Overview
Randomly Terminated Games are a subset of repeated games, an important topic in game theory which explores situations where players interact with each other through a sequence of plays or "games." In the specific context of randomly terminated games, the sequence does not have a predetermined endpoint. Instead, the game continues until a random event or process leads to termination. This makes the duration of the game uncertain, creating a rich area for strategic interactions and analysis.
Characteristics
These games maintain the basic premise of repeated interactions where the same game (stage game) is played over multiple periods. However, what distinguishes randomly terminated games from their finite and infinite counterparts is that players do not know when the game will end. The end could come as the result of a random draw each period or the occurrence of a certain stochastic event. The probability of continuation is typically common knowledge among the players.
Strategic Implications
In randomly terminated games, the strategic considerations of players are influenced by the persistent threat of the game ending unexpectedly. As a result, they have to weigh their short-term actions against the potential benefits of maintaining particular strategies over a potentially prolonged, yet unknown, period.
The typical goals, like cooperating, cheating, or punishing, are affected by the shadow of the future, but the shadow is highly variable and uncertain. This uncertainty can lead to different strategic behavior compared to games with a known endpoint.
Analysis of Equilibria
Analyzing such games involves understanding the equilibria which are sustainable when players must take into account the random termination. Key concepts like Nash equilibrium, subgame perfect equilibrium, and other equilibrium refinements are used within this context, but the analysis may be complicated by the randomness of the game's duration.
Relation to Real-world Situations
Many real-world interactions can be modeled as randomly terminated games. Relationships in business, like contracting with the possibility of bankruptcy, or international treaties with uncertain political climates, mimic the structure of a randomly terminated game.
In such scenarios, entities might behave differently than they would in a game with a known time horizon. For example, the possibility of a sudden end to a business due to external economic factors might influence risk-taking behavior or investment in reputation.
Conclusion
Randomly Terminated Games represent a fascinating area within the study of repeated games, where uncertainty and randomness play a significant role in shaping strategies and outcomes. They allow for more nuanced models that better reflect many unpredictable aspects of strategic interactions in the real world. The study of these games is crucial to develop a more comprehensive understanding of how individuals and organizations behave over time when faced with the unknown.