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Vladimir Gurvich

Publications and source records attributed to Vladimir Gurvich.

At least 37 records · Page 2Linked to original sources
arXiv2023

On Nash-solvability of n-person graphical games under Markov's and a priori realizations

We consider graphical $n$-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in mixed strategies, we introduce probabilistic distributions in all non-terminal positions. The corresponding plays can be analyzed under two different basic assumptions: Markov's and a priori realizations. The former one guarantees existence of a uniformly best response of each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in mixed strategies. The classical Nash theorem is not applicable, since Markov's realizations may result in the limit distributions and effective payoff functions that are not continuous. The a priori realization does not share many nice properties of the Markov one (for example, existence of the uniformly best response) but in return, Nash's theorem is applicable. We illustrate both realizations in details by two examples with $2$ and $3$ players and also provide some general results.

math.CO↗
arXiv2023

Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles

In 1975 the first author proved that every finite tight two-person game form $g$ is Nash-solvable, that is, for every payoffs $u$ and $w$ of two players the obtained game $(g;u,w)$, in normal form, has a Nash equilibrium (NE) in pure strategies. This result was extended in several directions; here we strengthen it further. We construct two special NE realized by a lexicographically safe (lexsafe) strategy of one player and a best response of the other. We obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form $g$ is given explicitly. Yet, in applications $g$ is frequently realized by an oracle $\cO$ such that size of $g$ is exponential in size $|\cO|$ of $\cO$. We assume that game form $g = g(\cO)$ generated by $\cO$ is tight and that an arbitrary {\em win-lose game} $(g;u,w)$ (in which payoffs $u$ and $w$ are zero-sum and take only values $\pm 1$) can be solved, in time polynomial in $|\cO|$. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in $|\cO|$. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.

cs.GT↗
arXiv2023

Deterministic n-person shortest path and terminal games on symmetric digraphs have Nash equilibria in pure stationary strategies

We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u,v) is a move whenever (v,u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n-person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.

cs.GT↗
arXiv2022

Supercentenarian paradox

Consider the following statement: $B(t, Δt)$: a $t$ years old person NN will survive another $Δt$ years, where $t, Δt\in \mathbb{R}$ are nonnegative real numbers. We know only that NN is $t$ years old and nothing about the health conditions, gender, race, nationality, etc. We bet that $B(t, Δt)$ holds. It seems that our odds are very good, for any $t$ provided $Δt$ is small enough, say, $1 / 365$ (that is, one day). However, this is not that obvious and depends on the life-time probabilistic distribution. Let $F(t)$ denote the probability to live at most $t$ years and set $Φ(t) = 1 - F(t)$. Clearly, $Φ(t) \rightarrow 0$ as $t \rightarrow \infty$. It is not difficult to verify that $Pr(B(t, Δt)) \rightarrow 0$ as $t \rightarrow \infty$, for any fixed $Δt$, whenever the convergence of $Φ$ is fast enough (say, super-exponential). Statistics provide arguments (based on an extrapolation yet) that this is the case. Hence, for an arbitrarily small positive $Δt $ and $ε$ there exists a sufficiently large $t$ such that $Pr(B(t, Δt)) < ε$, which means that we should not bet... in theory. However, in practice we can bet safely, because for the inequality $Pr(B(t, Δt)) < 1/2$ a very large $t$ is required. For example, $Δt = 1/365$ may require $t > 125$ years for some typical distributions $F$ considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good, either because the chosen testee NN is not old enough, or for technical (or, more precisely, statistical) reasons -- absence of a testee. This situation is similar to the famous St.Petersburg Paradox.

math.PR↗
arXiv2022

Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms

Let $\mathcal{A} = \{A_1, \ldots, A_m\}$ and $\mathcal{B} = \{B_1, \ldots, B_n\}$ be a pair of dual multi-hypergraphs on the common ground set $O = \{o_1, \ldots, o_k\}$. Note that each of them may have embedded or equal edges. An edge is called containment minimal (or just minimal, for short) if it is not a strict superset of another edge. Yet, equal minimal edges may exist. By duality, (i) $A \cap B \neq \emptyset$ for every pair $A \in \mathcal{A}$ and $B \in \mathcal{B}$; (ii) if $A$ is minimal then for every $o \in A$ there exists a $B \in \mathcal{B}$ such that $A \cap B = \{o\}$. We will extend claim (ii) as follows. A linear order $\succ$ over $O$ defines a unique lexicographic order $\succ_L$ over the $2^O$. Let $A$ be a lexicographically maximal (lexmax) edge of $\mathcal{A}$. Then, (iii) $A$ is minimal and for every $o \in A$ there exists a minimal $B \in \mathcal{B}$ such that $A \cap B = \{o\}$ and $o \succeq o'$ for each $o' \in B$. This property has important applications in game theory implying Nash-solvability of tight game forms as shown in the old (1975 and 1989) work of the first author. Here we give a new, very short, proof of (iii). Edges $A$ and $B$ mentioned in (iii) can be found out in polynomial time. This is trivial if $\mathcal{A}$ and $\mathcal{B}$ are given explicitly. Yet, it is true even if only $\mathcal{A}$ is given, and not explicitly, but by a polynomial containment oracle, which for a subset $O_A \subseteq O$ answers in polynomial time whether $O_A$ contains an edge of $\mathcal{A}$.

math.CO↗
arXiv2022

On Nash-Solvability of Finite Two-Person Tight Vector Game Forms

We consider finite two-person normal form games. The following four properties of their game forms are equivalent: (i) Nash-solvability, (ii) zero-sum-solvability, (iii) win-lose-solvability, and (iv) tightness. For (ii, iii, iv) this was shown by Edmonds and Fulkerson in 1970. Then, in 1975, (i) was added to this list and it was also shown that these results cannot be generalized for $n$-person case with $n > 2$. In 1990, tightness was extended to vector game forms ($v$-forms) and it was shown that such $v$-tightness and zero-sum-solvability are still equivalent, yet, do not imply Nash-solvability. These results are applicable to several classes of stochastic games with perfect information. Here we suggest one more extension of tightness introducing $v^+$-tight vector game forms ($v^+$-forms). We show that such $v^+$-tightness and Nash-solvability are equivalent in case of weakly rectangular game forms and positive cost functions. This result allows us to reduce the so-called bi-shortest path conjecture to $v^+$-tightness of $v^+$-forms. However, both (equivalent) statements remain open.

cs.GT↗
arXiv2021

On Nash-solvability of finite $n$-person deterministic graphical games; Catch 22

We consider finite $n$-person deterministic graphical (DG) games. These games are modelled by finite directed graphs (digraphs) $G$ which may have directed cycles and, hence, infinite plays. Yet, it is assumed that all these plays are equivalent and form a single outcome $c$, while the terminal vertices $V_T = \{a_1, \ldots, a_p\}$ form $p$ remaining outcomes. We study the existence of Nash equilibria (NE) in pure stationary strategies. It is known that NE exist when $n=2$ and may fail to exist when $n > 2$. Yet, the question becomes open for $n > 2$ under the following extra condition: (C) For each of $n$ players, $c$ is worse than each of $p$ terminal outcomes. In other words, all players are interested in terminating the play, which is a natural assumption. Moreover, Nash-solvability remains open even if we replace (C) by a weaker condition: (C22) There exist no two players for whom $c$ is better than (at least) two terminal outcomes. We conjecture that such two players exist in each NE-free DG game, or in other words, that (C22) implies Nash-solvability, for all $n$. Recently, the DG games were extended to a wider class of the DG multi-stage (DGMS) games, whose outcomes are the strongly connected components (SCC) of digraph $G$. Merging all outcomes of a DGMS game that correspond to its non-terminal SCCs we obtain a DG game. Clearly, this operation respects Nash-solvability (NS). Basic conditions and conjectures related to NS can be extended from the DG to DGMS games: in both cases NE exist if $n=2$ and may fail to exist when $n > 2$; furthermore, we modify conditions (C) and (C22) to adapt them for the DGMS games. Keywords: $n$-person deterministic graphical (multi-stage) games, Nash equilibrium, Nash-solvability, pure stationary strategy, digraph, directed cycle, strongly connected component.

cs.GT↗
arXiv2021

Shifting paths to avoidable ones

An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced path of the same length, and proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths are shifts of each other if their union is an induced path with $k+1$ vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.

math.CO↗
arXiv2021

Logical contradictions in the One-way ANOVA and Tukey-Kramer multiple comparisons tests with more than two groups of observations

We show that the One-way ANOVA and Tukey-Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher-Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with k > 2 groups of observations, even under the INAH assumptions, with the same significance level $α$, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis $H_0^{A}: μ_1 = \ldots = μ_k$, while the TK one, $H_0^{TK}(i,j): μ_i = μ_j$, is not rejected for any pair $i, j \in \{1, \ldots, k\}$; (ii) the TK test rejects $H_0^{TK}(i,j)$ for a pair $(i, j)$ (with $i \neq j$) while ANOVA does not reject $H_0^{A}$. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any $k \geq 3$ and in case (ii) for some larger $k$. Furthermore, in case (ii) ANOVA, being restricted to the pair of groups $(i,j)$, may reject equality $μ_i = μ_j$ with the same $α$. This is an obvious contradiction, since $μ_1 = \ldots = μ_k$ implies $μ_i = μ_j$ for all $i, j \in \{1, \ldots, k\}.$ Similar contradictory examples are constructed for the Multivariable Linear Regression (MLR). However, for these constructions it seems difficult to verify the Gauss-Markov assumptions, which are standardly required for MLR.

math.ST↗
arXiv2021

More about Exact Slow $k$-Nim

Given $n$ piles of tokens and a positive integer $k \leq n$, the game Nim$^1_{n, =k}$ of exact slow $k$-Nim is played as follows. Two players move alternately. In each move, a player chooses exactly $k$ non-empty piles and removes one token from each of them. A player whose turn it is to move but has no move loses (if the normal version of the game is played, and wins if it is the misére version). In Integers 20 (2020) 1-19, Gurvich et al gave an explicit formula for the Sprague-Grundy function of Nim$^1_{4, =2}$, for both its normal and misére version. Here we extend this result and obtain an explicit formula for the P-positions of the normal version of Nim$^1_{5, =2}$ and Nim$^1_{6, =2}$.

math.CO↗
arXiv2020

Balanced flows for transshipment problems

A transshipment problem (G, d, λ) is modeled by a directed graph G = (V, E) with weighted vertices d = (d_v | v \in V) and directed edges λ= (λ_e | e \in E) interpreted as follows: G is a communication or transportation network, e.g., a pipeline; each edge e \in E is a one-way communication line, road or pipe of capacity λ_e, while every vertex v \in V is a node of production d_v > 0, consumption d_v < 0, or transition d_v = 0. A non-negative flow x = (x_e \mid e \in E) is called weakly feasible if for each v \in V the algebraic sum of flows, over all directed edges incident to v, equals d_v; or shorter, if A_G x = d, where A_G is the vertex-edge incidence matrix of G. A weakly feasible flow x is called feasible if x_e \leq λ_e for all e \in E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities x_e > λ_e are allowed. However, such an excess is viewed as unwanted (dangerous) and so we minimize the excess ratio vector r = (r_e = x_e / λ_e | e \in E) lexicographically. More precisely, first, we look for the weakly feasible flows minimizing the maximum of re over all e in E; among all such flows we look for those that minimize the second largest coordinate of r, etc. Clearly, |E| such steps define a unique balanced flow, which provides the lexmin solution for problem (G, d, λ). We construct it in polynomial time, provided vectors d and λare integer. For symmetric digraphs the problem was solved by Gurvich and Gvishiani in 1984. Here we extend this result to directed graphs. Furthermore, we simplify the algorithm and proofs applying the classic criterion of existence of a feasible flow for (G, d, λ) obtained by Gale and Hoffman in late 1950-s.

math.CO↗
arXiv2020

Recognizing distributed approval voting forms and correspondences

Each voter $i \in I$ has $α_i$ cards that (s)he distributes among the candidates $a \in A$ as a measure of approval. One (or several) candidate(s) who received the maximum number of cards is (are) elected. We provide polynomial algorithms to recognize voting forms and voting correspondences generated by such voting schemes in cases when either the number of candidates or the number of voters is equal to $2$. We prove that for two voters, if $α_2 \geq α_1-2\geq 0$ then the unique voting correspondence has distinct rows. We also characterize voting forms with distinct rows.

math.CO↗
arXiv2020

Metric and ultrametric inequalities for resistances in directed graphs

Consider an electrical circuit $G$ each directed edge $e$ of which is a semiconductor with a monomial conductance function $y_e^* = f_e(y_e) = y_e^s / μ_e^r$ if $y_e \geq 0$ and $y_e^* = 0$ if $y_e \leq 0$. Here $e$ is a directed edge, $y_e$ is the potential difference (voltage), $y_e^*$ is the current in $e$, and $μ_e$ is the resistance of $e$; furthermore, $r$ and $s$ are two strictly positive real parameters common for all edges. In particular, case $r = s = 1$ corresponds to the Ohm law, while $r = \frac{1}{2}, s =1$ may be interpreted as the square law of resistance typical for hydraulics and gas dynamics. We will show that for every ordered pair of nodes $a, b$ of the circuit, the effective resistance $μ_{a,b}$ is well-defined. In other words, any two-pole network with poles $a$ and $b$ can be effectively replaced by two oppositely directed edges, from $a$ to $b$ of resistance $μ_{a,b}$ and from $b$ to $a$ of resistance $μ_{b,a}$. Furthermore, for every three nodes $a, b, c$ the inequality $μ_{a,c}^{s/r} + μ_{c,b}^{s/r} \geq μ_{a,b}^{s/r}$ holds, in which the equality is achieved if and only if every directed path from $a$ to $b$ contains $c$. MSC classes: 11J83, 90C25, 94C15,94C99

math.CO↗
arXiv2020

On the degree sequences of dual graphs on surfaces

Given two graphs $G$ and $G^*$ with a one-to-one correspondence between their edges, when do $G$ and $G^*$ form a pair of dual graphs realizing the vertices and countries of a map embedded in a surface? A criterion was obtained by Jack Edmonds in 1965. Furthermore, let $\boldsymbol{d}=(d_1,\ldots,d_n)$ and $\boldsymbol{t}=(t_1,\ldots,t_m)$ be their degree sequences. Then, clearly, $\sum_{i=1}^n d_i = \sum_{j=1}^m t_j = 2\ell$, where $\ell$ is the number of edges in each of the two graphs, and $χ= n - \ell + m$ is the Euler characteristic of the surface. Which sequences $\boldsymbol{d}$ and $\boldsymbol{t}$ satisfying these conditions still cannot be realized as the degree sequences? We make use of Edmonds' criterion to obtain several infinite series of exceptions for the sphere, $χ= 2$, and projective plane, $χ= 1$. We conjecture that there exist no exceptions for $χ\leq 0$.

math.CO↗
arXiv2020

Computational Hardness of Multidimensional Subtraction Games

We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove an existence of a game in this class such that solving the game is PSPACE-hard. The results are based on the construction introduced by Larsson and Wästlund. It relates subtraction games and cellular automata.

cs.CC↗
arXiv2019

Avoidable Vertices and Edges in Graphs

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as \textit{OCF}-vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs. We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

math.CO↗
arXiv2019

Sprague-Grundy Function of Matroids and Related Hypergraphs

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph $NIM$ for several classes of hypergraphs. In particular we characterize all $2$-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class.

math.CO↗