Dynamic game

What Is Dynamic Game?

A dynamic game is a model within game theory where players make decisions over multiple stages or periods, and later actions can be influenced by previous actions taken by themselves or other players. Unlike a single, simultaneous decision, the sequential nature of dynamic games introduces the element of time and the ability for players to observe and react to prior moves, leading to complex strategic interaction. This type of game falls under the broader financial category of game theory, which is a framework for analyzing situations involving interdependent choices among rational decision-making agents. The outcomes in a dynamic game often depend on the entire sequence of choices made over time, not just isolated decisions.

History and Origin

The foundational concepts of game theory, which underpin the study of dynamic games, can be traced back to the work of Hungarian-born mathematician John von Neumann and German-born economist Oskar Morgenstern. Their seminal book, "Theory of Games and Economic Behavior," published by Princeton University Press in 1944, established game theory as a distinct field of mathematical inquiry applicable to economics and social organization.⁴ This groundbreaking work introduced formal models for analyzing strategic situations, initially focusing on two-person zero-sum games.³

While the initial focus was on static games, the framework laid by von Neumann and Morgenstern paved the way for later developments in understanding sequential decision-making. The rigorous mathematical principles outlined in their book provided the groundwork for subsequent researchers to explore more complex, multi-stage interactions that define a dynamic game.² The field expanded significantly in the 1950s, with further refinements and applications by other mathematicians and economists, moving beyond simple conflicts of interest to encompass a wider range of behavioral relations and the complexities of interdependent choices over time.

Key Takeaways

• Sequential Decision-Making: In a dynamic game, players make decisions over a sequence of periods, with later decisions often informed by observations of earlier actions.
• Information Asymmetry and Observation: The ability to observe opponents' prior moves is a crucial characteristic, allowing players to adapt their strategies.
• Time-Dependent Outcomes: The final outcome of a dynamic game is a function of the entire path of decisions taken over time, not just a single set of simultaneous choices.
• Equilibrium Refinements: Concepts like subgame perfect equilibrium and backward induction are essential tools for analyzing dynamic games, as they account for credible threats and promises over time.
• Real-World Applicability: Dynamic games are used to model competitive scenarios in economics, business, politics, and military strategy where timing and responses are critical.

Formula and Calculation

While there isn't a single universal "formula" for a dynamic game, the analysis often involves defining the game in an extensive form. This typically uses a game tree, which is a diagram showing the sequence of moves, the information available to players at each decision point, and the resulting payoff matrix for all players at the end of each possible sequence of moves.

The primary solution concept for many dynamic games is the subgame perfect equilibrium (SPE), which is found using a technique called backward induction. Backward induction involves solving the game from its end backward to its beginning:

1. Identify the last decisions: For every subgame (a part of the larger game that starts at a single decision node and includes all subsequent nodes), determine the optimal action for the player making the last decision, assuming they are rational.
2. Substitute payoffs: Replace the subgame with the payoffs resulting from these optimal last decisions.
3. Iterate: Move to the preceding decision nodes and repeat the process, with each player making their optimal choice given the optimal choices expected from subsequent players.

This iterative process helps identify a set of strategies such that no player can improve their payoff by unilaterally changing their strategy, even when considering future responses within any subgame.

Interpreting the Dynamic Game

Interpreting a dynamic game involves understanding how players' incentives change over time due to the sequential nature of decisions and the information available. The key is to look beyond immediate gains and consider the long-term implications of each move, as well as the anticipated reactions of other players. For example, a company might initially price its product low not just for current sales, but to deter a competitor from entering the market dynamics.

The concept of a dynamic game highlights the importance of credible commitments and threats. A player's promise or warning only holds weight if it is rational to carry out, even if doing so is costly in the short term. The ability to model these sequential interactions, including the anticipation of future moves and responses, allows for a more nuanced understanding of strategic behavior than static models. This often involves considering the concept of sequential games and how they differ from simultaneous ones.

Hypothetical Example

Consider a scenario where two competing airlines, Air Alpha and Beta Airways, are deciding whether to offer a new route. This is a dynamic game because Air Alpha, being the larger carrier, decides first, and Beta Airways observes Air Alpha's decision before making its own.

Scenario:

• Stage 1: Air Alpha's Decision
  • Air Alpha can choose to enter the new route or not enter.
• Stage 2: Beta Airways' Decision (if Air Alpha enters)
  • If Air Alpha enters, Beta Airways can choose to enter the route as well (compete) or not enter (cede the market).

Payoffs (in millions of dollars for (Air Alpha, Beta Airways)):

• Air Alpha Does Not Enter:
  • (0, 0) – Neither airline gains from the new route.
• Air Alpha Enters:
  • Beta Airways Competes: (5, 3) – Both gain, but profits are split.
  • Beta Airways Cedes Market: (10, 0) – Air Alpha gains all, Beta gains nothing from this route.

Applying Backward Induction:

1. Beta Airways' Decision (Stage 2): If Air Alpha has already entered, Beta Airways faces a choice:
   • Enter and get 3 million.
   • Cede market and get 0 million.
   • Beta Airways will rationally choose to compete (3 > 0).
2. Air Alpha's Decision (Stage 1): Knowing Beta Airways will compete if it enters, Air Alpha now considers its options:
   • Do Not Enter: Payoff for Air Alpha is 0.
   • Enter (and anticipate Beta Airways will compete): Payoff for Air Alpha is 5 million.
   • Air Alpha will rationally choose to enter (5 > 0).

In this dynamic game, Air Alpha enters the market, and Beta Airways responds by competing. This illustrates how anticipating the rational response of the second player influences the first player's optimal investment strategy.

Practical Applications

Dynamic games are widely applied in various fields beyond theoretical economics, particularly where sequential decision-making and anticipation of rivals' moves are crucial. In the financial markets, they can model how firms set prices in oligopolies, considering competitors' reactions over time. For instance, in an industry where one dominant firm moves first (a leader) and other firms respond (followers), a dynamic game can predict market outcomes.

Government agencies also utilize dynamic game theory. For example, the Federal Communications Commission (FCC) has employed game theory principles in designing and conducting spectrum auctions. These¹ auctions are inherently dynamic, as bidders observe each other's bids and adjust their strategies in subsequent rounds, leading to complex strategic interactions aimed at maximizing utility and revenue. This application helps ensure efficient allocation of public resources by structuring the rules of engagement for multiple parties with conflicting interests. Dynamic games are also valuable in risk management to analyze how different risk mitigation strategies might be adopted or countered by other market participants or regulatory bodies over time.

Limitations and Criticisms

Despite their analytical power, dynamic games, like other game theory models, operate under certain assumptions that can limit their real-world applicability. A primary criticism is the assumption of perfect expected utility and unbounded rationality among all players. In reality, individuals and organizations often exhibit bounded rationality, meaning their decision-making is limited by available information, cognitive constraints, and time. They may not always calculate optimal strategies through complex backward induction, especially in games with many stages.

Furthermore, dynamic game models can become incredibly complex in real-world scenarios involving many players, imperfect information, or significant uncertainty. The reliance on complete and perfect information about other players' payoffs and rationality is a strong simplification that may not hold true. Critics also point out that human behavior is often influenced by emotions, heuristics, and biases, which are not typically accounted for in traditional rational game theory models, challenging the predictive power of such models in some contexts. The field of behavioral economics, for instance, seeks to incorporate these psychological factors into economic models, offering alternative perspectives to purely rational models of decision theory.

Dynamic Game vs. Static Game

The fundamental distinction between a dynamic game and a static game lies in the timing and information structure of players' decisions.

In a static game, players make their choices simultaneously, or at least without knowing the choices of others. This is like a sealed-bid auction, where all bids are submitted at once. In contrast, a dynamic game involves a sequence of moves, where players have the opportunity to react to what has transpired previously. This allows for concepts like commitment, deterrence, and reputation to play a significant role, which are not present in static game models. The ability to form strategies that account for future reactions is a defining characteristic of dynamic games.

FAQs

What is the primary difference between a dynamic game and a static game?

The primary difference is the timing of decisions. In a dynamic game, players make decisions sequentially, often with knowledge of prior moves, while in a static game, decisions are made simultaneously or without knowledge of others' choices.

Why is backward induction important in dynamic games?

Backward induction is a key solution method for dynamic games because it helps identify rational strategies by starting from the end of the game and working backward. This process ensures that decisions made at each stage are optimal, considering the anticipated rational responses of all players in future stages.

Can dynamic games model real-world business scenarios?

Yes, dynamic games are frequently used to model real-world business scenarios such as price wars, market entry and exit decisions, negotiations, and research and development (R&D) competitions. They are particularly useful when understanding the timing of actions and reactions among competitors is critical.

Are there any limitations to dynamic game theory?

Yes, common limitations include the assumption of perfect rationality, which may not reflect real-world human behavior. Dynamic games can also become very complex and difficult to analyze in situations with many players or imperfect information, making practical application challenging.

How does information play a role in a dynamic game?

Information is crucial in a dynamic game because players' decisions are influenced by what they know about previous moves by other players. This allows for adaptive strategies, where players can respond to observed actions, unlike in static games where decisions are made without such real-time information.