PROFESSOR BRAITHWAITE'S inaugural lecture, here published in book form,1 is a trial run at a Platonic definition of the concept of dianemetic justice; or, as he himself would put it, a rational reconstruction of the concept "sensible-prudent-and-fair". Aristotle left it that dianemetic justice was an equality and a matter of ratios. A just distribution of hosa merista tois koinonousi tes politeias2 (Greek for "Co-operators' Surplus") was one in which each had an equitable share, no one having either more or less than he should. Professor Braithwaite goes further and replaces Aristotle's ordered scale of the-more-and-the-less in which only imprecise and unhelpful answers could be given, by a numerical scale in which he can frame the question "Exactly how much is a fair share?" and propound his own solution.
Two musicians, one a trumpeter---Matthew---the other a piano player---Luke---, live in adjoining rooms, separated only by a thin partition. There is only one hour available every evening for music. Each party can decide either to play or not to play on any given evening, and depending on the decisions of them both, the result will be either undisturbed piano-playing or undisturbed trumpeting or complete silence or cacophony. Each party has an order of preference for the four possibilities: Matthew's is undisturbed trumpeting, undisturbed piano-playing, cacophony, silence: Luke's is undisturbed piano-playing, undisturbed trumpeting, silence, cacophony. Furthermore, Professor Braithwaite arranges the preferences of each on a scale; he says how many times more Matthew prefers trumpeting to cacophony than cacophony to silence. He also makes courses of action---or, as they are known in the Theory of Games, strategies---measurable: thus besides saying that Matthew decides to trumpet or  Matthew decides to be silent, he attaches a meaning to saying that Matthew decides on a course of two-thirds trumpeting and one-third being silent, namely that Matthew decides to trumpet on average two-thirds of the evenings and be silent the remainder.
Having made both preferences and courses of action measurable, Professor Braithwaite proceeds to apply the Theory of Games. Each party is able to adopt any one of a whole range of strategical policies, from that of never playing his instrument, through playing his instrument on a certain proportion of the nights and being silent for the remainder, to playing his instrument every night. The result will be sometimes silence, sometimes a solo, sometimes cacophony, and the relative frequency of these will depend on the policies the two parties adopt. Since we know the preference scales of both of them, we can assess what value each puts on the outcome of every possible combination of their respective strategies. It then emerges, as might have been expected, that the adoption of certain strategies on their respective parts produces results that are more in the interests of them both than are others, but that of these more favourable results no single one is more favourable to them both than all the others. It would pay them both to reach an agreement whereby for a certain proportion of the nights Matthew trumpeted and for a certain proportion Luke played the piano. But though the proportions are connected with one another, and it is in both their interests that interlocked proportions should be used (so that on those nights when Luke played the piano Matthew was silent, and vice versa), their interests as to what the proportions should be are opposed. Luke would like it to be a lot of piano-playing to little silence on his part, and a lot of silence to little trumpeting on Matthew's part, whereas Matthew would want it the other way about. Professor Braithwaite suggests that they should settle this question by competing directly for relative advantage, and having settled how much advantage one or other of them should have over his opponent should then fix the agreed proportions of playing time which each should have, so as to preserve the relative advantage. In this way he determines what awards he, as Knightbridge Professor of Moral Philosophy, would make if Luke and Matthew agreed to refer their dispute to him.
The award will not be popular with fashionable philosophers. For one thing it is mathematical: for another, the problem posed is extremely artificial. The moral philosopher, it will be complained, descends from his ivory tower and mounts the rostrum, but instead of handing down a judgement which could apply to an actual situation he unrolls with a flourish a scroll of tangents to a parabola with the aid of which he propounds his solution to a problem that could arise only between two fellows of a college who were given to music  and lived on adjoining staircases. Much of this criticism is just: the problem is very artificial, and no attempt is made to allay the reader's suspicion that it is often the elegancies of the geometry of conics rather than an intuitive sense of natural justice that has guided Professor Braithwaite's steps. One has a horrible fear that an army of directrices might appear round the corner of the next page. But then all philosopher's problems are artificial and are bound to be; and much of the feeling against mathematics is a prejudice, due partly to the reaction against Russell, more deeply to the Zeitgeist with its fear of what the mathematical sciences might be able to tell us about ourselves. Vague fears of committing the naturalistic fallacy have made philosophers unwilling to believe that anything in morality can be achieved more geometrico. But even Aristotle drew lines, and if we do the same, so long as we are careful on other accounts and do not neglect the complexity of actual situations, I cannot see that we are wrong.
In order to apply the Theory of Games it is necessary to make both courses of action and preferences measurable. The difficulty in making courses of action measurable is serious. Yet it must be done. The key theorem of the theory of games is that of the General Strict Determinateness of Zero-sum Two Person Games,3 which states that if two parties are directly competing with each other, then if each party is allowed not only to adopt any single one of the courses of action open to him, but to combine different courses of action in different proportions to form a "mixed" strategy, there is a definite and unique outcome which each party, though acting in opposite directions, can be sure of obtaining, irrespective of what the other party does. Hence in any situation which can be viewed as a competitive one between two parties, this would seem to yield the rational solution: each party, if he behaves sensibly, can get it, and since what is more favourable from the point of view of the one party is necessarily less favourable from that of the other4 neither party will have any power to enforce a more favourable, nor any incentive to allow a less favourable, solution than the "rational" one already determined. Thus there is a "rational" solution, if it is possible to combine different courses of action to form a "mixed" strategy, but not in general otherwise. Once we can find a way of representing combinations of different courses of action in different proportions, every competitive situation will admit of a unique, and in some sense rational, solution. The theory of games uses probability weights as the way in which different courses of action may be combined in different proportions. In a mixed "strategy" we assign a probability to each course of action  and then in that strategy play every course of action with its appropriate probability. Professor Braithwaite uses this method too, and comes down firmly for the frequency interpretation of the concept of probability; in the long run the number of occasions I do play my musical instrument must bear to the number of occasions I do not some definite and ascertainable proportion, if my policy is to be sensible, prudent and fair. This will not do. Moral problems are not like games: for the moralist, similar situations are not qualitatively identical; human actions and desires are not repeatable in the way that scientific experiments are, because human beings have memories and tempers and can get bored, whereas chemical substances do not. Repetition in itself creates a difference; and a long run of actions is different in kind from the actions by themselves, considered individually. If our neighbour makes a nuisance of himself playing his trumpet on one occasion only, we overlook it---de minimis; it is only if he does it persistently that his actions take on a moral quality and we suspect him of doing it to annoy. Per contra, the rule of precedents often applies: on its first occurrence the action is liable to be challenged, but once it has been carried out without objection being made, the right to do it has been established, and on subsequent occasions may not easily be impugned. Either way, continued courses of action are not statistical classes of qualitatively identical individual actions, and Braithwaite's frequency interpretation of probability cannot be applied. Games differ from serious occasions in just this, that in games almost all the factors which make up the complexity of real life are deemed irrelevant; only a few features are relevant, so that when the same game is played on different occasions, there are resemblances and can be statistical classes; whereas in serious situations many, infinitely many, factors are relevant, so that serious situations do not readily resemble one another, and are not immediately amenable to the techniques of the theory of games.
A similar objection can be made to the method whereby preferences are made measurable. Given an order of preferences, say Matthew's of rating trumpeting solo above cacophony and cacophony above silence, we can determine how many times more he prefers trumpeting to cacophony than cacophony to silence, by considering the two alternatives, one of certainly having cacophony, the other of having either solo trumpeting or silence, and then finding what the probability of solo trumpeting in the latter option has to be for neither of the two alternatives to be preferable to the other: then the preferabilities will be inversely proportional to the probabilities. Here again, Professor Braithwaite adopts a frequency interpretation of probability, and though he covers himself against objections (p. 8), again objections must be made. Repetition does not normally leave preferences unaltered. If in the last forty- three evenings I have been  playing my piano myself or hearing my neighbour perform upon his trumpet, for that reason alone I maybe glad of an evening's peace and quiet. Contrariwise, my neighbour, if he be a rule-bound creature, may, through having trumpeted on twenty-six of the previous forty-three evenings, have formed the habit of trumpeting, and be on the way to becoming an incurable trumpeter, who wants to trumpet far more than he wants me not to play on the piano. Preferences are characteristically altered by indulgence, whereas it is a crucial assumption for the applicability of the frequency theory that they are not. All our earlier objections to the statistical treatment of human affairs apply again: like actions, preferences must be considered as particulars, and cannot safely be construed or measured merely as instances of general types.
These objections are not fatal. In the case of preferences we do not need the frequency interpretation of probability, since most people do not mean, or do not think themselves to mean, by the word "probable" what the frequency theory says that they mean. It is enough, for preferences to be made measurable, that we should be able to ask a man a series of questions: "Do you prefer a 50-50 chance of solo trumpeting or silence to a certainty of cacophony?" "a 3-2 chance?" "a 2-1 chance?" "a 3-1 chance?" and that he should be able to return answers to our questions. He need only be able to understand our-rough- numerical assessments of probability. Provided he has a concept of probability which he can apply to single events, and can give those probabilities a rough numerical value, he can give his differences of preferences rough numerical values too. That this concept of probability is only a subjective estimation is (pace Von Neumann and Morgenstern: Theory of Games, p.19) no objection, because it is only a subjective scale of preferences we are trying to set up. Nor, for obvious reasons, are the objections of the behaviourists to asking people questions (instead of observing them behave) of any force here. Thus with preferences, the obstacles to their being made measurable can be overcome. With courses of action it is more difficult. The method of probability-weighting is much more artificial, and seems open to insurmountable objections. Frequencies, however, are not the only way of measuring out courses of action. Obviously when sharing out material goods, but often also when dividing out possibilities of action, some other basis of measurement suggests itself. Two fishermen wanting to fish the same stretch of water might divide it by length of bank. Nations wanting to reach international agreement about wireless wavebands might measure them in metres or in kilocycles. In Professor Braithwaite's example the natural metric is not times but time. We need not consider a run of evenings but can divide each hour, or whatever other period is convenient, between Matthew and Luke in the relevant ratio. Duration  rather than frequency is often the natural metric: but it should be emphasized that no method of measurement is necessarily fair---should wavebands be measured in wavelengths or kilocycles?---and that often probability-weighting is the most neutral, and sometimes- with all-or-none activities-the only, way of making courses of action measurable.
Professor Braithwaite reaches his solution by dividing the problem into two parts. First the parties compete for relative advantage, then they co-operate to obtain much better results, while still maintaining the relative advantage determined by the previous, competitive stage. In our case, Matthew has the advantage over Luke, because Matthew, by trumpeting, can ensure that the thing he really detests---silence---never occurs, whereas Luke by not playing in order to avoid cacophony merely lets Matthew have things all his own way. Matthew's preference scale gives him a thicker skin than Luke, and if they are both beastly to each other, it will hurt him less than it will hurt Luke. Therefore, says Professor Braithwaite, Matthew ought to have a bigger share of the music time than Luke, and he gives his two awards (according to whether the playing strategies of Matthew and Luke can be completely interlocked or not) that Matthew should play 59/104ths of the evenings and Luke 23/68ths of the evenings in the one case, and in the other that they should divide every forty- three evenings in the ratio of twenty-six, on which Matthew should trumpet and Luke be silent, to seventeen, on which Luke should play and Matthew be silent. It seems a bit thick. Sensible and prudent perhaps; but certainly not generous, nor self-evidently just.
It is clearly sensible to co-operate. Any agreed strategies will produce results in general better than strategies chosen by each party independently, and the ones recommended are peculiarly sensible in that they demand less mutual confidence than any others. By the same token they are the prudent ones for each party: each party, acting on the assumption that the other party is motivated by the same general principles as himself, will see that he would be foolish to accept a less favourable, and cannot hope to obtain a more favourable, distribution of playing time. This needs amplifying. If one of the performers was a simple maximizing animal, then the other, knowing this, could play on it. He could count on his not retaliating if it would cost him anything to do so. If Luke simply wanted to maximize his own interests, then if Matthew trumpeted every night Luke would simply be silent, trumpets being better than cacophony. It needs a man of spirit to act to his own despite, as Professor Braithwaite requires Luke to act, and produce cacophony, which he detests above all things, simply in order that Matthew should not be allowed to get away with undisturbed trumpeting every night. Only if both parties are human beings and are prepared to retaliate and not merely  maximize, does Professor Braithwaite's solution apply. Then it has the charm that it represents the situation where the amount of nose either party has to cut off himself in inflicting retaliation is just equal to the spite done to the other's face. This is the agreement it is worth holding out for, knowing that with any less favourable agreement retaliatory action will hurt the other party more than it will hurt you, but that from there on non-co-operation will hurt you more than it will hurt him.
Thus the outcome point A (Diagram V, p. 29), which represents Professor Braithwaite's provisional recommendation, is stable not only in the sense that it would be against each party's interests to swing his own strategy line round, thereby provoking the other to retaliate---this is true of any point on the parabola between its points of contact with the vertical and the horizontal tangents---but it is stable in the additional sense that here alone even if each party was willing to damage his own interest for the sake of damaging the other's also, in order to "learn" him, neither party can inflict any hurt on the other without suffering as much hurt himself. Not only interest but even a sober pride would hesitate to retaliate under those circumstances; the power to hurt at outcome point A is equally balanced, and neither party can hope by violent retaliation to force the other party to mend his strategies and be more co-operative, nor offset his own loss by the enjoyment of the other's discomfiture.
The rational reconstruction of sensible-and-prudent is convincing; but of fairness still open to dispute. It does not strike us as intuitively fair, because we have many different moralities, and though it is adequate for one conception of fairness within one morality, it conflicts with others. The egalitarian in us will find it repugnant, and will demand without more ado that the available time be divided equally between the claimants irrespective of any other considerations: the Christian will be unable to stomach the mechanism of threats.
Professor Braithwaite's model is richer than he realizes. Much of his geometrical representation we can accept without accepting all his assumptions on the way it should be interpreted. That the "isorrhopes" (contour lines of constant relative advantage) should be parallel to the axis of the parabola is acceptable, for various reasons I will not go into now---Medeis eiseto ageometretos---but that each party should have aims independent of the preferences and aims of the other, and should aim solely to maximize his own desires, and that the outcome point ought to be fixed by the most unfavourable isorrhope either party can inflict on the other, are assumptions quite unwarranted and false. There are other methods of abstracting a solution from an impasse, and we can represent many ways in which preferences and aims might be interlocked, and with their aid we can obtain models of many of our differing moralities. 
Professor Braithwaite does allow that men's preferences may be influenced by the preferences of other men, but wrongly assumes (p. 14) that preference scales can be adjusted so as to have the parties aiming in directions perpendicular to (i.e. independent of) each other. Brother Matthew and Brother Luke, living in adjoining cells in a monastery, would not find Professor Braithwaite's arbitration helpful. Brother Matthew, whose natural man inclines towards singing Bach arias in lusty German, is quite determined that Luke shall be able to chant uninterruptedly the purer forms of plainsong. Luke on the other hand is equally determined that his liking for Palestrina shall not deter Matthew from roaring to his heart's content. The Franciscan morality operates on the same preference scales to produce different solutions from those of Professor Braithwaite. The directions in which each monk is trying to move the outcome point have been interchanged. Matthew is trying to move it to the right, so as to maximize the satisfaction of Luke's preferences, while Luke is trying to shift it upwards for Matthew's sake. Similarly the various intermediate forms of Benevolent moralities, in which men are prepared to accord some weight to the other man's preferences (unlike Professor Braithwaite's pair) while still anxious also to achieve their own ends (unlike the Franciscans) can be represented by smaller deflections in the parties' aims. If Matthew cared for Luke but only a little he would wish to shift the outcome point not vertically upwards but slightly to the right of the vertical. As he came to love Luke more in comparison with himself, so the inclination to the vertical of the direction in which he wished to maximize would increase, until we reach the Franciscan case of complete selflessness where Brother Matthew loves Brother Luke all right, but does not love himself at all. The me-first morality of the amoral man, who is solely concerned to get as far as possible what he wants, is represented by having the directions of aim perpendicular to each other; and then if the two parties are equally matched the outcome will be Professor Braithwaite's solution, and if one is stronger than the other and more ready to retaliate he will get things all his own way. The Nietzschean morality of the immoral man, whose primary aim is to make as large an impact on the world as possible by inflicting as much pain on other people as possible, is represented by swinging the directions of aim the opposite way to that of the Benevolent or Franciscan moralities, so that Fiend Luke's aim is to minimize Fiend Matthew's satisfactions and vice versa.
Among these moralities the one within which Professor Braithwaite is explicating the concept of fairness has a peculiarly dominating position, much as Euclidean geometry has among other geometries. It is the minimum morality expected of us merely as rational agents, the necessary condition for community with the rest  of the human race. It is not an over-stringent one. It does not seek to alter men's whole attitudes or preferences, only to put restrictions on their exercising them at the expense of others. Though the rights of others are recognized, they are not paramount; self-interest is legitimized, and a man is entitled to do as well for himself as he can, having reasonable regard for the interests of others. It is roughly the standard of behaviour the voice of conscience enjoined upon the Jews, and was determined by recta ratio for the ancients. Men are expected to obey rules and accept certain restrictions on their behaviour---it would be unfair, as well as foolish, for Matthew to trumpet all the time---but they are not expected to be entirely altruistic and always sacrifice their own interests to the interests of others. They should not take more than an eye for an eye, but are not required to deny themselves or turn the other cheek; only to recognize the existence of other people, and treat them with reasonable consideration. All the same there is something repulsive about this morality. It is not so much the way the aspirations of the parties are independent of each other as the procedure of first competing for relative advantage and only then co- operating to better the result, It seems all wrong that Matthew should be so much advantaged by Luke's aversion to cacophony. The lex talionis is too much to the fore. Too much depends on the power of each party to hurt the other. Relative advantage can be obtained not only by bettering one's own position absolutely but equally by worsening one's opponent's. It is an unlovely form of justice which commends the latter. I prefer the justice of Joseph who "being a just man, and not willing to make her a public example, was minded to put her away privily,"5 that is, he was prepared to secure his own interests even at Mary's expense, by breaking off the engagement, but was not prepared to make a scandal for the sake of damaging Mary's interests and thereby increasing his relative advantage over her. Luke in a similar frame of mind would not play the piano when Matthew was trumpeting, enduring cacophony himself for the sake of preventing Matthew enjoying his trumpet: he would instead pursue the prudential strategy, the one best designed to secure his own interests, and would not depart from this merely in order to retaliate on Matthew. Pursuing this line of argument, and accepting the slope of the isorrhopes as giving the measure of equal increments of utility, I should base my awards not on the isorrhope through T11 (Diagram V, p. 29), but on the (unmarked) isorrhope through the intersection of the horizontal and the vertical tangents of the parabola, that is to say through the outcome point which would obtain if both parties played prudential strategies. This solution I find much more attractive than Professor Braithwaite's. For one thing it favours, for a change, the long-suffering Luke. For  another it does not depend on who has got the thickest skin: that is to say, this point is invariant under an interchange of T11 and T22 ,6 if Matthew hated cacophony and Luke silence, the award would be unaltered. Thirdly, in choosing this isorrhope no use is made of threats: if neither party realized that the musical interference was attributable to a person, and each assumed it was a natural phenomenon, then each would pursue a prudential policy; on discovering that it was a colleague who was making the noise, and in coming to an agreement with him to remedy matters the status quo ante would be a natural starting point for arranging how matters should be improved. A negotiation which began by each party seeing how it could hurt the other most, would be unlikely to proceed much farther, whereas one which started by considering how the present position could be bettered for both parties would have a fair chance of success. lf Dr. Matthew and Professor Luke are fellows of one of those colleges where the fellows are not friends but not enemies either, and treat one another in a distant though not discourteous fashion, maintaining a correct attitude towards colleagues without being much concerned for them or their affairs, then I should recommend that neither should threaten to retaliate against the other and that they should not compete for relative advantage at all but should only co-operate in improving the situation which each could have obtained for himself anyhow and which was all he would have obtained had his colleague turned out to be not a human being but merely a natural phenomenon.
Thus my external morality. It is hard, as are all legalistic moralities in which people do not care for one another and extend to one another only that consideration which is due to all men merely for being human, but it is less harsh than Professor Braithwaite's. There is no vindictiveness, only indifference. The morality of the spirit can be represented by an inclination in the directions in which the two parties aim; not, as in the Franciscan morality, so that each should aim solely at the interests of the other---selflessness can be carried too far: apart from other objections, it often becomes a new form of selfishness: in this case, Brother Matthew and Brother Luke are still competing, though in a contorted way, and a cynic might say that now they had entered for the sanctity stakes---but rather that both should aim in the same direction, namely that of the isorrhopes. This, presumably, is the way to represent each loving the other as himself. This is the consummation of collaboration, where all opposition and competition have been banished, and the minds of both are entirely at one. When Miss Matthew and Mr. Luke finally decide to live the rest of their lives together, then in their flat  they will spend every evening trumpeting. Luke will welcome this not because in accordance with the spirit of this age he holds submissiveness the chief of the husbandly virtues, but because sharing pleasures as well as goods with his wife he recognizes he can take more pleasure in her intense enjoyment of trumpeting than either of them can in his more modulated performances on the piano. Both husband and wife agree that they want to shift the outcome point as far as possible up the isorrhopes. It so happens in diagram V that the point of the convex hull farthest up the isorrhopes is T21 and so if husband and wife have an identity of aim represented by this direction, T21 is the best result they can achieve. On another occasion when the preferences were different, the concessions might be the other way.
In his fuller treatment of the topic, I hope that Professor Braithwaite will consider these three difficulties: (i) The Theory of Games is too artificial ever to be applicable to moral situations; (ii) no one method of measurement is necessarily fair; (iii) we have many moralities, and therefore different conceptions (as opposed to concepts) of what is fair. For my part I doubt whether the first two can be met, or whether the Theory of Games ever can be much of a tool for solving real moral problems. Nevertheless the attempt is valuable, not for the solution reached but for the approach used and the assumptions which have to be made, and the different moralities with different solutions which are obtained by altering the assumptions. It is a new way of looking at morals. The Theory of Games is never likely to provide a calculus of morals; but may well provide models on which we may sharpen our logical teeth and develop our moral sense.
Corpus Christi College,
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