The Mathematical Impossibility of Fair Systems

Arrow's Impossibility Theorem

The Basic Problem

In 1951, economist Kenneth Arrow proved something shocking: there is no perfect voting system. More precisely, any ranked voting method that tries to convert individual preferences into a collective decision must violate at least one principle we'd consider essential to fairness.

This isn't a matter of not being clever enough—it's mathematically impossible, like trying to find the largest integer.

Arrow's Conditions (What We Want)

Arrow identified five reasonable conditions a fair voting system should satisfy:

1. Unrestricted Domain (Universality): The system should work for any possible set of individual preferences—voters can rank candidates in any order they wish.

2. Non-Dictatorship: No single voter should always determine the group outcome regardless of others' preferences.

3. Pareto Efficiency (Unanimity): If every single voter prefers option A over option B, the system should rank A above B in the final result.

4. Independence of Irrelevant Alternatives (IIA): The relative ranking between two options should only depend on voters' preferences between those two options—adding or removing a third option shouldn't change whether the group prefers A to B.

5. Transitivity: If the group prefers A to B, and B to C, it should prefer A to C (the results should be logically consistent).

The Theorem

Arrow proved that with three or more alternatives, no rank-order voting system can simultaneously satisfy all five conditions. You must sacrifice at least one.

Why This Matters: Real Examples

The Spoiler Effect (IIA Violation)

Imagine an election: - 40% prefer: Progressive > Moderate > Conservative - 35% prefer: Conservative > Moderate > Progressive
- 25% prefer: Moderate > Progressive > Conservative

The moderate might win in a head-to-head against either opponent. But in a plurality vote, the conservative wins with 35% because the progressive "spoils" the moderate's chances by splitting the left-leaning vote. Adding or removing the progressive changes who wins between moderate and conservative—violating IIA.

This happened in the 2000 U.S. Presidential election, where many argue Nader's presence affected the Gore-Bush outcome.

Condorcet Paradoxes (Transitivity Violations)

Consider three voters choosing between A, B, and C: - Voter 1: A > B > C - Voter 2: B > C > A - Voter 3: C > A > B

Using majority rule for pairwise comparisons: - A beats B (voters 1 & 3) - B beats C (voters 1 & 2) - C beats A (voters 2 & 3)

We get a cycle: A > B > C > A. There's no consistent "winner"—the collective preference is intransitive, even though each individual's preferences are perfectly logical.

The Apportionment Problem

A related but distinct impossibility involves dividing seats in a legislature among states or districts based on population.

The Requirements (What Seems Reasonable)

The U.S. Constitution requires representatives be apportioned by population, which seems straightforward. But we also want:

1. House Monotonicity: If the total number of seats increases, no state should lose seats
2. Population Monotonicity: If state A grows faster than state B, A shouldn't lose seats to B
3. Quota Rule: Each state's share should be either the lower or upper whole number of its exact proportional share

The Impossibility Results

Balinski-Young Theorem (1980s): No apportionment method can simultaneously satisfy quota and avoid the population paradox (where a faster-growing state loses representation).

Real Historical Examples:

• Alabama Paradox (1880s): Under the Hamilton method, when the House size increased from 299 to 300 seats, Alabama lost a seat despite populations remaining constant.

• Population Paradox (1900s): Virginia grew faster than Maine but would have lost a seat to Maine under certain methods.

• New State Paradox: Adding Oklahoma as a state in 1907 would have changed seat distributions among existing states.

Current Compromise

The U.S. currently uses the Huntington-Hill method, which violates the quota rule to avoid paradoxes. No method avoids all problems—we choose which flaw we can live with.

Why These Results Are Profound

1. The Problems Are Structural, Not Solvable

These aren't bugs to be fixed with better design. The contradictions are embedded in the mathematics itself. Like the uncertainty principle in physics, this is a fundamental limit on what's possible.

2. Every System Makes a Hidden Choice

Since perfect fairness is impossible, every voting or apportionment system reflects a choice about which fairness criterion to violate:

• Plurality voting: Violates IIA (spoiler effects)
• Instant Runoff (Ranked Choice): Also violates IIA and can fail monotonicity (getting more votes can make you lose!)
• Borda Count: Vulnerable to irrelevant alternatives and strategic voting
• Approval Voting: Forces binary choices, losing preference intensity information

3. Strategic Manipulation Is Inevitable

The Gibbard-Satterthwaite theorem (1973) extends this further: any reasonable voting system with three+ alternatives can be strategically manipulated—sometimes voters benefit by voting dishonestly.

4. Implications for Democracy

This doesn't mean democracy is futile, but it does mean:

• We should be humble about claims that any system is "perfectly fair"
• Debates about electoral systems involve genuine tradeoffs, not right/wrong answers
• The stability of democracy depends partly on shared norms beyond pure mathematics
• Context matters—different systems may be better for different situations

Practical Responses

1. Choose Your Compromise

Understanding the tradeoffs helps select appropriate systems: - Plurality: Simple but prone to spoilers; works okay with two parties - Ranked Choice: Reduces spoilers but can have non-monotonicity - Score Voting: Avoids some paradoxes but assumes cardinal utilities - Condorcet Methods: Find majority-preferred winners when they exist

2. Reduce Dimensionality

Many paradoxes require three+ alternatives. Two-party systems (despite other flaws) avoid some mathematical impossibilities. Primary systems effectively reduce choices in stages.

3. Accept Imperfection

The search isn't for perfect systems but for good-enough ones that people accept as legitimate. Social stability and shared values matter as much as mathematical properties.

4. Context-Dependent Solutions

• Small committee decisions might use different methods than national elections
• Some contexts prioritize consensus (Condorcet methods)
• Others prioritize simplicity and public understanding (plurality)

The Deeper Meaning

Arrow's theorem reveals something profound about collective decision-making: individual rationality doesn't automatically aggregate into collective rationality. Just because each person has clear, consistent preferences doesn't mean the group will.

This connects to broader limits on formalization—like Gödel's incompleteness theorems showing limits on mathematical proof systems, or the halting problem showing limits on computation. Some problems have no algorithmic solution.

For democracy and representation, this means governance is inherently an art, not just a science. Mathematics can illuminate the tradeoffs, but cannot provide a formula for perfect fairness. The legitimacy of institutions ultimately rests on more than their mathematical properties—on shared values, transparent processes, and mutual acceptance of necessary compromises.

The impossibility isn't a reason for despair—it's a call for informed humility in institutional design.

Okay, let's delve into the mathematical impossibility of fair apportionment and the broader concept of Arrow's Impossibility Theorem, which explains why achieving a truly "fair" voting system is inherently problematic.

Part 1: The Impossibility of Fair Apportionment (The Apportionment Problem)

The apportionment problem arises when you need to divide a fixed number of items (typically seats in a legislature) among a set of groups (typically states or districts) based on population size. The key difficulty is that population sizes rarely divide perfectly into the number of items to be allocated. This leads to fractional shares and the need to round. The rounding process, however, inevitably creates imbalances and can lead to paradoxical results that violate seemingly intuitive notions of fairness.

The Core Problem: Rounding and Discrepancies

Imagine you have 100 seats in a legislature to allocate to three states: A, B, and C. Here's a hypothetical scenario:

• State A: Population = 1,050,000; Ideal Share of Seats = 52.5
• State B: Population = 700,000; Ideal Share of Seats = 35.0
• State C: Population = 450,000; Ideal Share of Seats = 22.5

The total population is 2,200,000. We calculate the "ideal" share of seats for each state by dividing its population by the total population and multiplying by the total number of seats (100). The problem is these ideal shares are almost never whole numbers. We need to round them to whole numbers to allocate the actual seats.

Apportionment Methods: A History of "Solutions" (and Their Flaws)

Over time, various methods have been proposed to address the apportionment problem. Each method has its own logic and potential for biases. Here are a few key examples, along with their inherent flaws:

1. Hamilton's Method (Vinton's Method):

   • Process:

     1. Calculate the standard quota for each state (as shown above).
     2. Give each state its lower quota (the integer part of its standard quota).
     3. Assign the remaining seats (if any) one at a time to the states with the largest fractional parts (remainders) until all seats are allocated.

   • Example:

     • State A: Lower quota = 52; Remainder = 0.5
     • State B: Lower quota = 35; Remainder = 0.0
     • State C: Lower quota = 22; Remainder = 0.5

     Initially, A gets 52, B gets 35, and C gets 22 (total 109). Since we have 1 seat still, it goes to A since it has the largest remainder. Thus A = 53, B = 35, C = 22.

   • Problems:

     • Alabama Paradox: Increasing the total number of seats can decrease the number of seats a state receives. This is counterintuitive because a larger legislature should, in principle, increase representation for everyone.
     • Population Paradox: A state can lose a seat to another state even if its population grows faster than the other state's population. This violates the principle that growth should be rewarded.
     • New States Paradox: Adding a new state can change the number of seats allocated to existing states.

2. Jefferson's Method:

   • Process:

     1. Choose a divisor (a modified population per seat). This is usually an integer.
     2. Divide each state's population by the divisor.
     3. Round each quotient down to the nearest whole number.
     4. If the total number of seats is not equal to the total number of seats to be allocated, adjust the divisor and repeat steps 2 and 3 until the total number of seats is correct.

   • Problems:

     • It always favors larger states. Smaller states tend to be underrepresented relative to their population.

3. Webster's Method (Method of Greatest Divisors):

   • Process:

     1. Choose a divisor.
     2. Divide each state's population by the divisor.
     3. Round each quotient to the nearest whole number (instead of always down or up).
     4. Adjust the divisor until the total number of seats is correct.

   • Problems:

     • While it's considered more balanced than Jefferson's, it still has potential to violate the population paradox, although it's less likely.

4. Hill-Huntington Method (Method of Equal Proportions):

   • Process: This method uses a geometric mean to determine the priority for allocating seats. It assigns a priority number to each state based on its population divided by the geometric mean of the number of seats it currently has and the number of seats it would have if it received the next seat.

     • The geometric mean of n and (n+1) is sqrt(n(n+1)).

   • Problems:

     • Still not perfectly fair. Some argue it favors larger states (though less so than Jefferson's).
     • It is currently used by the US Congress.

The Impossibility Result:

What all these examples show is that there's no apportionment method that can simultaneously satisfy a reasonable set of fairness criteria. These include:

• Quota Rule: A state's allocation should be either its lower quota (the integer part) or its upper quota (the integer part + 1). It shouldn't be dramatically different from its "fair" share.
• Avoiding Paradoxes: The Alabama, Population, and New States paradoxes should be avoided.
• Population Monotonicity: If state A's population grows faster than state B's, and no other changes occur, state A should not lose seats to state B.

A result often attributed to Balinski and Young (although related results exist earlier) essentially says: No apportionment method can satisfy both the quota rule and avoid all the paradoxes.

This mathematical impossibility is a key reason why debates about apportionment are so contentious and often lead to legal challenges. Any method chosen will inevitably lead to some form of perceived unfairness.

Part 2: Arrow's Impossibility Theorem (The General Voting Problem)

Arrow's Impossibility Theorem is a more general result that applies to any voting system used to rank multiple alternatives (e.g., candidates in an election). It states that it is impossible to design a social welfare function (i.e., a voting rule) that satisfies all of the following desirable conditions:

The Conditions (Axioms) of Arrow's Theorem:

1. Universal Domain (Unrestricted Domain): The rule must be able to handle any possible set of individual preferences (rankings) over the alternatives. Voters can have any preference ordering they want. The voting system must be able to produce a social ranking for every possible combination of individual rankings.
2. Non-Dictatorship: There is no single voter whose preferences automatically become the group's preferences, regardless of what everyone else thinks. No one person's preferences should completely determine the outcome.
3. Pareto Efficiency (Unanimity): If every voter prefers alternative A to alternative B, then the group preference must also prefer A to B. If everyone agrees on the ranking of two alternatives, the outcome should reflect that agreement. This is a very weak and seemingly obvious criterion of fairness.
4. Independence of Irrelevant Alternatives (IIA): The social ranking of two alternatives (A and B) should depend only on how individual voters rank those two alternatives, and not on how they rank any other "irrelevant" alternative. If, for example, everyone prefers A to B, introducing a new candidate C should not change the group's preference of A over B. This is perhaps the most controversial of the conditions.

The Impossibility Conclusion:

Arrow's Impossibility Theorem states that if there are three or more alternatives, no voting rule can simultaneously satisfy all four of these conditions. In other words, any voting system that satisfies Pareto efficiency, non-dictatorship, and the universal domain, must violate the independence of irrelevant alternatives (IIA).

Why IIA is the Usual Victim (and Why it Matters):

IIA is usually the condition that gets violated in real-world voting systems. This means that the presence or absence of "irrelevant" candidates can influence the outcome of the election between two other candidates. This can lead to strategic voting and unexpected results.

Examples of Voting Systems and Their Violations:

• Plurality (First-Past-the-Post): Voters choose their favorite candidate. The candidate with the most votes wins.
  • Violates IIA: Imagine three candidates A, B, and C. A wins with 40% of the vote, B gets 35%, and C gets 25%. If C drops out, B might win, even though voters' preferences between A and B haven't changed.
• Instant Runoff Voting (Ranked Choice Voting): Voters rank the candidates in order of preference. The candidate with the fewest first-place votes is eliminated, and their votes are redistributed to the voters' next preferred candidate. This process is repeated until one candidate has a majority.
  • Violates IIA: The "spoiler" effect. A candidate with little chance of winning can change the outcome between two leading candidates, even if the voters' preferences between those two leaders remain the same.
• Borda Count: Voters rank the candidates. Each candidate receives points based on their ranking (e.g., highest ranked gets the most points). The candidate with the most points wins.
  • Violates IIA: The ranking of other "irrelevant" alternatives directly influences the scores, and thus the outcome, of the relevant alternatives.

Implications of Arrow's Theorem:

Arrow's Impossibility Theorem is a profound result with significant implications for political science, economics, and decision-making in general. It tells us:

• No Perfect Voting System Exists: There is no universally "best" or perfectly "fair" voting system. Any system we choose will have potential flaws and can lead to outcomes that some people consider unfair.
• Trade-Offs are Inevitable: When designing a voting system, we must make trade-offs between desirable properties. We must decide which criteria are most important to us and be willing to accept violations of other criteria.
• Strategic Voting: The impossibility theorem encourages strategic voting. Voters may not always vote for their true favorite, but instead vote strategically to try to influence the outcome in their favor.
• Context Matters: The "best" voting system for a particular situation may depend on the specific context, including the number of voters, the number of alternatives, and the desired properties.

In Conclusion:

Both the apportionment problem and Arrow's Impossibility Theorem highlight the inherent difficulties in achieving perfectly fair allocation or decision-making processes. They demonstrate that mathematical constraints can limit our ability to create systems that satisfy all of our intuitive notions of fairness. Understanding these limitations is crucial for designing more robust and transparent systems and for engaging in informed discussions about the fairness and legitimacy of democratic processes. It forces us to critically examine the properties of different systems and to be aware of the potential for unintended consequences and strategic manipulation.