1 Deterministic graphical finite -person games
Nash-solvability (NS) of these games is discussed in . Here we use the concepts, definitions, and notation from this paper.
NS holds for and may fail for ; see respectively Sections 5 and 2 of . The first NE-free example was constructed for in the preprint  and published in  111Vladimir Oudalov refused to co-author paper  published in Russia. Then, a much more compact example for appeared in [2, Section 3]. In this example is the worst outcome for player 2 but for players 1 and 3 is better than (at least) two terminal outcomes. Somewhat surprisingly, computations show that such two players exist in each NE-free -person DG game (for all ). In other words, condition (C22) given in Abstract implies NS. We call this conjecture ”Catch 22”. It is stronger than conjecture ”(C) implies NS”, since condition (C22) is weaker than (C).
The latter means that every player strictly prefers each terminal play to each infinite one, which looks natural and can be easily interpreted [5, Section 3.2] for examples. If (C) does not imply NS then there exists a DG game that has a cyclic NE (with outcome ) but has no terminal NE (with an outcome ) although can be reached from the initial position [2, Proposition 3]. This property looks strange but not impossible.
To formulate (C22) more accurately denote by the number of terminals that are worse than for player . Then, (C22) does not hold if and only if for (at least) two players.
Digraph is called bidirected if each non-terminal move in it is reversible, that is, if and only if unless or is a terminal position. We conjecture that every -person DG game on a finite bidirected digraph is NS.
2 Deterministic graphical multi-stage (DGMS) finite -person games
It is well-known that every digraph is uniquely partitioned into the strongly connected components (SCC). An SCC will be called;
terminal if it has no outgoing edges (exits);
interior if it is not terminal and contains a directed cycle;
transient if it is neither terminal not interior.
Obviously, each transient component consists of a single non-terminal vertex. Contracting every SCC to a single vertex, one gets an acyclic digraph. Conversely, given an acyclic digraph , let us substitute some of its non-terminal vertices by strongly connected digraphs that contain directed cycles. They will form the interior SCCs of the obtained digraph , while each of the remaining non-terminal vertices of will form a transient SCC in .
It seems logical to assign a separate outcome
(resp., ) to each terminal (resp., interior) SCC, thus,
getting the set of outcomes
, which define a DGMS game. Merging the latter outcomes in it we obtain a DG game. Obviously, merging outcomes of a game form respects NS. Hence, an -person DGMS game may have no NE when . The main examples of [7, 2] show this.
Yet, any two-person DGMS game has a NE, which can be determined by a slightly modified backward induction (BI) procedure [4, Section 1.5]; see also [MN21B, Section 4.1]. It was proven that finite -person DG and DGMS game forms are tight, for all . (Note that merging outcomes respects tightness.) Yet, tightness is equivalent with NS only when ; for tightness is neither necessary nor sufficient for NS. Note also that the modified BI works only for and even in this case the obtained NE may be not subgame perfect, unlike the NE obtained by the standard BI.
Obviously, contracting each terminal SCC of a DGSM game to a single vertex one does not change the game. Finally, note that a DGSM game with a unique interior SCC is DG game.
Conditions (C) and (C22) can be adapted to DGMS games as follows.
For any player every interior SCC is better than each terminal SCC .
To formulate denote by the number of terminal SCC that are worse than for player . Then, (C’22) does not hold if and only if there are (at least) two distinct players and two, not necessarily distinct, interior SCC and such that and .
Obviously implies . We conjecture that implies NS of finite -person DGMS games (for ). .
3 Subgame perfect NE-free DG games
NE in a finite -person DG game is called uniform if it is a NE with respect to every initial position . In the literature uniform NE (UNE) are frequently referred to as subgame perfect NE. By definition, any UNE is a NE, but not vice versa. A large family of -person UNE-free DG games satisfying (C) can be found in [5, Section 3.3] for , and even for in [1, the last examples in Figures 1 and 3].
Every NE-free game contains a UNE-free subgame [2, Remark 3]. Indeed, consider an arbitrary NE-free DG game and eliminate the initial position from its graph . The obtained subgame is UNE-free. Indeeed, assume for contradiction that has a UNE . Then, has a NE, which can be obtained by backward induction. The player beginning in chooses a move that maximizes his/her reward, assuming that is played in by all players. Clearly extended by this move forms a NE in , which is a contradiction.
Thus, searching for a NE-free DG game (satisfying (C) or (C22)) one should begin with a UNE-free DG game trying to extend it with an acyclic prefix. This was successfully reallized in [7, 2], where condition (C22) (not to mention (C)) was waved. However, under these assumptions all tries failed.
The authors was partially supported by the RSF grant 20-11-20203.
-  Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Kazuhisa Makino, On Nash equilibria and improvement cycles in pure positional strategies for Chess-like and Backgammon-like n-person games, Discrete Math. 312:4 (2012) 772–788.
-  Endre Boros, Vladimir Gurvich, Martin Milanic, Vladimir Oudalov, and Jernej Vicic, A three-person deterministic graphical game without Nash equilibria, Discrete Applied Math. 243 (2018) 21–38; https://arxiv.org/abs/1610.07701 .
-  Vladimir Gurvich, A four-person chess-like game without Nash equilibria in pure stationary strategies, Business Informatics 1:31 (2015) 68–76.
-  Vladimir Gurvich, Backward induction in presence of cycles, Oxford Journal of Logic and Computation 28:7 (2018) 1635–1646.
-  Vladimir Gurvich and Mariya Naumova, On Nash-solvability of n-person graphical games under Markov’s and a priori realizations; https://arxiv.org/abs/2104.07542
-  Vladimir Gurvich and Mariya Naumova, Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles; https://arxiv.org/abs/2108.05469
-  Vladimir Gurvich and Vladimir Oudalov, A four-person chess-like game without Nash equilibria in pure stationary strategies, https://arxiv.org/abs/1411.0349 [math.CO], 2014.