Iterated Prisoner's Dilemma

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tags: - colorclass/evolutionary game theory ---see also: - Morality As Cooperation - Evolutionary Game Theory - Cooperation

Iterated Prisoner’s Dilemma (IPD)

The Iterated Prisoner’s Dilemma (IPD) is an extension of the classic Prisoner’s Dilemma (PD), where the game is played repeatedly by the same participants. This setup allows for the development of strategies that consider the history of interactions, leading to insights into how cooperation can evolve and be sustained over time.

Classic Prisoner’s Dilemma

To understand the IPD, it’s essential first to grasp the basics of the classic Prisoner’s Dilemma:

- Setup: Two players can either “Cooperate” (C) or “Defect” (D). - Payoff Matrix:

$$\begin{array}{ccc}& C& D\\ C& (R,R)& (S,T)\\ D& (T,S)& (P,P)\end{array}$$

where: - $R$ (Reward) is the payoff when both players cooperate. - $T$ (Temptation) is the payoff for defecting while the other cooperates. - $S$ (Sucker’s payoff) is the payoff for cooperating while the other defects. - $P$ (Punishment) is the payoff for mutual defection.

Typical values: $T>R>P>S$, ensuring defection is a dominant strategy in a single-shot game.

Iterated Prisoner’s Dilemma

The IPD extends the single-shot PD by having the same players repeatedly play the game, allowing strategies to evolve based on previous interactions.

Key Concepts

1. Repetition: The game is played for an indefinite number of rounds, or with a known probability of continuation after each round. 2. History-Dependent Strategies: Players can adopt strategies that depend on the entire history of play, not just the current round.

Strategies in IPD

Several well-known strategies illustrate the potential for cooperation in the IPD:

1. Tit-for-Tat (TFT) - Description: Start with cooperation, then mimic the opponent’s previous move. - Strengths: Simple and promotes mutual cooperation. It retaliates against defection but forgives after cooperation. - Weaknesses: Can be exploited by strategies that defect occasionally.

2. Tit-for-Two-Tats - Description: Similar to TFT but only defects if the opponent defects twice in a row. - Strengths: More forgiving, reducing the likelihood of mutual defection spirals. - Weaknesses: Can be exploited more easily than TFT by occasional defectors.

3. Grim Trigger - Description: Start with cooperation and continue to cooperate unless the opponent defects once, after which defect forever. - Strengths: Strong deterrent against defection. - Weaknesses: Unforgiving, can lead to permanent mutual defection after a single mistake.

4. Random Strategies - Description: Players mix cooperation and defection randomly according to a predetermined probability. - Strengths: Unpredictable and can sometimes exploit fixed strategies. - Weaknesses: Generally less effective in fostering sustained cooperation.

Evolutionary Dynamics

1. Replicator Dynamics: Models how the frequency of strategies changes over time in a population based on their success.

$${\dot{x}}_i=x_i\left(f_i-\bar{f}\right)$$

where $x_i$ is the frequency of strategy $i$, $f_i$ is the fitness (payoff) of strategy $i$, and $\bar{f}$ is the average fitness of the population.

2. Strategy Evolution: In populations, successful strategies that foster cooperation can become more prevalent. Strategies like TFT can thrive in environments where players interact repeatedly.

Conditions for Cooperation

1. Shadow of the Future: The longer the expected duration of interaction (i.e., higher probability of continuation), the more likely cooperation is to emerge. Players value future payoffs more, making retaliation for defection more significant. 2. Communication and Reputation: When players can communicate or have reputations, cooperative strategies are more likely to emerge and be maintained. 3. Population Structure: In structured populations (e.g., networks or groups), clusters of cooperators can protect themselves from defectors, promoting the stability of cooperation.

Implications of IPD

1. Understanding Social Behavior: IPD models help explain the emergence and stability of cooperation in human and animal societies. They show how trust, reciprocity, and punishment can sustain cooperative behaviors. 2. Designing Institutions: Insights from IPD can inform the design of policies and institutions that promote cooperation, such as creating environments where interactions are repeated and reputation matters. 3. Artificial Intelligence: In AI and multi-agent systems, IPD strategies are used to develop cooperative agents that can interact beneficially with humans and other agents in complex environments.

Conclusion

The Iterated Prisoner’s Dilemma provides a powerful framework for understanding the dynamics of cooperation. By allowing for repeated interactions, it reveals how cooperation can evolve and be maintained even in the presence of incentives to defect. The study of IPD has profound implications for various fields, from evolutionary biology and economics to artificial intelligence and social sciences.

> relate IPD to MAC

Relating Iterated Prisoner’s Dilemma (IPD) to Morality as Cooperation (MAC)

The Iterated Prisoner’s Dilemma (IPD) provides a robust framework for understanding the dynamics of cooperation, which is central to the Morality as Cooperation (MAC) theory. By examining how individuals can develop cooperative strategies over repeated interactions, the IPD helps illustrate the evolutionary and strategic foundations of moral behavior as posited by the MAC theory. Here’s how the IPD relates to and supports the MAC framework:

1. Evolution of Cooperation

MAC Theory: Morality evolved as a set of behaviors and norms that solve cooperation problems. These norms are adaptive solutions to recurring social dilemmas faced by our ancestors.

IPD Insight: The IPD demonstrates how cooperation can emerge and be sustained over time through repeated interactions. In a single-shot Prisoner’s Dilemma, defection is the dominant strategy, but in the IPD, strategies like Tit-for-Tat show that cooperation can be an evolutionarily stable strategy when interactions are repeated indefinitely.

- Example: In an IPD setup, players who adopt a cooperative strategy like Tit-for-Tat can thrive, as they benefit from mutual cooperation and retaliate against defection. This mirrors real-world scenarios where individuals who cooperate, punish defectors, and forgive occasional mistakes tend to foster stable cooperative relationships.

2. Mechanisms of Moral Behavior

MAC Theory: Different types of cooperation (e.g., kin selection, reciprocal altruism, indirect reciprocity) give rise to various moral norms.

IPD Insight: The IPD provides concrete mechanisms through which these types of cooperation can evolve and be maintained:

- Reciprocal Altruism: The IPD shows how strategies based on reciprocity (e.g., Tit-for-Tat) can support mutual cooperation. Individuals remember past interactions and condition their future behavior on the cooperation or defection of others, aligning with the MAC concept of reciprocal altruism.

- Moral Norms: Norms of reciprocity and fairness arise from this mechanism. For instance, the moral expectation to “return favors” aligns with reciprocal strategies observed in IPD.

- Indirect Reciprocity: IPD with reputation mechanisms extends the basic model to include scenarios where individuals cooperate based on others’ reputations, not just direct interactions.

- Moral Norms: Norms around honesty, reputation, and trustworthiness develop, as individuals are more likely to cooperate with those who have a history of cooperative behavior.

3. Sustaining Cooperation and Moral Norms

MAC Theory: Moral norms are culturally transmitted and enforced to sustain cooperation within societies.

IPD Insight: The IPD illustrates how cooperation can be stabilized through repeated interactions, reputation, and punishment:

- Reputation and Indirect Reciprocity: When players consider reputation, they are more likely to cooperate to maintain a good standing, which can be extended to understanding the role of reputation in human societies.

- Moral Norms: Societal norms that reward good behavior and punish bad behavior (e.g., through social approval or ostracism) are analogous to the enforcement mechanisms in IPD.

- Punishment and Altruistic Punishment: The IPD can include strategies where players punish defectors, even at a cost to themselves. This altruistic punishment helps sustain cooperation by deterring defection.

- Moral Norms: Norms that support punishment for wrongdoing (e.g., legal systems, social sanctions) ensure that cooperation remains beneficial in the long term.

4. Cross-Cultural Consistency and Variation

MAC Theory: While there are universal moral norms based on common cooperation problems, cultural variations arise from different social and environmental contexts.

IPD Insight: The adaptability of IPD strategies to different contexts reflects the cultural variability observed in moral norms:

- Adaptation to Context: In different environments, strategies that promote cooperation can vary. For example, harsher environments might favor more stringent punishment strategies, while more stable environments might support more forgiving strategies.

- Moral Norms: This adaptability explains why certain moral norms are universal (e.g., prohibitions against murder) while others vary (e.g., norms around sharing resources).

Conclusion

The Iterated Prisoner’s Dilemma (IPD) and the Morality as Cooperation (MAC) theory are closely related. The IPD provides a formal framework that illustrates how cooperative behaviors and strategies can evolve and stabilize through repeated interactions, directly supporting the MAC theory’s assertion that morality evolved to solve cooperation problems. The mechanisms of reciprocity, reputation, and punishment in the IPD align with the development and enforcement of moral norms in human societies. Thus, the IPD not only enhances our understanding of the evolutionary basis of morality but also offers practical insights into sustaining cooperation and moral behavior in diverse cultural contexts.