Algorithmic Homogenization

At a recent Algorithmic Fairness meeting, there was some discussion of algorithmic homogenization. The concern, as expressed, for example in Kleinberg and Raghavan (2021) is that
“the quality of decisions may decrease when multiple firms use the same algorithm. Thus, the introduction of a more accurate algorithm may decrease social welfare—a kind of “Braess’ paradox” for algorithmic decision-making”.

Now, no new model is needed to exhibit such a paradox for algorithmic decision making. The prisoner’s dilemma will do the job for us. Consider the instance of the dilemma in the table below.

	C	D
C	(1,1)	(-1,5)
D	(5,-1)	(0,0)

Here are two algorithms that our players can choose from to play the game. The first is a silly algorithm. It selects a strategy uniformly at random from among those available. The second is a `better’ algorithm. Its selects a strategy uniformly at random among those are rationalizable (meaning they are a best response to some mixed strategy of the opponent). Why is the second better? Holding the strategy of the opponent fixed, the second will deliver a higher expected payoff than the first. This is because strategy D, defect, is the only rationalizable strategy in the prisoner’s dilemma.

If both players use the inferior algorithm, total expected payoff will be 5/4. If both players use the better algorithm, total expected payoff will be 0. Thus social welfare is lower when both players switch to the better algorithm. Why doesn’t each player stick with the silly algorithm? If I know my rival is playing the silly algorithm, I am better off from switching to the better algorithm.

While this example makes the point, it does so unconvincingly because it is not tied to a compelling context. KR(2021) does not share this feature because it relies on algorithmic hiring as motivation. There are two firms competing to hire individuals and they may deploy algorithms to screen candidates. Unlike the example above, the algorithm does not choose each firm’s actions but provides information only about candidate quality. In other words, the algorithm makes predictions not decisions.

The idea that better information in a competitive context can make the players worse off is not a new one. Nevertheless, it is always useful to understand the precise mechanism by which this plays out in different contexts. The same is true in this case and I direct the reader to the KR(2021), but I would be remiss in not also mentioning this closely related paper by Immorlica et. al. (2011). As an aside, there is an obvious connection between homogenization and algorithmic price fixing (see Greenwald and Kephart (1999)) that is a subject for a future post.

Next, I consider a feature absent in KR(2021), I think critical. Wages. To see their presence will make a difference in how we view algorithmic homogenization, suppose two firms competing for a worker. The worker’s type is their productivity, denoted . If a firm hires the worker for a wage of they earn a profit of . Neither firm knows the worker’s type, but each receives a conditionally independent noisy signal of their type. We can think of the signal as being delivered by some mythical algorithm. Conditional independence is to be interpreted as the firms using different algorithms. Upon receiving their respective signals, each firm submits to the worker a take-it-or leave-it wage offer. The worker will select the firm that offers the highest wage. What’s just been described is a sealed bid first price auction in a common values setting. In equilibrium each firm will submit a wage that is below their estimate of worker productivity conditional on their signal because of the winner’s curse effect. On the other hand, suppose both firms use the same algorithm to estimate worker productivity that is error free, i.e., they receive perfect information about the worker’s productivity. Now, we have Bertrand competition and each firm offers a wage of $\latex t$. In this toy model, algorithmic monoculture does not affect efficiency, but it does affect the distribution of surplus. In particular, the worker benefits from homogenization! For a privacy slant with the same set up see Ali et. al (2023).

Irrespective of whether algorithmic homogenization will improve or decrease efficiency, we might still be worried about because of the possibility of systematic errors. In the meeting referenced earlier, Ashia Wilson asked her audience to imagine what might happen if a company like HireVue, for example, were to occupy a dominant position in the recruitment market (this would allow HireVue to effect wages, but that is a different story). HireVue and companies like it, claim that they provide accurate signals about a job candidates fit (there is a discussion to be had about whether this what they should be doing) with a given employer rapidly and at scale. Suppose the prediction algorithm that HireVue uses is indeed more accurate (in aggregate) than the alternative, but exhibits systematic errors for some well defined subset of job seekers. For example, it consistently underestimates the quality of fit for candidates whose last names have to many consonants (or over estimates this)? Depending on the nature of the alternative, this might be a concern. Does this call for an intervention or regulation? Why is it not in HireVue’s interest to discover such errors and correct them? If a company like Hirevue is rewarded per person placed, then, such a systematic error would lower their placement rate and thus their revenues.

I

Sydney N. Afriat, 1925-2021

Sydney Afriat passed away on December 31, 2021. I learnt this in the course of a hunt for one of his papers (more on this later). Unsurprising given his vintage, but disconcerting that I could recall no notice or announcement of this event. Not even Google’s knowledge panel for Afriat records his death. Eventually, I found a one paragraph bulletin at the University of Ottawa and the Econometric Society web site lists him in the deceased section of the Society’s fellows. For someone who was once described as belonging “to that select group of economic theorists who have become a legend in their own time”, it seems a shame.

Sydney N. Afriat was born in 1925 in Mogador (now known as Essaouira) in Morocco to a merchant family. One might conclude that would make Afriat francophone, but, no. There was a British presence in Morocco until the 1912 Treaty of Algeceiras when control passed to the French in return for Egypt passing to the British. Afriat’s grandmother, though Maghrebi, had been born in England and had settled in Mogador upon marriage. Thus Afriat, was sent to boarding school in England and spent his holidays in Mogador. The family was well to do, as evinced by the fact that they could afford a middle name for Afriat. The `N’ stood for Naftali, after a rabbinical martyr of the 18th century. Sydney was an anglicaztion of Sellam, first adopted by Afriat’s great uncle. For more on Afriat’s background see his fragmentary autobiography.

Afriat went up to Pembroke College, Cambridge, to read Mathematics in 1943, graduating in 1949. The six year duration was because of war time service in the Aerodynamics Division of the National Physical Laboratory at Teddington. Subsequently he obtained a D. Phil in Mathematics at Oxford. This was followed by a peripatetic professional life spent at Cambridge, Jerusalem, Princeton, Rice, Yale, Purdue, UNC- Chapel Hill, Waterloo, Ottawa, Bilkent before running aground at the University of Siena. The memorial note at the University of Ottawa offers the following oblique explanation:

“Sydney Afriat had a difficult character. His relationships with his colleagues were sometimes complicated, but the value of his scientific contributions was recognized by all.”

I reproduce one anecdote, first related elsewhere on this blog:

“Sydney Afriat arrived in Purdue in the late 60’s with a Bentley in tow. Mort Kamien described him as having walked out of the pages of an Ian Flemming novel. Why he brought the Bentley was a puzzle, as there were no qualified mechanics as far as the eye could see. In Indiana, that is a long way. Afriat would take his Bentley on long drives only to be interrupted by mechanical difficulties that necessitated the Bentley being towed to wait for parts or specialized help.”

That same post has a discussion of his Theorem.

Now to the paper I was hunting for. Its an investigation of the properties of a system of inequalities of the form . Inequalities of this form arise in a variety of places. They can be interpreted as the dual to the problem of finding a shortest path in a network. Their feasibility is related to Rockafellar’s cyclic monotonicity condition. They arise in revealed preference and mechanism design. All the results one needs for the various applications appear in that paper.

Robust Contracting & Linear Programming

Recently Ashwin Kambhampati, Juuso Toikka and myself were engaged on a project that grew out of Carroll’s robust contracting paper . A byproduct is a new proof of Carroll’s main result which is reproduced here. It shows how his main result is a consequence of linear programming duality. 

Let be the set of possible states and be the output in state . Without loss we may assume that . An action for an agent is a pair where is a probability distribution over and a cost. Denote by the set of known actions of which a typical element is written as  

 Let be the contract that the principal chooses. Each component of in represents the share of output that goes to the agent. Actually, the requirement that is not needed. The expected payoff to the agent under contract from choosing the known action is .  Let . Thus, represents the agent’s expected payoff from choosing an action in and denote by $t^* \in A_0$ the index of the optimal known action.

 Nature’s goal is to choose an action to offer the agent so as to minimize the principal’s payoff.  Nature’s choice can be formulated as the following linear program:

subject to

This LP is feasible because Nature can always choose an action in . 

First, we analyze Nature’s problem under a linear contract which promises share in each state of the world to the agent. Let 

 

 Then, Nature’s optimization problem is

subject to

A contract is called admissible if  

Lemma: If is an admissible contract, then,  

Proof: The dual to the linear contract problem

subject to

Given it follows that .

An optimal  solution must satisfy one of the following:

1)

2)

3)

4)

#1 and 2 are ruled out by admissibility. In #3

In #4, there must be at least two binding constraints:

Subtracting one from the other yields and therefore, , which puts us back in #3. QED

Now, let us turn to the general contract . The dual to Nature’s optimization problem problem  is (call it dual2):

subject to

Theorem: If and then,

Proof: Let be an optimal solution to (dual2). As in the linear case,  

Multiply each constraint in (dual2)  by and add them up. This yields the following:

Divide through by :

Keeping in mind that , it follows that

QED

Updating Polya on Style

From time to time, I return to Polya’s `How to Solve It‘ for advice to give my students. In spite of the passage of time it remains as trenchant and as useful as ever. One thing, however, I would amend (not emend); his second of three rules of style:

The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.

My proposal:

The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then stop.

My colleague Santosh Venkatesh (whose Probability Theory book has displaced Feller on my shelf) is more cynical:

The second rule of style is to control yourself when, by chance, you have two things to say; say nothing, nobody wants to hear what you have to say.

L’Affaire Gino

Like Tilman Borgers I believe that all behavioral economics and social psychology books should be housed in the self-help section of the bookstore. Indeed, Tilman tells me, that when bookstores existed he made it a point to move such books out of the Economics section and place them in the section they properly belonged to. Unsurprisingly, I enjoy tales of behavioral scientists behaving badly (and turn a blind eye on my own tribe). This post is prompted by one such instance.

In August of 2012, PNAS published a paper by Shu, Mazar, Gino, Ariely and Bazerman. It was an amalgamation of two independent projects: a field experiment and a laboratory study. A portion of the abstract, reproduced below, summarizes the paper’s motivation and point:

“Many written forms required by businesses and governments rely on honest reporting. Proof of honest intent is typically provided through signature at the end of, e.g., tax returns or insurance policy forms. ………..Using laboratory and field experiments, we find that signing before—rather than after—the opportunity to cheat makes ethics salient when they are needed most and significantly reduces dishonesty.”

Questions about the veracity of the data in the field experiment portion of the paper were raised two years ago in this post. The paper’s authors accepted that the data were indeed suspect and the paper retracted. However, Ariely, the co-author of the paper responsible for acquiring the data of the field experiment has yet to provide a satisfactory account of how the doctored data came into existence.

Recently, this post raised questions about the data reported from the lab experiment which was supplied by Gino. That post goes on to identify suspect data in other papers to which Gino contributed data. The Chronicle of Higher Education summarizes the `state of play’ as of June 17th of this year.  Andrew Gelman offers a meditation on  mendacity in academe in the same outlet.

The discussion prompted by these events has focused on fraud and how to eradicate it.  Uri Simonsohn, Joe Simmons, and Leif Nelson of the blog post mentioned earlier frame it this way:

“Addressing the problem of scientific fraud should not be left to a few anonymous (and fed up and frightened) whistleblowers and some (fed up and frightened) bloggers to root out. The consequences of fraud are experienced collectively, so eliminating it should be a collective endeavor.”

I believe there is another issue, separate from fraud, that these events highlight. I will argue that even if the paper were fraudulent, ex-post, it was not worth ferreting out. Second, absent fraud, I argue that the significance of the paper was overestimated (the paper’s authors described it as landmark in their replication study).

The PNAS paper argues that when you sign makes a difference and this choice could have consequential implications. How one evaluates this claim depends upon one’s prior. First possibility is that one’s prior is that when one signs makes no difference whatsoever. 

Under this prior, any compelling evidence that supports a difference between signing at the start or at the end will cause one to question one’s prior. Note, the direction of the difference does not matter as long as there is one. I suspect, though, that had the authors concluded that signing at the end was better than signing at the start, the paper would have been ignored. Given the stated prior, this would be incorrect and therefore, reveals something faulty in the way research is assessed.

Now, suppose one were to engage in fraud to establish a difference. Why should we care? If no one acts upon the finding, it was irrelevant. If many do act upon it, recall one’s prior, no harm is done.  In short, given the prior, the outcome of a fraudulent study is inconsequential.  The puzzle, then, is why, ex-post, one would allocate scarce resources to ferreting out this particular fraud. 

There are other costs of fraudulent activity. For example, the resources (both monetary and attention) that were diverted to the authors of the study. It is unjust and other worthy projects may have been deprived of those same resources. I argue, given the stated prior, absent fraud, the paper should not have garnered the attention or approbation that it did because it was incomplete. If one rejects the  assumption that when one signs makes no difference, it raises not one alternative, but many. Signing before completing the form, midway through, three-quarters of the way through etc etc.  With all these possibilities, by luck alone, one might conclude that signing at some place other than at the end of the form makes a difference. The paper’s authors did not report on these possibilities, which, in my judgement, makes the paper incomplete. 

What if one had started with a different prior? That is, when one signs makes a difference. Under this prior, concluding that signing at the start as opposed to the end makes a difference would only confirm what we know. The only relevant question would be which of the many places one could sign that would have maximum impact. This is not the question investigated by the authors.

The Irrelevance of Automated Bidding

Around the mid 2010’s Google introduced automated bidding. Other platforms have followed suit.

Rather than bidding directly for an `eyeball’, an advertiser delegates the bidding to the platform. In order to inform the bids that the platform will submit on their behalf, the advertiser submits two numbers to the platform. One is their budget and the second is their ROI target which can be thought of as  . Hence, the ROI is the inverse of the cost per click.

Some observers have remarked that auto-bidding is strange because one asks the auctioneer themselves to bid on one’s behalf. Others have been inspired to focus on the design of auctions when bidders have an ROI constraint. This, I think, is misguided. 

First, just because the auctioneer’s chosen bidding language uses an ROI target does not mean that a ROI constraint enters bidder’s preferences. One should never confuse the message space of a mechanism with the preferences of the agents. 

Second, once a bidder has submitted a budget and an ROI target, the subsequent auction is an irrelevance. Why? Suppose I submit a budget of . Then, my ROI target, says that the platform must deliver  clicks. For example, at a budget of $100 and an ROI target of 2, I am telling the platform that I will give them $100 in return for 200 clicks. Now, the platform, has access not to a finite number of clicks but a flow. They can, given time, satisfy every bid. In short, the platform will get your $100. The only issue is when. The automated auction is merely an elaborate device for determining the rate at which different bidders receive a click. One can think of far simpler procedures to do this. For example, round robin or deplete budgets at a uniform rate.

What I Learnt from Chat GPT

While my colleagues have been using Chat GPT to determine what it knows about important things, such as sunk costs and elasticity of demand, I was curious to learn what it knew about me. Here is a snippet:

“He is currently a Professor of Economics at the University of Pennsylvania, where he has been a faculty member since 2018. Prior to that, he was a Professor of Economics at the University of Chicago Booth School of Business, where he also served as the Dean from 2012 to 2019.”

I have, alas, never been a Professor of Economics at Booth, nor have I served as dean either at Booth or anywhere else. One might be tempted to dismiss this as another example of how GPT gets it wrong. However, if one realizes that GPT works by associating some block of text with another, it tells me, that given the keywords associated with my name, GPT predicts I should have been a Dean, at Booth, anyway. And not for one term, which is typically 5 years but 7 years. Did some great scandal bring my office to an end? Or, was I tempted away by an even higher office? We will never know. However, headhunters everywhere, take note, that powerful AI thinks I am dean material!

The Top 7

At dinner, some weeks ago, among my companions, was a cardinal of the profession. The cardinal lamented the fixation on the top 5. This cardinal, rebelled against it by declining to use the term in letters of reference, promotion and tenure. The cardinal urged a focus on the content of the work rather than the ranking of the outlet. While sympathetic to the sentiment, I think the proposal mistaken in some contexts. Specifically, for promotion and tenure letters, one is writing for `outsiders’. The intended audience is not part of the specialty. The letter serves as a marker of credibility for the outsider. Declining to rate the outlets diminishes the credibility of a supportive letter. I advocate, replacing the offending term by `top 7′. For example, `The candidate has three publications in the top 7 outlets of the profession’. Just as one never needs to say which are the top 5 journals, there is no reason to say what the top 7 are. Those who know, know. Those who don’t, are clearly uninformed

A Modest Proposal to Improve Basketball

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

Top 5 Tyranny and its Consequences

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

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

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

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

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