Prisoners’ Dilemma Part 3– The Predictions

I expected to find that some of the more complex, devious strategies would do better at the game than their “nicer” counterparts. How could they not? Obviously, strategies focused on seeking opportunities to further their own success, even at the expense of others are, by definition “good competitors”.

I expected to find the nicest players to be among the biggest losers. The more willing a strategy is to play nice, and be forgiving in the wake of an opponents treachery, the more open it leaves itself to exploitation affording its opponents a large payoff and subjecting itself to a heavy penalty. As I mentioned earlier this also seems to fit common western wisdom harmonizing with the “Nice guys finish last” sentiment.

Surprisingly, happily, I was very, very wrong.

 

Vocabulary

As I began to develop strategies to load into the simulation I was surprised by how well they could be characterized according to their “temperament”. To help give me the ability to generalize results as well as talk about them, I used the following structure and vocabulary to describe their behaviors:

“Nice” strategies prefer to cooperate and will never be the first to play “Defect”. After an opponent plays “Defect”, nice strategies retaliate, hold grudges, or do anything else they see fit.

Nice strategies are further classified as:

Naive Nice to the extreme… Play “Cooperate” on every turn, no matter what the other player does.
Vengeful After the first opponent-throw “Defect” , vengeful strategies protect themselves and try to recoup the losses of that betrayal by playing nothing but “Defect” for the remainder of the game.
Forgiving A vengeful strategy that returns playing “nice” if opponent shows some degree of contrition. A very forgiving strategy requires only one “Cooperate” from its opponent to return to playing nice. Less forgiving strategies may require more.

“Naughty” strategies are those that are capable of being the first player to play a “Defect”. These strategies try to take advantage of the large payoff that occurs when they can play a “Defect” against their opponents “Cooperate”

Naughty strategies can be further described as:

Selfish Plays “Defect” on every turn, regardless of what the other player plays.
Devious Plays “Cooperate”, while looking to be the first to play a “Defect” against opponents “Cooperate”. Will likely play “Defect” for the rest of the game to protect itself against being exploited by retaliation from the other player.
Repentant At some point after playing a “Defect”, a repentant strategy will try to coax its opponent into mutual cooperation by playing “Cooperate”. If successful, it may then start to play deviously, or stay “nice” for the remainder of the game.

 

Non-Zero Sum Game

One of the first things that struck me as interesting that I hadn’t considered at the start is that Prisoner’s Dilemma is not a zero-sum game. A zero-sum game is any competition in which one player or team benefits at the expense of the other player or team. One player is the “winner” and the other player the “loser”. The magnitude of the benefit of winning exactly equals the expense of losing. Most of the games and competitions that we are consciously exposed to are zero-sum games: Football, Soccer, Baseball, even often Business are all considered zero sum in that either one team wins or the other does. It is equally correct to think of a particular team scoring a basket or making a goal as it is to think of the opposing team giving up a basket or goal. The zero-sum method of thinking about a career advancement opportunity is, “I didn’t get the promotion because my (former) peer did.”

Prisoner’s Dilemma is a non-zero sum game in that the benefit due the winner of each game is not at the expense of the loser. Consider, for example, the game where both players play “Cooperate”. Each player is paid $300. One player did not benefit at the expense of the other. Instead, the players, together, gain $600 at the expense of the “banker”… an entity who is not even part of the competition. I found that many of the mistakes in thinking about Prisoner’s Dilemma stem from that fact that it is NOT a zero-sum game. It is no accident that most of the “competitions” we participate in Las Vegas are also not zero-sum games, but rather pay winning, not at the expense of other players, but at the expense of the “house”. So unaccustomed are we to thinking about the world in terms of non-zero-sum games that it is relatively easy to distort the innate feel for risk and rewards our “gut” affords each of us.

 

Strategy of Opponents

The success of any strategy is highly dependent on the other strategies against which it is competing. This is obvious, but I was surprised by the degree to which this was true. When I correlated the performance of a strategy with both the strategy itself and the other strategies present, I found that the correlation of all the OTHER strategies was much stronger than with the strategy itself. In other words, knowing about all the other strategies against which it will be competing made it easier to predict how well a strategy would perform than knowing anything about the strategy itself!

I saw this over and over again in the simulations. The end result would change dramatically depending on exactly which strategies were chosen to compete. I found plenty of examples of A vs. B vs. C where A was the winner. But when I added D, D would perform very poorly and not be the winner… It might even be the loser. But its affect on the system would still be felt leaving B (for example) as the winner.

To some extent, this is simply to be expected. If one comes up with a new strategy that is simply better at the game than previous strategies it will be the winner. Similarly if a new strategy is dreamt up that is unusually lethal to a particular strategy we’ll also find in competitions that involve this new strategy and its target, that its target will perform unusually poorly, while it will do unusually well.

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