What are the key differences between cooperative and non-cooperative games in microeconomics?

Cooperative games involve players forming coalitions to achieve collective outcomes, with binding agreements and shared payoffs. Non-cooperative games focus on individual strategies and decision-making, where agreements are not binding and players independently pursue their own interests. In cooperative games, the concept of fairness and distribution is central, whereas, in non-cooperative games, the Nash equilibrium is often analyzed.

How are the outcomes determined in a cooperative game in microeconomics?

Outcomes in a cooperative game in microeconomics are determined by how the players form coalitions and distribute the collective payoff. The common solution concepts include the Core, the Shapley Value, and the Nash Bargaining Solution, which reflect fairness and stability in allocation among the participating players.

What role does communication play in cooperative games in microeconomics?

Communication facilitates the sharing of information and strategy coordination among players in cooperative games. It enables players to form coalitions, negotiate agreements, and determine fair resource distribution, enhancing the potential for mutually beneficial outcomes.

How are resources allocated among players in a cooperative game in microeconomics?

In a cooperative game in microeconomics, resources are allocated among players using solution concepts like the core, the Shapley value, and the nucleolus. These solutions ensure a fair distribution based on contributions, stability, and individual rationality, aligning with the collective goals and agreements made by the coalition of players.

What is a cooperative game in microeconomics?

A cooperative game in microeconomics involves players forming coalitions and negotiating binding agreements to achieve mutual benefits. Players can collaborate and share the collective payoff, and the focus is on how to distribute these benefits among the coalition members.

What is a key benefit of cooperative games in microeconomics?

They focus solely on competition to maximize individual gains.

What is the Shapley Value in cooperative games?

A principle for maximizing the coalition’s total wealth through individual sacrifices.

What is required for a distribution to be in the Core of a cooperative game?

The distribution should maximize the largest player's payoff at the expense of others.

What differentiates cooperative games from non-cooperative games?

In cooperative games, there's no focus on resource allocation.

What is the Shapley Value used for in cooperative game theory?

To calculate the risk levels of different coalitions.

How is the Shapley Value for a player in a coalition calculated?

By averaging all possible coalition values

Cooperative Game: Theory & Meaning

cooperative game

A cooperative game is a strategic game in which players collaborate, forming coalitions to optimize a shared benefit, unlike non-cooperative games where players act independently. Key principles include the allocation of resources and the distribution of the payoff among players based on contribution, often analyzed using concepts like the core, Shapley value, and bargaining solutions. Understanding these concepts helps predict outcomes in various real-world scenarios, such as business alliances, political negotiations, and resource management.

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Cooperative Game Theory Definition

Cooperative game theory is an important branch of microeconomics and game theory that studies how groups of agents collaborate to achieve a common objective. Unlike non-cooperative games, where each player acts independently, cooperative games focus on coalition formation, which allows agents to form binding agreements and work together. In cooperative games, the central question revolves around the allocation of resources, benefits, or costs resulting from forming coalitions. Mathematical tools and concepts are employed to analyze these processes and outcomes.

Cooperative Game: A cooperative game is a type of game theory model where players can form coalitions and make binding commitments to share resources and payoffs.

Core Concepts of Cooperative Game Theory

Several core concepts are fundamental to understanding cooperative game theory. Here are some key features you should be familiar with:

Coalitions: Groups of players who cooperate to achieve a common goal. In such cases, the focus is on how the group performs rather than individual decisions.
Payoff Distribution: It determines how profits or benefits should be distributed among players within a coalition.
Shapley Value: A method used to fairly allocate the total payoff among players based on their marginal contributions to the coalitions.
Core: A set of allocations ensuring no subgroup of players would benefit more by breaking away and forming a separate coalition.

Consider three companies, A, B, and C, that wish to develop a new technology. By working together, the combined effort will increase the potential outcome. In this cooperative game, the allocation of resulting profits can be evaluated through the Shapley Value. Suppose the combined profit achieved by the coalition of all three companies is $300, and the contributions by formations are as follows:

Coalition	Value
A alone	$50
B alone	$40
C alone	$30
A and B	$150
A and C	$120
B and C	$110
All three	$300

The Shapley Value will equitably assign profits as follows:For A: $0.5 \times 50 + 0.3 \times 100 + 0.3 \times 50 = 100$For B: $0.4 \times 100 + 0.3 \times 60 + 0.3 \times 50 = 90$For C: $0.3 \times 60 + 0.2 \times 40 + 0.3 \times 60 = 110$

Cooperative Game Meaning and Basics

Cooperative game theory is a significant area within microeconomics that focuses on how players can form alliances or coalitions to optimize collective payoffs. This approach differs from non-cooperative games, where the emphasis is on the strategic decisions of individuals. By using cooperative game theory, you can analyze scenarios where collaboration yields better results than acting alone.

Understanding Coalitions in Cooperative Games

In the context of cooperative games, coalitions refer to groups of players that join forces to achieve mutual objectives. Key considerations in coalition formation include:

The potential value generated by a coalition as opposed to individual efforts.
The distribution of the coalition's payoffs among its members.
The strategic calculations used by players to decide on coalition formation and retention.

Mathematically, you represent a coalition's total value using the characteristic function $v(S)$, where $S$ is a subset of players (or coalition), and $v(S)$ is the payoff the coalition receives. Different coalitions can be formed, and their values can be calculated to determine the optimal collaboration strategy.

To further illustrate cooperative games, let's explore a detailed example that demonstrates the math behind coalition value. Suppose there are four players: A, B, C, and D, who have the following coalition values:

Coalition	Value $v(S)$
{A}	5
{B}	6
{C}	8
{D}	7
{A, B}	14
{A, C}	15
{A, D}	13
{B, C}	16
{B, D}	14
{C, D}	18
{A, B, C}	24
{A, B, D}	22
{A, C, D}	25
{B, C, D}	27
{A, B, C, D}	30

The optimal coalition can be found by analyzing these values and determining combinations that maximize the total payoff. For example, when considering the grand coalition {A, B, C, D} which has a payoff of 30, each player's contribution will be pivotal in calculating allocations such as Shapley Values.

To calculate the Shapley Value for a player in a coalition, the principle of marginal contribution is used. Let's calculate the Shapley Value for player A in the earlier example:Shapley Value formula: $ \phi_i(v) = \frac{1}{|N|!} \sum_{S \subseteq N \setminus \{i\}} |S|!(|N| - |S| - 1)![v(S \cup \{i\}) - v(S)] $Here, $N$ is the set of all players. By calculating this for player A considering all permutations of coalition formations, you find A's Shapley Value. Such calculations ensure that player A is fairly compensated according to their contribution across different coalitions.

Examples of Cooperative Games in Microeconomics

In microeconomics, cooperative games provide a framework for understanding how individuals or firms can collaborate to achieve outcomes that would be unattainable individually. Cooperation can lead to better resource allocation, increased efficiency, and, ultimately, higher payoffs for all involved parties. Let's examine some real-world examples that illustrate the application of cooperative game theory in economic contexts.

Joint Ventures and Partnerships

Joint ventures are cases where businesses come together to achieve a specific goal, such as entering a new market or developing a new product. Through cooperation, companies can pool resources, share risks, and leverage complementary strengths.Consider two firms, Firm X and Firm Y, collaborating to produce a new technology. By forming a joint venture, they can achieve a combined payoff higher than the sum of their individual efforts. This collaboration boosts innovation and reduces costs due to shared investments and expertise.

Suppose Firm X and Firm Y have individual payoffs of 100 each, but by cooperating, they can achieve a joint payoff of 300. The challenge lies in how to distribute this gain among them, respecting each firm's contribution to the collaboration. Using the Shapley Value, the payoff allocation can be calculated. Let's say:\[ \phi_X = 120 \] and \[ \phi_Y = 180 \]Through this distribution, each firm's profit reflects their marginal contribution, ensuring fairness in allocation.

An extension of cooperative games can involve dynamic coalition formation where new players might join or existing ones may leave coalitions over time. This variability adds complexity to payoff calculations, making tools like the Shapley Value even more critical.Imagine an evolving market where new firms continually introduce innovations. As coalitions form and realign, understanding each player's contribution and ensuring fair redistribution of collective gains keeps the collaborative effort sustainable and equitable.

Trade Agreements and Alliances

Countries often engage in trade agreements and alliances to enhance economic benefits beyond their borders. These arrangements can be analyzed as cooperative games, where member countries negotiate to achieve mutual benefits such as tariff reductions, improved trade terms, or shared resources.These agreements aim to maximize collective economic welfare while addressing individual member countries' specific interests and security concerns.

Consider a trade bloc comprising three countries: A, B, and C. Their combined trading capacity has a certain economic value represented by the coalition value convex hull of the set of all possible distributions. Let's assume:Country A: $v(A) = 70$Country B: $v(B) = 80$Country C: $v(C) = 90$Coalition Total Value: $v(A, B, C) = 300$The Shapley Value can help allocate these benefits fairly, ensuring each country's gains are proportionate to their contribution to the overall trade volume.

Ensure you consider externalities when forming coalitions; sometimes indirect benefits can significantly affect outcomes.

Techniques in Cooperative Games

Cooperative game theory employs various techniques to analyze and solve problems that involve coalition formation and resource allocation among players. Understanding these techniques is crucial for determining stable and fair outcomes in these games.

Shapley Value Technique

The Shapley Value is a cornerstone technique in cooperative games, utilized to distribute total gains fairly among players based on their individual contributions. It provides a solution concept that ensures each player's payout reflects their added value to the coalition. The Shapley Value calculates the weighted average of a player's contribution across all permutations of the coalition.The formula for Shapley Value is given by:\[ \phi_i(v) = \frac{1}{|N|!} \sum_{S \subseteq N \setminus \{i\}} |S|!(|N| - |S| - 1)![v(S \cup \{i\}) - v(S)] \]where:

$i$ is the player;
$S$ is the coalition excluding player $i$;
$v(S)$ is the value of coalition $S$;

Consider a cooperative scenario with three companies: A, B, C. They aim to develop a joint product. Their individual efforts yield profits of 10, 15, and 20, respectively, but together, they manage to earn 50. Using the Shapley Value, you distribute the total profit as follows:The grand coalition profit is calculated with respect to their individual contributions:

Coalition	Value
{A}	10
{B}	15
{C}	20
{A, B, C}	50

The Shapley Values would distribute the joint payoff among A, B, and C such that each company's contribution to the coalition is recognized proportionally.

The Shapley Value is respected for its properties of efficiency, symmetry, dummy player, and additivity, making it a robust technique for cooperative games.

Core Solution Concept

The Core is another key concept in cooperative game theory. It describes the set of possible distributions of total payoffs such that no subset of players could achieve better payoffs by breaking away and forming a separate coalition. A core solution exists only if the coalition is strong enough to satisfy all members.For a distribution $x$ to be in the Core, it must satisfy:

$ \sum_{i \in N} x_i = v(N) $
$ \sum_{i \in S} x_i \geq v(S) $ for all subcoalitions $S$

The core ensures stability within the coalition by ensuring all members receive at least as much as they could earn on their own.

Exploring the Core in detail reveals complexities, as it may not always be nonempty. For example, in market scenarios with external factors affecting valuation, ensuring a nonempty core requires strategic interventions. Potential analysis includes:

Examining market conditions that influence coalition values.
Using linear programming techniques to find core allocations.
Investigating cooperative games where traditional core solutions are impractical or impossible.

This exploration of the Core showcases its importance in achieving economically stable and equitable solutions in cooperative contexts.

cooperative game - Key takeaways

Cooperative Game Theory Definition: A branch of microeconomics and game theory that examines how groups work together, forming coalitions, to achieve a common objective and how to allocate resources, benefits, or costs among them.
Cooperative Game Meaning: A model in game theory where players can form coalitions and make binding commitments to share resources and payoffs.
Core Concepts: Include coalitions, payoff distribution, Shapley Value (method for fair allocation based on marginal contributions), and the Core (set of allocations preventing subgroup separation).
Examples of Cooperative Games in Microeconomics: Include joint ventures, partnerships, and trade agreements where collaboration enhances efficiency and payoff distribution is crucial.
Shapley Value Technique: A cornerstone method in cooperative games used to distribute total gains fairly among players by calculating their contributions to different coalitions.
Core Solution Concept: Represents payout distributions ensuring no subgroup benefits more by forming a separate coalition, focusing on stability and equitable solutions.

cooperative game

StudySmarter Editorial Team