Dutch Flower Auctions

If you have ever learned about auctions in a class, you will have heard about both "English" and "Dutch" auctions. The idea of an English auction is that you start at a low price and the bids keep increasing until the highest bid wins. The Dutch auction on the other hand starts at a high price which keeps dropping until one of the bidders says they are willing to pay the current price.

When you first hear about this auction, it is not clear why anyone would prefer the "Dutch" descending price auction, even if it officially gets the same expected revenue to the seller. But after finally seeing a video of how these auctions are run, it makes a lot more sense to me. My intuitive understanding now is that if you want to run a lot of auctions quickly, this is somehow simpler than it would be to run the auctions in an ascending-price way. (Something this video misses is that after the buyer wins, they submit how many flowers they want and the remainder get re-clocked. So that's why the clock keeps re-starting around 20 cents)

One closing thought: This NYT story suggests that the days of the clock auction may be numbered as buyers get larger and more of the time work directly with the sellers. I wonder what textbook authors will use to talk about the revenue-equivalence theorem then...

Are feelings additive?

Do you know the real definition of ambivalent? From Merriam-Webster:

1. simultaneous and contradictory attitudes or feelings (as attraction and repulsion) toward an object, person, or action

A lot of people interpret ambivalence as having no opinion at all. Based on this older story from NPR about the Ebola outbreak, maybe there is a reason for that. They describe a study where participants were asked to donate to charity to help a starving child, but were also provided information about varying numbers (0, 1, 6) of children who they would not be helping. In the questionnaire, they asked participants to rate the "warm glow" they got from thinking about helping the starving child, and found that how happy they felt was reduced when they also were told about children they weren't helping.

One particularly interesting comment in the study itself is that they tried to figure out if participants attempted to be internally consistent. Usually when you participate in a study if you realize you are being asked the same question a different way, you try to answer the same way you did before. However, this study did not find that people tried to correct their answers. I want to leave you with the closing question of the NPR story, "what should charities do given this understanding?"

The Toilet Paper Problem

During my first year in grad school I came across this paper on "The Toilet Paper Problem" which divides the population into "little choosers" and "big choosers." In trying to find it again to share with all of you, I also came across an essay which is perhaps more useful: "Toilet Paper Algorithms."

Math, answering the important questions in life.

Everything is a trade-off

I recently attended a project management workshop run by an Ann Arbor company (Menlo Innovations). While their company does software development, the techniques we learned are useful pretty much any time you can't do everything you need to, by yourself, in a day. The main takeaway was that by doing any particular thing, you are implicitly choosing not to do other things. Instead, you should try to make those choices explicit.

The tool he gave us for this purpose was to write every possible task on index cards along with estimated time to complete them. The idea is that now you have a physical representation of the trade-offs. You can choose card A which takes one week, or you can choose cards B and C which also add up to one week. He also talked a bit about what our brains are naturally good at:

• Making an initial plan is hard. Poking holes in a plan is easy. Therefore, you should do whatever you can to not work with a blank page.
• If you scale the size of the card to the length of the task (i.e., set your scale as one card = one day of work, so a task that takes half a day you use half a card for), our brains intuitively do the trade-offs.
• Taking away the technology leaves our brains free to just focus on the other aspects. There are definitely advantages to using technology some of the time. But most of the difficulty of project management is these trade-offs, not software giving you a better estimate of how long a particular task will take. Pretty much any estimate will be bad, but some estimates will be useful.

In implementing this approach for my thesis, I am finding that the most critical thing to be working on in the moment is obvious to me. But before I had this system I had no good way to lay out everything left for all of my projects. It would still be nice to have twice as many hours to get everything done, but I am finding a lot of peace in seeing how my current task specifically fits together with the bigger picture.

Comparing alternatives

A few years ago I took a systems engineering course and one of the topics we covered was how to compare different options you might need to choose between. There were two tips that stuck with me, so I thought I would share them here.

1. Compare actual alternatives. When you are considering different options, it is really easy to compare one specific possibility versus the entire world of other possibilities. Do we want pizza for dinner tonight? Well, pizza is cheap and easy, but it isn't exactly healthy, and there are things that are more delicious... But how many things are both healthier and more delicious at the same time as they cost the same amount of money and time? Pizza versus spaghetti however is an actual comparison.
2. Include an evolutionary alternative in addition to revolutionary alternatives. An evolutionary alternative involves doing more or less what you have been doing, but perhaps better. A revolutionary alternative involves a completely new way of accomplishing your goal. The idea being that typically revolutionary ideas will be a lot more costly to implement, and it is easy to get carried away with planning something totally new and lose sight of what you are really trying to accomplish. One example is the repair or replace decision for a car.

Do you have any other tricks for choosing the best alternative?

Next Generation 911

Yesterday the Diane Rhem Show had a piece on the 9-1-1 emergency phone system which itself was motivated by the recent NYT opinion column written by the chairman of the FCC. The basic point in both stories was that our 9-1-1 system is horribly out of date, and it will take a lot of effort and money to get it up to speed.

I have known for a while that cellphones were sort of a problem for 9-1-1 since it is hard to track the location the call is coming from. I had also heard that some areas are upgrading to accept text messages, though it was not until the story yesterday that I learned at least one reason that would be desirable (to silently notify police of an intruder). The thing that I had not considered before is all the benefits we could have if 9-1-1 were fully modernized including pictures and video (send a picture of the intruder, have a paramedic be able to see the injury, probably countless other things).

One of the guests on the Diane Rhem show commented that he expects the number itself to be about the only thing that is still the same five years from now. According to David Furth from the FCC, the US system is still pretty good on a global scale, but we have a lot of work to do if we want the system to perform in the best way possible.

Stupid Users

Fundamentally, humans are error prone. I just came across the "e-counting game" where you ask a class to count the number of letter "e"s in a 400 word document and offer $20 to the first person who shouts out the correct number. The author mentions that if you're ok missing 10-25% of the "e"s, you're fine. But most times we try to have people inspect things we really want the missed number to be a lot closer to zero.

There are many ways people try to reduce the error rate. Standards (righty tighty, lefty loosey) can help some. So can visual instructions (A giant "Pull" sign on a door). But the best tools focus on how the user actually interacts with whatever it is. The Toyota Production System has "Poka-yoke" which asks you to design something that cannot be put together wrong (think of the children's shape sorter boxes). An example of good visual implications is that newer stoves often have the knobs in the same arrangement as the burners themselves.

In short, there are some tasks humans are naturally bad at. And so if you find yourself paying for something online and the website asks you to enter your account number twice, you really should actually type it out each time.

Multi-objective decision making

In optimization we like to provide The Optimal Solution and as an OR person I like to apply the same principles to my daily life. However, any time there are multiple goals, the meaning of The Optimal Solution get fuzzier. Deciding the best nut butter to buy is one example of multi-objective decision making. As a company, we can say that a particular decision is objectively better if it leads to higher profits. But once you have more than one goal (taste and cost in the nut butter example, though you could add others like effort to make your own or nutrition), things get trickier. 

The linked post tries to get at this by having a region of worthwhile choices and a region of not-worthwhile choices. That is more or less what most optimization algorithms do. Generally if you have multiple objectives, you give them all weights and choose the option that maximizes the weighted sum. However, those weights are fundamentally subjective. The Pareto frontier attempts to partially get around this by capturing all possible sets of weights and identifying the set of good solutions. But you still typically have to choose one of those options, which takes us back to a subjective decision.

Making a subjective decision is not a bad thing, and weights are a pretty good way to formalize the subjectivity. But it still helps to recognize that taste and cost aren't truly substitutes for each other.

What causes slowdowns on the interstate?

This week I attended a talk which talked about flow rate and traffic slowdowns. The speaker mentioned that when the number of cars per hour traveling on a highway reached a certain level, we find that sometimes the traffic will be able to happily keep moving at the speed limit, but usually it cannot. I found Figure 1 in this article which shows the phenomenon. A couple other things that can contribute to a traffic slowdown include:

• Drivers needing to make a decision.
• An existing slowdown. In this old link the author experimentally found that he could reduce the amount of backed up traffic by driving slower but at a constant speed.

Avoiding traffic jams is then about ensuring your traffic network has enough capacity, minimizing decision points, and helping recover from a slowdown if it happens.

How should Amazon handle bad sellers?

In the current issue of the Harvard Business Review, there is an article discussing some of the results from this working paper by Feng Zhu and Qihong Liu. They look at products on Amazon which are available only through a third-party seller. The authors try to understand how Amazon decides which new products to add. Specifically, for objects previously available from a third-party, you have a bunch of data which may be able to help distinguish one item from another.

In the past there have been two possible motivations proposed for platform owners: They might identify items which the third-party market is not handling well, or they might instead try to improve their profits as much as possible. The article makes an empirical case for the latter objective, which makes sense. The point of this post is to explore if entering markets as a customer service move also makes sense.

First we have to wonder, does the platform owner entering a market necessarily improve the customer experience? There are some cases I can think of where it would, but also cases where it would not. If the product is good, presumably the right distribution would lead to satisfied customers. In Amazon's case, they need to identify if the manufacturer (e.g., packaging, understaffed distribution office) or the third-party seller (only ships on Tuesdays) is to blame for the poor distribution. Working with the manufacturer directly could improve the customer service experience for those cases.

Bad products are a somewhat different situation. Should the platform owner allow the bad product to stay on their platform? It depends why the product is bad. Just yesterday Apple banned hundreds of apps which violated users privacy. Similarly, you can find lots of stories online of Amazon sellers being banned without recourse for alleged violations. But what about the Fizz Saver which is just a terrible product? Does it help or hurt Amazon to let this product continue to be sold on their platform?

One final question: How can we use algorithms to distinguish from a bad product and merely bad distribution? For example, in my hunt for an exceptionally bad product I found many things which got either good or bad reviews as a joke. Does the existence of these kind of products on the platform have any effect on Amazon?

Book Review: Poorly Made in China

I recently read Poorly Made in China by Paul Midler on the advice of a friend. It was selected as the best book of 2009 by The Economist, and with good reason. Not only is it informative, but also a highly entertaining read.

When this book came out, I was living in Hong Kong for a semester abroad. When I first got there, I was very reluctant to buy from the street markets. I was both learning how to negotiate prices for the first time, and having to be much more careful about quality. As the semester progressed I got better at negotiating and somewhat better at checking quality. But the key trick I learned was to buy things that were easy to assess the quality of (tea, toys with non-movable parts, etc.).

The main lesson I learned, which is reiterated in the book, is that appearance is put first in China. One of my favorite examples of this was a watch which I bought without examining it closely enough. I thought I was getting a 3-function watch (which showed the time, date, and if it was night or day) which I expected to break within a few months. Instead, I found after I bought it that only the time actually changed and it always said it was half day-half night and the 17th, and also broke within a few months. Not all of my purchases went this way (I got an excellent price on a hiking backpack which I still use from time to time), but over time I got more careful about which sorts of things I bought off the street.

Paul Midler talks about this culture as it relates to working with Chinese manufacturers making goods to be sold in the rest of the world. My strategy worked for two reasons: I was only committing to buy the object one time, and I bought things that were more robust to the culture. Importers can be strategic about which products they contract to manufacturers in China, but quality fade over time is a serious issue for any product (as is exemplified many times in the book). Hopefully the fact that more people are aware of the culture has helped companies make better sourcing decisions.

Models are not real.

My department has had two speakers this semester who said some variant of "models are not real." From my physics training, that statement is obvious. There are tons of jokes about physicists stating their ridiculous assumptions before they start to solve a problem. But the reason the assumptions seem outlandish is that they explicitly state the difference between their model and reality. As the complexity of the system grows beyond a hydrogen atom, even articulating those differences quickly becomes impossible. And that is ok. The point of a model is not to be real, it is to be useful.

So how is all this a problem? In engineering we really cannot state all the assumptions we make in our model. Instead we focus on stating assumptions we make that may be different than those other people in our field make. But if an entire field has been making the same invalid assumption, there is no obvious opportunity to identify it.

This is where empirical work and diversity can help. If we find that some observed phenomenon is not captured by our model, we can change the model. If someone who is new to the field asks questions about something we didn't even realize we were assuming, we can change the model. As long as we try to focus on what we want our model to do, we can try to include the important features of reality to get meaningful results.

Ranking for simulation

This week I visited Georgia Tech and met with several faculty there. One of those people was Seong-Hee Kim who does work on a variety of topics including ranking and selection strategies for simulation. The idea behind selection via simulation is that we have a (manageable) set of alternatives and want to choose the best option. However, identifying some options which are not likely to be the best is much easier than actually identifying the best. Therefore, the goal is to get enough information about each alternative (in her example that was a few simulated outcomes) in order to identify some bad solutions. You then keep repeating the process (more simulated outcomes, eliminate more bad solutions) until you are left with only one option.

This method seems to be similar to satisficing (mentioned here) as a decision making strategy. With satisficing, we pick any alternative which matches our criteria. Said another way, we eliminate unacceptable alternatives until anything left is acceptable. There are certainly differences in the goals between the two problems... But if you paused either algorithm in the middle, you would probably have a similar looking set of alternatives. 

Bidding strategically on eBay

A while ago I discovered that eBay makes the last three months of sold as well as completed listings available (just click on the little "Advanced" tab on the right of the search button). Learning about this feature hugely changed how I shopped on eBay. While you can do a search of the current listings to try to get a sense of your options, you can get a much more accurate picture of what the market is doing by looking at what people have historically paid. Here are a few ways I use the historical data to decide how to bid:

• If I see a "buy it now" item that is around the lowest price the auctions typically end at, I can simply use that option.
• I can figure out what the market rate is for the item and set my proxy bidding maximum at that amount. If I get outbid, I just try the same price on a similar auction.
• I can figure out how often the item is available via eBay and if I should try to find a good deal there versus looking elsewhere.

If you aren't familiar with this feature, I highly recommend giving it a try. Also, if you have any other ideas for how to use this tool, feel free to share them in the comments.

Logistical planning

As I learned when searching Airbnb for somewhere to stay during the INFORMS Annual Meeting, the pope is visiting Philly this year. He will be there September 26-27 for the "World Meeting of Families Congress," and there has been an incredible amount of logistical planning to both ensure security and allow for the large numbers of people they expect to attend the pope's events.

This link covers a lot of the changes that have been planned which include: limited or no driving and parking (some of which started today!), reduced subway / trolley / bus routes, and extra security to get close to downtown at all. They also have planned for a lot of extra services including porta-poties, free wifi, and even 500 "teenage athletes" to help push around people in wheelchairs. And that doesn't even get into the "logistical nightmare" that would arise if a thunderstorm rolls in.

Everything should be long back to normal by the time I get there... But I am happy to take advantage of the extra capacity on Airbnb as a result of the visit.

Second-order bias in models

You may have heard about the controversy surrounding the SAT over the years since people of color have always done worse on the test. The test-writers have attempted more and more to avoid racial bias in the results. It is clear that the test used to have a significant racial bias. It is less clear if that this is still the case today.

This kind of question is studied in the field of "disparate impact." Salon.com just published this article talking about that field and the risks of having disparate impact especially when you trust algorithms to make the decisions. I would call the old SAT an example of first-order bias, and the remaining issues in the test mostly second-order bias.

The problem is clearly not simple to solve. However, hopefully with time the very data analysis techniques that currently lead to biased decisions can ultimately be used to avoid bias in outcomes.

On "Managerial Insight"

Professor Chris Tang asked the operations research community to come together and provide a definition of "Insight." In our field, most of the top journals ask authors to consider the potential managerial insights available as a result of their paper. Ostensibly clearly stating the insights will save the reader some effort in figuring out how they can use the results in the paper, and perhaps will also ensure that the paper has real-world implications.

When a fellow PhD student first introduced me to the idea of insight, I understood it to simply be the practical implications of your work. Since then I have learned a lot about how OR is used in practice. It seems that occasionally our models are directly implemented. Most of the time though, there is some reason decision makers would not want to directly implement the output of the model. Maybe the optimization problem is missing some important piece that keeps it from being directly implementable. Maybe the model is far too complex to be worth solving exactly. Or maybe the whole goal of the model is simply to provide the decision maker with a collection of options to decide between.

Given this understanding, I now think of insights as the results that can help get a better understanding of the problem even without actually implementing the model. For illustrations of insight, physics provides some good examples:

• Heavier objects do not fall faster just because they have more mass.
• Doors open more easily when you push near the handle instead of near the hinge.
• Going uphill takes more energy than going downhill.

Behind each of these examples are equations which we may or may not care about in any particular situation. But the insight provided is accessible and useful whether or not we need to make a quantitative decision.

One additional point. While insights may come from the results of optimization models, sometimes simply formulating the model in a clever way can help provide insights. To use an OR example, Michael Trick recently posted on the topic of complete enumeration as an argument for complexity. He points out that just because complete enumeration is one way to find the optimal solution does not mean the underlying problem is hard. In the same way, clever formulations of hard problems can bypass a lot of unnecessary complexity.

Dear EPA

Last week I got a letter which asked me to loan my car to the EPA for their research on emissions from privately owned vehicles. The car would be used between 1 and 4 weeks, assuming it met their criteria. They provided a couple of incentive options and also stipulated that my car would be returned with a full tank of gas after driving it up to 300 miles per day, along with being covered by their insurance.

• If I wanted a loaner car from them (which appears to be insured by them), I would receive $20 per day.
• If I did not receive a loaner car from them, I would receive $50 per day. 

Based on my post on the true cost of driving, I wondered how the offers compare to the cost to me. The costs I will experience if I participate in this (in approximate order of cost to me) include:

• Wear and tear on the car and tires at a rate of up to 300 miles per day (possibly excluding weekends, possibly not).
• The loss of use of my vehicle for the testing period if I take the $50 per day option
• Miles towards my next oil change.
• Time out of my day to do drop-off and pickup.
• The risk that something bad will happen to the car that may cost me additional time and money (which I would hopefully be appropriately compensated for).

The benefits I will experience (in approximate value for me) include:

• $50 or $20 per day for participating.
• A loaner car if I choose that option.
• A full tank of gas when I get my car back.
• If the loaner car is insured by them, I may avoid some risk due to accidents during the time of the study.

If I normally drove close to 300 miles a day, this problem would be easy to solve by choosing to get a loaner car (assuming there is no limit on the number of miles on it). Most days though, the car they would be using of mine drives 30 miles, so no easy out there. For me personally since we have two cars, I can quickly say that the loss of use of the vehicle is less than $30 a day, so I would go without if I participated at all.

We can quickly see that they are offering a minimum of 17 cents per mile driven if they drive it the full 300 miles every day they pay me for. From the information available from AAA, we can try to figure out if that leaves any extra money as an incentive for me to participate. Tires and maintenance (including oil changes) add up to 6 cents per mile. But what about the loss in value due to additional miles?

AAA estimates depreciation at 24 cents per mile based on numbers that are close to true for me, so there's definitely a risk of wiping out all the potential profit. But, how much of that has to do with the age vs. the actual miles driven? I decided to use Edmunds and Kelley Blue Book to estimate what the change in value to my car would be purely based on the miles driven. I tested both for the present, and since I didn't entirely trust the numbers, I also tried projecting out 2 years to when I might end up selling the car (I did this by checking the present value of a car 2 years older than mine with estimated mileage numbers). When I initially tried entering a change of 4000 miles to the odometer reading (a guess of the total miles that would be added), Edmunds gave me no change in value of the car at all. So instead I am basing the numbers in the table on a jump of 10,000 to the odometer just to avoid any weird artificial cutoffs.

	Edmunds	Kelley Blue Book
Effect on current value (cents / mile)	4.09	4.43
Effect on future value (cents / mile)	1.79	4.02

In short, that gives an upper bound of 5 cents per mile for depreciation due only to mileage. Therefore, we are left 6 cents a mile (or almost $20 a day) for participating. That once again seems like more than the value of having the second car to me, so I guess I should get the form in the mail!

Why do kids under 2 fly free?

If you look online for tips on flying with kids, you will inevitably find the nearly-universal advice to buy a ticket for children of any age. Even the FAA tells you in no uncertain terms that holding your child in your lap is not safe in the case of an accident. So why is it still an option?

In the FAA statement 10 years ago, they explained that this is a public policy issue. Their analysis concluded that if parents had to buy an extra plane ticket, more families would choose to drive rather than fly which would result in additional fatalities. While many people decry the FAA's decision for putting additional children at risk, I applaud that they work with the reality of how people make decisions (and used data to make that choice).

Beyond the lap-child issue, I have wondered if the drive to the airport actually carries a higher risk than the flight itself. According to the data here, in 2010 the risk was 1.1 deaths per 100 million miles driven. Flight data is somewhat harder to extract. The number cited in the link covers private airplanes separately (reasonable) but also terrorism and suicide (which we would probably want to include). That link suggests a fatality rate of 0.07 per billion passenger miles, which works out to 158 miles flown has the same risk as 1 mile driven. If we instead look at the National Travel Safety Board data we see an even more encouraging picture since many years do not have any accidents with fatalities among airliners (the scheduled commercial flights most of us are taking).

All in all, I think most people do not worry about safety when deciding if they will fly or drive to their destination. It is well known that our brains are not well suited for assessing modern risks. And so it is important for public policy to carefully include how people respond to policies in their decision making.

Conflict Minerals

One of the blogs I frequent just had a piece on companies trying to trace their supply chain all the way to the source. This initiative is because of changes in the law which require them to determine if they are using conflict minerals. The authors mentioned that the total costs of trying to be compliant with the new law has been about 709 million dollars and 6 million staff hours (and that only 24% of the companies are actually fully compliant so far).

709 million dollars is a lot of money, and I wondered if a) it might have been more effective to put it directly into aid, and b) how that number compared to the actual value of conflict minerals. From this link though, it looks like the reforms have been a lot more effective than I initially expected. The "enough project" estimated that in 2008 (prior to the new law) 185 million dollars went to armed groups via conflict minerals. They also attribute a lot of the progress since then to the law. This source estimates that revenues have decreased by 65%. They do think that some of the violent groups have shifted focus to gold since conflict gold hasn't suffered in value nearly as much as the other conflict minerals.

In short, while tracing an entire supply chain is incredibly costly, the law has been pretty effective at improving the situation.

OR and Analytics

In May of this year INFORMS (the primary professional society for Operations Research) had a vibrant conversation on their discussion board questioning what the field of Analytics is and how it does or does not overlap with OR. My favorite piece from this exchange came from Patrick Noonan who suggested that decision makers should answer two central questions:

1. "What should I do, given what I believe?" 
2. "What should I believe, given what I observe?

While you will frequently need to iterate between the two questions (and this combination is in fact what Professor Noonan describes as Analytics), I do think it is useful to think of decision making as being made up of these two pieces. The first question is a really succinct description of OR, and the second question sums up what statistics is good for.

Despite the overwhelming popularity of Analytics currently, I do think there is significant value in looking at these two questions in isolation as well as together. A big emphasis in Lean is data collection to understand your current state before you implement changes. Articulating what you wish you knew can be a useful exercise before even stating your problem. At the same time, examining your data carefully is critical to doing anything useful with it at all. Otherwise you might conclude that flashlight apps have the most valuable ads.

Hopefully this explanation helps any of my friends whose eyes glazed over when I said I do Operations Research!

Implementing model solutions

I just came across this blog post which talks about another layer of doing math modeling. The basis of my blog is that you should try to make sure you solve the right problem, otherwise you can't hope to get a good solution. What the blogger points out is that even if you solve the right problem, that does not guarantee that the solution will be implemented correctly. His example comes from doing retail stock forecasting. He describes several of the ways his customers may use the information gleaned from the software to make inappropriate decisions.

There are similar issues in my research area. I work on "the newsvendor problem" where you sell newspapers during the day. You start by choosing a quantity of newspapers y to buy at the beginning of the day for c dollars each, and sell them for r apiece. If you run out early, you have a lost sales cost of e, and if you have leftovers you can recycle them for v each. Empirical studies find that even when participants are trained about the optimal order quantity, they tend to over-order.

So how to resolve these issues? Effective communication skills are a key starting point. By asking the right questions of practitioners, modelers can ensure both that they solve the right problem and do their best to make sure that the solutions are taken seriously.

A case for paying friends $0.50 per mile

Giving friends money for gas is the default offer in college. The logic goes something like this: "Most of the costs to drive other than gas are either negligible or fixed, so paying for gas is close to fair." Yet, the cost of gas works out to around 15 cents per mile while the government estimate of the cost of driving is 57.5 cents per mile. The quote from the IRS website is that these costs include "the fixed and variable costs of operating an automobile, including depreciation, insurance, repairs, tires, maintenance, gas and oil." The IRS also says that moving or medical miles have a rate of 23 cents per mile to cover strictly the "variable costs such as gas and oil."

Based on the law, that suggests paying friends 23 cents per mile if they are making a special trip for you. However, an issue arises when you actually look at the list of "fixed costs." Clearly repairs due to normal wear and tear, depreciation, tires, and maintenance all are highly dependent on the actual number of miles driven. The only thing on the explicit list that is strictly a fixed cost is insurance. But, what is insurance actually for? Insurance covers (hopefully most) costs in the event of an accident. While registration costs are truly fixed since you either keep your car legally registered or not, insurance costs are an attempt to reduce risk.

It may be possible to separate the truly variable costs from the truly fixed and get a more accurate rate to pay friends. But I would argue that the government rate is a much better approximation of the variable cost of driving than simply paying for gas.