215 research outputs found
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
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
-person positional game with perfect information modeled by a finite acyclic
directed graph (digraph) and the output is a profile 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 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 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 of the players can be reduced from
to , (ii) if the number of the terminals can be reduced from to ,
and most important, (iii) whether it is possible to get a similar example in
which the outcome corresponding to all (possibly, more than one) directed
cycles is worse than every terminal for each player.
Yet, it is known that (j) cannot be reduced to , (jj) cannot be
reduced to , 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 and , introduce the set of outcomes of the game . Two players, Alice and Bob, have the sets of strategies
and that consist of all monotone non-decreasing mappings and , respectively. It is easily seen that
each pair produces at least one {\em deal}, that is, an
outcome such that and . Denote by the set of all such deals related to . The obtained mapping
is a game correspondence. Choose an
arbitrary deal to obtained a mapping , which is a game form. We will show that each such game form is
tight and, hence, Nash-solvable, that is, for any pair of
utility functions of Alice and of Bob, the obtained monotone bargaining game
has at least one Nash equilibrium in pure strategies. Moreover, the same
equilibrium can be chosen for all selections . We also
obtain an efficient algorithm that determines such an equilibrium in time
linear in , although the numbers of strategies
and are exponential in . 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 -Nim
Given piles of tokens and a positive integer , we study the
following two impartial combinatorial games Nim and Nim. In the first (resp. second) game, a player, by one move, chooses at least
and at most (resp. exactly) 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 and . For game Nim we also characterize its P-positions for the
cases and
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 -Sperner if for every two hyperedges the
smallest of their two set differences is of size one. We present several
applications of -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 -Spernerness, thresholdness, and -asummability of
their vertex cover, clique, dominating set, and closed neighborhood
hypergraphs. Furthermore, we apply a decomposition property of -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 variables is a mapping , where , and are arbitrary
finite sets. Function is called {\em separable} if there exist
functions for , such that for every
input the function takes one of the
values . Given a discrete function , it is an
interesting problem to ask whether 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 variables is known only for
. In this paper we will show that a slightly more general recognition
problem, when is not fully but only partially defined, is NP-complete for
. We will then use this result to show that the recognition of fully
defined separable discrete functions is NP-complete for .
The case is well-studied in the context of game theory, where
(separable) discrete functions of variables are referred to as (assignable)
-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
variables for any , thus generalizing the above result known for .
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
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