Prisoner's Dilemma
Prisoner's Dilemma is a game theory thought experiment that explores the nature of cooperation, defection and maximal return.
Single-Shot Prisoner's Dilemma
A classic example of the prisoner's dilemma ("PD") is presented as follows:
- Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies (defects from the other) for the prosecution against the other and the other remains silent (cooperates with the other), the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?
| Prisoner B Stays Silent | Prisoner B Betrays | |
|---|---|---|
| Prisoner A Stays Silent | Each serves 6 months | Prisoner A: 10 years Prisoner B: goes free |
| Prisoner A Betrays | Prisoner A: goes free Prisoner B: 10 years |
Each serves 5 years |
In this game, regardless of what the opponent chooses, each player always receives a higher payoff (lesser sentence) by betraying; that is to say that betraying is the strictly dominant strategy. For instance, Prisoner A can accurately say, "No matter what Prisoner B does, I personally am better off betraying than staying silent. Therefore, for my own sake, I should betray." However, if the other player acts similarly, then they both betray and both get a lower payoff than they would get by staying silent. Rational self-interested decisions result in each prisoner being worse off than if each chose to lessen the sentence of the accomplice at the cost of staying a little longer in jail himself (hence the seeming dilemma).
Applications to EM
A payoff matrix can be devised for any situation where this may arrive. In this case, let's consider a situation where there is a killer, a vanilla mafia (for the sake of points assignment), and a villager going into night, and the killer and mafia must decide who to kill to get the best outcome for themselves.
| Mafia | |||
|---|---|---|---|
| Cooperate | Defect | ||
| Killer | Cooperate | 40, 33 | 80, 0 |
| Defect | 0, 65 | 0, 0 | |
Game theorists may recognize that EM is not a "true" PD setup, because there is no distinction between the type of defection (e.g., defection when there had been cooperation by the side that was shot versus mutual defection). None the less, it's obvious that for the best goal for both players, cooperation is required.
Iterated Prisoner's Dilemma
However, single-shot prisoner's dilemma is not really the case that we have in a game such as mafia. In any single game, there are multiple nights (with essentially the same cast of characters) -- and the game can be replayed, so there are also multiple games. Some may hold a grudge during the course of the game (and graveyard/post-game chat) and let it go, and some may hold one longer.
As a result, we need to consider what happens when the same group of people work together on a regular basis.
Let's consider a situation where we may see an iterated pattern of behavior: illicit drug deals.
- Once upon a time in California, the police could not search a suspected drug dealer standing in a parking lot where drugs were frequently sold. The law required that they see the suspected drug dealer exchange something, presumably money and drugs, with a suspected customer. The drug dealers and their customers found a way to prevent the police from interfering in their business. The dealer would drop a plastic bag of white powder in the ornamental ivy beside the parking lot in a usual spot. The customer would, at the same time, hide an envelope full of money in a drain pipe on the other side of the lot. These actions were performed when the police were not looking. Both then walked with their best "I'm not up to anything" stride, exchanged positions, and picked up their respective goods. This is quite a clever system as long as the drug dealer and the customer are both able to trust each other. (Daniel Ashlock. "Evolutionary Computation for Modeling and Optimization." 2005. ISBN: 978-0387221960.)
Let's turn this into a prisoner's dilemma. There are two players (the drug dealer and the buyer). Each player has two options, co-operate or defect.
- Drug Dealer
- Cooperate: Supply the correct amount of drugs for the transaction.
- Defect: Supply a smaller amount of drugs (such as "no drugs"), tell the cops, etc.
- Buyer
- Cooperate: Supply the correct amount of money for the transaction.
- Defect: Supply a smaller amount of money (such as "no money"), tell the cops, etc.
Single-shot prisoner's dilemma tells us that the best course of action - the most return with the least risk - is for both players to defect, leaving the drug dealer with no money, and the buy with no drugs. But this is the real world: the drug dealer wants to make money, and needs repeat business to do so - he wants, no, needs his buyers to come back. The buyer wants his drugs - not a poor substitute or dilution - and have the dealer willing to sell to him again again. So the best shot of action, in this case, is generally for both of them to cooperate.
Applications to EM
In Epic Mafia, there are several opportunities where different factions need to work together to win. A few of them are listed below (and eventually will be discussed more in depth).
- Single-shot PD
- (Fool, Lyncher, or Survivor) and (Mafia or Killer)
- Amnesiaic and anyone
- Iterated PD
- Mafia and Killer
- Warlock and (Mafia or Town)
