September 30, 2000
Documents supporting application for promotion to Professor of Richard Hudson

Would Time Travellers Affect Security Prices? Financial Economics and Physics
by Richard Hudson


Abstract

Financial markets in a world with time travel would look very different from ours. But would time travellers come to our time, making our markets look like theirs? This paper discusses this issue and related matters such as the problem of prediction in financial economics, the nature of security prices, the social and mental nature of financial reality, and the relation of Financial Economics to Physics. It presents a solution to the problem of bilking behaviour of time travellers, and gives a definite answer to the title question.


Introduction

Since H.G. Wells (1895/1991), the advantages of time travel(1) for investors have been clear.(2) For the time traveller, risk would be replaced with certainty. Active investment policies would be far superior to passive. There would be no need for holding a diversified portfolio. Stock picking and market timing would make sense.

A model of equilibrium capital asset pricing in a time-travel world is presented in Reinganum (1986). Reinganum then uses this model to make some strong claims about the relation of economics to physics. In this paper, I will concentrate on the difficulties inherent in attempting to model human behaviour in a world far different from our own and on the necessarily intentional (mentalistic) aspect of financial economics.

Equilibrium capital asset pricing in a world with time travel

Reinganum's (1986) model of capital asset pricing is based on an arbitrage argument, and he arrives at a simple result: whatever the equilibrium prices for individual securities are, they will be unchanging over time and returns (nominal returns) will be zero on all securities throughout time.

For a time traveller, all investments are risk-free since she will have perfect knowledge of prices throughout time. In fact it makes no sense to talk of risk in a world of certainty. Reinganum (1986) claims time travellers will run around time, buying "underpriced" securities and selling (or shorting) "overpriced."(3) Security prices are really a function of supply and demand. Time travellers will supply or demand virtually infinite quantities of mispriced securities thus driving their prices to equilibrium levels. The equilibrium price will be the price that obtains in all temporal markets - otherwise arbitrage opportunities would still exist for time travellers. The existence of unchanging prices, of course, means rates of return, even in nominal terms, will be zero.

Because of the strange temporal nature of time travel, the adjustment in prices due to the trading activity of time travellers would be instantaneous, as seen by the time-bound. Time travellers from all time would be present en masse attempting to realize their arbitrage profits: the very first time traveller would meet multitudes from her future (and even multiple copies of herself) trying to engage in the same transactions she is trying to complete.

Economics and Physics

Financial markets in a time travel world would look far different from what we see. We don't see zero rates of return, zero inflation, and unchanging security prices. Instead we see lots of volatility in financial markets. Reinganum (1986) concludes that this means that time travel must be impossible. It is impossible, because if it were ever to exist, the effect would be seen already in our financial markets. So Economics, then, can show Physics the way in the complex issue of time travel. (Some physicists discuss time travel in a very serious way since nothing in Physics seems to outlaw it and some theories seem to require it.)(4) Reinganum (1986) claims: "This particular problem [i.e., the problem of time travel] shows that the tools of economics, a social science, can make a strong prediction about the physical universe."(5)

Whenever we claim that Economics can prove something to Physics, we must get worried. Economists have a hard time proving anything to anyone, even to themselves.(6) The question to ask here is whether anything could prevent a time travelling world's affecting the security prices we see, moving them to the time travel (unchanging) equilibrium prices.

Paradoxes and bilking behaviour

Reinganum does not talk about the paradoxes of time travel, and claims that there is no need to do so: since time travel is impossible, we need not worry about trying to understand the paradoxes which would exist if it were possible. But it is hard to discuss time travel without dealing with the paradoxical situations to which it seems inevitably to lead.

Lewis (1976) claims that any change to the past, even just being there when one had not been there "before," is paradoxical. But, as Brown (1992) notes, the really paradoxical situations come from actions which undermine their own circumstances (p. 436). Non-self-defeating actions, he believes, present no real problem. Horwich (1987) points out the difficulty of knowing whether time travellers had been present already in the past - e.g., the time travelling Horwich, who has no memory of ever having been punched in the nose, may have gone back in time and punched his earlier self on the nose: the fact he has no memory of this does not mean it didn't happen - he may merely have forgotten (p. 117). And, as Lewis (1976) notes, fully consistent causal loops, where the elements on the loop are caused by elements "earlier" on the loop, but later in time, present no real problem (except the problem of their origin).(7) Naturally, "mere-Cambridge changes" (see Ni, 1992) do not present a problem.

Paradoxical or self-refuting actions by time travellers seem possible because many different simple actions which we all may perform from time to time - such as killing babies or grandfathers(8) - can be self-refuting when undertaken by time travellers.(9) As Lewis (1976) notes there is an ambiguity in the word "can" when one speaks of what time travellers "can" do. He says of the time traveller Tim that (p. 151): "He can and he can't [kill Grandfather], but under different delineations of the relevant facts" - namely, he has ability and motivation, but, given the fact that Grandfather survived and the past cannot be changed, the time travelling Tim has no hope of succeeding. The problem of time travel is naturally linked to the old question of free-will versus determinism. Brown (1992) claims determinists would claim time travellers cannot do other than what they did, while compatibilists would claim time travellers can change the past but always choose not to. Horwich (1987) notes that there may be some consistency constraints on causality and that backward causality may simply be such that self-refuting actions are impossible.(10)

Horwich (1987), however, rejects his own argument about consistency constraints because he can see no mechanism that would make them work. The only two mechanisms he proposes are a "psychological quirk of time travellers" (p. 123) that makes them not want to take self-defeating actions, or "an endless string of improbable coincidences" (p. 123) which makes all attempts at self-defeating actions fail. However, he claims that our knowledge of human nature is such that we know some time travellers would attempt self-refuting actions, and he claims that it would make more sense to assume that time travellers would not fail when they tried to engage in bilking behaviour.(11) Horwich's conclusion is that time travel to the local past must be impossible for some physical reason. However, if travel to one's local past is possible, then something must prevent time travellers' taking self-refuting actions.

Investing in the past and bilking behaviour

Horwich (1987) does not formally justify his "bilking inclination" - he is not using any particular model of human behaviour. In finance, however, we do have a model of behaviour. Time travelling investors would seem to have the same motivations as the time-bound: they would be wealth maximizers. They would also seem to have the same abilities to engage in transactions on the stock market as the time-bound.

Self-defeating prophecies are, of course, common in finance literature. One reason that security price changes might be unpredictable is that if they were predictable, people would transact immediately rather than in the future, thereby defeating the prediction (see Samuelson (1965)) . There is a general problem in the relationship of knowledge to action in financial economics. The rational actor has beliefs and desires and tries to realize his desires by acting on his beliefs. According to Rosenberg (1992), all of economics depends on the rationality notion of belief-desire talk, or, as it is commonly called, "folk psychology." Beliefs and desires are examples of "intentional" states, where the term "intentionality" is that introduced by Brentano (1874/1995) to refer to the mark which divides the mental from the physical. As Davidson (1974/1980) points out, we understand human action as rational, i.e., based on beliefs and desires. Actions not based on beliefs and desires either aren't actions (and the philosophers have long lists of non-actions, such as moving your leg when a doctor hits you on the knee with a rubber hammer) or they are incomprehensible. Purely physical beings - e.g., rocks - don't seem to have beliefs and desires and don't act. We explain their movements in totally non-intentional terms, but intentionality is essential for economic explanation (see Rosenberg, 1992, chapter 5 "Economics and Intentional Psychology" or Hausman and McPherson, 1996).

In finance, the investor's beliefs and desires are simple. Beliefs are expectations of cash flows to be received and of their risk. Desires are to maximize wealth given risk preferences, which vary among investors. Time travelling investors, however, are, as investors, totally homogeneous. They all know the amount of cash flows they will receive from investment, and none of these cash flows is risky. They all have the same desire to maximize wealth, but in a world of certainty, they can be said to have exactly the same "risk preference."

The actions of the time traveller, in acting like the time bound (i.e., in wealth maximizing), are in themselves self-defeating. The time traveller, in looking at the past, sees changing prices and picks the best moment to buy. But all time travellers from all time are looking at the same data and are making the same decisions. Because the last trade registered (by the time-bound) is at an advantageous price for the time traveller, when she attempts to trade, she finds herself beside a multitude of time travellers, also trying to trade. But demand (for underpriced securities) then is infinite for any price below the time travel equilibrium price. Thus any time traveller, in attempting to enter into a transaction will find that the prices at which she can transact are not the prices she sees in her data, but are instead the time-travel equilibrium prices (i.e., the unchanging prices).

The problem here is not limited to time travellers. Prices one can observe are always for trades which were actually made. Usually, the price on the very next trade will not be far different from that on the last trade. But since there are only a finite number of shares available at any particular price, if the quantity demanded (or offered) is large, then large price changes are not unusual.

The time traveller, then, probably cannot transact at the favourable prices she sees when she looks at her data tape of past stock market prices. The time-bound, of course can trade at those prices - they can because they actually did. The time traveller, facing the time travel equilibrium prices, refuses to transact because she sees no arbitrage profits - she will not even time travel since she is time travelling only to realize arbitrage profits. Successfully transacting kills the incentive to transact, thereby making the transaction impossible, just as successfully killing one's infant self (the philosophers' favourite example of a paradox) makes it impossible to time travel to do so. Rational time travellers, who are even more homogeneous than ordinary investors, will anticipate the actions of other time travellers, and will realize there is no point in attempting to use time travel to invest. Their failure to transact is not due to any improbable coincidences or psychological quirks.

The unsatisfactory nature of this argument

What I have presented above is a possible scenario where time travellers would decide on their own, using their own utility functions, not to engage in bilking behaviour. This scenario is highly artificial and depends on perfect rationality of all investors. It is similar to an argument in Arntzenius (1990), which attempts to show the possibility of constructing theories of tachyons which prevent the paradoxes' occurring, without, however, actually claiming that no other theory is possible. It is an answer to Horwich's doubts that any argument can be made to outlaw bilking behaviour, but it may not be a very convincing argument.

Unfortunately, the above argument is of a similar form to the old Economics joke about the impossibility of $20 bills lying on sidewalks.(12) If time travelling investors are the super-rational types who appear in Economics textbooks, then none of them will time travel.(13) But we all know that investors are really a bunch of nuts, and some damn fool will go ahead and time travel and try to invest. Since time travellers will come from all future time, there will be bound to be huge numbers of damn fools trying to invest in the past (our present). If so, then Reinganum is right: time travelling investors would attempt to change the past, and since we don't see any of them, time travel is impossible.

Problems with bilking: motivations and abilities

Horwich's (1987) argument against time travel to the recent past depends on time travellers having the same abilities and motivations as the time-bound.(14) But time travellers obviously differ from the time-bound in some respects. Perhaps, in Lewis's (1976) example, nobody ever actually shot Grandfather during his lifetime because they did not realize what a horrible person he was. Tim, however, does know the consequences of Grandfather's actions, and so travels back in time to kill him before he can do his worst deeds. In Reinganum's (1986) example, although the time travellers may have the same ability as us to buy and sell stock, they would differ from us in their abilities to pick winners. Brown (1992), p. 433, proposes that time travellers and "retro-causal engineers" might somehow be different from us so that common sense assumptions about abilities and motivations, which may well apply to us, might not apply to them. He notes (p. 437) that "... a defender of backwards causation might argue that those who know or are in a position to know facts about the outcome of their efforts that we don't ordinarily know, differ from us in a way relevant to assessing their capacities."(15)

A major difference between time travellers and the time-bound is that time travellers know the future. There are many social institutions and activities which seem to depend on our not knowing the future. I will concentrate on the stock market, but gambling, too, depends on the future not being known, and most decision problems are problems because of uncertainty about the future. In fact, one could adopt Reinganum's (1986) view and argue that if time travel existed, we could not have the kinds of social institutions we have, but since we do have these institutions, time travel is impossible.

But instead of making time travel impossible, it is possible that the time traveller's origin in the future, and consequent knowledge of the future, may serve to restrict her action in our present, preventing bilking behaviour here. It is possible that a time traveller finding herself in our time, differing from us only in her time of origin and her knowledge of the future, will not act in the same way we would if we had her knowledge: she will not have the same abilities and motivations as us.

What is different about time travellers? In part they have different knowledge than do the time-bound: they know what security prices will be. But they also have a different social identity from us: they come from different societies than we do.

Reinganum concentrates on knowledge, and he assumes motivations and abilities will be the same as ours. But motivations and abilities will depend largely on one's social identity.

Social identity

Our financial markets form part of our social reality which we have constructed over centuries of development. According to Searle (1995), social reality is thoroughly linguistic and intentional. It can be modelled by the structure "X counts as Y in C."(16) For example, this dollar bill (X) counts as money (Y) in America (the context, C).(17) In social reality, things are what they are because they are believed or accepted as being what they are. I remember arriving in Germany a few years ago after a long absence and having the shock of discovering that a 10 DM coin - $6 at the time - I had saved from my last trip was no longer legal currency. The coin still existed physically, but it was worthless: nobody would accept it in trade. On the other hand, minor changes in the magnetic medium where my bank keeps its records on me, count as money: money has many different physical realizations. Although economics talks of real goods and services, the main emphasis is on their value, and value is an intentional (mental) term. All of the objects of finance - for example, all finance data (security prices, dates of transactions, SEC filings, accounting statements, newspaper or other public information, etc.) - are linguistic and intentional. Objective statements can be made about these objects: we can all agree on the price of Microsoft stock at the close of trading yesterday. The objects themselves, however, are what they are only because people take them as being what they are. Finance is irretrievably mentalistic.

Predicting what time travellers will do, then, implies predicting what they will think and what they will value. We believe we know what we would do on financial markets if we knew what prices would be in the future. But the markets we have are part of our social reality. Indeed, these markets are really quite recent in human history and, despite "globalization," do not exist everywhere in today's world. Even in large parts of Europe, there were no stock markets or private property for most of the century (prior to 1990). Finance theory has been developed mainly in America in this century, and despite its highly abstract nature, it has been a reflection, as well as a shaper, of twentieth century American social reality.

Time travellers coming from societies far different from our own may not even be able to understand what financial markets are or how they work. Most of our fellow citizens have no understanding of markets, so how would someone from a culture of the distant future? And even if time travellers were to come to understand what a financial security was, they may have no interest in money: they may not value what we value. Their society may be too rich to worry about some dollars from the past, or they might be more interested in love or in truth and beauty, or they might be concerned with saving their eternal souls and feel that wealth would get in the way. They may be fundamentally different from us in motivation: i.e., they may not be wealth maximizers or they may have a different view of what constitutes wealth. It is possible, even, that utility maximization is characteristic only of people living in certain societies,(18) and human rationality will express itself differently in the future. In any of these cases, time travellers from the future would not bother investing in our markets.

We simply cannot tell what time travellers will think or value: economic science surely does not permit us to predict the mental lives of others, and no other science (sociology?, political science?, psychology?) seems capable of such predictions.

Even if time travellers were to understand markets and they desired wealth, there is another aspect of social reality which makes it difficult to predict their actions. For an act to be the act it is, it must be carried out by authorized people in the right circumstances.(19) This is an important difference between social reality and physical reality. In Lewis's (1976) example, Tim should be able to fire the gun and kill Grandfather (although he can't because he hadn't). But while time travellers may be physically able to do many things, ownership of stocks and bonds is determined by social (and legal) conventions. Only authorized people can perform certain acts in social reality. If an unauthorized person performs an act, that act is invalid and has no legal standing. If I sign a contract with you, selling you the Brooklyn Bridge for $10,000, and you pay me, you still don't own the bridge (I don't own it either). When stocks and bonds are traded, cash is handed over for ownership rights to future cash flows. Ownership rights form part of social reality, and these rights are regulated by society. Time travellers may not be able legally to enter into contracts buying and selling securities, or they may not be able to benefit by investing in our markets.

Time travellers may have difficulty trading in our time due to our laws. While we don't specifically forbid investment by time travellers, unless time travel is invented in the next few years, any time traveller we encounter would be an illegal alien, or an underage minor, not authorized to sign contracts. If, say, time travel is invented in 2020 by someone born in 1990, that person would be eight years old today (in 1998) according to his birth certificate, regardless of his apparent physical age. Minors cannot enter into contracts. If time travel is invented in 2100, then no time traveller could even prove citizenship, and all would be underage. (The fight for the rights of the unborn would acquire new meaning in such a world.)

Time travellers would also have trouble with the IRS. Taxes may indeed wipe out much of their gains. For example, the investment strategy recommended in the Restaurant at the End of the Universe of investing a small sum in an interest bearing account and time travelling far into the future to collect compound interest wouldn't work due to taxes. Unfortunately, rates of return on fixed income securities are close to zero in real terms before taxes, and after tax real rates of return are almost surely negative(20) (of course, it might be hard to figure out how to pay taxes and how to reinvest funds when one is not there).

It is true that time travellers might be able to hide their identity, acquire false Social Security numbers, etc. But if so, why should they waste time investing: if they're going to act illegally, why not just steal the money instead?

Time travellers may have trouble in their own time as well. Trades on financial markets are zero-sum games. Time travellers will buy underpriced securities and sell overpriced. Each trade they enter into is a trade that had not happened. (We will abstract somewhat from the paradoxical nature of changing the past, and simply assert that the changes occur.) This means that individuals in the past who (prior to the time travel) continued to hold onto their stock, and who thus would have received the gains, will now not receive them. (Tenses get confusing when talking of time travel.) As the time travellers get rich, those who deal (dealt?) with them get poorer. From the perspective of the time traveller's own time, some citizens (who may be the heirs of these poor suckers) will become poorer than they were "before" (?) the time traveller travelled to the past. These are not Pareto optimal changes. Since permitting time travelling investment activity is not Pareto optimal for societies whose wealth allocations have already been determined, it is possible all future governments will pass laws forbidding time travellers to invest in the past. These laws would be passed before the first time trip as time travellers from the future reveal the necessity of such laws. And laws in a time-travel world can easily be enforced: the police have merely to time-travel to the site of the crime and either arrest the criminal in the act, or arrive a few seconds earlier and prevent the crime.(21)

Conclusion

A world with time travel would look very different from ours. Its financial markets would be very strange. Many social institutions would be different from ours. But the existence of our institutions as they are today proves nothing about the possibility of time travel.

Time travellers may be super-rational types who rationally anticipate that they won't be able to transact at the prices they see in their data series, so they won't even try to invest in the past. On the other hand they may be as nutty as the rest of us, in which case they would go to the past and invest. It is possible that they not have the same motivations as us - they may not care about wealth - and they may not have the same abilities - they may not be able to enter into transactions in our time. On the other hand, they might have the motivations and abilities and they might trade.

While people construct a social reality of laws, institutions, customs, language, etc., we have no science to predict the direction social reality might take. Finance deals with social reality, with people's beliefs and desires. It is thoroughly intentional. But financial economics can't tell what people will do. Time travellers might affect security prices. But they might not.

The science which we call "social science" is not particularly good at predicting anything at all. But then, so what? The physicists can't even tell us if the universe will continue expanding or not, and they don't know what the nature is of most of the matter in the universe. And some of them - as opposed to us - actually believe in time travel.


Appendix: Many worlds

David Lewis (1976) outlines a position similar to Everett's "many world" hypothesis. According to this position, all probabilities are frequencies. Individual events have probabilities because for each event, the universe splits, and everything which might happen actually does happen in one or more of the many worlds which result (the probability of the event's occurring is the percentage of total worlds in which it occurs). If the many worlds interpretation is correct, time travellers are not really travelling to their pasts and futures: they are travelling to other worlds, similar to their own, but yet different.

Time travelling investors have arbitrage opportunities only because they know all prices through time. The prices they see in their records will not, however, apply to the worlds they travel to for their investment activity. As soon as the time traveller makes her first investment, she will have changed prices. Although her arrival may have minor effects on the future at first, as Lewis (1976) notes, these effects may increase with time(22) - particularly if her stock market trading affects the allocation of resources in the economy. Depending on how "similar" worlds are, and on how sensitive prices are to new trading, the price information held by the time traveller may be of little or no use. If the time traveller follows one of Reinganum's (1986) active investment strategies - investing, then jumping to the future to pick up profits, then returning to the past to invest again - she will be switching worlds so often that her initial data will lose all validity. Even a time travelling investor following a more passive strategy of a single trip to the past will find that her "data" on prices is quickly rendered invalid by her own buying and selling activities. The time traveller simply will not have risk-free arbitrage opportunities: she will not know all prices through time for any of the worlds she may end up in.

Even if, somehow, the time traveller who goes to another world does manage to realize some arbitrage profits, she may not be able to increase her overall consumption. She cannot ever travel back to her own world and she does not know the future of her new world. The changes her trading activity has brought to the economy may have important social and political effects on the world, perhaps even causing wars, revolutions, etc. which had not happened and which may threaten her future wealth.

The "many worlds" interpretation of time travel, by introducing risk to the time traveller's investment opportunities, means that the actions of time travellers would not drive rates of return to zero. In fact, despite what Lewis (1976) says,(23) a world with time travel might look very much like our world, or, at least, its financial markets might look like ours.


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Endnotes:

1. In this paper, I use the term "time travel" to refer to what David Lewis (1976), p. 145, calls "science fiction" type of time travel, where the time traveller uses some apparatus to move from one time to another, in somewhat the way we move around space. I also assume the time traveller can costlessly move about time in either direction, as desired.

2. Wells (1895/1991), p. 7, where the "Very Young Man" says: "Just think! One might invest all one's money, leave it to accumulate at interest, and hurry on ahead!"

3. Reinganum (1986) actually talks about carrying securities around to different times, the way one would buy commodities in cheap markets, physically carrying them to sell in dear. This is not possible with securities, but the same effect can be achieved through taking either long or short positions.

4. See Gödel (1949a and 1949b) for the first serious physical argument for time travel. Thorne (1994) and Coveney & Highfield (1990) give more popular treatments of the physical issues. Savitt (1995) and Halliwell, Pérez-Mercader, Zurek (1994) give several useful readings dealing mainly with the problem of an arrow of time. Carroll, Farhi, and Guth (1992); Chown (1992); Deser, Jackiw, 't Hooft (1992); Gott (1991); Morris, Thorne, Yurtsever (1988); Redmount (1990); and Tipler (1977) give a taste of some of what physicists write on time travel. Malament (1985) uses an economic argument to argue against time travel in a Gödel universe: the amount of fuel to successfully complete a trip is more than the total amount of matter existing.

5. Reinganum (1986), p. 12. Reinganum claims his argument - that time travel is incompatible with our economic institutions - shows that time travel is "pure fantasy," that "it never has existed and furthermore never will exist," and that he is presenting "proof of the impossibility of time travel," and that his "proof will show that current economic conditions rule out the possibility of past, present, or future time machines."(All quotes from Reinganum (1986), p. 10).

6. DeLong and Lang (1992) propose a positive answer to their question "Are All Economic Hypotheses False?".

7. See Lewis (1976), 148-149: "The parts of the loop are explicable, the whole of it is not. Strange! But not impossible, and not too different from inexplicabilities we are already inured to."

8. Philosophers get very violent in their examples. The main example used by Lewis (1976) is Tim, the time traveller, who goes back to kill his evil Grandfather. Horwich (1987) prefers to kill his infant self, since it is easier to know who he is (was) than to know if Grandfather really was his grandfather.

9. Horwich (1987), p. 92, uses the expression "the bilking argument" to refer to conscious acts undertaken to undermine their possibility: "The rough idea is that any backward causation hypothesis would be necessarily refuted by the following experiment: repeatedly wait to observe the presence (or absence) of the alleged effect E, and then try to prevent (or produce) the subsequent, alleged cause L."

10. Horwich (1987), 119: "The fact that closed causal chains would be subject to consistency conditions is not at all remarkable."

11. Horwich (1987). Horwich's problem is that p. 122: "bilking attempts are usually attempts to perform actions that are ordinarily quite easy to perform." See also p. 125: "And we know enough about human motivation (specifically, about the factors that might produce this bilking inclination) and the kinds of phenomena that could cause this plan to fail (amnesia, gun-jamming, brilliant surgeons, etc.) to claim that any such correlation would be an improbable coincidence."

12. See McCloskey's (1990) version (p. 112), where he talks of $500 bills.

13. Samuelson (1965) notes that his proof that properly anticipated prices fluctuate randomly says nothing about how markets actually work (p. 783): "You never get something for nothing. From a nonempirical base of axioms you never get empirical results. Deductive analysis cannot determine whether the empirical properties of the stochastic model I posit come at all close to resembling the empirical determinants of today's real-world markets."

14. Brown (1992), p. 436, notes that most writers on time travel simply assume "that the time traveller and the retro-causal engineer don't differ from us in any way that would justify denying them capacities we take ourselves to have in similar circumstances."

15. I agree with Brown, but my claim here differs from what Brown ascribes to hard determinists and compatibilists. I make no claim about freedom or determinism, and I state that the time traveller may decide not to make changes to the past in the case I discuss simply because he will not want to. This is different from Brown's compatibilist who would say the time traveller could, but does not, make changes to the past, or the determinist who would say the time traveller may want to but be physically unable to change the past.

16. Searle (1995), p. 43-51, and chapters 4 and 5, "The General Theory of Institutional Facts" parts 1 and 2.

17. See Searle (1995) p. 31-57 (particularly p. 37-43) and Nadeau (1996) for the intentional nature of money.

18. As Nietzsche (1889/1969) noted, p. 55: "Der Mensch strebt nicht nach Glück; nur der Engländer thut das."

19. As Searle (1995) notes, p. 54: "If you are the chairman, then saying in appropriate circumstances "The meeting is adjourned" will make it the case that the meeting is adjourned ... The same words said by the wrong person or in the wrong circumstances will have no such effect."

20. Brealey and Myers (1991), p. 131, cite Ibbotson Associates, Stocks, Bonds, Bills, and Inflation: 1989 Yearbook (Chicago: Ibbotson Associates, 1989), to show that the average annual return on Treasury bills over the period 1926-88 is 3.6% in nominal terms (0.5% in real terms). It is trivial to show that at any tax rate of 15% or more on interest income, the real after-tax return is negative.

21. Lewis (1976) makes disparaging remarks about attempts to find "chaperones" who will miraculously prevent time travellers' engaging in bilking behaviour (p. 149, "To imagine such a chaperone, as some authors do, is a boring evasion, not needed to make Tim's story consistent."). I would maintain that the above argument, based on a theory of social decision making is not the type of "chaperone" Lewis is discussing.

22. Lewis (1976), p. 152: "Suppose that at the possible world of Tim's story the space-time manifold branches; the branches are separated not in time and not in space, but in some other way. Tim travels not only in time but also from one branch to another. ... The other branch diverges from the first when Tim turns up in 1920; there Tim kills Grandfather and Grandfather leaves no descendants and no fortune; the events of the two branches differ more and more from that time on."

23. Lewis (1976), 145: "Time travel, I maintain, is possible. The paradoxes of time travel are oddities, not impossibilities. They prove only this much, which few would have doubted: that a possible world where time travel took place would be a most strange world, different in fundamental ways from the world we think is ours." See also, p. 148: "As I said at the outset, the time traveller world would be a most strange one."