Sequential games

What Are Sequential Games?

Sequential games are a type of strategic interaction in game theory where players make decisions in a specific order, one after another, with each subsequent player fully aware of the choices made by those who moved before them. This contrasts sharply with simultaneous games, where players act without knowledge of others' current decisions. The defining characteristic of sequential games is this flow of information, which allows later players to adapt their strategies based on observed actions, making them central to understanding dynamic scenarios in finance and economics.³³

This turn-based structure allows for a richer analysis of strategic interactions and how the order of play influences outcomes. Understanding sequential games is crucial for analyzing situations where companies, investors, or policymakers engage in a series of contingent moves, rather than isolated, independent actions.³²

History and Origin

The foundational concepts of game theory, which underpin the study of sequential games, were formalized by mathematician John von Neumann and economist Oskar Morgenstern. Their seminal work, Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is widely considered the groundbreaking text that established game theory as an interdisciplinary field.²⁹, ³⁰, ³¹ While early game theory often focused on static models, the need to understand dynamic interactions, where decisions unfold over time, spurred the development of sequential game models.²⁸ This evolution led to the incorporation of tools like decision trees and solution concepts such as backward induction, which are essential for analyzing sequential decision-making processes.²⁷

Key Takeaways

• Sequential games involve players making choices in a specific, known order, with later players fully informed of previous moves.
• They are a core concept within game theory, providing a framework for analyzing dynamic strategic interactions.²⁶
• The order of moves and the flow of information significantly influence players' optimal strategies and potential outcomes.
• Solving sequential games often involves techniques like backward induction to determine optimal strategies and predict equilibrium outcomes.
• These games offer a valuable lens for understanding real-world scenarios in finance, business, and policy where actions and reactions are intertwined.²⁵

Interpreting Sequential Games

Interpreting sequential games involves analyzing the decision points and potential paths players might take, considering the information available at each stage. Unlike simultaneous games, where players choose actions without knowing what others are doing, sequential games allow for a player's strategy to be contingent on observed prior moves.²⁴

The primary method for solving sequential games with perfect information is backward induction. This process involves starting at the very end of the game tree—the final decision nodes—and working backward to determine the optimal action for the player making that last decision. Once the optimal last move is identified, the analysis proceeds to the second-to-last decision point, and so on, until the initial decision point is reached. By applying backward induction, analysts can identify the subgame perfect Nash equilibrium, which is a set of strategies that constitutes a Nash equilibrium in every subgame of the original game. This solution concept ensures that all players' strategies are credible, even for parts of the game that might not be reached if earlier players stick to their optimal paths. This iterative process helps predict the rational outcome of the strategic interaction.

Hypothetical Example

Consider two companies, Company A and Company B, deciding whether to invest in a new, costly technology. The game is sequential: Company A decides first, and then Company B makes its decision after observing Company A's choice.

• Scenario 1: Company A invests. If Company A invests, it gains a competitive advantage but incurs high costs. Company B, observing this, must decide whether to invest and compete directly or not invest and lose market share.
  • If Company A invests and Company B also invests, both earn a modest profit of $10 million due to intense competition and high investment costs.
  • If Company A invests and Company B does not, Company A earns a large profit of $50 million, while Company B earns $0.
• Scenario 2: Company A does not invest. If Company A does not invest, it avoids costs but foregoes potential gains. Company B, seeing this, can then decide.
  • If Company A does not invest and Company B invests, Company B earns a large profit of $50 million, and Company A earns $0.
  • If Company A does not invest and Company B also does not invest, both earn a modest profit of $5 million (maintaining the status quo).

Using backward induction:

1. Company B's decision (if Company A invested): Company B observes Company A invested. Company B can invest for $10 million or not invest for $0. Company B's optimal decision is to invest.
2. Company B's decision (if Company A did not invest): Company B observes Company A did not invest. Company B can invest for $50 million or not invest for $5 million. Company B's optimal decision is to invest.

Now, we move back to Company A's initial decision:

• If Company A invests, it knows Company B will respond by investing, leading to a $10 million profit for Company A.
• If Company A does not invest, it knows Company B will respond by investing, leading to a $0 profit for Company A.

Given this, Company A's optimal decision making is to invest in the new technology, as it yields a higher profit ($10 million) than not investing ($0).

Practical Applications

Sequential games are highly applicable in various real-world financial and economic contexts, where the timing and observation of actions are critical.

• Corporate Strategy and Mergers & Acquisitions: Companies frequently use sequential game theory to model competitive scenarios such as market entry, pricing decisions, and product launches. For instance, a firm might decide whether to enter a new market, knowing that an incumbent firm will respond. Similarly, in mergers and acquisitions, a bidding war can be seen as a sequential game where each successive bid is influenced by the previous one. Ins²³ights from game theory can significantly improve business decisions with game theory.
• ²² Negotiations and Bargaining: Labor negotiations, trade agreements, or even simple buying and selling can be modeled as sequential games where each offer or counter-offer influences the next move.
• Financial Modeling and Investment Decisions: In complex financial modeling, sequential games can analyze scenarios like tender offers in corporate takeovers or the exercise of option pricing where the holder's decision depends on the underlying asset's price movements over time. The²¹ application of game theory in finance is extensive, covering areas from asset pricing to capital structure decisions.
• ¹⁹, ²⁰ Regulatory Frameworks: Regulators may anticipate how firms will react to new policies or rules, treating the interaction as a sequential game to design more effective regulations. This considers how regulations influence market dynamics.

Limitations and Criticisms

While powerful, the application of sequential games, and game theory in general, has several limitations and criticisms:

• Assumption of Rationality: A fundamental criticism is that sequential games often assume players are perfectly rational, always acting to maximize their own utility. In reality, human decision making can be influenced by emotions, cognitive biases, incomplete information, or bounded rationality. The¹², ¹³, ¹⁴, ¹⁵, ¹⁶, ¹⁷, ¹⁸ concept of bounded rationality suggests that individuals make "good enough" decisions rather than optimal ones due to limitations in time, information, and processing power. Thi⁹, ¹⁰, ¹¹s can lead to outcomes that deviate from game-theoretic predictions.
• Complexity and Information Asymmetry: Real-world scenarios can be vastly more complex than the simplified models used in game theory. Accurately mapping out all possible moves and knowing the exact payoffs and information sets for every player can be challenging, especially when information asymmetry exists.
• ⁸ Difficulty in Capturing Behavioral Nuances: Game theory, particularly its traditional forms, may struggle to incorporate human elements like trust, loyalty, fairness, or altruism, which can significantly impact strategic choices and outcomes. [Be⁷havioral economics](https://diversification.com/term/behavioral-economics) attempts to address these shortcomings by integrating psychological insights into economic models.
• Predictive vs. Descriptive: Game theory is more descriptive of how rational actors should behave rather than how real-world actors do behave. Its ability to predict outcomes accurately can be limited when players do not adhere to purely rational choice theory or when the game is not perfectly defined.

##⁶ Sequential Games vs. Simultaneous Games

The fundamental difference between sequential games and simultaneous games lies in the timing of decisions and the information available to players.

In sequential games, the ability of later players to observe prior moves creates a dynamic where strategic foresight is paramount. Players can plan their actions by looking ahead to how others will react, and then "reasoning backward" to determine their own optimal initial move. In contrast, simultaneous games require players to anticipate what others might do, without the benefit of direct observation, making concepts like dominant strategies and mixed strategies more central.

##⁴ FAQs

What is the most important characteristic of sequential games?

The most important characteristic of sequential games is that players make decisions in a specific order, and each player is fully aware of the actions taken by those who moved before them. This perfect information about previous moves allows for adaptive strategies.

How are sequential games typically analyzed?

Sequential games are typically analyzed using a method called backward induction. This involves starting at the end of the game and working backward to determine the optimal strategy for each player at every decision point, leading to a subgame perfect Nash equilibrium.

Can sequential games have multiple Nash equilibria?

Yes, like other games in game theory, sequential games can have multiple Nash equilibria. However, backward induction often helps to refine these to a "subgame perfect" equilibrium, which is more robust as it ensures credibility in every subgame.

What is a common example of a sequential game in finance?

A common example of a sequential game in finance is a corporate takeover bid, where one company makes an offer, and the target company or other potential bidders respond in sequence. Another example is a company deciding on a pricing strategy, with competitors reacting to that initial price.

##³# Are zero-sum games always sequential?

No, zero-sum games can be either sequential or simultaneous. A zero-sum game simply means that one player's gains are exactly balanced by another player's losses. Chess is a sequential zero-sum game, while matching pennies is a simultaneous zero-sum game.¹, ²