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Register a new userA new quantum game framework: When player strategies are via directional choices
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For two-player quantum games, a Nash equilibrium consists of a pair of unitary operators. Here we present a scheme for such games in which each players strategy consists of choosing the orientation of a unit vector and Nash equilibria of the game are directional pairs. Corresponding classical games are then recovered from constraints placed on each players directional choices.
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The mean value theorem of calculus states that, given a differentiable function $f$ on an interval $[a, b]$, there exists at least one mean value abscissa $c$ such that the slope of the tangent line at $c$ is equal to the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. In this article, we study how the choices of $c$ relate to varying the right endpoint $b$. In particular, we ask: When we can write $c$ as a continuous function of $b$ in some interval? Drawing inspiration from graphed examples, we first investigate this question by proving and using a simplified implicit function theorem. To handle certain edge cases, we then build on this analysis to prove and use a simplified Morses lemma. Finally, further developing the tools proved so far, we conclude that if $f$ is analytic, then it is always possible to choose mean value abscissae so that $c$ is a continuous function of $b$, at least locally.
Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. While theoretical approaches to the problem have hit some limits, a recent research direction initiated by Duetting et al. (2019) consists in building neural network architectures to find optimal auctions. We propose two conceptual deviations from their approach which result in enhanced performance. First, we use recent results in theoretical auction design (Rubinstein and Weinberg, 2018) to introduce a time-independent Lagrangian. This not only circumvents the need for an expensive hyper-parameter search (as in prior work), but also provides a principled metric to compare the performance of two auctions (absent from prior work). Second, the optimization procedure in previous work uses an inner maximization loop to compute optimal misreports. We amortize this process through the introduction of an additional neural network. We demonstrate the effectiveness of our approach by learning competitive or strictly improved auctions compared to prior work. Both results together further imply a novel formulation of Auction Design as a two-player game with stationary utility functions.
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