Mathematics

The Crazy of Game Theory: Prisoner’s Dilemma

Have you ever come across the concept of the Prisoner’s Dilemma? It’s had a profound impact on various fields such as modern mathematics, economics, biology, and politics. There exist several versions of the Prisoner’s Dilemma, but I’d like to focus on a particular variant: the non-iterated version.

Picture this: two individuals find themselves apprehended by the authorities for a specific crime. They’re held in separate rooms for questioning. Now, they’re presented with a critical decision. They can either both admit to the crime, leading to a reduced sentence as a Reward for Cooperation, or they can both point fingers at each other, resulting in a severe prison term as Punishment for Defection. But here’s where it gets truly intriguing. If one chooses to accuse while the other chooses to confess, the accuser goes free, unburdened by prison, while the confessor faces the harshest penalty.

Now, when you look at the table (bearing in mind that the assigned points are arbitrary, merely emphasizing the severity of the consequences — a 0 implies an extremely unfavorable outcome, whereas a 5 signifies the best possible scenario), imagine yourself as the decision-maker in the first column, and consider the choices of the other person laid out in the first row. It might be tempting to contemplate accusing (defecting) because it yields the highest potential reward. However, this hinges entirely on what the other person opts for. If the other person is thinking along similar lines (a reasonable assumption, given they’re likely evaluating the situation logically), they’re bound to conclude that defecting is their optimal strategy as well. This implies that you’re almost certain to face a significant penalty for defecting (1 point).

The question then becomes: can you somehow influence the other person to choose cooperation? After all, if both individuals cooperate, the outcome is better for both (3 points). The conundrum arises from the fact that you’re unable to communicate or persuade each other. Thus, the entire decision-making process rests solely on speculation regarding the other person’s probable course of action.

So, what’s the most rational course of action here? Do you opt to defect (accuse) or cooperate (confess to the guilt)? While both choices offer equal benefits if both parties cooperate, this isn’t the equilibrium in this case. Let’s break this down logically. This is because, on average, you’re more likely to face a harsher penalty if you choose to cooperate. If you choose to cooperate and the other person chooses to defect, you receive 0 points. If both choose to cooperate, you obtain an average of 3 points. Consequently, cooperating averages to 1.5 points.

In contrast, if you opt to defect and the other person also chooses to defect, you receive 1 point. If you choose to defect and the other person chooses to cooperate, you gain a whopping 5 points. On average, accusing results in 3 points. This means that, on average, accusing leads to a higher payoff.

Hence, in this non-iterated version, it appears that choosing to accuse is the more advantageous option. However, in an iterated version where knowledge of past decisions of the opponent is available, being cooperative can yield benefits. We’ll delve deeper into this in the future. For now, take a moment to reflect and savor the world of mathematics!

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