A Game Theory of Thrones

Written for the Foyles blog.

(Note: I only talk about the premise of Game of Thrones. No spoilers here.)

I love the 'Brief Introduction To' series. Unsurprisingly it does exactly what it says on the tin. They don't take long to read, and you inevitably feel a lot smarter at the end. The last one I read was A Brief Introduction to Game Theory, which led onto some wider reading, and I decided to test what I learned in the geekiest way I could think of.

One of the most basic elements of game theory is that the result of a game is the pay-off. There are too many types of pay-off, and too many that are specific to individuals and individual circumstances to be listing them here. But the list of pay-offs are usually bracketed by the two extreme solutions, becoming King (literally or metaphorically) or becoming dead. This is called a zero-sum game. Chess is a zero-sum game. A zero-sum game is essentially a game that has a clear winner and a clear loser. It's called a zero-sum game because the value of loss and gain together is always zero. (In a chess example, one side wins (+1) and one side loses (-1), which added together gives you zero.)

In the case of Game of Thrones, becoming King usually entails being murdered. As Cersei Lannister proclaims: "When you play the game of thrones, you win or you die. There is no middle ground." Game theorists love diagrams, so here's a brief diagram:

The column represents one player, the row represents the other. It is clear from this diagram, the odds of winning don't seem to be great. As one of my friends so eloquently put it: "It takes a particular type of mentality to want to run a country, because you always lose. Even if you win, you lose." This is an open game, where the players have individual drives, the rules are not fixed, and the pay-off is not the same for all participants. The possible outcomes are not as simple as 1 and 0. For this we need to be able to name the pay-offs and quanitfy them.

So, it's clear from the offset that this is a game to the death. The two characters you may well presume are important in the prologue are dead and gone by the end of their warm-up chapter. Then in the first chapter Mr. Martin introduces us to the family who, for want of a better and pragmatically short description, are good: the father figure explains to his son the importance having to behead your enemies yourself in the name of honour and responsibility.

You may be asking yourself how a family that teaches the necessity for beheading can be 'good'? If you've read the book, maybe you're also very conscious of the fact that very little this family does is morally good. But they are okay, honestly... but only relatively. Locked in a pitch black and deep dungeon, the smallest glimmer of light seems bright. This book tries very hard to show us that good and bad are fantasies and redemption can only be seen in motive and success. Every character is playing their own game.Very Hobbesian so far.

There's Ned Stark, a man motivated by family values and powered by honour, and his wife, Catelyn Stark, a woman with disparate motivation and far too much drive. Both of them believe in reciprocation as the best method to get through life. Their idea of societal reciprocation is very feudal, so therefore fairly unethical, but the story is primarily one of the upper classes, so let's gloss over this point.

Let's take a moment to look at what the basic games are. There are three basic types of game: the zero-sum, the positive-sum, and the negative-sum, which respectively result in one winner and one loser, both sides winning, and both sides losing. There is a difference between the type of game being played, and the result of the game being played. You can play Monopoly with the intentions of winning, but end up finishing the game with no clear winner.

Ned tries to find the middle ground for every problem (positive-sum), while Catelyn spends a lot of time trying to right wrongs in the 'eye for an eye' fashion (negative-sum). However, you cannot simply judge someone on the game they play, that would be the same as judging Nigella Lawson on her cooking methods and not what came out the oven in the end. We have to look at the fact that between them they strike the metaphorical match that burns their world out of recognition, and that they bring about game-changing strategies in their initial refusal to play the titular game.

The problem with these two is that they are playing for emotional gain, while almost every other character is geared towards material gain. You can't play a game of chess while trying to make the other side agree to stick to their side of the board, can you? The aim of the game has been forced upon you. Which points to the most important characteristics of the rules in their world: honour and shame. And shows us why the Starks are the best in human life that their world has to offer, since their stakes are virtuously honourable and shameless.

The other central family, the Lannisters, on the other hand have such a huge, disjointed family with so many immoral twists and turns between them that their fight for honour is merely an indignant shield from a world of judgement. Their stakes in this game of thrones are actually higher because they have no common virtues with which to bind them and only supports of conspiracy holding them up. Triumphalism and resentment powered by shadenfreude. For the Lannisters, there are no self-imposed rules, only a direction towards success. They embody almost every relationship that characterises a zero-sum society: resentment, mistrust, envy, humiliation, shadenfreude, and lies.

Because the Lannisters are not a close-knit group who play the game in and amongst themselves, as much as against the other families, they are essentially playing a negative-sum game in the eyes of the Starks. They will never win as a family, only as individuals.

But how do you give value to their actions? While this is clearly a zero-sum game (in that there is only one crown, so only one winner), not everyone involved is playing with that as their desired pay-off. Lord and Lady Stark just want to settle down to the fire-side life in their winter castle.

There are two ways of measuring the pay-off (if you're interested in the lingo, the number given to pay-off is 'utility', and the two ways of measuring utility are 'ordinal' and 'cardinal'). The first is for games where the outcomes are ranked, rather than measured as in - the voting system of England is measured by first past the post. The second is for games where the ratio between results is important. For example, the alternate vote means that the proportion of representatives in parliament directly reflects the ratio of results.

Ed and Catelyn Stark don't want to play the zero-sum game; they have family values, so losing isn't an option. The best they can manage is the only option. Because everyone else is playing a zero-sum game we require the second way of measuring pay-off. The way this is usually pictured is with each players move being a lottery. The chances of success in each move are measured as a percentage, with the sum of all choices equal to 100%. This following is going to be a very basic way of looking at it, pretending that this is a game of one move and skipping past the equations and algorithms that economists use to figure this type of thing out:

Let's say the point of being a King is to gain possession of a pie, and possession of pie is equivalent to the amount of satisfaction in the games conclusion. If this is too silly and you need to visualize it as having a real value, imagine it as a huge golden pie. As King you can't hoard the pie, you need to reward others with pieces of pie. But to remain King you need to hold onto a majority of pie. So the King wins at least half.

Let's say the whole pie is split into 10 pieces. The King automatically gets 5 pieces. Lord and Lady Stark will require a King who gives them enough pie to chase their family dreams, and so their proportion of pie is dependent on how much everyone else receives. If a Lannister becomes King the remaining half pie will be consumed in internal family squabbles, along with a will and purpose to destroying their ideological enemies, the Stark family. Failing that, the Lannisters require a King who would give them at least 3 pieces of pie, therefore more satisfaction than the Starks. The Starks require a non-Lannister King who would permit them at least 2.5 pieces of pie, so that everyone is equally satisfied by the existence of this King.

Trying to picture a non-zero-sum game can be a bit of a headache, which is why people so often wish political descisions could be solved in a boxing ring, and wars fought by leaders on a paintball site. Unfortunately this is the sort of idealised vision of the world that relies on the idea that everyone is playing co-operatively, and one thing that Game of Thrones underlines so well, is that the game played for power is not fair and most certainly non-co-operative. Of course, a book about feudal lords and ladies arm-wrestling over territories might not be such entertaining reading.

An honour/shame society is by nature a zero-sum society, reducing, as it does, the pay-off to win (honour/1) and lose (shame/0), suggesting that while the proverb, "War is the sport of kings," may be true, for everyone else it's a matter of saving face.