should start by defining what I’m talking about. That’s fair to you, right?

Game Theory Optimal (GTO) is a concept that is usually applied to card games like poker, so we’re going start by viewing Game Theory Optimal through a poker lens and then transition it to a 40K lens.

Let’s start with a definition from Fedor Holz (one of the best poker players in the world):

“Game Theory Optimal is the mathematically perfect way to play that makes it impossible to be exploited by opponents. In other words, if you play a perfect GTO strategy, your opponents could do no better than break even against you — and that would only be if they played a perfect GTO strategy themselves.”

That’s honestly a pretty good definition, but know I’m going to lose some people with that because it sounds pretty nerdy. Let’s simplify it even further.

Game Theory Optimal is playing in a way that is optimized versus your opponent’s optimal decision.

Let me translate this into simple, straightforward 40K terms:

Every decision you make is optimized to maximize your probability of winning, regardless of what choices your opponent makes in response.

3 Pillars of Game Theory Optimal 40K

1. Always Assume Your Opponent Will Make the Optimal Play.
2. Play To Optimize Your Probability Of Winning
3. There Is A Correct Choice For Every Decision. Try To Find It.

Evaluating the Expected Value of a Play

Let’s talk about a key concept that every player uses to make decisions, even if they don’t consciously think about it like this: Expected Value (EV)

In the simplest form, EV is the (probability of each outcome) x (value of each outcome).

Charge, Or Do An Action?

Let’s talk about a practical example: You have a unit of Terminators. You can deepstrike them into your opponent’s deployment zone. You have the secondary Establish Locus. If you just deepstrike them and do the Establish Locus action, you gain 4VP.

The other option for your unit is to attempt a 9″ charge onto their home objective. Let’s say you’re playing a standard Take And Hold mission, so that would deny your opponent 5VP.

Let’s evaluate the EV of doing the action versus attempting the charge.

The EV of doing Establish Locus is 100% chance of success x 4VP, so 4VP. The EV of charging onto the objective is 28% chance of success x 5VP, so 1.4VP. Charging would be a -2.6 (4 – 1.4) VP decision. In a VP vacuum, don’t do that.

Now let’s explore what the situation would have to be for you to attempt that charge.

• You can re-roll charges
  • The probability of a successful 9″ charge with re-roll charges is 48%, so the EV of attempting a re-rolling charge is 2.4VP
  • Verdict: Establish that Locus
• You can 6″ deepstrike and charge
  • The EV of attempting the charge is 72% x 5VP, so 3.6VP.
  • Verdict: Establish that Locus
• You also have Storm Hostile Objective, so earn 9VP if you make the charge
  • The EV of attempting the charge is 28% x 9VP, so 2.5VP
  • Verdict: Establish that Locus

Now let me propose a situation: The score outside of this play is a 81-86 loss for you. You are playing Take And Hold (so holding the objective at the end of your turn is worth 5VP) and you have Storm Hostile Objective/Establish Locus. If you do the action, you score 4 points guaranteed but will lose. If you charge onto the objective, you will earn 9VP and win. Therefore, if the chance of making the charge is even >0% by the slightest amount, the correct play is to attempt the charge.

That’s why you should always keep in mind one of the key concepts of Game Theory Optimal 40K: Plays should be evaluated through the lens of how they affect your likelihood to win, not through the lens of VP/material/etc. Let’s talk more about this in the next section.

Do You Like Pie? I Like Pie. Let’s Talk About Pie.

Let’s introduce a key concept using a tasty metaphor.

You’re Not Fighting For VP. You’re Fighting for “Probability Pie”.

At every moment in a game of 40K, each player has a certain probability to win the game. Those probabilities add up to 100%*. There is a 100% chance that 1 of the 2 players playing wins the game, and that 100% is made up of the sum of each player’s probability to win. We can view the total 100% probability as a circle, or a pie. Each player’s probability of winning a game is their slice of that pie.

* There is a tiny sliver of probability that the game ends in a tie, but let’s ignore ties for now so I can use an even 100% as the total probability instead of 99.5% or whatever.

You and your opponent are battling over “probability pie” in every single decision. Make the right play? You’ve just made your slice of probability pie a little big bigger (well technically you just kept your slice of GTO Probability Pie from shrinking, not made it bigger, but we’ll get into that later). Make a blunder? You’ve shrunk your slice of probability pie and made your opponent’s slice bigger.

But Mike, We Can’t Measure Probability Pie

Just because we are currently incapable of measuring Probability Pie from looking at a board state doesn’t mean it’s not a real thing. Looking at games through this lens helps you optimize your decisions.

Imagine that above each game, there was a floating sign that showed each player’s Probability Pie. That would really effect how you play, right? Just because you don’t know the exact Probability Pie doesn’t mean it’s not impacting the game, and the player who has a more accurate “Probability Pie estimator” has a huge advantage during a game.

You Don’t Know The Final Score. This Is A Big Deal.

In some types of competitive games, you and your opponent are in a race to a certain milestone to determine the winner. 40K is not like that. 40K is not a race to a certain score, but a race to score more points than your opponent. It’s crazy to start a race without knowing how far you have to run to win the race, but that’s what we do every time we play a game of 40K.

This is an interesting idea I’ve literally never seen discussed in 40K, so let’s explore it. Let’s start by saying something really obvious: At the start of the game, you don’t know what the final score will be.

That means at the start of the game, you don’t know how many points you will need to score.

That means at the start of the game, you don’t know how VP translate into Probability Pie.

If you score 3 points on an early secondary instead of 4 points, how much does that change your chances of victory? Is it a 5% increase in your chance to win? 1%? 0.1? You have genuinely no idea. I have no idea either! If the final score is 88-87, I guess that extra 1VP was game-winning. If the final score is 88-10, that extra 1VP was completely irrelevant.

Here is the calculation you’re making subconsiously when sending out a unit to die but score you points: Is the Probability Pie of [my board state without that unit but with X more VP] higher than the Probability Pie of [my board state with that unit]?

As the game starts early, you can afford to play purely for VP. As the game goes on, and you understand how closely VP will be tied to the final score, you start to transition to playing for Probability Pie. The only reason you play for points and not Probability Pie is because humans are not capable of calculating exact Probability Pie, but you should always be trying to optimize your slice of Probability Pie regardless.

You Can’t Make Good Plays, Just Avoid Mistakes

This is something important to understand as you evaluate your own play. I talked about this concept briefly in my mindset article, but it’s very relevant here.

With every choice you make, there is an optimal decision. That is indisputable. The only reason we sometimes think there are several equally good plays is because we aren’t innately good enough at processing strategy games. If 2 supercomputers were playing 40K against each other, they wouldn’t shrug and say “There are merits for both plays, it doesn’t matter” like humans (including me) sometimes do. It would find the optimal play, and it would make that play.

As a result of the fact that there is an optimal choice for every decision, whenever you make any decision, you cannot gain any value. The best you can do is not make a mistake compared to the optimal line, and if you don’t choose the optimal line, you make the smallest mistake.

40K is not a competition between 2 players making good plays. 40K is a competition between 2 players constantly leaking EV, and trying to make the least sub-optimal plays.

If you see a clever line, you didn’t make something good happen, you just didn’t make the mistake of not making that line. This not only aligns with the reality of playing games more accurately than how most people view the game, but also keeps you constantly growing as a player and prevents you from developing a big ego at the table.

Avoid The Biggest EV Mistakes

This is something that’s glaringly true for poker where you can measure the chips lost in a decision. It’s also true for 40K, even though we can’t currently measure the value of a certain mistake.

If solvers played 40K, they would probably on average evaluate mistakes like this:

Missed a line to tag multiple of the opponent’s units? -2% winrate EV Loss.

Missed screening your backfield to allow the opponent to deepstrike a unit and flip your home objective? -10% winrate EV Loss.

Completely misunderstand the matchup and played aggressively when you should have played conservatively? -40% winrate EV Loss.

While it’s important to reduce how common your mistakes are, it’s also important to reduce the magnitude of your mistakes.

And once you’ve gotten better at intuitively avoiding the massive mistakes, you can focus your clock time and mental effort on optimizing marginal decisions.

I Played Perfectly And Lost. Explain THAT, Mike!

I’ve seen comments online like “This army sucks, I played perfectly and still can’t win”. A take like this is so delusional that even interacting with someone like this is a cognitive hazard.

Tactics Are Just Heuristics to Attempt to Emulate Winrate Optimization

This is a cool little concept I want to talk about briefly.

If there was a solver that could play 40K, it wouldn’t understand the concept of “Screening”. It wouldn’t understand the concept of “Trading”. It wouldn’t understand the concept of “Deploy conservatively”, and etc.

It would probably do something like screen its backfield because a board state where the opponent is allowed to go for plays where it can deepstrike into its backfield is -EV compared to a board state where the opponent can’t deepstrike behind you, but that’s not actually because it wants to “screen its backfield” as an intrinsically valuable thing. Humans have to understand that concept because we’re not capable of instantly running a billion simulations to see what happens if we screen or don’t screen our backfield in-game, so our tactics are basically heuristics as we make our best guess for how a solver would approach most situations.

A solver would not deploy all of its units on the line against a shooting army because it intrinsically wants to hide behind terrain, but rather because it ran a billion simulations and found out that a board state where its units are hidden from opponent’s guns gives it a higher chance to win than a board state where it deploys all of its units in the open to get shot. But that’s not because it cares about the concept of using terrain intrinsically.

I think this is a very important concept to keep in mind to not go on auto-pilot. In practical terms, I see players sometimes implement “good 40K principles” even in situations where it doesn’t matter against the opponent’s army. Don’t make what seem like good plays because you know they are good plays, make them because in that specific board state they are beneficial to your chance of winning.

Not getting shot is not intrinsically valuable. No 40K tactics are intrinsically valuable. When deciding whether or not to implement them, consider their impact on each player’s probability to win. Don’t “hide from their guns” because you “know” hiding from their guns are good, consider each situation on its individual merits.

Don’t Play Against Your Opponent, Play Against An Optimal Opponent

This is a key concept if you want to level up as a player.

“Solvers” (programs that solve for optimal strategies in games by playing against themselves) don’t know who their opponent is. When they implement a strategy, they’re assuming perfect play from their opponent. You should be doing the same, because the better you get, the more often you’ll play against players that are closer to solver lines than your Average Joe.

Let’s Talk About a Practical Example

Sometimes I’ll see lines in a game that will work as long as my opponent doesn’t notice a “big brain” counter. Examples of this can be a play to charge for distance that will give me a huge benefit, as long as the opponent doesn’t see the implications of where I want to pile-in and consolidate. If they see that line, and see the counter, they can heroically intervene with a random unit just to base my models and prevent my movement shenanigans.

A human player might view that as a good play and go for it. And in the traditional sense, it is a good play. It’s a “pro move”, most players won’t see it, and you gain a ton of value if it works.

From a Game Theory Optimal play, a play that requires your opponent to not see what is happening and make a counter play is complete trash. And truthfully, that play is complete trash. It’s not a “pro move” that gets countered by another “pro move”. It’s a bad move from a solver perspective because both sides are aware at all times of all plays and their values.

Plays that rely on your opponent not responding appropriately are -EV plays and should be avoided. And that’s actually where solvers and good competitive players are in complete agreement: 40K is an open information game. Assume your opponent isn’t forgetting your rules when making plays. And when you’re playing optimally like a solver, you’re able to share information freely because your gameplan doesn’t rely on your opponent forgetting your niche rules.

Maximize Your Chance To Win, Even If It Feels Weird

Let’s talk about the biggest “EV leak” I see in most player’s games: Playing a normal game of 40K from that point forward when playing a normal game of 40K is a sure loss.

In most high-level games you see between strong players, both players deploy pretty conservatively. This is because both players have an innate understanding of the game theory value of deployment decisions, even if they don’t think about it in those terms. This fundamentally comes from the fact that the downside of going second if you deploy aggressively is usually worse than the benefit you gain from deploying aggressively if you go first.

You can view the EV of deployment in most games largely like this:

• Probability Of Winning If You Deploy Aggressively/Go First: 55%
• Probability Of Winning If You Deploy Aggressively/Go Second: 40%
• Probability Of Winning If You Deploy Conservatively/Go First: 50%
• Probability Of Winning If You Deploy Conservatively/Go Second: 50%

The EV of deploying conservatively is higher (0.5 Wins > 0.475 Wins), so it makes sense to deploy conservatively.

But those assumptions are reliant on two equally matched players playing two equally matched armies on a fair mission. That is often not the case.

For example, let’s say there is a massive disparity in army strength or player skill. In that case, the EV of deployment might look more like this:

• Probability Of Winning If You Deploy Aggressively/Go First: 5%
• Probability Of Winning If You Deploy Aggressively/Go Second: 0%
• Probability Of Winning If You Deploy Conservatively/Go First: 0%
• Probability Of Winning If You Deploy Conservatively/Go Second: 0%

In that case, the EV of deploying conservatively is 0. The EV of deploying aggressively is 0.025. In that instance, the only logical choice is deploy super aggressively and hope to go first!

There are also situations where the Probability Pie is so unfairly allocated, that there is a negligible chance of winning regardless of what you do if you go second. In those situations, you should only consider your EV if you go first, because your EV is basically 0 regardless of what you do if you go second. And if deploying aggressively increases your EV if you go first, then do that, because you have 0 EV either way if you go second.

Situation like this are pretty rare, but they do happen sometimes. There is a specific game I can think of that changed how I view deployment.

It was early 10th Edition, and I was running Chaos Knights versus early index Eldar when they were still an absolute monster. I had beaten 2 Eldar opponents already that tournament, but my opponent this game was Zaak Kerstetter. He’s one of the best players in the world. I wasn’t going to be able to outplay Zaak like I had done to my previous opponents.

I deployed normally, prepared to play a normal game of 40K. I went first. I instantly lost. My first turn did nothing, Zaak killed any War Dogs I had staged forwards, and I was in the exact same position on the following turn but now with fewer War Dogs. “Playing a normal game of 40K versus early index Eldar with only 1500 points of your army” is not a gameplan that leaves you with any Probability Pie.

If you are in a really bad matchup, deploy like it.

Flip The Table If You’re Losing

If you’re going to lose, ragequit. I’m serious.

Now, I don’t mean ragequit in the sense of get salty or be rude to your opponent. Please don’t do that. Rather, I mean ragequit in the sense of “stop playing 40K”.

Let’s simplify the strategy for a monster mash Daemons player as two tactics:

• Play normal 40K.
  • This can have a Probability Pie slice from anywhere between 0% and 100%, depending on how far you are into the game and the matchup/current board state.
• Just shove big monsters forward and try to scam your opponent with invuln saves.
  • This always has a 30% chance of winning you the game, and resets your slice of Probability Pie to 30%.

Most good Daemon players don’t just deploy on the line and shove Greater Daemons forward, because that strategy has about a 30% winrate. I don’t play that way at all, and don’t advocate for playing Daemons like that as a default.

But the important thing to keep in mind is that the “shove everything forward and try to scam with invuln saves” line is always there in your back pocket. And, most importantly, you should use this strategy whenever your slice of Probability Pie drops below 30%.

If your opponent has a big points lead and you estimate they win 80% of the time, you know what time it is? “Greater Daemons be upon ye” time. Shove everything and see if you can scam them on dice.

Not all armies can “ragequit” like Daemons can ragequit, but I’m sure every army has some sort of similiar line they can take. Let me translate this into a generally useful heuristic: If You’re Winning, Reduce Variance. If You’re Losing, Increase Variance.

There Is No Such Thing As A Mutually Beneficial Decision

It is impossible for a decision to be mutually beneficial in a game of 40K. Literally completely impossible. If a decision changes your probability of winning, and the probability of winning is a zero sum game, then that decision helps one player and hurts the other. This is one of those things that no one thinks about in these terms, but is an important tool to help catch EV leaks in your game.

If your opponent decides not to make an aggressive play, and you are relieved that they didn’t make an aggressive play, one of you is making a mistake. One of you is either poorly estimating the EV of that play, or poorly estimating each player’s current slice of Probability Pie.

This is going to be a very short section, but it’s something worth mentioning.

If 40K Is A Zero Sum Game, I Should Never Help My Opponent Right?

I’m annoyed I even have to write this section, but I’m going to include it just so no one gets the wrong impression.

You should 100% be helping your opponent in game. It’s part of the social contract of 40K.

I want to draw a line between the game of 40K and the experience of playing 40K.

40K is a zero sum game. The experience of playing 40K is a non-zero sum game. Playing 40K should be a mutually beneficial experience. Let’s say you gain 15 “Happiness EV” just from spending a few hours rolling dice and playing a cool game, gain 10 “Happiness EV” from winning, and lose 10 “Happiness EV” from losing. In that instance, regardless of whether you win or lose, you’re better off from having played that game! Add in the “Happiness EV” of gaining experience at a hobby you care about, showing off your painted models, etc, and it’s obvious that 40K is a mutually +EV experience.

Even from a purely selfish perspective, you have to weigh the value of actions in terms of both winning your current game and winning future games.

Let’s say you have a 50% chance of winning the game you’re playing. If you pull a really dirty play (cheat, “gotcha” your opponent, etc), you increase your chance of winning that game you’re playing up to 75%. The EV of being a jerk in that instance is 0.25 Wins. Sounds like being a jerk is free EV, right?

Wrong. The good players in the community hear you’re a miserable opponent and avoid practicing with you. This plateaus your growth as a player. Being invited to practice nights with the area’s top players would have increased you from a 50% to a 60% winrate player. At your next 6-round GT, your Wins EV is 3 Wins instead of 3.6 Wins. You gained 0.25 Wins of EV by pulling a dirty but cost yourself 0.6 Wins of EV by pulling that move, meaning that dirty move lowered your EV.

Or, let’s say you straight up cheated to get that EV. Your local TO hears about your cheating, and bans you from the GT. Now instead of getting 3.6 Win EV from “playing GT after practicing with good players”, you’re at 0 Win EV because you can’t even play. Sucks to suck!

Add in all of the other social rewards that come from being a good opponent, and being a good opponent is a no-brainer.

I hope no one reads this and thinks I’m a cynical jerk who views every interaction as a way to maximize my EV. That couldn’t be further from the truth! Rather, I just want to not give people who are self-centered a chance to use my article a chance to justify anti-social behavior.

Practical Examples

Let’s talk about one of the most common ways I put Game Theory Optimal principles into practice when I play.

Threat Ranges Are A Scam

Big 40K doesn’t want you to know this, but threat ranges are fake. They’re a scam, sold to you by experienced players, to trick you into giving up board control.

Here is an extremely common situation: Your opponent has a very scary melee unit. You place your key unit [Movement Of Opponent’s Scary Unit] + 12.1″ away from the opposing scary unit. You pay yourself on the back, and flash your opponent a smug smile. You’re pre-measuring threat ranges. You’re a smart player. Good for you.

Well, you might be playing smart 40K, but you also might be playing awful 40K. Let’s explore this concept a bit more.

Let’s say you can be 2 inches closer in an optimal staging spot for the following turn. If you’re in the optimal space for you, your opponent has a possible 10″ charge. But for the opponent to even attempt that charge, they have to step out completely in the open. If they fail that charge, you shoot their unit dead for free. To even attempt that play is hugely -EV for your opponent. Why are you giving up value in case your opponent attempts a play that gives you Probability Pie?

It’s even funnier when a unit has access to advance and charge, and the opponent stands Movement + 6″ advance + 12.1″ away. What the hell are you doing? Why are you giving up so much board state in case your opponent chooses to go for a play with a <1% chance of working?

If you set up a -EV decision for your opponent, you want them to go for it. Now, if you do that, you have to be incredibly confident about Probability Pie and the impact of the play working/not working on that Probability Pie. But I am very confident that if solvers played 40K against each other they wouldn’t show as much respect for threat ranges as humans do.

I’ll let you in on the secret: All the time I tell my opponent “I’m placing my model here, if you go for the charge you’ll need to roll a 10”. A funny thing about that: They never go for it. This is disappointing. I really wish they would.

Mike, I Don’t Believe In All This Game Theory Stuff. I Just Play The Game.

That’s cool. Don’t let anyone tell you how to live your life.

But here is the cool thing about math and science: It does not matter at all whether you believe in them. Science* and math are true regardless of whether you believe in them. And, more importantly, they control the world around you. Game theory principles control every game you play, regardless of whether you believe in them or think about them at all.

* Unfortunately, I know a statement like that might be coded along certain political issues. While science is always true, the way humans interpret or present science are not always true. Please don’t make assumptions about any beliefs I have or interpret that in any way outside of 40K. I’m just talking about a game here.

Final Thoughts

Now go out there and treat yourself to a heaping slice of Probability Pie. You’ve earned it.

I could keep going for hours about this stuff, but I’ll stop here because I’ve already spent way too much time on this. If this gets a positive reception I might explore these concepts more in the future. Feedback is very welcome!

Interested in hanging out in the best Warhammer community I’ve ever seen? Come join the Warphammer Discord today! https://discord.gg/RydUCYeH

As always, have fun, stay safe, and may the Dark Gods bless your rolls!