Game Theory

Game Theory Parables for Game Designers

The Prisoner's Dilemma

The Prisoner's Dilemma is perhaps the most famous, and easily the most misinterpreted, example from modern Game Theory. As always, it begins with a story.

Two players are arrested, and accused of committing a crime. They are moved into separate rooms to be interrogated by the police, so there is no opportunity for communication. Each prisoner is given the same option: Betray their partner (blame them for the crime), or stay Quiet.

If one player stays Quiet, while the other Betrays them, the Betrayer will receive only 1 year in prison, while the one who stayed Quiet will receive 10 years. If both Betray each other, they will each receive 5 years in prison, while if they both stay Quiet, they will only receive 2 years each.

What should each player do?

It is tempting to say that both players should stay Quiet — this leads to the cooperative outcome that gives each player a better outcome than if they mutually Betray each other. Game Theory, however, says that both players should play Betray, with the end result that each receives 5 years in prison. This seems counter-intuitive to many people: can’t the players see that if they both stayed Quiet instead of both Betraying each other, they would each get 2 years, not 5?

Here is how Game Theory comes to its solution, and why the wishful thinking mentioned above must remain wishful thinking.

To think about what one player should do (Betray or Quiet), they need to think strategically. That means think about what the other player would do. If player one knew that player two was playing Betray, what should player one do? They could respond with Betray and get 5 years, or stay Quiet and get 10. In this case, Betray is the better option.

What if player two chose to stay Quiet? Player one could respond with Betray and get 1 year, or stay Quiet as well and get 2. Given the choice, Betray is the better option.

Thus, regardless of what player two does, player one is better off to play Betray. This means that it does not matter what player one does, player two should always play Betray.

The reasoning is the same for player two: they should always play Betray. The end result is that both players end up with 5 years in prison each. Indeed, this is the unique Nash Equilibrium for this game.

In Game Theory, the Betray strategy is said to Strongly Dominate the Quiet strategy: it gives a better outcome for the player no matter what the other player does.

Many people see this as a “Paradox of Rationality:” can’t the players see that they would each do better if they would both switch to Quiet instead of Betray? They would each get 2 years in prison, not 5. Based on this line of thinking, it is common to reject Game Theory’s analysis, and try to find a way to make cooperation happen in the Prisoner’s Dilemma.

The problem with this is that both players staying Quiet is not stable when players are rational. If one player knows that the other will stay Quiet, then it is in their best interests to switch to Betray, as that gets them 1 year in prison instead of 2. Of course the other player can think this way too, switching to Betray instead of Quiet, and they end up back at 5 years each.

One reaction to this supposed paradox focuses on the separation of the players. If they were only allowed to talk to each other, the story goes, they could agree to stay Quiet. Since they are both rational, and 5 years in prison is worse than 2, they would stick to their guns and stay Quiet.

Talk is cheap, though, and with nothing to bind the players to their promises, they are each incentivized to break them. In the absence of a mechanism to enforce the agreement, players cannot be counted on to live up to their end of the deal.

A tempting response is to say that players shouldn’t be self-interested, that they should care about the time that the other player spends in prison as well. This unfortunately misunderstands what “self-interest” encompasses in rational choice. To see this, the assumption of rational player behaviour needs to be examined.

What is the moral for game designers that comes from all of these mistaken attempts to “solve” the Prisoner’s Dilemma? Simply that if a designer ends up with a Prisoner’s Dilemma somewhere in their game, they need to change the game itself to get players to cooperate. There are many ways to do this, including playing the game multiple times in a row; read about them here.

Sam Hillier