How Games Twist Probability - Extra Credits Gaming

Welcome to Unnamed Game Show we won't name for trademark reasons. The game where you the player can win fabulous prizes or a goat. Now, behind one of these doors is a Half-Life 3 beta key. And behind the other two is, you guessed it, goats. So, pick a door, any door, and what's behind it is your prize. Okay, since I can't actually hear you because this is a gag in a YouTube video, I'm going to assume that you picked door number one. But before we show you your fate, I'm also going to reveal what was behind door number three. It's a goat. Getting nervous? Well, I'll let you switch to door number two if you'd like. Go ahead. Take a moment. What's the right move? Thanks so much to Brilliant for continuing to support our passion for lifelong learning.

Ready to have your mind blown? You should always switch. But why? And how that's relevant to video games we'll get to in a moment, because that little exaggerated example up top was the famous Monty Hall problem, a brain teaser loosely based on the game show Let's Make a Deal, which Monty originally hosted. And now that I've finished with my very poor impersonation of that, I'll explain the problem a little more robustly. Imagine a game where players are offered to pick between three doors. One of them has a good prize behind it, and the other two have duds. When a player picks a door, the game, knowing what's behind all three doors, immediately opens one of the other doors they didn't pick to reveal a dud. And then the game asks if the player wants to change their choice. In this situation, the game always reveals a dud, and there's no attempt by the game to trick the player. So, what should the player do? Setting aside the fact we kind of already spoiled that one, most people's immediate gut answer is probably something like, "Well, it doesn't really matter because I got a a chance either way." And that was certainly my answer at first, but it's actually incorrect. The truth is, the

player should always switch doors. Why is this you ask? Well, there's actually a lot of ways of looking at this, so let's run through them. The first is just to enumerate all the possibilities. So, for the sake of argument, let's say you picked door number one. In that situation, there's only three possible outcomes. Possibility number one, door one is the prize and door two and three are duds. Possibility two, door two is the prize and door one and three are duds. Or possibility three, door three is the prize and door one and two are duds. In possibility one, you win if you stay, but in possibility two and three, you win if you switch. So, you've actually got a 2/3 chance to win by switching, but only a 1/3 chance to win by keeping your original pick. Because as counter-intuitive as it seems, the player's winning chance depends solely on whether they've initially picked the prize. And whatever the odds are of initially picking the prize is, their odds of winning by switching are its inverse. But here's a simpler way of looking at it. Let's say instead of going through all of that rigmarole of

having the game reveal a door, it just offered you a simpler choice while all the doors were closed. Let's say you picked door number one, and the game then offered you the option of opening both doors two and three instead. Because that's actually what's happening under the hood. It's just obfuscated by the mechanics that are layered on top of it. If that was how it was phrased, and you knew the game wasn't trying to trick you, and that the prize was randomly distributed, you'd immediately go, "Of course I'll take doors two and three." And if you don't believe me that that's how it works under the surface, let's try imagining this with a thousand doors. Where, like in the original example, all but one unpicked door ends up revealed. You start by picking a door, and the game says you can either keep that one or have the other 999 doors. It's pretty easy to pick what to do here, as it's clearly not a 50/50 choice. So, why does this matter when we're designing games? Why is the Monty Hall problem important? Well, because humans are both great and terrible at probability. And how good or bad we are

at it in any given case depends as much on human psychology and how the probability problems are presented as it does the problems themselves. As designers, it's vital to understand that and use it in our design. Sometimes this can be to make simple problems more interesting or play on the player's desires. For example, critical hits play on our desire for power, making us feel like a mighty hero getting lucky. Even though from a design perspective, in 90% of games, crits are pretty easy to convert to a straight damage formula and balance against. Let's say for the sake of argument, your character hits for 100 damage, but 10% of the time you do double damage. Behind the scenes, we're just balancing enemy health as if you're doing 110 damage with every swing. Even more humorously, investing in your character's critical damage skills in many games is sub optimal because any overage damage is useless. If you hit an enemy that has five health with 800 points of damage, you're just wasting 95% of the attack. And if you put all of your resources into critical hits so that you're normally doing something along the lines of two damage, but on

occasion hitting for 100, you're strictly worse versus a five health enemy than if you'd set yourself up to just hit for five consistently. But as often, this idea that humans understand probability differently depending on how it's presented is also essential for what James calls design UX. Good designers make their design easily parsable by human beings so that players can focus on making the interesting decisions that come out of the design rather than making blind guesses at how the design actually works or worse still, operating under a completely false understanding of the design. Now, that doesn't always mean you want to make your design as straightforward as possible. After all, game shows that use things like the Monty Hall problem are much more exciting for everyone because no one knows the right decision to make in the moment. And so making the decision to switch or not feels full of peril. So it's important to realize where the engagement comes from in your design and only make your design deceptive or obtuse when the suspense of a blind guess is more thrilling than trying to make the most rational choice.

In the Monty Hall example, the engagement comes from wrestling with how to make the best decision. So that decision is abstracted and made more complicated. However, we know that in many games, probability estimates aren't really where the fun comes in. They're just there to make larger systems work. Leading us right back to the idea of understanding the psychology of your players as they approach a probability problem in your game. Because when we Monty Hall it up, it's not the probability problem itself that's engaging. It's how it makes you feel. Is it supposed to make you feel nervous? You could lose everything by picking the wrong door. Is it supposed to make you feel hopeful? Okay, the situation looks dire, but if you can just win this coin flip, maybe you've got a chance. Is it supposed to make you feel powerful by giving you the possibility of high rolling all the way? Boo-yah! Natural 20. So when you next look at probability problems in a game you're designing, remember clarity when you want to make the player think, obfuscation when you want to make the player feel. That's the real lesson of the Monty Hall problem. And if you're the kind of person that

would love to get a better handle on all sorts of probability concepts, it just so happens we can recommend quite the brilliant tool for that. You know by now that many of us in the EC community are passionate lifelong learners. Heck, you know, you're probably one of them yourself. You know, folks that are curious about a bunch of topics, but also want to learn about them on their own schedule. Which is why we think Brilliant, the interactive learning tool focusing on science, technology, engineering, and mathematics, you know, the whole STEM set, is perfect for learning at your own pace in a fun and interactive way. Replacing traditional lectures with hands-on lessons complete with awesome visual examples and a storytelling approach that really helps the information sink in. For instance, with so many people heading back to school, Jeff's been recommending Brilliant's math courses as a quick way to get caught up on what you or students you know might have forgotten about over the break. Seriously, it's way cheaper than a tutor and a lot more effective than study hall. And because I think math and history go together like peanut butter and jelly, I've been devouring their math history course, which is full of cool stories and specific mathematical challenges from the lives and research of dozens of famous

mathematicians. AKA delicious info that will stick in your brain. Hmm, and not food for your mouth. So, if you're a curious learner just starting out or a season pro, you should check out Brilliant to level up your learning by going to brilliant.org/extracredits and signing up for free. And the first 200 people that go to that link will also get 20% off an annual premium subscription, which added bonus helps our channel make shows like the one you just watched. Seriously, thanks so much for that. Thanks so much to Skyler Homes, Joseph Flame, Dominic Valenciana, Casey Mustya, Arclight Games, Angelo Valenciana, Alicia Bramble, and Ahmet Ziya Turk for being awesome legendary patrons.