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The prisoner's dilemma is a type of non-zero-sum game. In this game theory problem, as in many others, it is assumed that each individual player is trying to maximise his own advantage, without concern for the well-being of the other player. This Nash equilibrium does not lead to a jointly optimum solution in the prisoner's dilemma; in the equilibrium, each prisoner chooses to defect even though the joint payoff of the players would be higher by cooperating. Unfortunately (for the prisoners), each player has an individual incentive to cheat even after promising to cooperate. This is the heart of the dilemma.
In the iterated prisoner's dilemma cooperation may arise as an equilibrium outcome. Here the game is played repeatedly. Since the game is repeated, each player is afforded an opportunity to punish the other player for previous non-cooperative play. Thus, the incentive to cheat may be overcome by the threat of punishment, leading to a superior, cooperative outcome.
The classical prisoner's dilemma (PD) is as follows:
which can be summarized as:
| You Deny | You Confess | |
|---|---|---|
| He Denies | Both serve six months | He serves ten years; you go free |
| He Confesses | He goes free; you serve ten years | Both serve six years |
Let's assume both prisoners are completely selfish and their only goal is to minimize their own jail term. As a prisoner you have two options: to cooperate with your accomplice and stay quiet, or to betray your accomplice and confess. The outcome of each choice depends on the choice of your accomplice; unfortunately, however, you don't know the choice of your accomplice. Even if you were able to talk to him, you couldn't be sure whether to trust him.
If you expect your accomplice will choose to cooperate and stay quiet, the optimal choice for you would be to confess, as this means you get to go free immediately, while your accomplice lingers in jail for 10 years. If you expect your accomplice will choose to confess, your best choice is to confess as well, since then at least you can be spared the full 10 years serving time and have to sit out 6 years, while your accomplice does the same. If however you both decide to cooperate and stay quiet, you would both be able to get out in 6 months.
Confessing is a dominant strategy for both players. Given the other player's choice, you can always reduce your sentence by confessing. Unfortunately for the prisoners, this leads to a poor outcome where both confess and both get heavy jail sentences. This is the core of dilemma.
If reasoned from the perspective of the optimal interest of the group (of two prisoners), the correct outcome would be for both prisoners to cooperate, as this would reduce the total jail time served by the group to one year total. Any other decision would be worse for the two prisoners considered together. However by each following their selfish interests, the two prisoners each receive a lengthy sentence.
If you had an opportunity to punish the other player for confessing, then a cooperative outcome could be sustained. The iterated form of this game (discussed below) presents an opportunity for such punishment. In that game, if your accomplice cheats by confessing this time, you can punish him by cheating next time yourself. Thus, the iterated game builds in an opportunity for punishment absent in the classic one-period game.
The cognitive scientist Douglas Hofstadter (see References, below) once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was two people meeting and exchanging closed bags, with the understanding that one of them contains money, and the other contains an item being bought. Either player can choose to honor the deal by putting into his bag what he agreed, or he can defect by handing over an empty bag. In this exchange game, unlike in the PD, defection is always the best course.
In the same article, Hofstadter also observed that the PD payoff matrix can, in fact, be written in a variety of ways, as long as it conforms to the following principle:
where T is the temptation to defect (ie, what you get when you defect and the other player cooperates); R is the reward for mutual cooperation; P is the punishment for mutual defection; and S is the sucker's payoff (ie, what you get when you cooperate and the other player defects).
(It is also usually the case that (T + S)/2 < R, and this is required in the iterated case.)
The above formulae, then, ensures that, whatever the precise numbers in each part of the payoff matrix, it is always 'better' for each player to defect regardless of what the other does.
Following this principle, and simplifying the PD to the above 'bag switching' scenario (or an Axelrod-type two player game, see below), we get the following 'canonical' PD payoff matrix — that is, the one that is normally shown in literature on the subject:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | 3, 3 | 0, 5 |
| Defect | 5, 0 | 1, 1 |
In "win-win" terminology the table would look like this:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | win-win | lose much-win much |
| Defect | win much-lose much | lose-lose |
These particular examples, involving prisoners and bag switching and so forth, may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature which have the same payoff matrix. The prisoner's dilemma is therefore of interest to the social sciences such as economics, politics and sociology, as well as to the biological sciences such as ethology and evolutionary biology.
In political science, for instance, the PD scenario is often used to illustrate the problem of two states engaged in an arms race. Both will reason that they have two options, either to increase military expenditure or to make an agreement to reduce weapons. Neither state can be certain that the other one will keep to such an agreement; therefore, they both incline towards military expansion. The irony is that both states seem to act rationally, but the result is completely irrational.
Another interesting example concerns a well-known concept in cycling races, for instance in the Tour de France. Consider two cyclists halfway in a race, with the peloton (larger group) at great distance. The two cyclists often work together (mutual cooperation) by sharing the tough load of the front position, where there is no shelter from the wind. If neither of the cyclists makes an effort to stay ahead, the peloton will soon catch up (mutual defection). An often-seen scenario is one cyclist doing the hard work alone (cooperating), keeping the two ahead of the peloton. In the end, this will likely lead to a victory for the second cyclist (defecting) who has an easy ride in the first rider's slipstream.
Lastly, the theoretical conclusion of PD is one reason why, in many countries, plea bargaining is forbidden. Often, precisely the PD scenario applies: it is in the interest of both suspects to confess and testify against the other prisoner/suspect, even if each is innocent of the alleged crime. Arguably, the worst case is when only one party is guilty — here, the innocent one is unlikely to confess, while the guilty one is likely to confess and testify against the innocent.
In his book The Evolution of Cooperation (1984), Robert Axelrod explored an extension to the classical PD scenario, which he called the iterated prisoner's dilemma (IPD). In this, participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity; initial hostility; capacity for forgiveness; and so forth.
Axelrod discovered that when these encounters were repeated over a long period of time with many players, each with different strategies, "greedy" strategies tended to do very poorly in the long run while more "altruistic" strategies did better, as judged purely by self-interest. He used this to show a possible mechanism to explain what had previously been a difficult hole in Darwinian theory: how can seemingly altruistic behavior evolve from the purely selfish mechanisms of natural selection?
The best deterministic strategy was found to be "Tit for Tat", which Anatol Rapoport developed and entered into the tournament. It was the simplest of any program entered, containing only four lines of BASIC, and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, do what your opponent did on the previous move. A slightly better strategy is "Tit for Tat with forgiveness". When your opponent defects, on the next move you sometimes cooperate anyway with small probability (around 1%-5%). This allows for occasional recovery from getting trapped in a cycle of defections. The exact probability depends on the lineup of opponents. "Tit for Tat with forgiveness" is best when miscommunication is introduced to the game. That means that sometimes your move is incorrectly reported to your opponent: you cooperate but your opponent hears that you defected.
Tit for Tat was successful, Axelrod argued, for two main reasons. Firstly, it is 'nice': that is, it starts off cooperating and only defects in response to another player's defection, so it is never responsible for initiating a cycle of mutual defections. Secondly, it is provocable, always responding to what the other player does; it punishes another player immediately when they defect, but equally immediately it responds in kind if they start to cooperate again. Such clear, straightforward behaviour means that the other player can easily understand the logic behind Tit for Tat's actions, and can therefore figure out how to work alongside it productively. It is no coincidence, incidentally, that most of the worst-performing strategies in Axelrod's tournament were ones that were not designed to be responsive to the other player's choices; against such a player, the best strategy is simply to defect every time, because you can never be sure of establishing reliable mutual cooperation.
For the iterated PD, it is not always correct to say that any given strategy is the best. For example, consider a population where everyone defects every time, except for a single individual following the Tit-for-Tat strategy. 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. Simulations of populations have been done, where individuals with low scores die off, and those with high scores reproduce. The mix of algorithms in the final population generally depends on the mix in the initial population.
If an iterated PD is going to be iterated exactly N times, for some known constant N, then there is another interesting fact. The Nash equilibrium is to defect every time. That is easily proved by induction. You might as well defect on the last turn, since your opponent will not have a chance to punish you. Therefore, you will both defect on the last turn. Then, you might as well defect on the second-to-last turn, since your opponent will defect on the last no matter what you do. And so on. For cooperation to remain appealing, then, the future must be indeterminate for both players. One solution is to make the total number of turns N random.
Another odd case is "play forever" prisoner's dilemma. The game is repeated infinitely many times, and your score is the average (suitably computed).
The prisoner's dilemma game is fundamental to certain theories of human cooperation and trust. On the assumption that transactions between two people requiring trust can be modelled by the PD, cooperative behavior in populations may be modelled by a multi-player, iterated, version of the game. It has, consequently, fascinated many, many scholars over the years. A not entirely up-to-date estimate (Grofman and Pool, 1975) puts the count of scholarly articles devoted to it at over 2,000.
There are also some variants of the game, with subtle but important differences in the payoff matrices, which are listed below:-
Another important non zero-sum game type is called "Chicken". In Chicken if your opponent cooperates, you are better off to defect - this is your best possible outcome. If your opponent defects, you are better off to cooperate. Mutual defection is the worst possible outcome (hence an unstable equilibrium), but in the Prisoner's Dilemma the worst possible outcome is cooperating while the other person defects (so both defecting is a stable equilibrium). In both games, "both cooperate" is an unstable equilibrium.
A typical payoff matrix would read:
Chicken is named after the car racing game. Two cars drive towards each other for an apparent head-on collision - the first to swerve out of the way is "chicken". Both players can swerve to avoid the crash (cooperate) or keep going (defect). Another example often given is that of two farmers who use the same irrigation system for their fields. The system can be adequately maintained by one person, but both farmers gain equal benefit from it. If one farmer does not do his share of maintenance, it is still in the other farmer's interests to do so, because he will be benefiting whatever the other one does. Therefore, if one farmer can establish himself as the dominant defector - ie, if the habit becomes ingrained that the other one does all the maintenance work - he will be likely to continue to do so.
An Assurance Game has a similar structure to the prisoner's dilemma, except that the rewards for mutual co-operation are higher than those for defection. A typical pay-off matrix would read:
The Assurance Game is potentially very stable because it always gives the highest rewards to players who establish a habit of mutual co-operation. However, there is still the problem that the players might not realise that it is in their interests to co-operate. They might, for example, mistakenly believe that they are playing a Prisoner's Dilemma or Chicken game, and arrange their strategies accordingly.
Friend or Foe is a game show airing currently on the Game Show Network. It is an example of the prisoner's dilemma game tested by real people, but in an artificial setting. On the game show, three pairs of people compete. As each pair is eliminated, they play a game of Prisoner's Dilemma to determine how their winnings are split. If they both cooperate ("Friend"), they share the winnings 50-50. If one cooperates and the other defects ("Foe"), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the payoff matrix is slightly different from the standard one given above, as the payouts for the "both defect" and the "I cooperate and opponent defects" cases are identical. This makes the "both defect" a neutral equilibrium, compared with being a stable equilibrium in standard prisoner's dilemma. If you know your opponent is going to vote "Foe", then your choice does not affect your winnings. In a certain sense, "Friend or Foe" is between "Prisoner's Dilemma" and "Chicken".
The payoff matrix is
Friend or Foe would be useful for someone who wanted to do a real-life analysis of prisoner's dilemma. Notice that you only get to play once, so all the issues involving repeated playing are not present and a "tit for tat" strategy cannot develop.
In Friend or Foe, each player is allowed to make a statement to convince the other of his friend-ishness before both make the secret decision to cooperate or defect. One possible way to 'beat the system' would be for a player to tell his rival, "I am going to choose foe. If you trust me to split the winnings with you later, choose friend. Otherwise, if you choose foe, we both walk away with nothing." A greedier version of this would be "I am going to choose foe. I am going to give you X%, and I'll take (100-X)% of the total prize package. So, take it or leave it, we both get something or we both get nothing." Now, the trick is to minimize X such that the other contestant will still choose friend. Basically, you have to know the threshold at which the utility he gets from watching you get nothing exceeds the utility he gets from the money he stands to win if he just went along.
This approach has not yet been tried in the game; it's possible that the judges might not allow it.
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