Game Theory
Some thoughts on how Game Theory can apply to the real world
The moment Game Theory is mentioned, the first name that gets thrown up is that of John Nash, made famous by the movie “A Beautiful Mind”.
Game theory is essentially a framework used to produce mathematical models for potential outcomes when two sides engage in strategic interaction.
The examples often cited are those of two countries at war trying to determine when to attack or two companies engaged in competition for market share trying to determine which markets to go after.
The example that is often used as a starting point for discussions on Game Theory is the Prisoners Dilemma.
Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch ... If both prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is concerned only with his own welfare—with minimizing his own prison sentence.
This leads to four different possible outcomes for prisoners A and B:
If A and B both remain silent, they will each serve one year in prison.
If A testifies against B but B remains silent, A will be set free while B serves three years in prison.
If A remains silent but B testifies against A, A will serve three years in prison and B will be set free.
If A and B testify against each other, they will each serve two years.
Source: Wikipedia
Selfishly the best thing for each to do would be to testify. If the other does not one would escape, if the other does as well then at least they reduce their sentence by 1 year.
The penalty of remaining silent can be grave. Not only is there a risk of going to prison for 3 years but worse yet would be watching the other one get away. This is a case where there is no trust. This is taught as Nash Equilibrium.
What if both the prisoners are romantically involved?
Typical game setups involve only two people trying to guess what the other would do.
Recursion
Typical games involve finding out what the other person would do given certain information. This can take the form of recursion where you can go many levels deeper by running the same set of constraints over and over.
There is a very well-known sequence in the sitcom Friends where Rachel and Phoebe find out that Chandler and Monica are in a relationship. Neither wants to divulge the knowledge and it goes into hilarious recursion. I have dropped the link to the clip for those who do not know.
The other great illustration of recursion is the stock market. Keynes said, “Successful investing is anticipating the anticipation of the others”.
A person who buys a share at $10 is hoping to sell it to another at $20 who is hoping to then sell it to another person at $30 and so on. This obviously cannot go on forever as illustrated by the video above. But we often embroil ourselves in such games.
Rock, Paper Scissors
Anyone who has played rock paper scissors knows that you spend most of your time trying to guess what the other person is going to throw. This is the best example to try and understand the Nash Equilibrium.
Rock > Scissor
Scissor > Paper
Paper > Rock
So as this game is played over a prolonged number of cycles, two people would reach an equilibrium of the number of Rocks, Papers or Scissors that one throws. Each of the three options will come up a third of the time. In a two-person game, there always exists an equilibrium position that can be reached.
The problem with game theory is that it can provide us with the equilibrium position it cannot tell us how to get there. Also, the equilibrium may not be the best result as illustrated by the prisoner’s dilemma example. The equilibrium position is quadrant 4 and it is arguably not the best outcome for either.
Tragedy of the Commons
In England, before land became a property, all lands were considered commons and everyone in the village would be able to graze their sheep on the grass. Ideally, one would not want to overgraze the field. They would leave enough for the others. Let us say one of them were to graze the field slightly more assuming it would not hurt the others in the grand scheme of things.
But if everyone were to adhere to the same philosophy, the grass would be overgrazed and there would not be enough left. This is the tragedy of the commons.
If there are 5 factories making clothes and one decides it can use cheaper dyes and make its product cheaper while harming the environment, that alone would not destroy the environment. But, another factory follows suit and soon all 5 factories are using the cheap and harmful chemicals. Soon you get cancer from eating fish.
Silicon Valley has weaponised this by offering an unlimited leave policy to their employees. When you have unlimited leave you want to take fewer holidays than the people you are working with to be perceived as hardworking and fit for a promotion. There is always someone who is taking Zero days off and hence everyone ends up chasing this horrible goal.
Mechanism Design
Now let us say that the Silicon Valley CEO was not malicious.
How can this game be changed? Say, the people who took leave were given a $1000 incentive. Would that result in people taking leave? It changes the game mechanics but so long as the raise/promotion (Status) is more valuable than those $1000 people’s behaviour may not change.
Instead of a carrot; a stick might work here. Anybody who takes zero days off would be docked $1000 from their pay. The pain of losing is always greater than the joy of gaining and that change in the game design would certainly change behaviour.
Ultimately it is all about emotions
I had previously written about rationality or its absence thereof. Most of the problems that game theory tries to address come down to creating mind maps. A map of how another person would be thinking and what action one should take to get the best outcome.
But we are not rational at all. We are driven by emotions. We talk about the sentiment in the market quietly letting everyone know that the market is not rational.
Running a good game is about making sure that you understand the emotional state of the other as much as the logic of the other.
Hence game theory has been of little use in most applications apart from auction design and war where more than one person is involved in the decision-making process. Ultimately everyone has to agree on some logic and this can be modelled. Individual decision-making is always laced with emotions and hence harder to model.