Existence of equilibria in repeated games with long-run payoffs

Area of Science:

• Game Theory
• Mathematical Economics
• Decision Theory

Background:

• Nash equilibrium is a cornerstone of strategic game theory, crucial for analyzing rational decision-making.
• Its existence is well-established in finite games but less understood in infinite settings.
• Previous research has limitations in addressing infinitely repeated games with complex payoff structures.

Purpose of the Study:

• To prove the existence of Nash equilibrium in infinitely repeated games.
• To extend the applicability of Nash equilibrium to scenarios with a finite or countably infinite set of players.
• To analyze games where payoffs depend on long-term play rather than specific finite stages.

Main Methods:

• Combining techniques from stochastic games.
• Utilizing methods from alternating-move games with Borel-measurable payoffs.
• Developing novel mathematical approaches for infinite-horizon strategic interactions.

Main Results:

• Demonstrated the existence of Nash equilibrium in all infinitely repeated games under specified conditions.
• Established theoretical foundations for strategic analysis in unbounded game settings.
• Showcased the robustness of Nash equilibrium as a solution concept.

Conclusions:

• The existence of Nash equilibrium is confirmed for a broader class of games, including infinitely repeated ones.
• This finding has significant implications for understanding strategic behavior in complex, long-term interactions.
• The study advances the theoretical framework of game theory with broad applicability.