Sequential Games

Dive into the world of Microeconomics with an enlightening exploration of Sequential Games. This in-depth study unravels the definition, key elements, and the pivotal role of Sequential Games in economics. Enrich your understanding of game theory, learn effective solving methods, and delve into the fascinating concept of Nash Equilibrium in Sequential Games. Discover how these theories play out in real life, while navigating the nuances between sequential and simultaneous games. With a wealth of practical examples, this guide offers invaluable insight into the intriguing mechanics of Sequential Games.

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Table of contents

    Understanding Sequential Games in Microeconomics

    In microeconomics, Sequential games occupy a significant space, entailing a strategic interaction necessitating clear comprehension. So, let's embark on a journey to understand this concept thoroughly.

    Sequential Games Definition and Basics

    Let's first acquaint ourselves with what Sequential games are. Sequential games, a subset of game theory, are those where players make decisions one after another, with each player aware of the prior decisions made.

    In game theory, Sequential Games are essentially games where players take discrete action at different times, possessing full knowledge of the previous actions taken by other players.

    Understanding the concepts requires untangling some essential elements. To assist you, we use lists to organise information conveniently.

    • Players: This refers to the entities making decisions in the game. These may be individuals, firms or even countries.
    • Actions: Every move a player makes constitutes an action.
    • Payoffs: The outcome of the actions each player takes is referred to as their payoff. It is constrained by the sequence of actions and the game's overall structure.

    After understanding these elements, it becomes easier to comprehend the game's structure and strategy, which leads us to our next header.

    Understanding the Key Elements of Sequential Games

    Now that you are familiar with the basic concepts, it's time to delve deeper into the key elements of sequential games.

    One critical tool for understanding sequential games is the game tree or decision tree. This graphical representation illustrates the sequence of actions and payoffs, making it easier to visualise the strategic scenario.

    Game TreeA graphical representation using nodes (decision points) and edges (possible actions) to exhibit the possible outcomes of a game. The payoffs associated with each possible result are often shown at the end nodes.

    Another important concept in sequential games is the Nash Equilibrium.

    The Nash Equilibrium, named after John Nash, is a concept of game theory where the optimal outcome of a game is where no player can benefit by changing strategies while the other players keep theirs unchanged.

    In sequential games, the concept of Nash Equilibrium becomes Subgame Perfect Nash Equilibrium (SPNE), where no player can gain more by deviating from their current strategy.

    Let's take a simple example. Suppose two firms, Firm A and Firm B, are deciding whether to enter a new market. Firm A moves first and chooses between entering or not entering. If Firm A enters, Firm B next decides whether to enter or stay out. The payoffs are represented at the end nodes and vary depending on the actions of the two firms. The SPNE in this case would be the strategy that maximises the payoff for both firms given the other firm's decision.

    The Role of Sequential Move Games in Economics

    Having covered the definition and essential elements successfully in the prior sections, it's now interesting to explore the role of sequential games in economics. They are integral in studying strategic interactions, where the timing of decisions is crucial.

    • They serve as a fundamental analytical tool in industrial organisation to analyse firm behaviour in oligopolistic markets.
    • They aid in understanding contract theory, where the sequence of actions and commitment is essential.
    • In international economics, sequential games facilitate the analysis of trade negotiations and policies.

    Beyond economics, sequential games have broad applications in various disciplines, including political science, computer science, and biology. For example, in computer science, sequential games are used in the design of algorithms for sequential decision-making problems.

    How Sequential Games Drive Economic Decisions

    After understanding the role of sequential games in economics, it's crucial to comprehend how they can drive economic decisions. Via sequential games, one can analyze how different players (firms, countries, individuals) times their decisions strategically and impact the entire economic scenario.

    The sequence of actions can influence the strategies and payoffs in these games, impacting businesses' decision-making, market entry strategies, and pricing decisions. They can also influence countries' decisions regarding trade policies and negotiations. The sequential nature of these decisions makes them dynamic, thus influencing strategic interactions and shaping the economic landscape.

    Consider a market entry scenario. If a firm is considering entering a new market, it must anticipate how existing firms will react. If it expects aggressive competition from existing firms, it may choose not to enter. On the other hand, if it anticipates passive responses, it may decide to proceed. These sequences of strategic decisions, an inherent aspect of sequential games, significantly drive economic outcomes.

    Deep Dive into Game Theory: Sequential Games

    To further deepen your understanding of sequential games, it is crucial to examine them under the wider umbrella of game theory. This theoretical framework allows us to study interactive situations, predicting and analysing outcomes when players act strategically.

    Sequential Games Game Theory Explained

    Sequential games are a subcategory within the game theory, distinguished by the timing of actions or decisions by players. In sequential games, players make their decisions in a sequence – one after the other – not simultaneously. Each player observes the decisions made by previous players before deciding themselves, bringing a unique level of complexity.

    The framework of these games involves a few critical elements: players, strategies, and payoffs. Players do not need to be just humans or organisations; they can also be software in computer science context or even countries in an economic scenario. The strategies refer to the complete plan of action a player will follow throughout the game. The payoffs, finally, represent the results of the game—what each player receives at the end of the game as a consequence of the sequence of actions taken.

    Let's explore these elements using HTML tables:

    PlayersThey can be individuals, firms, countries or any entity making decisions in the game.
    StrategiesThese are the complete plan of action that a player will follow throughout the game, based upon the existing information and the uncertainty about the other players’ strategies.
    PayoffsThese represent the outcome that players receive as result of the sequence of actions taken within the game. These can be profits, utilities, welfare, etc.

    The Connection Between Sequential Games and Game Theory

    In game theory, the connection amongst sequential games, simultaneous games, static games, dynamic games, complete information games, and incomplete information games are vital. Sequential games, under the larger frame of game theory, fall under dynamic games as well as complete information games.

    Dynamic games are ones in which actions of players have implications across different periods, which is a characteristic inherent to sequential games. Complete information games are those in which each player knows or can infer the exact payoffs and strategies available to other players which is also applicable to sequential games.

    Solving Sequential Games with Effective Methods

    Solving sequential games entails finding the outcome of the game, or the series of actions (strategies) that players will optimally choose. The solution concept used in sequential games is the Subgame Perfect Nash Equilibrium (SPNE).

    The SPNE can be found using the method of backward induction. In layman's terms, the methodology uses the idea of "thinking ahead" where one begins at the end and then works backward to the starting decision. The outcome thus obtained is a chain of optimal decisions, thus making the equilibrium "subgame perfect".

    Let's consider an example to illustrate the concept. Think of a firm contemplating entering a market. It starts by considering the potential responses of its competitors if it were to enter the market. If the competitors decide to tackle aggressively thus reducing profits, the firm may decide not to enter the market. These chain of decisions when optimised leads to the Subgame Perfect Nash Equilibrium (SPNE).

    Sequential Games Can Be Solved Using Various Strategies

    One may use various strategies to solve sequential games and discern the Subgame Perfect Nash Equilibrium (SPNE). Von Neumann's procedure, the so-called minimax strategy, assures a nonnegative payoff to both players in any two-person zero-sum game, sequential or otherwise.

    Another commonly deployed solution in dynamic games is the Markov strategy. Here, a player's decision depends only upon the current position, not on how the game reached that position. This simplification often assists in complex game scenarios.

    In contrast, sometimes games follow a trigger strategy, where one player's strategy is contingent on the other player's previous behaviour. A classic example of this is the 'Tit for Tat' strategy: a player replicates the opponent's previous move. If the opponent cooperated, they cooperate in the next round, and if the opponent defected, they defect in the next round.

    A comprehensive understanding of these solutions forms a major part of solving sequential games effectively. They help you navigate through the seemingly complex terrain of decision sequences and move towards optimised results.

    Unravelling Sequential Games Nash Equilibrium

    Nash Equilibrium is a staple concept in game theory, providing a stable solution in non-cooperative games, including sequential games. Named after mathematician John Nash, it is the state of the game where no player can profitably deviate by unilaterally changing their strategy, given the strategies of the other players.

    The Concept of Nash Equilibrium in Sequential Games

    In sequential games, the prominent form of Nash Equilibrium used is the Subgame Perfect Nash Equilibrium (SPNE). SPNE refines the Nash Equilibrium concept by imposing the rationality of players in every subgame, after any history of play. Simply put, players following SPNE do not have regrets about their behaviour at any stage of the game, given the other players' strategies.

    To identify a SPNE in a sequential game, the technique of backward induction is often used. Players predict the end-stage actions of their opponents, then deduce their optimal strategy, and continuously step back until the first stage of the game is reached. This provides a roadmap of optimal moves, for each node of the decision tree. The series of moves leading to each player's best possible outcome, forms the game's SPNE.

    A Subgame is a part of the original game, starting at a single decision node and including all feasible future moves from that point. Every game is a subgame of itself.

    How Nash Equilibrium Influences Sequential Games?

    The Nash Equilibrium concept is pivotal in sequential games as it sheds light on the decision-making of players under strategic situations. The Subgame Perfect Nash Equilibrium, in particular, helps unravel the sequential nature of decision-making by stipulating no player can improve by deviating unilaterally at any stage of the game. This concept captures how the strategic interaction shapes the eventual outcomes of the game, given the rationality of players. It ensures consistency in players' plans by eliminating non-credible threats, and eliminates equilibria that rely on implausible threats.

    In the field of economics, Nash Equilibrium in sequential games also aids in understanding and predicting phenomena. For instance, it can offer explanations for firm behaviour in oligopolistic markets, bargaining scenarios, and even policy-making in political science.

    Examples of Sequential Games Nash Equilibrium

    To truly understand the functioning of Nash Equilibrium in sequential games, practical examples shed the necessary light. Examples provide a context-driven approach that theory cannot deliver. Let's deep-dive into the intriguing world of sequential games and analyse Nash Equilibrium.

    A classic example of sequential games is the Centipede Game. Imagine two players have a chance to take a progressively increasing pot of money. Either player can take the pot on their turn but if they pass it to the other player, the pot doubles. The game ends either when a player takes the pot and keeps the contents or after a certain number of rounds.

    In a sample Centipede Game of three rounds, if Player A passes the pot in the first round (pot doubles to $2), Player B can choose in the second round whether to take the $2 or pass it back to A (pot doubles to $4). Upon reaching A again in the third round, they can take the full $4 or pass it back to B (pot doubles to $8), in which case the game ends, and B gets the $8. The Subgame Perfect Nash Equilibrium in this example, derived using backward induction, is that Player A takes the pot in the very first round, securing $1.

    This outcome might perplex you, as the total payout could have been greater if both players cooperated. However, the Nash Equilibrium suggests each rational player will try to maximise their own payout rather than the total payout, leading to this outcome.

    Practical Illustrations of Nash Equilibrium in Sequential Games

    Sequential games and Nash Equilibrium permeate several aspects of our everyday life too, and the interpretation of these games extends far beyond just theoretical constructs.

    For example, consider two drivers coming at each other on a collision course. They must decide to swerve right or left to avoid collision. If both swerve in the same direction, collision is avoided. If both swerve in different directions, they collide. In this game, the Nash Equilibrium presents two solutions – both drivers swerve left or both swerve right.

    Another pertinent example exists in pricing strategies where two companies release a product simultaneously. Company A can opt for a high price or a low price, and so can Company B. Their choices directly affect each other's profits. The Nash Equilibrium here demonstrates the pricing strategy that each company should independently follow that leads to the optimal outcome given the other company's pricing strategy.

    Understanding the Nash Equilibrium equips us with the tools to analyse situations involving strategic interaction across diverse scenarios – economics, computer science, business strategy, politics, and everyday life.

    Sequential vs Simultaneous Games in Microeconomics

    In the field of microeconomics and game theory, the strategic moves of players can occur in sequence or simultaneously, shaping the structure of games as either sequential or simultaneous games. Both game types offer insightful lens to analyse strategic interactions and decision-making.

    Difference Between Sequential and Simultaneous Games

    In the realm of game theory, the distinction between sequential and simultaneous gameplay is paramount. Each depicts a different scenario or context of strategic interactions amongst players. Let's unveil their respective characteristics.

    A Sequential Game models an environment where players make their moves one after the other, in some pre-determined or recognisable order. Each player observes the actions of the players who moved before them, and then makes an informed decision. The perfect information about past actions influences the game's outcome, and reflects real-life examples such as chess, negotiation, or a firm responding to the pricing strategy of its competitor.

    On the other hand, a Simultaneous Game represents situations where players make their decisions at the same time. The catch is that they lack complete information about their rivals' choices while making their own decisions. The absence of information creates a strategic environment marked by uncertainty. Classic examples include rock-paper-scissors, bidding in auctions, or firms deciding on a price simultaneously without knowledge of competitors' pricing decisions.

    In contrast to the sequential game's representation as a game tree (also known as an extensive form), simultaneous games are typically represented using a normal form (often a matrix).

    The Normal Form presents a concise way to describe a simultaneous game. It illustrates the players, strategies, and payoffs in a matrix. In contrast, the Extensive Form is a tree representation, capturing the sequential aspect of a game, including the decision-making order and the specific information each player has at any decision point.

    Assessing the Key Differentials Between Sequential and Simultaneous Games

    With the basic understanding of sequential and simultaneous games set, let's delve deeper to tease apart their subtle yet crucial differences.

    • Decision-making Order: In sequential games, there is an explicit order in which players make their moves. This order can drastically impact the outcomes of the game. However, in simultaneous games, all players make decisions at the same time, leading to a lack of information and increased uncertainty.
    • Information Availability: Sequential games are marked by perfect information. Players have full knowledge of prior actions within the game before they make a decision. On the contrary, simultaneous games are characterised by imperfect information, where players make decisions without knowing the choices of the other players.
    • Representation: Sequential games can be visualised through game trees (extensive forms), illustrating the decision-making process over several stages. Simultaneous games are depicted using matrices (normal forms), displaying all possible outcomes of concurrent decisions.
    • Solution Concept: The solution concept often used in sequential games is Subgame Perfect Nash Equilibrium, which considers the actions of the players in each subgame. In simultaneous games, the standard solution concept is the Nash Equilibrium, where no player can benefit by altering their strategy while other players keep theirs constant.

    Examples Illustrating Sequential and Simultaneous Games

    Real-world examples often provide the most convincing illustration of complex game theory concepts. Let's examine the practical instances of both sequential and simultaneous games.

    The centipede game is a standard example of a sequential game. It's an extensive form game in which two players alternately get a chance to take the larger share of an increasing money pot. The catch? The game ends as soon as a player takes the money.

    RoundPlayerAction: Take or NotPot Size
    1ANot take$2
    2BNot take$4

    A prisoner's dilemma is a simultaneous game where two prisoners must independently decide to confess or remain silent. The payoff matrix for this game encapsulates all possible outcomes.

    Prisoner B
                | Confess  |  Remain Silent |
    Prisoner A  |-----------------------------|
    Confess     | (-7, -7) | (-1, -10)      |
    Remain Silent|(-10, -1) | (-2, -2)      |

    Note: The payoffs in the prisoner's dilemma are the sentences in years. Both receiving a 7-year sentence if they both confess might not seem optimal compared to both remaining silent. However, without knowing what the other would do, both confessing becomes the Nash Equilibrium in this simultaneous game.

    Real-life Instances of Sequential and Simultaneous Games

    Predicting real-life strategic interactions using microeconomic game theory provides pragmatic and crucial insights. Let's explore some instances of sequential and simultaneous games.

    Sequential games are commonplace in our everyday lives. For instance, consider a negotiation over a car sale. Here, the seller quotes a price, the buyer then decides whether to reject, accept, or counter. This back-and-forth continues until an agreement is reached or one party decides to walk away.

    Multiple political scenarios serve as illustrations for simultaneous games. The decision by two countries whether to invest in military spending for security or in public goods for citizens' welfare can be described as a simultaneous game. Neither country knows what the other will invest in, but their decision can impact the outcomes for both.

    In the case of the simultaneous game of military expenditure, the optimal outcome would be for both countries to invest in public goods. But due to uncertainty over the other's choice, both might end up investing in military spending.

    These instances underscore the prevalence and importance of understanding strategic interactions through sequential and simultaneous games. The study of these games serves as a powerful tool in predicting and interpreting decision-making processes across varied settings – from personal transactions to global relations.

    Real-life Examples of Sequential Move Games

    In the realm of game theory, sequential games often depict situations we encounter in our daily lives and professional settings. By understanding the structure and mechanics of these games, you can learn to predict outcomes and make better decisions. Essentially, the sequential game model is a valuable concept that unfolds real-world strategic interactions marked by a preset order of moves.

    Sequential Move Game Examples in Real Life

    Sequential games are everywhere - in your everyday tasks, marketplace interactions, corporate decisions, and geopolitical strategies. Let's delve into the details of these instances to grasp the vitality and practicality of sequential games.

    Market Competition: Consider two firms, Firm A and Firm B, in a marketplace. Firm A first decides the price of its product. After observing Firm A's decision, Firm B decides its product's price. Here, each firm's profit depends not only on its own price but also on the competitor's price. This interaction is a classic example of a sequential game, where later players make informed decisions based on their predecessors' actions. The possible strategies for each firm can be represented in a game tree, paving the way for determining the best decision course using the concept of backward induction.

    Game Tree: It is a graphical representation of a sequential game, depicting the players, their possible moves, and the resulting outcomes. It helps visualise a sequential game by structuring the decision-making process.

    Election Strategy: In the political arena, consider an election scenario. Here, one party first announces its manifesto. The opposition party, having observed the first party's agenda, then formulates and announces its manifesto. This real-life situation is a sequential move game, with both parties' payoff determined by voter response to their respective manifestos. Underlying this strategic interaction is the notion of credibility and commitment: an announcement once made, the party must follow through, even if a later advantageous strategy emerges. The order of actions in the game can greatly impact the election outcome, making the study of sequential games invaluable in political analysis.

    Football Penalty Shootout: Let's turn our attention to a lighter, yet captivating, real-life example of a sequential game: a penalty shootout in football. The sequence of play is strictly defined. The keeper must decide whether to jump left, right, or stay in the middle. Simultaneously, the attacker decides where to aim the shot - left, right, or centre. While timing may suggest this as a simultaneous game, it's considered a sequential game - the keeper makes the decision just a moment after the attacker, based on the latter's body language and past play behaviour. The sequential nature of the decision-making during a penalty kick illustrates how one player's strategy is influenced by the actions of another, giving the game its sequential characteristic.

    Practical Applications of Sequential Games in Everyday Scenarios

    Venturing into the scope of everyday scenarios, sequential games can prove highly relevant and applicable even beyond the realms of economics, politics, and sports.

    Buying a Car: Perhaps one of the most common examples of a sequential game is the simple act of buying a car. Here, the seller initially sets a selling price. The buyer, having observed the set price, then decides whether to accept, reject, or negotiate. This sequential structure of decision-making can continue as an iterative process until either the deal is sealed at a mutually agreed price or the negotiation breaks down. By understanding this interaction as a sequential game, the buyer and seller can anticipate each other's moves and develop an optimal negotiation strategy.

    Child-Parent Interaction: Sequential games can even be applied to family dynamics, specifically in child-parent interactions. Consider the classic situation of a child asking a parent for a favour, such as staying up late to watch a television show. The child sets the ball rolling by requesting, and the parent decides on granting permission based on various factors. In responding, the parent considers the child's past behaviour (history of obeying rules, completion of homework, conduct, etc.) much like a strategic player evaluates prior actions in a sequential game. This simple interaction is a sequential move game, with both parties' payoff determined by their strategies and interaction outcome. Understanding such processes as sequential games can offer insightful perspectives even in managing day-to-day relationships and interactions.

    Career Progression: Another everyday sequential game involves career progression and decisions. An employee first decides to invest in additional qualifications or skills to seek promotion. After observing the employee's effort, the employer decides on awarding the promotion. This interaction can be transformed into an extensive form game, with the employer's strategy contingent on the employee's previous actions. Thus, decisions regarding career growth and progression can be analysed and better understood through the lens of sequential games.

    Extensive Form Game: Also known as a game tree, it's a representation strategy of sequential games, illuminating the order of players' moves, their potential strategies, and corresponding payoffs. It allows players to investigate a game's chronological sequence and structure optimal strategies.

    These everyday scenarios underline how sequential games are deeply interwoven into our daily lives and decision-making processes. By recognising and understanding these games, you can develop strategies that bring about beneficial outcomes and enhance your decision-making skills in various environments.

    Sequential Games - Key takeaways

    • Sequential games, within the broader scope of game theory, are categorized as dynamic games and complete information games. They involve actions of players that have implications across different periods and each player has full knowledge of payoffs and strategies available to others.
    • The solution concept used in sequential games is the Subgame Perfect Nash Equilibrium (SPNE) which can be found using backward induction; this involves thinking ahead and working from the end of the game to the beginning.
    • Strategies for solving sequential games and determining the Subgame Perfect Nash Equilibrium (SPNE) include Von Neumann's minimax strategy, the Markov strategy and the trigger strategy. Understanding these tactics is crucial to effectively solve sequential games.
    • In sequential games, the Nash Equilibrium, specifically the Subgame Perfect Nash Equilibrium (SPNE), stipulates that no player can improve by deviating unilaterally from their strategy at any stage of the game. It plays a fundamental role in deciphering player's decision-making processes under strategic conditions.
    • Sequential games are distinct from simultaneous games in several ways including their decision-making order, availability of information, representation and solution concept. In sequential games, decisions are made in an explicit order with complete information, represented through a game tree, or extensive form, and often utilize Subgame Perfect Nash Equilibrium as a solution. Simultaneous games, on the other hand, feature concurrent decision-making with incomplete information, represented through a matrix, or normal form, and generally use the Nash Equilibrium as a solution concept.
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    Frequently Asked Questions about Sequential Games
    What are the rules of the Sequence game? Please write in UK English.
    In sequential games, players make their moves in turn, following a specific order. The actions of one player depend on the previous moves of other players. The sequence is determined by the rules of the game. Once players understand the sequence and anticipate other players' actions, they can formulate strategies.
    What is the goal of the Sequence game?
    The goal of a sequence game in microeconomics is to determine the optimal strategy of players who make their decisions in turns rather than simultaneously, taking into account future moves and responses of other players.
    What is an example of a sequential move game?
    A classic example of a sequential move game in microeconomics is chess. Here, players make moves in turn, each reacting to the opponent's previous move and considering the future moves they could make in return.
    Who has the advantage in a sequential game?
    In a sequential game, the player who has the first move generally has the advantage. This is because they can anticipate the possible reactions of their opponents and adjust their strategies accordingly.
    How can one find the Nash Equilibrium in sequential games?
    In sequential games, Nash Equilibrium can be found using the method of backward induction. This involves solving the game from its end, determining optimal strategies for each player given all possible subsequent player choices, and working backwards to the game's beginning.

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