The Nash Watch and an Introduction to Game Theory

For those of you who don’t know, I’ve been watching HBO’s Game of Thrones lately. I watched up to Season 4 Episode 2 when it released and have now re-watched (as of writing) into season 5. Personally, I don’t mind spoilers. But I know many of you do. So I will let you know which links have spoilers and which do not.

Game Theory is basically the math behind decision-making. David Levine of UCLA wrote this:

What economists call game theory psychologists call the theory of social situations, which is an accurate description of what game theory is about.

You can follow that link for a good primer on the Prisoner’s Dilemma which I’ll explain later. The link above is free of spoilers.

Side note: I’ll be pointing out examples from Game of Thrones but this isn’t specifically a Game of Thrones article. I just wanted to go for the easy pun (inspired here with some spoilers) and simple examples.


A Primer on Games200

Lets start with something simple. In this first example, inspired by Ben Sweeney here (very minor spoilers), we use Cersei Lannister’s famous quote (look at the gif).

In this game, we equate death with not being the sole king. Being the sole king means no pretenders, no rebellion, nothing like that. So the only good option is for the player to be the king and the other player to be dead. While this isn’t necessarily a decision-based game (who would choose death?), this is a simple example.

 King Death
 King  0,0 1,0
 Death 0,1  0,0

The first thing to learn about game theory is decisions and situations are broken down into what are called “games” (hence the name). These are most often represented with tables like the one above. We assume there are two players because there are two payoffs associated with each outcome.

8mwqjWait, what?

The two players, which I will refer to as Top and Left (they have progressive parents), are trying to maximize their payoff or reward (or utility, or money, or whatever else they value). The player on the left gets the left payoff, while the player on the top gets the right payoff. You want to be King and you want the other player to be dead. That is the desired outcome for both players.

So if Top is King and Left is dead, we end up in the second column on the third row, with a result of 0,1. Top gets 1, Left gets 0. For the sake of simplicity, higher numbers are better. That’s it. We’re not really going to worry about 10 being twice as “good” as 5. 10 is just bigger than 5.

So to recap: Game Theory follows games (and there are many, many variations). Each player can receive a payoff (usually referred to as utility but this can be anything). There are usually decisions to be made that lead to outcomes. And the worst part, we assume every player is rational and logical. I mention that because this thread on Reddit gives (with season 1 spoilers) a great overview of game theory within Game of Thrones except for some logical fallacies around this assumed rationality.


The Prisoner’s Dilemma

This first example is the foundation of game theory. I could not find an example relating to Game of Thrones but every course on game theory starts here.

If you have seen the movie A Beautiful Mind, think back to the bar scene. If you have not seen the movie, see it. Below is the bar scene in question. While the explanation is iffy, it’s close enough for our purposes here.

Anyway, on to the game.

silence turn
silence 5,5 -4,10
turn 10,-4 -1,-1

If the top player choose to turn and the left player chooses to remain silent, we arrive at the third column in the middle row, which reads -4,10. If both players choose silence, each player gets a reward of 5 something.

This game has what we call a Nash Equilibrium (spoiler free).

A pure-strategy Nash equilibrium is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile.

What this basically means is that a Nash equilibrium is the outcome in which both players cannot obtain a higher payoff by selecting a different action, as long as all the other players are also trying to maximize their reward. There’s more math to that than I’l explain but lets leave it there.

The idea behind this game is that if both players stay silent, they both get away with a payoff of 5. What if they want to improve their situation? They will defect for a shot at the payoff of 10. However, if one player defects, the other player will also defect to improve their payoff from -4 to -1. So while (-1,-1) is worse than (5,5), the natural progression of rational actors is to defect.

Want to see The Wire play this scenario out? Watch this (NSFW for language).


Stag Hunt

This link, with spoilers to the end of season 4, offers some more insight into The Game Theory of Thrones. This includes the Red Wedding and a battle, where Becky Ferreira describes both plot points as stag hunts. Without spoilers, I’ll explain what a stag hunt is.

A stag hunt is basically a reverse prisoner’s dilemma. Instead of dissenting, a higher payoff can be obtained by working together.

 stag  hare
 stag  5,5 0,1
 hare 1,0  1,1

Similar to the Prisoner’s Dilemma, the players in this game are not communicating and must individually reach a decision. Lucky for you, I won’t be taking into account things like likelihood to make a certain decision or cost (such as time) of a decision.

In this game, hunting stag requires two players. Hares only necessitate one player. So if both players hunt hares, they each get 1 utility. If one player hunts stag, he’ll go home empty-handed. If both players hunt stag, they’ll each go home with a higher payoff of 5 utility.

So the best thing for each player to do is hunt stag. Because they are not communicating, but they expect the other to act rationally, the outcome will most often be (stag,stag) with a payoff of (5,5).


the-wall-elevator-game-of-thrones-lord-snow-01The Tragedy of the Commons

This is less game theory and more a small discussion on rational decision-making. In the case of Game of Thrones, I’ll be discussing The Wall. To the left, you see a giant wall built to keep the White Walkers out. The White Walkers are bad. The Wall is so tall, they have an elevator to take the members of the Night’s Watch to the top, which you see in the image. The idea for this comes from this Reddit post.

This explanation is all hypothetical (at least to my knowledge). So it should be spoiler-free.

Joining the Night’s Watch is often a form of punishment offered as a choice, with the other choices being losing a limb or even your life. You can also choose to join of your own free will. The Night’s Watch protects the entire realm from the White Walkers; that is to say that everybody receives a reward from The Wall. (There are no mandates for any of the houses to send men to The Wall, but lets assume the houses face a choice between that or keeping them nearby in the following game.)

So if everybody receives an reward, regardless of who is manning The Wall, why would anybody contribute to it?

While this example isn’t perfect, I think it offers a pretty close explanation. In the game below, I introduce a third payoff: The Wall. However, there will be no decisions for this. Their rewards are based only on the decisions of House Top and House Left. So the payouts now follow this convention: (House Left, House Top, The Wall).

 Send men Keep men
 Send men  50,50,100 50,100,50
 Keep men 100,50,50 -100,-100,-100

So if nobody sends men to The Wall, the White Walkers will take The Wall and move south against the realm. If House Left sends men to the wall, they still get a reward but not as much as House Top gets.

The Tragedy of the Commons has been used to describe many things, including pollution and the extinction of the dodo bird. What it basically means is that if a shared, finite resource is available and there is no incentive to regulate individual use, the resource will quickly run out. Everybody benefits by being selfish. Greed is rewarded.

Another example is whaling. If two corporations are whaling in the same area, then whoever has the most whales at the end of the day will make more money. So they will compete with each other until all the whales are gone. Or they will agree to and abide by regulations, and then you deal with the potential for defection.


Multi-Stage Games

All of the above are single-stage games. This is what you would cover in the first class of a basic Game Theory course. There are also multi-stage games, and with these bring strategies. There are many variations of multi-stage strategies. I’ll mention two here because they personally fascinated me when I was studying game theory. The definitions come from this website, which I recommend for more information.

The first strategy is tit-for-tat. It’s a trigger strategy where a player retaliates and punishes another player for as long as the other player deviates.

A type of trigger strategy usually applied to the repeated Prisoner’s Dilemma in which a player responds in one period with the same action her opponent used in the last period.

This is a strategy wherein players copy each other. If, in a repeated prisoner’s dilemma, our star Left is playing a tit-for-tat strategy and Top decides to defect, then the next turn Left will defect. However, if Top decides to cooperate again, Left will follow suit and also choose to cooperate.

The second one is the Grim Trigger.

A trigger strategy usually applied to repeated prisoner’s dilemmas in which a player begins by cooperating in the first period, and continues to cooperate until a single defection by her opponent, following which, the player defects forever. Grim trigger is a severe trigger strategy since a single defection brings about an eternal end to cooperation, in contrast to the much more forgiving tit for tat.

An example of this is war and the concept of mutually assured destruction. Two nations, both armed with nuclear weapons, agree to not use those weapons. They’ll throw everything they can at the other except nuclear weapons. However, if one country launches their weapons, even accidentally, the other may implement their Grim Trigger response and launch all of their weapons. No more cooperation.

Want to see these in action? Play with this tool. That link sends you to a page where you can play in a repeated prisoner’s dilemma against 5 different personalities (strategies). There are other variations offered, as well.

If you want further reading that covers almost everything I mention here, follow this PDF download.

If you want access to free lectures on this subject, start on the Open Yale Courses page. I followed this in high school and it gave me a very good foundation.

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