Does Logical Decision Theories actually give meaningfully better recommendations on real world problems, particularly voting, frequently referenced?
One of the main reasons given for preferring logical decision theories (LDT), or particularly functional decision theory (FDT) is that agents do better in real world problems. Indeed, the article here on logical decision theory opens by discussing voting. I recently posted a discussion of a hypothetical where FDT agents perform worse, but I think when applying it in practice to the real world case of voting which is often given as a preference is actually better (see here for Eliezer Yudkowsky’s discussion of voting under decision theories where he argues for logical decision theory being better). Particularly, I think that for most people this discussion gets wrong what causal decision theory actually would recommend.
To begin (note, I spend a while going over how to model voting decisions and different utility to CDT modeling of decisions for a few paragraphs, and later discuss practical agent to agent comparisons), let us imagine what the expected utility is for an agent under CDT of voting in some election. Let’s say there are two candidates, like Yudkowsky, I will use the Simpson’s Kang and Kodos. If Kang wins, we have some expected outcome (O1), if Kodos wins we have some expected outcome (O2). Let’s say our agent is a Kang supporter and has a positive evaluation of O1 such that O1>O2.[1]
Our agent is evaluating the value of voting for Kang (A1) or not voting (A0).
In the simplest case, with no externalities an EDT agent would say: “we should vote if our evidential evidence indicates voting is more likely to lead to Kang winning” (i.e., if P(O1|A1)>P(O1|A0). A CDT agent would say “we should vote if there is a positive probability that our vote will cause Kang to win” (we can say this works out equivalently, if P(O1|A1)>P(O1|A0) we should vote).
If we are a simplistic agent, in both cases we should vote, as in either case the value is something greater than zero. But of course, realistically we are not so simple agents, and have some cost to voting. Taking one more step of complexity and stopping there is where I think Yudowsky (and others) go wrong. They correctly note for most real world scenarios the probabilistic effects of a single vote are de minus and humans have some cost associated with voting.
For the CDT agent, they expect there is some probability of their vote being pivotal (we can say P(pivotal)=P(O1|A1)-P(O1|A0) ). They also have some cost of voting (say E). So really, they should vote if P(pivotal) * (O1-O2) > E. That is to say, if the probability of their vote being pivotal, times the difference in expected outcomes caused by their decision to vote, is greater than the cost.
This leads to some sensible recommendations, (i.e., you should be more likely to vote the less costly it is to vote, the more impactful the outcome of the election and the more likely it is your vote will be pivotal). If I am a policy maker and what to increase voting, I should use the policy levers at my disposal to reduce E, and political campaigners should emphasize the impact of the election and the odds of voters impacting results to increase turnout. This is what we observe in the real world.
Where Yudowsky says CDT gets it wrong, however, is that as mentioned the P(pivotal) is vanishingly small. While I framed P(pivotal) as the difference in odds for voting or not, for a CDT agent this could also be reduced to the odds of the candidates tying but for your vote. Obviously, it is incredibly rare that major elections come down to single voters. For the EDT agent, they don’t have to make this reduction so fair slightly better under uncertainty, but still would value the difference in odds as very small. Yudowsky says this misses the mark. But what if we take our model one step further? Our agent is a person, people place real value on things other than strict outcomes.
When someone says “it is your civic duty to vote” they are appealing to a real value we can include in our utility functions—people value being members of civic society and participation. In addition, there are social benefits to voting in the form of signaling, people proudly display ‘I voted’ stickers all the time. This is not internally independent, the more contested an election and the more meaningful the outcomes, the more valuable signaling is.
We can say P(pivotal) is a function of the degree an election is contested and general voting population (number of people and associated behaviors). Similarly, we may say the value of signaling in an election is a function of social values and how contested an election can be.
So we can say a CDT agent under real world conditions should vote if P(pivotal) * (O1-O2) + personal utility (e.g. personal values on being a civically engaged person) + social utility (e.g., benefits from signaling you are civically engaged) > E
This again leads to additional sensible recommendations. If I think well of myself as a civically minded person and value civic contributions and have social connections for whom signaling my voting behavior will provide social benefits, that should all increase my odds of voting. Similarly, for policy makers and political activists, increasing the civic mindedness and social value placed on voting, publicly being seen to promote and reward those voting, etc can be seen as recommended way to increase agent’s decision to vote.
Bringing things together, where does FDT differ and what is the utility of these in practice?
As mentioned above, if I am a CDT agent deciding whether to vote, I have to answer the question “is my value from voting, which includes my personal values around voting and my expectations for the likelihood of my vote being pivotal and my expectations for different outcomes of the election greater than my anticipated cost of voting?” One’s expectations of the election being pivotal can be determined by examining polls and voting models and making empirically based estimates of the outcomes (these also likely affect one’s expectation of the value of signaling). This can be simply understood as trying to estimate what the causal outcome of my votes are likely to be. The empirical side of it is readily, concretely definable depending on my estimated values and while values and signaling effects must be individually parsed, they are usually fairly straightforward for most people.
So where does LDT/FDT differ? As Yud has put it instead of asking what your decision’s outcome “You ask what would happen if people like you voted.” This gives an obvious recommendation absent under CDT: “the more people that are similar to you, the more you should vote.” This recommendation does not match as neatly with intuition (at least not with my institution) and, in fact, implicitly seems to run counter to Yudowsky’s previous statement that under LDT it would still be irrational to vote if “if you don’t expect any of the elections to be close.” Under LDT if a lot of people like you (which might be heuristically judged by people voting for the same candidate as you) are voting, that would seem to provide more evidence that you should vote even more so the more dominant your side of the election is. Though this hinges on a fairly evident open question: who constitutes “people like you?” According to Yud, this is “just an empirical question” but is it really? His 2016 post on LessWrong gives multiple different ways you might think about who constitutes someone similar to you. There doesn’t seem to be any unified way for an agent to really make that estimate. Should we perhaps use polls or previous results, maybe just personal estimates of how many people might think similiarly to use under our theory of mind? Those questions seem unanswered, if answerable at all, and depending on how you understand it can lead to contradictory, unintuitive voting patterns.
Case 1 Voting for an Underdog:
So, I am an FDT agent. I want to estimate whether I should vote or not. I look at past polling data, 40% of Kang supporters voted in the past. 50% of Kodos voters voted. I expect that the odds of Kodos winning are, say, 80% at present and the odds of my vote being pivotal is 0.000001%. I know a handful of people who think of themselves as LDT agents, most of them have told me they decided not to vote. How can I calculate my EV from this? I don’t think there really is any clear way to quantify it, but let’s consider a few possibilities qualitatively; should I:
a. Say since I know other LDT agents decided not to vote assume we are similar and that LDT recommends not voting in this decision?
b. Say few people are like me, most people won’t reason the way I do, so the odds of the vote being different are de minus and not vote on those grounds?
c. See that most people with similar values to me are not voting and some are, seeing I identify with those who are voting more, but I therefore estimate that people similar to me are already voting so if people similarly to me voted the outcome would likely be the same. On that basis should I decide that being an agent that votes has little value and decide not to vote?
c. Say that there are a lot of Kang supporters not voting, to some extent we are similar (our voting behavior evidentially has some correlation with eachother), if they voted we would have a good chance of winning so I should by imagining the counterfactual where Kang had arbitrarily higher output and vote on that basis?
I don’t think there is any clear way of judging these scenarios.
CDT gives a pretty clear recommendation. My expected value for voting for Kang is 0.00000001*(O1-O2) + personal utility + social utility—E (cost of voting). If I evaluate that positively, under my personal internal values of personal utility and social utility and whatever the costs are, I should vote. If I do not, I shouldn’t.
Case 2 Protest Voting:
Let’s say Kang was running against Lisa in the primaries. As polls increasingly show turnout for her is low, she is not considered in the election. However, by some definition of people similar to me, if people similar to me were to turn out for her she could still have had some probability of winning and would be so preferable to Kang and Kodos that I chose to vote for her anyway. [2]This doesn’t seem very sensible. A CDT agent would value voting for her (despite knowing that in the specific case she had no chance of winning) if they expected the signaling benefit of their vote to be sufficiently greater to justify ‘throwing away’ their vote on a candidate who they know has already lost. There are cases where this might be justified, if one expects the odds of their vote being pivotal to be incredibly low, and the difference between O1 and O2 to be fairly negligible, then a protest vote can be considered somewhat rational under some signaling estimates. That calculation seems to me to be far more pragmatic and meaningful than the reasoning in FDT which requires implicitly treating you treating something you know to be false as if it could be true to determine whether to vote.
In this case, for a CDT agent, the expected value of voting can simply be evaluated as personal utility + social utility—E.
CDT seems to give clear recommendations that can be readily evaluated and do at least a serviceable job modeling real world behavior. FDT gives unclear recommendations, that to the extent they can be evaluated give less helpful recommendations. On that basis, it would seem to me that CDT actually wins out as a framework for considering whether it is rational to vote.
It may not be a very good test, in many cases. Perhaps modifying it to gauge confidence could be better?
One can imagine there are near infinite sets of things that might be true for whatever the secret knowledge is regarding. Only a subset actually is a true. What is the most probable prior may be wildly divergent from what is actually true. If you judge purely on how they confirm to your secret set that doesn’t tell you how good they are at forecasting in general, just that they happen to be wrong on that set.
If you gauge confidence, that might be better. If they are very confident about something you know to be wrong, it is unlikely that the prior probability lined up with reality. If they are only moderately confident, or believe it best explains the evidence they have but are fully aware it may be incomplete or not explain other evidence they lack, then it seems unreasonable to strongly hold a view based on them.