J. MOREH

RANDOMNESS,

GAME THEORY AND FREE WILL

ABSTRACT: Libertarians claim that human behaviour is undetermined and cannot be predicted from knowledge of past history even in principle since it is based on the random movements of quantum mechanics. Determinists on the other hand deny that macroscopic phenomena can be activated by sub-microscopic events, and assert that if human action is unpredictable in the way claimed by libertarians, it must be aimless and irrational. This is not true of some types of random behaviour described in this paper. Random behaviour may make one unpredictable to opponents and may therefore be rational. Similarly, playing a game with a mixed strategy may have an unpredictable outcome in every single play, but the strategy is rational, in that it is meant to maximize the expected value of an objective, be it private or social. As to whether the outcome of such behaviour is genuinely unpredictable as in quantum mechanics, or predictable by a hypothetical outside observer knowing all natural laws, it is argued that it makes no difference in practice, as long as it is not humanly predictable. Thus we have a new version of libertarianism which is compatible with determinism.

1.

THE

PROBLEM

The free will controversy is concerned with the question whether there are human decisions and actions which are unpredictable in the sense that they do not follow from past states of affairs. Determinists call such behaviour "random". Libertarians prefer the term "undetermined", though they base indeterminacy on the unpredictable random events of quantum mechanics.X The general idea of determinism is that "the future of the world is fixed in one unavoidable pattern" (Weatherford 1991, p. 3). Given a full description of the state of affairs at a point of time and the laws governing the interrelations between the elements of that state of affairs, the future is predictable with certainty by a being of immense intelligence standing outside the system, who commands this knowledge. It is important to stipulate that the observer is not part of the system. 2 Predictability in this sense will henceforth be referred to as "predictability in principle". Determinists deny that sub-microscopic events can affect macroscopic ones. Libertarians, however, affirm that the neural events in the brain which control decision making can be Erkenntnis 41: 49-64, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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influenced by sub-atomic events. According to van Iriwagen (1983, p. 127) all our decisions must be undetermined for us to have free will. He does not say, however, whether all decisions are triggered by subatomic movements. Libertarians insist on indeterminacy because they believe determinism is incompatible with moral responsibility. Determinists, on the contrary, assert that if our actions are not causally related to our pasts, we cannot be held responsible for them. Many determinists accept moral responsibility without metaphysical freedom, if grudgingly, as a practical necessity (Weatherford 1991, pp. 13, 160, 230-2). The weakness of the libertarian position is that its proponents do not attempt to explain how non-probabilistic behaviour (with which the participants in the controversy are usually concerned) can be activated by a sub-atomic stochastic process. (See however Popper's and Eccles's (1977) attempt in Section V below). This justifies its opponents in charging it with falling over the precipice of randomness (Weatherford 1991, p. 224). This paper brings a pragmatic approach to the free-will controversy. Human voluntary behaviour is broadly divided into two classes: nonprobabilistie behaviour which the contestants usually have in mind, and probabilistic, unpredictable behaviour which includes the randomized strategies of game theory. Some of this behaviour is aimed at a definite goal and is rational, e.g., mixed strategies in a game with one equilibrium point; others are unguided like groping in the dark, e.g., multiequilibrium games without coordination. A well-known game theoretician calls the latter situation "free will" (Shubik 1983, pp. 2, 16), but there is no reason to restrict the term to this type of random action and exclude purposive and rational behaviour. It is this randomness that introduces indeterminacy into human affairs. Determinists must agree that not all random actions are irrational. They may point out (as may libertarians) that such randomness is on a macroscopic level, and is not the genuine randomness of sub-microscopic events. My answer is that from a pragmatic point of view it does not matter how the randomness is produced, as long as the random events cannot be humanly predicted. It is true that the controversy has been shunted on to new tracks, as the very definition of libertarianism has been amended. This procedure is justified as follows: (1) It takes into account random behaviour which forms an important part of

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human experience. (2) It allows for indeterminacy in human affairs without resorting to speculative and ad hoc explanations of how submicroscopic events in the brain set off macroscopic events. (3) It is compatible with determinism, which is an essential premise in most sciences. Without claiming that each decision or act is undetermined as the libertarians do, there is enough randomness in human actions to introduce a great deal of indeterminancy in human existence as a whole. Unpredictability is a matter of degree. Irrespective of how indeterminacy has been introduced into human behaviour, it has not made it completely, unpredictable. If unpredictability is of the uncertainty type, limits can be set within which the action will take place. If it is of the risk type, then the closer to unity" is the probability of the occurrence of an act, the more it is equivalent to certainty. 3 It would therefore be desirable to make unpredictability and freedom fuzzy concepts (on fuzziness see e.g. Zimmerman 1992). This should help blur the distinction between determinism and libertarianism. Writers who attempt to reconcile determinism with human freedom are called "compatibilists". They claim that unless behaviour is constrained, such as compulsive or neurotic behaviour, it can be both determined and free. Some go so far as to claim that determinism is necessary for freedom, for without it actions would be random or chance, and would not follow from our nature. This was the position held by Hume. Mill added that without determinism our actions would be random and irrational (see Weatherford 1991, pp. 78-86). It will be clear that this paper is introducing a new type of compatibilism. Without invoking metaphysical indeterminism it countenances the view that the course of human history cannot be prophesied with certainty (see e.g. Popper 1966). The following is a brief outline of the rest of the paper: In Section II various types of probabilistic and non-probabilistic behaviour are presented. In Section III it is shown that the former types of behaviour are not just theoretical constructs, but are applied with various degrees of approximation often without the use of special randomizing devices. In Section IV the libertarian approach of resorting to quantum mechanics as a source of indeterminacy is examined and criticized. Adopting a pragmatic approach it is suggested that mixed strategies and other probabilistic behaviour be relied upon instead. In Section V a simple model of moral choice is put forward. It is shown, using simple games,

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that small changes in an agent's moral propensity can cause large unpredictable changes. The implications of random behaviour for moral responsibility are examined. The paper is summarized in Section VI. 2. P R O B A B I L I S T I C AND O T H E R B E H A V I O U R

This section is mainly devoted to describing probabilistic behaviour. Other kinds of behaviour will be mentioned by the way. (a) Randomized purposive behaviour occurs even in the animal kingdom. Some birds, e.g., the snipe (Holland 1955, p. 32) and some insects, e.g., flies, fly in complex zigzag paths to avoid being surprised by a predator. They have an in-built capacity for random movement. Suppose a predator, in the form of a spotted flycatcher chases a fly. The fly then changes its movement pattern to evade the predator. The bird "pursues it determinedly, wheeling and twisting as if on hairpin bends" (May 1993, p. 3). This is a complex game of mixed strategies (see below). Similarly a child, even an adult, who is chased by another may engage it evasive movements, so that pursuer and pursued enter into a game of mixed strategies. Humans are not given to reflex and instinctive random motions like some animals, but they do have, in varying degrees, the ability to randomize, even without the use of special devices such as throwing dice, drawing bits of paper out of a hat or using tables of random numbers. (b) Mixed or randomized strategies: Some games have dominant strategies, others have Nash Equilibria (NE) in pure strategies. 4 Some, however, have NE in mixed strategies, that is, the best reply of a player to the strategies used by the others is to mix his s moves with given probabilities. A game whether in pure or mixed strategies may be played only once or many times (see Section III). Game matrix 1 has neither dominant strategies nor NE in pure strategies, but it has a unique equilibrium in mixed strategies. Maximizing the expected utilities of both players requires that A should mix his strategies with probability 0.5 each, while B should play bl and b2 with probabilities 0.2 and 0.8 respectively. 6 B bl A

al a2

3, -1,

b2 2 1

-1, O,

Game matrix 1

3 0

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Equilibrium in mixed strategies: al, a2 with probabilities 0.5 each; bl, b2 with probabilities 0.2 and 0.8. A's expected utility is 0.2, B's is 1.5. For the purpose of Section V, the payoffs are in units of higher welfare. A has v = 1. His payoffs for move a2 are: - 1 , which is the weighted sum of - 2 H and 1L; 0, which is the weighted sum of - 2 H and 2L. In all other payoffs H = 0, so that higher welfare coincides with L.

(c) Multiple equilibria: While game matrix 3 has a single NE, some games have many (see Shubik 1983, p. 245, Exercise 9.5 for a game matrix which has 3 NE in pure strategies and 4 in mixed strategies). The players, if unable to coordinate their actions, would not know which strategies to play. As by assumption they have no alternative but to play, they might achieve at random some outcome or other. "Random" is here used in the sense of "aimless" or "unguided". Shubik (1983, pp. 2, 16) calls such situations "free will" because they are indeterminate. But there is no reason why one should not apply this term to the rational, though random action of game (b) above as well. To help solve the problem of multiple equilibria in pure strategies, Schelling (1960, pp. 54-55) put forward the concept of "focal point" or "salient point", a point with featui'es that commend it as the most likely equilibrium point. Since he wrote, game theoreticians have produced very sophisticated "refinements" to NE aimed at reducing the number of equlibria. 7 (d) In a zero-sum game, that is, strictly competitive game, a multiplicity of NE in mixed strategies does not give rise to such indeterminacy, for then each of the players has a continuum of solutions among which he is indifferent because they yield the same payoff (Allen 1959, p. 532). The player has only to pick one of them (see also (e) below). (e) Game matrix 2 is a game against Nature, here represented by the equity market. Investor A may be very pessimistic, that is, he believes that Nature is set to defeat him. He therefore plays the game as a zero-sum game. The solution is in pure strategies (al, b2). However, if A is not pessimistic but only ignorant of the probabilities of the various prospects, the game degenerates into a one-person game. For by the Principle of Insufficient Reason (Rawls 1971, pp. 168-9) he ascribes to them equal probabilities. Hence his behaviour is parametric (that is, it is expected to elicit no response from other players) and is

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equivalent to behaviour in the face of risk, since the different prospects have been assigned probabilities. Nature Prospect for shares

Good

Average

Bad

A

30 40

25 30

20 5

buy shares al buy shares a2

Game matrix 2 The payoffs are utilities of the estimates of future profits. It is seen that the expected values of the utilities of shares aa and a2 are 25 each. A is therefore indifferent among any proportions in which he may split his principal between them, including the extreme proportions of 100% in one of them and nil in the other. Consequently, he may use any die, fair or loaded to decide on how to split his principal between them. Similarly, if he is constrained to invest in only one type, it is immaterial whether the randomizing procedure is fair or not. (The same applies to the choice of mixed strategy in game (d) above). If the expected values of the utilities of al and a2 were unequal, he would choose the one with the higher expected value. The choice in this case is parametric behaviour. (f) The following game is based on Smart (1973, pp. 57-60) who intended to make a point in utilitarian ethics. I have changed the story (it had to do with breaking government regulations intended to save fuel in wartime) to make it more acceptable to common-sense morality. The game is not usually played, but it is well known in the ethical literature, has been quoted by many authors (e.g. Harsanyi, 1977 p. 52) and is relevant to the present discussion. There are n fishermen around a lake who have the same cost function. The social costs of fishing are increasing if only because of the depletion of the natural resource with the increase of the fishing activity. Maximizing the net sum of utilities for all fishermen, i.e., total benefit less total cost, requires that the number of fishermen fishing each period be m < n. All the fishermen are ethical to the extent that they aim at maximizing the net sum of benefits while giving each fisherman an equal chance. Each has 2 strategies: Fish, Do not fish, but they cannot coordinate a joint strategy to decide who will be the m fishermen who fish in a given period. The solution is that each gives himself a probability of fishing

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of m/n. We have here a multi-person game with randomized strategies, the aim of the players being to achieve a dual ethical goal: conservation of a natural resource and a fair opportunity for every fisherman. Not all game theoreticians are equally at ease with the concept of mixed strategies (see also next Section). Rasmusen (1989, pp. 70-73) has urged in connection with game (b) represented by game matrix 1, which is borrowed from him, that if A plays his optimal mixed strategy (0.5, 0.5) then B will be indifferent between his pure strategies bl and b2 (see footnote 6) and a continuum of mixed strategies. If B adopts any other strategy than (0.2, 0.8) he would cause losses to A, who would then change his strategy, so that the game would collapse. Rasmusen resorts to various interpretations of the game in which player B represents a large number of players of two different types. I do not think these interpretations are needed, for if B anticipates the collapse, he would avoid deviating from his optimal strategy unless he expects to benefit from the deviation. Moreover, if B's deviating strategy is known to A, the latter can play in such a way as to gain at the expense of B. In any case the question arises: who deviates first? Do we need another game to decide who does? While discussing probabilistic behaviour in this section, we have come across two types of non-probabilistic behaviour. The first is the pure strategies in games (c) and (e) the latter being considered a zero-sum game. The second is parametric behaviour, in game (e) interpreted as a one-person game where the expected value of the utility of a2 is assumed to exceed that of al so that a2 is chosen. Another kind of parametric behaviour not considered here is that of a single small consumer who cannot affect the market price. 3.

RANDOMNESS

IN

PRACTICE

The kinds of random behaviour described in the previous section are not just ideal constructs, but are approximated to in everyday life. I said above that people differ in their ability to randomize without the use of special devices. Kendall and Stuart (1958, pp. 208-211) present data which show that people differ in their ability to estimate the last digit in scale reading. The ones whose last digits are evenly distributed over the digits 0 to 9 have the best ability to randomize. They conclude: "There may be people whose psychological processes are so finely balanced that they can deliberately select random samples"

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(p. 211). The test for the ability to draw simple random samples is that in a large number of draws the objects of different relevant characteristics, e.g., size, are represented in about the same proportion as in the population as a whole. Those who tend to draw a certain size more frequently than others are biased like a loaded die. This bias does not always make their behaviour more predictable, especially if the bias changes over' time. To randomize with unequal probabilities without technical devices may be more difficult than simple random sampling (e.g., in game matrix 1 where B has to use his moves with probabilities 0.2 and 0.8). If the game is repeated many times the error may be perceived by the opponent (see Section II). For a few repetitions, however, the error may not be noticed. In cases where the agent is indifferent between a range of outcomes, as in games (d) and (e) of the last section, it does not matter whether he gives equal probability to all outcomes or he is biased, e.g., towards odd or even numbers. All that matters is that he should make a decision. I now describe some of the uses of mixed strategies in everyday life. There is the random selection of samples by the U.S. tax authorities (Kreps 1991b, p. 103 footnote, Rasmusen 1989, p. 72) and the drawing of prizes. In American football games the offensive team has to decide whether to run or to pass. What is important is to choose an action not expected by the other team (Rasmusen 1989, p. 70). Akin to this is the bluffing in poker mentioned by Kreps (1991b, p. 103 footnote) "If you hold a bad hand, you will sometimes bet heavily on it and sometimes not, choosing in each instance randomly between bluffing (betting) and not". He adds that the choice is such as to make the opponent indifferent between calling the bluff and giving in to it. The above are repeated games. Some authors, however, question whether in a one-off play of a game it is sensible to choose a move at random (with the required probability). But this is the rational thing to do. The alternative is choice in an aimless and unjustifiable way. Binmore (1988, p. 31) sees no difficulty. According to him "Jokes about finance ministers tossing coins to decide whether to devalue are misplaced. They can, and they always have appointed a committee of experts achieving precisely the same effect." It is random behaviour that introduces wilful indeterminacy into human affairs. This is different from the uncertainty or risk the agent faces in a game against Nature. In his criticism of the libertarians

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who ground free will on the random motions of sub-atomic particles, Weatherford (1991, p. 9) asks: "Is this the vaunted human liberty that it is a matter of chance whether I watch the game today or not?" The answer is "Yes, if mixed strategies are used, and without invoking subatomic particles".

4. THE LIBERTARIAN POSITION One serious weakness of the libertarian position is the following: 8 Libertarians make freedom of action, say parametric action, dependent on the random movement of sub-atomic particles without giving an adequate explanation of how this happens. Popper and Eccles (1977, pp. 540-1) can only speculate on the subject: "One may make use of quantum theoretical indeterminacy without committing oneself to the view that free-will decisions are probabilistic affairs". They draw an analogy from genetic mutations which are probabilistic but natural selection operates on them. They go on to say that free-will decisions suggest themselves as a set of probabilistic proposals of which only the acceptable ones are selected. The ad hoc character of this reasoning is pointed out by Honderich (1988, p. 305). Popper and Eccles would be hard put to it explaining free-will decisions even if they interpreted them as probabilistic as is done in this paper. For example, in game matrix 1 it is difficult to see how player B could harness quantum mechanical random events to decide to play, or to actually play, moves bl and b 2 with probabilities 0.2 and 0.8 respectively. Libertarians view human actions as undetermined but not random or uncaused. They base this distinction on the thesis that decision and actions are caused by the self which is a stable entity. To this Weatherford (1991, p. 13) answers that the self is not an unchanging entity, but consists of changing states. If it were unchanging, there is no reason why a certain decision was taken at time tl and not at time t2. An attempt has been made in this paper to adjust the libertarian position so as to make it more plausible. Random human behaviour is unpredictable (it does not matter if it is predictable in principle as long as it is not humanly predictable). It can be irrational, or rational aimed at promoting private or social objectives. 9 It introduces indeterminacy into human life. There is no need to invoke quantum mechanical indeterminacy which in fact raises more problems than it solves. Determi-

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nists should not object to randomness so interpreted, as it is consistent with determinism and is not necessarily irrational. The question now arises whether it is appropriate to hold people responsible for the consequences of their probabilistic behaviour. This will be discussed in the next section. 5.

MORAL

CHOICE

Introducing the moral dimension into strategic behaviour will make the very moral character of one's action dependent upon t h e strategies of one's opponents. Moreover, a slight change.in the agent's moral propensity such as acting on impulse may have a considerable effect on the outcome. The approach I shall adopt is that of common-sense morality (see Slote 1985, Moreh 1986, 1992). I shall put forward a simple model of moral behaviour. People are expected to give very high priority to morality, that is, to have a very high moral propensity. In fact this is not generally the case. For expository purposes, it is better to start with parametric behaviour. In a given situation, an agent has to choose one among various mutually exclusive courses of action, each of them yields generally both private utility for the agent and moral value. These will be measured in units to be called L (or lower goods) and H (or higher goods) respectively. Both H and L may be positive, zero or negative, and are assumed to be common knowledge. 1° To take an example, in a given situation, an agent may either tell the truth and achieve 10H and - 1 0 L or tell a lie and score - 2 0 H and 19L. It is assumed that in this particular situation these are the only alternatives open to the agent. His moral propensity (which reflects his moral character) is measured by v, the rate of exchange of H in terms of L which guides his behaviour. In his oft-quoted paper, Sen (1977, Sections 7 and 8) speaks of commitment to ends and ideals other than one's utility and sympathies, but he is silent on the degree of commitment, as if an individual is either entirely committed to ethical values or not committed at all. My concept of moral propensity v can be interpreted as a measure of commitment. Obviously the magnitude of v depends on the units of H and L. Apart from this it differs among agents, a higher v indicating a better moral character. In the case being discussed let v = 1. Then truthtelling yields 10 × 1 - 10 = 0 units of higher welfare (a weighted sum

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of H and L). Lying yields - 2 0 × 1 + 19 = - 1 units of higher welfare. Since truth-telling has a better outcome, the agent will tell the truth. If v drops to 0.95 it is easy to verify that a lie will be told (10 × 0.95 - 10 = - 0 . 5 is less than - 2 0 × 0.95 + 19 = 0). (To find the value of v for which the agent is indifferent between truth-telling and lying write 10v - 10 = - 2 0 v + 19. This gives v = 0.967. If v exceeds this value the agent will tell the truth, if it is below it he will lie). Note that if the agent was playing a game, truth-telling and lying would be strategies. Depending on the opponents' payoffs, he might have to play a mixed strategy, his telling the truth or a lie in any play being a probabilistic outcome. This will become clear in what follows. Let us go back to game (b) of Section II and game matrix 1. Let the payoffs be in units of higher welfare instead of private utility as hitherto assumed. Whenever H = 0 higher welfare is the same as L. This is assumed to be the case for all payoffs except A's payoffs in row a2. A has v = 1. A's payoff - 1 for (a2, bl) is assumed to be the weighted sum of - 2 H and 1L. His payoff for (a2, b2) which is zero is the weighted sum of - 2 H and 2L. The strategies and expected payoffs are as given in game matrix 1. A practises his moral choice by randomizing between two moves of which a2 produces negative moral value, the other, al, is morally neutral. Like taste in consumption theory, v is usually assumed exogenous. But it is not immutable. It can be changed by education, by repetition of an activity and by the demonstration effect. For a fuller treatment of these matters see Moreh (1986). I have added the possibility of slight random variations in v (as in a person's mood). These are like the "short-lived impulses" mentioned in Harsanyi (1991), p. 148. In the following, it is assumed A's v falls to 0.95. The effect on his behaviour is likely to be more extensive than if it had been parametric, as the elaboration of the example will show. Assume that when A's v was 1, there was a strategy a3 which yielded ( - 1 . 1 , 0) for (a3, bx) and (0,0) for (a3, b2), all payoffs being in units of higher welfare. It was weakly dominated by a2 and hence not played. Speaking of A's payoffs, let - 1 . 1 be the weighted sum of - 4 H and 2.9L; 0 the weighted sum of - 4 H and 4L. As before B's payoffs contain no moral value. Following the fall in A's v to 0.95, his payoffs in units of higher welfare for both a2 and a3 change. These are shown in game matrix 3. It will be seen that a2 will not now be played because it is dominated by a3. (Of course, as the game matrix has two columns, A cannot have more than

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two undominated strategies). Moreover b2 weakly dominates bl and the game is played in pure strategies: (a3, b2) with payoffs (0.2, 0). In comparison with the expected payoffs in game matrix 1, A's payoff remains the same while B loses. bl al a2 a3

3, -0.90, -0.90,

b2 2 1 0

-1, 0.10, 0.20,

3 0 0

Game matrix 3 Equilibrium in pure strategies (a3, b2). For the purpose of Section V, the payoffs are in units of higher welfare. A has v = 0.95. His payoffs for move a2 are: -0.90, the weighted sum of - 2 H and 1L; 0.10 the weighted sum of - 2 H and 2L. His payoffs for move a3 are: -0.90, the weighted sum of - 4 H and 2.9L; 0.20, the weighted sum of - 4 H and 4L. In all other payoffs H = 0, so that higher welfare coincides with L.

In the above example, v fell by 5%. However, examples can be devised in which changes in v as small as one likes cause large changes in the way a game is played: some moves might disappear and others emerge. The type of equilibrium, whether in pure or mixed strategies might change. (In practice, one has to allow for a threshold effect, below which a change in v does not come into operation). This shows that some human behaviour is more unpredictable than is envisaged by the participants in the controversy; yet it is not necessarily irrational. We now come to the subject of the moral responsibility of those who engage in strategic behaviour. In game matrix 3 the equilibrium point is in pure strategies. A plays a3 producing a negative moral value of - 4 H . This outcome partly depends on B's strategies and payoffs. For some different payoffs of B, A would have played at which is morally neutral. It would therefore seem arbitrary and unfair to blame A when the outcome of his actions depends partly on external circumstances. There is even more apparent unfairness in blaming A for certain outcomes in game matrix 1, for it is a matter of chance if in a given play of the game he uses a2 which produces negative moral value - 2 H . Apart from this, his optimal probabilities for al and a2 depend not on his payoffs but on those of B (see Footnote 6). Moreover even the expected value of the moral value he produces is influenced by B's strategies and payoffs. The question arises: do we have to exculpate A

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from some of the moral evil he produces because it depends partly upon chance and upon B's payoffs? It would seem that the free-will theory advocated here is very severe on the moral agent. The answer is that this problem and its solution are no different from what Nagel (1979) calls "moral luck". According to Nagel, the consequences of one's actions depend not only on one's intentions (as Kant would have it) but also on external circumstances beyond one's control. Among the examples Nagel cites is the following: A negligent driver would be merely called reckless or accused of manslaughter depending on whether a. pedestrian happens to be in his path (Nagel 1979, pp. 177, 178, the references are to the reprint in Watson). This can be classified as behaviour in the face of risk-or uncertainty (Cp. example (e) in Section II). The other examples that he mentions can similarly be regarded as behaviour in the face of risk or of uncertainty. It seems reasonable, however, to apply the concept of "moral luck" to strategic games, and this is in fact what common-sense morality does. On the arbitrariness of holding the moral agent responsible for more than his contribution to the consequences of an act, Nagel (1979, p. 184) says that to do otherwise, that is, to try to separate what a person does from the contribution of external circumstances would be to think of actions as events and of persons as things. In other words, if one keeps distinguishing what an agent has done from the effect of external factors including his past history and upbringing, nothing is left of himself, and nothing of his contribution to events. Determinists and libertarians alike have to come to terms with moral luck. 6.

CONCLUSIONS

A study of random human behaviour can throw light on the free-will controversy and help narrow the gap between the two sides of the controversy. It is recommended that the libertarian position be amended, but to accept this amendment requires adopting a pragmatic approach. Libertarians consider that moral responsibility entails that human decisions and actions be free, that is, unpredictable and undetermined. They claim that quantum mechanical events underlie human decisions and actions which thereby become unpredictable. They do not clearly explain how sub-microscopic events can trigger macroscopic events. Determinists also point out t.hat the indeterminacy of sub-atomic

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particles should level out by the law of large numbers, so that events in the macroscopic world should be deterministic. It is suggested in this paper that random human behaviour should replace sub-atomic events as a source of indeterminacy in human existence. Libertarians may object to this on the ground that "undetermined" does not mean "random". Actions are undetermined by past events, but are determined by the self, an unchanging entity and an originator. This is, however, difficult to justify, since the self is a series of changing states. Libertarians may also urge that what I am calling random is, unlike quantum mechanical events, not genuinely random, but can be predicted in principle. My answer is pragmatic: as long as it is not humanly predictable, it does not matter. Note, however, that unlike the libertarians, I am not claiming that every single human action is unpredictable. Determinists should approve of giving random behaviour this role as a source of indeterminacy, as it is predictable in principle and is not necessarily irrational. The proposed new version of libertarianism, like the existing one, is favourable to the view that long term predictions of human history cannot be made. With this determinists should not quarrel.

NOTES 1 There is a vast literature on the subject of free will. On the deterministic side of the debate see Honderich (1988) and Weatherford (1991). The latter contains an excellent survey of the controversy since ancient times. On the libertarian side see Popper and Eccles (1977), Popper (1982) and van Inwagen (1983). 2 Popper (1982, Ch. 2, 3) has pointed out that a predictor who is part of the system, like a scientist, cannot predict its future properties, and that a machine cannot predict the growth of its own future knowledge. Weatherford (1991, pp. 152-8) while agreeing with Popper on these points, says they do not preclude the possibility of the future being predictable by an observer situated outside the system, which is all that determinists require. 3 Henceforth it will be understood that predictability is a matter of degree. This should not reduce the force of the argument. 4 A strategy is dominant for a player if, whatever strategies other players adopt, it secures him the highest payoff, e.g., the strategies "defect" in a Prisoner's Dilemma. A pair of strategies (ai, b~) is a Nash Equilibrium or equilibrium point if, given that A chooses ai, B maximizes his payoff by playing bj, and vice versa, e.g., (a3, b2) in game matrix 3. In all the following games, the payoffs are in von Neumann-Morgenstern cardinal utility. 5 In this paper, when referring to people in general, "he" stands for "he or she", "his" stands for "his or her".

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6 The solution to game (b) is found as follows: Let p be the optimum probability of playing at and q that of playing bl. The expected value of A's utility is E(UA)= 3pq -- p(1 -- q) -- (1 -- p)q. Set OE(UA)/Op = 0, this gives the value of q. The optimal p is obtained by setting OE(UB)/Oq = 0. It will be seen that if B is playing optimally, that is q = 0.2, then A gets the same payoff from al and a2. The same applies to B i f A plays optimally. (Some authors find in this a difficulty, see next Section). A related point worth noting is that A's optimal probability p for al depends upon E(UB). Similarly, the optimal q depends upon E(UA), see also Rasmusen (1989, pp. 71-72). 7 See Rasmusen (1989, Ch. 4), Kreps (1991, pp. 417ff), Hey (1989, pp. 90-1). 8 On this subject see also Honderich (1988, Ch. 5), Weatherford (1991, pp. 112-122), van Inwagen (1983, pp. 191-7). 9 Fairness may require that choice be made in a random way, e.g., the assignment of a kidney machine to one of two patients both of whom need it for survival, is seen to be fair if they are given equal chances of getting it. See Diamond (1967, pp. 765-6) and Cp. Section 2 Case (f) above. 10 On the appropriateness of postulating two kinds of value, see Etzioni (1986), Moreh (1986).

REFERENCES Alien, R. G. D.; 1959 Mathematical Economics, 2nd ed., MacMillan, London. Binmore, K. 1988 'Modeling rational players: Part II', Economics and Philosophy, voI. 4, pp. 9-55. Diamond, P. A.; 1967 'Cardinal Welfare, Individualistic Ethics and Interpersonal Comparisons of Utility: Comment', Journal of Political Economy 75,765-6. Etzioni, A.; 1986 'The Case for a Multiple-Utility Conception', Economics and Philosophy 2, 158-183. Harsanyi, J. C.; (1977) 'Rule Utilitarianism and Decision Theory', Erkenntnis 11, 2553. Harsanyi, J. C.; 1991 'Equality, Responsibility and Justice as Seen from a Utilitarian Perspective', Theory and Decision, 31, 141-158. Hey, J. D. ; ed. 1989 Current Issues in Microeconomics, MacMillan, London. Holland, J.; 1955 Bb'd Spotting, Blandford Press, Poole, England. Honderich, T.; 1988 A Theory of Determinism: the Mind, Neuroscience and Life-Hopes, Clarendon Press, Oxford. Kendall, M. G. and A. Stuart; 1958 The Advanced Theory of Statistics, 1, Charles Griffith, London. Kreps, D. M.; 1990a A Course in Microeconomic Theory, Harvester Wheatsheaf, New York. Kreps, D. M.; 1990b Game Theory and Economic Modelling, Clarendon Press, Oxford. May, D.; 1993 'Saharan Dandy Shakes his Tail', The Times, 8th May, Weekend, p. 3. Moreh, J.; 1986 'Morality and Welfare', Theory and Decision, 21,209-230. Moreh, J.; 1992 'Economic Analysis, Common-Sense Morality and Utilitarianism', Erkenntnis, 37, 115-143. Nagel, T.; 1979 'Moral Luck', pp. 24-38 in Mortal Questions, Cambridge University

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Press, reprinted as pp. 174-186 in G. Watson (ed.) Free Will, Oxford University Press, 1982. Popper, K. R.; 1956, 1982 The Open Universe, Cambridge University Press. Popper, K.R.; 1966 The Open Society and lts Enemies, Princeton University Press, Princeton, N.J. Popper, K. R. and J. Eccles; 1977 The Self and Its Brain, Springer International, Berlin. Rasmusen, E.; 1989 Games and Information, Basil Blackwell, Oxford. Rawls, J.; 1971 A theory of Justice, Oxford University Press, London. Schelling, T. C.; 1960 The Strategy of Conflict, Oxford University Press, Oxford. Sen, A. K.; 1977 'Rational Fools:; a Critique of the Behavioural Foundations of Economic Theory', Philosophy and Public Affairs, 6, 317-344, reprinted in Sen, Choice, Welfare and Measurement, Basil Blackwell, Oxford, 1982. Shubik, M.; 1983 Game Theory in the Social Sciences, The MIT Press, Cambridge, Ma. Slote, M.; 1985 Common-Sense Morality and Consequentialism, Routledge and Kegan Paul, London. Smart, J. J. C.; 1973 An Outline of a System of Utilitarian Ethics, in Smart and Williams, Utilitarianism - f o r and against, Cambridge University Press. van lnwagen, P.; 1983 An Essay on Free Will, Clarendon Press, Oxford. Weatherford, R.; 1991 The Implications of Determinism, Routledge, London. Zimmerman, H.-J.; 1991 Fuzzy Set Theory and its Applications, 2nd ed., Kluwer Academic Publishers, Dordrecht. Manuscript submitted January 29, 1993 Final version received June 28, 1993 Dept. of Economics The Queen's University of Belfast Belfast BT7 1NN Northern Ireland