Sequential Games

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.

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.

Another important concept in sequential games is the Nash Equilibrium.

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.

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.

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.

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:

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".

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.

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.

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.

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).

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.

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)      |
            |-----------------------------|

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.

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.

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.

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.