EA Forum Reader

A variety of utility distribution models were tried, and it turned out not to matter very much.

The simulations by N. Tideman had methodological flaws and didn't measure the right thing, thus being approximately as useful as a random coin flip in this mathematician's view.

.

> All things considered, I think electoral reform, while probably not a “top tier” intervention, should be part of the longtermist EA portfolio.

My view is that electoral reform, specifically alternative voting methods, is by far the biggest “bang for the buck” utility-increasing reform for humanity. I began studying alternative voting methods in 2006, largely focused on the findings of a Princeton math PhD named Warren Smith, who created RangeVoting.org to publicize his research. What was very surprising and counterintuitive were the Bayesian regret calculations he performed via computer simulation, showing that an upgrade from plurality voting to score voting (aka range voting) nearly doubled the human-welfare increasing impact of democracy. Approval voting is just score voting on a 0-1 scale and it performs almost as well as score voting with a range like 0-5; and of course it has the advantage of using ordinary ballots and not requiring voting machine upgrades. As Smith puts it:

there are very few causes out there with this much "bang for the buck." Examine the numbers yourself. I do not believe religious causes can compete. Disaster relief cannot compete (in the long term; for large disasters in the short term, it can). Curing diseases also cannot compete except for the biggest killers. E.g, ending malaria or halving illiteracy each would cause an amount of good comparable to range voting, but would probably be more difficult to accomplish.

More recently, a Harvard stats PhD named Jameson Quinn performed his own computer simulations using slightly different modeling assumptions, and presenting the results inverted and normalized to voter satisfaction efficiency (VSE) rather than Bayesian regret. However he found similar results to Smith's.

That last point about the cost (the second component in "bang for the buck") is crucial. Aside from the one-time political campaigning cost, voting reform is essentially free. Unlike, say, mosquito nets or medicines which have to be manufactured and distributed.

Full disclosure: Smith, Quinn, and myself are all former board members of the Center for Election Science, but I am no longer associated with them and am speaking only for myself.

Approval voting is vulnerable to tactical voting.

All deterministic voting methods are vulnerable to tactical voting. But computer simulation and game theoretical analysis shows that approval voting is especially resistant to tactical voting. For instance, as Warren Smith and Jameson Quinn showed in their computer simulations, approval voting worked very well with wide variances in assumptions about voter behavior and the preponderance of tactical behavior. The difference was so large that in some of Smith's models, approval voting performed better with 100% strategic voters than instant runoff voting (aka "IRV", the most prominent ranked method) did with 100% honest voters. There is even a mathematical theorem that, given plausible models of voter strategy, approval voting always elects a Condorcet winner (a candidate who beats every rival by a head-to-head majority) whenever one exists. This is a very mild, some would say beneficial, reaction to strategy.

It's also notable that approval voting was shown generally favorable to ranked methods in the William Poundstone book Gaming the Vote, and is specifically advocated by Steve Brams, an NYU professor of political science and game theory, who wrote such page turners as Mathematics and Democracy.

It fails the later-no-harm criterion: approving a second candidate can hurt your favourite.

At the outset, I want to suggest eschewing properties as a means for evaluating voting methods, and instead focus on voter satisfaction efficiency as described above. Focusing on properties is kind of analogous to evaluating race cars based on characteristics such as horsepower and aerodynamics. You might intuitively think a car with 10% more horsepower will be better, but once you perform a statistically significant number of timed trials, you may find instead that other factors such as aerodynamics or tire quality (or even properties you never thought to consider) have a countervailing effect that causes the more powerful car to surprisingly lose. Voter satisfaction efficiency (or Bayesian regret) simultaneously measures the combined effects of all those properties, even ones you forgot to consider. And indeed, those figures I cited already account for later-no-harm (LNH); and yet of the five alternative voting methods compared, IRV did the worst, even though it was the only one of those five methods to satisfy LNH.

Having said that, let's consider this failure of the later-no-harm criterion (LNH) more specifically. I actually contend that failing LNH is a benefit, not a flaw. Here's a thorough analysis by Warren Smith (again, full warning, Smith's writing is interspersed with venom for his enemies, but the content itself is high quality).

To summarize, later-ho-harm means that given you've started by showing support for your favorite, X, it cannot hurt X to indicate support for a lesser liked candidate, Y. E.g. suppose you rank the Green in first place. Now it cannot hurt the Green to rank the Democrat (or Labour) in second place. IRV is the only commonly discussed method that satisfies LNH. By contrast, if you cast a vote for the Green with approval voting, casting a second vote for the Democrat (or Labour) could cause the Green (your actual favorite) to lose. IRV proponents often argue that this creates an incentive for "bullet voting" only for one's favorite candidate. I admit this does seem problematic on its surface, but that goes away once you inspect a bit deeper.

See, all that assumed that you started by ranking your favorite candidate honestly in first place. E.g. you ranked the Green #1, and now you're considering whether to rank your honest second favorite as #2. But your best strategy with IRV is, similar to FPTP, to rank your favorite frontrunner in first place, not necessarily your favorite overall. We see this often in USA primary elections, where people are afraid to vote for their favorite candidate because they fear she'll lose the general election. For instance, right now my aunt in Iowa favors Democrat Elizabeth Warren over Joe Biden, but voted for Biden because she feels he's more likely to win in the next round against Trump. To convert this to its IRV analog, with all three of those candidates running on a single ranked ballot together, she would insincerely/strategically rank Biden in first place, to help ensure Warren (her actual favorite) is eliminated, thus giving the presumably more competitive Biden the chance to square off against Donald Trump in the next round. IRV fails the favorite betrayal criterion.

And contrary to those concerned about bullet voting with approval voting, the best strategy is actually to approve everyone you prefer to the expected utility of the winner. Bullet voting is absolutely not the best general strategy. Case in point, imagine a Green Party supporter who normally casts a strategic vote for the Democrat. With approval voting, she still casts that strategic vote for the Democrat, but then she also casts a sincere vote for her true favorite, the Green, plus anyone else she prefers to the Democrat. This is the underlying game theory that makes approval voting so robust against strategy. There's even empirical data showing that approval voting often has less bullet voting than comparable elections with IRV.

Moving beyond strategy, to the fundamental social choice theory behind LNH, the problem with LNH is that it forces a voting method to ignore important preference data. E.g.

Imagine preferences like these:

35% LRC
34% RCL
16% CLR
15% CRL

C is eliminated first with 31%, and L wins with 51%.

But suppose just 2%, from the CLR faction, swap their LR preference, creating:

35% LRC
34% RCL
14% CLR
17% CRL

C is still eliminated, but now R wins with 51%.

A tiny change in preferences changes the winner from L to R.

But now suppose a comparatively massive change in preferences occurs. The LRC voters find out something terrible about R, causing them to lower R to 3rd place. And the RCL voters find out something super positive about L, causing them to elevate L to 2nd place.

35% LCR
34% RLC
14% CLR
17% CRL

An enormous shift in public opinion just took place, positive for L and harmful for R. But this huge change didn't change the winner from R to L, even though a tiny change of preferences moved the winner from L to R in the first place. LNH causes this fundamental distortion in sensitivity to preference information.

The winner, then, may not be the candidate with the most support, but the one that’s best at manipulating the system.

Absolutely true. But also consider that:

A. This happens with IRV too, and in some ways more so. Since you still have to worry about electability with IRV, a candidate like Biden can convince Warren's supporters to drop her for him just by running ads that make voters fear her to be unelectable. That cannot possibly happen with approval voting. The best they could do is convince Warren's supporters to also vote for Biden.

B. Approval voting behaves so much better in the general case (again, see the voter satisfaction efficiency results above) that it gives substantial margin for error, such that even if this effect you describe happens, approval voting plausibly still comes out ahead.

Admittedly, we have to see approval voting in the real world to get a better handle on such campaign dynamics that are beyond the capacity for a computer to simulate.

Approval voting radically re-interprets the common-sense notion of "having a majority"...For instance, approval voting sometimes selects a candidate even though a majority of voters would, in a head-to-head contest, prefer any other candidate. (This is the Condorcet loser criterion.)

Again, approval voting tends to elect Condorcet winners whenever they exist, in practice. And most ranked methods, including most Condorcet methods, are extremely vulnerable to tactics, meaning they may in practice be worse at electing Condorcet winners. Indeed, IRV can elect candidate X even though candidate Y is preferred to X by a huge majority and also has twice as many first place votes as X.

And to go more esoteric and technical, it is mathematically proven that a group may prefer candidate X even though a majority of voters in that group prefer candidate Y. See here and here. So it is not inherently wrong to avoid electing the Condorcet winner, or even the majority winner for that matter. Here are even some examples where it seems bad to elect the Condorcet winner. I contend the goal of social choice is to maximize expected utility, i.e. maximize the net utility of the group, not to elect "majority winners" per se. We can talk about the biological origins of that grey decision-making machine between our ears, and how that makes us effectively utility maximization machines, but I suspect the EA community is already fairly on board with this.

Indicating support or opposition for each candidate is more expressive than just having a single vote, but it is still binary and does not allow voters to express more nuanced preferences between different candidates.

Yes, this is the expressiveness issue. Ranked ballots do provide more expressiveness, but this is counteracted by two other factors: 1. their increased vulnerability to strategy, and 2. their generally decreased tabulation efficiency (particularly in the case of IRV). This is why approval voting generally outperforms the ranked methods in computer models. Even in the circumstances where some ranked methods perform better, it's only a marginal amount that I do not think justifies the complexity. Especially given ranked voting methods have been repealed in something like 60 U.S. cities, apparently largely on account of their complexity.

There is almost no track record of approval voting being successfully used in competitive elections. Where it was used, approval voting was often repealed later on - e.g. in Dartmouth alumni elections and in internal IEEE (Institute of Electrical and Electronics Engineers) elections (search for IEEE here).

But all indications are that it worked well in those situations, and it was repealed for "political" reasons, plausibly because it worked. Granted, that's little consolation if it manages to be repealed in American political contests. But there are good reasons to believe that's less likely to happen in government elections. E.g. it's harder to imagine a city like Fargo, where approval voting was adopted by a 64% majority, voting to repeal it by a majority via another ballot initiative. And given ranked voting has already been repealed so many times in the U.S., it's not clear that approval voting is riskier in this regard. We'll just to have find out empirically by seeing how it plays out in Fargo this June, and hopefully in other cities in the coming years.

My view is that the urgency of voting reform demands small experiments limited in their scope. If it turns out approval voting actually is better, and its simplicity allows it to scale faster, then I feel it will have been worth it. I think the urgency of issues like climate change and authoritarianism demands a massive scaling out of "electoral technology" that can increasingly democratize the USA in something like a decade. If approval voting fizzles out and/or IRV or other methods turn out to scale faster, it wasn't that expensive of an experiment.

the tendency to favour moderate candidates could also be considered a bias and is not universally viewed as a positive feature of a voting system.

Again, the key here is to look at it through the lens of utility efficiency, which already measures "bias" and any other distortionary factors in terms of human welfare, which is the ultimate metric we want to look at.

It seems to me that effective altruism has not examined approval voting (or alternatives) in sufficient detail.

Speaking as someone who has spent countless hours studying this subject since 2006, whereupon I soon did my first exit poll, I think it has been studied voluminously. Organizational elections, exit polling, computer simulation, and all manner of years-long debates between math PhD's and various engineers have taken place. An NYU professor of game theory and political science has worked on it for four decades and written multiple books on the subject. I've personally visited Kenneth Arrow, who won the Nobel Prize in economics for his study of voting theory going back to the 1950s. We'll still find out new things from what we see in municipal elections like those in Fargo, but I do believe the general arguments you raise have been analyzed inside and out.

In general, my impression is that discussions of voting reform suffer from the problem that people tend to pick their favourite method and then cherry-pick one-sided arguments in favour of it.

Perhaps, but what's relevant is the veracity of the arguments, not the personality traits of people who favor one system or another.

The Center for Election Science often talks about no favourite betrayal (which approval voting satisfies) and not much about later-no-harm (which it fails). FairVote doesn't talk much about no favourite betrayal and talks a lot about later-no-harm - because their favoured method (instant runoff voting) satisfies later-no-harm but fails no favourite betrayal.

I would put this the other way around. Favorite betrayal matters for important game theoretical reasons that I described in detail above, whereas LNH is effectively an "anti-criterion". And from those observations, I form a relative assessment of approval voting and IRV. I do not start with the system and then try to make the facts fit it. Indeed, Warren Smith initially ran his computer simulations with no particular foreknowledge of how it would turn out. He thought the winning system would vary depending on the assumptions (the "knob settings" of the program), but it turned out that score voting won in all 720 different permutations of those settings. And that led him to become a supporter of score voting. When I came across his work, I was initially skeptical and I sent him an email berating him. Then over the course of some email correspondence, he refuted my arguments and made me a convert.

A few years back, an associate of mine added a top two runoff to score voting and "STAR voting" was born. At first I was skeptical that it was a needless additional complexity. But then I saw that Jameson Quinn's voter satisfaction efficiency calculations were quite favorable to it, so I've shifted my thinking substantially.

Since there is (some degree of) consensus that plurality voting is bad, but no consensus on which alternative is best, we should focus on the reform proposals that are most viable. That’s arguably instant runoff voting (IRV, called ranked choice voting / RCV in the US), which is championed by FairVote.

I don't know if there is "consensus", but I do feel the facts objectively show approval voting to be superior to IRV (if not every ranked method) according to essentially every metric we have, from voter satisfaction to ballot spoilage rates to voting machine complexity. I also dispute that IRV is more viable. Approval voting was adopted by a 64% majority in Fargo, and is polling at 72% support in St Louis. I suspect we are about to see that approval voting is more politically viable, due to its simplicity and transparency.

Unlike approval voting, IRV has a track record in competitive elections and is much more in line with conventional notions of “majority”.

But again, it is mathematically proven that majoritarianism is not the right metric. Maybe you're talking more about public perception, but in that case I would again cite the 2-1 margin by which approval voting managed to pass in Fargo, and the 72% support it's getting in St Louis. If this majority failure is a risk in terms of political viability, we're not seeing it.

My personal favourite voting system would be a Condorcet method such as Ranked Pairs, but there are no large organisations advocating this, and it’s unlikely that Condorcet methods will be adopted.

I know I'm repeating myself, but approval voting may in practice be a better Condorcet method than real Condorcet methods. And it performs better under models of significant voter strategy. And of course is much simper and thus politically and logistically easier to implement and scale.

IRV seems clearly superior to plurality voting and has stood the test of time

It was used in nearly two dozen U.S. cities and repealed in all but one of them, then adopted in several cities again, and then repealed in four of them. So it's not clear what its long term staying power is yet.

For parliamentary, I think it’s best to use a form of proportional representation rather than (or in addition to) first-past-the-post in single-seat constituencies.

Whether PR systems can outperform the best single-winner systems, such as STAR voting and approval voting, is very much an open question and highly speculative. But it's also very difficult to adopt in the USA without first getting a system like score voting or approval voting which is capable of ending two-party domination. That's because federal law makes multi-winner districts illegal, and the two-party system will make that impossible to change.

Proportional representation tends to lead to multi-party systems that require cross-party collaboration and reduce the team sport mentality that drives US polarisation.

One could expect that a congress full of moderate/centrist candidates would be even less polarized than one that includes every faction from socialists to fascists.

A steelman of plurality voting is that it grants power to the largest coherent political coalition (coherent in the sense of being able to coordinate on a single candidate).

Computer simulation shows that it is extremely bad. And as a 41-year-old American, I've seen first-hand how much its two-party lock-in fosters binary tribal thinking that makes it impossible to view issues like impeachment or climate change through an objective non-partisan lens. And combined with Gerrymandering, it virtually nullifies democracy. As FairVote says, elections become so predictable that, "Under our current system, we can predict 379 of the 435 House seats — or more than 87 percent of the total — with high confidence."

For the problems-that-solve-themselves arguments, I feel like your examples have very "good" qualities for solving themselves: both personal and economic incentives are against them, they are obvious when one is confronted with the situation, and at the point where the problems becomes obvious, you can still solve them. I would argue that not all these properties holds for AGI. What are your thoughts about that?

I agree that it's an important question whether AGI has the right qualities to "solve itself". To go through the ones you named:

• "Personal and economic incentives are aligned against them"--I think AI safety has somewhat good properties here. Basically no-one wants to kill everyone, and AI systems that aren't aligned with their users are much less useful. On the other hand, it might be the case that people are strongly incentivised to be reckless and deploy things quickly.
• "they are obvious when one is confronted with the situation"--I think that alignment problems might be fairly obvious, especially if there's a long process of continuous AI progress where unaligned non-superintelligent AI systems do non-catastrophic damage. So this comes down to questions about how rapid AI progress will be.
• "at the point where the problems become obvious, you can still solve them"--If the problems become obvious because non-superintelligent AI systems are behaving badly, then we can still maybe put more effort into aligning increasingly powerful AI systems after that and hopefully we won't lose that much of the value of the future.

I'm not quite sure how high your bar is for "experience", but many of the tasks that I'm most enthusiastic about in EA are ones which could plausibly be done by someone in their early 20s who eg just graduated university. Various tasks of this type:

• Work at MIRI on various programming tasks which require being really smart and good at math and programming and able to work with type theory and Haskell. Eg we recently hired Seraphina Nix to do this right out of college. There are other people who are recent college graduates who we offered this job to who didn't accept. These people are unusually good programmers for their age, but they're not unique. I'm more enthusiastic about hiring older and more experienced people, but that's not a hard requirement. We could probably hire several more of these people before we became bottlenecked on management capacity.
• Generalist AI safety research that Evan Hubinger does--he led the writing of "Risks from Learned Optimization" during a summer internship at MIRI; before that internship he hadn't had much contact with the AI safety community in person (though he'd read stuff online).
• Richard Ngo is another young AI safety researcher doing lots of great self-directed stuff; I don't think he consumed an enormous amount of outside resources while becoming good at thinking about this stuff.
• I think that there are inexperienced people who could do really helpful work with me on EA movement building; to be good at this you need to have read a lot about EA and be friendly and know how to talk to lots of people.

My guess is that EA does not have a lot of unidentified people who are as good at these things as the people I've identified.

I think that the "EA doesn't have enough great people" problem feels more important to me than the "EA has trouble using the people we have" problem.

Ought (~$5000) and Rethink Priorities (~$500) have both done it, with bounties roughly what I indicated (though I'm a bit uncertain). Don't think either has completed the relevant hiring rounds yet.

Disclaimer: I'm the executive director for The Center for Election Science.

There’s some good stuff in this post.

Excessive political polarisation, especially party polarisation in the US, makes it harder to reach consensus or a fair compromise, and undermines trust in public institutions. Efforts to avoid harmful long-term dynamics, and to strengthen democratic governance, are therefore of interest to effective altruists. One concrete lever is electoral reform.

Great stuff.

Still, I think the downsides of plurality voting outweigh its advantages, and there is some degree of consensus among experts that plurality voting is not a good system. … Still, I think the downsides of plurality voting outweigh its advantages, and there is some degree of consensus among experts that plurality voting is not a good system.

Understated, but still good stuff. Also, experts really hate plurality. See one such meeting of experts where not a single person approved of plurality. As an aside, they favored approval voting with IRV following in second.

All things considered, I think electoral reform, while probably not a “top tier” intervention, should be part of the longtermist EA portfolio.

I might disagree on the degree, but my sentiment overlaps with its place in longtermism.

The post gets at the idea that anything is better than plurality and that we shouldn’t feel like we have to pick among systems. But then it ultimately picks among systems. This is the dilemma we find ourselves in. We have to advocate for something, and when we’re presenting an opportunity, it is only reasonable for those exploring to consider the options. If experts don’t then do the correcting, then errors will go ignored, and poor decisions will be made.

My full empathy goes to anyone who does the legwork to create a post here (it’s challenging and it’s putting yourself out there). But practically all the arguments raised in this well-meaning post are addressed in the previous posting [EA · GW], which also links to the approval voting criticisms article and limits of RCV. If after looking, you don’t see it addressed, please reply and I’ll see if I can find an answer. You’ll find answers about later-no-harm and bullet voting, proportional approaches, the reflection of candidate support, practicality, and much more.

I'm noticing that the arguments referenced here come almost exclusively from FairVote. As a caveat, this organization repeatedly argues that approval voting should not be used in virtually any circumstance (despite when experts clearly disagree and even prefer approval voting). They also failed to acknowledge any faults within an RCV election where virtually every RCV mistake occurred. It’s hard to take them seriously after that. This refers to the Burlington election where voters got a worse outcome for ranking their favorite first, candidates could have been harmed by getting more higher-preference rankings, voter segments would have gotten a better outcome staying at home, and the candidate who could beat everyone head to head was eliminated due to RCV’s tendency to vote split along the middle. They also discouraged others from looking at election data from alternative voting methods in a Wall Street Journal article.

FairVote frequently cites Dartmouth while omitting any other reason that might have caused voters to choose fewer candidates (like few candidates being on the ballot and an enormous number of write-ins). As one can also get from the approval voting criticisms article, even when a majority choose only one candidate, the remainder who choose multiple candidates can (1) have a material effect in choosing a different winner and (2) give support to candidates who would otherwise be invisible. These repeals were not “often” the case. This argument also fails to acknowledge all the cities that repealed RCV due to either complaints of complexity or flat-out bad winners being elected. There are also cities that take forever to implement RCV or don’t do it at all due to the cost of software and new machines. This is one big reason why Fargo and St. Louis wanted approval voting instead of RCV.

This isn’t to say that any FairVote reference is bad, just that it potentially warrants more investigation.

When we are looking at voting methods, a good track record isn’t merely recorded uses. We need to see how it performs in competitive elections that have a different plurality winner. And note that practically any alternative voting method will handle spoiler candidates who get little support. Want more data? Funding a research department for CES would go a long way.

It’s also important to remember that no voting method can guarantee a majority and that methods like RCV merely contrive a majority through eliminating candidates via vote splitting—sometimes by eliminating the best candidate. Metrics that you can use to see whether a good winner was elected involve looking at Condorcet matrixes and candidate utilities (note that using explicit Condorcet methods is not practical due to “tie-breakers” from cycles). It’s not enough to say that a method didn’t provide a “majority” and so that method must have chosen the wrong winner.

It’s also important to note that CES was seriously vetted for close to a year before a grant was awarded. This grant from two years ago wasn’t a rash decision. This is a system that has been studied academically since the late ’70s with one of its developers on the CES board of advisers.

Also, CES went into this space agnostic about the voting method. We took the time in our early years (before we had any funds) to really think about the alternatives, including practicality as a concern. This is not the same approach that other organizations have taken either going with a system merely because it either (1) has previous use, or (2) superficially approximates the setup of a separate proportional method (STV).

Ultimately, I think for EA to switch to a voting method that already has funding relative to approval voting (and has serious issues) would be a mistake. This is a space that is overall extremely underfunded relative to other election reform areas given its importance (see earlier analysis [EA · GW]). As an organization, we’ve already demonstrated how cost-effective we can be in a much shorter time frame compared to peers in our space. We hired staff and got approval voting in its first US city all within a year of initial funding.

Failing to provide further support or removing it at this crucial time wouldn’t be the optimal move here. It would just eliminate a promising alternative approach from being tested at all. If this is perceived as too much of a risk, then I would fear how this mentality would keep EA from pursuing other efforts where outcomes are even more unclear.

3) You mention that Google traffic data is still useful, even when few people use it. I am not familiar with that part of the app, but if it involves some form of prediction, it is important to note that Google has had years to get this right. With a pandemic, you have at best months(!), and on top of that the situation changes constantly.

Yes. This is another reason why working with someone like Google Maps or some other mapping app could be crucial, because they have accumulated domain / tribal knowledge that no one else might have. 

[Edit: I've received some private feedback that neither of us might be right here. The calculation for both (density of traffic and density of "infectiousness") may be quite straightforward. The reason traffic updates got so much better might just have been more data.]

2) Regarding the computation of the risk score: If you only use confirmed cases with voluntary sign up, you might not get enough data; if you use suspected cases by symptoms, you will get a lot of false positives due to worried people with the flu. In the absence of data on how to properly account for that, this is a very difficult problem.

These are significant challenges and I talk a bit about how they can be addressed in The Incentives Align and at the end of the section Example App Questionnaire. I imagine there would always be more confidence put in a confirmed case with a code than someone who just answers yes to having cold or flu symptoms recently. 

Also, for confirmed cases, as part of contact tracing, the CDC sometimes identifies a site of concern where a patient might recall that something particularly infectious happened before they were aware they were sick. For example: "Oh. I remember that a few days ago, I sneezed quite forcefully and unexpectedly at my favorite buffet, so I couldn't cover my nose. Oh, and then again on the way home on the BART! I'm so sorry." Tracing multiple paths backwards, you might get a lot of data from a single event. 

1) How likely are you to catch the virus at all just by being in the same area/frequenting the same shops as somebody infected? My impression from the Western cases so far was that it infections occurred generally with close contacts; this risk changes obviously when more infected people are around, but still should be estimated to decide whether such an app would be worth it.

I imagine the CDC could already have some estimates for this, which they might use in contact tracing. And it might turn out that contact tracing is enough to solve the problem. It seems to be working well right now in the U.S. 

But if not, a not-so-educated guess for a general outline of the calculation of risk for the app might be (1) Close contact. You were in the same location where a possibly infectious person spent some time within maybe 10 minutes of them. This is higher risk. (2) Semi-Close Contact. If the virus might live on surfaces for a few hours, the close contact risk distribution tapers off over a few hours and (3) Infectiousness. This isn't binary so the infectious person's distribution also peaks sometime around their first symptoms, and tapers off over a few days up until the tail reaches some maximum. 

A heat map that changes with time could reflect this information. And any individual's risk could be calculated by integrating over it. And, if the local situation is suddenly found to have had multiple people in it in the last few weeks, and the level of precision was possible, you could do multiple iterations of this given each user's (small) risk of contracting the virus from their interactions too, given an estimate of what we think the time might be between when a person contracts the virus to when they become infectious. This is harder, but if everything else works, it's possible.

Writing this out in words is long, but the actual summation/integral is not too complicated. It's a combination of science and hack-y guesses. But this seems true to me of almost all engineering.