Generalizing Gale's theorem on backward induction and domination of strategies

In 1953 Gale noticed that for every n-person game in extensive form with perfect information modeled by a rooted treesome special Nash equilibrium in pure strategies can be found by an algorithm of successive elimination of leaves, which is now called backward induction. He also noticed the same procedure, performed for the normal form of this game, turns into successive elimination of dominated strategies of the players that results in a single strategy profile (x_1,..., x_n), which is called a domination equilibrium (DE) and appears to be a Nash-equilibrium (NE) too. In other words, the game in normal form obtained from a positional game with perfect information is dominance-solvable (DS) and also Nash-solvable (NS). Yet, an arbitrary game in normal form may be not DS. We strengthen Gale's results as follows. Consider several successive eliminations of dominated strategies that begins with X = X_1 x ... x X_n and ends in X' = X'_1 x ... x X'_n. We will call X' a D-box of X. Our main (but obvious) lemma claims that for any i =1,..., n} and for any strategy x_i in X_i its projection to a D-box X' is dominated by a strategy x'_i in X'_i. It follows that any DE is an NE and, hence, DS implies NS. It is enough to apply the lemma in case when X' consists of a single strategy profile. The same lemma implies that the domination procedure is well-defined. A D-box X' is called terminal if it is domination-free, that is, it contains no pair of strategies such that one of them is dominated by the other. Any two terminal D-boxes X' and X" of X are equal. More precisely, there exist $n$ permutations \pi = (\pi_1, ..., \pi_n)$, with \pi_i : X_i to X_i for i in I, that transform X' into X", that is, \pi(X') = X" and the payoffs are respected. We also recall some published results on dominance-solvable game forms.Comment: 12 pages, 6 figures, first reported in 1971, dedicated to Yuriy Germeye

Backward induction in presence of cycles

For the classical backward induction algorithm, the input is an arbitrary $n$-person positional game with perfect information modeled by a finite acyclic directed graph (digraph) and the output is a profile $(x_1, \ldots, x_n)$ of pure positional strategies that form some special subgame perfect Nash equilibrium. We extend this algorithm to work with digraphs that may have directed cycles. Each digraph admits a unique partition into strongly connected components, which will be treated as the outcomes of the game. Such a game will be called a {\em deterministic graphical multistage}(DGMS) game. If we identify the outcomes corresponding to all strongly connected components, except terminal positions, we obtain the so-called {\em deterministic graphical}(DG) games, which are frequent in the literature. The outcomes of a DG game are all terminal positions and one special outcome $c$ that is assigned to all infinite plays. We modify the backward induction procedure to adapt it for the DGMS games. However, by doing so, we lose two important properties: the modified algorithm always outputs a {\em Nash equilibrium} (NE) only when $n = 2$ and, even in this case, this NE may be not {\em subgame perfect}. (Yet, in the zero-sum case it is.) The lack of these two properties is not a fault of the algorithm, just (subgame perfect) Nash equilibria in pure positional strategies may fail to exist in the considered game. {\bf Keywords:} deterministic graphical (multistage) game, game in normal and in positional form, saddle point, Nash equilibrium, Nash-solvability, game form, positional structure, directed graph, digraph, directed cycle, acyclic digraph.Comment: 8 page

A four-person chess-like game without Nash equilibria in pure stationary strategies

In this short note we give an example of a four-person finite positional game with perfect information that has no positions of chance and no Nash equilibria in pure stationary strategies. The corresponding directed graph has only one directed cycle and only five terminal positions. It remains open: (i) if the number $n$ of the players can be reduced from $4$ to $3$, (ii) if the number $p$ of the terminals can be reduced from $5$ to $4$, and most important, (iii) whether it is possible to get a similar example in which the outcome $c$ corresponding to all (possibly, more than one) directed cycles is worse than every terminal for each player. Yet, it is known that (j) $n$ cannot be reduced to $2$, (jj) $p$ cannot be reduced to $3$, and (jjj) there can be no similar example in which each player makes a decision in a unique position. Keywords: stochastic, positional, chess-like, transition-free games with perfect information and without moves of chance; Nash equilibrium, directed cycles (dicycles), terminal position.Comment: 9 pages, 1 figure, 1 tabl

Monotone bargaining is Nash-solvable

Given two finite ordered sets $A = \{a_1, \ldots, a_m\}$ and $B = \{b_1, \ldots, b_n\}$, introduce the set of $m n$ outcomes of the game $O = \{(a, b) \mid a \in A, b \in B\} = \{(a_i, b_j) \mid i \in I = \{1, \ldots, m\}, j \in J = \{1, \ldots, n\}$. Two players, Alice and Bob, have the sets of strategies $X$ and $Y$ that consist of all monotone non-decreasing mappings $x: A \rightarrow B$ and $y: B \rightarrow A$, respectively. It is easily seen that each pair $(x,y) \in X \times Y$ produces at least one {\em deal}, that is, an outcome $(a,b) \in O$ such that $x(a) = b$ and $y(b) = a$. Denote by $G(x,y) \subseteq O$ the set of all such deals related to $(x,y)$. The obtained mapping $G = G_{m,n}: X \times Y \rightarrow 2^O$ is a game correspondence. Choose an arbitrary deal $g(x,y) \in G(x,y)$ to obtained a mapping $g : X \times Y \rightarrow O$, which is a game form. We will show that each such game form is tight and, hence, Nash-solvable, that is, for any pair $u = (u_A, u_B)$ of utility functions $u_A : O \rightarrow \mathbb R$ of Alice and $u_B: O \rightarrow \mathbb R$ of Bob, the obtained monotone bargaining game $(g, u)$ has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be chosen for all selections $g(x,y) \in G(x,y)$. We also obtain an efficient algorithm that determines such an equilibrium in time linear in $m n$, although the numbers of strategies $|X| = \binom{m+n-1}{m}$ and $|Y| = \binom{m+n-1}{n}$ are exponential in $m n$. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.Comment: In this version we extend significantly Section 4. We add more classes of dual hypergraphs and show that for some of these classes the proof of the main theorem becomes much simpler than in genera

Slow kk-Nim

Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and mis\`ere version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \leq k}$ we also characterize its P-positions for the cases $n \leq k+2$ and $n = k+3 \leq 6$

On the computational complexity of solving stochastic mean-payoff games

We consider some well-known families of two-player, zero-sum, perfect information games that can be viewed as special cases of Shapley's stochastic games. We show that the following tasks are polynomial time equivalent: - Solving simple stochastic games. - Solving stochastic mean-payoff games with rewards and probabilities given in unary. - Solving stochastic mean-payoff games with rewards and probabilities given in binary.Comment:

On tame, pet, domestic, and miserable impartial games

Playing impartial games under the normal and misere conventions may differ a lot. However, there are also many "exceptions" for which the normal and misere plays are very similar. As early as in 1901 Bouton noticed that this is the case with the game of Nim. In 1976 Conway introduced a large class of such games that he called tame games. Here we introduce a proper subclass, pet games, and a proper superclass, domestic games. For each of these three classes we provide an efficiently verifiable characterization based on the following property. These games are closely related to another important subclass of the tame games introduced in 2007 by the first author and called miserable games. We show that tame, pet, and domestic games turn into miserable games by "slight modifications" of their definitions. We also show that the sum of miserable games is miserable and find several other classes that respect summation. The developed techniques allow us to prove that very many well-known impartial games fall into classes mentioned above. Such examples include all subtraction games, which are pet; game Euclid, which is miserable (and, hence, tame), as well as many versions of the Wythoff game and Nim, which may be miserable, pet, or domestic.Comment: Extra examples in Applications together with a new "Closing Remarks" section at the en

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs

A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of $1$-Spernerness, thresholdness, and $2$-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of $1$-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems.Comment: 31 pages, 9 figure

Separable discrete functions: recognition and sufficient conditions

A discrete function of $n$ variables is a mapping $g : X_1 \times \ldots \times X_n \rightarrow A$, where $X_1, \ldots, X_n$, and $A$ are arbitrary finite sets. Function $g$ is called {\em separable} if there exist $n$ functions $g_i : X_i \rightarrow A$ for $i = 1, \ldots, n$, such that for every input $x_1, \ldots ,x_n$ the function $g(x_1, \ldots, x_n)$ takes one of the values $g_1(x_1), \ldots ,g_n(x_n)$. Given a discrete function $g$, it is an interesting problem to ask whether $g$ is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of $n$ variables is known only for $n=2$. In this paper we will show that a slightly more general recognition problem, when $g$ is not fully but only partially defined, is NP-complete for $n \geq 3$. We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for $n \geq 4$. The case $n = 2$ is well-studied in the context of game theory, where (separable) discrete functions of $n$ variables are referred to as (assignable) $n$-person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of $n$ variables for any $n$, thus generalizing the above result known for $n=2$. Keywords: separable discrete functions, totally tight and assignable game formsComment: 25 page

Balanced flows for transshipment problems

A transshipment problem (G, d, \lambda) is modeled by a directed graph G = (V, E) with weighted vertices d = (d_v | v \in V) and directed edges \lambda = (\lambda_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 \lambda_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 \lambda_e for all e \in E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities x_e > \lambda_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 / \lambda_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, \lambda). We construct it in polynomial time, provided vectors d and \lambda are 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, \lambda) obtained by Gale and Hoffman in late 1950-s.Comment: 11 page