For analyzing strategic decision-making, game theory is a potent analytical instrument of economics studies. The prisoner's dilemma is among the most well-known examples of game theory. This is a game in which two suspects are apprehended and each is offered a deal: if one confesses and the other does not, the confessor is released and the other is given a lengthy prison sentence. If both parties confess, their sentences are reduced. If neither party admits guilt, they will both receive moderate sentences. We will discuss how to solve prisoner's dilemma in game theory assignments in this blog.

## 1. Identify the payoffs and strategies

The prisoner's dilemma is a classic problem in game theory in which two participants must decide whether to cooperate or defect. In this game, both participants have the option of cooperating or deviating, with the outcomes dependent on their decisions. Both players receive a relatively large payoff if they cooperate. However, if one player defected and the other player cooperated, the defector would receive the highest reward and the cooperator would receive the lowest. If both competitors defect, the payoff is relatively low for both.

The first step in solving prisoner's dilemma game theory problems for assignments is to identify the payoffs and strategies. Payoffs are the rewards or penalties associated with each possible outcome, whereas strategies are the actions that each participant can take.

For instance, suppose you are presented with a prisoner's dilemma problem in which two companies, A and B, must decide whether to collude or compete. If the two businesses collude, each will gain $5 million in profit. If one company colludes while the other competes, the colluder will earn $10 million while the competitor will only earn $1 million. If both businesses engage in competition, they will each gain $2 million in profit.

**In this scenario, the payoffs are:
**

- If both collude: A and B each earn $5 million
- If A colludes while B competes: A earns $10 million, B earns $1 million
- If A competes while B colludes: A earns $1 million, B earns $10 million
- If both compete: A and B each earn $2 million

**The strategies are:
**

- Collude
- Compete

## 2. Construct the payoff matrix

After identifying the payoffs and strategies for both players in Prisoner's Dilemma, the next stage is to create a payoff matrix. The payoff matrix is a table that displays the payoffs for each participant based on their selected strategy and the opponent's selected strategy.

**In a Prisoner's Dilemma game, the payoff matrix usually takes the following form:
**

Player A / Player B | Player A / Player B Cooperate | Defect |

Cooperate | R_{A}, R_{B} | S_{A}, T_{B} |

Defect | T_{A}, S_{B} | P_{A}, P_{B} |

In this matrix, R_{A/sub> represents the incentive for player A if both players cooperate, whereas SA represents the sucker's payoff for player A if player B defects. TA is the temptation payoff for player A if they defect while player B cooperates, whereas PA is the punishment payoff for player A if both players defect. The same notation pertains to the payouts for player B.
}

To complete the matrix, you must know the exact values of R, S, T, and P. These values can vary based on the particular scenario being approximated, but there are some frequently used values. For example, in the classic Prisoner's Dilemma game, the values are:

- R = 3 (reward for cooperation)
- S = 0 (sucker's payoff)
- T = 5 (temptation payoff)
- P = 1 (punishment payoff)

Once the appropriate values have been entered into the matrix, it can be used to analyze the game and determine the optimal strategy for each participant.

## 3. Identify dominant and dominated strategies

The next stage, following the construction of the payoff matrix, is to identify the dominant and dominated strategies.

A dominant strategy is one that is always optimal for a participant, regardless of the opponent's choice of strategy. In other words, a player will always select a dominant strategy because it offers the highest return regardless of what the other player does.

A subjugated strategy, on the other hand, is one that is always inferior to another strategy for a player, regardless of the strategy chosen by the opponent. In other words, regardless of what the other player does, a player will never choose a dominant strategy because it offers a lower payoff than another strategy.

To determine which strategies are dominant and which are dominant, we can compare the player payoffs associated with each strategy. If one strategy provides a player with a greater payoff regardless of what the other player does, then the former strategy is dominant and the latter strategy is dominated.

**For example,** let's consider the prisoner's dilemma game where Player A has two strategies, cooperate (C) and defect (D), and Player B also has two strategies, cooperate (C) and defect (D). The payoff matrix:

Player B: C | Player B: D | |

Player A: C | (3, 3) | (0, 5) |

Player A: D | (5, 0) | (1, 1) |

To determine Player A's dominant and subordinate strategies, we can compare the payoffs associated with each strategy. If Player B chooses to cooperate (C), the compensation for Player A is greater if they defect (D) rather than cooperate (C). This indicates that Player A's prevalent strategy is to defect (D), whereas Player B's is to cooperate (C).

Importantly, not all games have dominant or dominated strategies. In certain instances, there may be multiple optimal strategies or none at all. Therefore, it is essential to analyze the payoffs and strategies thoroughly to determine the most effective course of action.

## 4. Determine the Nash balance

Following the identification of dominant and dominated strategies, the Nash equilibrium must be determined. The Nash equilibrium is a solution concept in game theory that denotes a stable state of the game in which no player has an incentive to unilaterally deviate from their chosen strategy, given the strategy choices of the other players. In other words, it is a state in which each player employs the optimal strategy given the strategies of the other participants. The Nash equilibrium is a fundamental concept in game theory and is applied to the analysis of a wide variety of games.

To determine the Nash equilibrium, we must examine the payoffs of each player given their opponent's strategy. We begin by analyzing each player's strategy to determine if it is the optimal response to the opponent's. If so, we have discovered a potential Nash equilibrium. This procedure is repeated until a stable state is reached in which no participant has the incentive to deviate from their strategy.

Consider the game of prisoner's dilemma as an illustration. Assume that Players A and B have been apprehended by the police and must determine whether to cooperate or defect. If both participants cooperate, they each receive a relatively light sentence and a payout of three dollars. If both participants defect, they each receive a severe punishment and a payout of 1. Nonetheless, if one player cooperates while the other deviates, the defector receives a very light sentence while the cooperator receives a very heavy sentence, resulting in payoffs of 5 and 0 for the defector and the cooperator, respectively.

To determine the Nash equilibrium, we must examine the optimal response of each participant given the opponent's strategy. If Player A cooperates, then the optimal response for Player B is to defect, as this results in a greater compensation of 5 points. Likewise, if Player B cooperates, the optimal response for Player A is to secede, as this results in a greater payoff of 5 points. The Nash equilibrium of this game is therefore for both players to defect, resulting in payoffs of 1 for each player.

In conclusion, determining the Nash equilibrium requires analyzing each player's strategy to determine if it is the optimal response to the opponent's strategy. By identifying the Nash equilibrium, we can determine the optimal strategy for each participant and predict the outcome of the game.

## 5. Consider variations and extensions

There are numerous variations and extensions of prisoner's dilemma problems that you may encounter in your game theory assignments. Common examples include:

In this version of the prisoner's dilemma, the two participants play the same game repeatedly. This allows them to establish a reputation and consider the opponent's prior moves. In this scenario, strategies such as "tit-for-tat" (where a player copies their opponent's previous action) can be effective.

**Stochastic prisoner's dilemma:**In this variant of the game, the payoffs are subject to some randomness. For instance, a player may have a small possibility of receiving a larger payout if they choose the "defect" strategy. This adds an element of uncertainty that can alter the optimal course of action.**Public goods game:**a generalization of the prisoner's dilemma in which each player has the option of contributing to a public benefit (such as a charity or a public park). If sufficient participants contribute, everyone benefits. However, if too few participants contribute, the public benefit is not created, and everyone suffers the consequences. This game can be used to demonstrate cooperation and the tragedy of the commons.

In this game, one player is given a quantity of money and must propose a method for dividing it with the other player. The proposal is either accepted or rejected by the other participant. If they approve, the proposed division of the funds is implemented. If they decline, neither player will receive anything. This exercise can be used to illustrate the importance of equity and bargaining power in the decision-making process.

When confronted with variations and extensions of the prisoner's dilemma, it is crucial to thoroughly consider the differences and how they may impact the analysis. The fundamental principles of identifying payoffs, strategies, and Nash equilibrium continue to apply, but the optimal strategy may vary based on the particulars of the game. In addition, it is essential to specify which variation of the game you are analyzing and how it differs from the standard prisoner's dilemma in your analysis.

## 6. the concepts of real-world scenarios

The prisoner's dilemma is a classic game theory problem used to analyze situations where the outcome is contingent on the decisions of multiple participants. Although the game's structure is straightforward, it can be difficult to implement the concepts to real-world situations. However, it is possible to obtain insights that can inform strategic decision-making by understanding the fundamental principles and applying them to real-world scenarios.

One common application of prisoner's dilemma in the actual world is in business and economics. Consider, for instance, two companies competing for market share in the same industry. If one company chooses to lower prices, it may gain a short-term advantage, but it could also spark a price war that is detrimental to both companies in the long run. If both businesses work together to maintain higher prices, they may both realize increased profits. However, if one firm deviates and reduces prices while the other maintains higher prices, the firm that deviates may gain a significant competitive advantage.

International relations is an additional application of the prisoner's dilemma. Imagine two nations engaged in disarmament negotiations. Each nation has the option of cooperating and disarming or deviating and keeping their armaments. If the two nations work together, they may be able to reduce the danger of nuclear war and enhance global security. Nonetheless, if one country deviates and maintains its weapons while the other disarms, the country that deviates may acquire a significant strategic advantage over the other.

When implementing the concepts of the prisoner's dilemma to real-world scenarios, it is essential to consider the situation's specific context. Identifying the payoffs and strategies of the parties involved, as well as any external factors that may affect the outcome of the game, is crucial. By constructing a payoff matrix and analyzing dominant and dominated strategies, it is possible to determine the Nash equilibrium and comprehend each player's motivations.

In addition, variations and extensions of the prisoner's dilemma game must be considered. The iterated prisoner's dilemma, for instance, entails playing the game multiple times, which can result in different outcomes than a single-play game. The assurance game is a variation in which both players prefer to cooperate but may defect if they are uncertain about the other player's intentions.

In the end, applying prisoner's dilemma concepts to real-world scenarios requires meticulous analysis and strategic thought. By understanding the game's rules and considering the situation's specific context, it is possible to obtain insights that can aid in decision-making and lead to better outcomes.

## Conclusion

Solving prisoner's dilemma game theory problems for assignments requires identifying payoffs and strategies, constructing the payoff matrix, identifying dominant and dominated strategies, determining the Nash equilibrium, considering variations and extensions, and applying the concepts to real-world situations. By mastering these skills, you can become a more effective problem solver in game theory and achieve greater success.