Abstract. We dene the family of locally path-bounded digraphs, which is a

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1 Innite cyclic impartial games Aviezri S. Fraenkel and Ofer Rahat Department of Applied Mathematics and Computer Science Weizmann Institute of Science Rehovot 76, Israel Abstract. We dene the family of locally path-bounded digraphs, which is a class of innite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible, in a nite number of moves. This is done by proving that the Generalized Sprague- Grundy function exists uniquely and has nite values on this class.. Introduction We are concerned with combinatorial games, which, for our purposes here, comprise -player games with perfect information, no chance moves and outcome restricted to (lose, win), (draw, draw) for the two players. A draw position is a position in the game such that no win is possible from it, fraenkel@wisdom.weizmann.ac.il ofer@wisdom.weizmann.ac.il

2 but there exists a next move which guarantees, for the player making it, not to lose. You win a game by making a last move in it. A game is impartial if for every position in it, both players have the same set of next moves; otherwise it's partizan. Nim is impartial, chess partizan. A game is cyclic if it contains cycles (the possibility of returning to the same position), or loops (pass-positions). These notions, slightly changed here, can be found in [BCG8]. It is clear that a necessary (yet not sucient) condition for the existence of draw positions is that the game be cyclic. For Partizan cyclic games, see [Con78], [Sha79], [Fla8], [FrTa8], [Fla83]; nite impartial cyclic games are discussed only briey in [BCG8], [Con76]. Particular nite impartial cyclic games are analysed in [FrTa75], [FrKo87]. Innite impartial games are treated briey at the end of [Smi66], where both the \generalized Sprague-Grundy function", dened below, and its associated \counter function" were permitted to be transnite ordinals. Our purpose here is to dene a certain class of innite digraphs on which assumes only nite values, but the counter function may contain transnite ordinal values. The motivation for doing this is based, in part, on the following considerations. It is easier to compute with nite than with transnite ordinals. Often the structure of the digraph is such that the - function itself suces to provide a winning strategy, without the need of an additional counter function (x4). The -function always provides at least a nonlosing strategy; it's for consummating a win that the counter function may be needed. For consummating a win the \generalized Nim-sum" of a nite set of -values is required (x4). The generalized Nim-sum is based on the binary expansion of ordinals. It's easy to see that every ordinal, nite or transnite, has a unique expansion as a nite sum of powers of ordinals (based on the greedy algorithm and the fact that the ordinals are well-ordered see [Sie58 XIV, x9]). For example,! =!. We do not wish to enter here into the question of the computational complexity of computing with transnite ordinals. But it seems possible that it's easier to compare the size of ordinals with each other, which suces for counter function values, than to compute and work with their binary expansions, as needed for the -values. The connection between games and digraphs is simple: with any impartial game? we associate a digraph G = (V; E) where V is the set of positions of? and (a; b) E if and only if there is a move from position a to position b. It is called the game-graph of?. We identify games with their corresponding game-graphs, game positions with digraph vertices and game moves with digraph edges, using them interchangeably. It is thus natural

3 3 to dene a cyclic digraph as a digraph, nite or innite, which may contain cycles or loops. In x we provide basic tools needed for the statement and proof of the result (Theorem ), and x3 contains the proof. An example demonstrating Theorem is given in the nal x4.. Preliminaries The subset of nonnegative integers is denoted by Z, and the subset of positive integers by Z +. Given a digraph G = (V; E). For any vertex u V, the set of followers of u is F (u) = fv V : (u; v) Eg. A vertex u with F (u) = ; is a leaf. The set of predecessors of u is F? (u) = fw V : (w; u) Eg. A walk in G is any sequence of vertices u ; u ; : : :, not necessarily distinct, such that (u i ; u i+ ) E, i.e., u i+ F (u i ) (i Z + ). Edges may be repeated. A path is a walk with all vertices distinct. In particular, there's no repeated edge in a path. The length of a path is the number of its edges. If every path in G has nite length, then G is called path-nite. If there exists b Z such that every path in G has length b, then G is path-bounded. Denition. A cyclic digraph is locally path-bounded if for every vertex u i there is a bound b i (u i ) = b i Z such that the length of every (directed) path emanating from u i doesn't exceed b i. The integer b i is the local path bound of u i. Note that every path-bounded digraph is locally path-bounded, and every locally path-bounded digraph is path-nite. But neither of the two inverse relationships needs to hold. Our main result is concerned with locally path-bounded digraphs. Given a digraph G = (V; E). The Generalized Sprague-Grundy function, also called -function, is a mapping : V! Z [ fg, where the symbol indicates a value larger than any natural number. If (u) =, we say that (u) is innite. We wish to dene also on certain subsets of vertices. Specically:? F (u) = (v) < : v F (u). If (u) = and? F (u) = K, we also write (u) = (K). Next we dene equality of (u) and (v): if (u) = k and (v) = ` then (u) = (v) if one of the following holds: (a) k = ` < ; (b) k = (K), ` = (L) and K = L. We also use the notations V f = u V : (u) < ; V = V n V f ; (u) = mex? F (u) ; where for any nite subset S Z, the Minimum EXcluded value mex is dened by mex S = min(z n S) = minimum term in Z not in S:

4 4 It is also convenient to introduce the notation () (u) = mex? F (u) = mexf(v) < : v F (u)g: We need some device to tell the winner where to go when we use the -function. For example, suppose there is a token on vertex u (Fig. ). It turns out that it's best for the player moving now to go to a position with -value. There are two such values: one (the leaf) is an immediate win, and the other (v) is only a nonlosing move. This digraph may be embedded in a large digraph where it's not clear which option leads to a win. The device which overcomes this problem is a counter function, as used in the following denition. For realizing an optimal strategy, we will normally select a follower of least counter function value with specied -value. The counter function also enables us to prove assertions by induction. Denition. Given a cyclic digraph G = (V; E). A function : V! Z [ fg is a -function with counter function c: V f! J, where J is any innite well-ordered set, if the following three conditions hold: A. If (u) <, then (u) = (u). B. If there exists v F (u) with (v) > (u), then there exists w F (v) satisfying (w) = (u) and c(w) < c(u). C. If (u) =, then there is v F (u) with (v) = (K) such that (u) = K. Remarks. In B we have necessarily u V f ; and we may have (v) = as in C. To make condition C more accessible, we state it also in the following equivalent form: C'. If for every v F (u) with (v) = there is w F (v) with (w) = (u), then (u) <. If condition C' is satised, then (u) <, and so by A, (w) = (u) = (u). To keep the notation simple, we write (), (), (; ) etc., for (fg), (fg), (f; g), etc. The -function was rst dened in [Smi66]. It was found independently in [FrPe75]. The simplied version given above, and two other versions, appear in [FrYe86]. Since this function is not well-known, we repeated its denition above. The -function exists uniquely on any nite cyclic digraph, but its associated counter function exists nonuniquely. Here and below, when we discuss the existence of, we mean its existence as a nite ordinal (besides the special value ). We are now ready to state our main result.

5 5 Theorem. Every locally path-bounded digraph G = (V; E) has a unique -function with an associated counter function; and for every u V f, (u) doesn't exceed the length of a longest path emanating from u. 3. The Proof We wish to examine some properties of path-nite and locally pathbounded digraphs. To begin with, is it clear that for a path-nite digraph, if v F (u), then every path emanating from v is not longer than any path emanating from u? Perhaps it is clear, but it's also wrong: in a path-nite graph, every path originating at some vertex u and continuing to its ultimate end, terminates at a vertex v, where v is either a leaf or a predecessor of some w on the path. Thus a path of minimum length emanating from u in Fig. terminates at the leaf, whereas a path of maximum length beginning at u terminates at y. It has length 3. But a maximal-length path emanating from v F (u) clearly has length 4. u y v Figure. The numbers are -values. However, having embarked on a path u ; u ; : : :; u n of maximum length n, the maximum path length from any vertex u i encountered on it is n? i ( i n). If a digraph G = (V; E), possibly with innite paths, has no leaf, then the label on all the vertices is evidently a -function: A and B are satised vacuously, and C is satised with (u) = for all u V. If G has a leaf, then some of the vertices have a -function, such as the leaf and its predecessors, but possibly doesn't exist on some of the vertices. For the case where exists on a subset V V, we dene (u) = mexf(f (u)) : F (u) V g. Since F (u) may, nevertheless, be innite for any vertex u in a locally pathbounded digraph, it is not clear a priori that (u) exists. The following lemma takes care of this point.

6 6 Lemma. Let u be any vertex with local path bound b in a locally pathbounded digraph G = (V; E). Then (u) exists (i.e., it is a nonnegative integer), and in fact, (u) b. Proof. We consider two cases. (i) Suppose that u has nite -value m. Then (u) exists, and in fact, (u) = (u) = m by A. Moreover, there exists u F (u) with (u ) = m?, there exists u F (u ) with (u ) = m? ; : : : ; there exists u m F (u m? ) with (u m ) =. Then u; u ; : : : ; u m is a path of length m, so m b. (The path may continue beyond u m, but in any case (u) = m b.) (ii) Suppose that u has either no -value or value. It suces to show that if v F (u) \ V f, then (v) < b. Indeed, jf (u)j may be innite, and F (u) may contain vertices with no -value. But if (v) < b for all v F (u) \ V f, then clearly (u) exists and (u) b. Note that we cannot use the argument of case (i) directly on v, since a path from v may be longer than a path from u, as we just saw. So suppose there is v F (u) \ V f with (v ) = n b. As in case (i), there is a path v ; v ; : : : ; v n of length n with (v i ) = n? i (i f; : : : ; ng). Then u; v ; v ; : : : ; v n is a walk of length n + > b emanating from u. Hence it cannot be a path. But v i 6= v j for all i 6= j, since (v i ) 6= (v j ). Hence v j = u for some j f; : : : ; ng. The contradiction is that v j does and u doesn't have a nite -value. Thus (v) < b for all v F (u) \ V f, hence (u) exists, and in fact, (u) b. Proof of Theorem. Let V V be a maximal subset of vertices on which exists, together with an associated counter function c, subject to the following additions to B and C of Denition : () If (u) < and there is v F (u) \ V ; then there is w F (v) with (w) = (u); c(w) < c(u); (3) if (u) = ; then there is v F (u) with (v) = such that w F (v) \ V =) (w) 6= (u); where V = V n V. (In (3) we have (w) 6= (u), instead of w F (v) and (w) 6= (u) in C.) In addition we require: (4) If (u) = with (u) = l; then (v) l for all v V : The subset V is maximal in the sense that adjoining any u V into V violates either Denition, or () or (3) or (4). If V 6= ;, let u V. By Lemma, (u) = k exists for some k Z. It follows that there is a

7 7 minimum value m = minfk Z : u V ; (u) = kg. Let K = fu V : (u) = mg. Then V 6= ; =) K 6= ;. We consider four cases. Case. For every u K we have m (F (u)), where, consistent with (), (F (u)) = f (v) : v F (u)g = fmex (F (v)) : v F (u)g = fmexf(w) < : w F (v); v F (u)gg: Note that u K, v F (u) =) (v) 6= m by the denition of mex, so v F (u), (v) = m =) v V, in fact, v K. Thus putting (u) = for all u K satises C, and is also consistent with (3); and with (4) by the minimality of m. Furthermore, it doesn't violate A, and is consistent with B by (). This contradicts the maximality of V. We may thus assume henceforth that there exists u K such that (5) m = (F (u)): Case. There exist u K and v F (u) with (v) =, such that for every w F (v), either (w) 6= m, or w V with (w) 6= m. Putting (u) = is clearly consistent with C, and (3); and it doesn't violate A. In view of (), also B is satised. This contradicts the maximality of V. So we may assume that 8u K and 8v F (u) with (v) = ; 9w F (v) (with (w) = m or w V with (w) = m): We subdivide this into the following two cases: (6) 9u K such that 8v F (u) with (v) = ; or (7) 8u K and 8v F (u) with (v) = ; 9w F (v) with (w) = m; 9 no w F (v) \ V ; but 9w F (v) \ V with (w) = m: Case 3. (6) holds. We repeat that for any u K, since (u) = m, u has no follower with -value m. Suppose that there exists y F? (u) with (y) = m. Then by (), there exists v F (u) with (v) = m, contradicting (u) = m. Thus putting (u) = m is consistent with A. It is also consistent with (3): putting (u) = m could presumably increase (y) for some y F? (u)\v, and thus upset (3) for the value (z) = of some grandparent

8 8 z = F? (F? (y)) of y. Now by (4), (u) (z). If indeed (y) increased, then for the new value we have (y) > (u), so (y) > (z) and (z) = remains unaected. Consistency with C thus follows from (3) which becomes C when u is labeled m. Since y F? (u) =) (y) 6= m, as we saw at the beginning of this case, the potential adverse eect on any grandparent z of y considered above, cannot happen. We now show that also B holds. Suppose rst that F (u) V. For every v F (u) for which (v) > m, there exists w F (v) with (w) = m. This follows from A if (v) <, and from (6) if (v) =. It remains to dene c(u) suciently large so that c(w) < c(u). This will be done below. In view of the minimality of m and by (5), the second possibility is that for every v F (u) for which v V, we have (v) > m. For every such v there exists w F (v) with (w) = m by the denition of mex. Again we have to dene c(u) suciently large to satisfy c(w) < c(u). Let S = fv F (u) : (v) > mg [ fv F (u) : v V ; (v) > mg. We have just seen that for every v S there is w F (v) with (w) = m. Put T = fw F (v) : v S; (w) = mg. Let c(u) be the smallest ordinal > c(w) for all w T. Then also B is satised. This contradicts the presumed maximality of V. Note that the case F (u) V n V satises (6) vacuously, and so is also included in the present case. Case 4. (7) holds. If (7) holds nonvacuously, then as in Case, putting (u) = for all u K is consistent with C, (3) and the other conditions. This contradicts again the maximality of V. Hence K = ;, and so also V = ;. Whenever exists on a digraph, nite or innite, it exists there uniquely. See [Fra99]. Finally, if b is the local bound of u V, then (u) b by Lemma. Hence if u V f, then (u) b by A. 4. An Example We specify below a locally path-bounded digraph G = (V; E) on some of whose vertices we place a nite number of tokens. A move consists of selecting a token and moving it to a follower. Multiple occupancy of vertices is permitted. The player rst unable to move loses, and the opponent wins. If there is no last move, the outcome is a draw. For any r Z, a Nim-heap of size r is a digraph with vertices u ; : : :; u r and edges (u j ; u i ) for all i < j r. In depicting G (Fig. ), we use the convention that bold lines constitute a Nim-heap, of which only adjacent

9 9 edges are shown, to avoid cluttering the drawing. Thin lines denote ordinary edges. All the horizontal lines are thin, and from each vertex u on this horizontal line there emanates a Nim-heap H i of size i pointing upward. Each such u also connects via a vertical thin edge to a Nim-heap G i pointing downwards, of size b(4i+8)=3c; i Z. Each G i has a back edge to its top vertex forming a cycle of length i +. Thus G has a loop at the top of its Nim-heap (of size ). There is an additional back edge to the vertex u on the horizontal line, forming a cycle of length i + 3. H 3 3 H H H * H H H * * G 5 * G 4 3 G 3 G * G G G 6 G 7 Figure. The tail-end of a locally path-bounded digraph.

10 From any vertex u on the horizontal line there is a longest path, via G i, of length b(4i + )=3c, and the other vertices have shorter maximal length. Thus G is locally path-bounded. But it is not path-bounded, since i can be arbitrarily large. Sample Problem. Compute an optimal strategy for the 5-token game placed on the 5 starred vertices of G. To solve this problem, we introduce the generalized Nim-sum ([Smi66], [FrYe86], [Fra99]). For any nonnegative integer h we write h = P i hi i for the binary encoding of h (h i f; g). If a and b are nonnegative integers, then their Nim-sum a b = c, also called exclusive or, XOR, or addition over GF(), is dened by c i a i + b i (mod ); c i f; g (i ): The Generalized Nim-sum of a nonnegative integer a and (L), for any nite subset L Z, is dened by a (L) = (L) a = (L a); where L a = f` a: ` Lg. The Generalized Nim-sum of (L ) and (L ), for any nite subsets L, L, is dened by (L ) (L ) = (L ) (L ) = (;): Clearly the Generalized Nim-sum is associative and a a = for every a. Given any nite or innite game?, we say informally that a P -position is any position u from which the Previous player can force a win, that is, the opponent of the player moving from u. An N-position is any position v from which the Next player can force a win, that is, the player who moves from v. A D-position is any position u from which neither player can force a win, but has a nonlosing next move. The set of all P -, N- and D-positions is denoted by P; N and D respectively. For any nite multiset u = (u ; : : :; u n ) of vertices of G on which tokens reside, one token on each u i, we then have the result [Fra99]: Proposition. The P -, N- and D-labels of u = (u ; : : :; u n ) in any locally path-bounded digraph G are given by P = fu V : (u) = g; D = fu V : (u) = (K); = Kg N = fu V : < (u) < g [ fu V : (u) = (K); Kg: We are now ready to solve the above problem, by observing that the symbols appearing on Fig. are the -values of G. Simply check that they satisfy the conditions of Denition. In particular, B of Denition is satised if every vertex on the horizontal line with -value < gets a counter-value between! and!, and every vertex in the Nim-heaps with -value < is assigned a counter value <!, which is clearly feasible. For the 5 starred vertices we then have 3 4 (; ; ; 3; 4) = 4 (; ; ; 3; 4) = (4; 5; 6; 7; ), which contains, hence the position is

11 in N. Thus the player moving from this position can win by going to a position of Nim-sum, namely, pushing the token on the innity label to 4. Indeed the resulting Nim-sum is =. We remark that any tokens on two vertices with -value is a draw position, no matter where the other tokens are, if any. Also note that for realizing a win in this game we do not really need a counter function. Epilogue We have dened locally path-bounded digraphs, and shown that the generalized Sprague-Grundy function exists on such digraphs with nite, though not necessarily bounded, values. Of course local path-boundedness is only a sucient condition for the existence of. Any nite or innite digraph without a leaf, satises trivially (u) = for all its vertices u. A large part of combinatorial game theory is concerned, however, with digraphs which do have leaves. If we exclude digraphs without leaves, then Theorem is, in a sense, best possible. Consider the digraph G which consists of a vertex u, and F (u) = fu ; u ; : : :g, where, for all i Z, u i is the top vertex of a Nim-heap of size i, so (u i ) = i. Any path emanating from u has the form u; u i ; : : : for some i; its length is i +. Paths not emanating from u are shorter. Thus G is path-nite. But (F (u)) = f; ; : : :g, so (u) cannot assume any nite value. References. [BCG8] E. R. Berlekamp, J. H. Conway and R. K. Guy [98], Winning Ways (two volumes), Academic Press, London.. [Con76] J. H. Conway [976], On Numbers and Games, Academic Press, London. 3. [Con78] J. H. Conway [978], Loopy Games, Ann. Discrete Math. 3: Proc. Symp. Advances in Graph Theory, Cambridge Combinatorial Conf. (B. Bollobas, ed.), Cambridge, May 977, pp. 55{ [Fla8] J. A. Flanigan [98], Selective sums of loopy partizan graph games, Internat. J. Game Theory, {. 5. [Fla83] J. A. Flanigan [983], Slow joins of loopy games, J. Combin. Theory (Ser. A) 34, 46{ [Fra99] A. S. Fraenkel [999], Adventures in Games and Computational Complexity, to appear in Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI. 7. [FrKo87] A. S. Fraenkel and A. Kotzig [987], Partizan octal games: partizan subtraction games, Internat. J. Game Theory 6, 45{54.

12 8. [FrPe75] A. S. Fraenkel and Y. Perl [975], Constructions in combinatorial games with cycles, Coll. Math. Soc. Janos Bolyai, : Proc. Internat. Colloq. on Innite and Finite Sets, Vol. (A. Hajnal, R. Rado and V. T. Sos, eds.) Keszthely, Hungary, 973, North-Holland, pp. 667{ [FrTa75] A. S. Fraenkel and U. Tassa [975], Strategy for a class of games with dynamic ties, Comput. Math. Appl., 37{54.. [FrTa8] A. S. Fraenkel and U. Tassa [98], Strategies for compounds of partizan games, Math. Proc. Camb. Phil. Soc. 9, 93{4.. [FrYe86] A. S. Fraenkel and Y. Yesha [986], The generalized Sprague-Grundy function and its invariance under certain mappings, J. Combin. Theory (Ser. A) 43, 65{77.. [Sha79] A. S. Shaki [979], Algebraic solutions of partizan games with cycles, Math. Proc. Camb. Phil. Soc. 85, 7{ [Sie58] W. Sierpinski [958], Cardinal and Ordinal Numbers, Hafner, New York. 4. [Smi66] C. A. B. Smith [966], Graphs and composite games, J. Combin. Theory, 5{8. Reprinted in slightly modied form in: A Seminar on Graph Theory (F. Harary, ed.), Holt, Rinehart and Winston, New York, NY, 967.