Game Theory demystified
Shrouded in the shadows of forbidding mathematics oozing the menace of nuclear strategy, Game Theory can seem a potent yet mysterious field of academic research. It’s actually an extraordinarily simple idea and it’s important to understand what it’s useful and not useful for.
Not too many fields of academic research can boast a blockbuster Hollywood feature film about their origins. Game Theory can. A Beautiful Mind, starring Russell Crowe as the schizophrenic genius mathematician John Nash who won the “Nobel prize” in economics (it’s not, strictly speaking, a real one).
Born of the Cold War, used to make sense of the possibility that the world might be annihilated in a global thermonuclear war, to inform military strategy especially concerning the use of nuclear weapons in the shadowy halls of the Deep State, to illustrate the workings of genetic evolution and the emergence of cooperation in a violent and hostile world, there has always been a certain mystique about Game Theory. It has that sort of dark, cynical brutality about it which sends a shiver down your spine of something between fear and excitement. Complex, hieroglyphic, forbidding yet sparse and austere mathematics we’re told explains and governs our lives.
A great deal of the mystique of Game Theory is due to the manner in which it is taught. The novice quickly becomes lost in the logically circular mathematical entrails of this or that particular Game with fancy-sounding names: prisoner’s dilemma, battle of the sexes, zero-sum, repeated game, infinite horizon, finite sequential. This can make the Theory seem much more mind-boggling than it actually is. I want here to do something a little different, and rather than provide example after example introduce you to the structure of Game Theory as a theory of behaviour.
Origins: “Rational” Choice Theory
Game Theory originates in economics, specifically neoclassical economics and its theory of behaviour. “Rational” Choice Theory proposes that out of a set of various courses of action available to them, the individual selects that one which they consider most preferable. “Rational”, because a number of philosophers wouldn’t agree that such behaviour is rational. A more technical, but equivalent manner of stating “Rational” Choice Theory is to say that individuals choose out of their feasible set so as to maximise their “utility”, the “payoff” they get from a certain course of action.
The source of the particular dissatisfaction with this theory which gave rise to Game Theory is the impression it gives of individuals making decisions in complete isolation from and without reference to one another. They consider only the preferences they themselves have between the various available courses of action.
Nascence: von Neumann and Morgenstern
A genius Hungarian mathematician and a German economist associated with the famed wiener kreis formulated an extension of this theory which became known as the theory of strategic choice, or the Theory of Games. John von Neumann and Oskar Morgenstern painstakingly extended the concept of the “utility function” in their monumental work of 1944, Theory of Games and Economic Behaviour, so as to allow payoff on others’ decisions as well as one’s own. For instance, if we’re both bidding for a house, whether I get it depends on the bid you submit as well as the bid I submit.
When individuals maximised their utility, chose the most preferable available course of action, now their choice depended not only on the payoff, because the payoff wasn’t a given. Their decision depended on what others individuals did as well, which is reminiscent of the activity of strategizing in a game, of a certain form of play — hence “Game” Theory.
Behaviour in this theory became described by a “best response function”. My decision would be a best response to your decision in that I maximised my utility given that it is effected by your decision. This introduced a circularity into the “Rational” Choice theory of behaviour which is necessary, but bedevils the mathematics of it to this day: what I do depends on what you do which depends on what I do.
The problem with this was that economists (whose theory it still almost exclusively was) aren’t comfortable with a mathematical model of behaviour which can’t be “solved” to discover an equilibrium point, and von Neumann and Morgenstern didn’t really propose a concept which would permit this in every Game. So for a little while at least, Game Theory remained confined to their book, and little more than that.
Development: Nash
John Nash won the Nobel prize for, most notably, a two page letter in 1950 to the Proceedings of the National Academy of Sciences which came from his PhD thesis at Princeton University. He proposed a concept of equilibrium which came to be known as “Nash equilibrium” and, more importantly, he proved that one would exist in every single game where the set of alternative courses of action and the utility function had certain mathematical properties.
An equilibrium is a state from which there is no tendency to deviate. Now why would you deviate if you were maximising your utility and playing your best response? So, Nash proposed that in an equilibrium would exist in a Game if and only if all individuals, with no exceptions, were playing their best responses simultaneously. That is, everyone is playing their best response to everyone else’s best response to everyone else’s best response. Nash then proved that such a set of behaviours would exist in every game.
This sparked a revolution. For Games are quite complex mathematics because of the circularity inherent in the concept of best responses, and Nash had proved the existence of equilibrium in but one basic kind of game. There were boundless extensions to the types of games in which equilibria might be proved to exist and refinements to be made to the concept thereof which might be published. That activity has largely occupied the academic disciplines of Microeconomic Theory and Game Theory in its own right for the past seventy years, generating dozens of such concepts. Subgame Perfect, Bayes-Nash, Perfect Bayesian, Sequential Equilibria are but a few.
Game Theorists tend to get a bit slippery about the idea of equilibrium. The whole point of Game Theory is to “solve” Games to discover their equilibria. And why would we so solve them? Well to understand what’s going to happen in the situations the Games model of course.
But it’s one thing to say that an equilibrium exists, it’s another thing entirely to say that behaviours will converge toward it. Equilibria aren’t the same thing as “attractor points”. It requires a whole host of additional assumptions which aren’t particularly convincing to say that behaviours will converge toward it, and it’s equally plausible that the Game will spiral around in disequilibrium.
So Game Theorists tend to be coy about what will happen in the situation described by a game and if pressed say merely that if we happen to find ourselves in an equilibrium of a Game, we won’t depart from it. In reality, there’s no reason to suppose we will find ourselves in an equilibrium, and every reason to suppose that we will spiral around forever adjusting our behaviour to each other.
Uses and misuses
So that’s what Game Theory is: a theory of strategic interaction where individuals maximise their payoff when deciding what to do given what others are going to do. When they’re not proposing new solution concepts and proving their relevance Game Theorists try to “solve” these games for their equilibria and sort of (but not really) predict what’s going to happen in the situations they model.
What’s it useful for? Well ironically, given the heavy-duty complexity of the mathematics which goes into it, it’s most useful for relatively simple “back of the envelope” type thought experiments using the simplest concept within it — Nash equilibrium. This makes sense actually, the difficulty of the more complex concepts alone, let alone the application of their circular logic inhibits their extensive use in any sort of applied setting.
Let’s have a quick look at some examples of how Game Theory can be used in the concept of Mutually Assured Destruction and the work of John Maynard Smith, Robert Axelrod and Elinor Ostrom.
First take Mutually Assured Destruction. This is a solution to the famous Prisoners Dilemma Game which made Game Theory famous for its application during the Cold War. It’s also the plot of my favourite movie of all time: Dr Strangelove.
In this Game NATO and the USSR have two options: strike with nuclear weapons, or don’t. If NATO launches a “decapitating” strike, and utterly destroys the ability of the Soviet government to strike back, it eliminates an enemy with little cost to itself (Game Theory doesn’t very much allow for conscience to exist). Great, not striking isn’t an equilibrium, because NATO has an incentive to deviate and strike with nuclear weapons.
But here’s the problem. The USSR has exactly the same incentive. If it strikes first and utterly destroys NATO’s command structure it eliminates an enemy with little cost to itself. There’s an incentive for both NATO and the USSR to strike first, and so the equilibrium is that we end up in an apocalyptic irradiated world after thermonuclear war.
The way to get around this suggested by the Game Theorists is not to calm down, sit down and talk it out, but to ensure the ability to strike back with massive retaliation — to change the payoffs and make peace the equilibrium. This had real effects. It’s the reason the following mind-boggling things exist: the US “Minuteman Missileers”, the mysterious Soviet “Dead Hand” system, and the “letters of last resort” handwritten by Theresa May locked in a safe within a safe in a Royal Navy submarine somewhere (not even the MoD nor most of the crew know exactly where) on the bottom of the Atlantic Ocean.
Less frighteningly, John Maynard Smith, David Axelrod and Elinor Ostrom each made use of the Prisoner’s Dilemma Game to explain the conditions under which cooperation emerges from otherwise violent and hostile systems.
John Maynard Smith showed this in the context of biological systems where altruism involves inviting the risk of getting yourself killed, but in some cases in a way which sufficiently perpetuates the survival of the genes you carry for it to be. It is then “better” for the gene over the long run if you cooperate with others even if every now and then it leads to your death.
Robert Axelrod and Elinor Ostrom applied this to human cooperation and solving what’s known as the “free rider” and “tragedy of the commons” problems. Cooperation can be achieved once a group learns or figures out that it’s more beneficial over the long run to cooperate than be stuck in a stable but poor equilibrium. In each case, cooperation might not be a stable equilibrium, but if a long enough view is taken it might still be one. Thus cooperation emerges even when people are acting to maximise their own utility.
That’s how Game Theory is useful: as a “back of the envelope” type device which gives you a good way of running a thought experiment about this, that or the other environment where agents of some kind are having to make decisions when they’re all effecting each other’s payoffs.
Game Theory is misused if it’s thought to be a theory of systems as they actually exist or in any sense a general theory of human interaction. It’s too outlandish to be used except as an analogy. The calculations and logic required to determine one’s best response to everyone else’s best responses surpasses even the most brilliant Game Theorists sometimes. And there’s not really any interaction in the ordinary sense of the word within it — it’s a theory of how I would act when making decisions knowing that the payoff I get from those decisions will be effected by how you act. It’s not a theory of how we interact.
So Game Theory, a theory of strategic interaction, is not actually a theory of interaction in the normal sense of the word — communicating with each other. In fact, it’s most useful when it’s applied to situations in which interaction is not occurring except in a specialised sense of my actions being determined in a way which interacts with the way your actions are being determined.
So there we have it. Game Theory is a circular logic about what I will do given what you will do given what I will do… Once the mystique of high mathematics is stripped away and the nature of Game Theory is made clear, we find it to actually be a very simple idea which is a useful analogy for illustrating a few phenomenon and processes we see in the world.
