In ordinary decision theory, the name weak preference is used for a relation that could be defined as the complement of the inverse of strict preference, or as the union of strict preference with indifference. In ordinary decision theory, these two are equivalent. Typical symbols used for this relationship are ≽, ≿ and ⪰.

In my paper, I noted the conventional conception of weak preference as the aforementioned union; later, I defined it for my purposes such that

(X₁ ≽ X₂) ≡ [{X₁} ⊆ C({X₁, X₂})]

which is the complement of the inverse of strict preference.

Well, I've decided that I was just asking for trouble using that name and that symbol, because people would have trouble not thinking of it as the union of strict preference with indifference, and thence think that a distinct relation of indecision is precluded a priori. So I've switched to the name non-rejection and the symbol ⊀.

There will probably still be people who ask how this relation differs from that of weak preference, but that will be more an expression of their cleverness than of their confusion, and it will be easier to offer an explanation how the relation could be seen as weak preference, how it should not be seen as weak preference. (I may try to squeeze some preëmptive discussion into the paper, though I am bumping-up against size limits.)

Addendum (2009:04/18): I added a preëmptive discussion, so that even fewer people will get confused.

As noted earlier, I've been reading Subjective Probability: The Real Thing by Richard C. Jeffrey. It's a short book, but I've been distracted by other things, and I've also been slowed by the condition of the book; it's full of errors. For example,

It seems evident that black ravens confirm (H) All ravens are black and that nonblack nonravens do not. Yet H is equivalent to All nonravens are nonblack.

Uhm, no: (X ⇒ Y) ≡ (¬X ∨ Y) = (Y ∨ ¬X) = (¬¬Y ∨ ¬X) = [¬(¬Y) ∨ ¬X] ≡ (¬Y ⇒ ¬X) In words, that all ravens are black is equivalent to that all non-black things are non-ravens.[1]

The bobbled expressions and at least one expositional omission sometimes had me wondering if he and his felllows were barking mad. Some of the notational errors have really thrown me, as my first reäction was to wonder if I'd missed something.

Authors make mistakes. That's principally why there are editors. But it appears that Cambridge University Press did little or no real editting of this book. (A link to a PDF file of the manuscript may be found at Jeffrey's website, and used for comparison.) Granted that the book is posthumous, and that Jeffrey was dead more than a year before publication, so they couldn't ask him about various things. But someone should have read this thing carefully enough to spot all these errors. In most of the cases that I've seen, I can identify the appropriate correction. Perhaps in some cases the best that could be done would be to alert the reader that there was a problem. In any case, it seems that Cambridge University Press wouldn't be bothered.

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[1]The question, then, is of why, say, a red flower (a non-black non-raven) isn't taken as confirmation that all ravens are black. The answer, of course, lies principally in the difference between reasoning from plausibility versus reasoning from certainty.

After he read my paper, Anthony told me something that I already knew — that the Discussion section was brief, and the Conclusion sudden. I had been both sick of working on the paper, and having trouble thinking about it in natural language. So those parts were… lacking. Anthony's gentle remarks increased my sense that they were inadequate.

I have expanded and reörganized the Discussion section (cannibalizing part of the Conclusion), and added to the Future Work section between it and the Conclusion. In that context, the Conclusion should seem less sudden, and I have added some thoughts to it (as well as having taken one from it).

Anthony also suggested that the paper could be made more accessible by discussion of the historical background of the problem, and of real-world examples. But, as I told him, I fear to alienate experts by such discussion; and I have since learned that I am already bumping-up against the size-limits for a submission to the journal of my choice.

Anyway, the latest version of my paper is at

The mathematical expressions that appear (tiled) in the background of this 'blog; are from a paper on which I have been working (off-and-on) over a long time. I'm far from perfectly happy with the present state of the paper, but I've finally put-together something like a complete draft of it, in PDF, at

There are some temporary issues with presentation of this draft:

• The layout needs to be fixed, by the insertion of page breaks, so that things such as section headings are pushed to a next page, rather than orphaned.
• The OpenOffice formula editor does not support use of the relational symbols ≻, ≽, ≿, or ⪰. (For now, I am representing strict preference with pref and weak preference with wpref; later, I will output the paper as LAΤΕΧ and then tweak the formulæ to give me ≻ for strict preference and one of the other three for weak preference.)

At this point, I welcome comments, from experts and from non-experts, both on the underlying content and on how I have expressed myself.

Up-Date (2009:04/08): Professor Gamst of UCSD caught an error in what had been formula (10). It was easily patched. I have up-dated the on-line version of the paper.

Yester-day, I again encountered the slogan No Blood for Oil!, made popular in the run-up to the invasion of Iraq.

I greatly doubt that the people who embrace this slogan have much thought about what it would really mean to forswear the use of violence over economic resources. Only a tiny minority are truly prepared to do that.

This proposition should be obvious in the case of those who want the State to actively participate in decisions about the allocation of resources. The point in having the State determine what quantities will be produced or to whom or to what production will go, or at what prices goods or services will be sold is to use violence or a threat of violence, effected by police and by prison guards. Were the economic administration to be by friendly persuasion, it could be done without the State.

Removing the State as an active participant doesn't utterly remove violence from the equation. If one believes that individuals or communities may forceably defend acquisition, retention, or distribution of goods or of services, then one accepts the use of violence over economic resources. It doesn't matter whether the forceable defense is provided by the State, by private protection agencies, by mobs, or by rugged individualists.

So if one believes that the state should provide the poor with home heating oil, or control gasoline prices, or if one believes in forceably defendable private property in petroleum or in forceably defendable anarcho-socialistic management of petroleum, then one believes in trading blood for oil.

In order to genuinely reject such an exchange, one would have to be truly and utterly pacifistic about petroleum, as are the Amish (albeït that a petroleum pacifist might be violent about other things).

Now, I surely don't claim that the United States should ever use its military in an attempt to secure foreign sources of goods or of services. (We can set aside debate over what the actual relationship were of the invasion of Iraq to American dependence upon Middle Eastern petroleum.) But simple-minded slogans and ad hoc moralizations don't typically propel discourse or move convictions in a humane direction.

The Classic Monty Hall Problem

Andy is a contestant in a game. In this game, each contestant makes a choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments.

After Andy makes his choice, it is revealed to him that a specific tag that he did not choose is committed to an undesirable outcome. Andy is offered a chance to change his selection. Should he change?

Three Contestants

Andy, Barb, and Pat are contestants in a game. In this game, each contestant makes an independent choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments. In the event that multiple players select the same tag, outcomes are duplicated.

After all contestants make their choices, it is revealed that Andy, Barb, and Pat have selected tags each different from those of the other two contestants. And it is revealed that Pat's tag is associated with an undesirable outcome. Andy and Barb are each offerd a chance to change their selections. What should each do?

In the past, I've asserted that San Diego is seriously over-built. Hillcrest, the neighborhood in which I live, has many commercial sites that have been empty for some time. As an œconomist, I naturally wonder why the landlords haven't lowered the rent to a point where some party takes occupancy.

One possibility would be that the landlords were hoping for an up-turn in the economy. The possibility would make them more reluctant to commit to long-term leasing contracts at what would be present market-clearing rates, and might make them reluctant to even rent month-to-month, as there would be difficulties getting short-term renters out as quickly a more desirable renter might want to begin using the site.

Part of what would make it difficult to remove month-to-month renters would be local and state laws and regulations governing evictions. And other interventions in the market could be holding rents and vacancies at unnatural levels. For example, the structure of tax law might make sometimes make it advantageous to leave a property idle.

In any case, the numbers of vacancies, and durations of some of them, would be difficult or impossible to explain based simply on reference to market forces.

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308 Washington Street, half-a-block from where I live, was a Hollywood Video rental store when I first moved to Hillcrest. (Most or all of their wares were VHS tapes at that time!) At some point, they hived-off perhaps a fifth of the site for digital electronic game rentals. Eventually, this section became a separate store, 302 Washington Street, and then Hollywood Video vacated that section, which was then rented to a UPS franchise store. Some time in late 2007 or in 2008, Hollywood Video shut down the store at 308 Washington Street. Since then, the building has stood about four-fifths empty, with a large sign advertising its availability.

Finally, yester-day, I saw a crew in the site gutting things in what I take to be the first stage of a renovation.

The last time that someone was poking-around in 308 Washington Street, I asked Scott, who works at the UPS store, lives in the same complex as I, and frequents the same coffee house (Babycakes) if he knew who the party was, and he said that rumor held it to be PetCo, which we agreed would be cool. Last night, Art, who lives in the same complex and works part-time at the coffee house express hope that the new store would be, er, a Denny's Restaurant, and didn't like the idea of a pet supply store there, in spite of being a dog-owner. Art sees PetCo stores as essentially big boxes like Wal·Mart stores. Well, there's some truth to that. On the other hand, Denny's Restaurants are open all day, almost every day of the year, and can be associated with a lot of traffic. Our immediate neighborhood could become more congested and louder late at night with a Denny's Restaurant. But I don't think them a likely renter there.

[Up-Date (2009:03/26): Scott tells me that word remains that 308 is to be occupied by a PetCo store.]

The USPS doesn't have enough business, but prices will go up on 11 May. As I said, Basically, officials increase the price per letter in an attempt to off-set the cost per letter which increases as the number of letters decreases because of past price increases. It’s a death-spiral. Added to that now are the effects of the present economic down-turn, but most of the decrease in volume is just a continuation of established trend.

[…] administrative staff positions at the district level nationwide are being reduced by 15 percent. More than 1,400 mail processing supervisor and management positions at nearly 400 facilities around the country also are being eliminated and nearly 150,000 employees nationwide are being given the opportunity to take an early retirement.

Normally, a fair share of early retirees would plan on taking other jobs. In the present economic climate, benefits would have to be higher than otherwise to induce people to take earlier retirement.

In the entry in which I previously discussed tax incidence, I made the point that (setting aside transactions costs) the distributed burden of a per-unit tax is the same regardless of whether the tax is formally imposed upon the buyer or upon the seller or upon both. I didn't, though, explain who then actually bears the burden, because that was irrelevant to where I was going with the entry. But it has preyed upon my mind that I didn't explain that bit.

The answer is that the burden falls most heavily on whichever party is least flexible in response to the monetary price.

Imagine, for example, that buyers, driven by some internal compulsion, must always buy (or try to buy) the same quantity of something, regardless of its price. Price won't necessarily rise without limit, because of competition amongst sellers. If a per-unit tax is formally imposed upon buyers, well, they'd buy that same quantity and then pay the tax on top of what they'd been paying the sellers. Setting aside feelings of pity, or somesuch, the sellers would just shrug. If a per-unit tax is formally imposed upon sellers, then they can otherwise pass along the full cost to their buyers, and competition isn't going to help buyers because the sellers all face the same increase in their costs, so that their calculated optimal prices just increase by the amount of the tax.

On the other hand, if sellers somehow always had to sell the same amount while buyers were flexible, then the shoes would be on the other feet. If the tax is formally imposed upon sellers, then they'd just sell the same amount at the price where quantity demanded absorbed that amount. If the tax is formally imposed upon buyers, well, then buyers are at least going to slightly reduce their consumption unless the sellers cut their prices to fully-offset the tax, so the sellers do this if they must sell that same amount.

When the tax isn't actually per unit, the mathematics and the results can be somewhat messier, but the same underlying dynamics will decide the distribution of the burden.

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In that earlier entry, I said The next time that you hear or read of a politician arguing for employer-provided benefits, such as for health-care, consider the incidence. My primary point at that time was that there may be little or no difference in ultimate burden between a tax formally imposed upon employers and one formally imposed upon employees. But let's pursue the question.

Supposèd advocates of the interests of employees often call for requiring various sorts of employee benefits. At the same time, these alleged advocates tend to model the labor market as if workers have very little choice in their terms of employment. But if, indeed, workers don't have much flexibility, then it will be they who bear most of the cost of the mandated benefits — implicitly they are forced to buy the benefits, rather than spend their wages or salaries on other things.

The accuracy of that characterization of inflexibility will vary across regions, times, and employee-types. All else being equal, it will be less true in times when employers are competing heavily for workers, and more true in times when employees are struggling to find or to keep jobs. Thus, for example, in a recession, employees tend to bear more of the cost of benefits supposedly funded by employers.

In economics, when we write or speak of the incidence of a cost or of a benefit, we refer to the ultimate distribution.

Most people get the idea that, often, costs or benefits can be passed along, so that the party upon whom they formally fall isn't necessarily the final recipient. This is certainly true of taxes.

There's a standard result of microëconomics that, if one sets aside the effects of transactions cost (dat ol' debbil), then the incidence of a per-unit tax is same, regardless of whether it is formally placed on the seller or formally placed upon the buyer. Almost every first-term microëconomics course demostrates this result, although they usually dumb-it-down by failing to note that transactions costs can somewhat undermine the equivalence.

The argument goes as follows: Imagine that the quantity offered for sale fits some function S(P_s) where P_s is the price received by the seller, and that quantity that buyers seek fits some function D(P_b) where P_b is the price that buyers must pay. If the pre-tax price is just P, and sellers have to pay additional tax and tax-related costs of t_s, and buyers have to pay tax and tax-related costs of t_b, then

P_s = P - t_s

(tax-related cost subtracted because it is a reduction in the price that what the seller receives) and

P_b = P + t_b

(tax-related cost added because it is an increase in how much the buyer must pay). Algebraïcally,

P_b = P_s + t_s + t_b

and

P_s = P_b - t_b - t_s

And the market would equilibrate where

S(P_s) = D(P_b)

(At a lower P, D would be greater than S, and it would be in the interest of sellers to increase their prices or of buyers to offer a little more per unit; and, a higher P, S would be greater than D, and it would be in the interest of sellers to cut their prices a bit, or for potential buyers to cut the amount that they offered.) Which is to say that the equilibrium is

S(P_s) = D(P_s + t_s + t_b)

as if the buyer formally paid all the tax on the seller's received price, and

S(P_b - t_b - t_s) = D(P_b)

as if the seller formally paid all the tax on the buyer's paid price.

(The next time that you hear or read of a politician arguing for employer-provided benefits, such as for health-care, consider the incidence.)

Now, let's consider the incidence of a tax on carbon emissions (ignoring the question of whether there should be such a tax), in the absense of transactions costs, the incidence would be the same whether the state formally taxed the producer of the emissions, or taxed the consumer of each product associated with the emissions, based upon the amount of the emission associated with that product. But it is plainly less costly to place the formal tax on the producer than to have separate filings for each consumer.

Which brings me to this story:

Whatever measures are imposed to curtail carbon emissions, they can be conceptualized as a tax. And it's really, folks, not that the Chinese officials don't understand that the incidence would be the same (or very nearly the same) in the absence of transactions costs, nor that they don't recognize that the transactions costs would be lower if the tax were formally imposed upon producers. Rather, it's that the Chinese state

• knows that some consuming nations would avoid or evade the tax, lowering the incidence upon China,
• would be able to disguise some of its emissions for domestically consumed production as emissions for exported product, and
• would like to misdirect blame for any failure to reach agreement.