arXiv subjects

Reed Phillips

Publications and source records attributed to Reed Phillips.

6 recordsLinked to original sources

arXiv2026

The Quantum Query Complexity of Finding a Tarski Fixed Point on the 2D Grid

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice $\mathcal{L}^2_n$ on points $\{1, \ldots, n\}^2$ and where $(x_1, y_1) \leq (x_2, y_2)$ if $x_1 \leq x_2$ and $y_1 \leq y_2$. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on $\mathcal{L}^2_n$ is $\Omega((\log n)^2)$, matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.

cs.CC↗

arXiv2026

Dueling over Multiple Pieces of Dessert

We study the dynamics of repeated fair division between two players, Alice and Bob, where Alice partitions a cake into two subsets and Bob chooses his preferred one over $T$ rounds. Alice aims to minimize her regret relative to the Stackelberg value -- the maximum utility she could achieve if she knew Bob's private valuation. We show that if Alice uses arbitrary measurable partitions, achieving strongly sublinear regret is impossible; she suffers a regret of $\Omega\Bigl(\frac{T}{\log^2 T}\Bigr)$ regret even against a myopic Bob. However, when Alice uses at most $k$ cuts, the learning landscape becomes tractable. We analyze Alice's performance based on her knowledge of Bob's strategic sophistication (his regret budget). When Bob's learning rate is public, we establish a hierarchy of polynomial regret bounds determined by $k$ and Bob's regret budget. In contrast, when this learning rate is private, Alice can universally guarantee $O\Bigl(\frac{T}{\log T}\Bigr)$ regret, but any attempt to secure a polynomial rate $O(T^\beta)$ (for $\beta < 1$) leaves her vulnerable to incurring strictly linear regret against some Bob. Finally, as a corollary of our online learning dynamics, we characterize the randomized query complexity of finding approximate Stackelberg allocations with a constant number of cuts in the Robertson-Webb model.

cs.GT↗

arXiv2025

Tarski Lower Bounds from Multi-Dimensional Herringbones

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$ relation. In this setting, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. The goal is to find a fixed point of $f$ using as few oracle queries as possible. We show that the randomized query complexity of this problem is $\Omega\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)$ for all $n,k \geq 2$. This unifies and improves upon two prior results: a lower bound of $\Omega(\log^2{n})$ from [EPRY 2019] and a lower bound of $\Omega\left( \frac{k \cdot \log{n}}{\log{k}}\right)$ from [BPR 2024], respectively.

cs.CC↗

arXiv2024

The Randomized Query Complexity of Finding a Tarski Fixed Point on the Boolean Hypercube

The Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that every monotone function defined on a complete lattice has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$ relation. Specifically, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. A key special case of interest is the Boolean hypercube $\{0,1\}^k$, which is isomorphic to the power set lattice--the original setting of the Knaster-Tarski theorem. We prove a lower bound that characterizes the randomized and deterministic query complexity of the Tarski search problem on the Boolean hypercube as $\Theta(k)$. More generally, we give a randomized lower bound of $\Omega\left( k + \frac{k \log{n}}{\log{k}} \right)$ for the $k$-dimensional grid of side length $n$, which is asymptotically optimal in high dimensions when $k$ is large relative to $n$.

cs.CC↗

arXiv2024

Dueling Over Dessert, Mastering the Art of Repeated Cake Cutting

We consider the setting of repeated fair division between two players, denoted Alice and Bob, with private valuations over a cake. In each round, a new cake arrives, which is identical to the ones in previous rounds. Alice cuts the cake at a point of her choice, while Bob chooses the left piece or the right piece, leaving the remainder for Alice. We consider two versions: sequential, where Bob observes Alice's cut point before choosing left/right, and simultaneous, where he only observes her cut point after making his choice. The simultaneous version was first considered by Aumann and Maschler (1995). We observe that if Bob is almost myopic and chooses his favorite piece too often, then he can be systematically exploited by Alice through a strategy akin to a binary search. This strategy allows Alice to approximate Bob's preferences with increasing precision, thereby securing a disproportionate share of the resource over time. We analyze the limits of how much a player can exploit the other one and show that fair utility profiles are in fact achievable. Specifically, the players can enforce the equitable utility profile of $(1/2, 1/2)$ in the limit on every trajectory of play, by keeping the other player's utility to approximately $1/2$ on average while guaranteeing they themselves get at least approximately $1/2$ on average. We show this theorem using a connection with Blackwell approachability. Finally, we analyze a natural dynamic known as fictitious play, where players best respond to the empirical distribution of the other player. We show that fictitious play converges to the equitable utility profile of $(1/2, 1/2)$ at a rate of $O(1/\sqrt{T})$.

cs.GT↗

arXiv2021

The Number of Optimal Strategies in the Penney-Ante Game

In the Penney-Ante game, Player I chooses a head/tail string of a predetermined length $n\ge3$. Player II, upon seeing Player I's choice, chooses another head/tail string of the same length. A coin is then tossed repeatedly and the player whose string appears first in the resulting head/tail sequence wins the game. The Penney-Ante game has gained notoriety as a source of counterintuitive probabilities and nontransitivity phenomena. For example, Player II can always choose a string that beats the choice of Player I in the sense of being more likely to appear first in a random head/tail sequence. It is known that Player II has a unique optimal strategy that maximizes her winning chances in this game. On the other hand, for Player I there exist multiple equivalent optimal strategies. In this paper we investigate the number, $c_n$, of optimal strategies for Player I, i.e., the number of head/tail strings of length $n$ that maximize the winning probability for Player I assuming optimal play by Player II. We derive a recurrence relation for $c_n$ and use this to obtain a sharp asymptotic estimate for $c_n$. In particular, we show that, as $n\to\infty$, a fixed proportion $\alpha \approx 0.04062\dots$ of the $2^n$ head/tail strings of length $n$ are optimal from Player I's perspective.

math.CO↗