Ad auctions and cascade model: GSP inefficiency and algorithms

The design of the best economic mechanism for Sponsored Search Auctions (SSAs) is a central task in computational mechanism design/game theory. Two open questions concern the adoption of user models more accurate than that one currently used and the choice between Generalized Second Price auction (GSP) and Vickrey-Clark-Groves mechanism (VCG). In this paper, we provide some contributions to answer these questions. We study Price of Anarchy (PoA) and Price of Stability (PoS) over social welfare and auctioneer's revenue of GSP w.r.t. the VCG when the users follow the famous cascade model. Furthermore, we provide exact, randomized, and approximate algorithms, showing that in real-world settings (Yahoo! Webscope A3 dataset, 10 available slots) optimal allocations can be found in less than 1s with up to 1000 ads, and can be approximated in less than 20ms even with more than 1000 ads with an average accuracy greater than 99%.Comment: AAAI16, to appea

Extensive-Form Perfect Equilibrium Computation in Two-Player Games

We study the problem of computing an Extensive-Form Perfect Equilibrium (EFPE) in 2-player games. This equilibrium concept refines the Nash equilibrium requiring resilience w.r.t. a specific vanishing perturbation (representing mistakes of the players at each decision node). The scientific challenge is intrinsic to the EFPE definition: it requires a perturbation over the agent form, but the agent form is computationally inefficient, due to the presence of highly nonlinear constraints. We show that the sequence form can be exploited in a non-trivial way and that, for general-sum games, finding an EFPE is equivalent to solving a suitably perturbed linear complementarity problem. We prove that Lemke's algorithm can be applied, showing that computing an EFPE is $\textsf{PPAD}$-complete. In the notable case of zero-sum games, the problem is in $\textsf{FP}$ and can be solved by linear programming. Our algorithms also allow one to find a Nash equilibrium when players cannot perfectly control their moves, being subject to a given execution uncertainty, as is the case in most realistic physical settings.Comment: To appear in AAAI 1

Smoothing Method for Approximate Extensive-Form Perfect Equilibrium

Nash equilibrium is a popular solution concept for solving imperfect-information games in practice. However, it has a major drawback: it does not preclude suboptimal play in branches of the game tree that are not reached in equilibrium. Equilibrium refinements can mend this issue, but have experienced little practical adoption. This is largely due to a lack of scalable algorithms. Sparse iterative methods, in particular first-order methods, are known to be among the most effective algorithms for computing Nash equilibria in large-scale two-player zero-sum extensive-form games. In this paper, we provide, to our knowledge, the first extension of these methods to equilibrium refinements. We develop a smoothing approach for behavioral perturbations of the convex polytope that encompasses the strategy spaces of players in an extensive-form game. This enables one to compute an approximate variant of extensive-form perfect equilibria. Experiments show that our smoothing approach leads to solutions with dramatically stronger strategies at information sets that are reached with low probability in approximate Nash equilibria, while retaining the overall convergence rate associated with fast algorithms for Nash equilibrium. This has benefits both in approximate equilibrium finding (such approximation is necessary in practice in large games) where some probabilities are low while possibly heading toward zero in the limit, and exact equilibrium computation where the low probabilities are actually zero.Comment: Published at IJCAI 1

Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games

Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework enables a new proof of CFR even in the known setting, which is derived from a perspective of decomposing polytope regret, thereby leading to an arguably simpler interpretation of the algorithm. Our generalization to convex compact sets and convex losses allows us to develop new algorithms for several problems: regularized sequential decision making, regularized Nash equilibria in extensive-form games, and computing approximate extensive-form perfect equilibria. Our generalization also leads to the first regret-minimization algorithm for computing reduced-normal-form quantal response equilibria based on minimizing local regrets. Experiments show that our framework leads to algorithms that scale at a rate comparable to the fastest variants of counterfactual regret minimization for computing Nash equilibrium, and therefore our approach leads to the first algorithm for computing quantal response equilibria in extremely large games. Finally we show that our framework enables a new kind of scalable opponent exploitation approach

Decoding Hidden Markov Models Faster Than Viterbi Via Online Matrix-Vector (max, +)-Multiplication

In this paper, we present a novel algorithm for the maximum a posteriori decoding (MAPD) of time-homogeneous Hidden Markov Models (HMM), improving the worst-case running time of the classical Viterbi algorithm by a logarithmic factor. In our approach, we interpret the Viterbi algorithm as a repeated computation of matrix-vector $(\max, +)$-multiplications. On time-homogeneous HMMs, this computation is online: a matrix, known in advance, has to be multiplied with several vectors revealed one at a time. Our main contribution is an algorithm solving this version of matrix-vector $(\max,+)$-multiplication in subquadratic time, by performing a polynomial preprocessing of the matrix. Employing this fast multiplication algorithm, we solve the MAPD problem in $O(mn^2/ \log n)$ time for any time-homogeneous HMM of size $n$ and observation sequence of length $m$, with an extra polynomial preprocessing cost negligible for $m > n$. To the best of our knowledge, this is the first algorithm for the MAPD problem requiring subquadratic time per observation, under the only assumption -- usually verified in practice -- that the transition probability matrix does not change with time.Comment: AAAI 2016, to appea

Operation Frames and Clubs in Kidney Exchange

A kidney exchange is a centrally-administered barter market where patients swap their willing yet incompatible donors. Modern kidney exchanges use 2-cycles, 3-cycles, and chains initiated by non-directed donors (altruists who are willing to give a kidney to anyone) as the means for swapping. We propose significant generalizations to kidney exchange. We allow more than one donor to donate in exchange for their desired patient receiving a kidney. We also allow for the possibility of a donor willing to donate if any of a number of patients receive kidneys. Furthermore, we combine these notions and generalize them. The generalization is to exchange among organ clubs, where a club is willing to donate organs outside the club if and only if the club receives organs from outside the club according to given specifications. We prove that unlike in the standard model, the uncapped clearing problem is NP-complete. We also present the notion of operation frames that can be used to sequence the operations across batches, and present integer programming formulations for the market clearing problems for these new types of organ exchanges. Experiments show that in the single-donation setting, operation frames improve planning by 34%--51%. Allowing up to two donors to donate in exchange for one kidney donated to their designated patient yields a further increase in social welfare.Comment: Published at IJCAI-1