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Prisoner's Dilemma

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Imagine you are a criminal and you’ve been arrested along with your co-conspirator. The police take you both into separate rooms where you are unable to communicate with each other. The police do not have enough evidence to convict you on the main charge but do have enough to convict you on a lesser charge. While you are confined in the room, the police offer both of you the same deal—if you testify against your co-conspirator, you will get immunity! What do you do?

How do you approach this situation? Is there a correct decision? This classic thought experiment is called the Prisoner’s Dilemma and it's a classic game in game theory.

In economics, game theory is used to study interactions between rational, competing actors. Any interaction between players with decision points can be modeled as a type of game.

**Game theory** is a field of study concerned with analyzing and creating models for strategic games.

Fundamentally, Prisoner’s Dilemma is a very simple game in which two players make one decision simultaneously and without consulting each other. The scenario is two criminal co-conspirators who are being interrogated by the police in separate rooms and are both being offered the same deal--to confess to the crime and betray their co-conspirator in exchange for total immunity. The decision making in this scenario is driven by weighing the pros and cons of cooperation and competition.

The possible outcomes are as follows:

If you and your co-conspirator betray each other, both of you will serve two years

If you and your co-conspirator both remain silent, both of you will one year (from the lesser charge)

If you betray your co-conspirator but your co-conspirator remains silent, you will go free and your con-conspirator will serve five years

If your con-conspirator betrays you but you remain silent, your co-conspirator will go free and you will serve five years

A key factor here is that mutual cooperation--both remaining silent--results in a better outcome for both players than if both players confess and betray each other.

Prisoner’s Dilemma, and game theory, can provide a model and framework for how to balance cooperation and competition in various settings such as the pricing of goods and production of products. This game lends itself to analyzing oligopoly--where there are several firms in the same industry that have to weigh whether to cooperate or compete with each other.

For a strategic game, players will want to choose the strategy that maximizes their payoff. In order to determine the best option in Prisoner's Dilemma, let’s create a payoff matrix that shows the outcomes of each decision. Person A's options are shown in the left-most column, and Person B's options are shown along the top. Since the payoffs in this game are years in prison, each player wants the smallest number possible.

(Person A, Person B) | Cooperate (remain silent) | Betray (confess) |

Cooperate (remain silent) | (1, 1) | (5,0) |

Betray (confess) | (0,5) | (2, 2) |

A strategy is a complete plan of action, and the **optimal strategy** is the strategy that maximizes the individual's payoff. A strategy takes this form:

** "If my opponent cooperates, then I will _____, and if my opponent betrays, then I will _____." **

What should go in those blanks in order to maximize profit? If you ** knew **what choice the other player was going to make, what would you do? Notice that from Person A's perspective, if Person A takes as given that Person B cooperates, then Person A's choice looks like this. Either cooperate too, and get just one year in jail, or betray your co-conspirator and get immunity! Person A's outcome is better when they betray. Total immunity is better than one year in jail.

Also, if Person A takes as given that Person B betrays, then Person A's choice is to cooperate and get five years in jail because of your opponent's testimony against you, or betray them as well and get only two years in jail. Again, Person A's outcome is better when they betray. Just two years is better than serving five years. The same is true in reverse for Person B.

Now you know that the optimal strategy is:

**"If my opponent cooperates, then I will betray, and if my opponent betrays, then I will betray." **

The optimal strategy involves choosing the same action in both cases. This means that the choice of betray is a **dominant strategy** in the Prisoner's Dilemma.

A player has a **dominant strategy **when there is one action that will always result in a better outcome regardless of what the other person chooses.

In a game like this where both players have a **dominant strategy**, we can expect that the only rational outcome of the game is where both players play their dominant strategy.

The Prisoner’s Dilemma illustrates a counterintuitive concept. Since both players betray, the outcome is worse for both players than if they had cooperated, even though both actors used their optimal strategy. If person A and person B both betray, they both spend two years in prison. If they had both cooperated, they would each spent just one year in prison. However, it is not rational to choose to cooperate.

Collectively, when they both spend two years in prison, that's a total of four prison years. If both person A and person B chose to remain silent, the total number of years in prison would be just 2. So even the total social outcome is worse than if they had both cooperated.

This is because each person is only looking out for their own interests. When acting selfishly, the incentive to betray is strong enough to prevent cooperation. Consider a version of the prisoner’s dilemma where Person A knows what person B will choose to do. What is the optimal choice now?

The answer is still to betray. If Person A knows that Person B will betray them, the choice that will lead to less time in prison for Person A is to betray as well. If Person A knows that Person B will remain silent, the choice that will lead to less time in prison for Person A is still to betray.

In game theory, the outcome of a game is a Nash Equilibrium if no player can increase their payoff by changing strategies, assuming other players’ strategies remain the same. The term was created by John Nash, a mathematician and important figure in the study of game theory.

**Nash equilibrium **is when no player can improve their personal payoff by changing strategies, taking all other players' strategies as given.

A shortcut to determining the Nash equilibrium in a game is for all the players to reveal their choice to the other players. Knowing the other players' choices, do you want to change your own choice? If so, then with your new choice, do the other players want to change their choice? Once no player has an incentive to change their strategy, the Nash equilibrium is found.

Nash equilibrium and dominant strategy are terms that are often discussed together. Nash equilibrium describes an outcome in which no player can increase their individual payoff by changing strategies, assuming other players’ strategies remain the same. This outcome is "stable" in the sense that no one wants to change.

A dominant strategy is an action that always gives a higher payoff regardless of what the other player chooses. Not all games have a dominant strategy. And not all players in the game have a dominant strategy.

If a game has a dominant strategy for all players, then that outcome is automatically a Nash Equilibrium.

A game than does not have a dominant strategy can still have a Nash Equilibrium. In fact, not many games happen to have a dominant strategy, but *almost *every game has a Nash equilibrium. It is possible for a game to have multiple Nash equilibria, and it is possible to have none at all. However, it is common to have at least one outcome that can be considered stable in the sense of Nash equilibrium--that no one wants to change, given everyone else's strategy choices.

Thus, a Nash equilibrium may involve one or more players playing a dominant strategy, but not all Nash equilibria involve a dominant strategy. A game with no dominant strategy at all will *probably *still have a Nash equilibrium.

The Prisoner’s Dilemma is commonly used as a model for analyzing competing firms in a market. For example, consider two rival firms that are trying to decide how to price their competing products. If a company cuts prices in order to capture more market share, it will force the other company to do the same. In this example, cutting prices is analogous to betraying a co-conspirator.

To fully analyze this example, let’s construct another payoff matrix. The payoffs are the profit generating by sales under the given strategies. Both firms want to maximize their own profits.

(Company A, Company B) | Cooperate(keep price high) | Betray (reduce price) |

Cooperate (keep price high) | (500, 500) | (0, 750) |

Betray (reduce price) | (750, 0) | (250, 250) |

This game is a version of Prisoner’s Dilemma because for any given decision made by one company, it is in the best interest of the other company to reduce their price.

Let’s look at some more examples of the Prisoner’s Dilemma in economics. A popular, real-world example of the Prisoner’s Dilemma in business is advertising. The effectiveness of a company’s advertising efforts is partially determined by the advertising efforts of a rival company. If two rival companies choose to spend equal amounts of money on advertising, the advertisements from each company are canceled out by the other. In this scenario, both companies should reduce advertising spending to save money.

However, each company has a choice: they can “cooperate” and reduce advertising spending or they can “betray” and maintain or increase advertising which will provide extra profits through increased sales. This is similar to the Prisoner’s Dilemma because the total social benefit would actually be highest if both companies chose to cooperate and reduce advertising spending, but individual incentives go against doing so.

Prisoner’s Dilemma describes a game where two criminals have to decide between (1) cooperating and staying silent or (2) confessing and betraying each other.

This game shows that even though cooperation among both players produces both the best collective outcome and better individual outcomes than if they both betray, their individual incentives prevent cooperation.

A dominant strategy is one that leads to a better outcome when taking as given any particular strategy that the other player may choose.

Betraying the other person is a dominant strategy in the Prisoner’s Dilemma.

Nash equilibrium is when no player can improve their payoff by changing their strategy, taking all other players’ strategies as given.

A quick way to determine if a game is in Nash equilibrium is for all players to reveal their strategy. If, with this information, no player has an incentive to change their strategy, the game is said to be in Nash equilibrium.

In a game where all players have a dominant strategy, the outcome where all are playing their dominant strategy is a Nash Equilibrium.

Oligopoly pricing is an example of Prisoner’s Dilemma. A company can choose to “cooperate” by keeping the price high, or they can "betray" by reducing their price and stealing the whole market. The collective profit-maximizing outcome is for both companies to keep their prices high, but their individual incentive is to betray each other by reducing their price and stealing the market.

** knew** their opponents' choices, is a Nash equilibrium, and there can be more than one Nash equilibrium. However, in the Prisoner's Dilemma both players have a dominant strategy. Any outcome in which both players are playing a dominant strategy is automatically a Nash equilibrium. In the Prisoner's Dilemma, it happens that this outcome (betray, betray) is only Nash Equilibrium of the game.

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