Automated teller machines (ATM) play a major role in the world economy as they enable financial transactions and hence good exchanges and consumption. ATM transaction fees are incurred to cover the cost of running the network and these are often settled among the members including banks and cash machine operators. In this paper, we develop a novel biform game theoretic model for members to optimally invest in the ATM network and to share the cost. This biform game includes both a cooperative game theory mechanism for interchange fee sharing and a non-cooperative counterpart to model the fact that members also wish to maximize their utilities. While the proposed coopetition framework is applicable to general ATM networks, we focus the case study on the UK ATM network thanks to the accessibility of the data in addition to the notable stability issues that the network is currently experiencing as has been widely featured by the mainstream media. On the technical side, we prove the existence of a pure Nash equilibrium, which can be computed efficiently. We also show that, under some settings, the Shapley allocation belongs to the core and hence it is not only fair to all members but also leads to a stable ATM network. In addition, we show that the Shapley value allocation dominates the current mechanism in terms of social welfare. Finally, we provide numerical analysis and managerial insights using real data on the complete UK ATM network.

Automated teller machines (ATM) play a major role in the world economy as they enable financial transactions and hence good exchanges and consumption. ATM transaction fees are incurred to cover the cost of running the network and these are often settled among the members including banks and cash machine operators. In this paper, we develop a novel biform game theoretic model for members to optimally invest in the ATM network and to share the cost. This biform game includes both a cooperative game theory mechanism for interchange fee sharing and a non -cooperative counterpart to model the fact that members also wish to maximize their utilities. While the proposed coopetition framework is applicable to general ATM networks, we focus the case study on the UK ATM network thanks to the accessibility of the data in addition to the notable stability issues that the network is currently experiencing as has been widely featured by the mainstream media. On the technical side, we prove the existence of a pure Nash equilibrium, which can be computed efficiently. We also show that, under some settings, the Shapley allocation belongs to the core and hence it is not only fair to all members but also leads to a stable ATM network. In addition, we show that the Shapley value allocation dominates the current mechanism in terms of social welfare. Finally, we provide numerical analysis and managerial insights using real data on the complete UK ATM network.

We study online learning problems in which the learner has extra knowledge about the adversary's behaviour, i.e., in game-theoretic settings where opponents typically follow some no-external regret learning algorithms. Under this assumption, we propose two new online learning algorithms, Accurate Follow the Regularized Leader (AFTRL) and Prod-Best Response (Prod-BR), that intensively exploit this extra knowledge while maintaining the no-regret property in the worst-case scenario of having inaccurate extra information. Specifically, AFTRL achieves $O(1)$ external regret or $O(1)$ \emph{forward regret} against no-external regret adversary in comparison with $O(\sqrt{T})$ \emph{dynamic regret} of Prod-BR. To the best of our knowledge, our algorithm is the first to consider forward regret that achieves $O(1)$ regret against strategic adversaries. When playing zero-sum games with Accurate Multiplicative Weights Update (AMWU), a special case of AFTRL, we achieve \emph{last round convergence} to the Nash Equilibrium. We also provide numerical experiments to further support our theoretical results. In particular, we demonstrate that our methods achieve significantly better regret bounds and rate of last round convergence, compared to the state of the art (e.g., Multiplicative Weights Update (MWU) and its optimistic counterpart, OMWU).

This paper considers repeated games in which one player has a different objective than others. In particular, we investigate repeated two-player zero-sum games where the column player not only aims to minimize her regret but also stabilize the actions. Suppose that while repeatedly playing this game, the row player chooses her strategy at each round by using a no-regret algorithm to minimize her regret. We develop a no-dynamic regret algorithm for the column player to exhibit last round convergence to a minimax equilibrium. We show that our algorithm is efficient against a large set of popular no-regret algorithms the row player can use, including the multiplicative weights update algorithm, general follow-the-regularized-leader and any no-regret algorithms satisfy a property so called “stability”.

The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties—it always exists and lies in the core (if the core is non-empty), and it is unique. The nucleolus is considered as the most ‘stable’ solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n ≤ 15 ). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the ‘balancedness’ of at most (n-1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.

In this paper, we develop a model for dynamic pricing and inventory decisions for multiple substitutable and perishable products under a multiple-period lifetime. Retailers place orders at the beginning of the first period, and products will be sold at full price during that period. All leftover products will then be carried over to the subsequent periods and sold at discounted prices. Demands for leftover products are assumed to follow a linear stochastic model depending on the discounted prices of the substitutable products. The optimal order quantities and prices are obtained by maximizing the total expected profit over the lifetime, taking into account the revenue, the backorder cost and the holding cost. We provide analytical properties of the optimal policy such as the concavity of the value functions and then utilise these in the numerical scheme for finding the optimal prices and ordering quantities. Experimental results are reported through a case study of a high-street fashion company, demonstrating the benefits of considering pricing and inventory decisions simultaneously for substitutable products.

Title: Should newsvendors cooperate if the demand distribution is unknown?

In this paper, we consider a facility location problem where customer demand constitutes considerable uncertainty, and where complete information on the distribution of the uncertainty is unavailable. We formulate the optimal decision problem as a two-stage stochastic mixed integer programming problem: an optimal selection of facility locations in the first stage and an optimal decision on the operation of each facility in the second stage. A distributionally robust optimization framework is proposed to hedge risks arising from incomplete information on the distribution of the uncertainty. Specifically, by exploiting the moment information, we construct a set of distributions which contains the true distribution and where the optimal decision is based on the worst distribution from the set. We then develop two numerical schemes for solving the distributionally robust facility location problem: a semi-infinite programming approach which exploits moments of certain reference random variables and a semi-definite programming approach which utilizes the mean and correlation of the underlying random variables describing the demand uncertainty. In the semi-infinite programming approach, we apply the well-known linear decision rule approach to the robust dual problem and then approximate the semi-infinite constraints through the conditional value at risk measure. We provide numerical tests to demonstrate the computation and properties of the robust solutions.

This paper considers repeated games in which one player has more information about the game than the other players. In particular, we investigate repeated two-player zero-sum games where only the column player knows the payoff matrix A of the game. Suppose that while repeatedly playing this game, the row player chooses her strategy at each round by using a no-regret algorithm to minimize her (pseudo) regret. We develop a no-instant-regret algorithm for the column player to exhibit last round convergence to a minimax equilibrium. We show that our algorithm is efficient against a large set of popular no-regret algorithms of the row player, including the multiplicative weight update algorithm, the online mirror descent method/follow-the-regularized-leader, the linear multiplicative weight update algorithm, and the optimistic multiplicative weight update.

The linear production game is concerned with allocating the total payoff of an enterprise among the owners of the resources in a fair way. With cooperative game theory providing a mathematical framework for sharing the benefit of the cooperation, the Shapley value is one of the widely used solution concepts as a fair measurement in this area. Finding the exact Shapley value for linear production games is, however, challenging when the number of players exceeds 30. This paper describes the use of linear programming sensitivity analysis for a more efficient computation of the Shapley value. The paper also proposes a stratified sampling technique to estimate the Shapley value for large-scale linear production games. Computational results show the effectiveness of the proposed methods compared to others.

In this paper, we develop an extended dynamic programming (DP) approach to solve the problem of minimising execution cost in block trading of securities. To make the problem more practical, we add non-negativity constraints to the model and propose a novel approach to solve the resulting DP problem to near-optimal results. We also include time lags in the problem state to account for the autoregressive behaviour of most financial securities as a way of increasing problem sensitivity to variability of prices and information. The computation times achieved for the proposed algorithm are fast and allow for the possibility of live implementation. We demonstrate the benefits offered by the new approach through numerical analysis and simulation runs in comparison to the classic model without the non-negativity constraints.

Given a dataset an outlier can be defined as an observation that does not follow the statistical properties of the majority of the data. Computation of the location estimate is of fundamental importance in data analysis, and it is well known in statistics that classical methods, such as taking the sample average, can be greatly affected by the presence of outliers in the data. Using the median instead of the mean can partially resolve this issue but not completely. For the univariate case, a robust version of the median is the Least Trimmed Absolute Deviation (LTAD) robust estimator introduced in Tableman (Stat Probab Lett 19(5):387–398, 1994), which has desirable asymptotic properties such as robustness, consistently, high breakdown and normality. There are different generalizations of the LTAD for multivariate data, depending on the choice of norm. Chatzinakos et al. (J Comb Optim, 2015) we present such a generalization using the Euclidean norm and propose a solution technique for the resulting combinatorial optimization problem, based on a necessary condition, that results in a highly convergent local search algorithm. In this subsequent work, we use the $$L^1$$ norm to generalize the LTAD to higher dimensions, and show that the resulting mixed integer programming problem has an integral relaxation, after applying an appropriate data transformation. Moreover, we utilize the structure of the problem to show that the resulting LP's can be solved efficiently using a subgradient optimization approach. The robust statistical properties of the proposed estimator are verified by extensive computational results.

A descending clock auction (DCA) is for buying items from multiple sellers. The literature has focused on the case where each bidder has two options: to accept or reject the offered price. However, in many settings-such as the FCC's imminent incentive auction-each bidder may be able to sell one from a set of options. We present a multi-option DCA (MDCA) framework where at each round, the auctioneer offers each bidder different prices for different options, and a bidder may find multiple options still acceptable. Setting prices during a MDCA is trickier than in a DCA. We develop a Markov chain model for the dynamics of each bidder's state (which options are still acceptable). We leverage it to optimize the trajectory of price offers to different bidders for different options. This is unlike most auctions which only compute the next price vector. Computing the trajectory enables better planning. We reoptimize the trajectory after each round. Each optimization minimizes total payment while ensuring feasibility in a stochastic sense. We also introduce percentile-based approaches to decrementing prices. Experiments with real FCC incentive auction interference constraint data show that the optimization-based approach dramatically outperforms the percentile-based approach-because it takes feasibility into account in pricing. Both pricing techniques scale to the large.

The short-term unit commitment and reserve scheduling decisions are made in the face of increasing supply-side uncertainty in power systems. This has mainly been caused by a higher penetration of renewable energy generation that is encouraged and enforced by the market and policy makers. In this paper, we propose a two-stage stochastic and distributionally robust modeling framework for the unit commitment problem with supply uncertainty. Based on the availability of the information on the distribution of the random supply, we consider two specific models: (a) a moment model where the mean values of the random supply variables are known, and (b) a mixture distribution model where the true probability distribution lies within the convex hull of a finite set of known distributions. In each case, we reformulate these models through Lagrange dualization as a semi-infinite program in the former case and a one-stage stochastic program in the latter case. We solve the reformulated models using sampling method and sample average approximation, respectively. We also establish exponential rate of convergence of the optimal value when the randomization scheme is applied to discretize the semi-infinite constraints. The proposed robust unit commitment models are applied to an illustrative case study, and numerical test results are reported in comparison with the two-stage non-robust stochastic programming model.

A descending clock auction (DCA) is for buying items from multiple sellers. The literature has focused on the case where each bidder has two options: to accept or reject the offered price. However, in many settings--such as the FCC's imminent incentive auction--each bidder may be able to sell one from a set of options. We present a multi-option DCA (MDCA) framework where at each round, the auctioneer offers each bidder different prices for different options, and a bidder may find multiple options still acceptable. Setting prices during a MDCA is trickier than in a DCA. We develop a Markov chain model for the dynamics of each bidder's state (which options are still acceptable). We leverage it to optimize the trajectory of price offers to different bidders for different options. This is unlike most auctions which only compute the next price vector. Computing the trajectory enables better planning. We reoptimize the trajectory after each round. Each optimization minimizes total payment while ensuring feasibility in a stochastic sense. We also introduce percentile-based approaches to decrementing prices. Experiments with real FCC incentive auction interference constraint data show that the optimization-based approach dramatically outperforms the percentile-based approach--because it takes feasibility into account in pricing. Both pricing techniques scale to the large.

The minimum-cost spanning tree game is a special class of cooperative games defined on a graph with a set of vertices and a set of edges, where each player owns a vertex. Solutions of the game represent ways to distribute the total cost of a minimum-cost spanning tree among all the players. When the graph is partitioned into clusters, the generalized minimum spanning tree problem is to determine a minimum-cost tree including exactly one vertex from each cluster. This paper introduces the generalized minimum spanning tree game and studies some properties of this game. The paper also describes a constraint generation algorithm to calculate a stable payoff distribution and presents computational results obtained using the proposed algorithm.

The short-term unit commitment and reserve scheduling decisions are made in the face of increasing supply-side uncertainty in power systems. This has mainly been caused by a higher penetration of renewable energy generation that is encouraged and enforced by the market and policy makers. In this paper, we propose a two-stage stochastic and distributionally robust modeling framework for the unit commitment problem with supply uncertainty. Based on the availability of the information on the distribution of the random supply, we consider two specific models: (a) a moment model where the mean values of the random supply variables are known, and (b) a mixture distribution model where the true probability distribution lies within the convex hull of a finite set of known distributions. In each case, we reformulate these models through Lagrange dualization as a semi-infinite program in the former case and a one-stage stochastic program in the latter case. We solve the reformulated models using sampling method and sample average approximation, respectively. We also establish exponential rate of convergence of the optimal value when the randomization scheme is applied to discretize the semi-infinite constraints. The proposed robust unit commitment models are applied to an illustrative case study, and numerical test results are reported in comparison with the two-stage non-robust stochastic programming model.

The combinatorial optimization game where the characteristic function is generated by solving a combinatorial optimization problem forms an important class in cooperative game theory. In this paper, we introduce a new subclass of combinatorial optimization games namely the multidimensional integer knapsack game (MIKG). Finding the Shapley value as the solution for the payoff allocation problem is challenging and time consuming for MIKG because of the integer program structure of the characteristic function. We propose to utilize an algebraic approach of Gröbner bases for solving the integer programs, and generate an efficient way to find the Shapley value in these cooperative games. Some computational experiments with the MIKG are investigated where the numerical results show the advantages of the method compared to CPLEX solver.

The nucleolus offers a desirable payoff-sharing solution in cooperative games thanks to its attractive properties - it always exists and lies in the core (if the core is non-empty), and is unique. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e., the number of players $n \leq 20$). This, however, becomes more challenging for larger games because of the need to form and check the balancedness of possibly exponentially large sets of coalitions, each set could be of an exponentially large size. We develop a simplifying set of the Kohlberg criteria that involves checking the balancedness of at most $(n-1)$ sets of coalitions. We also provide a method for reducing the size of these sets and a fast algorithm for verifying the balancedness.