Marginal value (or lack thereof) of voting

By Vasco Grilo🔸 @ 2024-03-11T09:01 (+7)

Summary

• I built a model in Desmos which outputs the marginal value of voting for a party in $ as a function of the fraction of the votes it is expected to get (more).
• I concluded voting for parties expected to get few votes is generally worse than for one’s preferred party among the ones expected to get many votes (more).
• I have the impression I had better not vote given my opportunity cost (more).

Introduction

It makes sense to vote for a party which at the margin one thinks should have more votes. Having this in mind, I have voted for parties which have a negligible chance of winning the election, but whose ideas are more neglected. By doing this, I increase in expectation their representation, and thus contribute towards their ideas being at least more discussed. However, parties expected to get fewer votes will have less power to implement their ideas, which decreases the value of voting for them.

I built a model in Desmos which outputs the marginal value of voting in $ accounting for the 2 opposing effects I mentioned. Then I assessed whether I had better not vote at all. On this topic, you can check:

• If you care about social impact, why is voting important? by Robert Wiblin.
• Casting the Decisive Vote by Toby Ord.

Model

I estimated the value $V\left(x\right)$ of the benefits linked to a party expected to get a fraction $x$ of the non-null non-blank votes multiplying:

• A constant $k$ equal to 48.9 k$ (= 1/0.95/0.501*24.5*10^3) which is a measure of how much money each vote can influence. I determined it from:
  • The number of elections per year in Portugal, where I live, of 0.95 (= 1/5 + 3/4):
    • European elections happen every 5 years.
    • Local, legislative and presidential elections happen every 4 years.
  • The fraction of people who could vote that voted for a single candidate of 50.1 % (= 0.5142*(1 - 0.0263)) in the last legislative elections, given:
    • 51.42 % of people who could vote did so.
    • 2.63 % (= 0.0113 + 0.0150) of votes were blank or null.
  • The gross domestic product (GDP) of Portugal in 2022 of 24.5 k$.
• A function $f\left(x\right)=a{x}^b$ describing the total value of the ideas of the party if it had full power to implement them, but did not get all of the votes, where:
  • $a$ is the total value of the ideas of the party if it had full power to implement them, and got all of the votes.
    • $a$ depends on the party.
    • I would argue economic growth is a good proxy for social impact. So I express $a$ as an acceleration in the annual growth of the real GDP of the country for one to be indifferent between it and a certain party getting all of the votes. Nonetheless, $a$ is supposed to account for non-economic effects too, including the ideology of the party.
    • I guess $a$ could range from -1 pp to 1 pp, which would concern an annual growth in the real GDP of 1 % (= 0.02 - 0.01) to 3 % (= 0.02 + 0.01) for a baseline growth of 2 %.
  • $b>=0$ is the elasticity of the total value of the ideas of the party if it had full power to implement them, but did not get all of the votes, with respect to x.
    • I guess $b$ should be about the same for all parties.
    • I believe $b$ should be lower than 1 such that there are decreasing marginal returns of the total value of the ideas as x increases.
• A function $g\left(x\right)=0<=x<=0.5:g_1\left(x\right),0.5<=x<=1:g_2\left(x\right)$ translating the power of the party to implement its ideas, where:
  • $g_1\left(x\right)=2^{c-1}{x}^c$ is the power of the party to implement its ideas if it is expected to get less than half of the votes, and ranges from 0 to 0.5.
    • $g_1\left(0\right)=0$, i.e. the party has no power if it is expected to get no votes.
  • $g_2\left(x\right)=1-2^{c-1}{(1-x)}^c$ is the power of the party to implement its ideas if it is expected to get more than half of the votes, and ranges from 0.5 to 1.
    • $g_2\left(1\right)=1$, i.e. the party has full power if it is expected to get all of the votes.
    • $g_1\left(0.5\right)=g_2\left(0.5\right)=0.5$, i.e. the party has half full power if it is expected to get half of the votes.
    • I defined $g_2$ from the reflection of $g_1$ across the point $(x_0,y_0)=(0.5,0.5)$, which implies $g_2\left(x\right)=2y_0-g_1(2x_0-x)=1-g_1(1-x)$.
    • Consequently, the power of a party expected to get x of the votes equals 1 minus that of one expected to get $1-x$ of the votes. For example, for $c=2$, the power of a party expected to get 10 % of the votes is 0.02, and that of one expected to get 90 % (= 1 - 0.1) is 0.98 (= 1 - 0.02).
  • $c>=0$ is the elasticity of the power of the party to implement its ideas with respect to $x$ if it is expected to get less than half of the votes.
    • I would say $c$ should be roughly the same for all parties.
    • For $c>1$, $g$ is a S-shaped curve whose slope is null for both $x=0$ and $x=1$. In this case, g tends to a step function as $c$ increases, which means the power to implement ideas becomes more binary. If $c$ is super high, a party has negligible power for $x<0.5$, and practically full power for $x>0.5$.
    • I consider $c$ should be higher than 1 such that, as $x$ increases, there are increasing marginal returns of the power to implement ideas for $x<=0.5$, and decreasing marginal returns for $x>=0.5$. In other words, for $c>1$, the marginal increase in the power to implement ideas grows as $x$ approaches 50 %, which makes sense because having or not the majority of votes is increasingly on the table then.

The marginal value of voting for a party is $V^{\prime }\left(x\right)={\left(kf\right(x\left)g\right(x\left)\right)}^{\prime }=k\left(f^{\prime }\right(x\left)g\right(x)+f(x\left)g^{\prime }\right(x\left)\right)$, where $V^{\prime }\left(x\right)$, $g^{\prime }\left(x\right)$ and $f^{\prime }\left(x\right)$ are the derivatives of $V$, $g$ and $f$ with respect to $x$.

Which party should I vote for?

General properties of my model

According to my model, for any constant $a$, and my presumptions that $0<=b<1$ and $c>1$, the maximum marginal value of voting is reached for $0.5<=x<=1$, which is to say it is best to vote for a party expected to get at least half of votes. Below are $f\left(x\right)$, $g\left(x\right)$ and the marginal value of voting for $a=0.01$, and my non-resilient best guesses of $b=0.5$ and $c=2$. Under these circumstances, it is best to vote for a party expected to get half of the votes. It makes sense that the marginal vote is very valuable in this case, since it corresponds to the greatest chance of causing a party to have the majority. In agreement with this, $g^{\prime }\left(x\right)$ reaches its maximum for $x=0.5$ (as long as $c>1$).

On the other hand, I am surprised it is best to vote for a party expected to get the majority for all the combinations of $b$ and $c$ in my preferred ranges. My intuition was that, holding $a$ constant, parties expected to get fewer votes would fare better under some conditions. I guess I was imagining much steeper returns of the power of a party to implement its ideas for low $x$, i.e. underestimating $c$. Indeed, if $c<0.5$, the value of the marginal vote increases as a party is expected to get increasingly few or many votes (as $x$ tends to 0 or 1). Below is the marginal value of voting for $a=0.01$ and $b=0.5$ (as above), but $c=0.2$ (instead of 2).

I do not think the above is realistic, but it could still be worth voting for parties expected to get fewer votes if they would have higher total value if they got all of the votes, i.e. if they have higher $a$. Nevertheless, this would hardly be the case for parties expected to get very few votes. For instance, for a fixed value of $a$, $b=0.5$ and $c=2$ (as in my 1st graph), voting for a party expected to get 40 % of votes is 31.6 (= 619/19.6) times as valuable as one expected to get just 4 %, which means $a$ would have to be over 31.6 times as high for the least voted party of the 2 as for the most voted one for it to be worth voting for the former.

One may attempt to rescue the value of voting for parties expected to get fewer claiming one’s vote impacts not only the current election, but also forthcoming ones. I do not see this going through. For my preferred parameters (see 1st graph), the marginal value of voting has an increasing slope for $x<=0.5$. As a consequence, the marginal value of voting will tend to increase the most for parties expected to get many, but not the majority, of votes.

Applying my model to the portuguese elections

I computed the marginal value of voting for each of the 8 candidates in the 2024 Portuguese legislative election, which happened yesterday. As before, I assumed $b=0.5$ and $c=2$. Additionally:

• I guessed values for a with the procedure I described in the previous section, without doing any further analysis.
• I got my initial values for $x$ eyeballing on 4 March 2024 the public opinion polling for the election in question, although I do not know how reliable such pollings have been in the past. Then I divided the initial values by their sum of 94 % to get my final values for $x$, which add up to 100 %^[1].

The inputs and results are in the tables below. I got the marginal value of blank, null or not voting from the mean marginal value of voting for the 8 candidates weighted by x. The calculations are in this Sheet.

The results confirm parties expected to get fewer votes have to be way better than a positive party expected to get many votes for voting for the former to be worth it (see last paragraph of the previous section). I voted for PAN and Livre in the past, but the marginal value of voting relative to not voting for these is 30.5 % (= 39.3/129) and 26.7 % (= 34.5/129) as large as that of AD, which is the candidate in the opposition expected to get the most votes. This illustrates the much talked incentive to vote for one’s preferred party among the ones expected to get the most votes, which pushes for a two-party system.

Should I even vote?

I concluded voting for AD, which is the candidate for which the marginal value of my vote is the highest, would have an effect equivalent to increasing Portugal’s GDP by 129 $ relative to not voting. Is this high enough for me to vote? I do not think so:

• From a perspective of increasing nearterm human welfare:
  • The annual consumption per capita of households and non-profit institutions serving households (NPISHs) in Portugal in 2022 was 14.3 k 2015-$, whereas that of GiveDirecty’s recipients in 2023 was 286 2023-$, or 223 2015-$ (= 286*0.778). As a result, the consumption of GiveDirectly’s recipients is only 1.56 % (= 223/(14.3*10^3)) of that of households and NPISHs in Portugal.
  • Assuming welfare is proportional to the logarithm of consumption, as Open Philanthropy does, marginal increases in welfare are inversely proportional to consumption^[5]. It follows that increasing Portugal’s GDP by 129 $ is as good as donating 2.01 $ (= 129*0.0156) to GiveDirectly.
  • Furthermore, GiveWell’s cost-effectiveness bar is 10 times the cost-effectiveness of GiveDirectly, which suggests my most valuable vote is as impactful as donating to GiveWell’s funds just 0.201 $ (= 2.01/10).
  • I guess voting would take me 0.75 h, so I should skip it if I would instead do work whose cost-effectiveness in terms of donating to GiveWell’s funds is more than 0.268 $/h (= 0.201/0.75).
  • I have earned 20 to 24 $/h in my recent paid work, i.e. 74.6 (= 20/0.268) to 89.6 (= 24/0.268) times as much as the above, and my marginal earnings are going towards donations which I think are at least as impactful as GiveWell’s funds.
  • I have been donating to the Long-Term Future Fund (LTFF), and I think the best artificial intelligence (AI) safety and animal welfare interventions are more cost-effective than the best ones in global health and development. I have estimated corporate campaigns for broiler welfare increase nearterm welfare 1.71 k times as cost-effectively as GiveWell’s top charities, although I guess the difference in impact is smaller due to indirect effects.
  • For these reasons, it is better for me to earn more money and donate it in the time I would spend voting.
  • In reality, my marginal time is not being spent on paid work, so I would not be earning more because of not voting (at least in the nearterm). Nevertheless, I assume my time outside paid work is more than 1.34 % (= 1/74.6) as cost-effective as my paid work. If this was not the case, I would be leaving lots of impact on the table.
• From a perspective of increasing economic growth, as non-ideally proxied by real GDP:
  • GiveDirectly’s cash transfers to poor households in Kenya have a multiplier of 2.4, which points to a donation of 1 $ to GiveDirectly increasing local real GDP by 2.4 $.
  • GiveWell makes some grants which mostly increase income, such as to Evidence Action’s Deworm the World Initiative, and its cost-effectiveness bar is 10 times the cost-effectiveness of GiveDirectly. So some of the interventions funded by GiveWell have a multiplier of around 24 (= 2.4*10).
  • Relatedly, Copenhagen Consensus Center’s 12 best investment papers have benefit-to-cost ratios ranging from 18 to 125.
  • If I value my marginal time at 20 $/h, my vote costs 15 $ (= 0.75*20), hence having a multiplier of 8.60 (= 129/15).
  • So the upshot is that my marginal time has to be more than 35.8 % (= 8.60/24) as cost-effective as my paid work for me to skip voting. This means the case for skipping voting is much weaker from the perspective of increasing economic growth than from that of increasing nearterm human welfare, where I concluded my marginal time only had to be 1.34 % as cost-effective as my paid work.
  • On the other hand, increasing nearterm human welfare may well be a better heuristic for contributing to a better world than economic growth. I would also not be surprised if donating to the LTFF was a more cost-effective way to boost economic growth than GiveWell’s interventions focussed on increasing income.

Overall, I have the impression I had better not vote given my opportunity cost, and I did not yesterday. It was not an easy decision. I had planned to vote for PAN, and even posted on Facebook about why. Basically, my case was that:

• There are 10.1 M chickens in Portugal, roughly as many as people, living in pretty bad conditions.
• PAN is the party which takes animal welfare most seriously^[6].
• PAN only had 1 seat in the Parliament (out of 220), which means voting for it is very neglected.

I guess I was imagining voting for PAN would be around 10 times as beneficial as for AD, and that voting for this would be as valuable as increasing Portugal’s GDP by 200 $. If these had hold, voting for PAN would have had an effect equivalent to increasing Portugal’s GDP by 2 k$ (= 10*200), hence having a cost-effectiveness in terms of donating to GiveWell’s funds of 4.16 $/h (= 2*10^3*0.0156/10/0.75). I guess this would still not be high enough to justify voting based on direct effects alone (I typically ask for a salary of 20 $/h), but then I could see the indirect effects being sufficiently large to make voting worth it^[7]. Nonetheless, I concluded the direct effects of voting for PAN are just 1.96 % (= 39.3/(2*10^3)) as large as the just mentioned hypothetical ones.

I appreciate there is uncertainty in my parameters and model, but these decrease my confidence that I can make the right voting choice, thus decreasing the direct value of voting. To be honest, my current estimates for the 2 major candidates (AD and PS) have so little resilience that I sense my vote is roughly as likely to be positive as negative.

In addition, I am not compelled to vote for AD. A significant fraction of the indirect effects of voting for me have come from supporting minor candidates, which I perceive as a way of protesting against a very subpar political system, whereas AD was one of the major candidates to win the election. I feel like not voting has greater indirect effects in that regard. More importantly, I have already had some pretty interesting discussions with people from EA Lisbon about my decision not to vote, and I am looking forward to explaining my decision at family dinners too!

I realise voting takes little time^[8], so whatever I choose to do has a relatively minor effect. At the same time, I think altruism sharpens altruism, and I believe not voting is the more altruistic action in my case.

Replying to a comment of mine made 1 year ago discussing considerations like the ones I presented at the start of this section, Toby said:

I have a general presumption in favour of EAs acting as most people think morally responsible people should. In part because there is a good chance that the common-sense approach is tracking something important that our calculations may have lost sight of, in part because I don’t think we should be trying to optimise all aspects of our behaviour, and in part because it is a legible sign of moral earnestness (i.e. it is reasonable for people to trust you less if you don’t do the things those people see as basic moral responsibilities).

Factors along these lines have been my main reason for having continued to vote. I replied to the above that:

Makes sense. I think I have been voting mostly based on this, although I am not sure about whether it makes sense for me to do so.

I believe there is some truth to Toby’s point, but I am also wary of voting mostly on grounds that the majority of people think it is morally responsible to do so. I assume most people do not consider eating factory-farmed animals as morally reprehensible, whereas I do. It is also the case that only 51 % of people who could vote did so in the 2022 Portuguese legislative elections, so I assume at least half of the population does not consider voting a moral imperative (otherwise they would have voted), even if they believe it is morally desirable. Moreover, in my mind, it would be good if more people skipped voting with the goal of doing something more impactful from an impartial perspective. Similarly, I would encourage someone who wanted to decrease their greenhouse gas (GHG) emissions to drive a car if that allowed the person to earn more, and their marginal earnings were going towards effective donations^[9] (as mine are).

To be clear, I agree that safeguarding liberal democracy is quite important. For what it is worth, I was a member of PAN for 3 years. I was not very active, but during that time I discussed concepts related to effective altruism, suggested ideas for policies, took part in some meetings, and collected signatures. My decision not to vote does not mean at all that I would endorse going back to a dictatorship. In fact, I admire the people who took part in the Carnation Revolution “that overthrew the authoritarian Estado Novo government on 25 April 1974”, and could see myself having participated in some form. However, skipping voting in Portugal today has a very minor effect on increasing the chance of a totalitarian government at the current margin^[10]. Relatedly, continuing to eat factory-farmed animals on the basis that the food system would collapse if everyone decided to go plant-based overnight is a poor argument, because such an abrupt change is wholly unrealistic.

It is also worth noting that the marginal value of voting in my model is inversely proportional to the number of non-null non-blank votes (see parameter $k$). Accordingly, I would vote if I predicted sufficiently fewer people were going to participate in the elections. It would then be more neglected.

Here are some thought experiments which helped me internalise that not voting is fine:

• Would I join a forecasting exercise whose goal was predicting the best groups of people to rule the country if half of the population was going to participate in it, all predictions had the same weight, and I did not think I was particularly knowledgeable about which groups were best? Only if it was fun, because my contribution to improving the aggregate prediction would be super tiny.
• Would I be willing to spend 30 min to 1 h to save a few tens of cents? Only if there were significant indirect effects.

Acknowledgements

Thanks to Ana Borges, Marcel Graetz, Maria José Macedo and Miguel Galaz for feedback on the draft^[11].

1. ^{^}

    I neglected the votes for other candidates, but this does not change the takeaways.

2. ^{^}

    The candidates are ordered alphabetically.

3. ^{^}

    Coalition of 3 parties, Partido do Centro Democrático Social - Partido Popular (CDS-PP), Partido Popular Monárquico (PPM), and Partido Social Democrate (PSD).

4. ^{^}

    Coalition of 2 parties, Partido Comunista Português (PCP), e Partido Ecologista “Os Verdes” (PEV).

5. ^{^}

    The derivative of ln(x) is 1/x.

6. ^{^}

    See my shallow investigation in the 2nd comment in my Facebook post.

7. ^{^}

    For instance, my mother thinks I should vote (regardless of the candidate).

8. ^{^}

    Much less time than what I spent on this post, but hopefully it is valuable!

9. ^{^}

    The global emissions per capita were 4.7 t in 2022. In Founders Pledge’s 2018 climate report (see section 3.2), it was estimated the future work of Clean Air Task Force (CATF) would have a cost-effectiveness of 3.41 t/$. This suggests one can neutralise the emissions of a random person donating just 1.38 $ (= 4.7/3.41) to CATF.

10. ^{^}

     Note I accounted for this in my calculations by selecting the lowest a for Chega, which is the party I take to be threatening Portugal’s democracy the most.

11. ^{^}

     I ordered the names alphabetically.