In this episode, we check out the wonderful game of Citadels and we cover my entire understanding of soccer. Plus, of course, the game theory of mixed strategies!
Earlier this month, I had the amazing opportunity to sit down with the one and only Joseph Scrimshaw and talk about game theory. I know I talk about it a lot from my own academic point of view, but sitting across from a wonderful comedian turned everything into an incredibly interactive conversation. This is the kind of thing I wish I could do every day. Listen in, if only to find out how to deal with wild bears who want to play games with you! It might just save your life.
First, watch this.
Ridiculous, right? This is a record-breaking moment in the history of a game show that has been on TV since 1956. It’s also a perfect moment to take a look at some game probability as we determine exactly how ridiculous this moment actually is.
Let’s start with Wilbert. Determining the probability of an event occurring requires us to create a pretty simple ratio. On bottom of the fraction are the number of possible outcomes in a situation. On top, the number of favorable outcomes. In this case, we want to find out the number of possible spins of the wheel and the number of ways to get $1.00.
How many possible outcomes are there? There are twenty different spaces on the Wheel, starting at 5 cents and increasing by 5 all the way to $1.00. Your first spin can have any of those twenty results, as can your second spin. However, you only get a second spin if you don’t land on $1.00 the first time. There are 19 ways to get a second spin, and a second spin has 20 different results. We call this the sample space—the group of possible outcomes.
Okay, so how do you get a $1.00? You can either spin it on your first go, or get a total of $1.00 by adding up to spins. If your first spin is 15¢, the only way to get a dollar is with 85¢. 30¢ has to match with 70¢. For each spin, there’s only one possible second spin that nets $1.00 total. Which means there are twenty different favorable outcomes that lead to $1.00. Notice here a trick about probability: we consider 35¢ + 65¢ to be a different outcome than 65¢ + 35¢. Order matters.
So, not significantly different than rolling a 20 on a d20. But it didn’t just happen once… it happened five times in a row. What are the odds of that? Here’s a moment when you often stump students in a high school probability class (Hi there, former students! –Rich). There are five of them… maybe we should multiply by five? But that makes it more likely that the event occurs, which is wrong. So then maybe the next step is to divide by five… this is the point where I put an X next to the test answer, hope the next one is better, and plan to have a brief discussion about how probabilities are always horribly unintuitive.
When it comes down to it, the right answer is to build a probability tree. When we do this, we can start to put together the actual solution. In probability terms, this is a then question, as in event A occurs, then event B occurs. In then problems, we multiply the probabilities of each outcome together. Which means the odds of these five $1.00 spins happening in a row is roughly one in two and a half million.
How many stars are there in the sky? Would you believe that you only see between 2,500 and 5,000 that are visible to the naked eye at any one time? Think about one star in that vast expanse. One in two and a half million is another few orders of magnitude entirely. So, as much as I enjoy watching Drew Carey’s impersonation of Bob Barker, I’m not planning to see a Showcase Showdown like this one ever again.
As I wrap up final production on the Puzzle Keyring this week, I’ve spent a lot of time thinking about why I was inspired to make it in the first place. I enjoy puzzles and codes, of course, but there are different ways to show that passion. I could have made my own puzzles or designed an escape room. So why did I decide instead to create a teaching guide for new players to experience the joy of solving puzzles? Because I remembered how it felt to fail.
One day, I sat at a picnic table with three friends just outside Multnomah Falls. As we’d hiked up to see the waterfall, we’d been handed a series of beaded necklaces. The beads were of all sorts of different colors, and while we saw some repeated combinations, there wasn’t an overall pattern. Finally, we received a burned CD filled with Madonna songs. Mystified, we sat at the table and listened to the tracks, hoping there would be some secret code hidden inside, because the necklaces seemed terrifying.
We eventually figured it out—the colors were resistor code, the groupings were frequencies, the frequencies were notes that matched the Madonna songs. While I like to describe that solve sequence to friends as an example of how complicated and inspired these puzzles can be, I also can’t help but remember my frustration on that afternoon. We were playtesting a 36 hour puzzle event called Wartron, and the other two teams had tons more experience than our rag-tag little team. We knew they were ahead of us. We knew that our supervising member of Game Control was silently sitting with us, watching our struggle. We also knew there wasn’t much we could do about it.
Failure is a tricky thing. Our bodies train us to avoid it with fear and pain responses. In the real world, failure often has consequences as well. And yet, there are lessons that can only be learned through failure. Lessons about resilience, introspection, and self-development. We play games our whole lives just because they give us a safe space to learn these lessons. It’s almost a tragedy when all we do is succeed. We need to fail.
Playing and losing cooperative games is such a perfect way to experience failure. You and your team fail to stop the shadows from taking over the Round Table, or you let the Frost Dragons destroy your little town. Now comes the best part—you and your friends can talk about losing. Failure forces you to look at what you did and see how to do it better. How to find your own success. Maybe this time, you keep all of those explorers out of the center of your Island! (BTW, that’s exactly how you need to play Spirit Island. –Rich.)
So Wartron? My team and I floundered for hours in the face of codes and puzzles we’d never seen before. The other teams were so far ahead, they may as well have been in another game. Suddenly, things started to click. We learned new techniques and began to recognize similar mechanics. I could look at a puzzle and know how to the find the solution, and if I couldn’t, someone else on my team could. We found our second wind and eventually caught up to the other teams. It’s been years since Wartron, and I’ve never forgotten that satisfying thrill of pulling into the lead.
These days, I’m not too worried about getting stuck on a puzzle. Sure, some of that is from that moment where the switch flipped in my brain, but so much more is just from all of that failure. It taught me how to get back up. So now, when someone shows me a brutal cooperative game that just crushes us in the first round, I get amped up. The best thing anyone can learn from failure is that burning desire to set the game back up and play it again.
Have you played Splendor? If not, you really should. It’s a great game where you play a gem merchant, collecting gem resources to buy better and better mines and shops to sell your wares. Purchasing these mines and shops give you permanent gem resources, allowing you to buy better mines and shops and so on. The game is fantastic because of the huge ramp that becomes available to you during play. At the start, you strive for the most basic mines. By the end, you could buy those on a whim, but you’re on the lookout for better and better storefronts. The game ends when someone hits fifteen victory points, so its important for players to build a gem engine, gathering points as quickly as possible. When you do? It’s a very satisfying experience.
In theory, that’s how the game is supposed to work. The Splendor problem is that this isn’t the key to victory.
One day, I decided to play in a Splendor tournament in Portland. As I recall, the winner would earn some promo goods, but I admit that I lost focus on the prize very quickly. This was the day I learned about the Splendor problem.
Splendor is a game that has a flow in mind. The cards at the bottom have a low cost, but low victory point value. So you grab gems to acquire them, which takes a couple turns. Once you have enough of those, they help you grab the second level of cards. These help you take the third level, and you can finally grab the Lords at the top. The Lords require you to own certain colors of gem cards, so players are careful about what they acquire during the game. This flow is built into the setup of the game, the rule book, and the playmat. Sadly, it’s all wrong.
During the tournament, three of us played with the flow. The fourth player had been taught the real strategy. As soon as I saw it implemented, I recognized that the flow was wrong. Building a gem engine wasn’t the key to victory. Instead, players should use highly targeted efficiency. Skip level one cards entirely. Find a level two card that only requires one gem type. Reserve that card, granting you a wild Gold token. Take only that gem resource. Eventually, acquire that card. Do it again. Then move to level 3. Have fifteen victory points before anyone else hits six.
Why does it work? Beyond being ruthlessly efficient, this strategy also requires you to take most of a single gem type. Since other players need a wide variety of gems, they will find themselves starved for that gem type. This turns a vaguely competitive game into a different one, where efficient players are now controlling the resource market. It’s mean. I tried it once, instantly won, and gave up on the tournament.
The Splendor solution brings back the fun of the game and, I hope, the original intent.
The new expansion Cities of Splendor adds four variants to the game. Instead of Lords at the top, there are Cities. These either require a TON of gem cards to get, or a massive pile of victory points. Plus, now the game doesn’t exactly end at a victory point total. The game ends on the round when someone gets a City, and only players with a City are in the running to win the game.
When you add Cities into the mix, the game becomes a race to build a City as fast as you can. In this case, having a lot of gem resource cards is a huge advantage. Now, players have a realistic chance at victory no matter which strategy they employ.
I am, of course, very interested to see how this expansion plays out in reality. In my demo, players seemed very excited to play with the flow of the game. But it still may be possible for that alternative strategy to win out with certain Cities. We’ll see! Plus, with three other variations I haven’t even discovered yet, I think this expansion is actually going to put Splendor back on my play list for a while! That might be the most welcome change of all.
Last week was the culmination of mountains of research and development as my team and I launched the Emoticode, a three-day livestream on Twitch promoting the release of Activision’s Call of Duty: WWII. Over the course of those three days and eight hundred-thousand views, thousands of players competed to crack a series of forty-five ciphers, with solvers gaining a beta invite to the game. In the end, my devious ciphers stood up to the onslaught for almost fifty hours, and they would indeed have been doomed without me watching for the perfect moment to drop just the right hints.
At a baseline, each of the ciphers had to stand up to random brute-force attacks by hundreds of guessers for at least an hour. Meanwhile, they still needed to be solvable by fans who had potentially never cracked a cipher in their lives. Best of all, the cipher would be transmitted only by a series of blinking emotes in the Emoticode typewriter with no direction or guidance unless a hint was needed. A wide web of impenetrable difficulty until a solver can find the single, clever way to break through.
This is the kind of challenge I enjoy the most.
It turned out that the solution to this challenge was scaffolding. You probably remember this from your math classes—you studied addition, then learned that multiplication was just like adding, but faster! Your skills leveled up over time. When we build a proper scaffold, we allow players to build their skills. Which is why you should play Lords of Waterdeep before you take a crack at Agricola. But back to codes…
Our first ciphers were the simplest. Starting from a few hints, players needed to translate the Emotes into letters and numbers. In this alphabet, you might get emotes that translate to ?A??IN? and have to come up with the right solution. Since you know most of the letters, those ?s would have to be C, G, K, P, or Q. G is a good guess for the last letter, and soon we’re at !solve PACKING and victory. Of course, it wasn’t quite that easy, and it instantly got harder.
Before too long, we moved into simple substitution, like the Caesar or Atbash ciphers. Not much harder, but then we’d do some transposition, like a Rail Fence or Columnar Substitution. And then? We’d start doing both at the same time. I’d throw in Braille or Morse Code to keep things fresh, and then dig into some Binary or Hexadecimal digits. By the end of the Emoticode, our top solvers knew every single one of these types of ciphering systems, plus Vigenere, Playfair, and One-Time Pads. It was incredible to see these brand new solvers have a series of tricks they could bring out to try against my ciphers! On the other hand, if I’d started with a Rail Fence Hexadecimal Atbash puzzle, absolutely none of this would have ever occurred.
Scaffolding is something I learned as a teacher, but you often see it in board games as well. The extremely new Apocrypha has a pre-wrapped deck that you are meant to play without shuffling at all! A video guide helps lead you through the scenario. Once you play those early missions, additional rules and features can be tacked on to make things difficult in fun ways, not just through complexity and obfuscation. One of my favorite games in the world, Race for the Galaxy, fails to do this, and I have the worst time convincing new players to take a chance on the game.
By the end of the Emoticode experience, I’d written forty-five ciphers and spent almost that many hours moderating Twitch and Discord servers filled with people trying to solve my devious codes. A group of solvers banded together and are now looking for more puzzles and codes to solve as a new team! For a guy who teaches Cryptography, this was the perfect outcome. Along with my immediate twenty straight hours of sleep.
(And no, this was not a trial to get into the NSA. As I kept getting asked on Twitch every ten minutes.)
Yesterday was a serious reminder that all the games I played back in college are getting a revamp. First, the announcement from Fantasy Flight Games about the impending Fourth Edition of Twilight Imperium. Second, and much more fun for me, was the opportunity to play the recently updated Robo Rally.
Where to even start? Robo Rally is the chaotic programming game where players try to race their little robots around the factory to reach checkpoints before anyone else. It’s a huge, fun, ridiculous mess and we played it all the time. Were there problems? Absolutely. And instead of telling you what they were, I’ll leave it up to superhuman designer Richard Garfield.
“…when I play games with which I have finished the design I feel bad whenever I see a flaw. Generally when I force myself to play I enjoy myself, but that is my nature. When it came to RoboRally I really had trouble with the flaws, in part because it is one of my earliest designs and I care deeply about it. [My wife] suggested I redesign it using my 25 odd additional years of design experience. The more I thought about it the more I liked that idea so I began to think about a RoboRally reboot.”
As someone with game design aspirations of my own, this really resonated with me. Who wants to look back on something they know they did wrong, simply to look at what they did wrong? Self-forgiveness isn’t easy. Years of teaching, begging students to edit their work and learn from their mistakes taught me that this isn’t easy. So I admire Garfield’s efforts here to look back and fix a well-loved game that could honestly use a fix.
Robo Rally uses a series of programming cards to move around the board. Each turn, you draw nine cards and plan out a series of five register moves. When each player is done, everyone flips over their first register card, performing them in order. Sometimes, that means you step into another robot’s way unintentionally, and it pushes you into an unplanned spot. Other times, the board zips you around on a conveyor or crushes you in cannery row. Even in a perfect world, players can mismanage their program and end up in a totally weird place. Then it’s time to flip over the second card, and chaos gets magnified.
Of course, in the old version, everyone drew from the same deck. Which meant it was possible to draw nine left turns. Knowing this was not fun, Garfield redesigned the game so all players had their own deck.
Fans of deckbuilders will instantly recognize how this affects game probabilities. Since every deck has the same set of cards, hands of nine become more predictable. Sure, I may not always get a Move 3, but I can be all-but-certain that my hand will contain forward motion. In the old game, that wasn’t quite a safe bet.
Individual decks also allowed Garfield to update the damage system. In the old game, players might have one of their registers lock, meaning that the card they had in that particular register phase could stay there for turns and turns on end. Otherwise, damage just reduced the number of cards you could draw, making it even more likely that your turns would be entirely out of your control. The counter was to Shut Down, removing all damage by skipping your turn. Skipping a turn is, unfortunately, not fun.
However, in the time since then, we’ve seen plenty of deckbuilders introduce a variation on Curse or Waste cards. These useless filler cards just waste a bit of your time and make this turn a little tougher. Garfield used this idea to add Spam cards rather than damage. These feel like something a robot would hate, and allows for variations like Virus Spam or Trojan Horses, making what would otherwise be a simple damage counter into an interesting series of choices.
Reading the rundown of what Richard Garfield changed in Robo Rally is an opportunity to watch a master at work. I absolutely recommend that you check it out, and if you were a fan of the old game, try out the new one and see how it plays! It feels streamlined, simplified, and actually plays like the programatic robot combat racing game I’d always hoped to play.