Matthew W. Thomas

How to use GNU Screen

2020-01-16T00:00:00Z

Basic usage

Single terminal

To launch screen, just run

screen

Then run all of your commands like stata -b do filename. When you want to “minimize” the terminal, just press Ctrl+A followed by the D key.

You can then safely log out of ssh without killing the precess that you ran.

exit

When ssh back into the server, you can check back on the process by running

screen -r

This will reopen the terminal that you just closed.

Using tabs

You can create a new tab inside of a screen by pressing Ctrl+A followed by the C key. You will then have a tab 0 and a tab 1.

You can swap between tabs by pressing Ctrl+A followed by Shift+’ (actually Shift+").

You can close a tab by pressing Ctrl+D or just running exit.

Managing multiple screens

You can create a second screen by running

screen

you can list all open screens by running

screen -list

There are screens on:
	2239.tty.host	(Detached)
	2044.tty.host	(Detached)
	22287.tty.host	(Detached)
3 Sockets in /var/run/screen/S-user.

You can specify which screen you want to connect to using the name from the list.

screen -r 2044.tty.host

The pid (the number part) is also sufficient

screen -r 2044

You can close a screen using Ctrl+D

Real Induction

2021-03-08T00:00:00Z

Conventional proof by induction allows you to prove that a statement applies to an infinite sequence. The argument is that if a property holds on the first element, and always holds on the next element, then it must hold on all elements.

But what do you do when there is no next element?

Some notes on the arXiv show that there is an analogous version to proof by induction that can apply to uncountable sets like the Real numbers. Conventional induction cannot work on uncountable set because, by definition, you cannot reach all elements by iteratively stepping through them.

In the real induction, we get around this issue by breaking the space into countably many intervals. Formally, let $a < b$ be real numbers. We want to show that $[a, b] = S$ . That is, we want all elements of the interval to “satisfy” property $S$ . We define the subset $S \subseteq [a,b]$ to be inductive if:

$a \in S$ .
If $a \leq x < b$ , then $x \in S \implies [x,y] \subset S$ for some $y > x$ .
If $a < x \leq b$ and $[a,x) \subset S$ , then $x \in S$ .

The result is that a subset $S \subset [a,b]$ is inductive if and only if $S = [a,b]$ . So, if conditions 1-3 hold, then property $S$ is satisfied on $[a,b]$ .

Intuition and alternative

Most people do not know what real induction is. However, the logic behind it is very easy to understand. So, applying real induction directly is usually not the clearest way to write a proof. The underlying idea behind real induction can be found in many proofs. However, it is rarely referred to as real induction. Typically, people use a proof by contradiction that goes something like this:

Suppose, by way of contradiction, that there exists an element in $[a,b]$ that is not in $S$ . Then, there must be a first switching point. That is, there must be a minimal $x$ such that either $x \notin S$ or $(x,z] \notin S$ for some $z > x$ . Then, we need to prove

$x > a$ (using 1 and 2 from real induction). Therefore, $[a, x) \in S$ because $x$ is the first switching point.
If $[a, x) \in S$ then, $x \in S$ (using 3 from real induction). Therefore, $(x,z] \notin S$ for some $z > x$ .
There exists a $y > x$ such that $[x,y] \in S$ (using 2 from real induction).

Therefore, any point $w$ such that $w > x$ , $w \leq z$ , and $w \leq y$ is both in $S$ and not in $S$ . This is a contradiction.

Proving statements directly in this way is clearer to people unfamiliar with real induction. This is especially true when you express the above argument in terms of your problem.

Example

As an example, we will apply this method to prove a lemma in my current working paper. The proof in the paper does not use real induction directly. It instead uses the direct argument expressed in the previous section. In this section, we will apply real induction in order to demonstrate the method.

In this paper, we show that a particular candidate probability density is the distribution for the Nash equilibrium for a class of auctions. In this lemma, we want to prove that this candidate can be a probability density – i.e. it’s positive and integrates to one.

Assumptions

Consider the following assumptions on two functions $v: \mathbb{R}^2 \to \mathbb{R}$ (the value of the prize) and $c: \mathbb{R} \to \mathbb{R}$ (the cost of bidding). In the auction context, these assumptions basically say that bids are costly (A2), the prize attracts bids that are greater than zero and less than infinity (A3), and you would prefer to win the prize than lose it (A4).

Assumption 1 (A1, Smoothness)

The function $v(s; y)$ and $c(s)$ are continuously differentiable in $s$ and continuous in $y$ for all $s$ and $y \leq s$ .

Assumption 2 (A2, Monotonicity)

v'(s; y) < c'(s)

holds a.e., where we define $v'(s; {y}) := \frac{\partial v(s; {y})}{\partial s}$ .

Assumption 3 (A3, Interiority)

$v(0, 0) > c(0) = 0$ and $\lim_{s \to \infty} \sup_{y \leq s} v(s; {y}) < \lim_{s \to \infty} c(s).$

Assumption 4 (A4, Positive value on the margin)

v(s; s) > 0

Statement of the lemma

Lemma (Probability density) The solution, $g(s)$ , to

g(s) = \frac{1}{v(s; s)} \left( c'(s) - \int_0^{s} v'(s , y) g(y) dy \right)

is a probability density on some interval $[0, \bar{s}]$ .

Proof

The finite definite integral cannot diverge because the function is continuous and $g(0) = \frac{c'(0)}{v(0; 0)} > 0$ .

We still need to confirm that $g(s) > 0$ on the relevant interval $\{ s : \int_0^s \lvert g(y) \rvert dy \leq 1 \}$ . We will define the probability density to be zero outside this interval so that it integrates to one.

We show this by real induction on the interval $[0,T]$ where $T$ can be any value such that $\int_0^T \lvert g(y) \rvert dy \leq 1$ . We must prove the following statements:

$g(0) > 0$ .
For any $s \in [0,T]$ $, if $g(y) > 0$ for all $y \in [0,s]$ $, then there exists a $z > s$ such that $g(y) > 0$ for all $y \in [s,z)$ .
For any $s \in [0,T]$ $, if $g(y) > 0$ for all $y \in [0,s)$ $, then $g(s) > 0$ .

The first statement is true because $g(0) = \frac{c'(0)}{v(0; 0)} > 0$ . The second statement proceeds by continuity of $g$ (A1). The third statement comes from:

g(s) = \frac{1}{v(s; s)} \left( c'(s) - \int_0^{s} v'(s , y) \lvert g(y) \rvert dy \right)

g(s) \geq \frac{1}{v(s; s)} \bigg[ c'(s) - \lvert \max_{y\in[0,s]} v'(s; y) \rvert \underbrace{\left( \int_0^{s} \lvert g(y) \rvert dy \right)}_{\leq 1} \bigg]

g(s) \geq \frac{1}{v(s; s)} \underbrace{\left[c'(s) - \lvert \max_y v'(s; y) \rvert \right]}_{>0 \text{(A2)}} > 0.

Therefore, $g(s) > 0$ for any $s$ such that $\int_0^s \lvert g(y) \rvert dy \leq 1$ .

To complete the lemma, we must show that it is not possible for $\int_0^\infty \lvert g(y) \rvert dy \leq 1$ . We can do this in one step with Holder’s inequality.

c(s) = \int_0^s v(s; y) g (y) dy \leq \left(\int_0^s \lvert g(y) \rvert dy\right) \left( \max_{y \in [0,s]} v(s; y) \right)

so $\int_0^s \lvert g(y) \rvert dy \geq \frac{c(s)}{\max v(s; y)}$ which is assumed to be greater than one as $s$ approaches infinity (A3). By continuity, there exists an $\bar{s}$ such that $\int_0^{\bar{s}} \lvert g (y) \rvert dy = 1$ (A1).

How to Prevent Email Scraping

2021-03-10T00:00:00Z

If put your email in plaintext on your website, you will get spam from people trying to sell you SEO, development services, ghostwriting, etc. Emails get on spam lists through website scraping. The way that this works is simple. Scrapers typically look for strings like user@host.tld or mailto:user@host.tld in your HTML.

I used to use the Scrape Shield provided by Cloudflare. However, when I switched away from using Cloudflare a few months ago, I started to receive a lot of spam that could not be easily blocked. There are many solutions that you can find on the internet to stop people from scraping your email. The most common is to write your email as:

user [at] host [dot] tld

This is not ideal for three reasons:

People usually have to edit your email before using it (though GMail does this automatically 🧪).
This string is just as easy to scrape as user@host.tld. It’s just a little less common.
It’s not feasible for use in a mailto link.

This third issue is the most difficult to fix as there are many solutions that work for displaying an email if you don’t care about functional mailto: links. For example

<span>user@</span>
<span style="display:none;">HIDDEN JUNK</span>
<span>host.tld</span>

or the equivalent with CSS.

There are two solutions that I know of that fix the problem and allow mailto: links to work. The first is common and easy for scrapers to circumvent while the other is harder to circumvent, but requires javascript.

Solution 1: HTML entities

One simple and very common solution is to replace all characters in the mailto: links with HTML entities. For example user@host.tld becomes user@host.tld. You can then make an email link like so

<a
  href="mailto:&#x75;&#x73;&#x65;&#x72;&#x40;&#x68;&#x6F;&#x73;&#x74;&#x2E;&#x74;&#x6C;&#x64;"
  >email me</a
>

which renders as email me. As you can see, the browser translates these HTML entities completely for the end user. So this obfuscation does not create any issues for human visitors. However, it is very easy to transform all HTML entities before scraping. So, this offers only minimal protection.

Solution 2: Trivial javascript replace

I have tested a very simple method in javascript for the last few months. The idea is to replace your email with the correct string using javascript. A simplified version of my implementation follows.

<script>
  /* 1. define variables */
  var me = "name";
  var place = "host.tld";

  /* 2. find email link to replace */
  var elink = document.getElementById("mlink");

  /* 3. replace link href with variables  */
  elink.href = `mailto:${me}@${place}`;
</script>

<a id="mlink" href="#">email me</a>

So, we define the email in two parts, name and host.tld. We then combine these parts and put them into the email link. This solution is simple and resistant to scraping. The easiest way to scrape this document would be to execute the above javascript. However, scrapers do not execute javascript on arbitrary pages for several reasons. First, it is not performant to execute javascript on the thousands of websites that you scrape. Second, it opens scrapers up to various types of attacks and tracking that they would prefer to avoid.

At some point, scrapers may evolve to the point that such a simple method is not effective. However, I have found that I no longer receive spam after making this change. While I am generally against adding unnecessary javascript to pages, as it slows them down, I feel this snippet is both necessary and highly performant.

This javascript method is similar to the one used by CloudFlare’s builtin Scrape Shield, but is simpler. Cloudflare uses a hash that is decoded in javascript by using the last few characters as a “key” to unlock the hashed email.

Does it work?

I use the second method on my site and can confirm that it works. A few months ago, I ran an experiment to see if the spammers were really getting my email from my website. In the test, I replaced the email in the HTML of my site with a dummy address, 6adegxtzp@relay.firefox.com, which I still receive. However, I sort these into a special folder. When, the page loads, my script replaces this email with my main address. Human visitors with javascript enabled do not encounter this address.

Since making this change, I no longer receive spam on my main email. However, I receive a significant amount of spam though my Firefox relay address. This is despite the fact that I have never used this address anywhere else. This change took a few months though. So don’t expect results right away.

Mechanisms to Fund Open Source

2021-03-11T00:00:00Z

I recently came across a paper published in Management Science that proposes Quadratic Finance (QF), a a quadratic voting mechanism to allocate funds to public goods such as open source software. This has generated an understandable amount of excitement and there are already several cases in which it has been implemented.

This mechanism has also been referred to as Liberal Radicalism or LR.

I wanted to take the time to clarify a few common misconceptions about QF and explain what it actually is and is not. Moreover, implementations of QF have modified the mechanism in ways that are not efficient and often dominated by more common allocation approaches.

The basic problem with funding a public good is that people do not pay if they can take advantage of the good for free. So, collected payments/donations are always less than they should be. This is not a problem that QF solves. Several sources claim QF is “the mathematically optimal way to fund public goods in a democratic community” (Gitcoin). This is not entirely true. QF was not designed to solve the problem of funding public goods. It is not a mechanism for optimal fundraising. It was instead designed to solve the following problem:

How can we efficiently allocate resources to public goods given that we have unlimited access to resources.

Note that this is not a fundraising problem. It is a voting problem. QF was not designed to have the greatest revenue of any mechanism. It doesn’t tell you how to raise money. It instead tells you where you should put it once it is raised. QF is a system where people “vote with their wallets” to allocate resources. Yes, it raises funds as part of the process, but this is not the point.

This is a classic Economic problem that was technically solved by Vickrey, Clarke, and Groves in three separate papers. A quick summary of their mechanism (VCG) is essential for understanding what QF is about.

Classic VCG

The idea behind VCG is pretty simple. Suppose we are looking at funding webpack, a popular open source project. Suppose the value for person $i$ of webpack having an annual budget of $x$ is $v_i(x)$ where $v_i$ is a concave, differentiable, and increasing function^[1] with $v_i(0) \equiv 0$ . The efficient level of funding for webpack is the $x^\star$ that maximizes society’s payoff:

\left[ \sum_{i} v_i(x) \right] - x. \eqnum{1}

However, people only care about their own value – not the value of others. So, they would not be willing to give as much. To fix this, we conduct a three step process.

Ask everyone for their $v_i(x)$ .
Find the $x^\star$ that maximizes equation (1).
Give each agent a (possibly negative) transfer of $\left[ \sum_{j \neq i} v_j( x^\star ) \right] - x^\star$ .

When they add their own $v_i(x)$ to the transfer, their personal payoff is exactly the same as society’s (Equation 1). Because of this, they want $x^\star$ to be calculated correctly. So, they have no reason to lie about their value, $v_i$ .

The transfers noted above are very expensive. However, they can be made much cheaper because you can subtract any constant from these transfers. In fact, you can subtract any function that does not depend on $v_i$ .^[2] When the ideal function is chosen,^[3] the assumptions we made ensure the VCG mechanism raises funds from each person. The amount raised from person $i$ is.

\max_x \left( \left[ \sum_{j \neq i} v_j( x ) \right] - x \right) - \left( \left[ \sum_{j \neq i} v_j( x^\star ) \right] - x^\star \right) > 0

There is no incentive compatible efficient mechanism that raises more total funds (Krishna and Perry 1998).

Issues

There are two major issues with the VCG mechanism:

It requires too much information
It fails if bidders can work together (i.e. collude)
It can raise less than $x^\star$ and thus requires subsidization

VCG requires each participant to submit their exact valuation for each level of total funding for every open source project. This imposes a preparation cost for most participants as they likely do not know these valuations offhand. Issue 2 is somewhat overblown as the ways to cheat in VCG are generally NP-hard to compute. However, there are interesting restricted cases where the optimal collusion scheme can be computed in polynomial time.

The third issue is an important one that cannot be easily solved. Remember that there is no incentive compatible efficient mechanism that collects more revenue than VCG. So, collecting more funds requires sacrificing efficiency – an issue not dealt with here.

Quadratic finance

The aforementioned paper considers QF, a different allocation mechanism which solves the first issue with VCG (i.e. that it requires too much information) and it solves this brilliantly.

The QF mechanism allocates resources according to

x^\star = \left( \sum_i \sqrt{c_i} \right)^2 \eqnum{2}

where $c_i$ is an individual contribution made by each participant. So, participants no longer need to send their value for every level of funding. They just need to choose a contribution. This requires less information on the part of the contributors. It is also more private because less information is surrendered. The main point of the paper is that, in equilibrium, this $x^\star$ is the same as the one we considered in the VCG section.

The goal of QF is not to maximize $\sum_i c_i$ , but to instead find a function that achieves the socially optimal resource allocation, $x^\star$ .

The deficit of QR is

D_{QF} = x^\star - \sum_i c_i = \left( \sum_i \sqrt{c_i} \right)^2 - \sum_i c_i

which is positive when there is more than one contributor by Jensen’s inequality. Therefore, outside funding is always required for QR.

Issues

The issues with QF are similar to VCG.

It fails if bidders can work together (i.e. collude)
There is always a deficit and it is typically larger than with VCG

Collusion in QF is particularly easy. If you want to contribute $c_i$ and have friends who want to contribute nothing, then you can do better by contributing less and dividing your contributions amongst your friends.

The second problem is a big problem for applying both QF and VCG because the deficit is can be difficult to predict. Therefore, unlimited resources are needed guarantee that the costs are covered. This is somewhat reasonable for governments who can tax their citizens.^[4]

It is not reasonable when the funds are raised by a foundation or philanthropist. What do you do if $D_{QF}$ is larger than your grant funding, $G$ ? However, the authors note this problem and propose an approximate solution called CQF (also known as CLR)

x^\circ = \alpha \left( \sum_i \sqrt{c_i} \right)^2 + (1 - \alpha) \sum_i c_i

where $0 \leq \alpha \leq 1$ . When $\alpha = 0$ , CQF is the standard donation model with no subsidies. When $\alpha = 1$ , CQF is the same as QF and thus achieves the optimal allocation. At every $\alpha$ in between, CQF is a mix of the two. The deficit of CQF is

D_{CQF} = x^\circ - \sum_i c_i = \alpha D_{QF}.

So, CQF has a smaller deficit than QF. It results in a better^[5] allocation than private donations. However, CQF does not yield the efficient allocation unless $\alpha = 1$ .

Implementations

Implementations of QF have not actually implemented QF or CQF. They have instead implemented the following rule which I call NQF:

\tilde{x}^p = \left[ \sum_i c_i^p \right] + G \frac{\left( \sum_i \sqrt{c_i^p} \right)^2}{ \sum_p \left( \sum_i \sqrt{c_i^p} \right)^2}

where $G$ is the size of an outside grant and $p$ is an index for the open source project. A calculator for NQF is available with source code. This formulation guarantees that the deficit of the mechanism is exactly $D_{NQF} = G$ . This is a desirable property because it ensures that the round never goes over budget and that all of $G$ is used. However, this model is not ideal. In the next section, I’ll show a better way.

NQF lacks the desirable properties of QF and CQF. It not not efficient. It is a completely different mechanism.

The first thing to note is that a private good still gets a subsidy under NQF. Under VCG, QF, and CQF, any project with one contributor gets no subsidy. You can see in the equation above that this is not true for NQF because the subsidy for each project

G \frac{\left( \sum_i \sqrt{c_i^p} \right)^2}{ \sum_p \left( \sum_i \sqrt{c_i^p} \right)^2}

is always positive. This means that any individual who can list a project can fraudulently profit from the mechanism.

The second thing to note about NQF is funds allocated to project $p$ are decreasing in the contributions to all other projects. For example, when you donate to WebPack, you are taking funds away from all other projects. This generates inefficiency and is a major deviation from QF and CQF.

NQF does not necessarily beat private contributions. In fact, we show an example where $G$ is wasted entirely.

Example: Suppose there are two contributors and two projects. Contributor 1 likes both projects equally while Contributor 2 only likes Project 1. More concretely, assume

v_1^1(x) = v_1^2(x) = v_2^1(x) = 2 \sqrt{x}

while $v_2^2(x) = 0$ . Contributor 1 has no reason to contribute to Project 1 through the mechanism. He will give only to Project 2 such that the grant is split evenly between the two projects. This is despite the fact that Project 1 is more valuable to society. The quadratic nature of this mechanism actually exacerbates the problem because even a small contribution by Player 1 to Project 1 will divert a large amount of funding away from Project 2. Allocations can be found in the table below.

	NQF (G = 1)^[6]	Return Grant (G = 0)
Player 1	(0, 1/2)	(0, 1)
Player 2	(1/2, 0)	(1, 0)
Grant	(1/2, 1/2)	(0, 0)
Total	(1, 1)	(1, 1)
Optimal	(4, 1)	(4, 1)

The above table shows the contributions to Project 1 and Project 2 by each player and the grants. As you can see, the total contributions are the same between NQF and private contributions. So, in our example, NQF does not outperform private contributions even though a grant has been collected. Of course, both are outperformed by matching contributions or performing QF with any $\alpha > 0$ .

This points to a general issue with this mechanism. People who like both popular and unpopular projects have an incentive to divert resources to the unpopular ones. This is a classic voting issue and is exactly the sort of behavior that quadratic voting mechanisms were designed to prevent.

A better way

NQF has the nice property that the deficit is exactly equal to your grant funding. So you can never generate more deficit than you have funding for. However, it lacks all other nice properties of QF or CQF. It does not result in an approximately efficient allocation and it can be defrauded by a single agent. However, it is possible to get the one advantage of NQF ( $D = G$ ) without losing any of these properties. You just have to stop the mechanism when $D = G$ .

That is:

Run QF or CQF with your preferred choice of $\alpha$ .
Compute your deficit $D$ after every donation.
End the donation matching once $D = G$ .

This last step could be achieved by cutting off donations completely or by putting a “donation matching is currently disabled” message on the screens of potential donors. The choice of $\alpha$ would determine how long you can run the match.

These assumptions are from the QF/LR papers. VCG doesn’t need all of these assumptions. ↩︎
You can subtract any $h_i(v_{-i})$ where $v_{-i} = (v_1, \dots, v_{i-1}, v_{i+1}, \dots, v_n)$ . ↩︎
This ideal function is $\max_x \left[ \sum_{j \neq i} v_j( x ) \right] - x$ . ↩︎
As the paper points out, efficiency is lost when people consider how their choices affect their taxes. Though this goes away as the number of participants becomes large. It’s hard to see this as an issue if the mechanism played out on a national scale. ↩︎
‘Better’ in that it is closer to the efficient allocation ↩︎
The assumption that $G=1$ isn’t terribly important. It does not affect total contributions. Naturally, it’s possible to get above $(1,1)$ when $G > 2$ . However, contributors give nothing in this case. So, it is not terribly interesting. ↩︎

List of Econ Blogs

2021-03-12T00:00:00Z

This is a list I compiled of other blogs about Economics. If you would like to add or get added to this list, contact me, comment, or just submit a request to edit the file that generates this list.

—

A Fine Theorem

Academic Health Economists’ Blog

Alt-M

Angry Bear

Antonio Fatas

Arnold Kling

Bank Underground

Brad DeLong

Cafe Hayek

Calculated Risk

Capital Ebbs and Flows

Carola Binder

Causal Analysis in Theory and Practice

Causal Inference: the Remix

Cecchetti and Schoenholtz

Chris Blattman

Coordination Problem

Credit Writedowns

Croaking Cassandra

Crooked Timber

Dangerous Economist

Dan Millimet

Dani Rodrik

Dave Giles

Dean Baker

Digitopoly

Econbrowser

EconLog

Economic Principals

Economics UK

Economist’s View

EconoSpeak

Env and Urban Economics

Environmental Econ

Freakonomics

Fresh Economic Thinking

Greg Mankiw

Growth Economics

IMFBlog

Interfluidity

Jared Bernstein

Jayson Lusk

John Quiggin

John Taylor

Lane Kenworthy

Liberty Street

Macro Musings

macroblog

MacroMania

Magic, maths, money

mainly macro

Marc Bellemare

Marginal Revolution

Matthew W. Thomas

Miles Corak

Moneyness

Naked Capitalism

NEP-DGE Blog

No Hesitations

Noahpinion

Off the Charts

Owen Zidar

Peter Gordon’s Blog

Reality-Based Community

Richard Green

Robert Reich

Robert Waldmann

Roger Farmer

Stat Model and Social Sci

Stochastic Trend

Stumbling and Mumbling

Supply-Side Liberal

Tax Policy Blog

The Berkeley Econ Blog

The Big Picture

The Incidental Economist

The Irish Economy

The Money Illusion

Theory Class

Thomas Palley

Urbanomics

Worthwhile Canad Init

Econ Ipsum Now Available in TeX

2021-03-15T00:00:00Z

About two years ago, Maria Betto and I made Econ Ipsum, a Lorem Ipsum generator that uses words from Econometrica abstracts to produce Economic sounding nonsense. We never put any sort of tracking or analytics on the page. So, I don’t know how many people use it. However, I can tell from the number of api calls that it is more than originally expected.^[1]

One long running joke on the page is that the randomized articles have zero citations and are “not forthcoming”. That is, we remind that the random text is not a reference for any other article and is not scheduled for publication in any journal. These facts were intended to be self-evident. So, I was surprised when a student contacted me to ask if he could use some random paragraphs in a TeX package he was working on.

Naturally, I was ecstatic that someone liked Econ Ipsum enough to make a spinoff. So, I’m happy to announce that you can now use Econ Ipsum paragraphs in TeX via the econlipsum package by Jack Coleman. So, Econ Ipsum’s citation counter – intended to remain forever at zero – has now advanced to one.

The number of api executions is the maximum number of paragraphs requested by a single person divided by 100 (and rounded up). Each day (typically) has between 3 and 30 executions. On one day, there were 184 executions – which means that one person generated 18,400 paragraphs. ↩︎

Nonlinear War of Attrition with Complete Information

2021-03-22T00:00:00Z

These notes are about nonlinear war of attrition models with two players. The formulation we consider is a simplified version of Hendricks, Weiss, and Wilson (1988). These models are popular for modeling conflict that occurs in continuous time including wars, filibusters, bribes, and competition for standards and monopolies.

Basics

The war of attrition is a second price all-pay auction. That is, an auction where players pay a function of the second highest bid. The story is that two players enter into a battle royal to win some reward. Each moment that they fight, they must pay some cost. Each player chooses a secret time when they will exit the fight and forfeit the prize.

Because the last player wins as soon his opponent exits, the winner only fights until her opponent’s exit time. So, both players only face the cost of the lower exit time.

Model

There are two players, Player 1 and Player 2. Suppose player i exits at $t_i$ and the other player exits at $t_{-i}$ . Then, the payoff of player i is defined by:

U_i( t_i ; t_{-i} ) = \begin{cases} \ell_i(t_i) &\text{ if } t_i < t_{-i} \\ s_i(t_i) &\text{ if } t_i = t_{-i} \\ f_i(t_{-i}) &\text{ if } t_i > t_{-i} \end{cases}

where $\ell_i$ is the payoff of losing, $s_i$ is the payoff of a tie, and $f_i$ is the payoff of the winner. Note that the two players can have entirely different payoffs. In the most common formulation, $\ell_i(t) = -t$ and $f_i(t) = V_i - t$ .

We make one regularity assumption and another assumption which characterizes the war of attrition.

Assumption 1: The function $f_i(t)$ is continuous and $\ell_i(t)$ is continuously differentiable.

Assumption 2: Players like to win and war is costly. Specifically,

Winning is better than losing: $f_i(t) > \ell_i(t)$ for all t.
Winning is better than tieing: $f_i(t) > s_i(t)$ for all t.
War is costly when losing: $\ell'_i(t) < 0$ .
Costs are relevant: $\int_0^\infty \frac{-\ell'_i(z)}{f_i(z) - \ell_i(z)} dz = \infty$ .

Without loss of generality, we will say that $\ell_i(0) = 0$ . The second assumption guarantees that ties occur with probability zero in equilibrium. So, I will not mention them again. The last point of Assumption 2 prevents pathological behavior around infinity. Without this, it is possible that the players would never exit. It can be interpreted as a guarantee that the benefit of winning is not so large relative to the cost of the war.

Pure strategy Equilibria

If $\lim_{t \to \infty} f_i(t) < \ell_i(0)$ , there are asymmetric pure strategy Nash equilibria where one player, i, chooses a very large exit time t such that $w_{-i}(t) < 0$ . The other player best responds by playing zero because fighting enough to win is not worthwhile. Because the winner’s payoff does not depend on her own score, she is indifferent between all actions except zero. So, this is a Nash equilibrium.

Mixed strategy equilibria

In addition to the pure strategy equilibria, there is one Nash equilibrium in mixed strategies. The equilibrium will have full support over the real line. This is established in several steps.

If one player has a gap in their support, their opponent must have the same gap. Otherwise, the opponent would move density from the gap to just before it.
There are no gaps in either support. Otherwise, each player could do better by moving mass from the end of the gap to the beginning.
There are no atoms except at zero. Otherwise, the opponent would move mass from slightly below the atom to slightly above the atom. This would create a gap.
At most one player has an atom at zero. Otherwise, one player would prefer to move their atom slightly up.

We will see from the construction of the equilibrium that no player can have an atom and that the support cannot be bounded. However, these four points are enough to get us started.

In order for the player to be willing to mix, they must be indifferent between all points on the support. So, the following indifference condition applies:

P(t_{-i} < t_i) E[ f_i(t_{-i}) \vert t_{-i} < t_i ] + (1 - G_{-i}(t)) \ell_i(t) = u_i

where $u_i$ is the player’s (constant) payoff. This is the same as

\int_0^t f_i(x) dG_{-i}(x) + (1 - G_{-i}(t)) \ell_i(t) = u_i

where $G_{-i}$ is the equilibrium strategy distribution of the opponent. We want to solve for $G_{-i}$ . If we take derivatives of both sides with respect to $t$ , we get

f_i(x) g_{-i}(x) + (1 - G_{-i}(t)) \ell'_i(t) - g_{-i}(t) \ell_i(t) = 0

which simplifies to the following differential equation

- \ell'_i(t) = \left(f_i(t) - \ell_i(t) \right) \frac{g_{-i}(t)}{1 - G_{-i}(t)}.

This equation ensures that the marginal cost of participating in the war of attrition is equal to the prize value times the rate that your opponent exits (hazard rate). So, we are equating marginal cost with marginal benefit.

The differential equation has a known solution:

G_{-i}(t) = 1 - \exp \left( \int _0^t \frac{ \ell'_i(z) }{ f_i(z) - \ell_i(z) } dz \right).

One might be concerned that the above expression may not satisfy the properties of a distribution function. For example, it may be decreasing or $\lim_{t \to \infty} G_{-i}(t) \neq 1$ . This is not the case. The function is strictly increasing because the term inside the integral is negative. Assumption 2.4 guarantees that the distribution approaches one. However, it never reaches one at any time. So, the support is unbounded. Another way to see this is to note that the survival function is

S_{-i}(t) = \exp \left( \int _0^t \frac{ \ell'_i(z) }{ f_i(z) - \ell_i(z) } dz \right) > 0.

The solution is somewhat difficult to work with. So, in most applications, the prize is taken to be fixed.

Assuming fixed prizes

If we assume $f_i(z) = V_i + \ell_i(y)$ , where $V_i$ is a constant prize, then

So, the strategy reduces to

\begin{aligned} G_{-i}(t) &= 1 - \exp \left( \int _0^t \frac{ \ell'_i(z) }{ f_i(z) - \ell_i(z) } dz \right) \\ &= 1 - \exp \left( \int _0^t \frac{ \ell'_i(z) }{ V_i } dz \right) \\ &= 1 - \exp \left( \frac{ \ell_i(t) }{ V_i } \right). \end{aligned}

This equation is about as simple as the equilibrium of the linear war of attrition (where $\ell_i(x) = -x$ ). So, it is pretty popular. You can try other assumptions in the general equation to to get other expressions. For example, the equilibrium where players only pay a fraction of $\ell_i$ when they win is also relatively simple.

Tullock Lottery Contests with Direct and Covert Discrimination

2021-04-15T00:00:00Z

These notes summarize concepts introduced in Nti (1999), Fang (2002), Nti (2004), Epstein et al (2013), and Ewerhart (2017a) about revenue maximization in Tullock lottery contests.

Introduction

Contests are models of conflict where risk neutral players exert costly effort to win a prize. A lottery contest is any contest where there is not a deterministic relationship between effort and victory. For example, scores could be measured incorrectly or there could be some randomness in the relationship between effort and the final scores. Alternatively, you can interpret lottery contests as giving each player a slice of a prize. For example, a 5% probability of receiving the prize is the same as receiving 5% of the prize with certainty.

The most popular lottery contest is the Tullock contest.^[1] We discuss ways to increase competitiveness in Tullock contests by discriminating against the stronger player. Handicaps in sporting events are a real world example of such discrimination. The purpose of a handicap is to make a match more even so that both teams/players exert more effort.

Model

Suppose that the prize of contest is worth one to both players. So, the two players have the same value. However, their scores have different costs. In particular, we will say that the score is $k > 1$ times more costly for Player 2 than for Player 1.^[2] You can interpret this as saying that Player 1 is more skilled. So, it takes less effort for her to produce a high score.

The probability of Player 1 winning the prize in a Tullock contest when she chooses score $s_1$ and her opponent chooses score $s_2$ is:

p_1(s_1, s_2) = \frac{s_1^{r}}{s_1^{r} + (\delta s_2)^{r} }.

The parameter $\delta$ sets the level (and direction) of direct discrimination. If $\delta = 0$ $, then Player 1 always wins. If $\delta = \infty$ $, then Player 2 always wins. In general, $\delta > 1$ implies discrimination against Player 1. Note that the prize is symmetric when $\delta = 1$ . In this case, there is no direct discrimination.

The other parameter, $r$ , is more subtle. It affects the variance of the lottery in the contest. If $r \to \infty$ , the prize always goes to the player with the higher score.^[3] If $r = 0$ , the prize is allocated at random. This precision parameter changes the marginal returns to effort at different levels. For example, if $r < 1$ , then the lower effort player gets a disproportionately high return. While such a contest is fair, in the sense that the two players are not treated differently, it is actually rigged in favor of the weaker player.

The final payoffs are:

U_1(s_1, s_2) = \frac{s_1^{r}}{s_1^{r} + (\delta s_2)^{r} } - s_1

U_2(s_1, s_2) = \frac{(\delta s_2)^{r}}{s_1^{r} + (\delta s_2)^{r} } - k s_2

Assumptions

We consider pairs $r, \delta$ such that the following two conditions are satisfied.

No reversal: $\delta \leq k$
Pure strategy equilibrium: $\left( \frac{\delta}{k} \right)^{r} \geq r - 1$

The first assumption prevents excessive discrimination against Player 1 and is without loss of optimality. This is made for simplicity. Without it, we would have to consider more cases.

The second assumption is more important. It is a necessary and sufficient condition for there to be an equilibrium in pure strategies. It is always satisfied if $r \leq 1$ and is never satisfied if $r > 2$ . The condition is equivalent to

r \leq \bar{r}(\delta/k) = 1 - \frac{W \left( - (\delta/k) \log{\left( \delta/k \right)} \right)}{\log{\left( \delta/k \right)}}

where $W$ is the Lambert W function. The upper bound, $\bar{r}$ is a strictly increasing function.

It is not obvious, but this condition is also without loss of optimality.

Equilibrium

To find the Nash equilibrium, we need to find the best response functions for each of the players. We do this by maximizing each payoff function. Player 1’s first order condition is

\frac{r s_1^{r} (\delta s_2)^{r}}{s_1 \left(s_1^{r}+(\delta s_2)^{r}\right)^2} = 1

and Player 2’s first order condition is

\frac{r s_1^{r } (\delta s_2)^{r}}{s_2 \left(s_1^{r}+(\delta s_2)^{r}\right)^2} = k.

This is a system of two equations with two unknowns. Dividing the second by the first gives us $k = \frac{s_1^{\star}}{s_2^{\star}}$ . So the ratio of the scores does not depend on the discrimination parameters. Thus, any change that increases one also increases the other. With this, we can quickly solve the system of equations to get $s_1^{\star} = \frac{r (k \delta)^{r} }{\left( k^{r}+\delta^{r} \right)^2}$ and $s_2^{\star} = \frac{r (k \delta)^{r} }{k \left( k^{r}+\delta^{r} \right)^2}$ .

In order for the first order approach to be valid, we have to check the second order and boundary conditions.

The second order conditions require the second derivative of the payoffs to be negative at the equilibrium. They are satisfied if and only if $\left( \frac{k}{\delta} \right)^{r} > \frac{r - 1}{r + 1}$ . This inequality holds by Assumption 1.

The boundary conditions require that payoffs are non-negative at the proposed equilibrium. Otherwise, the player would prefer to chose a score of zero. Using $k = \frac{s_1^{\star}}{s_2^{\star}}$ , we can dramatically simplify the equilibrium payoffs:

U_1(s_1^{\star}, s_2^{\star}) = \frac{k^{r}}{k^{r} + \delta^{r}} - s_1^{\star}

U_2(s_1^{\star}, s_2^{\star}) = \frac{\delta^{r}}{k^{r} + \delta^{r}} - s_1^{\star}.

By Assumption 1, $U_2(s_1^{\star}, s_2^{\star}) \leq U_1(s_1^{\star}, s_2^{\star})$ . So, we only need a condition such that Player 2 has a positive payoff. Substituting $s_1^{\star} = \frac{r (k \delta)^{r} }{\left( k^{r}+\delta^{r} \right)^2}$ gives our lowest payoff

U_2(s_1^{\star}, s_2^{\star}) = \frac{\delta^{r}}{k^{r} + \delta^{r}} - \frac{r (k \delta)^{r} }{\left( k^{r}+\delta^{r} \right)^2}.

It’s not obvious that this is non-negative by Assumption 2. However, it just requires some factoring.

Contest design

Suppose that there is some contest designer who can choose $r$ and/or $\delta$ in order to maximize the revenue, $s_1 + s_2$ . Because we know that $k = \frac{s_1}{s_2}$ , we are actually trying to maximize $(1 + k^{-1}) s_1$ . Because $k$ is constant, maximizing $s_1$ is enough.

Direct discrimination

Suppose $r$ is fixed. For now, assume it is one. To maximize revenue, we need to maximize $s_1$ . Suppose that we can directly discriminate using $\delta$ . Then, our problem is

\max_{\delta} \frac{k \delta}{( k + \delta )^2}.

This maximum is obtained at $\delta^\star = k$ .

In this case, the revenue is

\begin{aligned} s_1 + s_2 &= \left( 1 + \frac{1}{k} \right) s_1 \\ &= \left( 1 + \frac{1}{k} \right) \frac{k^2}{( 2 k )^2} \\ &= \frac{1 + k}{4 k} \end{aligned}

If we allow $r$ to take any positive value, then the optimal delta will still be $\delta^\star = k$ . This result comes from the optimization problem

\max_{\delta} \frac{r (k \delta)^{r} }{\left( k^{r}+\delta^{r} \right)^2}

which has the following first order condition:

\frac{r^2 ( \delta k)^{r} ( k^{r} - \delta^{r} )}{ \delta ( k^{r} + \delta^{r} )^3 } = 0.

Covert discrimination

Suppose that the contest designer cannot discriminate directly. So $\delta = 1$ , and the designer chooses $r$ in order to maximize the revenue of the contest. Then, our problem is

\max_{r} \frac{r k^{r}}{( 1 + k^r )^2}.

If we take first order conditions and solve, we get

r = \frac{1}{\log(k)} \left( 1 - \frac{2}{1 + k^{r}} \right)^{-1}

which defines implicit function $r^\star(k)$ . However, these values cannot be attained unless $k$ is sufficiently large. Recall that there is no pure strategy equilibrium if $r > \bar{r}$ . So, there is no way to reach some of these values.

If you set up a Lagrangian as in Proposition 3 of Nti (2004), then you find that the constrained optimum is the minimum of the two pictured curves. No exact representation of the intersection is known. However, it is approximately $k = 3.509$ .

Overall optimum

We know that if both forms of discrimination are allowed, then $\delta^\star = k$ because this is the optimum for any $r$ . As you can see in the first plot, this means that $\bar{r} = 2$ . We will show that this is also the constrained optimum. Our optimization problem is

\max_{r \leq 2} \frac{r}{4}

which is obviously attained at $r = 2$ . So, the optimal revenue is $(1 + k^{-1}) \frac{2}{4} = \frac{1 + k}{2 k}$ .

Beyond pure strategies

Assumption 2, which guarantees that equilibria exist in pure strategies is without loss of optimality. So, the solutions in the previous sections hold for all $r \geq 1$ .

If payoffs are symmetric, Assumption 2 is violated if and only if $r > 2$ . In this case, Baye et al. (1994) shows that when there is a symmetric mixed strategy equilibrium in which both players receive a payoff of zero.^[4] This means that when $\delta = k$ ,

\begin{aligned} E[s_1] &= E \left[ \frac{s_1^{r}}{s_1^{r} + (k s_2)^{r} } \right] \\ k E[s_2] &= E \left[ \frac{(k s_2)^{r}}{s_1^{r} + (k s_2)^{r} } \right] \end{aligned}

Symmetry implies that that each player’s expected probability of winning is one half. Therefore, $E[s_1] = \frac{1}{2}$ , $E[s_2] = \frac{1}{2 k}$ , and $E[s_1 + s_2] = \frac{1 + k}{2 k}$ . This is the same maximum revenue as in the overall optimum in pure strategies. It’s not obvious that this is the maximum revenue. For example, one might imagine that the principal could ensure zero payoffs but have Player 1 win with probability greater than one half. In order to confirm that Assumption 2 is without loss of optimality, we need to answer a few questions.

Are equilibria revenue equivalent? Yes. Ewerhart (2017a) shows that the equilibrium is unique when $r \leq 2$ and Ewerhart (2017b) shows that equilibria are revenue and payoff equivalent when $r > 2$ .

Is $r \in (\bar{r}( \delta / k ), 2]$ ever optimal No. Wang (2010) shows that revenue is

$\frac{2 \delta}{r k} \left( \frac{1 + k}{2 k} \right) (r - 1)^{(r - 1)/r}$

on this interval which is weakly decreasing in $r$ . Therefore, $r = \bar{r}( \delta / k )$ yields weakly greater revenue.

Is $r > 2$ ever optimal? No. Alcade and Dahm (2010) show that if $r > 2$ , the Tullock contest is revenue and payoff equivalent to an all-pay auction. Revenue in this auction is $\left( \frac{\delta}{k} \right) \frac{1 + k}{2 k}$ . This is the same revenue as when $r = 2$ .

Is $\delta < k$ ever optimal? No. Note that the revenue in the last two points is increasing in $\delta$ .

The above shows an example of how revenue depends on $r$ . Note that the maximum is reached before $\bar{r}(\delta / k)$ when the players are very heterogeneous (orange) and at the upper bound when players are more homogeneous (blue).^[5]

Conclusion

This table summarizes the results that we have about each type of optimal contest. There is not much to summarize for covert discrimination because there aren’t any closed form expressions.

	$r,\delta$	Payoffs	Revenue
No discrimination	1, 1	$\frac{k^2}{(1+k)^2}, \frac{1}{(1+k)^2}$	$\frac{1}{k+1}$
Direct ( $r$ fixed)	$r,k$	$\frac{2-r}{4}, \frac{2-r}{4}$	$\frac{1+k}{4k}$
Covert ( $\delta = 1$ )	$r(k),1$
Unrestricted	$2,k$	0, 0	$\frac{1+k}{2k}$

The case of covert discrimination demonstrates an important precision tradeoff. Choosing a large value of $r$ increases competitiveness by awarding the prize more frequently to the player with the higher score. However, choosing a low value allows you to discriminate against the stronger player – which also increases competitiveness. The effect that wins out depends on the size of the asymmetry between players. When $k$ is large enough, this discrimination effect wins out. However, when direct discrimination is possible, there is no advantage to covert discrimination.

Tullock contests were introduced 1980 by Gordon Tullock. The other major lottery contest is the rank order contest (Lazear and Rosen 1981). ↩︎
The papers mentioned give different prize values instead of different costs. The two are equivalent. I prefer to say that one player is more skilled than to say that one player values the prize more. ↩︎
In this case, it is not a lottery contest. It converges to the all-pay auction – the main deterministic contest. ↩︎
The symmetric equilibrium is characterized in Ewerhart (2015). ↩︎
More figures like the above can be found in Chowdhury et al. (2020). ↩︎

Fix the R Library Bug on Windows 10

2021-04-16T00:00:00Z

This is a quick post about how to fix an annoying R bug on Windows 10 caused by OneDrive. Many users use OneDrive to sync their Documents folder. Unfortunately, R stores libraries in the Documents folder, and OneDrive creates sync locks that interfere with the installation and updating of packages.

The solution is to set the R_LIBS_USER environment variable to a folder that is not synced by OneDrive. This can be done by running this script or by following these steps:

Open the Run window (Win + R)
Paste rundll32 sysdm.cpl,EditEnvironmentVariables and hit OK.
Click New… under “User variables” (the top panel)
Paste variable name R_LIBS_USER
You can use Browse Directory… to choose any folder that is not synced with OneDrive (eg. C:\Users\username\R)

Now that you changed your library directory, you’ll need to reinstall your libraries. You could copy them from OneDrive, but some may be damaged.

Steps 1 and 2 can be replaced by searching for some part of “Edit environment variables for the current user” in the start menu. This is easier to remember, but this guide provides the shortest path.

Optimal Fair Contests

2021-09-30T00:00:00Z

I show that any efficient two player symmetric allocation rule between risk neutral, heterogenous players generates weakly less revenue than one of two contests. The revenue maximizing efficient symmetric contest is an all-pay auction with bid caps (Che and Gale 1998) when heterogeneity is low and a difference form contest (Che and Gale 2000) when heterogeneity is high.

Introduction

Contests are models of conflict in which risk neutral players exert costly effort to win a prize. When one player is more capable than another, contests become less competitive. This lack of competitiveness decreases total effort. A contest designer can use two common tools to increase the total effort: reserve bids (Bertoletti 2016) and direct discrimination (Ewerhart 2017; Franke, Leininger, and Wasser 2018).

Reserve bids (i.e. inefficient allocation rules) allow the contest designer to require either player to exert some minimum amount of effort to qualify for the prize. A large enough reserve bid can make the lack of competition from a competitor irrelevant. For example, a professional figure skater would not need to exert much effort to beat me in a figure skating competition. However, if winning a large prize requires a high score, then the professional is not really competing against me. He is competing against this reserve bid. A large enough reserve bid can ensure any amount of individually rational effort.

Discrimination allows the contest designer to discriminate against the stronger player to reduce or remove her advantage. Handicaps in sporting events are a common example. This requires the contest designer to know which contestant is more talented. Otherwise, the designer does not know who to handicap. If this information is known, direct discrimination can be used to level the playing field of any contest.

We consider an alternative way of maximizing revenue – “fair” design. The contest designer maximizes the revenue of a contest under complete information without withholding the prize or treating contestants differently. We show that under these conditions, the designer is unable to achieve the first best and must give a positive payoff to the stronger player. However, allowing an arbitrarily reserve bid or direct discrimination ensures that the first best revenue can be achieved.

This work takes a similar approach to Letina, Liu, and Netzer (2020). However, unlike this work, I do not allow the principal to engage in direct discrimination or modify/withhold the prize. This work on optimal contest design contributes to a literature on revenue dominance in symmetric efficient contests (Fang 2002; Franke, Kanzow, Leininger, and Schwartz 2014) by characterizing an efficient symmetric contest which weakly dominates all others and contributes to a much larger literature on contest design.

Model

Suppose that the prize of contest is worth one to both players. So, the two players have the same value. However, their scores have different costs. In particular, we will say that the score is $k > 1$ times more costly for Player 2 than for Player 1. You can interpret this as saying that Player 1 is more skilled. So, it takes less effort for her to produce a high score.

The probability of Player $i$ winning the prize in a contest when she chooses score $s_i$ and her opponent chooses score $s_{-i}$ is:

p_{i}(s_i, s_{-i}).

The final payoffs are:

U_1(s_1, s_2) = p_1 (s_1, s_2) - s_1

U_2(s_1, s_2) = p_2 (s_2, s_1) - k s_2.

We define two notions of “fairness” that are central to our analysis:

A contest satisfies efficiency iff $p_1 (x, y) + p_2 (y, x) = 1$ for all $x,y \geq 0$ .
A contest satisfies symmetry iff $p_1 (x, y) = p_2 (x, y)$ for all $x,y \geq 0$ .

Efficiency means that someone is always a winner. Symmetry means that both players are treated the same by the designer. We will see that these conditions do not individually constrain the designer’s revenue. However, when the two conditions are imposed together, the designer is meaningfully constrained.

Note that these conditions jointly imply $p_i(x,x) = 0.5$ .

Contest design

Suppose that there is some contest designer who can choose $p_1, p_2$ in order to maximize the expected revenue, $E[s_1 + s_2]$ given the (possibly mixed) strategies of the players. If strategies are pure, then the revenue is deterministic. Note that incentive compatibility implies

E[s_1 + k s_2] \leq E[p_1 (s_1, s_2) + p_2 (s_2, s_1)] \leq 1.

Therefore, expected revenue is bound above by $1 - (k - 1) E[s_2]$ and the first best is achieved iff $s_1 = 1$ and $s_2 = 0$ .^[1] We can immediately see that no contest which satisfies both symmetry and efficiency will achieve the first best. This is because both players receive a payoff of zero in the first best, but Player 1 can choose zero instead of one to receive a payoff of $p(0,0) = 0.5$ .

Reserve bid

If we allow for a reserve bid, but still enforce symmetry, is is easy for the designer to achieve the first best. For example, the following all-pay auction with a reserve bid achieves the first best:

p(x, y) = \begin{cases} 0 &\text{if } x < \max(y,1) \\ \frac{1}{2} &\text{if } x \geq 1, x=y \\ 1 &\text{if } x \geq 1, x > y. \end{cases}

This is because Player 1 is willing to meet the reserve bid of 1 and Player 2 is not.

One might wonder if the designer can still extract the full surplus with a smaller reserve bid. The answer is yes. In fact, $p(0,0) = 0$ is sufficient for the first best to be attainable under symmetry. That is, the designer need only be able to withhold the prize when both players play zero. To see this, consider the following contest which satisfies both properties except at zero.

p(x,y) = \begin{cases} 1 &\text{if } x-y >1 \\ x-y &\text{if } x-y \in(0,1] \\ 0 &\text{if } x = y = 0 \\ \frac{1}{2} &\text{if } x=y \neq 0 \\ 1+x-y &\text{if } x-y <0. \end{cases}

This contest has an equilibrium at the first best yet only denies the prize to players when they both exert no effort.

Direct discrimination

Direct discrimination is similar to a reserve bid. It is possible to mimic any reserve bid through direct discrimination by promising the prize to the weaker player whenever the reserve is not met. In particular, take either example from the previous section and set $p_1(x,y) = p(x,y)$ and $p_2(y,x) = 1 - p(x,y)$ . Such a contest is efficient and will be strategically equivalent to the original contest. For example, consider the aforementioned all-pay auction with a reserve bid. With this transformation,

p_1(x, y) = \begin{cases} 0 &\text{if } x < \max(y,1) \\ \frac{1}{2} &\text{if } x \geq 1, x=y \\ 1 &\text{if } x \geq 1, x > y \end{cases}

p_2(x, y) = \begin{cases} 0 &\text{if } y \geq 1, y > x \\ \frac{1}{2} &\text{if } y \geq 1, x=y \\ 1 &\text{if } y < \max(x,1). \end{cases}

This contest has an equilibrium at the first best.

“Fair” design

We now restrict the principal to use an efficient symmetric contest. In equilibrium, each Player must weakly prefer her equilibrium payoff over copying the strategy of her opponent. Therefore, the following weak incentive compatibility condition is necessary for equilibrium:

\frac{1}{2} - E[s_2] \leq E[U_1(s_1, s_2)].

Together with $E[U_2(s_1, s_2)] \geq 0$ this implies

\begin{aligned} \frac{1}{2} - E[s_2] &\leq E[U_1(s_1, s_2) + U_2(s_1, s_2)] \\ \frac{1}{2} - E[s_2] &\leq 1 - E[s_1 + k s_2] \end{aligned}

where the last line follows from efficiency. Rearranging this equation gives an upper bound on the revenue:

E[s_1 + s_2] \leq \frac{1}{2} + (2 - k) E[s_2]

which can be further bounded by

E[s_1 + s_2] \leq \begin{cases} \frac{1}{k} &\text{if } k < 2 \\ \frac{1}{2} &\text{if } k \geq 2. \end{cases}

I show by construction that this upper bound is tight. That is, there exist optimal contests which achieve these bounds.

The second bound comes from the fact that $E[s_2] \leq \frac{1}{2k}$ because Player 2 cannot win with probability more than one half. To see this, consider that the following two conditions must hold in equilibrium

\begin{aligned} \frac{1}{2} - k E[s_1] &\leq E[p(s_2, s_1) - k s_2] \\ E[s_2] &\leq E[s_1] + (k)^{-1} \left(E[p(s_2, s_1)] - \frac{1}{2}\right) \end{aligned}

and

\begin{aligned} \frac{1}{2} - E[s_2] &\leq E[p(s_1, s_2) - s_1] \\ E[s_2] &\geq E[s_1] - \left(E[p(s_2, s_1)] - \frac{1}{2}\right) \end{aligned}

which imply $E[p(s_2, s_1)] \leq \frac{1}{2}$ .

Case 1: $k < 2$ . This means that revenue cannot exceed $\frac{1}{k}$ . We can reach this upper bound with an all-pay auction with a bid cap at $\frac{1}{2k}$ as in Che and Gale (1998). That is

p(x,y) = \begin{cases} 1 &\text{if } \frac{1}{2k} \geq x > y \text{ or } y > \frac{1}{2k} \\ \frac{1}{2} &\text{if } x=y \\ 0 &\text{if } \frac{1}{2k} \geq y > x \text{ or } x > \frac{1}{2k}. \end{cases}

This has an equilibrium at $s_1 = s_2 = \frac{1}{2k}$ which achieves the upper bound for $k < 2$ .

Case 2: $k \geq 2$ . The above implies that revenue cannot exceed one half. Consider the following difference form contest as in Che and Gale (2000):

p(x,y) = \begin{cases} 1 &\text{if } x-y > \frac{1}{2} \\ \frac{1}{2} + x-y &\text{if } x-y \in \left[-\frac{1}{2},\frac{1}{2}\right] \\ 0 &\text{if } x-y < - \frac{1}{2} \end{cases}

The above contest has an equilibrium at $s_1 = \frac{1}{2}$ and $s_2 = 0$ which achieves the upper bound for $k \geq 2$ .

Note that these strategies must be pure because no mixed strategy can have expectation zero and no individually rational mixed strategy can have expectation one. ↩︎

Optimization of Functionals and the Calculus of Variations

2022-02-12T00:00:00Z

There are many settings in Economics, particularly in mechanism design, where one wants to fund a function that maximizes (or minimizes) some objective. It turns out this works similarly to regular calculus.

Motivation

Consider the expected revenue of a mechanism designed to sell a good to one agent who’s value of the good is distributed uniformly with support $[ 0, 1 ]$ . This expected revenue is known to be

R[p] = \int_0^1 p(v) (2 v - 1)\, dv

where $p(v)$ is a function that determines the probability of selling the good to the agent when her value is $v$ and $2 v - 1$ is the virtual surplus. The above equation, $R[p]$ , is a functional of $p$ . That is, it maps the function $p$ to a real number. Suppose that you wanted to find the revenue maximizing mechanism $p$ . Then, the problem you want to solve is

\max_p \int_0^1 p(v) (2 v - 1)\, dv

subject to $0 \leq p(v) \leq 1$ . If you look at this problem closely, you will note that the optimal function must be

p(v) = \begin{cases} 0 &\text{ if } v < 0.5 \\ 1 &\text{ if } v > 0.5 \end{cases}

where the value at one half does not matter. This is because $p$ is multiplied by some “slope”. When the slope is positive, we want $p$ to be as large as possible. When the slope is negative, we want $p$ to be as small as possible. Indeed, the functional derivative of $R[p]$ is

\frac{\delta R}{\delta p}(v) = 2v - 1

We will see that the functional derivative formalizes our intuition about this slope. It will allow us to solve optimization problems that are nonlinear.

On the reals

Optimization over function spaces is a natural extension of optimization over the reals. So, it’s important to remember exactly how optimization works on functions of real vectors. Consider the following function $g : \mathbb{R}^N \to \mathbb{R}$ .

g(X) = (x_1)^2 + (x_2)^2

where $X = (x_1, x_2)$ . The directional derivative in the direction $V = (v_1, v_2)$ is

\begin{align*} dg_V(X) &= \lim_{\epsilon \to 0} \left[ \frac{g(X + \epsilon V) - g(X)}{\epsilon} \right] \\ &= \nabla g(X) \cdot V \\ &= \begin{bmatrix} 2 x_1 \\ 2 x_2 \end{bmatrix} \cdot V \\ \end{align*}

If $X^\star$ is a minimum, there cannot be a direction, $V$ , such that $dg_V(X^\star) < 0$ . This would imply that there exists an $\epsilon > 0$ such that $g(X^\star + \epsilon V) < g(X^\star)$ . This contradicts the assumption that $X^\star$ is a minimum. In addition, there cannot be a $V$ such that $dg_V(X^\star) > 0$ because his would imply $dg_{(-V)}(X^\star) = - dg_{V}(X^\star) < 0$ .

Therefore, we know that for any $X^\star$ , $dg_V(X^\star) = 0$ for all $V \in \mathbb{R}^2$ . This is only possible if $\nabla g(X^\star) = (0, 0)$ which means $\frac{\partial g(X^\star)}{\partial x_1} = 2 x_1 = 0$ and $\frac{\partial g(X^\star)}{\partial x_2} = 2 x_2 = 0$ . So, the only local stationary point is at $(0,0)$ . However, stationarity is a necessary but not sufficient condition. We would need to test for convexity to ensure that this is a minimum. Though, it obviously is.

Functional optimization follows the same principles.

On continuous functions

Suppose there is a known continuous function $\phi$ that we want to approximate with another continuous function using least squares on the interval $[0, 1]$ . That is, you want to minimize the following functional

L[f] = \int_0^1 (f(x) - \phi(x))^2\, dx.

Of course, the minimum is going to be at $f^* = \phi$ . We will solve this by finding the Gateaux differential of $L$ with respect to some direction, $\psi$ . This is defined in the same way as the directional derivative on the reals

dL_{\psi}[f] = \lim_{\epsilon \to 0} \left[ \frac{L[f + \epsilon \psi] - L[f]}{\epsilon} \right].

So, the Gateaux differential of $L$ is

\begin{align*} dL_{\psi}[f] &= \lim_{\epsilon \to 0} \left[ \int_0^1 \frac{ (f(x) + \epsilon \psi(x) - \phi(x))^2 - (f(x) - \phi(x))^2}{\epsilon} \, dx \right] \\ &= \lim_{\epsilon \to 0} \left[ \int_0^1 \frac{ 2f(x) - 2\phi(x) + \epsilon\psi(x)}{\epsilon} \epsilon\psi(x) \, dx \right] \\ &= \int_0^1 2 (f(x) - \phi(x)) \psi(x) \, dx\\ \end{align*}

which is the inner product of the functional derivative,

\frac{\delta L}{\delta f}(x) = 2 (f(x) - \phi(x)),

and the direction, $\psi$ . Like in calculus on the reals, we need this to be zero for all directions. It turns out that, just like before, we only need to set the directional derivative to zero. This is due to the following lemma.

Fundamental lemma of calculus of variations If $g$ is continuous on $[a,b]$ and satisfies the equality

\int_a^b g(x) \psi(x) \, dx = 0

for all continuous functions $\psi$ such that $\psi(a)=\psi(b)=0$ , then $g(x) = 0$ for all $x \in [a,b]$ .

This means that $dL_{\psi}[f]=0$ if and only if $2 (f(x) - \phi(x)) = 0$ for all $x \in [0,1]$ . So, $f = \phi$ as anticipated.

Instructions for using Mathstodon fork of Mastodon

2023-12-30T00:00:00Z

This is a short guide to document my experience in enabling TeX support in Mastodon by installing Christian Lawson-Perfect’s Mathstodon fork. As of the time of writing, this fork is based on Mastodon 4.5.3 and there are no official installation instructions. This guide assumes that you already have a functional installation of Mastodon 4.5.3. If you do not, you should follow the Mastodon installation instructions. The instructions can likely be adapted for a fresh install.

All steps of this guide should be run as the mastodon user. If you are not logged in as the mastodon user, you can switch with sudo -su mastodon.

Set up MathJax assets

The fork expects a custom version of MathJax to be publicly accessible from /MathJax on the interface domain. To set this up, we need to download MathJax and place it in the correct location. First, we need to create a directory to store the assets:

mkdir -p ~/overrides/MathJax && cd ~/overrides

Then, download and extract the files:

wget -qO- https://www.matthewthom.as/gh/bin/mathjax-3.1.2-mathstodon.tar.gz | tar xzC ./MathJax

Finally, to make the files accessible, add the following to the https server block in your nginx config:

  location /MathJax/ {
    root /home/mastodon/overrides;
    add_header Cache-Control "public, max-age=14400, must-revalidate";
    add_header Strict-Transport-Security "max-age=63072000; includeSubDomains";
    try_files $uri =404;
  }

Install the fork

First, we want to add the fork as a git remote and switch to the correct branch:

# change to the mastodon directory
cd ~/live

# add the fork as a remote
git remote add mathstodon https://github.com/christianp/mastodon.git

# fetch the branches from the fork
git fetch mathstodon

# switch to the mathstodon-4.5 branch
git checkout mathstodon/mathstodon-4.5

The dependencies between the fork and main repo are the same at the time of writing, but I’d gave it a try anyway just to make sure.

# ruby dependencies
bundle install

# node dependencies
yarn install --frozen-lockfile

Then, we need to compile the assets:

RAILS_ENV=production bundle exec rails assets:precompile

The final step is to restart the Mastodon web service. For this, you need to exit the mastodon user and run the following command as root:

systemctl restart mastodon-web

Setting up KaTeX in Mastodon

2024-06-02T00:00:00Z

This is a short guide to document my experience in enabling TeX support in Mastodon by injecting KaTeX into the interface. I use this setup on https://econtwitter.net. This differs from the Mathsodon fork I used previously in that it does not require direct modification of the Mastodon code. I instead inject KaTeX into the interface using nginx at the proxy level. As a result, we can install Mastodon updates without breaking this feature. I do not cover the installation of Mastodon itself in this guide.

You can think of this as similar to a userscript or browser extension, but automatically applied to all users of your instance.

Install KaTeX static assets

The commands in this section should be run as the mastodon user. If you are not logged in as the mastodon user, you can switch with sudo -su mastodon. First, we need to create a directory to store the assets:

mkdir -p ~/overrides/katex && cd ~/overrides

Then, download and extract the files (feel free to replace the URL with the latest version of KaTeX):

wget -qO- https://github.com/KaTeX/KaTeX/releases/download/v0.16.27/katex.tar.gz | tar xzC ./katex

Then, make a file called custom.js in the ~/overrides directory with the following content:

// Render TeX by default
document.addEventListener("DOMContentLoaded", function () {
  var katexTootSelector =
    ".status__content .status__content__text:not(.katex-rendered)";

  // Define a function to render TeX in an element
  async function renderMathInToots(elm) {
    // Render the math in the element
    renderMathInElement(elm, {
      delimiters: [
        { left: "\\(", right: "\\)", display: false },
        { left: "\\[", right: "\\]", display: true },
      ],
      throwOnError: false,
    });

    // Add a class to the element so we don't render it again
    elm.classList.add("katex-rendered");
  }

  // On each mutation, render the math in the new elements
  const observer = new MutationObserver((mutations) => {
    // Render TeX in the new toots
    document.querySelectorAll(katexTootSelector).forEach(renderMathInToots);
  });

  // Observe the body for changes
  observer.observe(document.body, {
    childList: true,
    subtree: true,
    attributes: false,
    characterData: false,
  });

  // Render TeX in already loaded toots (there probably aren't any)
  document.querySelectorAll(katexTootSelector).forEach(renderMathInToots);
});

This short script calls KaTeX to render TeX in toots as they are loaded. We use a query selector to select toots that have not already been rendered by KaTeX. A mutation observer watches for changes to the DOM and renders TeX in new toots as they are loaded. The most fragile part of this script is the selector, which may need to be updated if the Mastodon interface changes.

Configure nginx

In order to load the KaTeX assets and the custom script, we need to configure nginx to inject the necessary HTML into the Mastodon interface. This requires the nginx http sub module which is included in debian, but may require installation on other distributions. As root, edit the /etc/nginx/sites-available/mastodon file in the following ways:

Add the following block to the main server block so that we can access the files that we just installed:

location /overrides/ {
  alias /home/mastodon/overrides/;
  add_header Cache-Control "public, max-age=0, must-revalidate";
  add_header Strict-Transport-Security "max-age=63072000; includeSubDomains";
  try_files $uri =404;
}

Add the following to the location @proxy block to inject the custom script:

  # Load KaTeX stylesheets and scripts
  sub_filter '</head>' '
<link rel="stylesheet" crossorigin="anonymous" href="/overrides/katex/katex.min.css" media="all" />
<script defer crossorigin="anonymous" src="/overrides/katex/katex.min.js" id="katex-script"></script>
<script defer crossorigin="anonymous" src="/overrides/katex/contrib/auto-render.min.js" id="katex-autorender"></script>
<script defer crossorigin="anonymous" src="/overrides/custom.js"></script>

</head>';

  # Only run the sub_filter once
  sub_filter_once on;

Finally, reload nginx with sudo systemctl reload nginx.

For security, you may want to add integrity checks to the script and stylesheet tags. I excluded them because your KaTeX version may differ. You can generate these with the following command:

while read -r file; do
  printf '\n%s\nintegrity="sha384-%s"\n' "$file" "$(openssl dgst -sha384 -binary < "$file" | openssl base64 -A)"
done <<EOF
/home/mastodon/overrides/katex/katex.min.css
/home/mastodon/overrides/katex/katex.min.js
/home/mastodon/overrides/katex/contrib/auto-render.min.js
/home/mastodon/overrides/custom.js
EOF

Custom CSS

I added the following CSS in the Mastodon admin interface (Administration > Server settings > Appearance) to make display mode math look better:

/* KaTeX
   ===================================== */
.status__content__text .katex-display {
  overflow: auto hidden;
  scrollbar-width: thin;
  /* Make space for the scrollbar */
  padding-bottom: 1em;
  padding-top: 1em;
  margin-bottom: 0;
  margin-top: 0;
}

I also hacked in some instructions for using TeX under the compose box.

/* Equation instructions in compose form
   ===================================== */
form.compose-form::after {
  content: "Inline LaTeX: \\( code \\) \00000a Display-mode: \\[ code \\]";
  white-space: pre-wrap;
  margin-top: 1em;
  font-family: monospace;
}

Economics of Accessibility in Consumption

2025-12-02T00:00:00Z

Markets tend not to accommodate disabled workers and consumers without outside intervention. There are numerous studies about disabled workers (e.g., Haveman and Wolfe 2000, Aizawa et al. 2024). However, I could only find one paper about how markets fail disabled consumers.^[1] I have been thinking about this issue for the last few years. I hope my thoughts are interesting both to economists (who may find a novel environment to explore) and to advocates for accessibility (who should find a different way of thinking about the problem). I will use blindness as my motivating example, but the ideas apply to disabilities more generally so long as those disabilities both (1) require accommodations that are either costly to the provider or diminish the experience for non-disabled consumers, and (2) are sufficiently uncommon.^[2]

1. Command-and-control successes and failures

Legal mandates are the standard tool for advancements in accessibility. While that seems to work for some clearly defined and enforced requirements (e.g., handicapped parking), mandates work less well when (1) enforcement is lacking or (2) requirements are harder to define.

For an example of low enforcement, Ameri et al. 2020 (that one relevant paper I mentioned) finds that Airbnb hosts are less likely to approve requests from guests with certain disabilities. They suggest that this may be because the ADA does not apply to most Airbnb hosts. Even if the regulations did apply, enforcing regulations for many small providers can be impractical.

For an example of less defined objectives, websites are arguably required to be accessible under the Americans with Disabilities Act,^[3] yet I have heard anecdotally from blind and visually impaired users that most of the web is inaccessible to the blind. There are also studies which show that the vast majority (94.8%) of top websites fail to meet requirements that can be automatically tested.

Legal mandates also do not encourage innovation beyond the minimum requirements. Many of the biggest innovations that benefit the blind and visually impaired seem to have been developed because of their usefulness to sighted users. For example, OCR (optical character recognition) software was initially developed to digitize printed text for searchability, not for screen readers. The same is true of AI image recognition, self-driving cars, voice assistants, and audiobooks.

2. Model of the market

I start with a quote from one of the many companies that has presented the market case for accessible websites for the blind and visually impaired alongside the legal mandate.

Organizations who fail to meet WCAG 2.2 Level AA requirements are not providing an accessible, ADA-compliant website nor are they complying with international accessibility laws. Non-compliance can result in potential legal action, including accessibility lawsuits and demand letters that can be time-consuming, expensive, and damaging to your reputation.

More importantly, website accessibility opens the door to an often overlooked market. For businesses, this can lead to more revenue and increased growth which results in more customer engagement, satisfaction, and brand loyalty.

– AudioEye

This argument is plausible. Offering an accessible website would enable a company to sell to more customers. Maybe these added sales more than cover the cost of making the website accessible. However, it seems like companies do not make their websites accessible unless they are forced to. Do they just not understand the market opportunity?

Consider another quote on the opposite extreme from Domino’s Pizza in their petition to the Supreme Court to overturn a ruling in circuit court that their website must be made accessible to the blind.

Other defendants eliminate online offerings instead of attempting compliance—a choice that ultimately hurts all consumers, including people with disabilities. Many businesses and non-profits lack the resources required to overhaul their websites and mobile apps. Faced with the threat of ADA liability, they may decide to jettison online content.

– Petition to the Supreme Court by Domino’s Pizza

This argument is also plausible. Maybe the cost of making a website accessible is so high that a company would rather shut their website down than provide an accessible one. Maybe the reason we see so few accessible websites is because the companies literally cannot run accessible products profitably.

Either of these possibilities could hold for some markets and some definition of accessibility. However, the most common situation is probably one that is in-between. The cost of making a website accessible is not so high that firms would prefer to shut down, but high enough that firms will not make their website accessible unless compelled to do so.

2.1. Basic model

Suppose that there is a monopolist seller providing a good which costs $c \geq 0$ to produce. A fraction, $\mu \in (0,1)$ , of consumers are disabled and require an accommodation to consume the good. If the seller does not provide the accommodation, they can sell $(1-\mu) Q(p)$ units of the good at price $p$ , where $Q(p)$ is the demand function. If the seller does provide the accommodation,

It must pay fixed cost $K \geq 0$ to provide the accommodation;^[4] and
It can sell $(1-\mu) Q(p)$ units of the non-accessible good at price p and $\mu Q(r)$ units of the accessible good at price r.

You may object that the seller can charge different prices for the accessible and non-accessible versions of the good. It will not matter in this model, because the prices will be the same anyway. However, we explore settings where this matters in Section 3.

2.2. Solving the model

When the seller does not provide the accommodation, we have a standard monopoly problem. The seller sets the price, $p^*$ , to maximize the profits of selling to non-disabled consumers,

\text{Profits} = \max_{p} \underbrace{(p - c)}_{ \text{per-unit profit} } \underbrace{(1 - \mu) Q(p)}_{\text{units}}. \eqnum{1}

When the seller does provide the accommodation, they will set the two prices to maximize profits from both disabled and non-disabled consumers,

\text{Profits} = \max_{p,r} (p - c) (1 - \mu) Q(p) + (r - c) \mu Q(r) - K,

which can also be written as

\text{Profits} = \max_{p} (p - c) Q(p) - K. \eqnum{2}

The seller would rather shut down than provide the accommodation if (2) is less than zero, and the seller will provide the accommodation if the profits from (2) exceed those from (1). There are three cases.

Case 1: The seller will provide the accommodation if

\max_{p} (p - c) Q(p) \geq \frac{K}{\mu}. \eqnum{3}

Case 2: The seller will not provide the accommodation, but prefers to provide the accommodation than to shut down if

\frac{K}{\mu} \geq \max_{p} (p - c) Q(p) \geq K. \eqnum{4}

Case 3: The seller will not provide the accommodation, and prefers to shut down than to provide the accommodation if

K \geq \max_{p} (p - c) Q(p). \eqnum{5}

2.3. Interpretation

The maximized part of the inequalities in (3), (4), and (5) is the profit that the seller would make if everyone were non-disabled. So, the firm’s decision depends on the relationship between the company’s profits, the cost of accommodating disabled consumers, and the fraction of consumers who are disabled.

Case 1 is the good scenario described by AudioEye where it is profitable to provide the accommodation because the disabled users are valuable enough to more than cover the cost of the accommodation. Case 3 is the bad scenario described by Domino’s where the cost of accommodating disabled users is so high that the company would rather shut down than provide the accommodation.

Domino’s identified one example of Case 3 where Berkeley discontinued a website which did not make any money rather that make it accessible to the blind. We can see that Case 3 always holds if profits are zero. I consider this further in Section 2.5.

Case 2 is the middle scenario where disabled users are not accommodated because they are a minority with specific needs that are not appreciated by the larger majority group of non-disabled users. In this case, the company would provide the accommodation if everyone were disabled, but since only a small fraction are disabled, the it is not profitable to accommodate them.

About 2.5% of the US population has a visual disability.^[5] They access websites less frequently than non-disabled users. So, let’s say the share of disabled website users is $0.0138 \sim 1/72 = \mu$ . That is similar to this estimate from Stack Exchange. We can substitute this back of the envelope calculation into (4) to obtain

72 K \geq \max_{p} (p - c) Q(p) \geq K.

If we think about a medium sized retailer that estimates the cost of making their website accessible to be about $35,000, then they would be in Case 2 if the profits they make from their website are between $35,000 and $2,520,000. That’s a wide range that many firms likely fall into.

In the opposite direction, if a firm’s website is among the 94.8% of websites that are not accessible, we can infer that they believe the cost of making their website accessible is at least 1.38% of their profits from the website. So, for every $1,000,000 that a firm expects to profit from their website, they believe the cost of making their website accessible is at least $13,800. That seems plausible.

2.4. This is not a traditional market failure

The market in this model does not maximize total surplus for multiple reasons. First, the seller is a monopolist; so, it is not competitive. Moreover, there is a fixed cost to provide the accommodation, which is a barrier to entry.

However, the bigger issue is not that this market fails to maximize total surplus. The issue is that disabilities are fundamentally ethically different from the preferences that markets typically aggregate. To see this, consider the following two situations:

A small group of users (1% of the population) is blind and must use screen readers to access websites. A firm chooses to not make its website compatible with screen readers because they assert that the cost of doing so ($30k) is too high relative to the revenue they would gain from the small group of blind users ($10k). Upset by this, the government mandates that all websites must be compatible with screen readers.
A small group of users (1% of the population) is eccentric and will only use Netscape Navigator to access websites. A firm chooses to not make its website compatible with Netscape Navigator because they assert that the cost of doing so ($30k) is too high relative to the revenue they would gain from the small group of eccentric users ($10k). Upset by this, the government mandates that all websites must be compatible with Netscape Navigator.

Do you view these situations as equivalent? A market does. A market does not distinguish between a customer who cannot do something and a customer who chooses not to. The Math is unaffected. Most everyone would want these two situations to be treated differently, but a market will not do this without intervention.

2.5. Robust policy: exempt noncommercial offerings

Consider the problem of setting a fee that a firm could pay to be exempt from disability regulation. If we were to set such a fee, we might want to set it so that all firms in Case 3 (who would rather shut their product down than provide the accommodation) would pay the fee and remain open, but all firms in Case 1 or 2 (who would provide the accommodation if everyone were disabled) would provide the accommodation rather than pay the fee. We will use the value of this fee to make the case for exempting noncommercial offerings. The formal analysis helps define what a noncommercial offering is and demonstrates the issue with extending such an exemption to small commercial offerings.

Suppose that we know the profits of a firm and the proportion of customers with relevant disabilities. The optimal fee is $(1 - \mu) \max_{p} (p - c) Q(p)$ , i.e., the full profits that the firm would make from non-disabled consumers. With this fee, the firm faces cost, $K$ , of accommodating (the left-hand side of Equation 5) and a benefit equal to the full profits they could earn from the entire market (the right-hand side of Equation 5). So, they will provide the accommodation exactly when (5) does not hold. If the fee were any lower, firms with a cost of accommodating less than the full profits would pay the fee rather than provide the accommodation. If it were any higher, firms with a cost of accommodating greater than the full profits would rather close than pay the fee.

This means that the ideal fee is one that ensures the firm is not allowed to earn any profits if they do not provide the accommodation. In practice, there is no reason to implement such a fee. A firm with no revenue is noncommercial and moreover, a firm with only enough revenue to recoup their costs is also noncommercial. If a company intended to be commercial, but the cost of accommodating disabled users is so high that they cannot make any profit, then they can still provide their offering if they do so noncommercially.

Note that this analysis excludes dynamics. Thus, it does not apply to a product which is currently unprofitable but intends to in the future. It would apply, however, to the open course website that Berkeley discontinued. Of course, we still want these noncommercial offerings to be made accessible, but this can more effectively be achieved through rewards rather than penalties. For example, Berkeley could have received a grant to make their open course website accessible.

You can also use grants to supplement penalties for commercial offerings. For example, you could offer grants to smaller commercial offerings to encourage them to provide accommodations.

2.6. Minimal penalty

Penalties should be large enough to push firms in Case 2 to provide accommodations. This means that a firm choosing not to accommodate must expect to pay at least $K - \mu \max_{p} (p - c) Q(p)$ , the accommodation cost minus the profits from selling to disabled consumers. Because enforcement is not perfect, the actual penalty charged should be at least this number divided by the probability of enforcement. Any penalty greater than this also works. The cost of accommodating, $K$ , divided by the enforcement probability is appealing because it requires estimating fewer parameters.

3. Extensions

The preceding analysis is very simple. I consider some extensions that help contextualize the model and raise additional issues.

3.1. Different demands

There is no reason to think that disabled and non-disabled consumers value the good equally or in the same way. For example, an image gallery could be more valuable to sighted users than blind users. On the other hand, the opposite might be true of a podcast. This will affect the interpretation slightly when the price for the accessible and non-accessible versions of the good are allowed to be different. Forcing the prices to be the same reduces the firm’s incentive to accommodate, but the basic insights still hold.

For this section, we modify the model from Section 2.1 so that the demand for disabled customers is given by a different demand function, $Z(r)$ :

An accommodating firm can sell $(1-\mu) Q(p)$ units of the good at price $p$ for the non-accessible version $\mu Z(r)$ units of the good at price $r$ for the accessible version.

3.1.1. Different prices

If the prices can be different, then we can get equivalence by substituting a transformed $\mu$ parameter. In particular, the analysis of this model is identical to that of Section 2 with $\mu$ replaced by

\mu' = \frac{\mu \max_r Z(r) (r-c) }{(1-\mu) \max_p Q(p) (p-c) + \mu \max_r Z(r) (r-c)}.

That is, in Section 2, we interpreted $\mu$ as the fraction of actual consumers who are disabled. What really matters is the fraction of profits that come from disabled consumers. When the demand functions are equal, the share of customers and the share of profits are the same.

3.1.2. Same prices

If we require that the prices be the same, we are not affecting the profits of a firm that chooses not to accommodate, but we are constraining the profitability of a firm that chooses to accommodate. The firm can choose to set the prices for the accessible and non-accessible products to be equal in either case. However, forcing the prices to be the same introduces an additional restriction on profits. Putting these same words in terms of math, by the triangle inequality,

\max_p{(p-c) (1-\mu) Q(p) + (p-c)\mu Z(p)} \leq \left[ \max_p{(p-c) (1-\mu) Q(p)} \right] + \left[ \max_r{(r-c) \mu Z(r)} \right].

Thus, when prices must be the same, firms have less incentive to accommodate. Most of the analysis works in one direction. A firm that meets the non-accommodating case conditions (Case 2 or Case 3) for differentiated prices will still be in that same case. However, it is possible for a firm to meet the accommodating condition (Case 1) for differentiated prices and actually be in Case 2.

The argument to exempt noncommercial offerings still holds with the added detail that you should also exempt a product which no disabled customers want— i.e., $Z(p^\star)=0$ . Moreover, the minimal penalty from Section 2.6 is not necessarily sufficient. However, the cost of accommodating, $K$ , is always sufficient.

3.2. Incorporating competition

Until this point, I have omitted the possibility that there may be competition in the market for non-disabled customers, but an opportunity for monopoly in serving disabled customers. One might suspect that a firm would be more likely to accommodate if doing so gives it access to a market which is less competitive than its current one. This is not true in general, but there are circumstances under which the intuition holds.

If the participants can charge different prices to disabled and non-disabled customers, a competitor in the non-disabled market does not affect the firm’s choice to accommodate because this competition reduces the firm’s profits from accommodating and not accommodating by the same amount. It does affect the firm’s choice to close because there is less benefit to remaining in the market. Thus, it is possible for accessibility requirements to make a market less competitive. If all firms but one exit a market, the behavior of the remaining monopolist is described by Section 2.

The more interesting case is the one where firms must charge the same price to disabled and non-disabled customers. To answer this question, we modify the model from Section 2.1 so that our firm can sell $(1-\mu)Q(p;y)$ units of the good to non-disabled customers where $y$ is the price set by a competitor. It will not matter how the competitor sets prices, but you can think of them as having the same demand. If we suppose the competitor does not accommodate, an accommodating firm can sell $\mu Q(p;\infty)$ units of the good to disabled customers. Then, the firm accommodates if and only if

\max_p{(p-c) (1-\mu) Q(p;y)} + K \leq \max_p{(p-c) (1-\mu) Q(p;y) + (p-c)\mu Q(p;\infty)}.

As in Section 3.1.2, if the firm accommodates when they cannot set separate prices, they would also accommodate if they could set separate prices because by the triangle inequality,

\max_p{(p-c) (1-\mu) Q(p;y) + (p-c)\mu Q(p;\infty)} \leq \left[ \max_p{(p-c) (1-\mu) Q(p;y)} \right] + \left[ \max_r{(r-c) \mu Q(r;\infty)} \right].

We already established that when disabled and non-disabled customers have the same demand and the firm can set separate prices, competition for non-disabled customers has no effect on the firm’s decision to accommodate. Adding the restriction that prices must be the same can only reduce the firm’s profit from accommodating and thus its incentive to accommodate.

However, if the disabled and non-disabled customers have different demands, it’s possible for competition in the market for disabled customers to increase the firm’s incentive to accommodate. For example, if disabled users have less demand for the good, the increased market power for disabled customers can help to offset this and result in an increased incentive to accommodate.^[6]

Disclaimer

The views expressed in this article are those of the author and do not necessarily reflect those of the Federal Trade Commission or any individual Commissioner.

This paper is Ameri et al. 2020 which is a Management paper demonstrating that Airbnb hosts are less likely to approve requests from guests with certain disabilities. ↩︎
These two conditions are sufficient, but they are not necessary. There are circumstances where an inexpensive accommodation for a very common disability may not be provided. E.g., if the disabled population is so low-income that they are not valuable customers. ↩︎
The legal requirements are interpreted differently in different circuit courts (see this blog post for more information). ↩︎
We could also consider per-unit costs, but they do not have much effect. In standard settings, per-unit costs alone cannot explain the lack of accommodations. ↩︎
See Cornell’s 2023 Annual Disability Status Reports which are based on the American Community Survey for more information. ↩︎
For example, consider the case where $Z(p;\infty)=Q(p;y)$ . Then, you can use the triangle inequality to show that if the firm has an incentive to accommodate without a competitor in the non-disabled market, then they have an incentive to accommodate with a competitor as well. ↩︎