An Impulse-Regime Switching Game Model of Vertical Competition

We study a new kind of nonzero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impulse-regime switching game between the two firms, focusing on explicit threshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple potential Nash equilibria, which we obtain thanks to a verification-based approach. We exhibit three candidate types of equilibria depending on the ultimate number of switches by the downstream firm (zero, one or an infinite number of switches). We illustrate the diversification effect provided by vertical integration in the specific case of the crude oil market. Our analysis shows that the diversification gains strongly depend on the pass-through from the crude price to the gasoline price.

Maker–Breaker percolation games I: crossing grids

Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n. Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ. If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

A probabilistic verification theorem for the finite horizon two-player zero-sum optimal switching game in continuous time

In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected backward stochastic differential equations with interconnected barriers, we show that this game has a value and an equilibrium in the players’ switching controls.

Some New Characterizations of Graph Colorability and of Blocking Sets of Projective Spaces

Let $G=(V,E)$ be a graph and $q$ be an odd prime power. We prove that $G$ possess a proper vertex coloring with $q$ colors if and only if there exists an odd vertex labeling $x\in F_q^V$ of $G$. Here, $x$ is called odd if there is an odd number of partitions $\pi=\{V_1,V_2,\dotsc,V_t\}$ of $V$ whose blocks $V_i$ are \(G\)-bipartite and \(x\)-balanced, i.e., such that $G|_{V_i}$ is connected and bipartite, and $\sum_{v\in V_i}x_v=0$. Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game.

Anti-Codes in Terms of Berlekamp's Switching Game

We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the \(n\)-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$. Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.

Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game

We work with a unifying linear algebra formulation for nowhere-zero flows and colorings of graphs and matrices. Given a subspace (code) $U\leq{\mathbb{Z}_k^n}$ – e.g. the bond or the cycle space over ${\mathbb{Z}}_k$ of an oriented graph – we call a nowhere-zero tuple $f\in{\mathbb{Z}_k^n}$ a flow of $U$ if $f$ is orthogonal to $U$. In order to detect flows, we view the subspace $U$ as a light pattern on the $n$-dimensional Berlekamp Board ${\mathbb{Z}_k^n}$ with $k^n$ light bulbs. The lights corresponding to elements of $U$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. The core result of this paper is that the subspace $U$ has a flow if and only if the light pattern $U$ cannot be switched off. In particular, a graph $G$ has a nowhere-zero $k$-flow if and only if the ${\mathbb{Z}}_k$-bond space of $G$ cannot be switched off. It has a vertex coloring with $k$ colors if and only if a certain corresponding code over ${\mathbb{Z}}_k$ cannot be switched off. Similar statements hold for Tait colorings, and for nowhere-zero points of matrices. Studying different normal forms to equivalence classes of light patterns, we find various new equivalents, e.g., for the Four Color Problem, Tutte's Flow Conjectures and Jaeger's Conjecture. Two of our equivalents for colorability and existence of nowhere zero flows of graphs include as special cases results by Matiyasevich, by Balázs Szegedy, and by Onn. Alon and Tarsi's sufficient condition for $k$-colorability also arrives, remarkably, as a generalized full equivalent.