The Story So Far…
Two players are presented with an opportunity. Each may remain loyal to the other player or betray him. Neither player will interact with the other in any way ever again. There is no out-of-game way to be rewarded or punished.
What could happen?
(1) If both remain loyal, each of them gets the “Cooperation” reward.
(2) If both betray the other, each of them gets the “Betrayal” reward.
(3) If one betrays the other while the other remains loyal, the betrayer gets the “Traitor” reward while the loyal one gets the “Sucker” reward.
This is the prisoner’s dilemma.
In a strict prisoner’s dilemma, the rewards are staggered with the “Traitor” reward best, followed by the “Cooperation” reward, then “Betrayal,” and finally “Sucker” the worst. In mathematical terms, T > C > B > S.
The Prisoner’s Dilemma In Mathematics
the Nash equilibrium for this game is for each to betray the other. The logic is an extension of the amoral player presented in my last post and looks like this:
(1) If you remain loyal, I am better off betraying you. In this way, I get the “Traitor” reward which is the best of the rewards I can get when you remain loyal.
(2) If you betray me, I am better off betraying you. In this way, I get the “Betrayal” reward which is the best of the rewards I can get when you betray me.
(3) Therefore, I should betray you.
(4) Since you also know this, you should also betray me.
(5) Therefore we betray each other.
This logic is well-illustrated at by this video from Kahn Academy.
The Prisoner’s Dilemma In The Social Sciences
Avinash Dixit and Barry Nalebuff present a classic arms race as an immediate example of the prisoner’s dilemma in their article at The Concise Encyclopedia of Economics. Two nations could Cooperatively prosper if neither invested in weapons and instead devote their resources to more positive objectives. Instead, the two nations both choose Betrayal and the arms race continues.
Dixit and Nalebuff go on to assert “On a superficial level the prisoners’ dilemma appears to run counter to Adam Smith’s idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group’s cooperation is not in the interests of society as a whole…One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests…The most common path to cooperation arises from repetitions of the game.”
In game designer terms: We must understand when and why our players cooperate before we can either encourage and discourage in the pursuit of engaging decisions. A path for us is to have this decision point occur repeatedly throughout our game.
The Mathematical View of Multiple Iterations
Game theory sticks to its amoral guns, even in the face of iterative play. Its argument looks like this:
(1) We know that I am best off betraying you in a single iteration because you have no opportunity to retaliate.
(2) You know this too.
(3) Therefore, we should betray one another in the final round.
(4) Therefore, I should betray you in the second-to-last round.
(5) Therefore, you should betray me in the second-to-last round.
(6) Iteratively, we betray one another in every round.
This issue is fixable. Key to this logic is based is the assumption that the players know which iteration will be the last iteration. When players are denied this knowledge, the starting point assumed in step (3) of the logic above is invalidated.
In game designer terms: Players should be presented with this decision point repeatedly but our game should obscure the number of times it will do so.
Peace War Games
Even in a group that doesn’t play the same game repeatedly, there exists a metagame memory of previous behavior which is carried into future games. I would suggest that designers can expect players to therefore gravitate toward Peace War Game behavior.
Peace War Games are iterated versions of the prisoner’s dilemma, extended out to larger numbers of players. Players are cast as nations, with each turn’s decision being whether to choose “peace” or “war” with each neighbor. According to Wikipedia, peace makers became richer over time, falling behind only the “Genghis Khan” strategy of constant aggression in which war supplied a steady stream of resources. The player response to Genghis Khan is an interesting one–the “Provocable Nice Guy.” This player selects peace always until attacked. When several Provocable Nice Guys work in consort, they promote one another while reigning Genghis Khan in.
Because they are iterative and because they involve groups of players, peace war games give us the most insight into how players are likely to act when faced with prisoner’s dilemmas in our games.
To Be Continued…
Our next column looks at the design of Friend or Foe rewards. Which types of rewards incentivize what types of behavior? See you Friday!
For those of you interested in further reading, Homo Ludditus makes several interesting arguments in its Nobody Understands “Prisoner’s Dilemma” article. While it contains several points I cannot agree with, it does make some solid ones as well. Certainly worth the time it will take you to read and digest it.
Have you played a game with a Friend and Foe mechanism? What did you think of it? Have you written one? How did your players respond to it? Share with your fellow readers in the comments below. And if you’re enjoying what you’re reading, create an account with WordPress and follow this blog. You keep reading. I’ll keep writing.