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We initiate the study of distortion in stable matching. Concretely, we aim to design algorithms that have limited access to the agents' cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare. Following the recent agenda of Amanatidis et al. (2021), we seek to find the best possible tradeoffs between the distortion of an algorithm and the amount of information elicited via means of cardinal value queries. As a warm-up, we show that purely ordinal algorithms (i.e., algorithms that only use the preference rankings of the agents but no value queries), including the classic Deferred Acceptance (DA) algorithm of Gale and Shapley (1962), have unlimited distortion. Despite this strong impossibility, we observe that a simple randomized version of the DA algorithm achieves a distortion of $2$, and show that this is in fact optimal among all randomized stable matching algorithms. We then consider algorithms that also use value queries on top of the ordinal ranking information. Again, it is easy to see that the same bound of $2$ can be achieved with only $1$ value query per agent, and thus we focus on the question of improving upon this bound. To this end, we provide two strong lower bounds, namely that any stable matching algorithm requires $Ω(n)$ non-adaptive value queries or $Ω(\log n)$ adaptive value queries per agent to achieve a distortion smaller than $2$, where $n$ is the number of agents. Finally, we show an asymptotically tight bound of $Θ(\log (1+n\varepsilon)/\varepsilon)$ on the number of queries that are necessary and sufficient to achieve near-optimal distortion, i.e., a distortion of $1+\varepsilon$, for any $\varepsilon >0$.

The tournament equilibrium set (TEQ), proposed by Schwartz in 1990, is a classical voting rule based on pairwise majority comparisons. Its definition is recursive: an alternative is judged by the alternatives that beat it, called its dominators, and by which of those dominators TEQ itself selects. Deciding whether a given alternative belongs to TEQ was known to be NP-hard and to lie in PSPACE. We close this gap by proving that the problem is PSPACE-complete. To our knowledge, this is the first PSPACE-completeness result for membership in a tournament solution. The hardness persists even when every nonempty subtournament has a unique minimal retentive set, the property conjectured by Schwartz to hold for all tournaments. Moreover, every such subtournament in our hard instances contains an alternative with at most three dominators. We prove that a single alternative with at most four dominators suffices both to guarantee uniqueness and to find a TEQ winner in $O(N^2)$ time. Thus, on our hard instances, finding one winner is efficient even though recognizing a specified winner is PSPACE-complete. We also study how the complexity changes with the dominator bound. If every nonempty subtournament has an alternative with at most one dominator, the entire TEQ can be computed in quadratic time; allowing two dominators makes membership NP-hard. Under the same condition with four dominators, all $t$ winners can be computed in $O(2^t N^2)$ time, where $N$ is the number of alternatives.

In the submodular secretary problem with a cardinality constraint, $n$ elements arrive in uniformly random order, and an algorithm must irrevocably accept or reject each one on arrival so as to maximize a nonnegative submodular function over sets of at most $k$ elements. For monotone objectives we settle the problem: the optimal competitive ratio tends to exactly $1-1/e$ as $k\to\infty$, and its worst case over all budgets $k$ is exactly $1/e$. The bound $1-1/e$ cannot be exceeded even with unlimited computation, and it is achieved by a single algorithm with at most $4n+1$ value queries; previously the best ratio without a shortlist was $1/e-o(1)$, with exponentially many queries. For general objectives, where the best known ratio was about $0.107$, we show that the optimal limiting ratio lies strictly between $1/e$ and $1/2$: we achieve $1/e+3.5\cdot10^{-6}$ for large $k$, and no algorithm achieves more than $0.497053$. We also achieve $0.2677$ at every budget. The main new ingredient of our algorithm is eligibility attenuation: an element that passes the price test at time $t$ is accepted with probability $e^{-τ}$, where $τ$ is the total earlier time during which it would have passed, which makes the probability that it has already been selected exactly $1-e^{-τ}$.

Fictitious play (FP) is one of the earliest and most fundamental learning procedures in games, where each agent independently chooses a best response to the empirical distribution of opponents' past actions by maximizing the expected payoff with respect to it. However, learning agents may not always be expected-utility maximizers: they may use nonlinear evaluation functionals to account for the uncertainty induced by the opponents, e.g., via risk measures. In this work, we study nonlinear fictitious play (NL-FP), in which each pure action is evaluated by a general, possibly nonlinear functional of the payoff distribution induced by the opponent's empirical belief. To analyze NL-FP, we first identify a single-crossing property (SCP) of evaluation functionals that preserves the classical geometry of FP in $2\times2$ games. Furthermore, we conduct a novel geometric analysis of the convergence behavior of NL-FP in $2\times2$ games, even when the SCP fails. Beyond $2\times2$ games, we show that the convergence of NL-FP in $2\times n$ ($n\ge3$) games may fail even for continuous-time NL-FP, when the two-action player uses conditional value-at-risk (CVaR) in a $2\times3$ game. In addition, we identify a finite-crossing condition that ensures convergence of the selected continuous-time dynamics and the discrete-time iterates in general $2\times n$ games, which also provides new insights into the global landscape of continuous-time classical FP in such games.

Kalah is a classic two-player game in the mancala family of sowing games. Computers have solved several standard-size starting positions, but the complexity of the game on boards of growing size has remained open. We study the natural generalization in which the number of pits per side is part of the input and an arbitrary position is given, with the stone counts given in unary. We show that deciding whether a designated player can force a win from such a position is PSPACE-complete. The hard positions have Alice to move first and both stores initially empty. The same positions are hard under either common convention for capture opposite an empty pit, and with or without the bonus rule, which gives a player another turn when the last stone is sown into their own store. We introduce Line Kalah as an intermediate problem, a game on a line in which stones that leave the line effectively disappear and Alice must save each of her target pits before Bob captures it, and we reduce quantified Boolean formulas to it. The reduction uses a set of sequential timer piles whose last stones approach a group of targets, forcing Alice to choose between two captures in order to set the truth value of a variable, and a final capture gadget to test a clause chosen by Bob (only one is needed adversarially). Since either player may move any pile at any time, the stages are placed so that premature timer firings protect earlier potential targets. For a true formula, Alice's strategy against arbitrary opposing play either gains enough time to save the remaining targets (after Bob deviates from the intended play) or reaches the final test with a correctly encoded, protected input. Finally, the line positions are embedded in Kalah with both stores empty. A separate Bob pit holds the score adjustment, so the winner takes a strict majority of the stones.

We provide an efficient implementation of combinatorial Bernoulli factories. Such a factory turns coins of unknown biases $x$ into a random vertex of a polytope $P$ whose mean is exactly $x$. For every affine slice of the cube (the class of polytopes that admit a factory), we give a single algorithm whose expected number of coin flips and running time at every interior $x$ are polynomial in the dimension and in $1/{δ_*}$, where ${δ_*}=\min_i\{x_i,1-x_i\}$ is never supplied to the algorithm. For perfect matchings this gives $O(n^{8}{δ_*}^{-2}\log(n/{δ_*}))$ flips, while the factory of Niazadeh, Paes Leme, and Schneider [STOC 2021] spends at least $n^{2n-1}$ flips in expectation at the uniform doubly stochastic matrix. For other polytopes, our method rounds exactly given a point deep inside $P$ and the promise that $x$ is far from the boundary of $P$.