Finitely repeated prisoners’ dilemma experiments without a commonly known end

Using a symmetric two-player prisoners’ dilemma as base game, each player receives a signal for the number of rounds to be played with the same partner. One of these signals is the true number of rounds R while the other is R − 5. Thus both players know that the game has a finite end. They both know that the opponent knows this, but the finite end is not commonly known. As a consequence, both mutual defection and mutual cooperation until the second last round are subgame perfect equilibrium outcomes. We find experimental evidence that many players do in fact cooperate beyond their individual signal round.     

On computational complexity of membership test in flow games and linear production games

Let Γ≡(N,v) be a cooperative game with the player set N and characteristic function v: 2^N→R. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the flow game (and equivalently, for the linear production game), the core is always non-empty and a solution in the core can be found in polynomial time. In this paper, we show that, given an imputation x, it is NP-complete to decide x is not a member of the core, for the flow game. And because of the specific reduction we constructed, the result also holds for the linear production game. Received: October 2000/Final version: March 2002     

Van der waerden and ramsey type games

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy. By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK _N ^k , and the first player wins if he has selected allk-tuples of ann-set. LetR _k*(n) be the least integerN so that the first player has a winning strategy. We prove (W* (n))¹/n → 2,R ₂*(n)<(2+ε) ⁿ andR _k ^* n<2 ^nk / ^k! fork ≧3.     

Correlated equilibrium payoffs and public signalling in absorbing games

An absorbing game is a repeated game where some action combinations are absorbing, in the sense that whenever they are played, there is a positive probability that the game terminates, and the players receive some terminal payoff at every future stage.  We prove that every multi-player absorbing game admits a correlated equilibrium payoff. In other words, for every ε>0 there exists a probability distribution p _ε over the space of pure strategy profiles that satisfies the following. With probability at least 1−ε, if a pure strategy profile is chosen according to p _ε and each player is informed of his pure strategy, no player can profit more than ε in any sufficiently long game by deviating from the recommended strategy. Received: April 2001/Revised: June 4, 2002     

Chromatic uniqueness of certain complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k ²/4 + i + 1.     

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).     

Holomorphic two-spheres in complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, 4)

In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent.     

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x ₁,…,x _n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I _n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I _n,d )=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I _n,d )≤⌊(n−d)/(d+1)⌋+d.     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

On two Hamilton cycle problems in random graphs

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G _n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G _n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n. Research supported in part by NSF grant CCR-0200945. Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.     

Estimates of the number of singular points of a complex hypersurface and related questions

It is well known that the number of isolated singular points of a hypersurface of degree d in ℂP^m does not exceed the Arnol’d number A_m(d), which is defined in combinatorial terms. In the paper it is proved that if b _m−1 ^± (d) are the inertia indices of the intersection form of a nonsingular hypersurface of degree d in ℂP^m, then the inequality A_m(d)<min{b _m−1 ⁺ (d), b _m−1 ⁻ (d)} holds if and only if (m−5)(d−2)≥18 and (m,d)≠(7,12). The table of the Arnol’d numbers for 3≤m≤14, 3≤d≤17 and for 3≤m≤14, d=18, 19 is given. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 180–190. Translated by O. A. Ivanov and N. Yu. Netsvetev.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v _∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v _∞ (p) (and will converge from above to v _∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v _∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG _∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v _n (p)− v _∞ (p) is of the order of magnitude of . In that case the value of SG _∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛^∞={X _k }^∞ _k=1, one defines for each n : V _n (𝔛^∞) ≔E∑ⁿ _k=1 |X _k+1 − X _k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛^∞, V _n (𝔛^∞) can be of the order of magnitude of n ^1/2−ε, for arbitrarily small ε>0. Received January 1999/Final version April 2002     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

Acyclick-strong coloring of maps on surfaces

A coloring of graph vertices is called acyclic if the ends of each edge are colored in distinct colors and there are no two-colored cycles. Suppose each face of rank not greater thank, k ≥ 4, on a surfaceS ^N is replaced by the clique on the same set of vertices. Then the pseudograph obtained in this way can be colored acyclically in a set of colors whose cardinality depends linearly onN and onk. Results of this kind were known before only for 1 ≤N ≤ 2 and 3 ≤k ≤ 4. Translated fromMatematicheskie Zametki, vol. 67, No. 1, pp. 36–45, January, 2000.     

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

Bounds on the Total Restrained Domination Number of a Graph

Let G = (V, E) be a graph. A set S í V{S \subseteq V} is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ _tr (G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ _tr (G) ≤ n − diam(G) − r + 1.     

Weierstrass points on trigonal curves I: The ramification points

LetC be a smooth curve of genusg≥5. Assume thatP is a Weierstrass point onC which first non-gap is equal to 3. The gap sequence atP is completely determinated by numbersn and ε satisfying (g−1)/3≤n≤g/2 and ε is 1 or 2 as follows. Given suchn and ε, the corresponding gap sequence is (1, 2, 4, 5,…, 3n−2, 3n−1, 3n+ε, 3n+3+ε, …, 3(g−n−1)+ε). We say thatP is of then-th kind andP is of type I (resp. II) if ε=1 (resp. 2). Because a curve of genusg≥5 has at most one linear systemg1/3, it follows that the Weierstrass points onC with first non-gap equal to 3 are of the same kind.     

A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation

In the case of the Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with a small parameter ɛ multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost ɛ-uniformly (that is, the convergence rate of this scheme weakly depends on ɛ) is constructed. When ɛ is not very small, this scheme converges with an order of accuracy close to two. For the construction of the scheme, we use the classical monotone (of the first order of accuracy) approximations of the differential equation on a priori adapted locally uniform grids that are uniform in the domains where the solution is improved. The boundaries of such domains are determined using a majorant of the singular component of the grid solution. The accuracy of the scheme is improved using the Richardson technique based on two embedded grids. The resulting scheme converges at the rate of O((ɛ⁻¹ N ^−K ln² N)² + N ⁻²ln⁴ N + N ₀⁻²) as N, N ₀ → ∞, where N and N ₀ determine the number of points in the meshes in x and in t, respectively, and K is a prescribed number of iteration steps used to improve the grid solution. Outside the σ-neighborhood of the lateral boundary near which the boundary layer arises, the scheme converges with the second order in t and with the second order up to a logarithmic factor in x; here, σ = O(N ^{−(K − 1)}ln² N). The almost ɛ-uniformly convergent finite difference scheme converges with the defect of ɛ-uniform convergence ν, namely, under the condition N ⁻¹ ≪ ɛ^ν, where ν determining the required number of iteration steps K (K = K(ν)) can be chosen sufficiently small in the interval (0, 1]. When ɛ⁻¹ = O(N ^{K − 1}), the scheme converges at the rate of O(N ⁻²ln⁴ N + N ₀⁻²).