Mixed strategy solutions for quadratic games

Mixed strategy solutions are given for two-person, zero-sum games with payoff functions consisting of quadratic, bilinear, and linear terms, and strategy spaces consisting of closed balls in a Hilbert space. The results are applied to linear-quadratic differential games with no information, and with quadratic integral constraints on the control functions.     

A linear quadratic optimal control problem with disturbances—An algebraic Riccati equation and differential games approach

A linear quadratic optimal control problem with coexisting initial and persistent disturbances is studied. Upper and lower values and relevant algebraic Riccati equations (ARE for short) are introduced. Various relations among these values are presented. The solvability of the resulting AREs is shown to be closely related to the solvability of the original optimal control problem. A formula is obtained for the solution of one of the AREs, which is of nonstandard form. Several unexpected features of the original problem are revealed from the standpoint of differential games. Some known results on the so-calledH optimal control problem are recovered.This work was partially supported by the Chinese NSF under Grant 19131050, the Chinese State Education Commission Science Foundation, the SEDC Foundation for Young Academics, and the Fok Ying Tung Education Foundation.     

Bankruptcy games

Bankruptcy problems are considered from a game theoretic point of view. Solution concepts from cooperative game theory are studied for bankruptcy games. A necessary and sufficient condition for a division rule for bankruptcy problems to be a game theoretic rule is given. A new division rule which is an adjustment of the proportional rule is given. This rule coincides with theT-value for bankruptcy games. Properties of the new rule are treated and a set of characterizing properties is given.

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Zusammenfassung In dieser Arbeit werden Bankrottprobleme von spieltheoretischer Warte aus behandelt; insbesondere werden Lösungskonzepte der kooperativen Spieltheorie für Bankrottspiele untersucht. Eine notwendige und hinreichende Bedingung wird angegeben dafür, Daß eine Aufteilungsregel für Bankrottprobleme spieltheoretischer Natur ist. Ferner wird eine neue Aufteilungsregel angegeben, welche eine passende Modellierung der Proportionalitätsregel ist. Diese Regel fällt mit dem-Wert für Bankrottspiele zusammen. Schließlich werden Eigenschaften dieser neuen Regel untersucht und eine Axiomatisierung angegeben.

     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Weak Barriers in Nonlinear Potential Theory

We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for -harmonic functions under the usual assumptions on , and for Cheeger p-harmonic functions in metric spaces.      

ℓ-norm and its Application to Learning Theory

We investigate connections between an important parameter in the theory of Banach spaces called the -norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the -norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the -norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces.     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Some Congruences on S (X)

Some congruences on S (X) are investigated in this paper. A unique atom in the sublattice [C(E), C (E)] is determined for some spaces X and S-equivalences E and X.AMS Subject Classification (1991): 20M20 54H15     

Sufficient conditions for Nash equilibria inN-person games over reflexive Banach spaces

Sufficient conditions are obtained for the existence of Nash equilibrium points inN-person games when the strategy sets are closed, convex subsets of reflexive Banach spaces. These conditions require that each player's cost functional is convex in that player's strategy, weakly continuous in the strategies of the other players, weakly lower semicontinuous in all strategies, and furthermore satisfies a coercivity condition if any of the strategy sets is unbounded. The result is applied to a class of linear-quadratic differential games with no information, to prove that equilibrium points exist when the duration of these games is sufficiently small.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

A Unified Fixed Point Theory in Generalized Convex Spaces

Let B be the class of ＇better＇ admissible multimaps due to the author. We introduce new concepts of admissibility （in the sense of Klee） and of Klee approximability for subsets of G-convex uniform spaces and show that any compact closed multimap in B from a G-convex space into itself with the Klee approximable range has a fixed point. This new theorem contains a large number of known results on topological vector spaces or on various subclasses of the class of admissible G-convex spaces. Such subclasses are those of O-spaces, sets of the Zima-Hadzic type, locally G-convex spaces, and LG-spaces. Mutual relations among those subclasses and some related results are added.     

Testing optimality for quadratic 0–1 unconstrained problems

This paper analyses a necessary and sufficient optimality condition for quadratic pseudo-Boolean unconstrained problems. It is proved that in general testing any necessary and sufficient optimality condition is a difficult task for anyNP-hard problem. An-optimality condition is derived together with an approximation scheme to test it.This work has been supported by the Progetto Finalizzato Trasporti 2, 93.01799.PF74.Professor Paolo Carraresi died unexpectedly on March 5, 1994. At the time of his death this paper had been completed. While undertaking the final revision, the other two authors were reminded just how much they were indebted to Professor Carraresi after many years of common work together.     

Nash Equilibria of Risk-Sensitive Nonlinear Stochastic Differential Games

This paper considers a class of risk-sensitive stochastic nonzero-sum differential games with parametrized nonlinear dynamics and parametrized cost functions. The parametrization is such that, if all or some of the parameters are set equal to some nominal values, then the differential game either becomes equivalent to a risk-sensitive stochastic control (RSSC) problem or decouples into several independent RSSC problems, which in turn are equivalent to a class of stochastic zero-sum differential games. This framework allows us to study the sensitivity of the Nash equilibrium (NE) of the original stochastic game to changes in the values of these parameters, and to relate the NE (generally difficult to compute and to establish existence and uniqueness, at least directly) to solutions of RSSC problems, which are relatively easier to obtain. It also allows us to quantify the sensitivity of solutions to RSSC problems (and thereby nonlinear H-control problems) to unmodeled subsystem dynamics controlled by multiple players.     

On Some Structural Properties of Banach Function Spaces and Boundedness of Certain Integral Operators

In this paper the notions of uniformly upper and uniformly lower -estimates for Banach function spaces are introduced. Further, the pair (X, Y) of Banach function spaces is characterized, where X and Y satisfy uniformly a lower -estimate and uniformly an upper -estimate, respectively. The integral operator from X into Y of the form

Improvement of the Delsarte Bound for τ-Designs When It Is Not the Best Bound Possible

In this paper the problem for improvement of the Delsarte bound for -designs is investigated. Two main results are presented. Firstly, necessary and sufficient conditions for improving the bound are proved. We define test functions with the property that they are negative if and only if the Delsarte bound D(phmmat;, ) can be improved by linear programming. Then we investigate the infinite polynomial metric spaces and give exact intervals, when the Delsarte bound is not the best linear programming bound possible. Secondly, we derive a new bound for the infinite PMS. Analytical forms of the extremal polynomials of degree + 2 for non-antipodal PMS and of degree + 3 for antipodal PMS are given. The new bound is investigated in different asymptotical processes for infinite PMS. When and n grow simultaneously to infinity our bound is better than Delsarte bound.     

Punishment strategies make the α-coalitional game ordinally convex and balanced

We present a sufficient condition under which the -coalitional game associated with a strategic form game is ordinally convex and balanced. The n-person prisoners dilemma games satisfy this condition.The author is grateful to Mikio Nakayama, Yukihisa Utsumi, William Thomson, an anonymous referee and an associate editor for helpful comments and suggestions.     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

Novel approach to accelerating Newton's method for sup-norm optimization arising inH ∞-control

We present algorithms for solving general sup-norm minimization problems over spaces of analytic functions, such as those arising inH control. We also give an analysis and some theory of these algorithms. Part of this is specific to analytic optimization, while part holds for general sup-norm optimization. In particular, we are proposing a type of Newton-type algorithm which actually uses very high-order terms. The novel feature is that higher-order terms can be chosen in many ways while still maintaining a second-order convergence rate. Then, a clever choice of higher-order terms greatly reduces computation time. Conceivably this technique can be modified to accelerate Newton algorithms in some other circumstances. Estimates of order of convergence as well as results of numerical tests are also presented.This work was partially supported by the Air Force Office of Scientific Research and the National Science Foundation.