The award-winning HBO series based on probably one of the greatest fantasy sagas in history, A Song of Ice and Fire, surely has its fair share of plot twists and unexpected deaths. Although in the moment of watching your favourite character being butchered moments after he found someone, who seemed to be, the love of his life you think that this is happening for no other reason than to hurt your feelings, a pragmatic (and, to be frank, stone-hearted) viewer might be able to spot a pattern of actions and/or events that led the character to his cruel demise. There is another reason for the existence of logic behind, what, at first, seems to be completely insane, beside the genius of George R. R. Martin, and the reason is game theory. To be more precise, not the game theory per se, but the very goal this remarkable branch of mathematics is trying to achieve.
Most of the encounters between two or more decision-makers, be it a person, a government or a cosmic entity, can be modelled, with appropriate assumptions, as a game (hence, the game theory). It is only natural to apply such tool to a novel full of military decisions, political intrigues and good old interpersonal conflicts. To demonstrate the power of the method, let us go back to one of the very first conflicts that have sparked the chain of events leading to the chaos that unfolded later on in the story: the capture of Lord Eddard Stark.
Many of the game theory models seem too simplistic to depict the real-life situations. I have two counterarguments to this common complaint. Firstly, there are numerous examples of when simple models have proven to be very powerful in predicting, or explaining, how and why a certain event unfolds. Secondly, we are dealing with the world of fiction, and so, quoting probably the most frequently quoted American writer of all times, Mark Twain: “Reality is often stranger than fiction, because fiction has to be believable”. Thus, without further ado, let us introduce the game. The two players of this game are: the Northern houses led by the House Stark and the Southern houses, who are led by the House Lannister after the death of King Robert Baratheon. The Lannisters have Ned in captivity and, naturally, have the following three courses of action: release Ned Stark, send Ned Stark to the Wall and execute Ned Stark, to which the North can reply either by attacking the South or keeping the peace. This game has a relatively low amount of possible actions and, thus, possible scenarios, so we will start analysing it by assigning a numerical value to each of the potential outcomes.
The process of assigning a number to the outcome is governed by what is called utility theory. A utility is the numerical value of a player’s satisfaction (or dissatisfaction) with the result of the game. In some cases the utility can be measured fairly objectively, e.g. in terms of the monetary gain, in other cases, it is not that straight-forward. Our case is the latter. However, even though it is impossible to perfectly assess the utility of the North in case, for example, the peace is kept and Ned is released, we could make a guess whether this scenario is more or less preferable to one of the other possible scenarios. In other words, we rank all the possible outcomes in the order of preference for each of the players. This brings us to the first couple of assumptions we have to make for the model to make sense: the players can perfectly distinguish the preferable outcome out of the two and no two outcomes are exactly equally preferable. We have a choice of three actions for the South and a choice of two actions for the North. Thus, there are six different scenarios that can unfold once both players make their decision. Remember the South calls first and the North replies. Let us list all six possibilities and, firstly, assign the values for the utility of the South for each of them. For the sake of simplicity, we will be simply assigning natural numbers between 1 and 6.
Some justification for this ranking is as follows. It is always a good idea to start by choosing the two extremes. Clearly, the worst-case scenario (1) for the Lannisters would be Ned walking away freely and starting a war. On the other hand, keeping the Iron Throne without the battle and discrediting Lord Stark, who happened to discover all their dirty secrets by forcing him to join the Night’s Watch is the best outcome the “lions” can hope for (6). Whilst one might think the death of Ned Stark together with keeping peace in the realm would be more preferable, but the sudden death of the Warden of the North, who was Lannister’s prisoner would raise an eyebrow or few, and might lead to a chaos in Westeros, thus, we give this scenario a (5).
Another horrible outcome for the “lions” is letting Ned go, but he manages to persuade all other houses (including Lannisters’ bannermen) that his findings are true, which, indeed, leads to the removal of the House Lannister from the seat of power (if not a death penalty) (2). The South has a relatively strong army with a more than capable war commander, Tywin Lannister, and can certainly cause some trouble would the war unfold. If one of the best military minds of the realm, Ned Stark, was sent to the Wall before he could discredit the Lannisters, I doubt that the whole army of the North would follow the Young Wolf into the battle that promises to be brutal. Hence, I rank this scenario at (4). And, finally, all-out-war against the North, even without their Warden, is costly, and the Lannisters do realise that (3).
Now let us look at the utility of the North.
We follow the same algorithm as before. The best-case scenario for the North would be a peaceful removal of the Lannisters from the Iron Throne under the wise leadership of Lord Eddard Stark (6). The least preferable outcome is, of course, keeping the peace after the Lannisters execute Ned (1). Winning the war against the South under with Ned alive and most of the big houses joining the cause would not be a trouble (5). Winning the war while Ned is at the Wall, however, might cause some difficulties, but the army of the North is larger and northmen are known to be fierce warriors, thus a (4). The North, and the Starks, in particular, would be happy to see Ned live even if he has to spend the rest of his life at the Wall. This would seem to be unfair, but not a cause to go to an all-out war (3). On the other hand, the death of Ned Stark is a calamity that significantly decreases the North’s military potential and, thus, a (2) to a war without Lord Eddard Stark.
Now that we have all the possible outcomes ranked out for both players, let us combine them into a representative table.
Each entry (a,b) represents the utility of two players under the certain outcome: a – utility of the South, b – utility of the North. The South moves first and is to decide on the move whilst having a perfect information about all the values (including utilities of the North). This is another assumption that we have to state. Furthermore, we assume that all players are rational and, thus, want to maximise their respective utilities.
Now, a careful reader might have spotted that sending Ned to the Wall (utilities marked with green colour below) leads to a preferable outcome in both scenarios that can happen depending on the North’s response. This is called a strictly dominant strategy in game theory terminology. The two other strategies are called (surprise-surprise) strictly dominated. By the laws of game theory (and of common sense), a rational player should always use the dominant strategy, i.e. the South should send Ned to the Wall. Coincidentally (or not), this is exactly what Tywin Lannister planned to do, and we all know that this man can be called anything, but not irrational.
To sum up, we can say two things: firstly, the first stand-off between the Starks and the Lannisters is representative and shows how a simplistic game theory model can help to analyse a complex political conflict. Secondly, King Joffrey is a moron with no other agenda beside his lust for cruelty (and some daddy issues).
*Disclaimer: This is not an article with yet another hypothesis (although they do get called theories, they are technically hypotheses) of how things will unfold in upcoming books and series of Game of Thrones, but the strategic analysis of the event that has already happened. However, I, personally, do enjoy investigating these, so here are few of my favourite youtube channels making predictions and unravelling easter eggs in both the show and the books: