Decisions Matter: Game Theory in
Billions – Part I

Game Theory. A term we have repeatedly heard on Billions.

Chuck talks about his strategy to make Pete Decker sing in Season 1, Episode 3 Yum Time:

Spyros: That’s what I like to call “the prisoner’s dilemma.”

Chuck: No, you don’t like to call it that. That’s what it’s called. Started as a thought experiment, game theory in the ’50s. Does no one ever check you on this bullshit?”

Wendy talks to Axe about Krakow (without naming him) in Season 2 Episode 1 Risk Management:

source: Showtime

“And the manager played some heavy game theory on me, boxed me into a spot, essentially put himself in a position to win no matter what I said.”

Axe tries to get Taylor play in the Alpha Cup in Season 2 Episode 3 Optimal Play:

I had a feeling, pulled your resume, saw some game theory strengths, looked into it a little bit more.”

What is game theory?

Game theory, in a nutshell, is a mathematical methodology that provides a language to model, analyze and understand strategic interactions in which an a player’s choices potentially affect the interests of the other players. One important assumption is that the players are rational. That is, they have clear preferences over outcomes and choose the optimal (best) outcome for themselves from a set of choices through analyzing the costs and benefits and taking into account what other players might be doing.

Sounds familiar?

source: Showtime

While the methodology is widely used in economics, political science, psychology, computer science and biology, game theory can be applied to any setting where actors interact strategically and purposefully to maximize their own benefits. Examples abound: Think about nuclear negotiations between states, product markets in which several firms compete, provision of public goods, or political campaigns in which candidates run to win a political seat. That said, we, ordinary human beings, also use game theory in our daily lives, in every interactive situation where we seek to do what is best for ourselves: in the games we play, e.g. chess, checkers, poker as well as board games like my all-time favorite Risk, or in our own personal negotiations with a car dealer, our boss, or our significant other.

source: Showtime

And you may ask, and rightly so, how come I am a know-it-all about this.

Well, as I told Billions cast and creators at Billions PaleyLIVE New York last year, I am extremely lucky that, next to having my favorite actor in it and taking place in my favorite place on earth, Billions deals with what I am doing for a living! Yes, I have been teaching game theory both at undergraduate and graduate level for more than a decade. And so this long overdue post(s) in which I try to illustrate how the show makes brilliant use of game theory is my personal tribute to all those collectively giving us the smartest show on TV. Hope you enjoy it.

I vividly remember the day we saw Billions Pilot. We watched it over some yummy French Toast, courtesy of Lewisto, for breakfast on New Year’s Day 2016. We had a plan to write a collective post about our first impressions of the show and I was dying to write about my first take on the characters, their relationships as well as my predictions about what could be in store for them in the show…  However, I  left it aside to focus on one single aspect of Billions that fascinated me beyond words… WRITING.

Here is an excerpt from our “first impressions” post on January 4, 2016:

source: Showtime

“What is it that impresses me the most about the writing? Well, you know I am a college professor. Yet you don’t know what I teach. I teach game theory, a field in social sciences that, in a nutshell, studies strategic interactions among people in business, politics, law, sports, etc. Believe me, you all use it, even without knowing, in your daily interactions as well as in games like poker, chess or risk! And, if I am correct in my thinking, with Billions, we are in for one of the best cat-and-mouse stories in the history of TV thanks to… game theory! I can easily say I have not seen any show to date in which game theory has played such a central role in developing the plot. I know this is just the pilot; however, if the rest of the show develops along these lines, Billions will be one of the most thrilling shows ever!”

Now, am I right, or am I right? 😀

Backward Induction

The first Billions moment that made me smile is the scene we meet Axe at Axe Capital. Mick Danzig, a portfolio manager and his rookie analyst Ben Kim tell Axe that Lumetherm is being bought by Electric Sol. Its price was $41, it is now trading at $35 and they envision a 17% jump when the deal closes. Danzig plans to buy 2 million shares for the main fund with which Axe seems to agree. However, things change when Danzig shares new piece of information: Scott Kazawitz is rumored to become the new chairman of the company.

Axe knows Kazawitz is an animal. He currently controls Electric Sol and he also owns 19.3% of Lumetherm backdoored through his stake in Southern Wind. Southern Wind had a block trade (meaning they sell a lot of their shares) through Merrill Lynch last Thursday at 12:52 pm – lunch time – which tells Axe they wanted people to miss it. Axe thinks Kazawitz is using this potential merge as a ploy to temporarily prop up Lumetherm and use the block trade to get out of Southern Wind and  Lumetherm…  which means Axe Capital needs to get out of this immediately! So… Short!

Ha! What Axe talks about when he talks about looking at it backward is nothing but backward induction in game theory. It is the process of reasoning backwards in time, from the end of a problem or a situation, in this particular case, from Kazawitz’s name being floated around as the new chairman to determine a sequence of optimal actions taken earlier by Kazawitz.

Short term loss for long term gain

Next we see Axelrod boys getting game theory education at dinner table. While dad explains what “pissing contest” means to Dean and Gordie, he chooses to manipulate them into one 😀 He asks Dean about the seventh president: “Andrew Jackson.” Next question is for Gordie: What about Jackson’s home state? Gordie seems to have no idea and wants to talk about the Yankees… until he gets Dean dare him to a bet for 10 push ups: “Border areas near the Carolinas.”

A rational actor may be willing to incur some cost today if he anticipates large enough benefits tomorrow that would compensate for that loss. Gordie, by letting Dean think of him as stupid, fools his brother and sends him to do 10 push ups. Dad explains to his first born:

“He knows his customer and he sets you up. He is willing to look stupid short term for long term gain.”

Axe himself follows a similar strategy later in Episode 5 The Good Life when he tells Wags to sell  5% of all positions across the board immediately and then sell sector by sector starting with Telecom and plays hooky for a few  weeks. He is willing to look like a man having a “bullshit midlife meltdown” in his people’s eyes in addition to having a bit of loss by selling 5% of all positions until… he appears at Axe Capital to reap the benefits once the news break that the CEO of Mundia Tel, a telecom company, and office presidents in six countries were caught cooking the books triggering a big dip for telecom stocks. For some reason, I filed this gif of Axe’s arrival under GreekGod1.gif 😀 

source: Showtime

Signaling Games

Chuck Rhoades, holding a score of 81 – 0 in insider trading cases, would not open a case file against Bobby Axelrod if he were likely to fail. Chuck wants to take “Mike Tyson in his prime” only when he is “gettable.” And he has his window of opportunity when he finds out Axe may be buying a beach mansion only 1% of the 1% can afford.

One may be surprised at first when Chuck sends Axe a message telling him not to buy the mansion. If Axe does not buy, it will be politically infeasible for Chuck to go after Axe who has a “man of the people” public image thanks to his post 9/11 generosity to NYPD and FDNY. Is Chuck trying to avoid a case against Axe because his wife works at Axe Capital? Absolutely not. Chuck wants to put Axe in jail if there is a case. So he sets Axe up in a perfect textbook signaling game. Chuck reasons that an innocent Axe, a disciplined player, would pass on the house whereas a guilty Axe would buy the house just to show Chuck that he has nothing to hide.

There are two players in a standard signaling game: There is a “sender” who takes an action and a “receiver” who, upon observing the sender’s action, tries to identify the type of the sender and take his best action accordingly. In this particular game, Axe is the “sender.” If he passes on the house, he signals to Chuck he is innocent and Chuck passes on the case. On the other hand, if Axe buys the house, he signals to Chuck he is guilty, and Chuck opens a case file against him.

We expect Axe, a rational and a disciplined player, to pass on the house, don’t we? But he ends up buying the house signaling Chuck he is guilty! You may ask why. Well, because even rational players may take irrational actions at times. In this particular case, it is Elmo, the family dog, who triggers Axe to buy the house on an impulse.

source: Showtime

Elmo, who used to run on high wire and piss on the furniture to claim his turf earlier in the episode, is back from the vet “fixed.” Elmo is the perfect metaphor to convince Axe that he would never let anyone “fix” him: Neither Chuck who has told him to walk away from the house nor Chuck Sr’s friend who has now made a higher bet on the house. So Axe goes ahead with a $63M take-it-or-leave- it cash offer and gets a call from Hall letting him know that the US Attorney’s Office is opening a case file on him.

Prisoners’ Dilemma

As Chuck shares his plan to make Pete Decker spill the beans about Axe with Bryan and Spyros, he brings up Prisoners’ Dilemma, a strategic game explaining why two rational players may end up digging each other’s grave while a mutually better outcome is within their reach. I usually introduce this game on the first day of classes along the same lines that Chuck does 😀

source: Showtime

Two criminals from the same gang are arrested. Each one is in solitary confinement and cannot communicate with the other. Prosecutors do not have enough evidence to convict both so they simultaneously offer each prisoner the following: Each can either keep silent or betray by testifying that the other committed the crime. If both criminals betray the other one, they both serve a 2-year sentence. If criminal 1 betrays and criminal 2 keeps silent, criminal 1 will be released and criminal 2 serves a 3-year sentence and vice versa. if they both choose to keep silent, then both serve a 1-year sentence.

Now what do you think happens?

Suppose you are criminal 1. Given that criminal 2 keeps silent, what is the action that gives you the best outcome? Betrayal. You go out of prison, and the other one stays in for 3 years. Now, given that criminal 2 betrays, what is the action that gives you the best outcome? Betrayal. At least you get the lesser prison sentence. The same applies to criminal 2 given whatever you choose to do. That is, regardless of what the other side does, betrayal always gives the better outcome – we call this dominant strategy in game theory. Both criminals play their dominant strategies and serve a 2-year sentence. You may find this video which illustrates what I have just told you helpful.

Developed in 1950 by Merrill Flood and Melvin Dresher, two mathematicians at the RAND Corporation, to explain the Cold War, Prisoners’ Dilemma is applicable to many real life situations through the concepts of competition and cooperation. One quick example from sports is use of performance-enhancing drugs, a sad reality in sports. While it is obvious all professional athletes would be better off if no athlete used them, the problem is that, regardless of what other athletes do, an individual athlete would be tempted to consume them and outperform everyone else. Using these drugs is the dominant strategy and so everyone is tempted to use them.

Back to the US Attorney’s office. Chuck tells Decker they had another guy who did the same Pepsum trade in the room two hours ago and he was very cooperative. It now comes to who flips on Axe or “first in gets the lollipop!” Chuck’s plan does not work since Decker is not convinced about the other guy being cooperative. He sees Chuck’s bluff and leaves the room with no sign of caving in.

I am a rational player who wants to maximize the number of her readers and it is statistically less likely for people to read a post going too long. So I stop here to talk about more games, in particular the ones with Wendy  getting involved in with her two favorite boys, next week! Stay tuned!

Next: Decisions Matter: Game Theory in Billions – Part II

10 thoughts on “Decisions Matter: Game Theory in
Billions – Part I”

  1. Can I have you as math teacher, please?
    You are awesome, creative and statistically rational above average.

    You left me thinking though about the sender and receiver, I would have thought the other way, like Chuck sending a sign not to action and Bobby receiving it and trying to act at his rationale best. Which of course he fails, because everytime someone puts emotional before rational is 33% more likely to fail.

    Thank you for sharing the video. I have to say that always the prisoner’s dilemma does not make sense as it implies betrayal, which something hard for me to deal with. Of course the marketing example sums it better for someonr like me, whose morals intervene while understanding that despite what we all think, likelyhood is a math thing because it is based on a “universe”
    And I belive there eould be a math formula to incorporate moral deviation and or distress as the extremes right? Those that are total psychopaths and the total “shells” that can not see the world out of their own mind frame.

    Again, you should be teacher online….I am too far away from you to visit one of your classes 🙂

    Have a great, happy birthday! 🙂

    1. Haha. You nail it. I’ve always been a math girl 🙂

      Sender and Receiver are just the vocabulary we use in signaling games. Chuck kicks off the game by telling Axe not to buy the house. Axe is the sender because he takes an action (buy or not buy the house) and depending on his action Chuck, the receiver, can (and sometimes cannot) identify the sender’s type and act accordingly.

      Game theory assumes that actors are completely rational. That is, they typically do not act on emotions or distress or morality. They make their cost/benefit analysis, take into account what the other side may be doing, and takes the best action possible for themselves. That said, there are ways to incorporate morality, altruism, or whatever it is that makes you happier into your utility function.

      Game theory also tells us, like in Axe’s case to purchase the house, players can sometimes take irrational actions. Because they are not robots, they are human beings. In short, game theory approximates the real world and predicts what could happen assuming the players are rational. That said, in many games, the players are not individuals but entities like states, organizations, businesses that do not have emotions, they have full rationality and so it is much easier and more accurate to model their behavior to predict what will happen, say, in an international crisis situation, or in some bargaining between to firms, etc. Taylor is also a good example of a fully rational robot-like individual. They know when to completely detach from emotion. We know Axe also most of the time can do it, but he sometimes cannot. I think the main reason he needs Wendy by her side is for the times he cannot detach himself from emotion and when these emotions make him take wrong decisions.

      And thanks so much for your kind birthday wishes!

    1. Oh no! Anyone can enjoy the show without knowing game theory because, as I highlight in the post, we all use it all the time in our lives. We are always in situations where our action will affect the interests of others which we call a “game” from when we negotiate with a car dealer, when we ask for a raise to our boss to when we play chess or cards or box games and to when we bargain with our significant other.

  2. Maybe this is why I love the show as well – I am also a math teacher – high school – AP Statistics being my specialty, and so I too am very familiar with game theory, although not to the depth you are. Impressive rundown – can’t wait for part 2.

    I think you also hit on why certain episodes stand out for me – like S2Ep11 when Chuck snares Axe through the Ice Juice -such a masterful game of chess that Chuck played there, and I totally didn’t see it coming. When I went back to watch season 2 I noticed they first subtlety introduced Ice Juice through Ira in the second or third episode – just a little aside at dinner with Ira and Chuck and a couple ladies. Of course it would play a huge role in the entire scope of the season. But you’d never know it from when it was first introduced – just brilliant writing from Koppleman and Levien. Have you asked them if they have any background in game theory?

    1. Thank you so much for reading and for your kind words! I thought you might enjoy the post. I’ve always been a math girl since grade 1 and math teachers are my heroes!

      Billions is the smartest show I have ever seen on TV. No question about it. I was lucky to chat a bit about game theory with Mr. Koppelman when I bumped into him once midtown Manhattan (they were filming the tobacco shop scene in Season 2 that day). He is fascinated by game theory and he certainly reads a lot. Besides, he and Mr. Levien made Rounders – a major poker movie, and I believe both of them are poker players, too. Poker players know and use game theory — which I will talk about in the context of poker tournament in Season 2 — so they are certainly pretty knowledgable about the topic and it stands out in their writing!

      Since I have so much to say about how they use game theory in the show, this will be a three-part series. The last one will be all about Chuck’s long game which is certainly the smartest we have ever seen in the show. Season 2 is sort of the movie “Sixth Sense” that you feel the urge to watch from the beginning to find the clues. I saw Chuck’s game coming but probably because I was totally obsessed with the show and I knew he would use Boyd in some capacity but of course I did not know how. The entire “Chuck playing with Boyd” storyline is fascinating.

      And Golden Frog Time is my favorite episode as well. Mind blowing!

  3. I hope that invitation to sit in on one of your classes is still open! I’d love to see you in your element, and get taught game theory in person!

    The prisoner example is a very good one. However, where I come from if you talk, you are a rat, and rats don’t last that long on the outside, so it’s better to keep your mouth shut, do your time, and come out with your honor in tact! Dollar Bill is a good example of this. He would never sell out Axe, He’d never be the rat.

    PS: I’m hoping Showtime releases at least the 1st episode a bit early like they have in the past. We all need that extra time to dissect every episode!

    1. You have an open invitation!

      Haha I almost feel like Lara is talking 🙂 That is exactly why, in some settings, people prefer not to “betray” or “rat.” Because everyone tries to make the best decision for themselves and if they know there is no life after betrayal — not in a literate sense if we are not talking about the mafia, but for example if you know you will never be able to find a job ever again once you rat — then you prefer not to betray because it is very costly. That is why all strategic players first make their cost/benefit analysis and take into account what others may be doing and then they make the best decision for themselves and the best decision could vary from person to person. The prisoner example is the typical way to teach about situations where people end up in a suboptimal situation while a better outcome is within reach. I think the use of drugs in sports example that I give in the post is a good one from real life.

      I am currently reading The Black Edge by Sheela Kolhatkar. It is my prep for Season 3 since I want to learn about the legal process when there is an indictment. Of course, the Axe case is different from Cohen’s in the sense that they were never able to find anything on Cohen, he was very very careful and he kept himself insulated. He was never personally charged with a crime. However, the FBI used “Prisoner’s Dilemma” exactly in the same way I talk about in the post to make two former SAC employees talk about the dirty business SAC was involved in, and it worked. The two people were brought to SEC separately, kept in separate offices. They were supposed to be friends but they sold each other out in minutes, not in hours, when they were told that they could get a lesser punishment if they did that. I believe they were already rich enough to have a good life if they could save their ass from this mess and so they just went with it.

      I am very hopeful they will release the first episode early and maybe they will again show the first two episodes in movie theaters before the premiere on TV. I remember you caught it in NY last year and told me all about it 🙂 Fingers crossed!

  4. merçi pour vos belles explications!!J’ai le DVD de la saison 2,je regarde mais comme mon Anglais est mauvais,je ne comprend pas beaucoup!
    Je trouve qu’en ce moment il n’y a pas beaucoup de choses sur Damian,je sais qu’il tourne la saison 3,mais il me semble que lorsqu’il tournait Homeland,il y avait plus d’informations sur lui,des photos, j’aimerais plus d’informations sur lui,des nouvelles photos!pardonnez moi,je sais que je suis exigeante,mais cela me manque,vous savez a quel point j’aime Damian!Et vous aussi je vous aime! Monique

    google translate: Thanks for your nice explanations !! I have the DVD of season 2, I look but as my English is bad, I do not understand much!
    I think at the moment there is not much about Damian, I know he’s shooting season 3, but it seems to me that when he was shooting Homeland, there was more information about him, photos, I would like more information about him, new photos, forgive me, I know I’m demanding, but I miss you, you know how much I love Damian! And you too I love you! Monique

    1. Hi Monique: You are right, Damian is not out and about as much as he used to be, say, last year, in New York. I think the reason is the following: He had the kids last year with him in the city as Helen was shooting Fearless and had to go out with them all the time – to events, museums, etc. And because he spent the entire six months in NY, he also participated in a few events like Gotham Film Awards, International Emmy Awards and so he was more visible – we had a lot of pictures! It seems he is constantly flying back and forth between London and New York this year to spend more time with family. And that is probably why we do not see him out and about much. They are shooting Billions for another month or so and I think we will see him at talk shows as the premiere date gets closer – March 25! Much love to you, we love you, too, very much!

      Update: Sone new cast promo photos from Billions – enjoy: https://www.damian-lewis.com/2018/02/02/23206.php

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