Tag Archives: game theory

On the optimality of self-quarantine

Is self-quarantine early in an epidemic optimal, either individually or for society?

Individual incentives are easier to analyse, so let’s start with these. Conditional on catching a disease, other things equal, later is better. The reasons are discounting and the advances in treatment. A delay of many years may increase the severity conditional on infection (old age weakens immunity), but such long time intervals are typically not relevant in an epidemic.

Conditional on falling ill within the next year (during which discounting and advances in treatment are negligible), it is better to catch the disease when few others are infected, so hospitals have spare capacity. This suggests either significantly before or long after the peak of the epidemic. Self-quarantine, if tight enough, may postpone one’s infection past the peak.

Another individually optimal choice is to get infected early (also called vaccination with live unattenuated virus), although not if immunity increases very little or even decreases. The latter means that one infection raises the probability of another with the same disease, like for malaria, HIV and herpes, which hide out in the organism and recur. Cancer displays similar comebacks. For viral respiratory diseases, as far as I know, immunity increases after infection, but not to 100%. The optimality of self-quarantine vs trying to be infected early then depends on the degree of immunity generated, the quality of the quarantine, whether the disease will be eradicated soon after the epidemic, and other details of the situation.

Individual optimality also depends on what the rest of the population is doing. If their self-quarantine is close to perfect, then an individual’s risk of catching the disease is very low, so no reason to suffer the disutility of isolation. If others quarantine themselves moderately, so the disease will be eradicated soon, but currently is quite infectious, then self-isolation is individually optimal. If others do almost nothing, and the disease spreads easily and does not generate much immunity, then an individual will either have to self-quarantine indefinitely or will catch it. Seasonal flu and the common cold (various rhinoviruses and adenoviruses) are reasonable examples. For these, self-quarantine is individually suboptimal.

Social welfare considerations seem to weigh in favour of self-quarantine, because a sick person infects others, which speeds up the epidemic. One exception to the optimality of self-quarantine comes from economies of scale in treatment when prevalence is not so high as to overwhelm the health system. If the epidemic is fading, but the disease increases immunity and is likely to become endemic, with low prevalence, then it may be better from a social standpoint to catch the disease when treatment is widely available, medical personnel have just had plenty of experience with this illness, and not many other people remain susceptible. This is rare.

Herd immunity is another reason why self-quarantine is socially suboptimal for some diseases. The logic is the same as for vaccination. If catching chickenpox as a child is a mild problem and prevents contracting and spreading it at an older age when it is more severe, then sending children to a school with a chickenpox epidemic is a smart idea.

Reducing the duration of quarantine for vulnerable populations is another reason why being infected sooner rather than later may be socially optimal. Suppose a disease is dangerous for some groups, but mild or even undetectable for most of the population, spreads widely and makes people resistant enough that herd immunity leads to eradication. During the epidemic, the vulnerable have to be isolated, which is unpleasant for them. The faster the non-vulnerable people get their herd immunity and eradicate the infection, the shorter the quarantine required for the vulnerable.

For most epidemics, but not all, self-quarantine is probably socially optimal.

Prefereeing increases the inequality of research output

Why do top researchers in economics publish almost exclusively in the top 5 journals? Random idea generation and mistakes in the course of its implementation should imply significant variance of the quality of finished research projects even for the best scientists. So top people should have more of all quality levels of papers.

Nepotism is not necessary to explain why those at top universities find it easier to publish in top journals. Researchers at the best departments have frequent access to editors and referees of top journals (their colleagues), so can select ideas that the editors and referees like and further tailor the project to the tastes of these gatekeepers during writing. Researchers without such access to editors and referees choose their projects “blindly” and develop the ideas in directions that only match gatekeeper tastes by chance. This results in much “wasted work” if the goal is to publish well (which may or may not be correlated with the social welfare from the research).

In addition to selecting and tailoring projects, those with access can also better select journals, because they know the preferences of the editorial board. So for any given project, networking with the gatekeepers allows choosing a journal where editors are likely to like this project. This reduces the number of rejections before eventual acceptance, allowing accumulating publications quicker and saving the labour of some rounds of revision of the paper (at journals that reject after a revise-and-resubmit for example).

A similar rich-get-richer positive feedback operates in business, especially for firms that sell to other firms (B2B). Top businesspeople get access to decisionmakers at other organisations, so can learn what the market desires, thus can select and tailor products to the wants of potential customers. Better selection and targeting avoids wasting product development costs. The products may or may not increase social welfare.

Information about other business leaders’ preferences also helps target the marketing of any given product to those predisposed to like the product. Thus successful businesspeople (who have access to influential decisionmakers) have a more popular selection of products with lower development and marketing costs.

On the seller side, firms would not want their competitors to know what the buyers desire, but the buyer side has a clear incentive to inform all sellers, not just those with access. Empirically, few buyers publish on their websites any information about their desired products. One reason may be that info is costly to provide, e.g. requests for product characteristics reveal business secrets about the buyer. However, disclosure costs would also prevent revealing info via networking. Another reason buyers do not to publicly announce their desired products may be that the buyers are also sellers of other products, so trade information for information with their suppliers who are also their customers. The industry or economy as a whole would benefit from more information-sharing (saving the cost of unwanted products), so some trading friction must prevent this mutually beneficial exchange.

One friction is an agency conflict between managers and shareholders. If managers are evaluated based on relative performance, then the managers of some firms may collude to only share useful information with each other, not with those outside their circle. The firms managed by the circle would benefit from wider sharing of their product needs, because outside companies would enter the competition to supply them, reducing their costs. However, those outside firms would get extra profit, making their managers look good, thus lowering the relative standing of the managers in the circle.

Popularity inequality and multiple equilibria

Suppose losing a friend is more costly for a person with few contacts than with many. Then a person with many friends has a lower cost of treating people badly, e.g. acting as if friends are dispensable and interchangeable. The lower cost means that unpleasant acts can signal popularity. Suppose that people value connections with popular others more than unpopular. This creates a benefit from costly, thus credible, signalling of popularity – such signals attract new acquaintances. Having a larger network in turn reduces the cost of signalling popularity by treating friends badly.

Suppose people on average value a popular friend more than the disutility from being treated badly by that person (so the bad treatment is not too bad, more of a minor annoyance). Then a feedback loop arises where bad treatment of others attracts more connections than it loses. The popular get even more popular, reducing their cost of signalling popularity, which allows attracting more connections. Those with few contacts do not want to imitate the stars of the network by also acting unpleasantly, because their expected cost is larger. For example, there is uncertainty about the disutility a friend gets from being treated badly or about how much the friend values the connection, so treating her or him badly destroys the friendship with positive probability. An unpopular person suffers a large cost from losing even one friend.

Under the assumptions above, a popular person can rely on the Law of Large Numbers to increase her or his popularity in expectation by treating others badly. A person with few friends does not want to take the risk of losing even them if they turn out to be sensitive to nastiness.

Multiple equilibria may exist in the whole society: one in which everyone has many contacts and is nasty to them and one in which people have few friends and act nice. Under the assumption that people value a popular friend more than the disutility from being treated badly, the equilibrium with many contacts and bad behaviour actually gives greater utility to everyone. This counterintuitive conclusion can be changed by assuming that popularity is relative, not a function of the absolute number of friends. Total relative popularity is constant in the population, in which case the bad treatment equilibrium is worse by the disutility of bad treatment.

In order for there to be something to signal, it cannot be common knowledge that everyone is equally popular. Signalling with reasonable beliefs requires unequal popularity. Inequality reduces welfare if people are risk averse (in this case over their popularity). Risk aversion further reduces average utility in the popular-and-nasty equilibrium compared to the pooling equilibrium where everyone has few friends and does not signal (acts nice).

In general, if one of the benefits of signalling is a reduction in the cost of signalling, then the amount of signalling and inequality increases. My paper “Dynamic noisy signaling” (2018) studies this in the context of education signalling in Section V.B “Human capital accumulation”.

The smartest professors need not admit the smartest students

The smartest professors are likely the best at targeting admission offers to students who are the most useful for them. Other things equal, the intelligence of a student is beneficial, but there may be tradeoffs. The overall usefulness may be maximised by prioritising obedience (manipulability) over intelligence or hard work. It is an empirical question what the real admissions criteria are. Data on pre-admissions personality test results (which the admissions committee may or may not have) would allow measuring whether the admission probability increases in obedience. Measuring such effects for non-top universities is complicated by the strategic incentive to admit students who are reasonably likely to accept, i.e. unlikely to get a much better offer elsewhere. So the middle- and bottom-ranked universities might not offer a place to the highest-scoring students for reasons independent of the obedience-intelligence tradeoff.

Similarly, a firm does not necessarily hire the brightest and individually most productive workers, but rather those who the firm expects to contribute the most to the firm’s bottom line. Working well with colleagues, following orders and procedures may in some cases be the most important characteristics. A genius who is a maverick may disrupt other workers in the organisation too much, reducing overall productivity.

Privacy reduces cooperation, may be countered by free speech

Cooperation relies on reputation. For example, fraud in online markets is deterred by the threat of bad reviews, which reduce future trading with the defector. Data protection, specifically the “right to be forgotten” allows those with a bad reputation to erase their records from the market provider’s database and create new accounts with a clean slate. Bayesian participants of the market then rationally attach a bad reputation to any new account (“guilty until proven innocent”). If new entrants are penalised, then entry and competition decrease.

One way to counter this abusing of data protection laws to escape the consequences of one’s past misdeeds is to use free speech laws. Allow market participants to comment on or rate others, protecting such comments as a civil liberty. If other traders can identify a bad actor, for example using his or her government-issued ID, then any future account by the same individual can be penalised by attaching the previous bad comments from the start.

Of course, comments could be abused to destroy competitors’ reputations, so leaving a bad comment should have a cost. For example, the comments are numerical ratings and the average rating given by a person is subtracted from all ratings given by that person. Dividing by the standard deviation is helpful for making the ratings of those with extreme opinions comparable to the scores given by moderates. Normalising by the mean and standard deviation makes ratings relative, so pulling down someone’s reputation pushes up those of others.

However, if a single entity can control multiple accounts (create fake profiles or use company accounts), then he or she can exchange positive ratings between his or her own profiles and rate others badly. Without being able to distinguish new accounts from fake profiles, any rating system has to either penalise entrants or allow sock-puppet accounts to operate unchecked. Again, official ID requirements may deter multiple account creation, but privacy laws impede this deterrence. There is always the following trilemma: either some form of un-erasable web activity history is kept, or entrants are punished, or fake accounts go unpunished.

M-diagram of politics

Suppose a politician claims that X is best for society. Quiz:

1. Should we infer that X is best for society?

2. Should we infer that the politician believes that X is best for society?

3. Should we infer that X is best for the politician?

4. Should we infer that X is best for the politician among policies that can be `sold’ as best for society?

5. Should we infer that the politician believes that X is best for the politician?

This quiz illustrates the general principle in game theory that players best-respond to their perceptions, not reality. Sometimes the perceptions may coincide with reality. Equilibrium concepts like Nash equilibrium assume that on average, players have correct beliefs.

The following diagram illustrates the reasoning of the politician claiming X is best for society: M-diagram of politics In case the diagram does not load, here is its description: the top row has `Official goal’ and `Real goal’, the bottom row has `Best way to the official goal’, `Best way to the real goal that looks like a reasonable way to the official goal’ and `Best way to the real goal’. Arrows point in an M-shaped pattern from the bottom row items to the top items. The arrow from `Best way to the real goal that looks like a reasonable way to the official goal’ to `Official goal’ is the constraint on the claims of the politician.

The correct answer to the quiz is 5.

This post is loosely translated from the original Estonian one https://www.sanderheinsalu.com/ajaveeb/?p=140

Economic and political cycles interlinked

Suppose the government’s policy determines the state of the economy with a lag that equals one term of the government. Also assume that voters re-elect the incumbent in a good economy, but choose the challenger in a bad economy. This voting pattern is empirically realistic and may be caused by voters not understanding the lag between the policy and the economy. Suppose there are two political parties: the good and the bad. The policy the good party enacts when in power puts the economy in a good state during the next term of government. The bad party’s policy creates a recession in the next term.

If the economy starts out doing well and the good party is initially in power, then the good party remains in power forever, because during each of its terms in government, it makes the economy do well the next term, so voters re-elect it the next term.

If the economy starts out in a recession with the good party in power, then the second government is the bad party. The economy does well during the second government’s term due to the policy of the good party in the first term. Then voters re-elect the bad party, but the economy does badly in the third term due to the bad party’s previous policy. The fourth government is then again the good party, with the economy in a recession. This situation is the same as during the first government, so cycles occur. The length of a cycle is three terms. In the first term, the good party is in power, with the other two terms governed by the bad party. In the first and third term, the economy is in recession, but in the second term, booming.

If the initial government is the bad party, with the economy in recession, then the three-term cycle again occurs, starting from the third term described above. Specifically, voters choose the good party next, but the economy does badly again because of the bad party’s current policy. Then voters change back to the bad party, but the economy booms due to the policy the good party enacted when it was in power. Re-election of the bad is followed by a recession, which is the same state of affairs as initially.

If the government starts out bad and the economy does well, then again the three-term cycle repeats: the next government is bad, with the economy in recession. After that, the good party rules, but the economy still does badly. Then again the bad party comes to power and benefits from the economic growth caused by the good party’s previous policy.

Overall, the bad party is in power two-thirds of the time and the economy in recession also two-thirds of the time. Recessions overlap with the bad party in only one-third of government terms.

Of course, reality is more complicated than the simple model described above – there are random shocks to the economy, policy lags are not exactly equal to one term of the government, the length of time a party stays in power is random, one party’s policy may be better in one situation but worse in another.

Tradeoff between flashiness and competitive advantage in sports

Sports equipment is often brightly coloured, with eye-catching shape, such as for bicycle frames. Sometimes flashiness is beneficial, for example improving the visibility of a bike or a runner on the road, or a boat on the water. However, in sports where competitors act directly against each other (ballgames, racquet sports, fencing), eye-catching equipment makes it easier for opponents to track one’s movements, which is a disadvantage. For a similar reason, practical military equipment is camouflaged and dull-coloured, unlike dress uniforms.

Athletes would probably gain a small advantage by using either dull grey clothing, perhaps with camouflage spots, or equipment that matches the colour of the sports arena, e.g. green grass-patterned shoes and socks for a football field, blue or red for a tennis court. Eye-deceiving colouring would be especially useful in competitions based on rapid accurate movement and feints, such as fencing or badminton.

Another option for interfering with an opponent’s tracking of one’s movements is to use reflective clothing (mirror surfaces, safety orange or neon yellow) to blind the rival. This would work especially well for outdoor sports in the sunshine or in stadiums lit by floodlights.

One downside of dull clothing may be that it does not inspire fans or sponsors, so wearing it may reduce the athlete’s income from merchandise and advertising. A similar tradeoff occurs in real vs movie fighting. Blindingly bright equipment does not have this disadvantage.

Another downside of camouflage may occur if it replaces red clothing, which has been found to give football teams a small advantage. The reason is psychological: red makes the wearers more aggressive and the opponents less.

Golf as a cartel monitoring device for skilled services

Many explanations have been advanced for golf and similar costly, seemingly boring, low-effort group activities. One reason could be signalling one’s wealth and leisure by an expensive and time-consuming sport, another may be networking during a low-effort group activity that does not interfere with talking.

An additional explanation is monitoring others’ time use. A cartel agrees to restrict the quantity that its members provide, in order to raise price. In skilled services (doctors, lawyers, engineers, notaries, consultants) the quantity sold is work hours. Each member of a cartel has an incentive to secretly increase supply to obtain more profit. Monitoring is thus needed to sustain the cartel. One way to check that competitors are not selling more work hours is to observe their time use by being together. To reduce boredom, the time spent in mutual monitoring should be filled somehow, and the activity cannot be too strenuous, otherwise it could not be sustained for long enough to meaningfully decrease hours worked. Playing golf fulfills these requirements.

A prediction from this explanation for golf is that participation in time-consuming group activities would be greater in industries selling time-intensive products and services. By contrast, if supply is relatively insensitive to hours worked, for example in capital-intensive industries or standard software, then monitoring competitors’ time use is ineffective in restricting their output and sustaining a cartel. Other ways of checking quantity must then be found, such as price-matching guarantees, which incentivise customers to report a reduced price of a competitor.

Platform providers fake being popular

Crowdfunding platforms, stock exchanges and other providers of two-sided markets want to appear popular, because having more buyers attracts more sellers and vice versa. The platform’s revenue is usually proportional to the number of users, because it charges a commission fee on trades or advertisers pay it to show ads to users. The exchange’s marginal cost of a user is close to zero, giving it an incentive to fake a high volume of trades, a large limit order book and a small bid-ask spread.

The platform’s cost of posting a great volume of outstanding buy and sell orders at a small spread is that many investors try to trade at these favourable bid and ask prices. Either the market maker has to take the other side of these attempted transactions or is found fraudulent. Taking the other side results in a large loss if some investors are better informed than the exchange.

The platform could falsely display a large trading volume, but keep the order book honestly small by adding fake trades at prices between the bid and the ask only, so no investor’s real limit order is ignored. This seems difficult to detect, unless one side of the limit order book is empty (e.g. no buyers) and at least one at-market order on the other side (e.g. a sell) is outstanding. In this case, any trades occurring would have to satisfy the at-market order. However, the platform or real investors can then take the other side of the at-market order at a very favourable price to themselves, which discourages at-market orders. A large trading volume with a thin order book is still slightly suspicious, because it requires that crossing buy and sell orders between the bid and ask prices arrive almost simultaneously, in order to be matched without appearing on the order book for long, and without triggering the real limit orders. Displaying the fake buys and sells on the order book risks attracting actual matching trades, which the platform would have to honour (at a cost).

Without automated quote matching, there are no at-market orders, for example on the Funderbeam crowdfunding platform. Instead, everyone either posts a limit order or picks an order from the other side to trade with, e.g. a buyer chooses a sell. Investors can pick an order with a worse price (higher sell or lower buy) on the other side, which frequently occurs on Funderbeam. Choosing a worse price is irrational, unless the traders in question are colluding, so the asset is effectively not changing ownership. Reasons to carry out such seemingly irrational trades are to manipulate price and volume, e.g. price can be raised or reduced by targeted trades outside the bid-ask interval. Volume can only increase after added trades, rational or not, but such seemingly greater activity is exactly what benefits the stakeholders of the platform. The employees of the market maker have a natural motive to fake-trade between themselves to make their firm look good, even without any inappropriate pressure from their boss.

Another way to attract issuers and investors is to demonstrate successful initial public offerings, meaning that the funds are raised quickly (good for issuers) and the price of the newly listed stock (or other asset) goes up, which benefits investors. Adding fake capital-raisers is difficult, because potential investors will check the background of the supposed issuer. Inserting spoof investors into an actual funding campaign is costly, because real money would have to be invested. One way to manipulate popularity upward is to simultaneously add a fake issuer and fake investors who satisfy its funding need. The idea is to not leave time for real investors to participate in the campaign, by pretending that the capital-raiser achieved its target funding level before most investors could react. This is easier in markets with a small number of real investors and without an auto-invest feature. However, the real investors who were supposedly pre-empted may still research the supposedly very popular issuer.

A costless way to briefly boost the popularity of a real fundraising campaign is to add fake investors after the target funding is achieved, and forbid issuers from increasing the target or accepting funds from those who subscribed after the goal was reached. Any campaign meeting its target can then be made to look heavily oversubscribed. However, if the issuers are informed in advance of the restriction not to increase the target, then they may find an alternative unrestricted platform to raise funds. On the other hand, if the restriction is not mentioned beforehand, then it will likely anger the issuers who will then create negative publicity for the platform. Competition between exchanges thus curtails their manipulation incentives.

The platform can motivate real investors to raise their bids when the campaign reaches its target by rationing demand: bidders in an oversubscribed share issue get only a fraction of what they wanted to buy. Anticipating this, buyers will increase their requested quantities so that the fraction of their new bid equals their actual demand. This makes the campaign look oversubscribed and creates a feedback loop: if other investors increase their quantities, then rationing reduces the fraction of a given investor’s demand that will be satisfied, so this investor raises her or his requested amount, which in turn makes others increase theirs.

If investors know of the bid rationing in advance, then they may select a rival market provider without this restriction, but if rationing takes them by surprise, then they may leave and publicly criticise the platform. Capital-raisers compare exchanges, so if many market providers inflate demand and the issuers pay attention to the level of oversubscription (instead of the fraction of campaigns reaching the target, which is what should matter to the capital-raiser), then the biggest inflator wins. Of course, platforms may not want to reveal unsuccessful campaigns (e.g. Funderbeam does not), so public data on the fraction of issuers who achieved their funding goal is unlikely to exist.

Theoretically, the feedback from bid rationing to increased quantity demanded could lead to infinite amounts requested. A countervailing incentive is that with positive probability, other investors do not honour their commitment to buy, in which case a given investor may be required to buy the amount (s)he demanded, instead of the lower amount (s)he actually wanted. If there is no commitment to buy (for example, on Funderbeam the bids are only non-binding indications of interest), then the danger of overcommitting is absent, so the rational choice seems to be requesting an infinite amount. Investors do not indicate infinite interest, so either they are irrational or some other penalty exists for outbidding one’s capability to pay.