An equilibrium of a multi-stage game in which players are informed about the strategy choices of opponents at previous decision nodes. Formally, all past play is common knowledge at the start of each stage. In a closed-loop equilibrium, the information structure allows the choice of a player at a decision node to depend on how the opponents have played the game up to that point. The strategy choices therefore evolve as progress is made through the stages of the game. In policy games, this is often referred to as the case of discretion. It should be contrasted to an open-loop equilibrium, which occurs when the past choices of other players are not observed.
Historical Context
Closed-loop equilibrium has its roots in the broader field of game theory, which was formalized by mathematicians such as John von Neumann and Oskar Morgenstern in the mid-20th century. The concept has since been extended to various strategic interactions, particularly in economics, where it is used to model decision-making processes in dynamic environments.
Types and Categories
Types
- Perfect Information Closed-Loop Equilibrium: Every player is fully informed of all previous actions by all other players at each decision node.
- Imperfect Information Closed-Loop Equilibrium: Players have incomplete information about some past actions but have enough information to make informed decisions.
Categories
- Policy Games: Decision-making frameworks where policymakers adapt their strategies based on the evolving actions of other policymakers.
- Economic Models: Used to study interactions in markets where firms adjust their strategies over time based on competitors’ actions.
Key Events and Developments
- 1944: Publication of “Theory of Games and Economic Behavior” by John von Neumann and Oskar Morgenstern.
- 1950s: Nash Equilibrium introduced by John Nash.
- 1980s: Increased focus on dynamic games and their applications in economics.
Detailed Explanations
Closed-loop equilibrium considers that each player makes decisions based on the complete history of the game. This is formally represented as follows:
- Let \( \Gamma = {N, H, {A_i}{i \in N}, {u_i}{i \in N} } \) be a multi-stage game where:
- \( N \) is the set of players.
- \( H \) is the set of all possible histories of the game.
- \( A_i \) is the set of actions available to player \( i \).
- \( u_i \) is the utility function of player \( i \).
At each stage \( t \), each player \( i \) chooses an action \( a_i^t \in A_i \) based on the observed history up to that point, \( h_t \in H \).
Mathematical Models
The strategy for each player \( i \) in a closed-loop equilibrium can be denoted as \( s_i(h_t) \), which maps from the history \( h_t \) to an action \( a_i \).
Charts and Diagrams
graph TD;
Start -->|History| Decision_Node_1;
Decision_Node_1 -->|Choice A| Decision_Node_2A;
Decision_Node_1 -->|Choice B| Decision_Node_2B;
Decision_Node_2A -->|Outcome 1| End;
Decision_Node_2B -->|Outcome 2| End;
Importance and Applicability
Closed-loop equilibria are crucial in fields where decisions are sequential and interdependent:
- Economics: Models competition in dynamic markets.
- Finance: Analyzes trading strategies based on historical data.
- Political Science: Studies strategic interactions in international relations.
Examples
- Cournot Duopoly: Firms adjust their output based on previous period outputs of their competitors.
- Monetary Policy: Central banks adjust policies based on past economic indicators.
Considerations
- Information Availability: Accurate and comprehensive history data is essential.
- Computational Complexity: Calculating closed-loop equilibria can be complex and computationally intensive.
Related Terms
- Open-Loop Equilibrium: Decisions are made without observing past actions.
- Dynamic Game: A game with sequential decision-making.
- Nash Equilibrium: A strategy profile where no player can benefit by unilaterally changing their strategy.
Comparisons
| Feature | Closed-Loop Equilibrium | Open-Loop Equilibrium |
|---|---|---|
| Information Structure | History is known | History is not known |
| Adaptability | High | Low |
| Complexity | More Complex | Less Complex |
Interesting Facts
- Dynamic Strategy Evolution: In a closed-loop equilibrium, strategies can evolve dynamically, reflecting more realistic scenarios in competitive environments.
Inspirational Stories
- Nash’s Insights: John Nash’s work on equilibrium concepts provided the foundation for understanding strategic interactions, earning him the Nobel Prize in Economic Sciences in 1994.
Famous Quotes
- “The best way to predict the future is to create it.” — Peter Drucker
Proverbs and Clichés
- “Actions speak louder than words.”
Expressions, Jargon, and Slang
- Backward Induction: A method used in game theory to solve multi-stage games by analyzing decisions from the end of the game to the beginning.
- Strategy Profile: A combination of strategies chosen by all players in the game.
FAQs
What is a closed-loop equilibrium?
How does closed-loop equilibrium differ from open-loop equilibrium?
Where is closed-loop equilibrium used?
References
- von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
- Nash, J. (1950). Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences.
- Fudenberg, D., & Tirole, J. (1991). Game Theory. MIT Press.
Summary
Closed-loop equilibrium is an essential concept in game theory, providing a framework for understanding dynamic and strategic interactions in multi-stage games. By considering the entire history of the game, players can make informed decisions that lead to optimal outcomes. This concept has profound implications in various fields, including economics, finance, and policy-making, where adaptive and strategic decision-making is crucial.
