There are two well-known sufficient conditions for Nash equilibrium in two-player games: mutual knowledge of rationality (MKR) and mutual knowledge of conjectures. MKR assumes that the concept of rationality is mutually known. In contrast, mutual knowledge of conjectures assumes that a given profile of conjectures is mutually known, which has long been recognized as a strong assumption. In this note, we introduce a notion of "mutual assumption of rationality and correctness" (MARC), which conceptually aligns more closely with the MKR assumption. We present two main results. Our first result establishes that MARC holds in every two-person zero-sum game. In our second theorem, we show that MARC does not in general hold in n-player games.
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Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.
The Shapley value equals a player's contribution to the potential of a game. The potential is a most natural one-number summary of a game, which can be computed as the expected accumulated worth of a random partition of the players. This computation integrates the coalition formation of all players and readily extends to games with externalities. We investigate those potential functions for games with externalities that can be computed this way. It turns out that the potential that corresponds to the MPW solution introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339-356), is unique in the following sense. It is obtained as a the expected accumulated worth of a random partition, it generalizes the potential for games without externalities, and it induces a solution that satisfies the null player property even in the presence of externalities.
Propensity score matching is commonly used to draw causal inference from observational survival data. However, its asymptotic properties have yet to be established, and variance estimation is still open to debate. We derive the statistical properties of the propensity score matching estimator of the marginal causal hazard ratio based on matching with replacement and a fixed number of matches. We also propose a double-resampling technique for variance estimation that takes into account the uncertainty due to propensity score estimation prior to matching.
A central challenge in the verification of quantum computers is benchmarking their performance as a whole and demonstrating their computational capabilities. In this work, we find a universal model of quantum computation, Bell sampling, that can be used for both of those tasks and thus provides an ideal stepping stone towards fault-tolerance. In Bell sampling, we measure two copies of a state prepared by a quantum circuit in the transversal Bell basis. We show that the Bell samples are classically intractable to produce and at the same time constitute what we call a circuit shadow: from the Bell samples we can efficiently extract information about the quantum circuit preparing the state, as well as diagnose circuit errors. In addition to known properties that can be efficiently extracted from Bell samples, we give two new and efficient protocols, a test for the depth of the circuit and an algorithm to estimate a lower bound to the number of T gates in the circuit. With some additional measurements, our algorithm learns a full description of states prepared by circuits with low T-count.
Large Vision-Language Models (VLMs) have demonstrated impressive performance on complex tasks involving visual input with natural language instructions. However, it remains unclear to what extent capabilities on natural images transfer to Earth observation (EO) data, which are predominantly satellite and aerial images less common in VLM training data. In this work, we propose a comprehensive benchmark to gauge the progress of VLMs toward being useful tools for EO data by assessing their abilities on scene understanding, localization and counting, and change detection tasks. Motivated by real-world applications, our benchmark includes scenarios like urban monitoring, disaster relief, land use, and conservation. We discover that, although state-of-the-art VLMs like GPT-4V possess extensive world knowledge that leads to strong performance on open-ended tasks like location understanding and image captioning, their poor spatial reasoning limits usefulness on object localization and counting tasks. Our benchmark will be made publicly available at //vleo.danielz.ch/ and on Hugging Face at //huggingface.co/collections/mit-ei/vleo-benchmark-datasets-65b789b0466555489cce0d70 for easy model evaluation.
Two CNF formulas are called ucp-equivalent, if they behave in the same way with respect to the unit clause propagation (UCP). A formula is called ucp-irredundant, if removing any clause leads to a formula which is not ucp-equivalent to the original one. As a consequence of known results, the ratio of the size of a ucp-irredundant formula and the size of a smallest ucp-equivalent formula is at most $n^2$, where $n$ is the number of the variables. We demonstrate an example of a ucp-irredundant formula for a symmetric definite Horn function which is larger than a smallest ucp-equivalent formula by a factor $\Omega(n/\ln n)$ and, hence, a general upper bound on the above ratio cannot be smaller than this.
Quantum computing shows great potential, but errors pose a significant challenge. This study explores new strategies for mitigating quantum errors using artificial neural networks (ANN) and the Yang-Baxter equation (YBE). Unlike traditional error correction methods, which are computationally intensive, we investigate artificial error mitigation. The manuscript introduces the basics of quantum error sources and explores the potential of using classical computation for error mitigation. The Yang-Baxter equation plays a crucial role, allowing us to compress time dynamics simulations into constant-depth circuits. By introducing controlled noise through the YBE, we enhance the dataset for error mitigation. We train an ANN model on partial data from quantum simulations, demonstrating its effectiveness in correcting errors in time-evolving quantum states.
By abstracting over well-known properties of De Bruijn's representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi's approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.
Human participants play a central role in the development of modern artificial intelligence (AI) technology, in psychological science, and in user research. Recent advances in generative AI have attracted growing interest to the possibility of replacing human participants in these domains with AI surrogates. We survey several such "substitution proposals" to better understand the arguments for and against substituting human participants with modern generative AI. Our scoping review indicates that the recent wave of these proposals is motivated by goals such as reducing the costs of research and development work and increasing the diversity of collected data. However, these proposals ignore and ultimately conflict with foundational values of work with human participants: representation, inclusion, and understanding. This paper critically examines the principles and goals underlying human participation to help chart out paths for future work that truly centers and empowers participants.
Several extensions of the equal division value and the equal surplus division value to the family of games with a priori unions are proposed in Alonso-Meijide et al. (2020) ``On egalitarian values for cooperative games with a priori unions.'' TOP 28: 672-688. In this paper we provide new axiomatic characterizations of these values. Furthermore, using the equal surplus division value in two steps, we propose a new coalitional value. The balanced contributions and quotient game properties give rise to a different modification of the equal surplus division value.
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