Game Theory 110 — The Prisoner’s Dilemma

An altruistic strategy is best over the long haul, but not in the beginning, and not for short term interactions.

Readership: All
Reader’s Note: In this post, game theory is about decision making strategies, not about the PUA kind of game, although they can and do overlap.
Some of the game theory in this post is rather taxing to wade through, but it is well worth your time because it contains some very important conclusions. Also, since the knowledge is cumulative and each section builds on previous sections, you cannot really save time by skimming or skipping. Sorry! Just take my word that you’ll be much wiser after studying this.
A while back, I wrote a primer in game theory, Introduction to Game Theory 101 (2018 February 11). If you have not yet read this post, I urge you to do so before continuing on.
Length: 1,400 words
Reading Time: 5 minutes

Basic Game Theory

In the first part of this post, I will go over some generalized game theory, including Positive-Sum game, Zero-Sum game, and especially Negative-Sum game.  The last part of this post will cover one Negative-Sum game of particular interest — The Prisoner’s Dilemma. In future posts, I’ll be using this theory as a springboard to examine theories of intersexual dynamics in dating and marriage.

Positive Sum or “Win Win” Games

The Positive-Sum game is a game of cooperation and partnership. The partners think in terms of “we”, and “us”, and seek mutually beneficial outcomes.

In a Positive-Sum Game, the interests of the players are not in direct conflict. Thus, there are some outcomes which would benefit all players. Examples include most recreational activities (hiking, camping, having a barbecue), athletic non-competitive sports (e.g. weight lifting, gymnastics, kayaking), and typical business transactions.

Constant Sum

A game is Constant-Sum if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. Examples include competitive sports in which only one player or team is declared the winner (e.g. baseball, basketball, volleyball, etc.).

A Constant-Sum game can be converted into a Zero-Sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.

Zero Sum or “Win Lose” Games

In a Zero-Sum Game, the players’ interests are in direct conflict such that there is a winner and a loser. The zero sum game is typically a game of chance, competition, skill, power, and dominance. Examples include card games, and athletic competitive sports (e.g. wrestling, boxing, etc.).

Many non-competitive activities are sometimes cast in a competitive playoff for a fun challenge and entertainment. For example, dancing troupes and science fairs hire judges to pick out the best of show. Construction workers will split into teams and see who can complete the work the fastest or best.

In social intercourse, informal competitions are held to establish social acumen. Most people enjoy this as long as this is done in good will or in fun. However, playing the zero-sum game in serious matters tends to divide people into factions. Some people (i.e. risk-takers) who view this as a challenge are willing to raise the stakes and go all in on it, while others (i.e. risk adverse) view it as unwise, or as a potential threat, and withdraw. Women have been known to sift out risk-adverse men when they play, “Let’s you and him fight!” Women who habitually play the zero-sum game in serious matters are typically deemed by marriage-minded men to be unworthy as a partner, and are thus avoided and rejected from LTR’s, especially if she has an arrogant or presumptuous attitude in the matter.

Negative Sum games

In a Negative-Sum game, the interests of the players are not contingently opposed to each other, such that everyone could lose.

The Negative-Sum game is a game of compromise, sacrifice, and perhaps trust, and the goals depend on the context, the history of interaction, and the personalities of the individuals. Sometimes the game is played for one’s own benefit, and other times, it can be played for the mutual benefit of certain figures who are involved.

A well-studied example of the Negative-Sum game is the Prisoner’s Dilemma. We’ll go over this next.

The Prisoner’s Dilemma [1]

The Prisoner’s Dilemma is a standard example of a game analyzed in game theory that shows why two completely “rational” individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it, “The Prisoner’s Dilemma” (Poundstone, 1992), presenting it as follows:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to (1) confess to the major crime and receive a reduced sentence, and in doing so, betray the other by testifying that the other committed the crime, or (2) to remain silent. The possible outcomes are:

• If A and B both confess to the major crime (i.e. each betray the other), each of them serves 2 years in prison.
• If A confesses to the major crime and rats out B, but B remains silent, then A will be set free and B will serve 3 years in prison (and vice versa).
• If A and B both remain silent, then both of them will only serve 1 year in prison (on the lesser charge).

The following conditions also apply.

• Each prisoner has no clue of which choice the other prisoner might make.
• It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get, and that their decision will not affect their reputation in the future.

A matrix of possibilities and outcomes is shown below.

It is better to Defect!

Here, regardless of what the other decides, each prisoner gets a better reward by betraying the other (“defecting”). The reasoning (from the perspective of prisoner A) involves a deductive argument by dilemma:

1. B will either cooperate or defect.
2. If B cooperates, A should defect, because going free is better than serving 1 year.
3. If B defects, A should also defect, because serving 2 years is better than serving 3.
4. So either way, A should defect.

Parallel reasoning will show that B should defect.

Because betraying a partner offers a greater reward than cooperating with him, all purely rational self-interested prisoners would betray the other, and so the only possible outcome for two purely rational prisoners is for them to betray each other.

The important thing to understand from this result is that pursuing an individual reward logically leads both of the prisoners to betray, even though they would collectively serve less time if they both kept silent.

The Iterated Prisoner’s Game [2]

An extended, “iterated” version of the Prisoner’s Dilemma also exists. In this version, the classic game is played repeatedly between the same prisoners, and consequently, both prisoners continuously have an opportunity to reward or penalize the other for previous decisions.

The iterated prisoners’ dilemma game is fundamental to some theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behavior in populations may be modeled by a multi-player, iterated version of the game. Prisoner’s Dilemma tournaments have been held to test algorithms and strategies through competition.

If the game is played exactly N times and both players know this, then it is always game theoretically optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof is inductive: one might as well defect on the last turn, since the opponent will not have a chance to later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on. The same applies if the game length is unknown but has a known upper limit. Therefore, if the number of times the game will be played is known to the players, then (by backward induction) two classically rational players will betray each other repeatedly, for the same reasons as the single shot variant.

Of note, this proof holds to be the only correct answer within standard economic theory.

In a non-economic game of infinite or unknown length there is no fixed optimum strategy. This opens up a plethora of game strategic possibilities. Unlike the standard Prisoners’ Dilemma, in the Iterated Prisoners’ Dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. The superrational strategy in the Iterated Prisoners’ Dilemma with fixed N is to cooperate with a superrational opponent, and in cases of a large N, experimental results on strategies agree with the superrational version, not the game-theoretic rational one.

For cooperation to emerge between game theoretic rational players, the total number of rounds N must be random, or at least unknown to the players. In this case ‘always defect’ may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.

I will discuss some detailed strategies of the Iterated Prisoners Dilemma in some upcoming posts.

Conclusions

The optimal (points-maximizing) strategy for the one-time Prisoner’s Dilemma game is simply defection. As explained above, this is true whatever the composition of opponents may be. However, in the Iterated Prisoner’s Dilemma game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations.

For example, consider a population where everyone defects every time, except for a single individual following the Tit for Tat strategy (i.e. making the same move that the opponent did in the last round). That individual is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy for that individual is to defect every time. In a population with a certain percentage of always-defectors and the rest being Tit for Tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.

Follow up posts will discuss the evolution of winning strategies in The Prisoner’s Dilemma, as applied to Dating and Marriage.

References

1. Infogalactic: Game Theory
2. Infogalactic: Prisoner’s Dilemma

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