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Lower the height of the basket and impose a restriction on the height of players, say 5 foot 8. This will increase the pool of available players and eventually improve the quality of play compared to the status quo where we are restricted to choosing players who are at least 6 feet.

In a 2020 paper in the Journal of Economic Literature, Heckman and Moktan argue that the obsessive focus on top 5 publications has a deleterious effect on the profession. In addition to documenting the impact a top 5 publication has on career outcomes, they argue that their hold distorts the incentives of junior faculty. For example, junior faculty may scrap a possibly good idea if it can’t get published in a top-five or focus on ideas with a ready made audience on the relevant editorial boards.

In addition to insufficient experimentation/exploration, there are two other consequences we should expect. As departments outsource their promotion and tenure decisions to the editorial boards of the top 5, we should anticipate an increase in the balkanization of individual departments. One’s colleagues have less of an incentive to engage with one’s own work and conversely. There should also be a decline in the willingness of faculty to contribute to the needs of the department. After all, a department is now merely an ensemble of special interest groups.

A second consequence should be an increase in attempts to subvert the primary goal of the peer review process: to provide disinterested, but, informed assessments of work. One need only look at Computer Science (CS) to see that such a possibility is not far fetched. Unless Economists are immune to the temptations that plague other mortals, we should anticipate the same. Nihar Shah at CMU surveys the problems that beset peer review in CS. Like Economics, there is a small subset of prestige venues for one’s work. Acceptance into these venues affects grants and tenure. The increased stakes have spawned collusive refereeing rings (mutual appreciation societies would be a less perjorative term) that subvert the goal of disinterested, but, informed review. There is also a strong suspicion that there are coalitions of scientists who agree to include each other as co-authors on multiple papers (risk sharing) so as to maximize the chances that they will make it in.

Nihar’s paper discusses various strategies to inoculate peer review against strategic behavior, but none are perfect. The fundamental problem is that the rewards to acceptance in these select venues exceed the expected penalties one might face.

These issues are part of a larger question: what is the optimal organization of scientific activity? The literature on contests, moral hazard and mechanism design focus on the individual component, ignoring other aspects such as rewarding discovery vs verification, incentivizing sharing, exploration and the decision to enter scientific work. For example, entry may involve high up front investments in specialized skills. Who will make these investments if the ex-post rewards from doing so are concentrated on a tiny number?

Peter Bickel taught me a nifty, little, literally, proof of the minimax theorem. Peter says it was inspired by his reading of David Blackwell‘s vector generalization of the same. Here goes.

Let be the payoff matrix of zero-sum game. Here is the payoff to the row player from playing when the column player plays . Let be $k$-simplex.

For each , let This is the set of payoffs that the row player can achieve when playing mixed strategy . By convexity, it must be an interval whose endpoints depend on . We will only be interested in the left most endpoint, call it . Thus, we can assume that (strictly speaking should be subset) Similarly, for each , let . By the same observations we may suppose that

For any and we have Now, choose which exists by continuity and compactness. Similarly, choose Then, is the value of the game.

No Farkas or separating hyperplane. Now, minimax is equivalent to duality. So, one might wonder whether such a proof could be used to obtain the duality theorem of linear programming. Assuming that the primal and the dual had optimal solutions, yes. However, the case where one is infeasible or unbounded would cause this kind of proof to break.

PS: Then, Olivier Gossner pointed out that perhaps because I wanted to believe so much that it was true, that I had overlooked something. See the comments. He was right. I tried to make the proof work but could not. A proof of minimax that avoids Farkas and/or the separating hyperlane is not outlandish. There is Guillermo Owen’s proof, via induction, for example. By the way this was not the first such elementary proof. There was a flurry elementary proofs of the minimax in the 40s and 50s in response to a challenge of von Neumann. Indeed, Peck (1958) writes in the introduction to his paper:

*“There are so many proofs of this theorem in the literature, that an excuse is necessary before exhibiting another. Such may be found by examining the proof given below for the following: it uses no matrices, almost no topology and makes little use of the geometry of convex sets; it applies equally well to the case where only one of the pure strategy spaces is finite; also there is no assumption that the payoff function is bounded.”*

That an induction proof should exist is unsurprising given David Gale’s inductive proof of the Farkas lemma. In fact, Gale’s proof can be seen as an instantiation of Fourier’s method for solving linear programs, again, no Farkas or separating hyperplane.

The majorization relationship arises frequently in Economic applications, most recently in the context of information design (see Kleiner, Moldovanu and Strack (2021) for instance). This post highlights the connection between optimization problems involving the majorization relationship and polymatroid optimization problems.

The majorization relationship is a partial order on vectors. If is an vector, denote the largest component by . Let and be two vectors in dimensions. The vector is majorized by the vector if

In the typical application one is given a non-negative vector and the goal is to identify a vector that is majorized by to optimize a linear function of . The problem is easily formulated as a linear program. Suppose . Then, our goal is to maximize subject to

A classic result of Hardy, Littlewood and Polya states that is majorized by iff there is a doubly-stochastic matrix such that . By the Birkhoff-von Neumann theorem it means that the set of vectors majorized by is the convex hull of all permutations of , i.e., a permutahedron. Permutahedra are polymatroids (see, for example, Kuipers, Vermeulen and Voorneveld (2010)). Therefore, we can reformulate the linear program above as a polymatroid optimization problem which can be solved via a greedy algorithm. Dahl (2010) gives the underlying submodular function that describes the polymatroid. For each let . Note that depends only on the cardinality of and not itself. Given this, the optimization problem above can be expressed as

subject to

There is also a related `minorization’ problem that is of interest (see Akhil and Sundaresan (2016) for a survey):

subject to

Dahl (2001) characterizes the extreme points of the feasible region of this program. Morton, von Randow, Ringwald (1985) show that its feasible region can be described as a polymatroid with monotonicity constraints. Set In words if is the largest index such that the set is contained in . Otherwise . Set The function is non-decreasing and submodular on . The feasible region of the minorization problem can be expressed as

Charactertizing the structure of the optimal solution is straightforward. It involves the `pooling’ of some variables. Variables are partitioned into sets consisting of a consecutive sequence of indices. Within each of these sets the variables are equal. Variables from sets with lower’ indices will of course have higher values than variables from sets with larger indices.

For the more general problem of optimizing a linear function over the intersection of a polymatroid and an affine space, see Hartvigsen (1998).

The goal of this post is to highlight a feature of the Bayesian Persuasion problem that seems useful but as far as I can tell not explicitly stated anywhere.

Let be the finite set of states and the finite set of actions the receiver can take. Elements of each are denoted by and , respectively.

The value to the receiver (he/him) of choosing action in state is . The payoff to the sender (she/her) if the action that the receiver takes in state is is denoted . Sender and receiver share a common prior over .

The persuasion problem can be formulated in terms of choosing a distribution over state and action pairs such that an obedience constraint holds for the receiver. Call this the obedience formulation. The more popular formulation, in terms of finding a concave envelope is, as will be seen below, is equivalent.

The sender commits to a mapping from the state to an action recommendation. Given a recommendation, the receiver can update their prior over states. Once the algebraic dust settles, it turns out all that matters is the joint probability of action and state. Thus, the sender’s problem reduces to choosing , the joint probability of action and state . The sender’s optimization problem is:

subject to

The first constraint is the obedience constraints (OC) which ensure that it is in the receiver’s interest to follow the sender’s recommendation.

The second ensures that the total probability weight assigned to actions recommended in state matches the prior probability of state being realized.

The difficulty is dealing with the OC constraints. Many relax them using the method of Lagrange multipliers and try to pin down the values of the multipliers. We take a different approach.

For each action , let be the polyhedral cone defined by:

By the Weyl-Minkowski theorem, is characterized by its generators (extreme rays), which we will denote by . The generators are non-negative and non-zero and can be normalized so that they sum to 1 which allows us to interpret them as probability distributions over the state space. For any generator (probability distribution) in , action is the best response of the receiver. Many of the qualitative features of a solution to the persuasion problem that have been identified in the literature are statements about the generators of .

As each vector in can be expressed as a non-negative linear combination of its generators, we can rewrite the constraints as follows:

If one adds up the constraints, we see that

In other words, the prior distribution can be expressed as a convex combination of other distributions. In this way arrive at another popular formulation of the persuasion problem which is in terms of finding a decomposition of the prior into a convex combination of possible posteriors. Notice, the set of possible posterior distributions that need to be considered are limited to the generators of the corresponding cones . We illustrate why this is useful in a moment. For now, let me point out a connection to the other magical phrase that appears in the context of persuasion: concavification.

First, express the persuasion problem in terms of the weight assigned to the generators:

subject to

Letting be the vector of dual variables, the linear programming dual of the persuasion problem is:

subject to

This dual problem characterizes the sender’s optimal value in terms of a concave envelope (since Kamenica & Gentzkow (2011) this is the most popular way to state a solution to the persuasion problem). Notice, the approach taken here shows clearly that the receiver’s preferences alone determine the set of posterior beliefs that will play a role. This idea is implicit in Lipinowski and Mathevet (2017). They introduce the notion of a posterior cover: a collection of sets of posterior beliefs, over each of which a given function is convex. When the given function is the best response correspondence of the receiver, this reduces precisely to the cone of the obedience constraints.

To demonstrate the usefulness of the conic representation of the obedience constraints, lets examine a persuasion problem with one dimensional state and action space described in Kolotillin and Wolitzky (2020). To simplify notation assume that and . Write as and as .

The goal of KW2020 is to provide a general approach to understanding qualitative properties of the optimal signal structure under two substantive assumptions. The first is that the sender’s utility is increasing in the receiver’s action. The second is called aggregate downcrossing, which implies that the receiver’s optimal action is increasing in his belief about the state.

1)

2) For all probability distributions over ,

A result in KW2020 is that it is without loss to assume each induced posterior distribution has at most binary support. We show this to be an immediate consequence of the properties of the generators of each .

Given condition 1, the sender only cares about the downward obedience constraints. We show that the adjacent downward constraints suffice.

Note that for any ,

The first term on the right hand side of the equality is non-negative by virtue of the adjacent obedience constraint holding. The second term is non-negative by aggregate downcrossing. Hence, each is described by a single inequality:

It is straightforward to see that each generator takes one of the following forms:

1) A single non-zero component with weight 1 assigned to a state where .

2) Two non-zero components one assigned to a state where and one to a state where . The value assigned to component is while the value assigned to component is

Note that items 1 and 2 correspond to Lemma 1 & Theorem 1 of KW2020 and follow immediately from the properties of the cone of the OC constraints.

Serious infectious diseases are a prime example of a public bad (non-exclusive and non-congestible). We limit them by restricting behavior and or getting individuals to internalize the externalities they generate. For example, one could mandate masks in public places. To be effective this requires monitoring and punishment. Unpleasant, but we know how to do this. Or, one could hold those who don’t wear masks responsible for the costs they impose on those whom they infect. Unclear exactly how we would implement this, so impractical. However, it is still interesting to speculate about how one might do this. Coase pointed out that if one could tie the offending behavior to something that was excludable, we would be in business.

To my mind an obvious candidate is medical care. A feature of infectious diseases, is that behavior which increases the risk of infection to others also increases it for oneself. Thus, those who wish to engage in behavior that increases the risk of infection should be allowed to do so provided they waive the right to medical treatment for a defined period should they contract the infection. If this is unenforceable, perhaps something `weaker’ such as treatment will not be covered by insurance or the subject will be accorded lowest priority when treatment capacity is scarce.

How exactly could such a scheme be implemented? To begin with one needs to define which behaviors count, get the agent to explicitly waive the right when engaging in it and then making sure medical facilities are made aware of it. We have some ready made behaviors that make it easy. Going to a bar, gym and indoor dining. The rough principle is any activity with a $$ R_0 > 1 $$ whose access is controlled by a profit seeking entity. The profit seeking entity obtains the waiver from the interested agent as a condition of entry (this would have to be monitored by the state). The waiver releases the profit entity from liability. Waiver enters a database that is linked to health records (probably the biggest obstacle).

The efficacy of lockdowns was debated at the start of the pandemic and continues to this day. Sweden, famously, chose not to implement a lockdown. As Anders Tegnell, remarked:

`Closedown, lockdown, closing borders — nothing has a historical scientific basis, in my view.’

Lyman Stone of the American Enterprise Institute expresses it more forcefully:

The quotes above reveal an imprecision at the heart of the debate. What exactly is a lockdown? Tegnell uses a variety of words that suggest a variety of policies that can be lumped together. Stone is careful to say there is no evidence in support of *strict* lockdowns. Suggesting that `less’ strict lockdowns might be beneficial. So, lets speculate about the other extreme: **no lockdown** relaxed or otherwise.

The presence of infected individuals increases the cost of certain interactions. If infected individuals don’t internalize this cost, then, in the absence of any intervention we will observe a reduction in the efficient level of economic activity.

How will this manifest itself? Agents may adopt relatively low cost precautionary actions like wearing masks. On the consumer side they will substitute away from transactions that expose them to the risk of infection. For example, take out rather than sit down and delivery versus shopping in person. In short, we will see a drop in the demand for certain kinds of transactions. Absent subsidies for hospital care, we should expect an increase in price (or wait times) for medical care further incentivizing precautionary actions on the part of individuals.

The various models of network formation in the face of contagion we have (eg Erol and Vohra (2020)) all suggest we will see changes in how individuals socialize. They will reduce the variety of their interactions and concentrate them in cliques of `trusted’ agents.

On the supply side, firms will have to make costly investments to reduce the risk of infection to customers and possibly workers. The ability of firms to pass these costs onto their customers or their suppliers will depend on the relative bargaining power of each. Restaurant workers, for example, may demand increased compensation for the risks they bear, but this will occur at the same time as a drop in demand for their services.

To summarize, a `no lockdown’ policy will, over time, resemble a lockdown policy. Thus, the question is whether there is a *coordinated* lockdown policy that is superior to an uncoordinated one that emerges endogenously.

One of the delights of pursuing a relatively young discipline is that one meets its pioneers. As one grows old in the discipline, so do the pioneers, who eventually pass into `the undiscovered country from whose bourn no traveler returns.’ Overlooked, at least by me, was that one also meets, in the chrysalis stage, those who will eventually lead the discipline into the next century. It was the untimely passing of William Sandholm on July 6th of this year, that brought this to mind.

I first met Bill in 1998. I had just moved to MEDS and he was on his way out as a new minted PhD. He, a shiny new penny and myself on the way to becoming so much loose change.

Within a decade, Bill rose to prominence as an authority on Evolutionary Game Theory. His book, “Population Games and Evolutionary Dynamics” became the standard reference for population games. The concept of evolutionary implementation can be credited to him.

Bill was also a provider of public goods. He wrote and made freely available software for working with evolutionary dynamics, served on panels and editorial boards.

As I recall Bill, I am reminded of a line from Mary Chase’s play, Harvey uttered by the main character Elwood Dowd:

Years ago my mother used to say to me, she’d say, ‘In this world, Elwood, you must be’ – she always called me Elwood – ‘In this world, Elwood, you must be oh so smart or oh so pleasant.’ Well, for years I was smart. I recommend pleasant. You may quote me.

Bill was both.

Will widely available and effective tests for COVID-19 awaken the economy from its COVID induced coma? Paul Romer, for one, thinks so. But what will each person do with the information gleaned from the test? Should we expect someone who has tested positive for the virus to stay home and someone who has tested negative to go to work? If the first receives no compensation for staying home, she will leave for work. The second, anticipating that infected individuals have an incentive to go to work, will choose to stay home. As a result, the fraction of the population out and about will have an infection rate *exceeding* that in the population at large.

In a new paper by Rahul Deb, Mallesh Pai, Akhil Vohra and myself we argue that widespread testing alone will not solve this problem. Testing in concert with subsidies will. We propose a model in which both testing and transfers are targeted. We use it to jointly determine where agents should be tested and how they should be incentivized. The idea is straightforward. In the absence of widespread testing to distinguish between those who are infected and those who are not, we must rely on individuals to sort themselves. They are in the best position to determine the likelihood they are infected (e.g. based on private information about exposures, how rigorously they have been distancing etc.). Properly targeted testing with tailored transfers give them the incentive to do so.

We also distinguish between testing at work and testing `at home’. An infected person who leaves home to be tested at work poses an infection risk to others who choose to go outside. Testing `at home’ should be interpreted as a way to test an individual *without* increasing exposure to others. Our model also suggests who should be tested at work and who should be tested at home.

On the 3rd of July, 1638, George Garrard wrote Viscount Wentworth to tell him:

The Plague is in Cambridge; no Commencement at either of the Universities this year.

On October 2nd of that same year, Cambridge canceled all lectures. Even if history does not repeat (but historians do), one is tempted to look to the past for hints about the future.

From the Annals of Cambridge (compiled by Charles Henry Cooper ) we learn that the plague combined with the residency requirements for a degree at Oxford, prompted a rush of Oxford students to Cambridge to obtain their Masters of Arts degree. We know this from an anonymous letter to Oxford’s Chancellor:

…..many of Batchelor of Arts of Oxford came this Year for their Degrees of Masters of Arts here, which this Year they could not obtain at Oxford, which I endeavored to prevent……..

This prompted a complaint to Cambridge. Its vice-chancellor replied,

I Pray receive this assurance from me, and I doubt not but the Practice of our University will make it good……

Oxford, in the meantime, maintained country homes for its scholars where they could hide from the Black Death. The plague lowered property values which allowed the colleges to expand their land holdings.

What effect on the intellectual life of the University? Anna Campbell’s 1931 book entitled `The Black Death and Men of Learning‘ estimates that about a third of European intellectual leaders perished during the plague and Universities were in a precarious position.

James Courtenay, writing in 1980 with access to more detailed data about Oxford suggests a less bleak outcome.

The mortality rate was not particularly high , either of brilliant or of marginal scholars and masters. The enrollment levels across the next few decades do not seem to have been seriously affected.

He notes an argument for a drop in the quality of higher education but that would have been a response to a drop in the quality of primary education.

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