Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterise the vertices of these polytopes. We define the combinatorial problem “Another completely labeled Gale string” whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful

On finding another room-partitioning of the vertices

Let T be a triangulated surface given by the list of vertex-triples of its triangles, called rooms. A room-partitioning of T is a subset R of the rooms such that each vertex of T is in exactly one room in R. We prove that if T has a room-partitioning R, then there is another room-partitioning of T which is different from R. The proof is a simple algorithm which walks from room to room, which however we show to be exponential by constructing a sequence of (planar) instances, where the algorithm walks from room to room an exponential number of times relative to the number of rooms in the instance. We unify the above theorem with Nash’s theorem stating that a 2-person game has an equilibrium, by proving a combinatorially simple common generalization

Finding a Second Hamiltonian cycle in Barnette Graphs

We study the following two problems: (1) finding a second room-partitioning of an oik, and (2) finding a second Hamiltonian cycle in cubic graphs. The existence of solution for both problems is guaranteed by a parity argument. For the first problem we prove that deciding whether a 2-oik has a room-partitioning is NP-hard, even if the 2-oik corresponds to a planar triangulation. For the problem of finding a second Hamiltonian cycle, we state the following conjecture: for every cubic planar bipartite graph finding a second Hamiltonian cycle can be found in time linear in the number of vertices via a standard pivoting algorithm. We fail to settle the conjecture, but we prove it for cubic planar bipartite WH(6)-minor free graphs

Complexity of the gale string problem for equilibrium computation in games

This thesis presents a report on original research, extending a result published as joint work with Merschen and von Stengel in Electronic Notes in Discrete Mathematics [4]. We present a polynomial time algorithm for two problems on labeled Gale strings, a combinatorial structure introduced by Gale [11] that can be used in the representation of a particular class of games. These games were used by Savani and von Stengel [25] as an example of exponential running time for the classical Lemke-Howson algorithm to find a Nash equilibrium of a bimatrix game [16]. It was therefore conjectured that solving these games was a complete problem in the class PPAD (Polynomial Parity Argument, Directed version, see Papadimitriou [24]). In turn, a major motivation for the definition of PPAD was the study of complementary pivoting methods, such as the Lemke-Howson algorithm. Our result, unexpectedly, sets apart this class of games as a case where a Nash equilibrium can be found in polynomial time. Since Daskalakis, Goldberg and Papaditrimiou [6] and Chen and Deng [5] proved that finding a Nash equilibrium in general normal-form games is PPAD-complete, we have a special class of games, unless PPAD = P. Our proof exploits two results. As seen in Savani and von Stengel [25] [26], we represent the Nash equilibria of these special games as Gale strings. We then give a graph where the perfect matchings correspond to Nash equilibria via Gale strings, and we exploit Edmonds’ polynomial-time algorithm for a perfect matching in a graph [7]. The proof given in Casetti, Merschen and von Stengel [4] covered only the case of even-dimensional Gale strings; here we extend the result to the general case. Merschen [19] and V´egh and von Stengel [28] expanded on our ideas, proving further results on the index of Nash equilibria (see Shapley [27]) in the framework of “oiks” introduced by Edmonds [8] and Edmonds and Sanit`a [9]

The BG News July 22, 1971

The BGSU campus student newspaper July 22, 1971. Volume 56 - Issue 4https://scholarworks.bgsu.edu/bg-news/3617/thumbnail.jp

NASA patent abstracts bibliography: A continuing bibliography. Section 2: Indexes (supplement 14)

This issue of the Index Section contains entries for 3512 patent and applications for patent citations covering the period May 1969 through December 1978. The Index Section contains five indexes --- subject, inventor, source, number, and accession number

Oriented Euler complexes and signed perfect matchings

This paper presents “oriented pivoting systems” as an abstract framework for complementary pivoting. It gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign. A special case are the Nash equilibria of a bimatrix game at the ends of Lemke–Howson paths, which have opposite index. For Euler complexes or “oiks”, an orientation is defined which extends the known concept of oriented abstract simplicial manifolds. Ordered “room partitions” for a family of oriented oiks come in pairs of opposite sign. For an oriented oik of even dimension, this sign property holds also for unordered room partitions. In the case of a two-dimensional oik, these are perfect matchings of an Euler graph, with the sign as defined for Pfaffian orientations of graphs. A near-linear time algorithm is given for the following problem: given a graph with an Eulerian orientation with a perfect matching, find another perfect matching of opposite sign. In contrast, the complementary pivoting algorithm for this problem may be exponential