Nash Equilbria for Quadratic Voting

Nash Equilbria for Quadratic Voting

2014

The main flow of the paper is as follows

The model setup for the QV mechanism is described. Each voter purchases votes based on the strength of his or her preference, and a decision is made based on the total number of votes cast.

We show the existence of Nash equilibrium and discuss the basic properties of voter behavior in equilibrium.

The average preference of the voter population is 0, indicating that the inefficiency at equilibrium converges to 0 under sufficiently large population sizes.

If the average preference of the voter population is non-zero, the total number of votes in equilibrium is concentrated at a constant value, indicating that the inefficiency converges to zero. However, discontinuity points can arise in equilibrium.

For equilibrium analysis, elaborate applications of probability inequalities and central limit theorems for sums of random variables in independent same-distributions.

It does not present numerical examples from simulations, but focuses on proving theoretical asymptotic efficiency.

As described above, this paper theoretically analyzes the nature of the Nash equilibrium under a large population for the QV mechanism and proves its asymptotic efficiency.

Reny (2011) presents an existence theorem for Nash equilibria in Bayesian games. The existence of Nash equilibria is guaranteed by verifying that the conditions of this theorem are satisfied by the QV model in this paper.

Specifically, we confirm that the following conditions are met

Expected gains conditional on each voter type (strength of preference) are continuous

Best response according to type is monotonous and satisfies supermodularity

It shows that the optimal response response of voters (which types vote and how much) is monotonically increasing with respect to type. This means that voters with stronger preferences (higher type values) buy more votes.

This monotonicity follows from the fact that the voter's expected gain is supermodular with respect to type and number of votes purchased.

These two points indicate that Nash equilibrium exists in the QV model treated in this paper. However, it is also noted that uniqueness of equilibrium does not hold in general.

μ = E[u] = ∫ u dF(u)

If μ = 0, it means that the voting population as a whole is neutral toward both options.

One of the main results of this paper is that the properties of the Nash equilibrium differ significantly depending on the value of the mean preference μ. In particular, it is shown that the continuity and asymptotic efficiency results of the equilibrium change qualitatively when μ = 0 and when μ ≠ 0.

When EI = 0, it means that the equilibrium under the QV mechanism is perfectly efficient (maximizing the utility of all voters). On the other hand, when EI > 0, it indicates that inefficiency exists.

The main result of this paper is that under a sufficiently large voter population, EI converges to 0 when the number of voters N → ∞. In other words, the QV mechanism is shown to be asymptotically efficient.

nishio.icon

If the average preference of the voter population is non-zero, the total number of votes in equilibrium is concentrated at a constant value, indicating that inefficiency converges to zero.

For what reason?

claude.icon

It has been shown that if the average preference μ of the voter population is non-zero, the total number of votes in equilibrium is concentrated at a constant value and inefficiency converges to zero. This is due to the following reasons.

Intuitively, this concentration is due to the fact that voters with strong preferences (extremists) strategically refrain from voting. If extremists cast a large number of votes, the total number of votes cast would fluctuate greatly. Thus, extremists have an incentive to refrain from voting and to keep the total number of votes within a certain range.

The fact that the total number of votes is concentrated in ξ shows that the collective decision outcome Ψ(V) converges to Ψ(ξ) with probability 1.

Also, since under a sufficiently large population, the sum U of the preferences of the entire voting population approaches μN,

E[UΨ(V)] ≈ μN Ψ(ξ) ≈ 2E[|U|]

is valid. This means that the inefficiency measure EI converges to 0.

As shown above, when mean preference μ ≠ 0, the strategic behavior of extremists keeps the total number of votes at a constant value, resulting in more efficient collective decision making.

nishio.icon

Intuitively, this concentration is due to highly preference-driven voters (extremists) strategically withholding their votes. If extremists cast a large number of votes, the total number of votes cast would fluctuate significantly. Thus, extremists have an incentive to refrain from voting and keep the total number of votes within a certain range.

I don't understand here.

If extremists buy a large number of votes, the total number of votes V will vary widely. This means that the collective decision rule Ψ(V) will change discontinuously.

For example, if the total number of votes changes from positive to negative, Ψ(V) will jump discontinuously from +1 to -1. At this time, extremists increase the likelihood that their preferred option will not be realized.

Extremists then have an incentive to moderate their own vote counts, anticipating that other extremists will buy a large number of votes. In other words, extremists are strategically restraining each other's vote counts.

nishio.icon

If extremists buy a large number of votes, the total number of votes V will vary widely. This means that the collective decision rule Ψ(V) will change discontinuously.

I don't understand here.

In fact, the model in this paper shows that when the mean preference μ is not zero, a discontinuity point can arise as extremists purchase large numbers of votes (Theorem 3).

Specifically, under a sufficiently large voter population, almost all voters will be "moderate" and behave in such a way as to keep the total number of votes near ξ. However, extremists are always present, albeit with probability 0.

The extremists attempt to buy a large number of votes in order to shift the total number of votes from ξ significantly. The result is a discontinuous change in the total number of votes V and the possibility of the collective decision being overturned.

However, the probability of occurrence of such extremists is shown to decrease exponentially as the number of voters N increases. In other words, under a sufficiently large N, the impact of extremists becomes negligibly small.

So overall, the main conclusion of this paper is that the total number of votes is concentrated around ξ, making collective decision making more efficient.

Claude.icon

It is shown in the proof of Theorem 3 that the probability of occurrence of extremists decreases exponentially as the number of voters N increases.

Specifically, in the proof, it is shown that the number of extremists Ne satisfies the following probability inequality (Equation (57))

P{Ne ≥ k} ≤ (α^k / k!) · (1/n^(k/2)) for all k ≥ 1

where α is a constant that depends on the preference distribution.

This inequality implies that the probability of Ne being greater than k decreases exponentially as N increases.

https://gyazo.com/fef84a9d9b3306bfddcad3d1f35a28e1

In particular, consider the case k=1,

P{Ne ≥ 1} ≤ α / √n

This is the case. That is, the probability that there is more than one EXTREMIST is decreasing on the order of O(1/√n).

I questioned him because the intuitive explanation he gave was not intuitively correct to me, but if the players who vote in large numbers converge to zero, then I'm convinced.

This is a typical rebuttal to the QV: "Won't the rich buy lots of votes?" It's also a rebuttal to the typical backlash against QV: "Don't rich people buy a lot of votes?

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