abramdemski comments on Voting-like mechanisms which address size of preferences?

[abramdemski]

Oooh, is it really that simple?

Expected state of affairs if I don’t buy one more vote (ignoring what else I could have done with the money):

${\sum }_i{P}_i{U}_i$, where $P_i$ is the probability of candidate $i$ winning if I do nothing, and $U_i$ is the utility of candidate $i$ winning.

Expectation if I do buy one more vote:

$(1-ϵ)\left({\sum }_{i\ne x}\right(P_i{U}_i\left)\right)+(ϵ+P_x)(U_x-c)$, where $x$ is the candidate under consideration, $ϵ$ is the probability there would have been a tie w/​o this one extra vote, and $c$ is the utility adjustment for losing your money in the case of a win. (I’m going to go ahead and pretend $c$ is precisely the monetary cost, for convenience.)

So this is worth it when the expected benefits outweigh the expected cost:

$c(P_x+ϵ)<ϵ(U_x-{\sum }_{i\ne x}(P_i{U}_i\left)\right)$

So the price at which we’re indifferent would be:

$c=\frac{ϵ(U_x-{\sum }_{i\ne x}(P_i{U}_i\left)\right)}{P_x+ϵ}$

Hm, this is weird.

• We were looking for $P_x$ to disappear, indicating that the probability of a candidate no longer factors into our considerations. It didn’t disappear. In fact, improbable candidates now look better, because we know we get to avoid the cost of paying for votes (guaranteed refund).

• We were looking for $c$ to be proportional to $U_x$. So $c=ϵU_x$ would be fine.$c=ϵ(U_x-{\sum }_i{P}_i{U}_i)$ would also be fine; that just means $c$ is proportional to a normalized utility, where we normalize compared to the baseline of expected election outcomes. But instead we’re normalizing compared to the expectation minus the component from candidate $x$, which is weird—it means we’re not really normalizing at all (because this adjustment will be different for different values of $x$).

[sxae]

We were looking for $P_x$ to disappear, indicating that the probability of a candidate no longer factors into our considerations.

I’m surprised that we are looking for $P_x$ to disappear entirely, I’m not sure I understand that. Quadratic voting shines when you have lots of votes with the same voting token pool, because you force people to allocate resources to decisions they really care about. It’s absolutely not meant to decide one decision—it’s meant to force people to allocate limited resources over a long period, and by doing so reveal their true valuation of those decisions. I would therefore fully expect $P_x$ to play a part in every agent’s considerations, as they must consider the probability of success in each vote in order to plan allocation of voting tokens for every other vote.

[abramdemski]

Interesting. But then how do you argue that it gives approximately correct results? As I understand it, Weyl sees the argument as just: votes end up being roughly proportional to utility (under a lot of differenc scenarios/​assumptions). When this condition holds, the quadratic vote is a good representation of the utilitarian value of the different options.

So, the reason I think we’re looking for $P_x$ to disappear entirely is because $P_x$ is a function of $x$! It’s fine if $c=\alpha +\gamma \frac{U_x-\sqrt{\zeta }}{BB\left(7\right)}$ or whatever, so long as none of those extra terms are a function of $x$, so that in the end $c$ is still proportional to (some normalization of) $U_x$.

You’re effectively arguing that it’s OK to divide by $P_x+ϵ$, because this just represents voters rationally investing less in cases where $P_x$ will probably win anyway. But this means the quadratic vote systematically undervalues the candidates who are seen as the probable winners!

Quadratic voting shines when you have lots of votes with the same voting token pool, because you force people to allocate resources to decisions they really care about. It’s absolutely not meant to decide one decision—it’s meant to force people to allocate limited resources over a long period, and by doing so reveal their true valuation of those decisions.

OK, but this should mean the quantity we are willing to spend on an election is overall adjusted up or down based on properties of that particular election (IE, how much the issue matters to us). This should not mean a dependence on $P_x$; a dependence on $P_x$ distorts the vote, compromising it as a representation of collective utility.

[sxae]

Interesting, you make some great points here and I don’t think I have any good refutations to any of them. Perhaps if we play around with the auction structure by which we take away and refund these tokens?