Israel Journal of Mathematics, Volume 5, Issue 3 (1967)

CONTRACT LAW A N D THE VALUE OF A G A M E BY

JOHN R. ISBELL ABSTRACT

For the special case of games with linearly transferable utility, a treatment preserving the main features of the controversial treatment in the author's doctoral dissertation is newly derived from a model for negotiation and play that is more elaborate than most such models. The crucial point of the derivation is that the author's special bargaining theory is not needed; the usual Zeuthen-Nash-Harsanyi bargaining theory gives the same result. The main novelty in the model that makes this possible is the replacement of customary informal uses of "enforceable agreements" by explicit contract law. The problems of contract law for cooperative games seem to be very complex, and the present work makes only a bare beginning on them. A characteristic function and value are derived. Introduction. This paper presents a value for certain n-person cooperative games (viz. those with linearly transferable utility) which, though technically new, could have been derived by trivial steps from two different places in the literature. There are two main points to the paper. Background: my previous work in cooperative games [4, 5] begins by changing utility theory and goes forward on radically different lines from anyone else's work; and, sound or not, it has not been followed. Main point 1: here I concede almost everything to the opposition, using their utility theory, their (Zeuthen-Nash-Harsanyi [1]) bargaining theory, and (though this is agreement, not concession) Shapley's formula, which can be based on a model due to Harsanyi [2], for deriving a value from a characteristic function. Nevertheless the value obtained is consistent with my previous scattered remarks [4, 5] on what a value should do, and inconsistent with the evaluations most recently proposed by Harsanyi [3], Selten [9], and Shapley [11 ]. The crucial turn of course lies under the word "almost". I depart from the custom of treating the coalition as a sort of religious order acting like a single person. I introduce a simple explicit contract law. It is actually too simple; for playing a game cooperatively, one would want a more sophisticated law, and its development appears to involve knotty problems. But the simple contracts used here seem to suffice for evaluation, in much the same way that ordinary mixed strategies suffice for a matrix game, because the means proposed secure certain results regardless of what the other players do. Main point 2: for the games considered, one can speak of "the" other value, for Harsanyi [3], Selten [9], and Shapley [11 ] assign the same values to these games. Received December 20, 1965, in revised form June 22, 1966. 135

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[July

The point is that the other value, to speak picturesquely, swindles certain players. Obviously there is room for considerable controversy about such a point. At any rate, the middle section of this paper concentrates ontwo three-person games which we evaluate differently. Selten's derivation of a value and other considerations lead us to examine a couple of related games. I have profited from a confrontation with Selten on this question, and shall try to indicate some of his points about the examples. But the Selten value seems to require support by a theory at least as detailed as the simple theory given in section 1 below taking account of contract law. Toward that, Selten has made one point quite clear: he follows ScheUing [8] in stressing unilateral commitments and feels they should have a legal standing if my contracts do. Lacking a detailed development, one can still note that if commitments are to have the force suggested by Schelling, there must be possible legal proceedings compelling "specific performante", the actual carrying out of promised actions. In contrast ,I propose to exclude specific performance, securing contracts only by penalty clauses for pecuniary (utility) indemnities. Section 3 of the paper has the only non-trivial theorems, but is much the least interesting; it gives some results on "games with infinitely many players". The ideas of this paper were worked out at the 1965 Jerusalem Game Theory Workshop, sponsored by The Hebrew University and the Israel Academy of Sciences and Humanities. I am much indebted to J. C. Harsanyi, L. S. Shapley, Martin Shubik and R. M. Thrall for constructive criticism there. The writing was supported by the National Science Foundation. 1. Value. For the simplest definition of the proposed value, we consider games in normal form. Though for most game theory, and for everything in this paper, the normal form suffices, we shall usually find it convenient to speak of the extensive form. If F is an n-person finite game in normal form with payoff function h, define h-(xt, ...,x~) for each (pure) strategy n-tuple ~ = (xl ..... x~) by ht'(O = hi(O + 1/n(m - Y, hj(~)), where m is the joint maximum maxY.hl(~). Let F - be the game derived from F by replacing the payoff h with h-. We may follow an unpublished manuscript of Harsanyi and call F - the upraised game. It is evidently constant-sum. We define the upraised characteristic function v- = Vr of F as the Neumann-Morgenstern characteristic function of F - and the contract value d)- of F as the Shapley value of F-. What does ~b- evaluate ? To begin with, there is substantial agreement on the Shapley value for constant-sum transferable-utility games. As far as I know, none of the numerous values for more or less general games that have been proposed since Shapley's original paper [10] has differed on these games. A number of arguments leadingto it are known. Note Shapley's axiomatic argument, adapted to the constant-sum setting in [4], and note Harsanyi's bargaining model [2].

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Accordingly, we shall here take the step from v- to ~ - as established, and examine the passage from F to v-. A specification of the assumptions involved that would be adequate for, say, the designer of a game-theoretic experiment, would be extremely long. Roughly, we need the usual assumptions after von-Neumann-Morgenstern [7], amplified by explicit legal assumptions (see below), and supplemented by a symmetry or democracy assumption to the effect that the players recognize each other as peers. That last assumption enters in two ways: in the (standard) bargaining theory to be applied to the model, and also in the structure of the model, where we seem to need a special meeting of all players. Of course no positive or negative concern of the players with each other's welfare is implied. In Zeuthen's and in Harsanyi's presentation of the bargaining theory [1], a player's resistance to a concession is supposed to depend monotonically on the ratio of the cost (to him) of the concession to the cost (to him) of a breakdown of negotiations, and by the same rule for all players. As for the special meeting of all players, it serves to conclude indefinitely long negotiations which may have already determined the outcome; but players considering whether to delay or not in the previous negotiations have the definite prospect of a last grand meeting, into which they will go with the support derived from previous agreements or the independence secured by previous disagreements, whichever they prefer. Concerning the legal apparatus of the model, some preliminary remarks. Indisputably one wants contracts to be sufficiently definite so that a trial court can in a finite time determine whether the contract has been violated. That sounds like a question from recursive function theory; and I think the analogy is sound, remote though it seemsfrom ordinary business practice. An analogue in contracts to the endless passage from n to n + 1 that gives rise to arithmetic may be found in this remark: if a contract Cn (between A and B, say) could affect the prospects of the players, then so could a contract Cn+ t ! (between A and C) binding its parties to enter no contract of the form Cn. We may hope that the analogy is unsound. At any rate, the apparatus here proposed gives it no footing. This model admits those and only those contracts C such that, in consideration of certain side payments x among the set S of parties to C, all players in S agree (1) to relinquish all their turns to move in the game tree to a designated agent who is to play a specified mixed strategy a and (2) to sign no other contract; any member i of S violating (1) or (2) is to pay a specified vector indemnityfi to the rest of S. Enforcement is paternalistic; without plaintiff, defendant, or trial, the court will assess the indemnities provided for. Time taken to move or to negotiate is customarily ignored in game theory and can be ignored here, as long as we have three successive periods: the first for general negotiation, the second for an attempt for all players to agree on an outcome, the third (if necessary) for play and assessment and payment of indennities. There is nothing to add to what has been said on the first period, and almost nothing on the third; but the second period bargaining requires a determination c f the outcome if no agreement is reached. Now the players may be legally committed to play certain strategies, and may even have inconsistent commitments. In the second period I want them actually committed to a certain

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course of play. Thus the second period begins wi h each player handing to a clerk a statement of a mixed strategy on the following finite set: the union of his set of pure strategies in the game tree and his set of contracts. (Playing a contract means complying with its play clause, and that would determine his play completely. This "early" commitment to a strategy is objectionable if one thinks of it as early, but the players are supposed to be through negotiating, ready to play, only making a last attempt to secure a jointly optimal outcome by compromise.) Then if the second period results in agreement on an outcome, that is the outcome; if not, play is governed by the strategies given to the clerk. One remark seems needed, on cancelling contracts. Formally, we allowed the set of all players to cancel or render irrelevant previous contracts. In effect any superset of the set of parties to a contract C can cancel C by drawing up a new contract D (a direct violation of C) and offsetting the indemnities associated with C by the side payments associated with D. The model is now completely specified, by the second and third paragraphs before this one. It is not specifically a bargaining model, though because of the rudimentary contract law it is a defective model for play. I remark that the customary model for cooperative play, from [7] on, amounts to the first period of this model, with a contract law about a millennium more primitive, followed by play. Assuming Zeuthen-Nash-Harsanyi bargaining theory, any set S of players can secure jointly in this model v-(S). (Since the same will hold for the complement of S, it will follow that v-(S) is all that S can secure.) After the completely trivial proof, we shall note how the model leads to a result so different from the result of Harsanyi's similar model in [3]. Proof: our (second period) bargaining problem, for any threat payoff (tl ..... tn), has the solution (t 1 + a ..... t, + a), where na is the excess m - Eti[l ].Thus the first-period problem for S is(t) the maximin problem, familiar from the Neumann-Morgenstern theory, for the upraised game F - ; and its solution is the solution v-(S) of that problem [7]. How ? Well, Harsanyi does not permit S to choose the best coordinated mixed strategy a and still act as a number of separate persons in the bargaining. He does not even permit them to play uncoordinated strategies and bargain as separate persons. In Harsanyi's model, the number corresponding to v-(S) (call it u(S)) comes from S and its complement choosing the best coordinated strategies and bargaining as two ("corporate") persons. (Strictly speaking, the present model does not permit them to bargain separately or corporately as they may prefer; but it would not matter if it did, since Zeuthen-Nash-Harsanyi bargaining theory for games with transferable utility never makes combination gainful.) Accord ingly the rather strange function u that emerges is not a plausible characteristic function (not superadditive). Harsanyi puts no stress on u and describes his model (1) The fact that the problem for "S", which is not a legal individual, is fairly simple follows from their ability to secure each other's continued cooperation by large indemnities.

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as a model for bargaining, not for play. The presentation in [3] goes on, in effect reproducing Harsanyi's previous [2] bargaining-model derivation of the Shapley value. 2. Crucial examples. There are two rather simple three-person games, for one of which the contract value seems prima facie sound and the Selten value(z) strange, with the reverse holding for the other. It is not only as a debating tactic that I present the latter first; it is simpler. In fact, there is no need to bother about the details of moves. There are three players, Adams, Brown, Cox. If Adams wishes it, Brown will receive 90 units. Otherwise no one gets anything. Call the game Fgo. The Selten value is (45, 45, 0). Surely this seems obviously right. But the upraised characteristic function gives Adams (precisely, {Adams}) 30, Brown 30, Cox 0, and so on, and the contract value is (40, 40, 10). Several objections may be made to this contract value. First, what has Cox to do with the game ? (With a little jargon, the objector can give an answer instead of a question; Cox is a dummy, and has nothing to do with the game.) The model already described provides an answer. The indicated contract for Cox to try to negotiate with (e.g.) Adams will call for Cox to pay Adams 15 units, and for Adams to deny Brown the 90, paying Cox a prohibitively large indemnity if he violates. If this is legal, and accomplished, Brown cannot afterward hope to split 45-45 with Adams; Cox has bought a full one-third share in the enterprise of stealing 90 dollars, or whatever it is. Cox can expect a gross return of 30, net of 15. The reader can easily work out the whole analysis. There is a second-order objection that I have met often enough to justify a comment on it. "Why don't Adams and Brown close ranks and get 45 instead 407" They may. OrAdams and Cox may close ranks. That would get Adams 45. The second serious objection is, if Cox can do this, what about Davis ? We were discussing a three-person game. It seems to me that the reminder that the enterprise might consist of stealing 90 dollars almost suffices to answer this. It is notorious that in applying game theory to the description of actual conflict situations, often the hardest part is to say what game is being played. There is a legitimate question, what happens if there are very many players in the situation of Cox. Some answers are given in Section 3 below. Of course there are other legitimate questions on various levels (Is this model for cooperative play socially useful ?); not for this paper. A counterquestion: how does Selten justify the value (45, 45, 0)? By axioms. A dummy is a player having no moves, and the same payoff at every outcome. Two axioms require dummies to get nothing and to exert no influence [9]. Harsanyi's model secures that value by requiring Cox in effect to marry Adams and lose his legal identity if he makes any agreement with Adams at all. (2) Selten's paper [9] discusses a number of values, but the main results (according Selten's Introduction) concern this one and its constant -- sum specialization.

to

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Presumably some readers arc now at least provisionally convinced. I must address readers who consider dummies outside the community or object to a pluralistic society. The next game Fs4 has merits for that purpose and will lead to other points too. The moves are as before, but the payoff is (0, 54, 36) if Adams wishes, otherwise (0, 0, 0). Here the contract value is the rather obvious (30, 30, 30). The reader can check the analysis, but a reader who stayed with Selten for the previous example will give no credit to the contract value. An ad hoc argument for (30, 30, 30): each player may be supposed to ask for a "fair share". Adams' claim is ironclad; without him, nothing can be done. Brown's claim is at least as good as Cox's. Cox's claim is that the other players cannot get anything without giving Cox more than 30; if they wish to divide 90, they can do it foolishly and overpay Cox, or they can treat it as a project requiring unanimous agreement and share equally. The Selten value is (33, 33, 24). All arguments for it (that I know of) depend somehow, and I think must depend, on a judgment that the 54-36 security which Adams-Brown have against irrational behavior, or skillful bargaining, or other deviant actions by Cox, is worth something, while the lower-level security which Adams-Cox have is worth nothing. (If it were worth something, Adams would have a higher value than Brown.) Of course the dividing line is 45, and an unfriendly critic can readily explain it as the product of a naive theory of coalition formation. I do not see how the friendliest critic can maintain the (33, 33, 24) value in a social context like that of Harsanyi's model(3) or my more detailed model in Section 1. If the players are bargaining, we may suppose Cox to say: "Your insurance against my deviations is interesting, and I wish I had insurance against your theory. I will not accept 24. If you wish to optimize, give me 30. If you value your insurance more, take it, and I will pocket my 36". If the context is arbitration, Cox may be more helpless, but otherwise his argument still seems sound. The more serious defense of (33, 33, 24), in line with the insurance idea, treats the value as some sort of average outcome of play. (With the best precedent; the idea is older [10] than the idea of a value as outcome of a definite procedure.) It is difficult to criticize a distribution of which only the mean is known, but we can look. Since an insurance policy paying 45 is worth 0 in this setting, the last 9 of AdamsBrown's 54 provides all their advantage. (We are comparing Fs4 with F45, defined in the obvious way. The Selten and contract values for F45 agree: (30, 30, 30).) This is 4½ units each, some of the time. Their gain in value is 3 each, in the mean. Clearly we cannot get such a result by giving Adams and Brown the extra 4½ only the one-third of the time that the A - B coalition may be supposed to form. (3) The working of Harsanyi's model for F involves three shotgun marriages, i.e. coalitions formed in order to lose.

1967]

C O N T R A C LAW T AND THE VALUE OF A GAME

141

We must assume that if Cox combines with Adams, he will have to compensate Adams for entering the less secure A - C partnership instead of A-B(4). This suggests that Adams can be 4½ better off two-thirds of the time. (For if BrownCox combine, the insurance available for Adams-Brown cannot help Adams.) But the possibility of Adams-Brown getting 60 instead of 54 has completely disappeared. The player excluded from a two-man coalition must behave deviantly with probability 1. And with invariable success. Perhaps a better explanation, though not for supporting Selten's theory, is that as this game is 20 ~o of the way from F4s to Fgo, where Cox would be a dummy, Cox is suffering from 20 Yo leprosy. There is a serious point here; is Selten's value for F54 required by linearity to be the weighted average that it is of the values of F,5 and of Fg0 ? In a way, yes; the value of F~4 is determined by the axioms. But the linearity notion involved in the axioms, a standard one, does not make Fs, a weighted average of the other two. .8 F4 s + .2 I"90 is a well-defined game M, played as follows. There is an initial chance move, after which either F,5 (with probability .8) or 1"9o(with probability .2) will be played. Selten's axioms require M to have the corresponding average value, (33, 33, 24), which happens to be the same as that of Fs,. The contract value also has this linearity property, and the contract value of M is (32, 32, 26). It differs from the Selten value just in proportion as the contract value of 1-'9o differs from the Selten value of Fgo. But M is not at all the same as F5,; the difference has some value according to my theory, none according to Selten's. The difference is just that the one windfall of 90, which in 1-'5, Adams can give or withhold, breaks into two parts in M. The parts happen to be mathematical expectations, but for game theory it would be the same if they were certainties, mere physical parts of the 90-unit pile. Adams can give 72, 36 each to Brown and to Cox; and he can give Brown 18 independently of what he does with the 72. Obviously Cox has no claim to equal standing in the game M. According to the Selten value, the detached 18 in M as compared with Fs, does not strengthen Adams and Brown. A remark is in order about linearity. The contract value has the property, and I have no argument against linearity; but neither do I regard it as a satisfactory axiom to support dubious theories with. If one drops the artificial hypothesis Of transferable utility (greatly increasing the difficulties), neither the Nash cooperative value for two-person games [6], the value for two-person games in my thesis [4], nor any value that I know of satisfies any substantial remnant of linearity. One would not expect equality, for the average of values need not be Pareto optimal in the average of games. But one might expect an inequality. But it fails. (4) 4~ is enough compensation, for Adams has no prospect in 1"54 more than 4½ better than in 145.

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How does the Selten value of F54 follow from the axioms ? The complete answer is very complicated (fourteen auxiliary games in Selten's proof [9]), but the qualitative point that Cox's value will be less than the others is easily established; and the argument brings out another feature of the theory that may enrich our dossier. Consider the game A in which Adams chooses (0, 0, 0) or ( - 3 6 , 0, -36). In F54 + A (a careful reader may translate via .5 Fs4 + .5 A, for correctness) one strategy for Adams yields ( - 3 6 , 54, 0); so Adams-Brown can certainly secure (9, 9, 0), and Cox is dearly weaker. (It is interesting to compare F54 + A and M; my theory agrees with Selten's that Cox is weaker in these games, but makes him quite a bit worse off in Fs4 + A. But this is not the place for fine points.) Of course that settles Fs4, for the Selten value of A is (0, 0, 0). Look at what the dummy taboo does in A. The contract value is (4, 4, - 8). The mechanism is plain; this is a game of extortion, the threat ( - 3 6 , 0, - 3 6 ) is not very persuasive, but a threat of ( - 1 8 , - 1 8 , - 3 6 ) may move Cox. Selten forbids it - - this may be socially desirable. It is not impartial. From Adams' point of view as well as Brown's, it prohibits best play. 3. Residual value. If F has players 1.... , n, let Dk(F) denote an n + k-player game constructed from F by adding dummy players n + 1.... , n + k, each of whom gets 0 at each outcome of F. Dk~b-(F) is the n-vector consisting of the first n components of q~-Dk(F). The residual value ~r (F) is limk~o~D~b-(F). The question of the easiest proof that q~" exists seems mildly interesting; as we see from the example A, the vectors DkqS-(F) need not be monotone decreasing. Here we shall describe a computation previously done by L. S. Shapley (unpublished) and indicate why it yields the residual value. Recall that the Shapley value ~b~(Dk(F)-) of the upraised game with k dummies, to the i-th player, is the average of the expressions v-(S u { i } ) - v-(S), averaged over the (n + k)! total orderings of the players, S being the set of predecessors of i. Let xi( j = 1,..., n) be the fraction ofallthe players preceding the j-th, So = {j ~ S:j < n} = So(x, i). Then v-(S u {i}) - v-(S) = 6k(X, i) is determined by the n-vector x and the indices i, k; v-(S), for example, is the value of a two-player constant-sum game with sum rn derived from F by aggregating players, So making the first player and getting at each outcome z the payoff Z[hi(z): i ~ So] + x i ( m - F,hi(z)). The other term involves k. So

(3.0)

qSi(Dk(r)-) = f 6k(X, i)dpk,

where /~k is a suitable atomic measure on the unit n-cube. The formulas define 6k(X, i) On the whole cube, piecewise uniformly continuous on a fixed set of n! pieces; then so is 6(x, i) = limk~o 6k(X, i), for the convergence is uniform. We conclude

1967]

CONTRACT LAW AND THE VALUE OF A GAME

(3.1)

~bir =

f

6(x, i)dl~,

where # is ordinary volume. For the functionals J"

tof

143

dl~k(Riemann sums) converge

d,.

To illustrate (3.1), the double integral for ~b~ (a = Adams) in the example A reduces to 2 Sos (36x - 72xZ)dx = 3; q~ is - 1 5 . Evidently the operators Dk, 4)" are positive-linear on games to games resp. vectors. From this we can get an interesting remark and a computational shortcut. The constant-sum extension of F, with any chosen constant sum c, is constructed [7] by adjoining a player who gets c - Zhi(z) at each outcome z. In general the contract value of the constant-sum extension differs from the residual value; for A it is (12, 0, - 2 4 , 12 + c). But: (3.2)

The constant-sum extension gives the residual value for fixed-threat games.

A fixed-threat, or "characteristic function" game is determined by giving a superadditive function v on the set of all sets of players. To play, each player names a set; those sets S named by all their members get v(S) (divided equally among them, say), and players i left out get v((i}). Now the characteristic functions, as set functions, are linearly generated [10] by the pure bargaining games Bs determined by v(T) = 1 for T ~_ S, v(T) = 0 otherwise. Thus every fixedthreat game F satisfies a relation F + Z2sB s = EI~sBs with non-negative coefficients 2s and/z s. (Strictly, + for games complicates the strategies; to justify the equation we must modify the sums of games by requiring a player to name the same S for each summand. Clearly this will not affect the values.) So it suffices to prove (3.2) for the games Bs. In the constant-sum extension of Bs with sum 1, the added player's value is the probability that in a total ordering he is between two members of S; this is (s - 1)/(s + I). The remaining 2/(s + 1) is shared equally by the members of S. In DR(Bs) , if the players are totally ordered, each player i of S has fig(X, i) < 1/k except for the first and last of them, who get together nearly the fraction of players not between two members of S. The expected value is again 2/(s + 1), and S shares it equally. REFERENCES 1. J. C. Harsanyi, Approaches to the bargaining problem before and after the theory of games: a critical discussion of Zeuthen's, Hick's, and Nash's theories, Econometrica, 24 (1956), 144-157. 2. J. C. Harsanyi, A bargaining model for the cooperative n-person game, Annals of Math. Study 40 Princeton, (1959), 325-355. 3. J. C. Harsanyi, A simplified bargaining model for the n-person cooperative game, International Economic Review 4 (1963), 194-220.

4. J. R. Isbell, Absolute games, Annals. of Math. Study 40, Princeton, N. J. 1959, 357-396. 5. J. R. Isbell, A modification of Harsanyi's bargaining model, Bull. Amer. Math. Soc. 66 (1960), 70-73. 6. J. Nash, Two-person cooperative games, Econometrica 21 (1963), 128-140.

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7. J. yon Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 2nd edn., Princeton, 1947. 8. T. C. Schelling, The Strategy of Conflict, Cambridge, Mass., 1960. 9. R. Selten, Valuation of n-person games, Annals of Math. Study 52, Princeton, 1964, 577-626. 10. L. S. Shapley, A value for n-person games, Annals of Math. Study 28, Princeton, N. J. 1953, 307-317. 11. L. S. Shapley, Values of large market games: status of the problem, Rand Memorandum RM-3957-PR, 1964. CASE-WESTERNRESERVE UNIVERSITY CLEVELAND, OHIO

NEIGHBORHOODS OF EXTREME POINTS(1) BY

I. NAMIOKA ABSTRACT

An examination of relationship between two neighborhood systems (relative to two linear topologies) of extreme points yields a unified approach to some known and new results, among which are Bessaga-Petczyfiski's theorem on closed bounded convex subsets of separable conjugate Banach spaces and Ryll-Nardzewski's fixed point theorem. §0. Introduction. Let C be a compact subset of a Banach space E. Then, of course, the norm topology and the weak topology agree on C. Now suppose that C is only weakly compact. Then the identity map: (C, weak)--. (C, norm) is no longer continuous in general. Nevertheless one may still ask how the set of points of continuity of this map is distributed in C. In particular, when C is convex as well as weakly compact, is the identity map: (C, weak) ~ (C, norm) continuous at any of the extreme points of C, i.e., do there exist extreme points of C which have weak neighborhoods (relative to C) of arbitrarily small diameter? The importance of an answer to such a question is demonstrated in Rieffel [7] and in note [6]. Professor J. L. Kelley also recognized the relevance of this question to Ryll-Nardzewski's fixed point theorem. The work of Lindenstrauss in [4] yields the following answer: if C is a weakly compact, convex subset of a separable Banach space, then there are " m a n y " extreme points of C, where the identity map (C, weak) ~ (C, norm) is continuous. This fact was proved by using deep Banach space techniques due to Kadec and Lindenstrauss. In the present article, we shall generalize this result in various directions. The main theorem of this paper (Theorem 2.3) is stated in somewhat obscure, if not pedantic, language, because we tried to combine all the generalizations into one theorem. However, we hope this is forgiven because of the diverse applications of the single theorem. Here are some of the consequences of the main theorem: each bounded subset of a separable, conjugate Banach space is "dentable" in the sense of [7]; each closed, convex, bounded subset of E is the closed convex hull of its extreme points, where E is either a separable, conjugate Banach space or a Frrchet space such that E** is separable relative to its strong topology. In addition, a slight generalization of Ryll-Nardzewski's fixed point theorem can easily be derived from the main theorem. Received February 27, 1967 (1) This research was partly supported by the U.S. National Science Foundation. 145

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The paper is organized as follows: Section §0, the present one, is the introduction. Section §1 contains the preliminary material, and section §2 is devoted to the main theorem. Our proof of the main theorem is independent of Lindenstrauss' work and is quite different in spirit. Category plays a large r61e throughout §§1-2. Section §3 gives applications of the main theorem. Our terminology and notation will be those of Kelley, Namioka, et al. [3]. Finally, we wish to thank R. Phelps for many enlightening discussions on the subject of the present paper. 1. Preliminaries. Let (E,~-) be a linear topological space, and let A be a subset of E. Then we denote by (A,~--) the space A with the topology induced by 3-. If p is a pseudo-norm on the linear space E, then ~-'~ denotes the pseudonorm topology on E given by p. The pseudo-norm p is lower 3"-semicontinuous if {x:p(x) 0 } sei nicht leer. Dann liegt (wegen U, = + oo auf IIz I[ = r) die abgeschlossene Hiille/~, yon E, ebenfalls in C, und somit besitzt ~b ein Maximum in einem Punkt von E,, etwa im Punkt (Zo, 5o). Nun gilt der Satz: HILFSSATZ 6. Hat d/ ein Maximum in z e E , , so gilt dort die Ungleichung (3.5)

0 z ~ k t~k < 0.

Dabei sind die ~ willkiirliche komplexe Zahlen. Beweis. Man sehreibe ~k + ir/k (mit reellen ~k, qk) fik ~ und beachte, dass ~k

+ r/k ~

k~

+ k~

(keine Summation)

also

k = I\

OZk

~Zk

)

"

Erreicht nun ~ in z o sein Supremum, so muss dort bei beliebiger Wahl der (k die Ungleichung

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A. DINGHAS

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gelten. Daraus folgt wegen DgO + D2;O= 4 02~p die Behauptung (3.5). Nachfolgender Hilfssatz gestattet die Anwendung des Hilfssatzes 5: HILFSSATZ 7. Man setze 1 Vo = l o g 1 -

Ilwll 2

Dann ist die (Hermitesche) quadratische Form

O2Vo positiv definit.

Bewds. Da in E, die Ungleichung I JI > 0 gilt, hat das lineare Gleichungssystem

nichttriviale L~Ssungenin E,. Nun ist

02Vo

02110 Ow. ~ ,

o e vo

02110

und somit

Das beweist (mit Rticksicht auf den Hilfssatz 3) die Behauptung. Nach diesen Vorbereitungen kann die Ungleichung (3.4) folgendermassen bewiesen werden: Aus (3.5) folgt (in Zo) 02V1 r ~

02U, "

und somit nach dem Hilfssatz 5 d.h. der Ungleichung (2.10)

I 02V1

02Ur --

~

Das liefert wegen (3.2) und (3.3) die Ungleichung

"

19671

OBER DAS SCHWARZSCHE LEMMA UND VERWANDTE $ATZE

165

exp V1 - lim E ~5 + [Jof,,,(s)dw(s)) )

( ["T ---

\2\p12

Jo

where the first inequality follows from (9) and the second from Fatou's lemma. COROLLARY 1. (10)

EI

Under the condition of Theorem 1.

:of(s)dw(s) i'=< ( p -

1)p12T(t'-2)/2

f: Elf(s)l'ds.

The proof follows from (9) by direct application of the H61der inequality to y~l E]:'[f(s) l'ds with a = p/(p - 2) and p = p/2. REMAgK: For p = 4, inequality (10) has been known ([1] p. 23, with 36 instead of (p - 1)p/2 = 9, and [4] in a weaker form). 2. Stochastic differential equations. Let E, be the Euclidean r-space. Let m(t, x), (t >=O, x e E,) assume values in E,; G(t, x), (t ~_ O, x e E,) will assume real r x q matrix values and w(t) will denote the q-dimensional Wiener process. The prime (') will denote the transpose of a vector or a matrix; for vectors I • [ will denote the Euclidean norm, for matrices G, [G I will denote the norm (trace GG') 1/2 ([2] p. 209). Assume that re(t, x) and G(t, x) are measurable functions of their variables, that (11)

Im(t,x)- m(t'y)l 0

[July u, v > O,

t > max (0, - u" v).

It can be shown for these examples that the assumptions on F(u,v) of both parts of Theorem 1 hold. From the last example it follows that (12)

fi

(Y+Yl+t+t), Y,,+I=Yt t > m a x ( O , - y l y f i

i#j

i=l,...,n)

/=1

is maximal when (y) is arranged in circular symmetrical order, and if no three elements of (y) have the same value, then the maximum is attained only if (y) is arranged in circular symmetrical order. We now turn to another result concerning a product which is a generalization of the following lemma of Duitin and Schaeffer. LEMMA I-2, p 522]. Let the set (y) > 0 of 2n nonnegative numbers be given except in arrangement. Then M

(13)

I-[ (Y2+-1" Y2~ + 0

t> 0

i=1

is maximal when (y) is arranged in decreasing order. Generalizing this result we obtain the following theorem.

THEOREM 2. Let (a)=(aa,...,am) and ( Y ) = ( Y t , ' " , Y 2 , ) be sets of nonnegative qumbers where (y) is given except in arrangement, then the product (14)

fi

[(y2~_ay2~) m + at(Y2~_ly2~) m-1 + ... + %]

f=l

attains its maximum when (y) is arranged in decreasing order.

Proof. The proof is by induction on m. m = 1 is the lemma of Duttin and Schaeffer. Let Yl be the maximal term in (y). If between Yl and Y2 there is a term, let us call it Y3, Y2 < Ya < Yl, we interchange Y2 with Y3 and consider the difference a = [(YlYa)m+ a,(ylya)m-1+ ... + am] [(Y2Y+)"+ al(Y,Y+) " - t + "'" + am] - [(Yly2)m + al(y,y2) " - t + "" am] [(Y3Y+)m + a l ( Y , y 4 ) ' - ' + "'" + am] = {[ax(YlY3)m-t+ "'" + am] [al(Y2Y+) m-1 + "'" + am] - -

[a~(y~y2) " - ~ + ... + am] [al(yzy+) m-x + ... + am]}

+ {(yay3)"[at(yzy+) "-~ + ... + a,,] + (y2y+)"[at(y~y3) ''-1 + ... + am] - (yly2)'[at(y3y,,) " - I + "'" + am] - (y~y+)'[al(y~y2)" + "'" + am]}.

1967]

THE INCREASE OF SUMS AND PRODUCTS

181

Let us look at the terms after the equality sign: By the assumption of induction the term in the first braces is nonnegative. The term in the second braces is also nonnegative because it is equal to ~, ak(yty2yay4) m-k [(Y~ -- yk4) (yak - y2k)] k=l

and we assumed that Ya > Y2, Y~ > Y4. Therefore A >=0 and we can rearrange the set (y) in such a way that two greatest numbers of (y) will appear in the same term of the product (14) without diminishing it. We continue the same process for the remaining terms of the product (14). This completes the proof of the theorem. The proof shows that if (y) > 0 and at least one of the ak, k = 1, ..., n, is posit ive, then the maximum is attained only in those cases in which neither Y2i- 1 < Yk < Y2~

nor Y2i < Y~ < Y2i-1

holds. REFERENCES

1. A. L. Lehman, A result on rearrangements, Israel J. Math. 1, No. 1 (1963), 22-28. 2. R. J. D u f ~ and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Sot:. 50 (1941) 517-528. TECHNION--ISRAELINSTITUTEOF TECHNOLOGY HAIFA, ISRAEL

SUBFAIR CASINO FUNCTIONS ARE SUPERADDITIVE BY

LESTER E. DUBINS(*) AI~TRACT It is shown that all subfair casino functions are superadditive on the unit interval. As the main step in showing that bold play in a primitive casino is optimal, and, consequently, that the utility U of the bold strategies in a primitive casino is a casino function, it was shown in [1, Chap. 6] that for a l l f a n d g in the closed unit interval, U satisfies: (1)

U(f + g) ~_ U (f) + U(g) i f f + g < l ,

and

(2)

U(f+g-1)_~V(f)+U(g)-I

if f + g _ - l .

It turns out to be very simple to prove TI-mOl~M 1. (**) Every subfair casino function U satisfies (1) and (2). Proof.*** We first show thatlU satisfies (1). As was observed in [1, Chap. 4, Sees. 2 and 3], (3)

U(f g) ~_ U(f)U(g),

and

(4)

U(f + g(1 - f ) ) ~ U ( f ) + U(g)(1 - U(f)),

for all f and g in the unit interval The only discontinuous subfair casino function is 0 for 0 V*(U*(x), U*(y)),

even if V is not monotone. Therefore, it may be supposed that x, y, and V*(x, y) are in the unit interval, but V(f, g) is not. Verify that because V*(x, y) is in the unit interval, so is V(U(f), U(g)). So by monotoneity of V, V(f,g) cannot be less than 0, and therefore must exceed 1. So V*(U*(x), U*(y))= 1 - V ( f , g ) < 0 < U*(V*(x,y)). This completes the proof of Lemma 1. Since (1) has been shown to hold for all subfair casino functions, the function

184

LESTER E. DUBINS

V(f,g) = f + g is in ~v'. In view of Lemma 1, so is the function f + g - 1, that is, (2) also holds for all subfair casino functions U. The proof of Theorem 1 is now complete. As is easily seen, the proofs of Theorem 1 and Lemma 1 apply not only to casino functions U but to all bounded solutions U to (3) and (4) omitting those U for which U ( f ) - 1 for 0 < f < 1. Inequalities (3) and (4) for casino functions U have intuitive interpretations that make them apriori plausible and, therefore, natural to conjecture. It would be nice to find such interpretations for (1) and (2).

REFERENCES

1. L. E. Dubins, and L. J. Savage, How To Gamble If You Must. McGraw-Hill, New York. 1965. UNIVERSITYOF CALIFORNIA, BERKELY,CALIFORNIA

A FUNCTIONAL METHOD FOR LINEAR SETS BY

R. KAUFMAN ABSTRACT

Kronecker sets for approximation by characters are constructed by an application of Baire's theorem to Banach spaces of differentiable functions. A compact subset E o f ( - oo,oo) is a Kronecker set [2, §5.2] if the exponential functions e~aX(- oo < 2 < oo) are uniformly dense in the continuous complexvalued functions of modulus 1 on E. Wik [3] has constructed Kronecker sets of Hausdorff dimension 1; in fact E can carry a positive measure subject to any prescribed continuity condition weaker than absolute continuity. However, the sets constructed in [3] seem to be very unevenly dispersed; we shall describe a function-space method that necessarily yields Kronecker sets with some degree of symmetry. Let p be a positive integer and rl,rz, r3,.., numbers such that (1)

0 < 2r~+1 < r~ < 1

(2)

sup

(1 __ 2 ( / / ) -1 > r . + 1, satisfy the inequalities //2 r,p >//;t(//)-1 > r.+1;

(3)

,~,(//)-1and r. can be made arbitrarily small by (2). For a number t > 0, //1e (0, I), and a complex number z of modulus I set

V(t,Z, rh)= { - ~o < x < oo, le '= - z I = 2//lt -1 or > 2n. Divide Y into disjoint closed subsets (with mutual distances > p > 0) on each of which h has oscillation ~. One can, therefore, limit oneself from the beginning to an initial section consisting of all ordinals < ~ and then let ~ be arbitrary large. In fact, the whole work can be carried in Zermelo-Fraenkel's set theory at the cost of encumbering, somewhat, the formulation. Those who still feel unsure may imagine that all the classes involved are subsets of 0, where 0 is some strongly inaccessible cardinal, and Ord = O. For the sake of convenience we introduce, a new symbol, " ~ " , and make the convention that ~ < go for all ordinals ~. go is not to be considered an ordinal, and symbols such as " ~ " , "fl",--- which are used to denote ordinals never denote OO.

A sequence X = <X~>~~ 0 (6 > ~ > 0) we have X~ = na O if D(f) = 0(9) and f ( X ) D_g(X) for all X e O(f). If {fi}i ~ff~ is family of functions then n i , tf~ is the function whose domain is n~iD(f~) and whose values are given by: ( At ~ff,) (X) --- n , ~tf,(x) F = ~ 0 and, a # U a < ~c9 whenever 7 < a and a p < a for all f l < ? . The members ot ft/'(Rg U {0}) is what Mahlo, [1], calls the n~-numbers.

n r,~(X) = D f th(y, Jp~(X)). The following proposition sums up the properties of the doubly-indexed array

n~,.(x). PROPOSITION 3. (i) I f It > 0 then ct = nx,u(X) for some 4, iff for all v < # = n~,~(X).

(ii) nz,(X) as a function of v is non-decreasing. (iii) I f It > v and na,~(X) < oo then the following three conditions are equivalent na,z(X) > n~,~(X), nz,~,(X) ¢ ft g`+t(X), 4 6fl ?+ X(X). (iv) I f It > 4 and 4 + 1 • X then n~,,x(X) ~ na.~(X ). (v) Assume that # > 4 and that either n~,~(X) < oo or n~,x(X) < oo. Then n~,~,(X) = n~,,x(X) iff # = n~,~,(X); each side implies that 4 is the least ordinal which is not in X. Proof. (i) is straightforward. (ii) follows from the fact that fp~(X) is decreasing with v, and from proposition

2(i). (iii) Each of the conditions 4~f1¢'+l(X), n a , ~ f p " + l ( X ) i s equivalent to 4 = n~,,(X). Now, zr~,(X) >_ n~.~(X) > 4. Hence 4 = n~,j,(X) implies zc~,,(X) = na,~(X). Consequently nx,,(X)> rcx.,(X) implies each of the other two con-

1967l

A GENERALIZATION OF MALHO'S METHOD

193

ditions. On the other hand n~,~,(X)= rcx,,(X) means, since 7rx,~(X)< 0% that na,~(X) ~ fp"(X), hence rcx,~(X)~fp v+l(X), implying 2 = rcx,v(X) = rca,~(X). (iv) If na,u(X) = ~ it is obvious. Otherwise let V be the first ordinal which is not in X. Then ~ < 2 and hence rc~,u(X) < ~ . Obviously ~ CfpU+1(X).Hence, from (ii) and (iii) it follows that rcr,,(X) is strictly increasing as a function of v, for all v tt, Hence nx,~(X)> p. Put ~ = 7ra,u(X). Now ~efp~(X)~_fpX+~(X), therefore ~=zcg, x(X). Since ~ > # we have rcu,x(X) =< ~ , ~ ( x ) = ~ = ~ , ~ ( x ) . (v) If 7ra,~(X) = =u,a(X) < m, then, putting ~ = rca,~(X), it follows by (i) that = n=,a(X). Hence ~ = # and we get # = 7ra,u(X). On the other side, if/~ = =a,~(X), then, again by (i), # = rcu,a(X) and therefore 7ra,u(X)= 7ru,a(X). It is clear that if 2 + 1 ___X then rca,,(X ) = 2, for all a. Hence, if rc;,,u(X) =/~ > 2 we have an ordinal < 2 which is not in X. Let ~ be the smallest one, By (iv) we have: ~,~(X) < 7rr,u(X). Since # < rc~,,~(X)one gets: # < rc~,~(X) < rc~,~(X) < 7~a,u(X) = ~t. Hence rcr,a(X) = 7~a,~(X) < m and, consequently, y = 2. The class of fixed points for the second index is F'(X) where F = (fpV), ~, on the other hand it is < ~. Hence ~eL(X). If ~ e L(X) then ~ = U ~ < ~ where ~ is a limit ordinal > 0 and ~ a strictly increasing sequence of members of X. Since ~ is regular we have ~ = ~. Put Y = {~: 2 < 7} u {~}. Then ~ = th(~, Y). Since Y ~ X it follows that if~ = th(fl, X) then fl >-_~. But we must have fl < ~. Hence ~ = th(~, X). Note that the argument of the first half of the proof shows that, for all

X, Jp(X) O L(Ord) c_ L(X). We define X to be closed in Y if L(X) r3 Y ___ Y. If X _ Y this actually means that X ts a subclass of Y which is closed in the order-induced topology, tin general it means that X is a closed subclass of X u Y. A function f is closed in Y if, for everygX e D(f), if X is closed in Y so is f(X). It is easily seen that if Xi is closed in Yfor all i e I so is Ni ,IXi • Consequently iff~ is a function which is closed in Yfor all i e I so is N~ ~1fi. I f f a n d # are closed in Y s o i s f o g . PROPOSITION 5. (i) If X is continuously decreasing sequence of classes of ordinals, each of which is closed in Ythen XDis closed in Y. (ii) If F is continuously decreasing sequence of functions, each closed in Y, then FD is closed in Y.

194

HAIM GAIFMAN

[July

ProoL Assume with no loss of generality that X = (X~)~ 2 consider fl = U ~ < ~ and Y = {4: fl < ~ < a}.) THEOREM (MAHLO). (I) I f Y ~_ Rg and rcu,~(Y)> lt, v then z~p,~(Y)(~h(Y). (II) If Y ~ Rg then the first ~ > 0 such that a ~fp~(Y) does not belong to h(Y). This follows from the generalization which we formulate and prove next. The idea of the following theorems is to indicate a sense in which the function h is "stronger" than other "decent" functions. Hereby, "stronger" roughly means that its application yields smaller classes of ordinals, whose members can, therefore, be considered as "larger." " D e c e n t " includes, among the rest, the function fp, as well as any function fp, fpA, and lots of others which are specified in this section. Theorem 1 implies that, under certain conditions, a regular a, which is in h(7), cannot be removed by an application of any function which arises out of a "decent function" by means of a process which involves composition of functions,

1967]

A GENERALIZATION OF MALHO'S METHOD

195

iterations of less than ~ times, and forming the diagonal of a continuously decreasing sequence of functions. Mahlo's theorem, which is implied by Theorem 1, deals with two special cases which are typical to the general state of affairs. The first case is that where = G.~(Y) >/~, v. Here, applying fp~ will still leave ~, but one more application of fp (or # + 1 applications of q) will remove it. In the second case ~ is the first fl such that fl = fpa(y). Here an application of the diagonal of 0. Let f be a LTF and let g be its restriction to subsets of 0~+ 1. We will have g#~ Q(~,g) for all ~ < 0~. Hence, since the domain of g is limited to subsets of ~ + 1 and p_0) is exactly the family of all (positive) homothcties of the unit ball B. Suppose that ~ were any family of closcd convex scts.Define S to be strictlyconvex with respect to c¢ if S c C for some C eC¢ and C e ~ S c C and • n bdC ~ f~ implies that S r3 bdC has exactly one clement Since B could have been a translateof any given compact convex body, wc have the following reformulation of Theorem 2. TrmOREM 2'. I f the set S is strictly convex with respect to any family of all positive homotheties of some fixed compact convex body, then S consists of exactly one point. 4. Proof of Theorem 2.

We introduce the real-valued function g defined by

g(x) = [1qCx) - x [[ = sup {11 y - x U: y ~ s} The function g is the supremum of the elementary functions x -~ Ity - x [I that are clearly convex, so g itself is convex in the sense of Section 1. It also satisfies the following Lipschitz condition (4.1)

- II x - y II --< gCx) -

gcy) g*(z), z ~ doing*} and its supporting hyperplane {(z, a): a = (z, x ) - g(x)} In other words, Og(x) is the projection onto R" of a face of grg*, in the sense of Section 2. We want of course to apply the generalized Straszewics theorem to grg*. However, this is an unbounded set, so we must find out how to truncate it in a good way. The following inequalities hold for g:

Its - x il < g(x) < g(0) + II - x 11 for all x in R" From them, we derive the corresponding inequalities for g*:

- g ( O ) < g * ( x ) < ( x , s ) - 0 we have

g(x) - g(y) < - ( - z , x - y ) = -

Iltx -

y ]l

so, by (4.1) we have equality, and q(x) = q(y). We will now show that the midpoint o f u and b(x) is in bdB, i.e. ½(u + b(x)) = 1 . Take 2 = q ( x ) - x above. Then = [q