9.3 Applications to voting power and cost allocation
3 min read•august 7, 2024
explores how players can work together to achieve better outcomes. This section focuses on two key applications: and . These tools help analyze influence in decision-making bodies and fairly divide shared costs among participants.
Voting measure each voter's ability to sway outcomes, while cost allocation methods divide expenses fairly. Both concepts use cooperative game theory principles to solve real-world problems in politics, business, and resource management. Understanding these applications deepens our grasp of coalition dynamics.
Voting Power Indices
Measuring Voting Power
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- Voting power index quantifies the influence of each voter in a weighted voting system
- Considers the probability of a voter being pivotal in changing the outcome of a vote
- Takes into account the different weights assigned to each voter
- Useful in analyzing the distribution of power in decision-making bodies (legislatures, committees, shareholders' meetings)
Banzhaf Index
- measures the proportion of coalitions in which a player is critical for the coalition to reach a majority
- Calculates the number of times a voter is a , meaning their vote changes the outcome from losing to winning
- Assumes each coalition is equally likely to form
- Normalized Banzhaf index divides each voter's raw Banzhaf power by the total power of all voters to get a percentage
Shapley-Shubik Index
- assigns power to voters based on their to forming winning coalitions
- Considers the order in which voters join a coalition, giving more weight to voters who are pivotal earlier in the process
- Calculates the fraction of permutations in which a voter is pivotal, averaged over all possible permutations
- Satisfies several desirable properties for power indices (, , )
Weighted Voting Games
- model decision-making bodies where each participant has a specific voting weight
- Represented by a quota (minimum total weight required to pass a motion) and a list of voters' weights
- Examples include the U.S. Electoral College, the EU Council of Ministers, and corporate boards with different classes of stock
- Power indices help analyze the actual influence of each voter beyond their nominal
Cost Allocation and Fair Division
Cost Allocation Problems
- Cost allocation involves dividing the cost of a shared resource or project among the participants
- Aims to find a fair and efficient way to allocate costs based on each participant's usage or benefit
- Examples include dividing the cost of a shared facility (airport, water system) or a joint venture (research project, advertising campaign)
- Cooperative game theory provides tools for modeling and solving
Airport Problem
- The is a classic cost allocation scenario where the cost of an airport runway must be divided among airlines with different aircraft sizes
- Larger aircraft require longer runways, which are more expensive to construct and maintain
- The , a solution concept from cooperative game theory, can be used to allocate the runway costs based on each airline's marginal contribution
- Other methods, such as the and the , have also been proposed for solving airport problems
Fair Division
- deals with the problem of dividing a set of resources among participants in a way that is considered fair by all parties
- Involves both divisible goods (cake cutting) and indivisible goods (room assignment, inheritance distribution)
- Fairness criteria include (no participant prefers another's allocation to their own), (each participant receives at least 1/n of the total value), and (each participant believes they received the same fraction of the total value)
- Procedures for achieving fair division include the , the , and the
Cooperative Game Theory
Coalitional Bargaining
- models situations where players can form coalitions to increase their bargaining power and achieve better outcomes
- Involves a that assigns a value to each possible coalition, representing the total payoff the coalition can obtain
- The is a key solution concept in coalitional bargaining, consisting of all allocations that no coalition can improve upon by deviating
- The Shapley value and the nucleolus are also used to determine fair allocations in coalitional bargaining situations
- Applications include political coalition formation, labor union negotiations, and resource allocation in multi-agent systems
Key Terms to Review (29)
Adjusted Winner Algorithm: The Adjusted Winner Algorithm is a method for fairly dividing resources or goods between two parties in a way that reflects each party's preferences. This algorithm ensures that both parties receive an allocation that is proportional to their reported valuations of the items being divided, ultimately promoting fairness in negotiations and resource distribution.
Airport Problem: The airport problem refers to a situation in game theory and economics where a resource, such as an airport, has limited capacity but is in high demand from multiple users. This scenario highlights issues of allocation and fairness in distributing access to the resource among competing parties, emphasizing the complexities in voting power and cost allocation when different stakeholders vie for limited slots or services.
Banzhaf Index: The Banzhaf index is a measure of the power of a voter in a voting system, calculated based on the number of times a voter's vote can change the outcome of a decision. This index is particularly useful in analyzing situations where voting weights are unequal, allowing for an assessment of how much influence each voter has in different scenarios. It highlights the strategic importance of individual votes in collective decision-making processes, particularly when considering coalition formation and power distribution.
Characteristic Function: The characteristic function is a mathematical representation used in cooperative game theory that assigns a value to every possible coalition of players, reflecting the total worth that the coalition can achieve by working together. This function helps in analyzing how cooperative behavior among players can lead to different outcomes and distributions of benefits. It plays a crucial role in understanding the dynamics of alliances and negotiations, providing a framework for assessing voting power and cost allocation among participants.
Coalitional bargaining: Coalitional bargaining refers to the process where a group of individuals or parties come together to negotiate and reach an agreement, often in the context of shared interests or goals. This form of negotiation is crucial when individual interests may not be aligned, as parties form coalitions to strengthen their negotiating power, especially in scenarios involving voting power and cost allocation.
Cooperative Game Theory: Cooperative game theory is a branch of game theory that studies how players can work together to achieve a better outcome for all participants, rather than acting solely in their self-interest. This approach emphasizes the importance of forming coalitions and agreements among players to maximize their collective payoffs. By analyzing the ways in which players can collaborate, cooperative game theory provides insights into situations where teamwork can lead to improved results, such as in collusion, voting scenarios, and negotiation processes.
Core: The core is a solution concept in cooperative game theory that identifies allocations of resources or payoffs among players such that no subset of players would benefit by breaking away from the grand coalition to form their own. This concept ensures stability in cooperative arrangements, as it reflects outcomes where participants cannot improve their situation by acting independently. The core is particularly relevant in scenarios involving voting power and cost allocation, negotiations, and various forms of bargaining.
Cost allocation: Cost allocation is the process of distributing indirect costs to various cost objects, such as products, departments, or projects, based on a predetermined method or formula. This concept is essential for understanding how shared expenses are assigned and helps organizations determine the true cost of their operations and make informed financial decisions.
Cost allocation problems: Cost allocation problems involve distributing costs among different participants or entities that share resources or benefits. This concept is crucial in various economic contexts, especially in determining how costs should be divided in cooperative scenarios, like joint projects or shared services, ensuring fairness and efficiency in resource use.
Divide-and-choose method: The divide-and-choose method is a fair division technique where one party divides a resource into parts and the other party chooses their preferred part. This method ensures that both parties have an incentive to act fairly, as the divider will seek to make the portions as equal as possible to avoid disadvantage. It is commonly applied in various scenarios, including voting power and cost allocation, ensuring that resources are allocated in a just manner.
Efficiency: Efficiency refers to the optimal allocation of resources to maximize output and minimize waste, ensuring that resources are utilized in a way that achieves the best possible outcomes. In various decision-making contexts, efficiency highlights how effectively resources are used, particularly when considering voting power and cost allocation, as it helps to determine fair and effective distribution mechanisms that maximize collective welfare.
Envy-freeness: Envy-freeness is a fairness concept in resource allocation where each participant receives a share of resources that they believe is at least as good as what others receive. This means no individual feels envy towards another's allocation, as everyone perceives their share as fair or better. It plays a crucial role in various applications, ensuring equitable outcomes in scenarios like voting and cost sharing.
Equitability: Equitability refers to the fairness and impartiality of a distribution process, ensuring that resources, benefits, or costs are allocated in a just manner among participants. This concept plays a crucial role in decision-making frameworks, particularly in contexts where voting power and cost allocation are involved, as it seeks to balance the interests of all stakeholders while promoting a sense of fairness and legitimacy.
Fair Division: Fair division refers to the process of allocating resources or benefits among individuals in a way that is considered equitable and just. This concept is important in various contexts, including voting power and cost allocation, where the goal is to ensure that all parties feel they have received a fair share based on their contributions or needs.
Marginal Contribution: Marginal contribution refers to the additional value or benefit that an individual player adds to a coalition in a cooperative game. It is a way to assess how much each member contributes towards the overall success of the group, particularly in contexts like voting power and cost allocation, where understanding individual impact is crucial for fair distribution and influence.
Moving-knife procedure: The moving-knife procedure is a method used to fairly divide a continuous resource among multiple parties, where one party controls a knife that moves over the resource, allowing others to claim portions of it. This method ensures that each participant can secure a share of the resource they value while minimizing conflict over allocation. It reflects principles of fairness and strategy, making it applicable in contexts like voting power and cost allocation.
Nucleolus: The nucleolus is a solution concept in cooperative game theory that focuses on distributing resources among players in a way that minimizes the dissatisfaction of the most dissatisfied player. It aims to find a stable allocation that ensures that no coalition of players would want to deviate from this distribution due to their dissatisfaction, thus addressing issues related to fairness and stability in resource allocation.
Null player property: The null player property refers to a situation in cooperative game theory where a player does not affect the overall value of any coalition they are part of. This means that regardless of whether they join or leave a coalition, the coalition's total worth remains unchanged. This concept is particularly relevant in understanding voting power and cost allocation, as it helps identify players who do not contribute to the success or payoff of groups, thereby clarifying their influence in collective decision-making processes.
Pivotal Voter: A pivotal voter is an individual in a voting situation whose vote can change the outcome of an election or decision-making process. This concept highlights the significance of each voter's decision in scenarios where choices are closely contested, emphasizing how one person's vote can sway results and influence collective preferences.
Power Indices: Power indices are mathematical tools used to measure the influence or voting power of different players within a collective decision-making process, particularly in contexts such as voting systems and cost allocation scenarios. These indices help to quantify how much sway an individual or group holds in determining outcomes, which can vary significantly based on the rules of the voting system and the distribution of votes. They provide valuable insights into fairness and representation in decision-making structures.
Proportionality: Proportionality refers to the principle that a certain outcome or resource allocation should be proportional to the input or contribution made by each party involved. This concept plays a vital role in ensuring fairness in systems such as voting power and cost allocation, where it is essential to balance influence and responsibility according to the stakes of each participant. By applying this principle, stakeholders can achieve a more equitable distribution of power and costs, reflecting their respective contributions accurately.
Shapley Value: The Shapley value is a solution concept in cooperative game theory that assigns a unique distribution of a total surplus generated by a coalition of players based on their contributions. It emphasizes fairness and ensures that each player's payoff reflects their individual contribution to the coalition's overall success. This concept has significant implications in areas such as voting power, cost allocation, and negotiations, highlighting how resources or benefits should be shared among participants.
Shapley-Shubik Index: The Shapley-Shubik Index is a measure of voting power in a cooperative game setting that evaluates the influence of each player based on their ability to change the outcome of a vote. This index calculates the proportion of all possible permutations of players in which a specific player is pivotal, meaning their vote can sway the decision from a loss to a win. It highlights not only the importance of individual votes but also how collective action and coalitions affect outcomes in scenarios such as political elections or cost allocation.
Swing voter: A swing voter is an individual who does not consistently vote for one political party and can be influenced to vote for different candidates in different elections. This unpredictability makes swing voters critical in elections, especially in closely contested races where their decisions can determine the outcome. They often represent a demographic that is not firmly aligned with the traditional party ideologies, making them a target for campaign strategies aimed at appealing to their diverse interests and concerns.
Symmetry: Symmetry refers to a situation in which two or more elements are arranged in a balanced and proportionate manner, creating an equality or equivalence among them. In the context of voting power and cost allocation, symmetry is crucial as it ensures that all participants have equal influence or responsibility, leading to fairer outcomes and more equitable distributions of costs or votes.
Voting power indices: Voting power indices are quantitative measures that assess the influence or power of individual voters or groups in a voting system, particularly in scenarios involving collective decision-making. These indices help analyze how much sway a voter has in determining the outcome of votes based on the structure and rules of the voting system. They are essential for understanding the dynamics of power distribution, especially in weighted voting systems where not all votes carry equal weight.
Voting Weights: Voting weights refer to the relative influence or power assigned to different voters or groups within a voting system. This concept is crucial in analyzing how decisions are made in collective settings, where unequal voting power can significantly impact outcomes and resource allocation. Understanding voting weights allows for deeper insights into the dynamics of power distribution, consensus-building, and the implications for fairness and equity in decision-making processes.
Weighted voting games: Weighted voting games are a type of voting system where each voter has a different amount of voting power, represented by weights assigned to their votes. These weights reflect the influence each voter has in the decision-making process, meaning that not all votes carry the same weight when determining the outcome. In this context, weighted voting games are often analyzed to understand the distribution of voting power and how it affects collective decision-making and cost allocation among participants.
τ-value: The τ-value, or tau value, is a measure used to assess the influence of individual voters in a voting game or decision-making process. It quantifies the extent to which a voter's participation can affect the outcome, reflecting their power and significance in a collective decision-making scenario. This metric is particularly relevant in contexts involving voting power and cost allocation, as it helps to analyze how resources and responsibilities are distributed among participants based on their voting influence.