There is a test for `smarts’ that Sir Peter Medawar was fond of. I often think of it when teaching equilibrium.

If you have ever seen an El Greco, you will notice that the figures and faces are excessively elongated. Here is an example.El_Greco_-_Portrait_of_a_Man_-_WGA10554The eye surgeon Patrick Trevor-Roper, brother to the historian Hugh offered an explanation. Readers of  certain vintage will recall the long running feud between Hugh Trevor-Roper and Evelyn Waugh. Waugh said that the best thing Hugh Trevor-Roper could do would be to change his name and leave Oxford for Cambridge. Hugh Trevor-Roper eventually  became Lord Dacre and left Oxford for Cambridge. But, I digress.

Returning to Patrick, he suggested that El Greco had a form of astigmatism, which distorted his vision and led to elongated images forming on his retina. Medawar’s question was simple: was Patrick Trevor-Roper correct?

The various US intelligence agencies have identified three ways in which the Russian state meddled with the recent US elections:

  1. Intrusions into voter registration systems.
  2. Cyberattack on then DNC and subsequent release of hacked material.
  3. Deployment of `fake’ news and internet trolls.

The first two items on this list are illegal. If a PAC or US (or green card holder) Plutocrat had deployed their respective resources on the third item on this list, it would be perfectly legal. While one should expect the Russian’s to continue with item 3 for the next election, so will each of the main political parties.

Why is `fake’ news influential? Shouldn’t information from a source with unknown and uncertain quality be treated like a lemon? For example, it is impossible for a user to distinguish between a twitter account associated with a real human from a bot. Nor can a user tell whether individual twitter yawps are independent or correlated.

Perhaps it depends on the distinction between information used to make a decision like which restaurant to go to and that which is for consumpiton value only (gossip). There appears to be no fake news crisis in restaurant reviews. There could be a number of reasons for this. The presence of non-crowd sourced reviews, the relatively low cost of experimentation coupled with frequent repetition and the fact that my decision to go to a restaurant does not compel you to do so comes to mind.

Political communication seems to be different, closer to entertainment than informing decision making.  If I consume political news that coincide with my partisan leanings because these enteratin me the most, it means that the news did not persuade me to lean that way (it follows that surpressing fake news should not change the distribution of political preferences). So, such news must serve another purpose, perhaps it increases turnout. If so, we should expect the DNC to be much more active in the deployment of `fake’ news and an increase in turnout.


In a CS paper, it is common to refer to prior work like [1] and [42] rather than Brown & Bunter (1923) or Nonesuch (2001). It is a convention I have followed in my papers with CS colleagues. Upon reflection, I find it irritating and mean spirited.

  1. No useful information is conveyed by the string of numbers masquerading as references beyond the statement: `authors think there are X relevant references.’
  2. A referee wishing to check if the authors are aware of relevant work must scroll or leaf to the end of the paper to verify this.
  3. The casual reader cannot be surprised by some new and relevant reference unless they scroll or leaf to the end of the paper to verify this.
  4. Citations are part of the currency (or drug) we live by. Why be parsimonious in acknowledging the contributions of A. N. Other? It shows a want of fellow feeling.

I suspect that the convention is an artifact of the page limits on conference proceedings. A constraint that seems quaint. Some journals, the JCSS for example, follows the odd convention of referring to earlier work as Bede [22]! But which paper by the venerable and prolific Bede does the author have in mind?


When discussing the allocation of indivisible objects, I point to randomization as a method. To emphasize it is not merely a theoretical nicety, but is used to allocate objects that are of high value I give the example of suicide lotteries. I first came across them in Marcus Clarke’s `For The Term of His Natural Life’. It counts as the first Australian novel. Its hero, an Englishman, Rufus Dawes is transported to Australia for a crime he did not commit. In the course of his tribulations, Dawes is shipped to a penal settlement on Norfolk Island, 800 miles east of Australia; even a prison needs a further prison. Robert Hughes, in his heart rending and eloquent account of the founding of Australia, called the `Fatal Shore’, describes Norfolk island in these words:

`….a place of breathtaking barbarity……. On Norfolk Island an Irishman named William Riley received 100 lashes for ”Singing a Song” (no doubt a rebel one) and 50 for asking a warder for a chew of tobacco. Deranged by cruelty and misery, some men would opt for a lifetime at the bottom of the carceral heap by blinding themselves; thus, they reasoned, they would be left alone.’

It is in this portion of his book, that Hughes recalls an eyewitness account of a suicide lottery of the type mentioned in Clarke’s novel. Here is Clarke’s succinct description of it:

The scheme of escape hit upon by the convict intellect was simply this. Three men being together, lots were drawn to determine whom should be murdered. The drawer of the longest straw was the “lucky” man. He was killed. The drawer of the next longest straw was the murderer. He was hanged. The unlucky one was the witness. He had, of course, an excellent chance of being hung also, but his doom was not so certain, and he therefore looked upon himself as unfortunate.

Clarke and Hughes deviate slightly upon the precise incentives that would drive participation in the scheme. As many of the convicts on Norfolk island were Irish, the scheme was concocted as a way to to circumvent the Catholic prohibition on suicide. Hughes suggests that, after the murder, witness and culprit would be shipped back to the mainland for trial. Conditions there were better, so for both there was brief respite and a greater opportunity for escape.

Its an arresting story, that one is loath to give up. But, one is compelled to ask, is it true? If yes, was it common? Tim Causer of King’s College London went back to look at the records and says the answers are `maybe’ and `no’. Here is his summing up:

`Capital offences committed with apparent suicidal intent are an important part of Norfolk Island’s history, but they need to be understood more fully. It should be recognised just how rare they were, that ‘suicide lotteries’ are embellishments upon actual cases of state-assisted suicide and repeating the myth only reinforces the sensationalised interpretation of Norfolk Island’s history.’

You can find the full account here.

Many people say (actually, just one) that the Republican’s have a plan to remove Trump from the Presidency, should he win in November using the 25th amendment. Section 4 of the amendment reads:

`Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.’

The VP is Pence. The President pro tempore of the Senate, is the senior senator of the majority party and Paul Ryan is the Speaker of the House.
The President can object. At which point, Congress resolves the matter, specifically,

`….two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.’


It is not often that Terry Tao gets into politics in his blog, but, as political observers like to say, normal rules don’t apply this year. Tao writes that many of Trump’s supporters secretly believe that he is not even remotedly qualified for the presidency, but they continue to entertain this possibility because their fellow citizens and the media and politicians seem to be doing so. He suggests that more people should come out and reveal their secret beliefs.

I generally agree with Tao’ sentiment and argument, but I have a quibble. Tao describe the current situation as mutual knowledge without common knowledge. This, I think, is wrong. To get politics out of the way, let me explain my position using a similar situation which Tao also mentions: The Emperor’s new clothes. I have already come across people casting the Emperor’s story in terms of mutual knowledge without common knowledge, and I think it is also wrong. The way I understand the story, before the kid shouts, each of the Emperor’s subjects sees that the Emperor is naked, but after observing everybody else’s reaction, each subject updates her own initial belief and deduces that she was probably wrong. The subjects now don’t think that the Emperor is naked. Rather, each subjects thinks that her own eyes deceived her.

But when game theorists and logicians say that an assertion is mutual knowledge (or mutual belief) we mean that each of us, after taking into account our own information including what we deduce about other people’s information, think the assertion is true. In my reading of the Emperor’s new cloths story this is not the case.

For an assertion to be common knowledge, we need in addition that everybody knows that everybody knows that the assertion is true, and that everybody knows that everybody knows that everybody knows that the assertion is true, and onwards to infinity. A good example of a situation with mutual knowledge and no common knowledge is the blue-eyed islanders puzzle (using the story as it appears Terrence’ blog and a big spoiler ahead if you are not familiar with the puzzle): Before the foreigner makes an announcement, it is mutual knowledge that there are at least 99 blue-eyed islanders, but this fact is not common knowledge: If Alice and Bob are both blue-eyed then Alice, not knowing the color of her own eyes, thinks that Bob might observe only 98 blue-eyed islanders. In fact it is not even common knowledge that there are at least 98 blue-eyed Islanders, because Alice thinks that Bob might think that Craig might only observe 97 blue-eyed Islanders. By similar reasoning, before the foreigner’s announcement, it is not even common knowledge that there is at least one blue-eyed islander. Once the foreigner announces it, this fact becomes common knowledge.

No mutual knowledge and no common knowledge are two situations that can have different behavioral implications. Suppose that we offer each of the subjects the following private voting game: Is the emperor wearing clothes ? You have to answer yes or no. If you answer correctly you get a free ice cream sandwich, otherwise you get nothing. According to my reading of the story they will all give the wrong answer, and get nothing. On the other hand, suppose you offer a similar game to the islanders — even before the foreigner arrives — Do you think that there is at least one blue-eyed islander ?  they will answer correctly.

There is an alternative reading of the Emperor’s story, according to which it is indeed a story about mutual knowledge without common knowledge: Even after observing the crowd’s reaction, each subject still knows that the Emperor is naked, but she keeps her mouth shut because she suspects that her fellow subjects don’t realize it and she doesn’t want to make a fool of herself. This reading strikes me as less psychologically interesting, but, more importantly, if that’s how you understand the story then there is nothing to worry about. All the subjects will vote correctly anyway and get the ice cream even without the little kid making it a common knowledge. And Trump will not be elected president even if people continue to keep their mouth shut.

When explaining the meaning of a confidence interval don’t say “the probability that the parameter is in the interval is 0.95” because probability is a precious concept and this statement does not match the meaning of this term. Instead, say “We are 95% confident that the parameter is in the interval”.  Admittedly, I don’t know what people will make of the word “confident”. But I also don’t know what they will make of the word “probability”

If you live under the impression that in order to publish an empirical paper you must include the sentence “this holds with p-value x” for some number x<0.05 in your paper, here is a surprising bit of news for you: The editors of Basic and Applied Social Psychology have banned p-value from their journal, along with confidence intervals. In fact, according to the editorial, the state of the art of statistics “remains uncertain” so statistical inference is no longer welcome in their journal.

When I came across this editorial I was dumbfounded by the arrogance of the editors, who seem to know about statistics as much as I know about social psychology. But I haven’t heard about this journal until yesterday, and if I did I am pretty sure I wouldn’t believe anything they publish, p-value or no p-value. So I don’t have the right to complain here.

Here are somebodies who have the right to complain: The American Statistical Association. Concerned with the misuse, mistrust and misunderstanding of the p-value, ASA has recently issued a policy statement on p- values and statistical significance, intended for researchers who are not statisticians.

How do you explain p-value to practitioners who don’t care about things like Neyman-Pearson Lemma, independence and UMP tests ? First, you use language that obscures conceptual difficulties: “the probability that a statistical summary of the data would be equal to or more extreme than its observed value’’ — without saying what “more extreme’’ means. Second, you use warnings and slogans about what p-value doesn’t mean or can’t do, like “p-value does not measure the size of an effect or the importance of a result.’’

Among these slogans my favorite is

P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone

What’s cute about this statement is that it assumes that everybody understands what “there is 5% chance that the studied hypothesis is true” and that the notion of P-value is the one that is difficult to understand. In fact, the opposite is true.

Probability is conceptually tricky. It’s meaning is somewhat clear in a situation of a repeated experiment: I more or less understand what it means that a coin has 50% chance to land on Heads. (Yes. Only more or less). But without going full subjective I have no idea what is the meaning of the probability that a given hypothesis (boys who eat pickles in kindergarten have higher SAT score than girls who play firefighters) is true. On the other hand, The meaning of the corresponding P-value relies only on the conceptually simpler notion of probabilities in a repeated experiment.

Why therefore do the committee members (rightly !) assume that people are comfortable with the difficult concept of probability that an hypothesis is true and are uncomfortable with the easy concept of p-value ? I think the reason is that unlike the word “p-value”, the word “probability” is a word that we use in everyday life, so most people feel they know what it means. Since they have never thought about it formally, they are not aware that they actually don’t.

So here is a modest proposal for preventing the misuse and misunderstanding of statistical inference: Instead of saying “this hypothesis holds with p-value 0.03” say “We are 97% confident that this hypothesis holds”. We all  know what “confident” means right ?

Platooning, driverless cars and ride hailing services have all been suggested as ways to reduce congestion. In this post I want to examine the use of coordination via ride hailing services as a way to reduce congestion. Assume that large numbers of riders decide to rely on ride hailing services. Because the services use Google Maps or Waze for route selection, it would be possible to coordinate their choices to reduce congestion.

To think thorough the implications of this, its useful to revisit an example of Arthur Pigou. There is a measure 1 of travelers all of whom wish to leave the same origin ({s}) for the same destination ({t}). There are two possible paths from {s} to {t}. The `top’ one has a travel time of 1 unit independent of the measure of travelers who use it. The `bottom’ one has a travel time that grows linearly with the measure of travelers who employ it. Thus, if fraction {x} of travelers take the bottom path, each incurs a travel time of {x} units.

A central planner, say, Uber, interested in minimizing total travel time will route half of all travelers through the top and the remainder through the bottom. Total travel time will be {0.5 \times 1 + 0.5 \times 0.5 = 0.75}. The only Nash equilibrium of the path selection game is for all travelers to choose the bottom path yielding a total travel time of {1}. Thus, if the only choice is to delegate my route selection to Uber or make it myself, there is no equilibrium where all travelers delegate to Uber.

Now suppose, there are two competing ride hailing services. Assume fraction {\alpha} of travelers are signed up with Uber and fraction {1-\alpha} are signed up with Lyft. To avoid annoying corner cases, {\alpha \in [1/3, 2/3]}. Each firm routes its users so as to minimize the total travel time that their users incur. Uber will choose fraction {\lambda_1} of its subscribers to use the top path and the remaining fraction will use the bottom path. Lyft will choose a fraction {\lambda_2} of its subscribers to use the top path and the remaining fraction will use the bottom path.

A straight forward calculation reveals that the only Nash equilibrium of the Uber vs. Lyft game is {\lambda_1 = 1 - \frac{1}{3 \alpha}} and {\lambda_2 = 1 - \frac{1}{3(1-\alpha)}}. An interesting case is when {\alpha = 2/3}, i.e., Uber has a dominant market share. In this case {\lambda_2 = 0}, i.e., Lyft sends none of its users through the top path. Uber on the hand will send half its users via the top and the remainder by the bottom path. Assuming Uber randomly assigns its users to top and bottom with equal probability, the average travel time for a Uber user will be

\displaystyle 0.5 \times 1 + 0.5 \times [0.5 \times (2/3) + 1/3] = 5/6.

The travel time for a Lyft user will be

\displaystyle [0.5 \times (2/3) + 1/3] = 2/3.

Total travel time will be {7/9}, less than in the Nash equilibrium outcome. However, Lyft would offer travelers a lower travel time than Uber. This is because, Uber which has the bulk of travelers, must use the top path to reduce total travel times. If this were the case, travelers would switch from Uber to Lyft. This conclusion ignores prices, which at present are not part of the model.

Suppose we include prices and assume that travelers now evaluate a ride hailing service based on delivered price, that is price plus travel time. Thus, we are assuming that all travelers value time at $1 a unit of time. The volume of customers served by Uber and Lyft is no longer fixed and they will focus on minimizing average travel time per customer. A plausible guess is that there will be an equal price equilibrium where travelers divide evenly between the two services, i.e., {\alpha = 0.5}. Each service will route {1/3} of its customers through the top and the remainder through the bottom. Average travel time per customer will be {5/3}. However, total travel time on the bottom will be {2/3}, giving every customer an incentive to opt out and drive their own car on the bottom path.

What this simple minded analysis highlights is that the benefits of coordination may be hard to achieve if travelers can opt out and drive themselves. To minimize congestion, the ride hailing services must limit traffic on the bottom path. This is the one that is congestible. However, doing so makes its attractive in terms of travel time encouraging travelers to opt out.

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