Game theory: an introduction
If you had an internet connection or a friend during the 2010s, you might have heard of a little TV series called “Game of Thrones”, which was notorious not only for its ability to desensitize viewers to gory deaths and spicy scenes characterized by sometimes dubious levels of consent, but also for the complicated scheming of its characters. Thousands were captivated by the chair-grabbing game the characters were playing and were curious to find out who would win and who would die, as per the general rules delineated by Cersei during Season One. Students of game theory can only dream of sparking such compelling discussions and drawing interesting real-world parallelisms as the HBO series did. But one of these scholars came up with a theory so interesting, he was awarded the Nobel prize for mathematics, so maybe they are worth something after all. Let’s do a deep dive and discover who were the first nerds to spend their afternoons debating which strategy would be the best to win a game, and at what games they were looking at before Game of Thrones came along.
The OG game theorists
Game theory sounds like a very fun thing until you realize it was developed by mathematics, and then it immediately loses its appeal. Still, it is a vibrant field of study, full of twists and turns, with lots of applications in everyday life: it’s the science of strategy.
Game theory wasn’t considered a proper area of study until a man published a paper, and this happened in 1928 with “On the Theory of Games of Strategy” by Von Neumann. In his essay he managed to prove there is a perfectly logical and rational way to solve a two-person, zero-sum game by using a mathematical theorem and in 1944 he published a book with Oskar Morgenstern called “Theory of Games and Economic Behavior”. Before Von Neumann and Morgenstern started bringing math into this, historians, politicians, organized crime lords and military strategists had already started thinking about certain problems in a “game-theory” sort of way. At its core, game theory is interested in finding the best strategy to win a game. The thing was, they didn’t mathematically determine which strategy was the most rational, which carried the best outcomes, and which was the most likely to happen. And this is what game theory can do.
What’s a game?
In game theory, a game is not necessarily an activity you do for fun, just like the Game of Thrones isn’t fun for the people who play it. The outcomes of a game depend on the interaction between two or more players. One of the key assumptions of game theory is that players are rational individuals, and they will adopt the strategy which is more likely to help them reach their desired outcome while trying to anticipate the adversary’s moves. Mathematicians create schemes and use equations to solve games, but it isn’t necessary to know advanced math to understand the basic concept underlying game theory.
Suppose you’re an ambitious noble in the Game of Thrones universe and you, too, want to sit on the most coveted chair in the continent and become King. In doing that, you must keep an eye out for other competitors, figure out what they’re going to do and anticipate their moves. You might even draw a graph, to keep track of the different outcomes and probabilities.
Let’s say one of your adversaries is an upright, noble person. There might be limits to what they’re willing to do, for example killing innocents, lying or bribing. They are offered three choices: getting the throne immediately by killing a pregnant, innocent woman, getting the throne in a couple of months by winning the throne fairly in a military campaign where they risk dying or losing, or getting the throne in a couple of years when the current king will have died, with the possibility that someone else might get there first. Even though the first option is the one that would get them to the throne faster, since they value moral integrity, this option is the one that they’re least likely to choose even if it is the thing that would get them to their goal more efficiently. Now imagine another noble takes all their decisions based on the outcome of the throw a pair of dice. They assign values to the different options so, for example, they decide that if the result of the dice throw is a number between 1–2 they will choose the first option, if it’s a number between 3–6 they will choose the second, and if it’s a number from 6 to 12 the third. If you want to predict which option they will choose, you have to know the odds of each throw and then know to what numbers they associated each possible choice [1].
Strategies: pure and mixed
In the book “Moral Calculations” author László Mérő explains that a pure strategy is one where a player’s actions are dictated by a single principle which will always bring the same outcome in identical situations. Batman uses a pure strategy in his fight against crime because, no matter the circumstance, he never kills. A mixed strategy instead is one where the player assigns different probabilities to each option and then chooses [2]. The noble who makes decision by throwing dice is adopting a mixed strategy: he assigns a set of numbers to each option, and then the throw will determine which one will be chosen. In most games, mixed strategies tend to be more “efficient” than pure strategies, just like in evolution, societies that are more genetically diverse are more likely to survive because they are more likely to adapt to the environment etc.
There aren’t only pure and mixed strategies in this world. In his book, Mero gives the example of a person who says that they will consult the horoscope once a year and then make decisions accordingly: this is neither a pure nor a mixed strategy, and yet many people use these sorts of strategies to take decisions. They are still important, but they’re not studied by game theory. Game theory can be used to understand and analyze only rational decisions, and this is one of its the main problems, which often gets overlooked.
Game theory as Neumann developed it has certain characteristics: the game must have a limited number of options for each move, be a zero-sum game (a player wins exactly what the other loses) and be characterized by complete information. This last condition is extremely important. It implies that every player not only knows the options available to them and their adversaries, but they also know each possible outcome of the game, their value scale and the one of their adversaries. Since it’s a zerosum game and players are assumed to be rational, each part tries to minimize their profit, knowing that the adversary will try to do the same.
If it sounds too good to be true, that’s because it is.
Game theory has been applied to almost everything: biology, psychology, computer science, political science, economics, etc. There are games that explain the strategy of nuclear deterrence during the Cold War, games that explain how the selection mechanism of evolution works, and that explain the raising of prices in the economy. These games though are “theorized” and “constructed” by people to try and explain complex phenomena. Game theory might have started as a mathematical thing, and it still is in some ways, but when we try to explain social interaction or economic phenomena with game theory and especially with some of these games, we often forget something.
These games were made by people, with specific backgrounds, educations and biases. In real life, perfect information simply doesn’t exist, and rationality is much more of a fragile concept than we think it is. You want an example? Let’s look at the most infamous game of game theory: the prisoner’s dilemma.
The prisoner’s dilemma
Steve and Barry get arrested by the police and accused of having committed a serious crime. There’s no evidence to incriminate them for that crime, but the police have enough evidence to charge them with another, minor crime. The police put them in separate cells and tells Steve:
“If you confess and snitch on your partner, we’ll let you go and close an eye to that small crime you committed, while your partner will serve ten years. This offer is valid only if your partner doesn’t say anything either: if Barry confesses too, you’ll both serve five years. If neither of you confesses, you’ll both serve one year because we don’t have proof to incriminate you. Oh and by the way, we just offered the same deal to your friend. Make your choice!”
Scholars say that math and logic show that there is a “best choice”, and that is, for both to confess. Why do they say this? Scholars think that Steve will reason in this way:
“If Barry confesses and I don’t, they get off with no jail at all and I serve ten years. Barry will obviously be tempted by this proposition, so I should probably confess just to avoid serving ten years. In that case we will both serve five years. But what if Barry doesn’t say anything while I confess? I got a get out of jail free card and serve zero years! And if I don’t confess, and for some reason my partner does the same, we will both serve one year. So, since Barry is probably making the same reasoning as me, the best thing to do is to confess, regardless of what he will do. Best case scenario, I get out tomorrow, worst case scenario, we do five years”
This reasoning always profoundly bothered me because that would mean that logic excludes cooperation. To cooperate in this game would mean making an irrational choice. But is there only one kind of rationality in this world?
Let’s get serious about this game
One of the basic assumptions of game theory is that all players are rational individuals, which means that they will try to maximize their gains — sometimes even at the expense of the other players. This means that they will likely follow the same reasoning, because the “rational” way of thinking is only one.
The solution given to the prisoners’ dilemma seems simple enough: Steve has no idea what Barry is going to do, but game theory tells us “what I’m thinking about, he’s probably thinking about too, because we’re rational people”. So, if he follows this logic, Steve realizes the most advantageous thing for him is to confess and he does so, hoping that Barry will not. Because if Barry doesn’t confess, Steve might not go to jail at all, but if Barry snitches on him while he decides to be loyal, he will spend more time in jail. The players construct a sort of list of pros and cons, and assigns a specific amount of points to each choice. Let’s consider the game more realistically.
Option one: many people see the prisoner’s dilemma as a non-cooperative game, but what we saw could still be considered a cooperation game. Non-cooperation among the players (Steve and Barry) translates to cooperation with the authority that is “setting up” the game (the police). If our two test subjects see the game in this way, the choice won’t be between “non-cooperation” and “cooperation” but between “cooperating with the partner in crime” vs “cooperating with the police”. If Steve and Barry have, for some reason, an aversion to the police and the judiciary system, they might be against the idea of cooperating with the police and they will keep those personal feeling under consideration when making their choice and their list of preferences, and vice versa. In this example, the most rational thing to do is influenced by the player’s feelings toward the police/their partner.
Option two: another problem with game theory (and this game specifically) is that it asks you to take a decision based on what the police tell you, but players don’t need to base themselves only on that information. They might add other things to that list of pros and cons, information they have that the police don’t. Let’s imagine that Steve and Barry are in a gang, or in the mafia. In this case, snitching on their partner (non-cooperation) would doom them once they get out of jail — possibly in jail too, if other members of the gang/criminal organization are there. Non-cooperation could at worst cost them their lives, at best a couple of fingers, leave them in a worse economic condition and with a destroyed reputation — which is fundamental in social environments characterized by high levels of organized crime. If Steve and Barry decide to see the problem in this way, the scenario changes: if they don’t cooperate with each other their personal safety is at risk, whereas if they cooperate they might get some jail, but they keep their reputations and their lives. If Steve cooperates (refuses to confess) while Barry betrays him (confesses), Steve will go to jail but his reputation and safety aren’t at risk while the same can’t be said for Barry. In this example, saying that the most rational thing to do is not cooperate sounds foolish, even though before it seemed perfectly logical.
Option three: confessing (non-cooperation) might look like a very rational idea now, from the interrogation room. But just as no one liked the kid who told the teacher that the class didn’t do their homework, nobody likes people who confess just to save themselves at the expense of their partners. This might be especially true in jail. If Steve and Barry confess and betray each other, choosing the “most rational option”, this could make their lives in jail very, very dangerous because they would gain a bad reputation. If they choose to confess they will get less jail time than they would have if only one of them had confessed, but if they spend those five years being brutalized by other inmates they might realize too late that they have actually chosen the least “rational” option. There was a hidden cost that they didn’t know about. In this example, confessing seems rational until the players learn more about the implications of their choice. This is often the case. Using asbestos to build seemed like a great idea … until it didn’t, because people were getting cancer. Burning fossil fuels seemed like a great idea in the 1800s until we learned that it wasn’t. Throwing plastic wrappers out of space shuttles or leaving dead satellites in orbit seemed harmless, until we learned that it wasn’t.
Option four: Steve and Barry refuse to play the game. They listen to the speech made by the policeman and then focus on the small, but very important part of their speech saying that the police doesn’t have enough evidence to make them serve more than a year. There is a “best-case scenario” in which Steve and Barry get away with their crime by serving one year each, but since scholars believe rationality means pursuing one’s best interest while ignoring the others, they believe that though this option is available, it gets overlooked in the pursuit of maximizing individual gains — and that’s a tragedy. But rationality doesn’t necessarily mean that, and in any case, the issue lays in the players’ perception of the game. The prisoner’s dilemma is a way of framing the situation, a trick used by the police to raise their chances of getting criminals to confess. If they had never asked the question at all, the criminals would have served one year each. And they still can, they just need to refuse to see the situation in the way the police are presenting it to them — they can refuse to play the game. In the prisoner’s dilemma, we think that the players are only the criminals, but the person framing the issue, in this case the police, is playing too.
A big problem with strategy and game theory rests on the assumption that the adversary sees the situation exactly as we do. If players perceive a problem as a “prisoner’s dilemma scenario”, then they will perceive the situation as a non-cooperation game where the most rational thing to do is pursue their best interest while completely ignoring the other’s. But what if player number two perceived the situation as a cooperation game instead, or worse, what if player number two wasn’t playing at all?
The prisoners’ dilemma: multiplayer edition
If you are familiar with discourses surrounding climate change and the tendency to exploit common goods, you have heard of the “common pastures” problem, which usually leads to “tragedy of the commons” theme [3]. To see the result of this game you just need to look around you in the real world, but we’ll summarize it here.
We have a village with a common pasture where people can leave their cows to graze. There are ten farmers who each own one cow, and they all let their cows graze in the same pasture. Soon, one or two of these farmers can afford to buy another cow, and then more farmers start to be able to buy more cows. At that point, all cows start becoming thin and go hungry because there isn’t enough grass for everyone. By the time the tenth farmer buys a cow, all the others have starved. According to game theory, all farmers behave in a rational way, because it is more advantageous for them economically speaking. Scholars define this as the “tragedy of the commons”.
But maybe, like in the case of our friends Steve and Barry, the “best-case scenario” could have been easily attained if only the farmers had seen the issue in a different way. Clearly, if all farmers follow their interests selfishly while ignoring the reality of the situation (the resources are limited) everyone will end up dying and starving, just like our criminals will both end up in jail. But buying another cow looks like the most rational thing only if you grow up and are socialized in a capitalist and highly individualistic society: to most indigenous people instead, this looks like a very foolish thing to do.
To some societies, even the main assumption of this game looks irrational. Why buy another cow in the first place, when one is enough to sustain you? We tend to forget that overproduction is one of the features of sedentary societies and capitalist economies, but these rules don’t apply to nomadic tribes or other economic models.
Which reasoning is the best?
One of the main problems of game theory in general and of the prisoner’s dilemma in particular, is that the people who theorized it assume that players, being “rational” individuals, will make the same type of reasoning and therefore will reach the same conclusion. What they didn’t consider is that the two players can be completely sane people possessing full control over their mental capacities and still, make a different type of reasoning that will lead them to two different choices. They didn’t consider that just as there are different people in the world, there are different kinds of rationalities.
The best outcome on each game, whether it be the prisoner’s dilemma, the chicken game, the dove and the falcon or the common pastures problem, depends on the player’s priorities, values and on other factors that are too numerous to fit into a graph, such as socioeconomic class, occupation, education, religion, upbringing and so on. It may also depend on the consequences that their choice will have for the community at large. Choosing to maximize one’s interest at the detriment of others and, potentially, of the community, isn’t the only rational choice to make, but one of the many. The best outcome for any player depends on their type of rationality.
For this reason, it is imperative for us to shift our perspective. When looking at game theory, especially at non-cooperative games such as the prisoner’s dilemma and the common pasture problem, we need to ask ourselves. Are we doomed to non-cooperation, because it is an inherently “irrational” thing since it goes against our immediate interests, or have we been looking at the problem wrong this whole time?
[1] A similar example can be found in László Mérő, “Moral Calculations”
[2] László Mérő, “Moral Calculations”
[3] “The tragedy of the commons”, Garret Hardin, 1968
