Game Theory

Chess boardAs briefly touched on in August, EC2066 is part of my studies this year. Microeconomics is the branch of Economics studying how individuals take consumption and production decisions and how their interaction affects the economy, considering we live in a world of limited resources.

Despite such a (simplistic) general introduction, I’d like to tell you about a particular topic – embedded within the realm of Micro and Mathematical Economics – which I believe influences (whether explicitly or implicitly) our daily lives: Game Theory (from now on GT).

You’ll probably agree with me the name of the subject is appealing in itself. Generally, playing a game translates into lots of fun and if you think GT is something along these lines… well… you got it right! Not sure? Listen to how Jeffrey M. Perloff introduces GT in his book Microeconomics: Theory and Applications with Calculus:

A camper awakens to the growl of a hungry bear and sees his friend putting on a pair of running shoes. “You can’t outrun a bear”, scoffs the camper. His friend coolly replies, “I don’t have to. I only have to outrun you!”

I bet you’ll appreciate how strategic competition lies at the very heart of the discipline.

John von Neumann

John von Neumann

The birth of GT as a unique field and formal discipline dates back to 1944, when the genius John von Neumann and the economist Oskar Morgenstern had their work published as Theory of Games and Economic Behaviour, based on von Neumann’s previous research (1928). Since then, many scholars – including last year Nobel laureate Jean Tirole – either studied or developed the theory further. The major theory’s refinement came in 1950, when John Forbes Nash, Jr conceived a stronger solution concept: the Nash equilibrium.

Now, what actually is GT? Professor Bernhard von Stengel (LSE – Department of Mathematics) provides a succinct description:

Game theory is the formal study of conflict and cooperation […] where ‘players’ interact, so that it matters to each player what the other players do. [It] provides mathematical tools to model, structure, and analyse such interactive scenarios. The players may be, for example, competing firms, political voters, mating animals, or buyers and sellers on the internet. The language and concepts of game theory are widely used in economics, political science, biology, and computer science, to name just a few disciplines. [It] helps to understand effects of interaction that seem puzzling at first. For example, the famous ‘prisoners’ dilemma’ […]. [It] treats players equally and recommends to each player how to play well, given what the other players do. This mind-set is useful in strategic questions of management, because ‘you put yourself in your opponent’s shoes’.

By the way, Professor von Stengel’s course MT3040 is available as an elective, should you be willing to acquire some foundations in GT during your undergraduate studies.
Take a look at his brief interview to see how the subject relates to current events at a global level!

It’s quite complicated to describe GT comprehensively in a book, let alone in one post! Hence, I’ll present (using words) a very simple and common game, hoping it’ll tickle your fancy and get you explore the subject on your own. In addition, I’ll take this opportunity to introduce some keywords used in GT. Prisoners' DilemmaMy aim is to show that even such an easy game can be used to derive some meaningful conclusions relative to strategic interaction and competitive behaviour. Thus, here’s the Prisoners’ Dilemma (see picture: payoff matrix).

Two thieves (players) are caught by the police and interrogated separately. If neither of them confesses, they both get 10 months in jail. If only one confesses, he gets 5 months and his fellow thief gets 25 months. If both confess, they get 20 months each. In situations like this, we say both players have a dominant strategy. Indeed, no matter which action the other player takes, it’s always better for them to confess. Not convinced? Let’s check this out.

If thief 1 confesses, thief 2 should confess (20 months is better than 25). If thief 1 keeps quiet, again, thief 2 should confess (5<10). By symmetry, thief 1’s best response to thief 2’s actions is always to confess. Thus, the outcome of this static game – i.e. played only once and simultaneously – is given by the pair of actions (Confess, Confess), where both get 20 months in jail.

John Forbes Nash, Jr

John Forbes Nash, Jr

At this point, you may ask “Why don’t they choose to cooperate and get 10 months each, which is certainly a better option?” Suppose they could talk to each other and agree on cooperation (i.e. Keep quiet) before the interrogation. Once in their respective rooms, waiting for the police officer, they’ll both think “I know my fellow thief will keep quiet, I’m better off confessing” or “My partner knows I’m going to keep quiet, what if he decides to squeal and cheat on me?” Either way, they cannot really trust each other. Therefore, again, they’ll end up confessing. We have a Nash equilibrium.

You might also object “What if they ‘work’ together and get caught again in the future? Will this convince them to cooperate?” This situation is slightly more interesting and it’s called dynamic game. To be precise, repeated game. As the name suggests, this kind of game involves the repetition of the stage game described above.

If their collaboration lasted for an indefinite period of time (i.e. infinitely/indefinitely repeated game), it would be profitable for them to collude*. Most importantly, cooperation would be sustainable; in other words, the promise to collaborate (‘Keep quiet’) would be credible, because it’s in both players’ interest to do so. In fact, deviating from such behaviour (‘Confess’ instead of ‘Keep quiet’) would trigger a punishment by the cheated player. The punishment is implemented by confessing and eventually harming both players (20>10), for each subsequent period after deviation. This is what we call the grim-trigger strategy – since defection from the implicit agreement triggers a grim prospect indeed!

On this basis, it pays to keep quiet, as long as no cheating occurred in the past. Therefore, unlike the static case, it’s now possible to sustain the pair of strategies (Keep quiet, Keep quiet), where players only get 10 months each.

Let’s now suppose the thieves will work together twice so that the stage game is played only twice (i.e. finitely repeated game). If we start our analysis from the last period (i.e. t=2) – a method called backward induction – we notice both players know they won’t collaborate thereafter, since there is no t=3,4,5,…,n. Is the promise of cooperation credible now? Or, is there any credible threat inducing them to collaboration? The answer is no.

In fact, there is no incentive for them to collude because none of them can punish the other, should he cheat. Why? Because there isn’t any period after t=2 where you can possibly punish your partner, the game ends at t=2. Therefore, in the last period, each player squeals and expects the other to do the same. Next, since they confess at t=2, why should they keep quiet in the first period? Again, there’s no way to punish the cheater because, independent of their actions at t=1, they defect in the last period. We can generalise this logical argument. If players know the reasonable finite value of t – reasonable since we expect thieves to have finite lifetime – we’d reach the same conclusion, by rolling back to t=1. Hence, this finitely repeated game’s outcome is the same as the initial static game’s one: (Confess, Confess). Whenever the stage game has only one Nash equilibrium, the unique perfect equilibrium of the finitely repeated version of the game is the repetition of the Nash equilibrium of the stage game itself.

In conclusion, we demonstrated why cooperation in the Prisoners’ Dilemma can only be sustained if the game is repeated indefinitely, such that players are uncertain about the end of the game and led to a collaborative resolution.

To me, this was a great achievement. Since the first time I heard about this game in high school, I always wondered why cooperation didn’t occur and why my eyes were always pointing at the ‘wrong’ cell of the payoff matrix. Now, I appreciate the reasoning behind those conclusions and I can see people playing games all the time in their daily lives.

Of course, there’s a lot more to say about GT but, as I said, this would require a slightly longer and more technical discussion. So, you’re free… for the moment.

I hope this compressed overview of GT will stimulate your curiosity and get you involved in some way! As always, feel free to share your views or extend our discussion, if you like ;-)

Oscar is studying for the BSc Economics and Finance independently in Italy.

*Actually, we should determine for what values of the discount factor this statement holds true. However, for our purpose, it’s sufficient to know that cooperation is reasonable in this case.

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