CYCLICAL GAMES WITH PROHIBITIONS
Alexander V. Karzanov\
Institute for Systems Studies
9, Prospect 60 Let Oktyabrya, 117312 Moscow, Russia;
and
Vasilij N. Lebedev
Institute of Physics and Technology, Moscow, Russia
Abstract. We consider a certain combinatorial game on a digraph for two cases
of the price function. For one case the game in question extends the cyclical game
studied in [1,2] which, in its turn, is a generalization of the well-known problem of
finding a minimum mean cycle in an edge-weighted digraph. We prove the existence of
optimal uniform stationary strategies for both cases and give algorithms to find such
strategies.
Key words. Digraph, Cyclical Game, Stationary Strategy, Mean Cycle.
Abbreviated title. Cyclical Games.
1. INTRODUCTION
Let G = (V, E) be a digraph whose vertex-set V is partitioned into two subsets A
and B, that is, A ∩ B = ∅ and A ∪ B = V . We admit loops but not multiple edges in
G; the edge leaving a vertex v and entering a vertex u is denoted by vu . Throughout
the paper we assume that G has no terminal vertices, that is, for each v ∈ V the set
E(v) of edges in G leaving v is nonempty. Let V (v) denote the set of end vertices u of
edges vu ∈ E(v).
Let us interpret A (B) as the set of positions of a first (second) player, White
(Black) say, and imagine that White and Black play the following game with opposite
interests. There is a chip which is placed at some initial position s ∈ V . The players
move the chip step by step along edges of G. If, at a current moment, the chip occurs
at a position v ∈ A (v ∈ B) then White (Black) chooses some leaving edge vu ∈ E(v)
to move the chip along vu to the next position u.
\
A part of this work was written while this author was visiting IMAG ARTEMIS,
Université Fourier, Grenoble, France and was supported by a grant from Mairie de
Grenoble.
1
Suppose that the game continues infinitely long; thus the chip passes an infinite
path in G. Denote by P := Ps the set of all infinite (directed) paths in G starting at
s. Let f : P → IR be a function; f (P ) is called the price of a path P ∈ P. The aim
of White is to maximize the price f (P ) of the path P made by the chip, while that of
Black is to minimize this value.
In what follows we consider certain price functions f which can be expressed in a
compact form. Moreover, f in question will provide the existence of optimal behaviors
for both players such that, after a finite numbers of moves, the chip will go along one
and the same cycle in G. A game (an instance of the above game specified by f ) having
such a property is called cyclical.
One special case of cyclical games was studied in [1,2]. Let each edge e ∈ E has
an integer-valued cost c(e) ∈ ZZ. For P = v0 v1 . . . vi . . . ∈ Ps (v0 = s) and an integer
t ≥ 1 denote by Pt the beginning part v0 v1 . . . vt of P . Define the cost c(Pt ) of Pt to be
P
(c(vi vi+1 ) : i = 0, . . . , t − 1), and the mean cost c(Pt ) of it to be c(Pt )/t. Then the
price f (P ) of P is defined as
(1)
f (P ) := limt→∞ c(Pt ).
The game with the price function (1) is called the mean cost game. This generalizes
finite positional games and so-called ergodic extensions of matrix games [5]. It turns
out that for each position v ∈ V there exists an optimal choice of a move which depends
neither on previous moves nor on an initial position s.
More precisely, we say that White applies a stationary strategy σA : A → V , where
σA (v) ∈ V (v) for v ∈ A, if whenever the chip occurs at the position v ∈ A, White moves
it every time to the same position σA (v). A stationary strategy for Black is defined
analogously. It is clear that if White applies a stationary strategy σA and Black does
a stationary strategy σB then the resulting path P consists of a beginning part L of at
most |V | edges, followed by an infinitely repeated cycle C, that is, P = LCC . . . C . . . .
Thus the price c(P ) of P is the mean cost c(C) of C. We say that a stationary strategy
is uniform if the player applies it independently of an initial position s.
In fact, the existence of uniform stationary optimal strategies for the mean cost
game can be deduced from a general theorem on stationary optimal strategies in
stochastic games (cf. [6]) whose proof relies on fixed point techniques. Another, more
direct, proof occurred in [1]. In [2] a structural theorem was stated which enables us
to observe such strategies explicitly. To formulate this, we need some terminology and
notation.
Let ε : V → IR be a mapping, called a potential. Define the ε-reduced cost function
c by
0
2
c0 (vu) := c(vu) + ε(v) − ε(u)
for vu ∈ E;
ε
we say that c0 is obtained from c by the ε-reduction, denoting this as c−→c0 . Observe
that an ε-reduction does not change, in essence, our game (since for Pt = v0 v1 . . . vt
we have c0 (Pt ) = c(Pt ) + (ε(v0 ) − ε(vt ))/t, and the second term in the sum becomes
infinitely small when t increases). We say that c0 is equivalent to c if c0 can be obtained
from c by some ε-reduction. For a : E → Q and v ∈ V define the value
M (a, v) := max{a(vu) : u ∈ V (v)}
if v ∈ A,
:= min{a(vu) : u ∈ V (v)}
if v ∈ B;
and three subsets of V (v):
V 0 (a, v) := {u ∈ V (v) : a(vu) = M (a, v)};
V + (a, v) := {u ∈ V (v) : a(vu) > M (a, v)};
V − (a, v) := {u ∈ V (v) : a(vu) < M (a, v)}.
Theorem 1 [2]. Let the price function be as in (1). Then there exists a rationalvalued function c0 on E equivalent to c so that for any v ∈ V the following hold:
(i) M (c0 , u) = M (c0 , v)
if u ∈ V 0 (c0 , v),
(ii) M (c0 , u) ≥ M (c0 , v)
if u ∈ V + (c0 , v),
(iii) M (c0 , u) ≤ M (c0 , v)
if u ∈ V − (c0 , v).
The numbers M (c0 , v), v ∈ V , are determined uniquely for G and c.
Moreover, it was shown in [2] that c0 in Theorem 1 can be chosen not huge, namely,
kc0 k ≤ 2nkck (where kak denotes the norm max{|ai | : i = 1, . . . , r} of a vector a =
(a1 , . . . , ar )), and that the denominator of each component of c0 does not exceed nn .
We say that an ordered partition (V1 , V2 , . . . , Vm ) of V is ergodic if:
(2) for i = 1, . . . , m the subgraph of G induced by Vi has no terminal vertices;
(3) for 1 ≤ i < j ≤ m there are no edges vu in G such that either v ∈ A ∩ Vi and
u ∈ Vj , or v ∈ B ∩ Vj and u ∈ Vi .
Theorem 1 has the following obvious corollary.
Proposition 1.1 [2]. Let c0 be as in Theorem 1. Let p1 < p2 < . . . < pm be all
different numbers among M (c0 , v), v ∈ V , and let Vi be the subset of vertices v of G
such that M (c0 , v) = pi . Then (V1 , . . . , Vm ) is an ergodic partition of V .
3
Theorem 1 (or Proposition 1.1) shows that if White follows a stationary “max”strategy σA by taking for each v ∈ A an edge vu with c0 (vu) maximum (that is,
u = σA (v) ∈ V 0 (c0 , v)) then he ensures his gain at least p(s), where p(s) is pi for i such
that s ∈ Vi . Similarly a stationary “min”-strategy σB for Black ensures his losses at
most pi . Thus
inf c(σA , σ
eB , s) = sup c(e
σA , σB , s),
e
σB
e
σA
where the infinum (supremum) is taken over all (not necessarily stationary) strategies
0
0
σ
eB (e
σA ) of Black (White), and c(σA
, σB
, s) denotes the price of the path starting at s
0
0
that arises when strategies σA and σB are applied. This implies the following discrete
minimax relation:
max min c(σA , σB , s) = min max c(σA , σB , s) = p(s),
σA
σB
σB
σA
where σA (σB ) runs over the set of all stationary strategies of White (Black).
The value p(v) is called the price of a position v ∈ V (when v is considered as
the initial position in the game). Since p(v) is the mean cost of a (simple) cycle, it
is a rational with the denominator at most |V |. From Theorem 1 it follows that the
problem of determining whether or not the price of a position v is more (less) than
or equal to a given number p ∈ Q belongs to N P ∩co-N P . However, at present no
polynomial algorithm is known for this problem. It should be noted that the proof of
ε
Theorem 1 given in [2] provides an algorithm for finding an ε-reduction c−→c0 with c0
as in this theorem. Lebedev [4] showed that this algorithm is not polynomial in worse
case (though it turned out to be fast enough in computer experiments).
Note that if A = ∅ then the game turns, in fact, into the well-known problem of
determining a minimum-mean cycle in G, which can be solved in strongly polynomial
time (e.g., an algorithm of Karp [3] has complexity O(|V ||E|)).
Proposition 1.1 has one combinatorial consequence mentioned in [2]. A bicolored
digraph G = (A, B; E) without terminal vertices is called ergodicly simple if for any
cost function c on E all vertices of G have the same prices. From Proposition 1.1 it
follows that G is ergodicly simple if and only if it has no ergodic partition (V1 , . . . , Vm )
with m > 1. In particular, consider G such that for each v ∈ A and u ∈ B at least
one of vu and uv is an edge of G. Then G has no ergodic partition with m > 1. This
ε
implies that for any c : E → ZZ there exists an ε-reduction c−→c0 such that the values
M (c0 , v) are the same for all v ∈ V (this generalizes the result of Moulin [5] that, in our
terms, any complete symmetric bipartite graph is ergodicly simple). An unexpected
fact is that the problem of determining whether a bicolored digraph is ergodicly simple
is N P -complete [4].
4
In the present paper we study a generalization of the mean-cost game as well as
one more kind of cyclical games, the so-called maximum cost game.
Let k : V → ZZ+ be a mapping such that k(v) < |E(v)| for any v ∈ V . We call
k(v) the number of prohibitions in a vertex v. This means that if, at a current moment,
the chip occurs at a position v of White (Black) then the other player, Black (White),
forbids using on this step the edges of some subset E 0 ⊂ E(v) of cardinality |E 0 | ≤ k(v)
(we also may interpret k(v) as the maximum number of elements of E(v) which can be
“spoilt” at the same time).
If the price function f is defined as in (1), we get the mean cost game with prohibitions.
For a : E → IR and v ∈ A, let Mk (a, v) denote the (k(v) + 1)th maximum among
the numbers a(e), e ∈ E(v), that is, Mk (a, v) := a(e) for e ∈ E(v) such that
|{e0 ∈ E(v) : a(e0 ) > a(e)}| ≤ k(v) and |{e0 ∈ E(v) : a(e0 ) < a(e)}| ≤ |E(v)| − k(v) − 1.
Similarly for v ∈ B define Mk (a, v) to be the (k(v) + 1)th minimum among a(e),
e ∈ E(v). The set V (v) is partitioned into three subsets:
Vk0 (a, v) := {u ∈ V (v) : a(vu) = Mk (a, v)};
Vk+ (a, v) := {u ∈ V (v) : a(vu) > Mk (a, v)};
Vk− (a, v) := {u ∈ V (v) : a(vu) < Mk (a, v)}.
At the present paper we prove the following theorem.
Theorem 2. Let the price function be as in (1), and let k → ZZ+ be a mapping such
that k(v) < |E(v)| for any v ∈ V . Then there exists a rational-valued function c0 on E
equivalent to c so that for any v ∈ V the following hold:
(i) Mk (c0 , u) = Mk (c0 , v)
if u ∈ Vk0 (c0 , v),
(ii) Mk (c0 , u) ≥ Mk (c0 , v)
if u ∈ Vk+ (c0 , v),
(iii) Mk (c0 , u) ≤ Mk (c0 , v)
if u ∈ Vk− (c0 , v).
The number Mk (c0 , v), v ∈ V , are determined uniquely for G, c and k.
Theorem 2 easily implies the following result.
Proposition 1.2. Let c0 be as in Theorem 2. Let p1 < p2 < . . . < pm be all different
numbers among Mk (c0 , v), v ∈ V , and let Vi be the subset of vertices v of G such that
Mk (c0 , v) = pi . Then for i = 1, . . . , m and v ∈ Vi the following are true:
5
(i) if v ∈ A then di+1 (v) + . . . + dm (v) ≤ k(v) < di (v) + . . . + dm (v);
(ii) if v ∈ B then d1 (v) + . . . + di−1 (v) ≤ k(v) < d1 (v) + . . . + di (v);
where dj (v) is the number of edges vu ∈ E(v) with u ∈ Vj .
The value pk (v) := pi is called the k-price of a vertex v ∈ Vi . When k ≡ 0
the mean cost game with prohibitions turns into the above “pure” mean cost game,
and Theorem 2 (Proposition 1.2) turns into Theorem 1 (Proposition 1.1). Theorem 2
and Proposition 1.2 explicitly exhibit optimal uniform stationary strategies for both
players. Here a stationary strategy for White means a choice of a subset E 0 (v) ⊆ E(v)
of k(v) + 1 edges for each v ∈ A and a choice of a subset E 00 (u) ⊂ E(u) of k(u)
edges for each u ∈ B so that whenever the chip occurs at a position v ∈ A (u ∈ B)
White is admitted to pick out for moving any edge among those in E 0 (v) which are
not forbidden by Black (respectively, forbids Black to use the edges in E 00 (u)). More
formally, by a stationary strategy for White we mean a mapping σA : V → 2V such
that σA (v) ⊆ V (v), |σA (v)| = k(v) + 1 if v ∈ A, and |σA (v)| = k(v) if v ∈ B. A
stationary strategy for Black is defined in a similar way.
Theorem 2 will be proved in Section 3 by use of an auxiliary theorem stated in
Section 2. The proof uses ideas of that of Theorem 1 given in [2], and it provides an
algorithm to find c0 as in Theorem 2 (as well as optimal stationary strategies and the
k-prices of the vertices).
In Section 4 we consider the game with another price function f , namely,
(4)
f (P ) := limt→∞ c(vt vt+1 ),
where c : E → ZZ and P = v0 v1 . . . vi . . . ∈ Ps . In other words, the price of P is the
maximum cost c(e) over the edges e which occur infinitely many times in P . The game
with the price function (4) is called the maximum cost game. We shall consider at once
the corresponding game with prohibitions. We prove that this game has the property
similar to that for the mean cost one: the game can be solved in uniform stationary
strategies (and therefore, it is cyclical). Moreover, the proof will provide a strongly
polynomial algorithm to find such strategies.
We shall use one observation concerning cyclical games with prohibitions. Given
a k as above, define the function k 0 on V by
(5)
k 0 (v) := k(v) if v ∈ A
:= |E(v)| − k(v) − 1 if v ∈ B.
6
Proposition 1.3. Let G0 = (A0 , B 0 ; E) be the graph with A0 := A∪B and B 0 := ∅, let
f be a price function, and let k 0 be as in (5). Suppose that the game with prohibitions
for G0 , s, k 0 and f has an optimal solution in stationary strategies σA0 and σB 0 of
White and Black respectively. Then σA and σB are optimal strategies for the game
with G = (A, B; E), s, k, f , where
σA (v) := σA0 (v) and σB (v) := σB 0 (v) for v ∈ A;
σB (v) := V (v) − σB 0 (v) and σA (v) := V (v) − σA0 (v) for v ∈ B.
Thus for the price functions (1) and (4) it suffices to consider only the assymmetric
version of the game with prohibitions in which for each position the choice of edges to
move belongs to White, while the choice of edges to forbid belongs to Black. The proof
of Proposition 1.3 appeals to obvious reasonings and we leave it to the reader.
2. AUXILIARY THEOREM AND ALGORITHM
Consider the assymetric version of the mean cost game with prohibitions with
A = V and B = ∅. One may assume that the graph G is connected. The proof of
Theorem 2 is based on an auxiliary theorem that we state in this section. For brevity
the symbol k in the terms Mk (·, v), Vk0 (·, v), Vk+ (·, v), Vk− (·, v) will be omitted.
Theorem 2.1.
Let c be a rational-valued function on E. Define
µ := µ(c) := max M (c, v);
ν := ν(c) := min M (c, v).
v∈V
v∈V
Let p ∈ Q be a number such that ν ≤ p ≤ µ. Then there exist a function c0 on E
equivalent to c and a partition (V 0 , V 00 ) of V (possibly V 0 = ∅ or V 00 = ∅) so that:
(6)
ν ≤ M (c0 , v) ≤ p for any v ∈ V 0 ;
(7)
p ≤ M (c0 , v) ≤ µ for any v ∈ V 00 ;
(8)
if v ∈ V 0 then (V 0 (c0 , v) ∪ V − (c0 , v)) ⊆ V 0 ;
(9)
if v ∈ V 00 then (V 0 (c0 , v) ∪ V + (c0 , v)) ⊆ V 00 .
Proof. We construct a finite sequence h0 , h1 , . . . , ht of functions on E such that h0 :=
c, each hi is obtained from hi−1 by an ε-reduction, and c0 := ht together with some
partition (V 0 , V 00 ) satisfies (6)-(9). In fact, an algorithm will be clearly seen behind the
proof; the ith iteration of the algorithm consists in forming hi from hi−1 .
7
Suppose that a function hi−1 (i ≥ 1) has been already determined. For brevity we
denote hi−1 by h, and denote M (h, v), V 0 (h, v), V + (h, v), V − (h, v) by M (v), V 0 (v),
V + (v), V − (v) respectively. The new function hi that we construct is denoted by h0 .
Introduce the following sets:
D0 := {v ∈ V : M (v) = p};
D+ := {v ∈ V : M (v) > p}.
If D+ = ∅ then c0 := h satisfies (6)-(9) with V 0 = V and V 00 := ∅. Thus one may
assume that D+ 6= ∅ (note that if the case D+ ∪ D0 = V takes place then c0 := h
satisfies (6)-(9) with V 0 := ∅ and V 00 := V , but we do not distinguish this case to get
a stronger version of the theorem (Proposition 2.3)). An edge e ∈ E is called tight
(exceeding, deficit) if h(e) is equal to (respectively, more, less than) p.
The function h0 arises from h by an ε-reduction of a special kind. We say that h0
is the L, δ-shift of h, where L ⊆ V and δ ∈ Q if
(10)
h0 (vu) := h(vu) − δ
if v ∈ L and u ∈ L
:= h(vu) + δ
if v ∈ L and u ∈ L
:= h(vu)
otherwise,
where L := V − L. Clearly h0 is obtained from h by the ε-reduction with the potential
ε defined as ε(v) := 0 if v ∈ L and ε(v) := δ if v ∈ L.
To determine L, we apply the following labelling process. At first we label the
set X0 := D+ . Suppose that we have already labelled sets X0 , X1 , . . . , Xj , where
X1 , . . . , Xj are pairwise disjoint nonempty subsets of D0 . Put Lj := X0 ∪ X1 ∪ . . . Xj .
Define Xj+1 to be the set of all still unlabelled vertices v in D0 such that there are at
least k(v) + 1 − |V + (v)| tight edges from v to Lj , that is,
(11)
Xj+1 := {v ∈ D0 − Lj : |V + (v)| + |V 0 (v) ∩ Lj | ≥ k(v) + 1}.
If Xj+1 = ∅, the labelling process is completed, and Lj is the resulting set L.
From the labelling process it is easy to see that D+ ⊆ L ⊆ D+ ∪ D0 , and
(12) v ∈ V belongs to L if and only if there are at least k(v) + 1 − |V + (v)| tight edges
from v to L, that is, |V + (v)| + |V 0 (v) ∩ L| ≥ k(v) + 1.
Now we explain how to determine δ.
8
(i) For v ∈ L let ρ(v) be the number of non-deficit edges from v to L, that is,
ρ(v) := |(V 0 (v) ∪ V + (v)) ∩ L|. If ρ(v) ≤ k(v), define π(v) to be the (k(v) + 1 − ρ(v))th
maximum among the numbers h(vu) for u ∈ L (by (12), there are at least k(v)+1−ρ(v)
exceeding edges from v to L). If ρ(v) ≥ k(v) + 1, put π(v) := +∞.
(ii) For v ∈ L let ω(v) be the number of exceeding edges from v to L, that is,
ω(v) := |V + (v) ∩ L|, and let λ(v) be the number of edges from v to L. If ω(v) + λ(v) ≥
k(v) + 1, define π(v) to be the (k(v) + 1 − ω(v))th maximum among the numbers h(vu)
for u ∈ L (by (12), there are at most k(v) − ω(v) non-deficit edges from v to L).
Otherwise put π(v) := −∞.
From (12) and the definition of π one can see that
(13)
π(v) > p if v ∈ L,
and π(v) < p if v ∈ L.
Consider three possible cases.
Case 1. π(v) is finite for at least one vertex v ∈ V . Put
(14)
δ := min{min{π(v) − p : v ∈ L}, min{p − π(v) : v ∈ L}}.
Case 2. L 6= V and π(v) is +∞ or −∞ for all v ∈ V . Put
(15)
δ := 1 + 2 max{|h(e)| : e ∈ E}.
Case 3. L = V (this case is possible only if D+ ∪ D0 = V ). Put δ := 1.
Let h0 be the L, δ-shift of h. In cases 2 and 3, the process of constructing h0 , h1 , . . .
is completed, and we finish with c0 := h0 . In case 1, we continue the process by forming
hi+1 from hi := h0 , and so on.
(To illustrate the above process, consider the graph G drawn in Fig. 1. In fragment
(A0), the numbers on edges e indicate their original costs c(e) =: h0 . Let k(x) = k(y) =
1 and k(z) = k(w) = 0; then M (c, x) = M (c, y) = M (c, z) = −1 and M (c, w) = 1.
Let p = 0. Fragment (Ai), i = 1, 2, 3, shows the numbers hi (e), e ∈ E, obtained on
ith iteration and the labelled set L = L(i + 1) on the next, (i + 1)th, one. After four
iterations we get the resulting function c0 := h4 and the partition (V 0 , V 00 ) with V 0 =
{x, y} and V 00 = {z, w}, as indicated in fragment (A4). One can see that M (c0 , x) = −1,
M (c0 , y) = M (c0 , z) = 0, M (c0 , w) = 1, and (6)-(9) hold.)
We have to prove that: (a) the sequence Q := (h0 , h1 , . . . , hi , . . .) is finite, and (b)
the final function ht =: c0 in Q satisfies (6)-(9) with some partition (V 0 , V 00 ).
9
First we prove (b) (assuming validity of (a)). Consider h := hi−1 for some i ≥ 1.
By induction on i one may assume that ν ≤ M (h, v) ≤ µ for all v ∈ V . If D+ = ∅ then
c0 := h satisfies (6)-(9) with V 0 := V and V 00 := ∅. Let D+ 6= ∅, and let L, δ and h0
arise from h according to the above labelling process. If L = V then (6)-(9) hold for
V 0 := ∅ and V 00 := V . Let L 6= V . Consider v ∈ V . Since δ > 0 (by (13)-(15)), the
definition (10) shows that for e ∈ E(v), h0 (e) ≤ h(e) if v ∈ L, and h0 (e) ≥ h(e) if v ∈ L.
Hence,
(16) M (h0 , v) ≤ M (h, v) ≤ µ for v ∈ L,
and M (h0 , v) ≥ M (h, v) ≥ ν for v ∈ L.
Fig. 1
To prove that
(17)
M (h0 , v) ≥ p for v ∈ L,
and M (h0 , v) ≤ p for v ∈ L,
10
denote by E 0 (E 00 ) the set of edges from v to L (respectively, to L).
(i) Let v ∈ L. Then h0 (e) = h(e) for e ∈ E 0 , and h0 (e) = h(e) − δ < h(e)
for e ∈ E 00 . If ρ(v) ≥ k(v) + 1 then M (h0 , v) ≥ p is obvious. If ρ(v) ≤ k(v) then
M (h0 , v) ≥ min{m1 , m2 } ≥ min{p, m2 }, where m1 is the ρ(v)th maximum among
h0 (e), e ∈ E 0 , and m2 is the (k(v) + 1 − ρ(v))th maximum among h0 (e), e ∈ E 00 . From
the definitions of π(v) and δ we observe that m2 = π(v) − δ ≥ p, whence M (h0 , v) ≥ p.
(ii) Let v ∈ L. Then h0 (e) = h(e) for e ∈ E 00 , and h0 (e) = h(e) + δ > h(e) for
e ∈ E 0 . If ω(v) + λ(v) ≤ k(v) then there are at most k(v) edges e ∈ E(v) for which
h0 (e) > p, whence M (h0 , v) ≤ p. Let ω(v) + λ(v) ≥ k(v) + 1. By (13), (14) and (10),
the (k(v) + 1 − ω(v))th maximum among h0 (e), e ∈ E 0 , is at most p. Since there are
exactly ω(v) edges e in E 00 for which h0 (e) > p, we conclude that M (h0 , v) ≤ p.
Thus (17) is true. Suppose that π(v) is +∞ or −∞ for all v ∈ V . Put V 0 := L,
V 00 := L and c0 := h0 . Then (6)-(7) follow from (16)-(17). Next, observe from (10) and
(15) that c0 (e) < p if v ∈ L and e ∈ E 00 , and that c0 (e) > p if v ∈ L and e ∈ E 0 . Hence,
(8)-(9) hold.
Now we prove that the sequence Q is finite. Consider two consequtive functions
h := hi−1 and h0 := hi for 1 ≤ i < |Q|. Let h0 be an L, δ-shift of h, and let L be the
union of sets X0 , X1 , . . . , Xr arising in the labelling process for h according to (11). For
j = 0, . . . , r define Uj to be the set of non-deficit edges from V − Lj to Xj , that is,
Uj := {vu ∈ E : v ∈ V − Lj , u ∈ Xj , h(vu) ≥ p},
where Lj := X0 ∪ X1 ∪ . . . ∪ Xj . We associate with Xj ’s and Uj ’s the sequence
S = α0 , β0 , α1 , β1 , . . . , αr , βr
of numbers such that αj = |Xj | and βj = −|Uj |. Similarly, let hi+1 be the L0 , δ 0 -shift
of h0 , and let (X00 , X10 , . . . , Xr0 0 ), (U00 , U10 , . . . , Ur0 0 ) and S 0 = α00 , β00 , α10 , β10 , . . . , αr0 0 , βr0 0
be the corresponding sequences for h0 .
The finiteness of Q is implied by the following lemma.
Lemma 2.2.
S 0 is lexicographically less than S.
Proof. Since X0 (X00 ) is the set of vertices v ∈ V with M (h, v) > p (respectively,
M (h0 , v) > p), and M (h0 , v) ≤ M (h, v) holds for any v ∈ V with M (h, v) ≥ p, we have
X00 ⊆ X0 . Thus one may assume that X00 = X0 . Let q be the maximum index such
that Xj0 = Xj for j = 0, . . . , q. We know that h0 (e) = h(e) for any edge e with both
ends in L and that h0 (e) > h(e) for any edge e from L to L (since δ > 0), therefore,
Uj ⊆ Uj0 for j = 0, . . . , q. Thus one may assume that Uj0 = Uj for j = 0, . . . , q. Let z
be a vertex achieving the maximum in (14). From the definition of π it follows that:
11
(18)
if z ∈ L then |V + (h0 , z) + V 0 (h0 , z) ∩ L| ≤ k(z);
(19)
if z ∈ L then |V + (h0 , z) + V 0 (h0 , z) ∩ L| ≥ k(z) + 1.
In particular, (18)-(19) imply that the case q = r = r0 is impossible. Indeed, if
q = r = r0 then L = L0 . But, according to the labelling process for h0 , z belongs to
L0 if and only if |V + (h0 , z) + V 0 (h0 , z) ∩ L0 | ≥ k(z) + 1 (cf. (12)), which is impossible
because of (18) and (19). Thus at least one of r and r0 is greater than q.
Suppose that S 0 is lexicographically more than or equal to S. If r = q, we may
assume by definition that Xq+1 := ∅ and αq+1 := 0. Since αj0 = αj and βj0 = βj for
0
0
j = 0, . . . , q, αq+1
> 0 and αq+1
≥ αq+1 . Moreover, from the maximal choice of q it
0
follows that there is a vertex z ∈ Xq+1
which is not in Xq+1 . We assert that there is an
edge zu such that u ∈ Xq , h(zu) < p and h0 (zu) = p, which contradicts with Uq0 = Uq .
For g : E → Q and j = 0, . . . , q let γj (g) denote |V + (g, z)| + |V 0 (g, z) ∩ Lj |. From
the labelling processes for h and h0 one has:
(20)
γq (h) ≤ k(z),
γq (h0 ) ≥ k(z) + 1
and γq−1 (h0 ) ≤ k(z)
(putting γ−1 (h0 ) := 0). Consider two cases.
(i) z ∈ L. Then h0 (e) ≤ h(e) for any e ∈ E(z). This implies γq (h0 ) ≤ γq (h),
contrary to (20).
(ii) z ∈ L. Let W := V 0 (h0 , z) ∩ Xq . We know that h0 (e) > h(e) for any edge e
from z to L. This implies that if W is nonempty then Uq0 6= Uq (as p = h0 (zw) > h(zw)
for w ∈ W ). Suppose that W = ∅. If q ≥ 1 then W = ∅ implies γq (h0 ) = γq−1 (h0 ),
contrary to (20). Thus q = 0. Then W = ∅ and γ0 (h0 ) ≥ k(z) + 1 imply that
|V + (h0 , z)| ≥ k(z) + 1, and therefore, M (h0 , z) > p, which is impossible by (17).
Thus Uq0 6= Uq . This contradiction proves the lemma.
The proof of Theorem 2.1 is completed.
•
••
The above proof obviously provides an algorithm for finding c0 as in Theorem 2.1.
However, Lemma 2.2 gives an upper bound for the number of iteration of the algorithm
which is only exponential in |V |.
In fact, the above proof enables us to get a slightly stronger version of Theorem
2.1; it will be used in the proof of Theorem 2 in the next section.
12
Proposition 2.3. Let c, µ, ν and p be as in the hypotheses of Theorem 2.1. Then
there exist a function c0 on E equivalent to c and a partition (V 0 , V 00 ) of V which satisfy
(6)-(9) and the following property:
(21) if the set D+ := {v ∈ V : M (c0 , v) > p} is nonempty then there exists a (simple)
cycle C in G such that:
(i) C contains at least one vertex in D+ ;
(ii) for each edge vu in C, c0 (vu) = M (c0 , v) ≥ p.
Proof. Let c = h0 , h1 , . . . , ht = c0 be the sequence determined in the proof of Theorem
2.1. We show that c0 satisfies (21). If t = 0 then M (c, v) ≤ p for all v ∈ V , and we
are done. Let t ≥ 1. Then ht is an L, δ-shift of h := ht−1 ; let L be the union of sets
X0 , X1 , . . . Xr arising in the labelling process for h. Since π(v) = ∞ for all v ∈ L = V 00
(as ht is the last member of the above sequence), we have:
(i) X0 = {v ∈ V : M (c0 , v) > p};
(ii) for any v ∈ V 00 , M (c0 , v) = M (h, v) ≥ p and if vu is an edge such that v ∈ V 00
and c0 (vu) = M (c0 , v) then u ∈ V 00 and c0 (vu) = h(vu).
Moreover, from the labelling process it follows that for any v ∈ Xj , j = 1, . . . , r, there
is an edge vu with u ∈ Xj−1 and h(vu) = p. Now, for each v ∈ V 00 choose an edge
ev = vu ∈ E(v) such that: (a) c0 (ev ) = M (h, v), and (b) if v ∈ Xj for j ≥ 1 then
u ∈ Xj−1 . Let U := {ev : v ∈ V 00 }. We know that both ends of each ev are in V 00 .
Let C be a simple cycle in the subgraph induced by U . Then the choice of edges ev for
v ∈ V 00 − X0 implies that C contains a vertex in X0 , whence (21) follows.
•
3. PROOF OF THEOREM 2.
First of all we prove two lemmas. As before, we assume that A = V and B = ∅.
We omit the symbol k in the term Mk (·, v).
Lemma 3.1. Let g be a function on E equivalent to c. Let µ (ν) be the maximum
(respectively, minimum) of M (g, v) for all v ∈ V . Then there exists a rational number
p such that ν ≤ p ≤ µ and the denominator of p is at most n := |V |.
Proof. For each v ∈ V choose an edge ev ∈ E(v) such that g(ev ) = M (g, v). Let
U := {ev : v ∈ V }, and let C be a cycle (considered as an edge-set) in U . Then
ν ≤ g(C) ≤ µ. Since g is equivalent to c, g(C) = c(C). Now the lemma follows from
the integrality of c and the fact that |C| ≤ n.
•
13
Lemma 3.2. Let g be a function on E equivalent to c and such that µ − ν ≤ 1/n2 ,
where µ, ν and n are defined as in Lemma 3.1. Then there exist a rational p and a
function h on E equivalent to c such that ν ≤ p ≤ µ and M (h, v) = p for all v ∈ V .
Proof. Let p be a rational with the denominator at most n such that ν ≤ p ≤ µ
(existing by Lemma 3.1). By Proposition 2.3 applied to g, there exist a function g 0
equivalent to g and a partition (V 0 , V 00 ) of V satisfying (6)-(9) and (21) (with g 0 instead
of c0 ).
Suppose that the set {v ∈ V : M (g 0 , v) > p} is nonempty. Consider a cycle C as in
(21). Since g 0 (e) ≥ p for any edge e of C and this inequality is strong for at least one
e, c(C) = g 0 (C) > p. The denominator of c(C) is at most n (as c is integral), hence
c(C) − p > 1/n2 . On the other hand, ν ≤ g 0 (C) ≤ µ (as ν ≤ M (g 0 , v) ≤ µ for all v ∈ V ,
by (6)-(7)), therefore µ − ν ≥ c(C) − p. Thus µ − ν > 1/n2 ; a contradiction.
Thus ν ≤ M (g 0 , v) ≤ p for all v ∈ V . Define f (e) := −g 0 (e) for e ∈ E and
k 0 (v) := |E(v)| − k(v) − 1. Applying to f and k 0 similar arguments as to g and k we
deduce that there exists a function f 0 such that Mk0 (f 0 , v) = p for all v ∈ V . Then
h := −f 0 is as required.
•
To prove Theorem 2, we now describe a process of splitting the graph G. For a
subset X ⊆ V let hXi denote the subgraph of G induced by X. Suppose that there have
been obtained a partition Π = (V1 , V2 , . . . , Vq ) of V and functions gi on Ei , i = 1, . . . , q,
where Ei is the edge-set of Gi := hVi i, such that the following hold:
(22) for i = 1, . . . , q, gi is equivalent (in Gi ) to the restriction of c on Ei ;
(23) for i = 1, . . . , q and v ∈ Vi , the number a(v) of edges from v to Vi+1 ∪ . . . ∪ Vq is
at most k(v), while the number of edges from v to Vi ∪ . . . ∪ Vq is at least k(v) + 1;
(24) ν1 ≤ µ1 ≤ ν2 ≤ µ2 ≤ . . . ≤ νq ≤ µq , where µi (νi ) is the maximum (respectively,
minimum) of values Mk0 (gi , v) (concerning Gi ) for all v ∈ Vi , and k 0 (v) := k(v) −
a(v).
(The process starts with q := 1 and g1 := c.) Consider a graph Gi such that
νi 6= µi . Two cases are possible.
Case 1. µi − νi ≤ 1/n2i , where ni := |Vi |. Then, by Lemma 3.2 (applied to Gi ,
gi and k 0 ), there exist a rational p and a function h on Ei equivalent to gi such that
νi ≤ p ≤ µi and Mk0 (h, v) = p for all v ∈ Vi .
Case 2. µi − νi > 1/n2 . Put p := (µi + νi )/2. By Theorem 2.1 (applied to Gi , gi
and k 0 ), there exist a function h on Ei equivalent to gi and a partition (V 0 , V 00 ) of Vi
satisfying (6)-(9) (with h and k 0 instead of c0 and k).
14
In Case 1 we replace gi by the new function h. In Case 2 we replace Π by the
new partition Π0 := (V1 , . . . , Vi−1 , V 0 , V 00 , Vi+1 , . . . , Vq ); define the function g 0 on the
edge-set E 0 of hV 0 i to be the restriction of h on E 0 , and similarly define g 00 on the
edge-set E 00 of hV 00 i. Then Π0 and the functions g1 , . . . , gi−1 , g 0 , g 00 , gi+1 , . . . , gq (after
a corresponding renumeration) maintain (22)-(24).
Eventually, after a polynomial (in |V | and logkck) number of steps, we get a partition Π = (V1 , . . . , Vq ) and functions g1 , . . . , gq satisfying (22)-(24) and
(25)
µi = νi
for i = 1, . . . , q.
Put pi := µi . By (22), gi can be obtained by an εi - reduction from the restriction
of c on Ei . Define the potential ε0 on V by ε0 (v) := εi (v) for i = 1, . . . , q and v ∈ Vi ,
and let g be the function on E obtained from c by the ε0 -reduction. In order to get a
function c0 as in Theorem 2 we correct g by “shifting” it as follows. For 1 ≤ i < j ≤ q
and an edge e from Vi to Vj (from Vj to Vi ), define the “discrepance” ∆(e) to be
max{0, pi − g(e)} (respectively, ∆(e) := max{0, g(e) − pj }. Let ∆ij (respectively, ∆ji )
be the maximum of ∆(e) over the edges e from Vi to Vj (from Vj to Vi ). Choose
numbers γ1 , γ2 , . . . , γq−1 , γq = 0 so that they satisfy
γi + γi+1 + . . . + γj−1 > max{∆ij , ∆ji }
ε00
for all 1 ≤ i < j ≤ q. Now execute the ε00 - reduction g −→c0 with ε00 defined by
ε00 (v) = γi + γi+1 + . . . γq
for v ∈ Vi , i = 1, . . . , q.
A straightforward check-up shows that, for any edge e ∈ E from Vi to Vj , i 6= j,
one has c0 (e) > pi if i < j, and c0 (e) < pi if i > j. This and (22)-(25) imply that c0
satisfies (i)-(iii) in Theorem 2.
This completes the proof of Theorem 2.
••
Remark. One can show that there exists a function e
c0 satisfying (i)-(iii) in Theorem 2 such that ke
c0 k < 2nkck + n, and the denominator of each component of e
c0 does
not exceed nn . Such a e
c0 can be obtained from c0 as above in the following way. For
each v ∈ V fix an edge ev ∈ E(v) such that c0 (ev ) = Mk (c0 , v); let U := {ev : v ∈ V }.
Consider the weak components Gi = (Vi , Ui ), i = 1, . . . , r, of the graph induced by U .
By Theorem 2 the values c0 (ev ) are equal to the same number pi for all v ∈ Vi ; let for
definiteness p1 ≤ p2 ≤ . . . ≤ pr . One can show that there exists an ε0 -reduction (which
is the composition of certain L, δ-shifts) of c such that the resulting function e
c0 satisfies
(i)-(iii) in Theorem 2 and has the following additional properties:
15
(i) for i = 1, . . . , r, each edge vu ∈ E with v, u ∈ Vi satisfies e
c0 (vu) = c0 (vu);
(ii) there is a tree T = (V, E) with V = {1, . . . , r} such that for each {i, j} ∈ E, i < j,
the value
min{ min{e
c0 (vu) − pi : uv ∈ E, v ∈ Vi , u ∈ Vj },
min{pj − e
c0 (vu) : uv ∈ E, v ∈ Vj , u ∈ Vi }}
is equal to 1 if pi < pj , and 0 if pi = pj .
For each {i, j} ∈ E let eij be an edge in G connecting Vi and Vj which achieves the
minimum in the above expression; put W := {eij : {i, j} ∈ E}. Observe that for any
two vertices v, u ∈ V there is a sequence v = v0 , v1 , . . . , vq = u of distinct vertices of G
such that vi and vi+1 are connected by an edge, say bi , either in U or in W . Now the
required property for e
c0 follows from the facts that |ε0 (vi+1 ) − ε0 (vi )| = |c(bi ) − e
c0 (bi )|,
and that |e
c0 (bi )| is a rational not exceeding kck + 1 whose denominator is at most n.
4. MAXIMUM COST GAME
In this section we prove the following theorem on the maximum cost game with
prohibitions.
Theorem 4.1.
Let the price function f be as in (4), and let k : V → ZZ+ be a
mapping such that k(v) < |E(v)| for any v ∈ V . Then the game for G = (A, B; E), c
and k has an optimal solution in uniform stationary strategies.
Proof.
According to Proposition 1.3, it suffices to consider the case A = V and B = ∅.
Let p1 < p2 < . . . < pm be all different values among c(e), e ∈ E. Fix some
i ∈ {1, . . . , m}, and let p := pi . The core of our proof is to show the existence of a
partition Πi = (Vi− , Vi+ ) of V (possibly Vi− = ∅ or Vi+ = ∅) such that for each vertex
v ∈ Vi+ (v ∈ Vi− ), considered as the initial position in the game, White ensures his gain
at least p (respectively, Black ensures his losses less than p) using a uniform stationary
strategy. More precisely, we prove the existence of a partition Πi , a stationary strategy
σA : V → 2V of White (that is, σA ⊆ V (v) and |σA (v)| = k(v) + 1) and a stationary
strategy σB : V → 2V of Black (that is, σB ⊆ V (v) and |σA (v)| = k(v)) so that:
(26) σA (v) ⊆ Vi+ for v ∈ Vi+ ;
and (V (v) ∩ Vi+ ) ⊆ σB (v) for v ∈ Vi− ;
(27) if v ∈ Vi+ and if P = v0 v1 . . . vt . . . ∈ Pv is an arbitrary path satisfying σA (that
is, if vt+1 ∈ σA (vt ) for any t) then f (P ) ≥ p;
(28) if v ∈ Vi− and if P = v0 v1 . . . vt . . . ∈ Pv is an arbitrary path satisfying σB (that
is, if vt+1 6∈ σB (vt ) for any t) then f (P ) < p.
16
For a vertex v and a stationary strategy σ of White (Black) define l(σ, v) to be
the minimum (respectively, define u(σ, v) to be the maximum) of f (P ) over all paths
P ∈ Pv satisfying σ. Then (27) and (28) can be rewritten as:
(29)
l(σA , v) ≥ p for all v ∈ Vi+ ;
(30)
u(σB , v) < p
for all v ∈ Vi− .
We proceed by induction on |V |. For v ∈ V denote by T (v) the set of vertices
u ∈ V (v) such that c(vu) ≥ p.
The process of forming Πi falls into several stages. First of all define the set
(31)
X := {v ∈ V : |T (v)| ≥ k(v) + 1}
Let y1 , y2 , . . . , yr be a maximal sequence of distinct vertices in X (:= V − X) such
that for each yi and Yi := X ∪ {y1 , . . . , yi }, i = 1, . . . , r:
(32) the number of edges from yi to Yi−1 and “non-deficit” edges from yi to V − Yi−1
is at least k(v) + 1, that is, |V (yi ) ∩ Yi−1 | + |T (yi ) ∩ Y i−1 | ≥ k(yi ) + 1
(taking Y0 := X). Put Y := Yr . For an infinite path P in G denote by I(P ) the set of
edges of G which occur infinitely many times in P .
Claim 1.
Suppose that Y = V . Let σA be a stationary strategy for White such
that σA (v) ⊆ T (v) for any v ∈ X and σA (yi ) ⊆ T (yi ) ∪ Yi−1 for i = 1, . . . , q. Then
l(σA , v) ≥ p for all v ∈ V .
(The existence of σA obviously follows from (31) and (32).)
Proof. Consider v ∈ V and a path P = v0 v1 . . . vt . . . ∈ Pv (v0 = v) satisfying σA .
Suppose that c(e) < p for all e ∈ I(P ). Let t be an index such that vj vj+1 ∈ I(P )
for any j ≥ t. Clearly vj 6∈ X for j ≥ t, therefore, vj ∈ Y − X. Take the minimum
number i such that yi = vj for some j ≥ t. Since c(vj vj+1 ) < p, vj+1 6∈ T (yi ), whence
vj+1 ∈ Yi−1 . Thus vj+1 = yi0 for some i0 < i; a contradiction.
•
Thus if Y = V then (26),(29) and (30) hold for Vi+ := V , Vi− := ∅ and a strategy
σA as in Claim 1.
Now suppose that Y 6= V . From the maximality of (y1 , . . . , yr ) it follows that
(33)
|V (v) ∩ Y | + |T (v) ∩ Y | ≤ k(v)
17
for any v ∈ Y .
Let z1 , z2 , . . . , zq be a maximal sequence of distinct vertices in Y such that for each
zi and Zi := Y ∪ {z1 , . . . , zi }, i = 1, . . . , q:
(34)
|V (zi ) ∩ Z i−1 | ≤ k(zi )
(taking Z0 := Y ). Put Z := Zq .
Claim 2. Let σB (v) ⊇ T (v) ∪ (V (v) ∩ Y ) for v ∈ Y and σB (zi ) ⊇ (V (zi ) ∩ Z i−1 ) for
i = 1, . . . , q. Then u(σB , v) < p for all v ∈ Z.
(The existence of σB obviously follows from (33) and (34).)
Proof. Consider v ∈ Z and an arbitrary path P = v0 v1 . . . vt . . . ∈ Pv (v0 = v)
satisfying σB . Suppose that there is an edge wu ∈ I(P ) with c(wu) ≥ p. Since
wu 6∈ σB (w) and T (v 0 ) ⊆ σB (v 0 ) for all v 0 ∈ Y , we have w ∈ Z − Y . We may assume
that wu is chosen so that w = zj with j maximum. Consider an edge v 0 w belonging to
I(P ) (such an edge obviously exists). It follows from the definition of σB that neither
v 0 ∈ V − Z nor v 0 ∈ Zj 0 with j 0 ≤ j are possible. Therefore, v 0 = zj 0 for some j 0 > j; a
contradiction.
•
From the maximality of (z1 , . . . , zq ) it follows that
(35)
|V (v) ∩ Z| ≥ k(v) + 1
for any v ∈ Z.
If Z = V then (26),(29) and (30) hold for Vi+ := ∅ and Vi− := V . Otherwise
put V 0 := Z. Define c0 to be the restriction of c on the edge-set of the subgraph
G0 of G induced by V 0 , and define k 0 to be the restriction of k on V 0 . By (35),
|V 0 (v)| ≥ k(v) + 1 = k 0 (v) + 1 for all v ∈ V 0 where V 0 (v) := V (v) ∩ V 0 . Since |V 0 | < |V |,
by induction, for G0 , c0 and k 0 there are a partition (Vi0+ , Vi0− ) of V 0 and stationary
0
0
strategies σA
and σB
satisfying (26),(29) and (30).
Finally, put Vi− := Z ∪ Vi0− and Vi+ := Vi0+ . Choose stationary strategies σA and
0
0
σB in G so that σA (v) = σA
(v) for v ∈ Vi+ ; σB (v) = σB
(v) for v ∈ Vi0− ; and σB (v) is
as in Claim 2 for v ∈ Z (σA and σB are chosen arbitrarily for the other vertices in G).
It is easy to see that the resulting partition and strategies satisfy (26),(29) and (30).
Now we finish to prove the theorem. Clearly ∅ = V1− ⊆ V2− ⊆ . . . ⊆ Vm− . Let J be
−
the set of indices i ∈ {1, . . . , m} such that the set Vi := Vi+ ∩ Vi+1
is nonempty (taking
−
i
i
Vm+1 := V ). Let σA (σB ) stand for a strategy σA (σB ) as in (26),(29) and (30) for a
∗
∗
corresponding i. Define σA
and σB
as
(36)
i+1
∗
i
∗
for i ∈ J and v ∈ Vi , σA
(v) := σA
(v) and σB
(v) := σB
(v).
18
∗
∗
We assert that for each i ∈ J and v ∈ Vi the equality l(σA
, v) = u(σB
, v) = pi
∗
∗
holds, and therefore, σA and σB are optimal stationary strategies for White and Black
respectively. First of all we observe from (26) that
(37)
∗
∗
σA
(v 0 ) ⊆ (Vj ∪ . . . ∪ Vm ) and V (v 0 ) ∩ (Vj+1 ∪ . . . ∪ Vm ) ⊆ σB
(v 0 )
for any j ∈ J and v 0 ∈ Vj .
Consider arbitrary i ∈ J, v ∈ Vi and a path P = v0 v1 . . . vt . . . ∈ Pv (v0 = v).
∗
. Then, by (37), each vertex of P is contained in some Vj
Suppose that P satisfies σA
with j ≥ i. Therefore, there are numbers j ≥ i and t such that vt0 ∈ Vj for all t0 ≥ t.
In other words, after t steps the chip is moved only within one set Vj according to the
j
strategy σA
(by (36)). In view of (29), this implies that f (P ) ≥ pj ≥ pi . Similarly,
∗
if P satisfies σB
then, by (37), after a finite number of steps the chip is moved only
j+1
within one set Vj , j ≤ i, according to the strategy σB
. Then, by (30), we have
f (P ) < pj+1 ≤ pi+1 . Now, since there is no edge e ∈ E with pi < c(e) < pi+1 , we
∗
∗
and σB
then f (P ) = pi .
conclude that if P satisfies both σA
This completes the proof of the theorem.
•
The above proof can be easily turned into a strongly polynomial algorithm for
finding optimal stationary strategies in this game.
REFERENCES
[1] A. Ehrenfeucht and J. Mycielski, “Positional strategies for mean payoff games,”
International Journal of Game Theory 8 (2) (1979) 109-113.
[2] V.A. Gurvitch, A.V. Karzanov and L.H. Khachiyan, “Tzyklicheskie igry i nakhoszdenie minimaksnykh srednikh tziklov v orientirovannykh grafakh,” Jurnal Vichislitelnoi Matematiki i Matematicheskoi Fiziki 28 (9) (1988) 1407-1417, in Russian.
[3] R.M. Karp, “A characterization of the minimum mean cycle in a digraph,” Discrete
Mathematics 23 (3) (1978) 309-311.
[4] V.N. Lebedev, “Cyclical games and related topics,” Ph. D. Thesis, Inst. of Physics
and Technology, Moscow, 1990.
[5] H. Moulin, “Prolongement des jeux à deux joueurs de somme nulle,” Bull. Soc.
Math. France 1976, Mem. 45.
[6] T. Partharsarathy and T.E.S. Raghavan, Some topics in two-person games (American Elsevier Company, NY, 1971).
19