Effects of a Partially Effective Vaccine

The race to publish COVID-19 related papers is on, and I am already behind. Instead, I will repurpose a paper by Eduard Talamas and myself on networks and infections which is due out in GEB.

It is prompted by the following question: if you are given the option to distribute—without cost to you or anyone else—a perfectly safe but only moderately effective vaccine for a viral infection, should you? That we’ve posed it means the answer must be no or at least maybe not.

Unsurprisingly, it has to do with incentives. When the risk of becoming infected from contact declines, individuals tend to be less circumspect about coming into contact with others. This is risk compensation, first suggested by Charles Adams  in 1879 and popularized by Sam Peltzman in the 1970’s.

Therefore, the introduction of a vaccine has two effects. On the one hand, it reduces the probability that an individual becomes infected upon contact. On the other hand, it decreases individuals’ incentives to take costly measures to avoid contact. If the second effect outweighs the first, there will be an increase in infections upon the introduction of a moderately effective vaccine.

These are statements about infection rates not welfare. Individuals make trade-offs. In this case between the risk of infection and the costs of avoiding it. Therefore, observing that an individual’s infection probability will increase upon introduction of a partially effective vaccine is insufficient to argue against introduction.

In our paper, Eduard and I show that the introduction of a vaccine whose effectiveness falls below some threshold could make everyone worse off, even when each individual is perfectly rational and bears the full cost of becoming infected. If the vaccine is highly effective, this outcome is reversed. This is because risky interactions can be strategic complements. An individual’s optimal amount of risky interactions can be increasing in the amount of risky interactions that others take.

To illustrate, call two individuals that engage in risky interactions partners. Every risky interaction that Ann’s partner Bob has with Chloe affects Ann’s incentives to have risky interactions with Chloe in two countervailing ways. It increases Chloe’s infection probability. But it also increases the probability that Ann is infected conditional on Chloe being infected—because if Chloe is infected, chances are that Ann’s partner Bob is also infected. Given that a risky interaction between Ann and Chloe only increases the probability that Ann becomes infected when Chloe is infected and Ann is not, the combination of these effects can lead to an increase in Ann’s incentives to engage with Chloe and her partners when Bob engages with Chloe.

One might ask, given the huge trove of papers on epidemiological models, this effect must have been identified before and discussed? No, or at least not as far as we could tell. This is because we depart from from a standard feature of these models. We allow agents to strategically choose their partners— instead of only allowing them to choose the number of partners, and then having matches occur uniformly at random.

Econometrica Submission Fees

This morning, a missive from the Econometrics society arrived in my in box announcing “two modest fees associated with the submission and publication of papers in its three journals.” As of May 1st 2020, the Society will assess a submission fee of $50 and a page charge of $10 per page for accepted papers. With papers on the short side running to around 30 pages and 10 page appendices this comes out to about $400. By the standards of the natural sciences this is indeed modest.

At the low end the American Meteorological Society charges $120 per page, no submission fee. In the middle tier, the largest open-access publishers — BioMed Central and PLoS — charge $1,350–2,250 to publish peer-reviewed articles in many of their journals, and their most selective offerings charge $2,700–2,900. At the luxury end of the market is the Proceedings of the National Academy which starts out at $1590 for 6 pages and rises upto $4,215 for a 12 page paper.

My colleague Aislinn Bohren has suggested rewarding referees with free page coupons: publish one page free for each five pages you referee. This may suffer the same fate as the Capitol Hill Baby Sitting co-operative.

In the short run the effect will be to drive papers to JET and GEB as not all academics have research budgets which will cover the fees. An alternative is to submit the paper for $50. If accepted, decline to have it published. Send it elsewhere and send a copy of the acceptance letter to one’s promotion and tenure committee. Voila, a new category in the CV: accepted at Econometrica but not published.

 

 

How Bibi changed the disagreement point

Time for some game theory about the recent development in Israeli politics. (You can read more about it here)

In the aftermath of the last election, no side could form a governing coalition. So the options were either a unity government, composed of the two major parties (Bibi Netanyahu’s Likud and Benny Gantz’ Kahol-Lavan) or yet another, fourth in a row, election. But with a year of election campaigns sowing hatred throughout the country, the ongoing coronavirus lockdown and an impending recession, nobody wants another election. So the two factions have to bargain on the terms of the unity government, with the possibility of another election being what game theorists call a disagreement point: this is what happens if negotiations break down.

Game theory teaches us that the disagreement point has a significant impact on the outcome, even when the parties reach an agreement, and that the better your position in the disagreement point, the better the bargaining outcome will be for you.

The disagreement point in the negotiation following the last election is impacted by two factors that did not exist in the previous rounds:

• The coronavirus came as a gift from the gods for Bibi. Frightened citizens usually stick with incumbent leaders in times of crisis, and in Bibi’s case, this tendency is magnified by the fact that he has been around for so many years. Even before the virus popped up, the greatest difficulty of the opposition was to convince the public that he is not the only person who is qualified for the prime-minister job. So an election will be advantageous to Bibi. Moreover, since the election will be postponed until after the lockdown, Bibi will likely stay in power for a long period even under the disagreement point.
• While he could not form a government, this time Gantz did manage to form a functioning majority coalition in the parliament. He was on the verge of electing a parliamentary speaker from his party, which would allow him to implement a law — the so-called “Bibi law” that forbids Bibi from being appointed for a  prime-minister due to the fact that he was indicted for bribery.

So, in terms of the disagreement point, Bibi has something good going for him and something bad going for him. So he made a threat: If you continue with your current plan to take over the parliament, I will abandon the negotiations and we will jump straight ahead to the disagreement point. Gantz, believing the threat, gave up on the plan. Instead, he promised to give the speakership to somebody from Bibi’s party and to join a government led by Bibi and, and he has split his own party for that purpose, bringing with him only half of it to the new coalition under Bibi.

So now Gantz made the disagreement point much better for Bibi and much worse from himself:
— If there is an election again, Gantz will be viewed as a fool who tore down his party for nothing, and he will likely not be elected.
— Gantz has made public that he too thinks Bibi should be a prime minister. This statement substantiates Bibi’s image as the only qualified person for the job. It also diminishes the moral argument against choosing an indicted person to prime-minister, which was the justification for Bibi’s law and a central theme in the campaign against him.
— Gantz currently took the speakership to himself. Joining Bibi’s government will force him to resign from the speakership, and he has already agreed to give it to Bibi’s party. After Bibi has the speakership, even before the government is formed, it is not possible to pass the Bibi law.

The alternative was to take over the parliament, uncover information about the government’s lack of preparation and incompetent handling of the corona crisis, and meanwhile negotiate secretly over the unity government. But without publicly supporting Bibi as the prime-minster until the deal is reached, all the while keeping the option of passing Bibi’s law if the negotiation fails. It’s hard to predict a bargaining outcome, but I think it is possible that under this plan, even if Bibi would have continued to be the prime minister for some time, we would see a meaningful division of power and a date for the termination of Bibi’s reign. Now there is no such hope.

Why did Gantz give up all his cards? I think the reason is that, because he believed in Bibi’s threat to abandon the negotiations, Gantz thought he had to choose between another election and a unity government. Indeed, Gantz conducted an election poll before he made this move, which suggests he was thinking in these terms.

Bibi’s threat, however, was empty, or, to use another game-theoretic jargon, not credible. The fact that the disagreement point would have become worse for Bibi precisely implies that, whatever he says now, he would have been willing to negotiate in order to escape it. The only difference is that the negotiation would be conducted in an environment that is less convenient to Bibi.
It is also not at all clear that the probability of reaching a deal has increased following Ganz’s move. In fact, an exultant Bibi might insist on terms that even Gantz will not be able to accept.

Many of Gantz’ voters are furious because they preferred another election over a Bibi government. But Gant’s voters who were willing to stomach a Bibi government should also be furious because Gantz created a bargaining environment in which he is desperate to achieve it.

Pass/Fail and Unraveling

With the move to on-line classes after spring break in the wake of Covid-19, my University has allowed students to opt to take some, all or none of their courses as pass/fail this semester. By making it optional, students have the opportunity to engage in signaling. A student doing well entering into spring break may elect to take the course for a regular grade confident they will gain a high grade. A student doing poorly entering into spring break may elect to take the course pass/fail. It is easy to concoct a simplified model (think Grossman (1981) or Milgrom (1981)) where there is no equilibrium in which all students elect to take the course pass/fail. The student confident of being at the top of the grade distribution has an incentive to choose the regular grading option. The next student will do the same for fear of signaling they had a poor grade and so on down the line. In equilibrium all the private information will unravel.

This simple intuition ignores the heterogeneity in student conditions. It is possible that a student with a good score going into spring break may now face straitened circumstances after spring break. How they decide depends on what inferences they think, others, employers, for example, make about grades earned during this period. Should an employer simply ignore any grades earned during this period and Universities issue Covid-19 adjusted GPAs? Should an employer conclude that a student with a poor grade is actually a good student (because they did not choose the pass/fail option) who has suffered bad luck?

Going On-line

In response to the Covid-19 virus a number of American Universities are moving instruction on-line. Some see this as great natural experiment to test the efficacy of virtual instruction (NO). Others believe it will speed the pace at which instruction  moves on-line (NO). The focus now is on execution at scale in a short period of time.  We would be better off canceling the rest of term and giving all the students A’s.

Here is what I predict will happen. Students will be dilatory in viewing lectures. Temptation and the difficulty of adjusting to new habits will be obstacles. When the exams approach, some will complain that they are unprepared because virtual is not as good as live, their instructor made a hash of things, the absence of live office hours, etc. etc. The exams will be take home without any proctors. While one’s own spirit is willing, there are doubts about the rectitude of one’s classmates.

On the other hand, during this period of exile, perhaps, there will emerge another Newton.

 

Who Should Quarantine?

An agent with an infectious disease confers a negative externality on the rest of the community. If the cost of infection is sufficiently high, they are encouraged and in some cases required to quarantine themselves. Is this the efficient outcome? One might wonder if a Coasian approach would generate it instead. Define a right to walk around when infected which can be bought and sold. Alas, infection has the nature of public bad which is non-rivalrous and non-excludable. There is no efficient, incentive compatible individually rational (IR) mechanism for the allocation of such public bads (or goods). So, something has to give. The mandatory quarantine of those who might be infected can be interpreted as relaxing the IR constraint for some.

If one is going to relax the IR constraint it is far from obvious that it should be the IR constraint of the infected. What if the costs of being infected vary dramatically? Imagine a well defined subset of the population bears a huge cost for infection while the cost for everyone else is minuscule. If that subset is small, then, the mandatory quarantine (and other mitigation strategies) could be far from efficient. It might be more efficient for the subset that bears the larger cost of infection to quarantine themselves from the rest of the community.

 

New Intermediate Microeconomics Book

Six years ago, I decided to teach intermediate microeconomics. I described my views on how it should be taught in an earlier post. The notes for that course grew into a textbook that is now available in Europe and in the US this April. I am particularly delighted at being able to sport Paolo Ucello’s `The Hunt’ upon the cover. The publishers, Cambridge University Press, asked me to provide an explanation for why I had chosen this, and it appears on the rear cover. Should you make your way to Oxford, be sure to stop by the Ashmolean Museum to see it, the painting of course, in all its glory. I day dream, that like Samuelson’s `Economics’, it will sell bigly.

The Incentive Auction for Radio Spectrum

Over a rabelaisian feast with convivial company, conversation turned to a twitter contretemps between economic theorists known to us at table. Its proximate cause was the design of the incentive auction for radio spectrum. The curious can dig around on twitter for the cut and thrust. A summary of the salient economic issues might be helpful for those following the matter.

Three years ago, in the cruelest of months, the FCC conducted an auction to reallocate radio spectrum. It had a procurement phase in which spectrum would be purchased from current holders and a second phase in which it was resold to others. The goal was to shift spectrum, where appropriate, from current holders to others who might use this scarce resource more efficiently.

It is the procurement phase that concerns us. The precise details of the auction in this phase will not matter. Its design is rooted in Ausubel’s clinching auction by way of Bikhchandani et al (2011) culminating in Milgrom and Segal (2019).

The pricing rule of the procurement auction was chosen under the assumption that each seller owned a single license. If invalid, it allows a seller with multiple licenses to engage in what is known as supply reduction to push up the price. Even if each seller initially owned a single license, a subset of sellers could benefit from merging their assets and coordinating their bids (or an outsider could come in and aggregate some sellers prior to the auction). A recent paper by my colleagues Doraszelski, Seim, Sinkinson and Wang offers estimates of how much sellers might have gained from strategic supply reduction.

Was the choice of price rule a design flaw? I say, compared to what? How about the VCG mechanism? It would award a seller owning multiple licenses the marginal product associated with their set of licenses. In general, if the assets held by sellers are substitutes for each other, the marginal product of a set will exceed the sum of the marginal products of its individual elements. Thus, the VCG auction would have left the seller with higher surplus than they would have obtained under the procurement auction assuming no supply reduction. As noted in Paul Milgrom’s  book, when goods are substitutes, the VCG auction creates an incentive for mergers. This is formalized in Sher (2010). The pricing rule of the procurement auction could be modified to account for multiple ownership (see Bikhchandani et al (2011)) but it would have the same qualitative effect. A seller would earn a higher surplus than they would have obtained under the procurement auction assuming no supply reduction. A second point of comparison would be to an auction that was explicitly designed to discourage mergers of this kind. If memory serves, this reduces the auction to a posted price mechanism.

Was there anything that could have been done to discourage  mergers? The auction did have reserve prices, so an upper limit was set on how much would be paid for licenses. Legal action is a possibility but its not clear whether that could have been pursued without delaying the auction.

Stepping back, one might ask a more basic question: should the reallocation of spectrum have been done by auction? Why not follow Coase and let the market sort it out? The orthodox answer is no because of hold-up and transaction costs. However, as Thomas Hazlett has argued, there are transaction costs on the auction side as well.

 

2.2.22 manifesto

This morning I sent the following message to the heads of the NSF and the ERC.

As Don Quixotian as this manifesto may be, it is in fact simple, sets incentives right, and requires little coordination among the two largest research funding agencies.

---

Dear Professors Cordova and Ferrari,

We are all aware of the escalation in journal publication costs for researchers, reaching thousands of dollars per article in  e.g. leading life sciences journals, and at the other extreme the phenomenon of completely fake journals, that publish un-refereed papers for huge fees.  This is public money going to private gate-keepers who provide little or even negative added value.

 

A relatively simple joint policy by the NSF and the ERC may overturn this situation:

 

1) Announce that as of 2.2.22, NSF and ERC grants will be awarded only to researchers all of whose journal publications as of that date appear in journals which are (i) open access, and (ii)  charge at most $100 for submission+publication;  in grant applications, researchers’ CVs will be required to mention explicitly, next to each publication, the publication cost.

 

2) In view of this new policy, encourage entire editorial boards of all scientific journals to migrate to one of the open and free journal systems  (e.g. OJS), and to re-establish there their journal under the name “The New Journal/Review of…”, respectively.  The NSF and the ERC may finance a small team to facilitate the technical transition of editorial boards.

 

Sincerely,

Aviad Heifetz

 

The game theory of the electoral threshold

Israel is holding an election Tuesday. It’s a multi-party system in Israel, based on proportional representation, and the main battle is between two blocs of parties. We call them the left-wing bloc and the right-wing bloc, and civics textbooks will tell you that this division is about policy: the Arab-Israeli conflict, economics and the role of religion. But let’s face it, nobody cares about public policy anymore. It’s all about Bibi. Parties that intend to form a coalition under Bibi belong to the right-wing bloc and parties that intend to rid the country of him belong to the left wing bloc.

In addition to the battle between the blocs, there is also a contest between the parties in each bloc, especially between one big party and several satellites. Bibi, in particular, is expected to use the final days of the campaign to siphon votes from satellites to his own Likud party, as he did in previous elections. Indeed, in recent days, Bibi and his social media fans said that the number of seats of the Likud party compared to the major opponent (called BlueWhite), and not the size of the blocs, will determine whether Bibi will form the government again. BlueWhite makes the same argument to left-wing voters.

But if Bibi is serious about cannibalizing his bloc, he could use a more fatal argument by appealing to the electoral threshold, that is the minimum share of total votes that a party has to achieve to be entitled to seats in the parliament. The current threshold 3.25%, which translates to roughly five seats in the 120-seat parliament. This sounds like a low threshold, but there are eleven parties in the current parliament, some of them have split into two, there are new parties that are serious contenders, and Likud and BlueWhite together are expected to win about half of the total votes. So many small parties are close to the threshold in the opinion polls. Since the two blocs are almost tied (with a small advantage to the right-wing bloc), the electoral threshold might end up playing a significant role in determining the outcome if many votes are cast to a party that doesn’t cross the threshold.

Enter game theory

It is difficult to do a game-theoretic analysis of anything election because we don’t have a good explanation for what drives voters to stand in line in the voting booth rather than go to the beach. Let me bypass this issue and just assume that you gain some utility from voting to your favorite party. So assume for simplicity that right-wing voters have two options, Likud or New Right (NR), which is one of the new parties in the right-wing bloc, whose leaders Bibi detests. According to the polls, 4% of the voters are New Right supporters. These guys will get utility 2 if they vote New Right and utility 1 if they vote Likud. And here is the twist: you only get this utility if the party you vote to passes the electoral threshold. If it doesn’t, you feel like your vote has been wasted and you get utility 0.

Whether or not an NR supporter will vote her preference depends on what she thinks the other supporter will do. In fact, NR supporters are involved in a population stag hunt game (sometimes called investment game). There are two Nash equilibria in the game: either everyone votes NR or everyone votes Likud. The first equilibrium requires players to take the risky action of voting NR, and if they all go along, they all get high utility 2. I have played variants of the investment game in class several times. Typically, players go for the safe choice, which in this case is Likud. If you do this, you guarantee yourself a utility 1 regardless of what the other players do.

How could the potential voters of NR coordinate on the risky equilibrium which gives a higher utility? I think the public opinion polls have a big role here. In the recent polls, NR has been consistently hovering above the threshold. The polls make it commonly known that there are sufficiently many potential NR voters, and also that they plan to go along with voting NR.

Suppose however that on Monday Bibi hints that his internal polls show that NR does not cross the threshold. What would our NR supporter do? She will likely know that there are no such polls since most Bibi supporters know that he is a lier. But perhaps other players will believe Bibi and switch to voting Likud, which will drive NR below the threshold. Or perhaps other players will worry that some other players will believe him. You see where this is going: to play the risky equilibrium, you need confidence that the other players are playing it too. Bibi can shake this confidence even if he can’t change the fact that NR has enough supporters to cross the threshold if they all collaborate.

By the way, I believe that established parties that are already represented in the parliament are less vulnerable to such an attack, since the confidence of their supporters in the risky equilibrium is stronger, as they played it once already.

Go ahead Bibi, make my day.