You can’t speak to each other and don’t know what the other person has decided. You’re friends, yes, but neither of you has any special loyalties towards one another, so you have no reason to think that Musa wants to protect you. Both of you are criminals, after all. What do you do?
This is called “The prisoner’s dilemma”.
Game theory was put into the mainstream in the 1950s by a mathematician called John Nash. This is the character that Russell Crow was playing in “A beautiful mind”
. But game theory isn’t about games in how we normally think about them. Instead, a game is;
Any interaction between multiple people in which each person’s payoff is affected by the decisions made by others.
Sure this applies to games like chess and poker. But it also applies to any situation where people get together and interact. If you have interacted with anyone today, you can probably analyze all the decisions you made using game theory. It’s incredibly wide-ranging and is used all the time by economists, political scientists, biologists, military strategists and psychologists, just to name a few.
Game theory has two main branches; cooperative and non-cooperative. Non-cooperative game theory has to do with competitive social interactions where there will be some winners and some losers. It’s often referred two as a zero-sum game, the winning of one side necessarily means the losing of another and vice-versa. The prisoner’s dilemma falls into this category.
Then there are cooperative games where each player has agreed to work together towards a common goal. This could be anything from a group of friends deciding how to split the cost of a road trip to a coalition of nations deciding how to divide the burden of solving climate change.
In game theory, a coalition is a group of players in a cooperative game. The primary question in cooperative games is how much each player should contribute to the coalition and how much they should benefit from it. It tries to determine what’s fair. It achieves this by applying these precepts;
- The contribution of each player is determined by what is gained or lost by removing them from the game. This is called their marginal contribution. If on our road trip you’re the only one with a car, your marginal contribution is a car, that’s what we lose by removing you from the coalition.
- Interchangeable players have equal value. If two parties bring the same things to the coalition, they should have to contribute the same amount and should be rewarded for their contribution equally.
- Dummy players have zero value. If a member of the coalition contributes nothing then they should receive nothing.
- If a game has multiple parts, cost or payment should be decomposed across those parts.
We’ll look at cooperative games another day, for now, let’s go back to the prisoner’s dilemma.
The dominant strategy
The prisoner’s dilemma contains all elements of a game. The two players are Thabo and Musa. There are two strategies available to them, confess or don’t confess and the payoff of the game range from going free to serving either 5, 10 or 20 years in prison. Moves are simultaneous, which means neither player knows the other player’s decisions and decisions are made at the same time.
Both prisoners are in separate rooms and won’t be let out until both have made their decisions. One common solution to simultaneous games is called the dominant strategy. This is defined as the strategy that has the best payoff, regardless of what the other player chooses.
You don’t know if Musa will confess or not so you consider your options. If Musa doesn’t confess then you’re better off confessing because that way you would go free.