By incorporating regret minimization, double oracle methods have demonstrated rapid convergence to Nash Equilibrium (NE) in normal-form games and extensive-form games, through algorithms such as online double oracle (ODO) and extensive-form double oracle (XDO), respectively. In this study, we further examine the theoretical convergence rate and sample complexity of such regret minimization-based double oracle methods, utilizing a unified framework called Regret-Minimizing Double Oracle. Based on this framework, we extend ODO to extensive-form games and determine its sample complexity. Moreover, we demonstrate that the sample complexity of XDO can be exponential in the number of information sets $|S|$, owing to the exponentially decaying stopping threshold of restricted games. To solve this problem, we propose the Periodic Double Oracle (PDO) method, which has the lowest sample complexity among regret minimization-based double oracle methods, being only polynomial in $|S|$. Empirical evaluations on multiple poker and board games show that PDO achieves significantly faster convergence than previous double oracle algorithms and reaches a competitive level with state-of-the-art regret minimization methods.

Abstract The success of crowdsourcing projects relies critically on motivating a crowd to contribute. One particularly effective method for incentivising participants to perform tasks is to run contests where participants compete against each other for rewards. However, there are numerous ways to implement such contests in specific projects, that vary in how performance is evaluated, how participants are rewarded, and the sizes of the prizes. Also, the best way to implement contests in a particular project is still an open challenge, as the effectiveness of each contest implementation (henceforth, incentive) is unknown in advance. Hence, in a crowdsourcing project, a practical approach to maximise the overall utility of the requester (which can be measured by the total number of completed tasks or the quality of the task submissions) is to choose a set of incentives suggested by previous studies from the literature or from the requester’s experience. Then, an effective mechanism can be applied to automatically select appropriate incentives from this set over different time intervals so as to maximise the cumulative utility within a given financial budget and a time limit. To this end, we present a novel approach to this incentive selection problem. Specifically, we formalise it as an online decision making problem, where each action corresponds to offering a specific incentive. After that, we detail and evaluate a novel algorithm, , to solve the incentive selection problem efficiently and adaptively. In theory, in the case that all the estimates in (except the estimates of the effectiveness of each incentive) are correct, we show that the algorithm achieves the regret bound of $\mathcal {O}(\sqrt {B/c})$ O ( B / c ) , where B denotes the financial budget and c is the average cost of the incentives. In experiments, the performance of is about 93% (up to 98%) of the optimal solution and about 9% (up to 40%) better than state-of-the-art algorithms in a broad range of settings, which vary in budget sizes, time limits, numbers of incentives, values of the standard deviation of the incentives’ utilities, and group sizes of the contests (i.e., the numbers of participants in a contest).

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”.