When Should We Forgive and Forget?

A Lesson from Game Theory

A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman & Hall/CRC Press).

REPEATED GAMES

The Prisoner’s Dilemma exemplifies a one-stage game, where there is no repercussion or continuation after a player chooses to confess or deny and the interplay ends. This is obviously not representative of most real-world scenarios, which often involve multi-stage interactions and decisions that are influenced by previous outcomes. This leads us to the realm of repeated games.

UNCERTAINTY AND REPEATED GAMES

In many real-world situations, it’s often unclear when the game will end, a scenario that can be modelled by rolling two dice after each round in a game. If a double-six is rolled, the game ends. For any other combination, you play another round, with the game continuing until a double-six is rolled. Your total score for the game is the sum of your payoffs.

THE SEVEN PROPOSED STRATEGIES IN REPEATED GAMES

In such games of repeated rounds with no defined end-point, several strategies emerge, which we can model for simplicity by assuming that there are two possible decisions for each player at every stage: to cooperate and ‘split’ in a Golden Balls game-style scenario (be friendly), or to be selfish (‘steal’ in a Golden Balls game-style scenario). In a repeated game, we can model this friendly/hostile choice into seven scenarios:

Always Friendly: This strategy involves being friendly every time.

Always Hostile: As the name suggests, this strategy involves being hostile every time.

Retaliation: This strategy requires you to be friendly as long as your opponent is friendly, but if your opponent is ever hostile, you should turn hostile.

Tit for Tat: This strategy starts with being friendly and then replicating your opponent’s previous move in subsequent rounds.

Random Approach: This strategy suggests tossing a coin for each move and deciding based on the outcome.

Alternate: This strategy alternates between being friendly and hostile.

Fraction: This strategy involves starting friendly and remaining so if the fraction of times your opponent has been friendly is more than half.

DETAILED ANALYSIS OF THE SEVEN PROPOSED STRATEGIES IN REPEATED GAMES

Understanding the dynamics of indefinite repeated games often involves exploring various strategies that players can adopt. Let’s go deeper into the seven strategies outlined:

Always Friendly: Here, the player adopts a cooperative approach, choosing to be friendly in every round. This strategy could lead to high payoffs when interacting with other friendly players but leaves the player vulnerable to exploitation by hostile players.

Always Hostile: This strategy is the opposite of the ‘Always Friendly’ approach. The player chooses to be hostile in every round, aiming to exploit friendly opponents. However, when encountering another hostile player or retaliatory strategies, the outcome can be less favourable.

Retaliation: The player starts friendly and remains so if the opponent does the same. However, if the opponent ever chooses to be hostile, the player shifts to a permanently hostile stance. This strategy can deter hostile behaviour from the opponent but might lead to an endless cycle of hostility if triggered.

Tit for Tat: This strategy is famous for its effectiveness and simplicity. The player starts friendly and then mimics the opponent’s behaviour from the previous round. It rewards cooperation and retaliates against hostility, but it also forgives after retaliation since it reverts to cooperation if the opponent does so.

Random Approach: The player’s choice of action is determined by a coin toss, making this strategy completely unpredictable. While this randomness might confuse the opponent, it also disconnects the player’s actions from the opponent’s behaviour, making it less effective in promoting cooperation.

Alternate: The player alternates between friendly and hostile actions. Again, this does not adapt to the opponent’s behaviour and thus may not be an optimal strategy.

Fraction: This strategy starts friendly and then assesses the overall behaviour of the opponent. If the opponent has been friendly more than half of the time, the player continues to be friendly; otherwise, they turn hostile. This strategy attempts to mirror the opponent’s overall conduct but might be less responsive to recent changes in behaviour.

DOMINANT STRATEGY IN REPEATED GAMES

Although there’s no dominant strategy in such games, simulations of tournaments where each strategy plays against every other have often shown Tit for Tat to emerge victorious. This strategy tends to win overall because it performs well against friendly strategies without being exploitable by hostile ones. The key attributes contributing to the success of Tit for Tat are its niceness, retaliation, forgiveness, and clarity.

A DEEPER DIVE INTO THE SUCCESS OF ‘TIT FOR TAT’ IN REPEATED GAMES

Repeated games offer an intricate canvas for strategic interactions. Although no strategy dominates all others universally in such scenarios, Tit for Tat often proves to be the most successful one overall in a variety of conditions. This effectiveness results from several of its unique characteristics:

Niceness: A Tit for Tat player starts off by being friendly or cooperative. By not being the first to defect, it encourages cooperative behaviour right from the start. It shows goodwill to its opponents, promoting an atmosphere of trust and mutual benefit.

Retaliation: Tit for Tat is not naive; it does not allow exploitation. If an opponent chooses to defect or act hostile, Tit for Tat will retaliate in kind in the next round. This principle of immediate retaliation provides a clear disincentive for opponents to defect, knowing that such behaviour will not go unpunished.

Forgiveness: Despite its willingness to retaliate, Tit for Tat is also forgiving. If an opponent returns to cooperative behaviour after a round of defection, Tit for Tat will reciprocate with cooperation in the next round. This characteristic allows for the possibility of restoring cooperation, even after bouts of hostility.

Clarity: Tit for Tat is an easy strategy to understand and predict. It does not use complicated rules or random behaviour. This clarity makes it easier for opponents to comprehend and adapt to Tit for Tat, encouraging long-term cooperation.

Tit for Tat also provides valuable insights beyond game theory. Its fundamental principles—niceness, retaliation, forgiveness, and clarity—are effective guidelines for a wide range of social, economic, and political interactions. They capture the essence of how to balance cooperation and self-defence, trust and scepticism, and how to promote stable and beneficial relationships even in a world of self-interested individuals.

REAL-WORLD APPLICATIONS AND EXAMPLES

In international relations, strategies like Tit for Tat are evident in diplomatic negotiations and trade agreements, where countries often reciprocate actions (both positive and negative). In the corporate world, companies frequently use a mix of cooperative and competitive strategies based on their competitors’ actions. Environmental agreements often see a blend of these strategies, where nations commit to certain standards and adjust their policies in response to the actions of others.

PSYCHOLOGICAL AND SOCIOLOGICAL ASPECTS

The success of strategies like Tit for Tat in repeated games reflects certain psychological and sociological truths. For instance, the effectiveness of Tit for Tat aligns with psychological principles of reciprocity and fairness, suggesting an innate human tendency to respond to cooperation with cooperation. Sociologically, these strategies highlight the importance of norms and trust in maintaining cooperative relationships within societies.

LIMITATIONS AND CRITICISMS

While strategies like Tit for Tat have been celebrated for their simplicity and effectiveness, they are not without limitations. In complex real-world situations, including those involving multiple players with varying objectives, these strategies can sometimes lead to suboptimal outcomes. For example, Tit for Tat can lead to endless cycles of retaliation in certain situations.

EMERGING TRENDS AND FUTURE RESEARCH

With the advent of artificial intelligence and machine learning, new strategies in repeated games are emerging. These technologies allow for the analysis of vast amounts of data to identify patterns and develop strategies that were previously too complex to model. Research is also focusing on how these strategies can be adapted in the digital world, particularly in areas like cybersecurity, where strategic interactions are frequent and complex.

CONCLUSION: THE EVOLUTION OF COOPERATION

The success of the Tit for Tat strategy in repeated games isn’t an isolated phenomenon. Whether in interpersonal relationships, business negotiations, or global diplomacy, the lessons from Tit for Tat are valuable, timeless, and universal. Robert Axelrod, in his book The Evolution of Cooperation, commends it for its willingness to retaliate tempered by forgiveness. He also commends its clarity and essential niceness. These principles, when applied in real life, can lead to more harmonious and successful interactions, paving the way for positive outcomes in various situations. Even so, the strategy, especially in a complex and interconnected world, is not without its limitations.