John Fearnley

John Fearnley, Markus Rabe, Sven Schewe et al.

We study the time-bounded reachability problem for continuous-time Markov decision processes (CTMDPs) and games (CTMGs). Existing techniques for this problem use discretisation techniques to break time into discrete intervals, and optimal control is approximated for each interval separately. Current techniques provide an accuracy of O(ε^2) on each interval, which leads to an infeasibly large number of intervals. We propose a sequence of approximations that achieve accuracies of O(ε^3), O(ε^4), and O(ε^5), that allow us to drastically reduce the number of intervals that are considered. For CTMDPs, the performance of the resulting algorithms is comparable to the heuristic approach given by Buckholz and Schulz, while also being theoretically justified. All of our results generalise to CTMGs, where our results yield the first practically implementable algorithms for this problem. We also provide positional strategies for both players that achieve similar error bounds.

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos

We study the computational complexity of finding a solution for the straight-cut and square-cut pizza sharing problems. We show that computing an $\varepsilon$-approximate solution is PPA-complete for both problems, while finding an exact solution for the square-cut problem is FIXP-hard. Our PPA-hardness results apply for any $\varepsilon < 1/5$, even when all mass distributions consist of non-overlapping axis-aligned rectangles or when they are point sets, and our FIXP-hardness result applies even when all mass distributions are unions of squares and right-angled triangles. We also prove that the decision variants of both approximate problems are NP-complete, while the decision variant for the exact version of square-cut pizza sharing is $\exists\mathbb{R}$-complete.

Argyrios Deligkas, John Fearnley, Alexandros Hollender et al.

We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD$=$P. In fact, we prove that computing any approximation better than $1/11$ is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.

Argyrios Deligkas, John Fearnley, Alexandros Hollender et al.

We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a $(1-δ)$-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant $δ> 0$, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow $\varepsilon$-approximate clearing instead of perfect clearing, for any constant $\varepsilon < 1/9$. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant $δ$: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.

John Fearnley, Paul W. Goldberg, Alexandros Hollender et al.

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain $[0,1]^2$ is PPAD $\cap$ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more "natural" counterpart to PPAD $\cap$ PLS and contains many interesting problems - is itself equal to PPAD $\cap$ PLS.

Thomas Spooner, John Fearnley, Rahul Savani et al.

Market making is a fundamental trading problem in which an agent provides liquidity by continually offering to buy and sell a security. The problem is challenging due to inventory risk, the risk of accumulating an unfavourable position and ultimately losing money. In this paper, we develop a high-fidelity simulation of limit order book markets, and use it to design a market making agent using temporal-difference reinforcement learning. We use a linear combination of tile codings as a value function approximator, and design a custom reward function that controls inventory risk. We demonstrate the effectiveness of our approach by showing that our agent outperforms both simple benchmark strategies and a recent online learning approach from the literature.