It’s very likely that you have, at some point, debated about what the perfect election system would look like. You probably believe that a fair system should respect, for example, the absence of a dictator. There are some other characteristics many people would agree a fair election system should have. The economist Kenneth J. Arrow (1921-2017) axiomatized some of these characteristics in his famous impossibility theorem (that he called “possibility” theorem) in his book ‘Social Choice and Individual Values’ and showed that no election system could satisfy all these fairly general and easy to agree principles. Let’s see what axioms he imposed.

Description of the axioms:

Universal Domain

All individuals i ∈ G have rational preferences over the set of alternatives X, but beyond that, they can have any set of rational orderings.

Weak Pareto Principle

If every member of G prefers a to b, then the social decision rule must prefer a to b.

Formally, if a ≿_i b ∀i ⇒ a ≿ b.

Independence of Irrelevant Alternatives

Suppose we have two different societies G and G’, but within G and G’ everyone has the same orderings of alternatives an and b. Then the social ordering between an and b in both societies must be the same, even if members of G and G’ have different rankings of other alternatives. This is the assumption relaxed in many real life voting systems.

No dictatorship

There is no particular individual i* ∈ G such that the preferences of i* determine the social ranking, regardless of other group members.

Arrow’s Impossibility Theorem

Suppose there are at least three alternatives. There is no social ranking function such that for any group G whose members all have rational preferences, the social ranking is a rational (complete and transitive) ranking and satisfies the Universal Domain, Weak Pareto Principle, Independence of Irrelevant Alternatives and No Dictatorship axioms.

Here’s an image about it with an example and intuition.

And here’s an image with a more formal treatment of the theorem.

Proof:

Does the result still hold if we relax some axioms?

Here’s where Gibbard-Satterthwaite Theorem enters the game. The theorem states the following.

Gibbard-Satterthwaite Theorem:

In every deterministic ordinal electoral system, at least one of the following conditions must hold:

1. The rule is dictatorial.

2. The rule limits the possible outcomes to only two alternatives.

3. The rule is not strategyproof (concept from Mechanism Design that means that players of a game have incentives to report their true preference).

4. The rule does not yield a single winner.

Muller-Satterthwaite Theorem:

Assume at least three alternatives. No resolute voting rule (voting rule that selects a single winner) satisfies strong monotonicity (which means that if a is elected, and then some voters like an even more than before, a should not lose, i. e., improving the winner’s position doesn’t hurt the winner), non-imposition (no alternative is ruled out a priori), and non-dictatorship. Equivalently, there is no social choice function that satisfies these properties.

Bad news for ranked electoral systems lovers, but at the same time it’s amazingly beautiful that we can prove these things mathematically. What do you think about it?

References

Arrow, K. J. (2012). Social Choice and Individual Values. Yale University Press. http://www.jstor.org/stable/j.ctt1nqb90

Geanakoplos, J. (2005). Three Brief Proofs of Arrow’s Impossibility Theorem. Economic Theory, 26(1), 211–215. http://www.jstor.org/stable/25055941

Shoham, Y., & Leyton-Brown, K. (2008). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press.

Monotone Comparative Statics (MCS) deals with how the solutions to an optimization problem change with its parameters. It can deliver interesting results under relatively weak assumptions. One of its main tools is Topkis’ Theorem, which allows us to derive comparative statics results without differentiability assumptions and without explicitly solving the model. In order to understand the theorem, we need to know some concepts first. In every concept I introduce, I will adjunct an image with an example for better understanding. The theorem will also have an image with an example and the intuition behind it. Let a,b ∈ R^n. We define:

Meet and Join

We define the meet and join of an and b as the vectors of minimum and maximum coordinate-wise comparison of an and b, respectively.

Strong Set Order
A set A ⊆ R^n is greater than a set B ⊆ R^n in the strong set order if, for every a ∈ A and b ∈ B, (a ∨ b)∈ A and (a ∧ b) ∈ B.

Lattice
A lattice is a set X ⊆ R^n such that (x ∨ y) ∈ X and (x ∧ y) ∈ X for every x,y ∈ X.

Increasing Differences
A function f has increasing differences in (x,θ) if whenever xᴴ ≥ xᴸ and θᴴ ≥ θᴸ,

f(xᴴ,θᴴ) − f(xᴸ,θᴴ) ≥ f(xᴴ,θᴸ) − f(xᴸ,θᴸ).

Supermodularity
A function f:X×Θ→R is supermodular in x if

f(x∨y,θ) − f(x,θ) ≥ f(y,θ) − f(x∧y,θ).

Topkis’ Theorem (1978)
Let x ∈ X and θ ∈ Θ. If X ⊆ R^n is a lattice, Θ ⊆ R^m and f:X×Θ→R has increasing differences in (x,θ) and is supermodular in x, then the solution set X*(θ) is increasing in the strong set order.

Single Crossing

What’s the minimal (necessary and sufficient) condition for Monotone Comparative Statics? Milgrom and Shannon (1994) answered this question. What we need is the Single-Crossing property:

A function f:X×Θ→ℝ is single-crossing in (x,θ) if whenever

xᴴ ≥ xᴸ and θᴴ ≥ θᴸ, we have

f(xᴴ,θᴸ) ≥ f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) ≥ f(xᴸ,θᴴ),

and

f(xᴴ,θᴸ) > f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) > f(xᴸ,θᴴ).

Increasing differences implies single-crossing, but the converse is not true because single-crossing means that the ranking between xᴴ and xᴸ can’t reverse, whereas increasing differences means that the increase in the function from going from xᴸ to xᴴ must weakly increase, which is a stronger condition.

Milgrom and Shannon’s Theorem (1994)

Topkis showed that increasing differences and supermodularity are sufficient conditions for Monotone Comparative Statics. Milgrom and Shannon showed that single-crossing is the minimal condition for Monotone Comparative Statics (it’s necessary and sufficient, if and only if).

If f is single-crossing in (x,θ), then the solution correspondence

X*(θ) = argmax f(x,θ)

ₓ∈X

is increasing in the strong set order.

Conversely, if X*(θ) is increasing in the strong set order for every choice set X⊆ℝ, then f is single-crossing in (x,θ).

Application: Consumer Theory

In the next image, we have an application of Topkis’ Theorem to Consumer Theory in both the nondifferentiable and differentiable function cases.

The differentiable case example comes from Amir, R. (2005).

References

Amir, R. (2005). Supermodularity and Complementarity in Economics: An Elementary Survey. Southern Economic Journal, 71(3), 636–660. https://doi.org/10.2307/20062066

Milgrom, P., & Shannon, C. (1994). Monotone Comparative Statics. Econometrica, 62(1), 157–180. https://doi.org/10.2307/2951479

Topkis, D. M. (1978). Minimizing a Submodular Function on a Lattice. Operations Research, 26(2), 305–321. http://www.jstor.org/stable/169636