Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies.
Ergodicity and the all-important difference between time averages and ensemble averages are difficult concepts that many students of economics have problems with understanding. So let me just try to explain the meaning of these concepts by means of a couple of simple examples.
Would you accept the gamble?
If you’re an economics students you probably would, because that’s what you’re taught to be the only thing consistent with being rational. You would arrest the arrow of time by imagining six different “parallel universes” where the independent outcomes are the numbers from one to six, and then weight them using their stochastic probability distribution. Calculating the expected value of the gamble – the ensemble average – by averaging on all these weighted outcomes you would actually be a moron if you didn’t take the gamble (the expected value of the gamble being 5/6*€0 + 1/6*€10 billion = €1.67 billion)
If you’re not an economist you would probably trust your common sense and decline the offer, knowing that a large risk of bankrupting one’s economy is not a very rosy perspective for the future. Since you can’t really arrest or reverse the arrow of time, you know that once you have lost the €1 billion, it’s all over. The large likelihood that you go bust weights heavier than the 17% chance of you becoming enormously rich. By computing the time average – imagining one real universe where the six different but dependent outcomes occur consecutively – we would soon be aware of our assets disappearing, and a fortiori that it would be irrational to accept the gamble.
[From a mathematical point of view you can (somewhat non-rigorously) describe the difference between ensemble averages and time averages as a difference between arithmetic averages and geometric averages. Tossing a fair coin and gaining 20% on the stake (S) if winning (heads) and having to pay 20% on the stake (S) if loosing (tails), the arithmetic average of the return on the stake, assuming the outcomes of the coin-toss being independent, would be [(0.5*1.2S + 0.5*0.8S) - S)/S] = 0%. If considering the two outcomes of the toss not being independent, the relevant time average would be a geometric average return of squareroot[(1.2S *0.8S)]/S - 1 = -2%.]
Why is the difference between ensemble and time averages of such importance in economics? Well, basically, because when assuming the processes to be ergodic,ensemble and time averages are identical.
Assume we have a market with an asset priced at €100 . Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be €100 – because we here envision two parallel universes (markets) where the assetprice falls in one universe (market) with 50% to €50, and in another universe (market) it goes up with 50% to €150, giving an average of 100 € ((150+50)/2). The time average for this asset would be 75 € – because we here envision one universe (market) where the asset price first rises by 50% to €150, and then falls by 50% to €75 (0.5*150).
From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen. Assuming ergodicity there would have been no difference at all.
Just in case you think this is just an academic quibble without repercussion to our real lives, let me quote from an article by Ole Peters in the Santa Fe Institute Bulletin from 2009 – On Time and Risk - that makes it perfectly clear that the flaw in thinking about “rational” decisions in terms of ensemble averages has had real repercussions on the functioning of the financial system:
In an investment context, the difference between ensemble averages and time averages is often small. It becomes important, however, when risks increase, when correlation hinders diversification, when leverage pumps up fluctuations, when money is made cheap, when capital requirements are relaxed. If reward structures—such as bonuses that reward gains but don’t punish losses, and also certain commission schemes—provide incentives for excessive risk, problems arise. This is especially true if the only limits to risk-taking derive from utility functions that express risk preference, instead of the objective argument of time irreversibility. In other words, using the ensemble average without sufficiently restrictive utility functions will lead to excessive risk-taking and eventual collapse. Sound familiar?
On a more economic-theoretical level the difference between ensemble and time averages also highlights the problems concerning the neoclassical theory of expected utility that I have raised before (e. g. in Why expected utility theory is wrong).
When applied to the neoclassical theory of expected utility, one thinks in terms of “parallel universe” and asks what is the expected return of an investment, calculated as an average over the “parallel universe”? In our coin tossing example, it is as if one supposes that various “I” are tossing a coin and that the loss of many of them will be offset by the huge profits one of these “I” does. But this ensemble average does not work for an individual, for whom a time average better reflects the experience made in the “non-parallel universe” in which we live.
Time averages gives a more realistic answer, where one thinks in terms of the only universe we actually live in, and ask what is the expected return of an investment, calculated as an average over time.
Since we cannot go back in time – entropy and the arrow of time make this impossible – and the bankruptcy option is always at hand (extreme events and “black swans” are always possible) we have nothing to gain from thinking in terms of ensembles.
Actual events follow a fixed pattern of time, where events are often linked in a multiplicative process (as e. g. investment returns with “compound interest”) which is basically non-ergodic.
Instead of arbitrarily assuming that people have a certain type of utility function – as in the neoclassical theory – time average considerations show that we can obtain a less arbitrary and more accurate picture of real people’s decisions and actions by basically assuming that time is irreversible. When are assets are gone, they are gone. The fact that in a parallel universe it could conceivably have been refilled, are of little comfort to those who live in the one and only possible world that we call the real world.
Our coin toss example can be applied to more traditional economic issues. If we think of an investor, we can basically describe his situation in terms of our coin toss. What fraction of his assets should an investor – who is about to make a large number of repeated investments – bet on his feeling that he can better evaluate an investment (p = 0.6) than the market (p = 0.5)? The greater the fraction, the greater is the leverage. But also – the greater is the risk. Letting p be the probability that his investment valuation is correct and (1 – p) is the probability that the market’s valuation is correct, it means that he optimizes the rate of growth on his investments by investing a fraction of his assets that is equal to the difference in the probability that he will “win” or “lose”. This means that he at each investment opportunity (according to the so called Kelly criterion) is to invest the fraction of 0.6 – (1 – 0.6), i.e. about 20% of his assets (and the optimal average growth rate of investment can be shown to be about 2% (0.6 log (1.2) + 0.4 log (0.8))).
Time average considerations show that because we cannot go back in time, we should not take excessive risks. High leverage increases the risk of bankruptcy. This should also be a warning for the financial world, where the constant quest for greater and greater leverage – and risks – creates extensive and recurrent systemic crises. A more appropriate level of risk-taking is a necessary ingredient in a policy to come to curb excessive risk taking.
If economics was an honest profession, economists would focus their efforts on documenting the waste associated with protectionist barriers for professionals. They devoted endless research studies to estimating the cost to consumers of tariffs on products like shoes and tires. It speaks to the incredible corruption of the economics profession that there are not hundreds of studies showing the loss to consumers from the barriers to trade in physicians’ services. If trade could bring down the wages of physicians in the United States just to European levels, it would save consumers close to $100 billion a year.
But economists are not rewarded for studying the economy. That is why almost everyone in the profession missed the $8 trillion housing bubble, the collapse of which stands to cost the country more than $7 trillion in lost output according to the Congressional Budget Office (that comes to around $60,000 per household).
Few if any economists lost their 6-figure paychecks for this disastrous mistake. But most economists are not paid for knowing about the economy. They are paid for telling stories that justify giving more money to rich people. Hence we can look forward to many more people telling us that all the money going to the rich was just the natural workings of the economy. When it comes to all the government rules and regulations that shifted income upward, they just don’t know what you’re talking about.
In case you’re in doubt, you might better have a look at e. g. what Harvard economist and George Bush advisor Greg Mankiw writes on the rising inequality we have seen for the last 30 years in both the US and elsewhere in Western societies:
Even if the income gains are in the top 1 percent, why does that imply that the right story is not about education?
I then realized that Paul [Krugman] is making an implicit assumption–that the return to education is deterministic. If indeed a year of schooling guaranteed you precisely a 10 percent increase in earnings, then there is no way increasing education by a few years could move you from the middle class to the top 1 percent.
But it may be better to think of the return to education as stochastic. Education not only increases the average income a person will earn, but it also changes the entire distribution of possible life outcomes. It does not guarantee that a person will end up in the top 1 percent, but it increases the likelihood. I have not seen any data on this, but I am willing to bet that the top 1 percent are more educated than the average American; while their education did not ensure their economic success, it played a role.
Matias Vernengo has a piece today on Naked Keynesianism giving us yet another clue to why IS-LM – and its different hybrids – still isn’t up to the task of helping students to grasp the essentials of modern monetary economies:
Wendy Carlin and David Soskice have an interesting post at Voxeu.org on “How should macroeconomics be taught to undergraduates in the post-crisis era.” They explicitly follow … a New Keynesian framework. Which is an ISMP model with price rigidities …
In their view, the monetary authority can move the economy to the equilibrium rate, but the equilibrium rate is not controlled by the authorities. I would suggest that the central bank actually has control of the key variable, which is not natural at all, and as Keynes suggested is conventional, the normal rate of interest. And that could be set at a level which will not lead to full employment, particularly for distributive reasons. In my view the essential feature of a more relevant macro model would be the elimination of the natural rate, the idea that the system has a tendency to bounce back to the trend …
And by the way, the model says nothing about how income distribution affects spending.Yet, the main reason behind the crisis has been the fact that workers had to get indebted, since income has stagnated. So income distribution has to be brought back to macroeconomics. Not enough Kalecki, with their New (but still neoclassical) version of Keynesianism, that is the problem with Carlin and Soskice.
(1) America is a land of opportunity. While rags-to-riches stories still grip our imagination, the fact of the matter is that the life chances of a young American are more dependent on the income and wealth of his parents than in any of the other advanced countries for which there is data. There is less upward mobility — and less downward mobility from the top — even than in Europe, and we’re not just talking about Scandinavia.
(2) Trickle-down economics works (a k a “a rising tide lifts all boats”). This idea suggests that further enriching the wealthy will make us all better off. America’s recent economic history shows the patent falsehood of this notion. The top has done very well. But median American incomes are lower than they were a decade and a half ago. Various groups — men and those without a college education — have fared even worse. Median income of a full-time male worker, for instance, is lower than it was four decades ago.
(3) The rich are the “job creators,” so giving them more money leads to more and better jobs. This is really a subset of Myth 2. But Romney’s own private sector history gives it the lie. As we all know from the discussion of Bain Capital and other equity firms, many made their money not by creating jobs in America but by “restructuring,” “downsizing” and moving jobs abroad, often using debt to bleed the companies of money needed for investment, and using the money to enrich themselves. But more generally, the rich are not the source of transformative innovations. Many, if not most of the crucial innovations in recent decades, from medicine to the Internet, have been based in large measure on government-financed research and development. The rich take their money where the returns are highest, and right now many see those high returns in emerging markets. It’s not a surprise that Romney’s trust fund invested in China, but it’s hard to see how giving the rich more money — through more latitude to escape taxation, either through low taxes in the United States or Cayman Islands hide-aways — leads to a stronger American economy.
(4) The cost of reducing inequality is so great that, as much as idealists would like to do so, we would be killing the goose that lays the golden egg. In fact, the engine of our economic growth is the middle class. Inequality weakens aggregate demand, because those at the middle and bottom have to spend all or almost all of what that they get, while those at the top don’t. The concentration of wealth in recent decades led to bubbles and instability, as the Fed tried to offset the effects of weak demand arising from our inequality by low interest rates and lax regulation. The irony is that the tax cuts for capital gains and dividends that were supposed to spur investment by the wealthy alleged job creators didn’t do so, even with record low interest rates: private sector job creation under Bush was dismal. Mainstream economic institutions like the International Monetary Fund now recognize the connection between inequality and a weak economy. To argue the contrary is a self-serving idea being promoted by the very wealthy.
(5) Markets are self-regulating and efficient, and any governmental interference with markets is a mistake. The 2008 crisis should have cured everyone of this fallacy, but anyone with a sense of history would realize that capitalism has been plagued with booms and busts since its origin. The only period in our history in which financial markets did not suffer from excesses was the period after the Great Depression, in which we put in place strong regulations that worked. It’s worth noting that we grew much faster, and more stably, in the decades after World War II than in the period after 1980, when we started stripping away the regulations. And in the former period we grew together, in contrast to the latter, when we grew apart.
As I have explained in detail elsewhere, the cost of these myths goes far beyond the damage to our economy, now and in the future. The fabric of our society and democracy is suffering.
Challenging conventional (neoclassical) wisdom on modern monetary theory, economics professor L. Randall Wray in this new book presents a thought-provoking analysis of how money is created and functions. Instead of the usual mainstream economic textbook mystifications of the nature of money, we are here given a theory that shows what’s really going on in modern monetary economies.
For those still having problems with understanding the nuts and bolts of what’s wrong with textbook versions of the savings and investment nexus, perhaps this little lecture might help:
Q: If consumers spend less and save more, does this mean investment must increase?
A: Absolutely not. Someone increasing their saving does not automatically imply that some firm will decide to buy more capital goods.
Q: But surely savings equals investment by identity in the national accounts.
A: Indeed. Total output = total income = total expenditure = Y. In the most simple model of a closed economy without government, income (Y) = consumption (C) + saving (S), but also expenditure (Y) = consumption (C) + investment (I). So S=I by definition. But here investment includes what is called ‘stockbuilding’ or ‘inventory accumulation’, which includes goods that firms wanted to sell but could not. To make this clear, lets split measured investment (I) into these two components: I=DK (buying new capital goods) +DS (stockbuilding). So if people consume less (C falls), but investment in new capital (DK) stays the same, measured investment rises because firms accumulate inventories of the goods that consumers did not buy (DS rises).
Q: But this situation cannot continue, as firms may be losing money.
A: Exactly. They will cut back on their output, incomes will fall, consumption may fall further, and savings will also fall, cutting back on the initial increase that we started with.
Q: When will this process stop?
A: When firms stop accumulating inventories i.e. when DS=0. Then, and only then, will S=DK.
Q: But how can this be? We have assumed that DK stayed the same, and we started with an increase in S?
A: You have not been paying attention. Each time firms reduce their output to match lower demand, incomes and savings fall. Eventually the initial rise in savings is reversed, because overall income has fallen.
Q: Got it. But textbooks make a big thing about aggregate savings equalling investment. If it is just an accounting identity, why is it important?
A: What the textbooks really mean is that we eventually end up with a position in which S=DK. And that is important, for the reasons we have just discussed. It is called the paradox of thrift. A desire by consumers to increase savings ends up just reducing output, and savings do not increase at all. (Of course they are still saving more of their income: S/Y has gone up, but because Y has fallen, not because S has increased.)
Q: But I thought with all this ‘just in time’ production stuff, firms did not hold many inventories any more.
A: Well we could short circuit the story by forgetting about inventories and having firms accurately forecast what demand will be, and therefore what their output should be. In practice what we call involuntary inventory accumulation can still be important when looking at quarterly movements in national output.
Q: But is it realistic to assume investment – I mean DK – stays the same if savings are initially higher? If there are more savings around, it becomes cheaper to borrow, which will encourage investment, right?
A: It might, but it might not. In particular, if output is falling, firms may be reluctant to add to their capital stock.
Q: But won’t interest rates keep falling until they do? After all, the asset market has to clear.
A: Savers have an alternative, which is to just keep their savings as money.
Q: But they will put the money in a bank, and the bank will lend it.
A: Maybe, but the bank may just decide to hold on to the cash.
Q: It seems to be really important what people do with their additional savings.
A: Perhaps. But I think the key point is that, most of the time, the person doing the saving is different from, and has different motives to, the person doing any investing. A highly complex financial system links the two. And in that system, there will be lots of opportunities for the additional savings to be parked as money.
Q: Money seems very important here. It is why the extra saving does not have to find its way into more investment.
A: I think that’s right.
Q: If people hold the extra savings as money, will that not increase money demand. What happens if the central bank keeps the money supply fixed?
A: People hold money not just as a way of saving, but also to buy and sell things. And if less is being consumed, there is less need for money on this account. It is difficult to predict what will happen to the total demand for money, which is why central banks nowadays focus on determining short term interest rates rather than the money supply.
Q: That’s not what it says in my textbook. It says the central bank fixes the money supply.
A: Yes I know. I’m afraid it’s a bit out of date. Don’t ask me why.
Q: So if the central bank determines the interest rate, why don’t they ensure the interest rate is low enough to encourage firms to buy more capital goods?
A: That is what they would like to do. There are two problems. First, it may take some time for the monetary authorities to work out what is happening, and what the right interest rate is. (I could talk about real and nominal rates here, but let’s leave that for another day.) Second, nominal interest rates cannot go below zero, and maybe we would need negative interest rates to persuade firms to raise investment enough.
Q: My textbook also says that the classical model assumes interest rates adjust so S=I, by which I assume they mean S=DK. Does that mean the classical model is wrong?
A: Only if you think it applies at all times, and that there is no other reason why output cannot fall. However if we assume that the monetary authorities eventually are able to chose the right interest rate, then the classical model is fine when thinking about economies over a long enough time horizon.
Q: This all seems like common sense. I feel a bit stupid not to have understood this before.
A: Don’t worry, you are not alone.
Roger Backhouse and Bradley Bateman has a nice piece in New York Times on the lack of perspectives and alternatives shown by mainstream economists when dealing with the systemic crises of modern economies:
Economists do much better when they tackle small, well-defined problems. As John Maynard Keynes put it, economists should become more like dentists: modest people who look at a small part of the body but remove a lot of pain.
However, there are also downsides to approaching economics as a dentist would: above all, the loss of any vision about what the economic system should look like. Even Keynes himself was driven by a powerful vision of capitalism. He believed it was the only system that could create prosperity, but it was also inherently unstable and so in need of constant reform. This vision caught the imagination of a generation that had experienced the Great Depression and World War II and helped drive policy for nearly half a century …
In the 20th century, the main challenge to Keynes’s vision came from economists like Friedrich Hayek and Milton Friedman, who envisioned an ideal economy involving isolated individuals bargaining with one another in free markets. Government, they contended, usually messes things up. Overtaking a Keynesianism that many found inadequate to the task of tackling the stagflation of the 1970s, this vision fueled neoliberal and free-market conservative agendas of governments around the world.
THAT vision has in turn been undermined by the current crisis. It took extensive government action to prevent another Great Depression, while the enormous rewards received by bankers at the heart of the meltdown have led many to ask whether unfettered capitalism produced an equitable distribution of wealth. We clearly need a new, alternative vision of capitalism. But thanks to decades of academic training in the “dentistry” approach to economics, today’s Keynes or Friedman is nowhere to be found.
And Philip Mirowski has an equally interesting article on opendemocracy.net explaining why neoclassical economists
don’t seem to have suffered one whit for the subsequent sequence of events, a slow-motion train wreck that one might reasonably have expected would have rubbished the credibility of lesser mortals.
Two articles well worth reading!
Den socialdemokratiska partistyrelsen har lagt förslag om nationella kvalitetslagar för ordning och reda i välfärden. Syftet är att förbättra kvaliteten och avsevärt begränsa vinsterna.
Detta är ett viktigt beslut. Socialdemokraterna går nu till val med löfte om vinstbegränsning.
Men beslutet rymmer också problem. Mest problematiskt är att partistyrelsen inte föreslår någon formell reglering. Sverige ska förbli det enda land i Europa som saknar regler för vinstuttag i välfärden. Inte ens skolan ska fredas från vinstintressen …
Den skarpa ideologiska skiljelinjen mellan borgerlig och socialdemokratisk politik har alltid handlat om synen på marknaden. Socialdemokraterna ser att marknadens räckvidd behöver begränsas, i syfte att värna gemensamma intressen som familjebildning, hälsa, social sammanhållning och miljö. Borgerligheten har i olika grad förespråkat marknadens frihet.
Ett problem med förslaget om nationella kvalitetslagar är det saknar en konsekvent idé om vad marknaden kan – och inte kan – göra. Det sägs å ena sidan att marknaden inte klarar att lösa välfärdens kvalitetsproblem. Å andra sidan bejakas i förslaget både vinstuttag och marknad. Det saknas tydlighet i såväl analys som värderingar.
Vissa åtgärder, som öppna böcker, syftar uppenbarligen till att förbättra marknadens funktion. Med mer information kan brukare lättare välja rätt och verksamheten kan kontrolleras.
Men andra åtgärder, som möjligheten att sätta stopp för nyetableringar, syftar tvärtom till att försämra marknadens funktion. En marknad utan fritt tillträde kan ju aldrig bli en effektiv konkurrensmarknad. Vinstens disciplinerande funktion går om intet när nya konkurrenter inte får komma intill. Istället får politiker makt att dela ut exklusiva rättigheter till lokal näringsverksamhet. Vore det inte rimligt att företag som ges exklusivt tillträde till en stängd marknad också har ett tak för vinsten? …
Staten [bör] behandla alla medborgare med samma omsorg och respekt … Det är en princip som jag ställer mig bakom, men jag menar att vi idag bör ställa oss frågan om det överhuvudtaget är möjligt att följa denna princip på en välfärdsmarknad, där marknadslogiken i långa stycken har ersatt de normer som bärs av demokratin. Vad händer med medborgarskapet, när staten, som har uppdraget att behandla alla medborgare lika, delegerar skolor och sjukhus till vinstdrivande företag, som har ett intresse av att skilja ut lönsamma medborgare från olönsamma? Och vad händer med socialdemokratins allra viktigaste mål: jämlikheten, när marknadskonkurrens och valfrihet driver fram bristande likvärdighet och segregation? …
Jag vill hävda att det aktuella förslaget om vinstbegränsning i välfärden, Nationella kvalitetslagar – för ordning och reda i välfärden, tyvärr i allt väsentligt bärs av marknadsliberala idéer.
De bärande värden som lyfts i förslaget om nationella kvalitetslagar är effektiv användning av skattemedel och att alla ska ha rätt att välja.
Dessa värden kan framstå som självklart positiva, men de står inte desto mindre i tydlig konflikt med de värden som socialdemokratin historiskt har hävdat.
Vi har hävdat idén att välfärdspolitik är en produktiv investering, inte en kostnad att minimera.
Vi har hävdat att idén att jämlikhet – alla människors lika värde – är ett överordnat mål för samhällsbygget.
Vi har hävdat att frihet – och valfrihet – har en gräns. Friheten måste vara hållbar. Den måste vila på en bädd av solidaritet.
Lena Sommestad, ordförande för S-kvinnor
Man kan inte annat än hålla med denna eminenta ekonomihistoriker i den analys hon gör. Frågan är nu om landets socialdemokratiska gräsrötter kan leva med partiledningens kompromissförslag. Risken är tyvärr stor att Socialdemokraterna – om man håller fast vid förslaget – förlorade nästa val redan i fredags.
An act of individual saving means — so to speak — a decision not to have dinner to-day. But it does not necessitate a decision to have dinner or to buy a pair of boots a week hence or a year hence or to consume any specified thing at any specified date. Thus it depresses the business of preparing to-day’s dinner without stimulating the business of making ready for some future act of consumption. It is not a substitution of future consumption-demand for present consumption-demand, — it is a net diminution of such demand …
If saving consisted not merely in abstaining from present consumption but in placing simultaneously a specific order for future consumption, the effect might indeed be different. For in that case the expectation of some future yield from investment would be improved, and the resources released from preparing for present consumption could be turned over to preparing for the future consumption … In any case, however, an individual decision to save does not, in actual fact, involve the placing of any specific forward order for consumption, but merely the cancellation of a present order. Thus, since the expectation of consumption is the only raison d’être of employment, there should be nothing paradoxical in the conclusion that a diminished propensity to consume has ceteris paribus a depressing effect on employment.
The trouble arises, therefore, because the act of saving implies, not a substitution for present consumption of some specific additional consumption which requires for its preparation just as much immediate economic activity as would have been required by present consumption equal in value to the sum saved, but a desire for “wealth” as such, that is for a potentiality of consuming an unspecified article at an unspecified time. The absurd, though almost universal, idea that an act of individual saving is just as good for effective demand as an act of individual consumption, has been fostered by the fallacy, much more specious than the conclusion derived from it, that an increased desire to hold wealth, being much the same thing as an increased desire to hold investments, must, by increasing the demand for investments, provide a stimulus to their production; so that current investment is promoted by individual saving to the same extent as present consumption is diminished.
J M Keynes General Theory
In an interview not so long ago, Steven Levitt (Mr Freakonomics) answered the question how he could tell the difference between a smart person and a not-so-smart person:
Well, one good indicator of a person who’s not so smart is if they vote in a presidential election because they think their vote might actually decide which candidate wins. . . . there has never been and there never will be a vote cast in a presidential election that could possibly be decisive.
Statistics professor Andrew Gelman aptly retorted:
I have no problem if Levitt wants to argue that it doesn’t matter who wins the election—-I disagree with him, given the evidence that the positions of the Democratic and Republicans in the U.S. are farther apart on economic issues than are left and right groupings in many other countries, but Levitt can make that argument if he wants to. And I have no problem if Levitt wants to say that, for a one in a million or one in ten million chance, he doesn’t think it’s worth it to vote … But I don’t think Levitt is making sense when he writes that “there never will be a vote cast in a presidential election that could possibly be decisive.” A low probability is not a zero probability …
This confuses a lot of people, so I can understand Levitt missing the point, but it’s important. Just cos you don’t know whether a particular vote was decisive, that doesn’t mean a single vote can’t be decisive. Or, to put it another way, when he says “virtually impossible,” I’d say “1 in a million to 1 in 10 million if you live in a swing state.”