Stackelberg security games form the backbone of systems like ARMOR, IRIS and PROTECT, which are in regular use by the Los Angeles International Police, US Federal Air Marshal Service and the US Coast Guard respectively. An understanding of the runtime required by algorithms that power such systems is critical to furthering the application of game theory to other real-world domains. This paper identifies the concept of the deployment-to-saturation ratio in random Stackelberg security games, and shows that problem instances for which this ratio is 0.5 are computationally harder than instances with other deployment-to-saturation ratios for a wide range of different equilibrium computation methods, including (i) previously published different MIP algorithms, and (ii) different underlying solvers and solution mechanisms. This finding has at least two important implications. First, it is important for new algorithms to be evaluated on the hardest problem instances. We show that this has often not been done in the past, and introduce a publicly available benchmark suite to facilitate such comparisons. Second, we provide evidence that this computationally hard region is also one where optimization would be of most benefit to security agencies, and thus requires significant attention from researchers in this area. Furthermore, we use the concept of phase transitions to better understand this computationally hard region. We define a decision problem related to security games, and show that the probability that this problem has a solution exhibits a phase transition as the deployment-to-saturation ratio crosses 0.5. We also demonstrate that this phase transition is invariant to changes both in the domain and the domain representation, and that the phase transition point corresponds to the computationally hardest instances.

In large extensive form games with imperfect information, Counterfactual Regret Minimization (CFR) is a popular, iterative algorithm for computing approximate Nash equilibria. While the base algorithm performs a full tree traversal on each iteration, Monte Carlo CFR (MCCFR) reduces the per iteration time cost by traversing just a sampled portion of the tree. On the other hand, MCCFR's sampled values introduce variance, and the effects of this variance were previously unknown. In this paper, we generalize MCCFR by considering any generic estimator of the sought values. We show that any choice of an estimator can be used to probabilistically minimize regret, provided the estimator is bounded and unbiased. In addition, we relate the variance of the estimator to the convergence rate of an algorithm that calculates regret directly from the estimator. We demonstrate the application of our analysis by defining a new bounded, unbiased estimator with empirically lower variance than MCCFR estimates. Finally, we use this estimator in a new sampling algorithm to compute approximate equilibria in Goofspiel, Bluff, and Texas hold'em poker. Under each of our selected sampling schemes, our new algorithm converges faster than MCCFR.

In AI research, mechanism design is typically used to allocate tasks and resources to agents holding private information about their values for possible allocations. In this context, optimizing payments within the Groves class has recently received much attention, mostly under the assumption that agent's private information is single-dimensional. Our work tackles this problem in multi-parameter domains. Specifically, we develop a generic technique to look for a best Groves mechanism for any given mechanism design problem. Our method is based on partitioning the spaces of agent values and payment functions into regions, on each of which we are able to define a feasible linear payment function. Under certain geometric conditions on partitions of the two spaces this function is optimal. We illustrate our method by applying it to the problem of allocating heterogeneous items.

We analyze symmetric protocols to rationally coordinate on an asymmetric, efficient allocation in an infinitely repeated N-agent, C-resource allocation problems. (Bhaskar 2000) proposed one way to achieve this in 2-agent, 1-resource allocation games: Agents start by symmetrically randomizing their actions, and as soon as they each choose different actions, they start to follow a potentially asymmetric "convention" that prescribes their actions from then on. We extend the concept of convention to the general case of infinitely repeated resource allocation games with N agents and C resources. We show that for any convention, there exists a symmetric subgame perfect equilibrium which implements it. We present two conventions: bourgeois, where agents stick to the first allocation; and market, where agents pay for the use of resources, and observe a global coordination signal which allows them to alternate between different allocations. We define price of anonymity of a convention as the ratio between the maximum social payoff of any (asymmetric) strategy profile and the expected social payoff of the convention. We show that while the price of anonymity of the bourgeois convention is infinite, the market convention decreases this price by reducing the conflict between the agents.

In cooperative games, a key question is to find a division of payoffs to coalition members in a fair manner. Nucleolus is one of such solution concepts that provides a stable solution for the grand coalition. We study the computation of the nucleolus of a number of cooperative games, including fractional matching games and fractional edge cover games on general weighted graphs, as well as vertex cover games and clique games on weighted bipartite graphs. Our results are on the positive side---we give efficient algorithms to compute the nucleolus, as well as the least core, of all of these games.

In many real-world auctions, a bidder does not know her exact value for an item, but can perform a costly deliberation to reduce her uncertainty. Relatively little is known about such deliberative environments, which are fundamentally different from classical auction environments. In this paper, we propose a new approach that allows us to leverage classical revenue-maximization results in deliberative environments. In particular, we use Myerson (1981) to construct the first non-trivial (i.e., dependent on deliberation costs) upper bound on revenue in deliberative auctions. This bound allows us to apply existing results in the classical environment to a deliberative environment. In addition, we show that in many deliberative environments the only optimal dominant-strategy mechanisms take the form of sequential posted-price auctions.

We join the goals of two giant and related fields of research in group decision-making that have historically had little contact: fair division, and efficient mechanism design with monetary payments. To do this we adopt the standard mechanism design paradigm where utility is assumed to be quasilinear and thus transferable across agents. We generalize the traditional binary criteria of envy-freeness, proportionality, and efficiency (welfare) to measures of degree that range between 0 and 1. We demonstrate that in the canonical fair division settings under any allocatively-efficient mechanism the worst-case welfare rate is 0 and disproportionality rate is 1; in other words, the worst-case results are as bad as possible. This strongly motivates an average-case analysis. We then set as the goal identification of a mechanism that achieves high welfare, low envy, and low disproportionality in expectation across a spectrum of fair division settings. We establish that the VCG mechanism is not a satisfactory candidate, but the redistribution mechanism of [Bailey, 1997; Cavallo, 2006] is.

We extend work by Christian et al. [Review of Economic Design 2007] on lobbying in multiple referenda by first providing a more fine-grained analysis of the computational complexity of the NP-complete Lobbying problem. Herein, given a binary matrix, the columns represent issues to vote on and the rows correspond to voters making a binary vote on each issue. An issue is approved if a majority of votes has a 1 in the corresponding column. The goal is to get all issues approved by modifying a minimum number of rows to all-1-rows. In our multivariate complexity analysis, we present a more holistic view on the nature of the computational complexity of Lobbying, providing both (parameterized) tractability and intractability results, depending on various problem parameterizations to be adopted. Moreover, we show non-existence results concerning efficient and effective preprocessing for Lobbying and introduce natural variants such as Restricted Lobbying and Partial Lobbying.

We consider the problem of selecting fair divisions of a heterogeneous divisible good among a set of agents. Recent work (Cohler et al., AAAI 2011) focused on designing algorithms for computing maxsum—social welfare maximizing—allocations under the fairness notion of envy-freeness. Maxsum allocations can also be found under alternative notions such as equitability. In this paper, we examine the properties of these allocations. In particular, We provide conditions for when maxsum envy-free or equitable allocations are Pareto optimal and give examples where fairness with Pareto optimality is not possible. We also prove that maxsum envy-free allocations have weakly greater welfare than maxsum equitable allocations when agents have structured valuations, and we derive an approximate version of this inequality for general valuations.

Distance rationalizability is an intuitive paradigm for developing and studying voting rules: given a notion of consensus and a distance function on preference profiles, a rationalizable voting rule selects an alternative that is closest to being a consensus winner. Despite its appeal, distance rationalizability faces the challenge of connecting the chosen distance measure and consensus notion to an operational measure of social desirability. We tackle this issue via the decision-theoretic framework of dynamic social choice, in which a social choice Markov decision process (MDP) models the dynamics of voter preferences in response to winner selection. We show that, for a prominent class of distance functions, one can construct a social choice MDP, with natural preference dynamics and rewards, such that a voting rule is (votewise) rationalizable with respect to the unanimity consensus for a given distance function iff it is a (deterministic) optimal policy in the MDP. This provides an alternative rationale for distance rationalizability, demonstrating the equivalence of rationalizable voting rules in a static sense and winner selection to maximize societal utility in a dynamic process.