Social Welfare Function & Arrow’s Theorem

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Questions and Answers

In the Adjusted Winner procedure, what is the significance of fractional allocation, and under what circumstances is it required?

Fractional allocation ensures a precise balance of total valuations, achieving equity. It’s required when whole items cannot create an equal division, allowing for a portion of an item to be allocated to equalize the total value received by each party.

Explain why achieving a fair division becomes more complex when dealing with more than two people, referencing specific properties (envy-free, equitable, efficient) that may be difficult to satisfy simultaneously.

With more than two people, it’s difficult to satisfy envy-free, equitable, and efficient allocations simultaneously because individual preferences and possible coalitions increase exponentially. Meeting one condition often undermines others, leading to trade-offs.

Describe the purpose of the Shapley value method and provide a brief explanation of how it determines the allocation of costs or benefits among participants.

The Shapley value fairly allocates costs or benefits among participants based on their marginal contributions to every possible coalition. It averages the contributions of each participant across all possible orderings to determine their share.

Outline the steps required to compute the Shapley value for a scenario involving three participants and explain why considering all possible orderings is essential.

List all orderings of participants. Compute each participant’s marginal contribution for each ordering. Average the contributions. Considering all possible orderings ensures impartiality, as it accounts for every sequence in which participants could contribute, preventing bias towards any individual or group. Signup and view all the answers

In the context of matching and assignment models, briefly describe the core mechanism that enables the Deferred Acceptance Algorithm to produce a stable outcome.

The Deferred Acceptance Algorithm achieves stability through a process where one side proposes, and the other accepts or rejects based on preference. Rejections are only temporary, and rejected proposers then propose to their next preferred choice. The algorithm terminates when no more changes occur and all assignments are stable. Signup and view all the answers

Explain the purpose and outcome of the Top Trading Cycles algorithm. Why does it result in a Pareto-efficient allocation?

The Top Trading Cycles algorithm is used in school choice to assign students to schools by allowing them to trade their current 'endowments' (initially their existing rights or priorities). It results in a Pareto-efficient allocation because students receive their most preferred schools without making anyone else worse off, as trades continue until all students are assigned. Signup and view all the answers

Differentiate between 'expectation damages' and 'reliance damages' in contract law, illustrating with an example how each type of damage compensates the harmed party.

Expectation damages compensate for the benefit the harmed party expected to receive from the contract (e.g., lost profits). Reliance damages compensate for costs incurred in reliance on the contract (e.g., expenses made preparing for the contract) whether or not there was a benefit. Signup and view all the answers

Contrast 'strict liability' with 'negligence' in terms of the conditions under which a defendant can be held liable for damages or harm.

Under strict liability, a defendant is responsible regardless of their precautions (i.e., if harm occurs, they pay). Under negligence, a defendant is liable only if they failed to take reasonable precautions to prevent the harm. Signup and view all the answers

Explain why the Condorcet voting method might be considered superior to simple plurality voting in certain elections. What specific flaw of plurality voting does Condorcet address?

The Condorcet method ensures the winner is preferred by a majority over every other candidate in a pairwise comparison, addressing plurality’s flaw where a candidate can win without majority support due to vote splitting among other candidates. Signup and view all the answers

Describe a real-world scenario where applying the Shapley Value method would be more appropriate than a simple proportional allocation.

In sharing the cost of a shared resource like an irrigation system, Shapley Value would be more appropriate. Farmers who benefit more from the system due to larger land sizes should contribute more. Shapley Value would fairly allocate cost compared to proportional allocation, which may not account for individual benefits of the irrigation system. Signup and view all the answers

Explain how Arrow's Impossibility Theorem demonstrates a fundamental challenge in designing fair voting systems. Briefly describe one of the conditions that it states cannot be simultaneously satisfied.

Arrow's Impossibility Theorem shows that no voting system can perfectly translate individual preferences into a collective decision while satisfying a set of fairness criteria. One such criterion is the Independence of Irrelevant Alternatives (IIA), which states that the ranking between two options should not change depending on the presence or absence of other options. Signup and view all the answers

Describe a scenario in which the Plurality Voting system might not accurately reflect the preferences of the majority of voters. What is one potential consequence of this?

In a scenario with more than two candidates, the plurality voting system can result in a winner who receives less than 50% of the votes. A potential consequence is that the elected candidate may not have broad support from the electorate, leading to dissatisfaction and potential instability. Signup and view all the answers

Explain how the Borda Count method attempts to improve upon Plurality Voting. What is a major drawback of the Borda Count method?

The Borda Count method attempts to improve upon Plurality Voting by considering the full ranking of each voter's preferences, assigning points based on rank. A major drawback is that it is susceptible to strategic voting, where voters may misrepresent their true preferences to manipulate the outcome. Signup and view all the answers

What is a Condorcet Winner? Explain why a Condorcet Winner might not always exist in an election.

A Condorcet Winner is a candidate who would win in a head-to-head election against every other candidate. A Condorcet Winner might not exist due to cyclic preferences, where voter preferences create a cycle such as A > B, B > C, and C > A, making it impossible for any candidate to beat all others in pairwise comparisons. Signup and view all the answers

Describe a situation where the 'Instant Runoff' (Hare) system might produce a different outcome than a Plurality Voting system. In what way might it be considered more representative?

In a situation with multiple candidates where no candidate receives a majority of first-place votes, the Instant Runoff system eliminates the lowest-ranked candidate and redistributes their votes until one candidate achieves a majority. This can lead to a different winner than Plurality. It can be considered more representative as it ensures the winner has support from a majority of voters, not just a plurality. Signup and view all the answers

Explain the concept of 'single-peaked' preferences in the context of social choice theory. Why are single-peaked preferences important for avoiding paradoxical voting outcomes?

Single-peaked preferences occur when voters have a clear 'best' option, and their satisfaction decreases as options move further away from this ideal point. Single-peaked preferences are important because they help avoid cyclical voting patterns such as the Condorcet Paradox, ensuring the Majority Ranking Rule produces a valid and stable outcome (Black's Theorem). Signup and view all the answers

In the context of fair division, explain the difference between an 'efficient' allocation and an 'equitable' allocation. Is it possible for an allocation to be efficient without being equitable, or vice versa?

An 'efficient' allocation means that it is impossible to make one person better off without making someone else worse off (Pareto efficiency). An 'equitable' allocation means each person receives the same perceived value. It is possible for an allocation to be efficient without being equitable; for example, one person could have most of the resources while no one can be made better off without reducing that person's resources. The reverse is also possible, if an equitable allocation is not pareto efficient. Signup and view all the answers

Briefly describe the 'Divide & Choose' method for fair division. What are its limitations, and in what scenarios is it most applicable?

In 'Divide & Choose', one person divides an item (e.g., a cake) into two pieces, and the other person chooses which piece they want. Its limitation is that it only works for two people and single divisible goods. It is most applicable when dividing physical assets where subjective valuation matters. Signup and view all the answers

Flashcards

Social Welfare Function (SWF)

A rule for combining individual preferences into a collective societal choice.

Arrow's Impossibility Theorem

No SWF can simultaneously satisfy Pareto Efficiency, Independence of Irrelevant Alternatives, Monotonicity and Non-Dictatorship.