The Harsanyi-Shapley solution and independence of irrelevant expansions

We characterize the Harsanyi-Shapley solution/value on essential and smooth games. An axiom – independence of irrelevant expansions of N T U games is introduced.The authors are grateful to two anonymous referees for very helpful suggestions and comments which make the paper much more readable. They are also grateful to Professor Peter Sudhoelters useful comments and the associate editor who is in charge of the paper.     

On finding curb sets in extensive games

We characterize strategy sets that are closed under rational behavior (curb) in extensive games of perfect information and finite horizon. It is shown that any such game possesses only one minimal curb set, which necessarily includes all its subgame perfect Nash equilibria. Applications of this result are twofold. First, it lessens computational burden while computing minimal curb sets. Second, it implies that the profile of subgame perfect equilibrium strategies is always stochastically stable in a certain class of games.I am grateful to J. Kamphorst, G. van der Laan and X. Tieman, who commented on the earlier versions of the paper. I also thank an anonymous referee and an associate editor for their helpful remarks. The usual disclaimer applies.     

Mixed strategy solutions forN-person quadratic games

Mixed strategy -equilibrium points are given forN-person games with cost functions consisting of quadratic, bilinear, and linear terms and strategy spaces consisting of closed balls in Hilbert spaces. The results are applied to linear-quadratic differential games with no information and quadratic integral constraints on the control functions.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

The core and the Weber set for bicooperative games

This paper studies two classical solution concepts for the structure of bicooperative games. First, we define the core and the Weber set of a bicooperative game and prove that the core is always contained in the Weber set. Next, we introduce a special class of bicooperative games, the so-called bisupermodular games, and show that these games are the only ones in which the core and the Weber set coincide.      

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.

---

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.

     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Given a family of real-valued functions defined in a normed vector space X, we study a class of -convex functions having a simpler representation for the --subdifferential. The case =X^* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X^*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.This work is based on research material supported in part by CONICYT-Chile through FONDECYT 101-0116 and FONDAP-Matemáticas Aplicadas II.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

On vector lattices of elementary Carathéodory functions

In this paper we deal with the vector lattice C(B) of all elementary Carathéodory functions corresponding to a generalized Boolean algebra B.This work was supported by grant VEGA 2/1131/21.     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.     

Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: -convexity (in the sense of Klötzler and Hartwig), -convexity and midpoint -convexity (in the sense of Hu, Klee, and Larman), -convexity and midpoint -convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.     

Fundamentals for symplectic $$ \mathcal{A} $$-modules. Affine Darboux theorem

In his [9–11], the first author shows that the sheaf-theoreti-cally based Abstract Differential Geometry incorporates and generalizes classical differential geometry. Here, we undertake to explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.      

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

Indivisibility of Class Numbers and Iwasawa λ-Invariants of Real Quadratic Fields

Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Ono's idea, we show that for any prime p>3, {0<D<X|h(D)0(mod p)}>> _p (X)/logX.