arXiv subjects
Publications and source records attributed to Edan Orzech.
We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree $\mathbin{!n}$ (the derangement number), and its minimal polynomial has Galois group $S_{\mathbin{!n}}$ and $\mathbin{!n}+1$ nonzero coefficients.
We present for every $n\ge4$ an $n$-player game in normal form with payoffs in $\{0,1,2\}$ that has a unique, fully mixed, Nash equilibrium in which all the probability weights are irradical (i.e., algebraic but not closed form expressible even with $m$-th roots for any integer $m$).
We consider zero-sum games in which players move between adjacent states, where in each pair of adjacent states one state dominates the other. The states in our game can represent positional advantages in physical conflict such as high ground or camouflage, or product characteristics that lend an advantage over competing sellers in a duopoly. We study the equilibria of the game as a function of the topological and geometric properties of the underlying graph. Our main result characterizes the expected payoff of both players starting from any initial position, under the assumption that the graph does not contain certain types of small cycles. This characterization leverages the block-cut tree of the graph, a construction that describes the topology of the biconnected components of the graph. We identify three natural types of (on-path) pure equilibria, and characterize when these equilibria exist under the above assumptions. On the geometric side, we show that strongly connected outerplanar graphs with undirected girth at least 4 always support some of these types of on-path pure equilibria. Finally, we show that a data structure describing all pure equilibria can be efficiently computed for these games.
In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play Nash equilibria in finite two player $n\times n$ games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play in the (unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by the players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games. These results highlight (i) the nontriviality of the assumption that players can play any mixed strategy and (ii) the disparity in resources that players may require to play Nash equilibria in normal-form games.
We consider team zero-sum network congestion games with $n$ agents playing against $k$ interceptors over a graph $G$. The agents aim to minimize their collective cost of sending traffic over paths in $G$, which is an aggregation of edge costs, while the interceptors aim to maximize the collective cost by increasing some of these edge costs. To evade the interceptors, the agents will usually use randomized strategies. We consider two cases, the correlated case when agents have access to a shared source of randomness, and the uncorrelated case, when each agent has access to only its own source of randomness. We study the additional cost that uncorrelated agents have to bear, specifically by comparing the costs incurred by agents in cost-minimal Nash equilibria when agents can and cannot share randomness. We consider two natural cost functions on the agents, which measure the invested energy and time, respectively. We prove that for both of these cost functions, the ratio of uncorrelated cost to correlated cost at equilibrium is $O(\min(m_c(G),n))$, where $m_c(G)$ is the mincut size of $G$. This bound is much smaller than the most general case, where a tight, exponential bound of $\Theta((m_c(G))^{n-1})$ on the ratio is known. We also introduce a set of simple agent strategies which are approximately optimal agent strategies. We then establish conditions for when these strategies are optimal agent strategies for each cost function, showing an inherent difference between the two cost functions we study.
Recall that a binary linear code of length $n$ is a linear subspace $\mathcal{C} = \{x\in\mathbb{F}_2^n\mid Ax=0\}$. Here the parity check matrix $A$ is a binary $m\times n$ matrix of rank $m$. We say that $\mathcal{C}$ has rate $R=1-\frac mn$. Its distance, denoted $\delta n$ is the smallest Hamming weight of a non-zero vector in $\mathcal{C}$. The rate vs.\ distance problem for binary linear codes is a fundamental open problem in coding theory, and a fascinating question in discrete mathematics. It concerns the function $R_L(\delta)$, the largest possible rate $R$ for given $0\le\delta\le1$ and arbitrarily large length $n$. Here we investigate a variation of this fundamental question that we describe next. Clearly, $\mathcal{C}$ has distance $\delta n$, if and only if for every $0<n'<\delta n$, every $m\times n'$ submatrix of $A$ has a row of odd weight. Motivated by several problems from coding theory, we say that $A$ has the unique-neighbor property with parameter $\delta n$, if every such submatrix has a row of weight $1$. Let $R_U(\delta)$ be the largest possible asymptotic rate of linear codes with a parity check matrix that has this stronger property. Clearly, $R_U(\cdot),R_L(\cdot)$ are non-increasing functions, and $R_U(\delta)\le R_L(\delta)$ for all $\delta$. Also, $R_U(0)=R_L(0)=1$, and $R_U(1)=R_L(1)=0$, so let $0\le\delta_U \le\delta_L\le1$ be the smallest values of $\delta$ at which $R_U$ resp.\ $R_L$ vanish. It is well known that $\delta_L=\frac12$ and we conjecture that $\delta_U$ is strictly smaller than $\frac12$, i.e., the rate of linear codes with the unique-neighbor property is more strictly bounded. While the conjecture remains open, we prove here several results supporting it. The reader is not assumed to have any specific background in coding theory, but we occasionally point out some relevant facts from that area.