We provide an optimization-based framework to perform counterfactual analysis in a dynamic model with hidden states. Our framework is grounded in the “abduction, action, and prediction” approach to answer counterfactual queries and handles two key challenges where (1) the states are hidden and (2) the model is dynamic. Recognizing the lack of knowledge on the underlying causal mechanism and the possibility of infinitely many such mechanisms, we optimize over this space and compute upper and lower bounds on the counterfactual quantity of interest. Our work brings together ideas from causality, state-space models, simulation, and optimization, and we apply it on a breast cancer case study. To the best of our knowledge, we are the first to compute lower and upper bounds on a counterfactual query in a dynamic latent-state model.

Recent work has leveraged the popular distributionally robust optimization paradigm to combat overfitting in classical logistic regression. While the resulting classification scheme displays a promising performance in numerical experiments, it is inherently limited to numerical features. In this paper, we show that distributionally robust logistic regression with mixed (\emph{i.e.}, numerical and categorical) features, despite amounting to an optimization problem of exponential size, admits a polynomial-time solution scheme. We subsequently develop a practically efficient cutting plane approach that solves the problem as a sequence of polynomial-time solvable exponential conic programs. Our method retains many of the desirable theoretical features of previous works, but---in contrast to the literature---it does not admit an equivalent representation as a regularized logistic regression, that is, it represents a genuinely novel variant of the logistic regression problem. We show that our method outperforms both the unregularized and the regularized logistic regression on categorical as well as mixed-feature benchmark instances.

The game of darts has enjoyed great growth over the past decade with the perception of darts moving from that of a pub game to a game that is regularly scheduled on prime-time television in many countries such as the United Kingdom, Germany, the Netherlands, and Australia, among others. It involves strategic interactions between two players, but to date, the literature has ignored these interactions. In this paper, we formulate and solve the game of darts as a dynamic zero-sum game (ZSG), and to the best of our knowledge, we are the first to do so. We also estimate individual skill models using a novel data set based on darts matches that were played by the top 16 professional players in the world during the 2019 season. Using the fitted skill models and our ZSG problem formulation, we quantify the importance of playing strategically—that is, taking into account the score and strategy of one’s opponent—when computing an optimal strategy. For top professionals, we find that playing strategically results in an increase in win probability of just 0.2%–0.6% over a single leg but as much as 2.2% over a best-of-31-legs match. Summary of Contribution: Dynamic zero-sum games (ZSGs) are of considerable interest, as they arise in many applications including sports, the management of communication networks, interdiction games, and heads-up poker—an important topic in modern artificial intelligence. In this study we consider the game of darts, which is growing increasingly popular around the world today. We formulate the game of darts as a ZSG and solve it iteratively by formulating each player’s best-response problem as a stochastic shortest-path (SSP) problem. We then solve these SSPs using standard dynamic programming methods. In solving the ZSG, we are able to accurately quantify the importance of top professionals playing strategically.

Recent work by Han and Van Roy [Han J, Van Roy B (2011) Control of diffusions via linear programming. Infanger G, ed. Stochastic Programming: The State of the Art, in Honor of George B. Dantzig (Springer, New York), 329–354] introduced a linear programming technique to compute good suboptimal solutions to high-dimensional control problems in a diffusion-based setting. Their problem formulation worked with finite horizon problems where the horizon, T, is an exponentially distributed random variable. We extend their approach to finite horizon problems with a fixed horizon T. We also apply these techniques to dynamic portfolio optimization problems and then simulate the resulting policies to obtain lower bounds on the optimal value functions. We also use these policies in conjunction with convex duality methods designed for portfolio optimization problems to construct upper bounds on the optimal value functions. In our numerical experiments we find that the primal and dual bounds are very close, and so we conclude, for these problems at least, that the linear programming approach performs very well.