Risk-Sensitive Mean-Field-Type Games with Lp-norm Drifts

We study how risk-sensitive players act in situations where the outcome is influenced not only by the state-action profile but also by the distribution of it. In such interactive decision-making problems, the classical mean-field game framework does not apply. We depart from most of the mean-field games literature by presuming that a decision-maker may include its own-state distribution in its decision. This leads to the class of mean-field-type games. In mean-field-type situations, a single decision-maker may have a big impact on the mean-field terms for which new type of optimality equations are derived. We establish a finite dimensional stochastic maximum principle for mean-field-type games where the drift functions have a p-norm structure which weaken the classical Lipschitz and differentiability assumptions. Sufficient optimality equations are established via Dynamic Programming Principle but in infinite dimension. Using de Finetti-Hewitt-Savage theorem, we show that a propagation of chaos property with 'virtual' particles holds for the non-linear McKean-Vlasov dynamics.Comment: 37 pages, 8 figures. to appear in Automatica 201

Non-Asymptotic Mean-Field Games

Mean-field games have been studied under the assumption of very large number of players. For such large systems, the basic idea consists to approximate large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications. However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networks with few entities. In this paper we propose a mean-field framework that is suitable not only for large systems but also for a small world with few number of entities. The applicability of the proposed framework is illustrated through various examples including dynamic auction with asymmetric valuation distributions, and spiteful bidders.Comment: 36 pages, 2 figures. Accepted and to appear in IEEE Transactions on Systems, Man, and Cybernetics, Part B 201

Lowest Unique Bid Auctions with Resubmission Opportunities

The recent online platforms propose multiple items for bidding. The state of the art, however, is limited to the analysis of one item auction without resubmission. In this paper we study multi-item lowest unique bid auctions (LUBA) with resubmission in discrete bid spaces under budget constraints. We show that the game does not have pure Bayes-Nash equilibria (except in very special cases). However, at least one mixed Bayes-Nash equilibria exists for arbitrary number of bidders and items. The equilibrium is explicitly computed for two-bidder setup with resubmission possibilities. In the general setting we propose a distributed strategic learning algorithm to approximate equilibria. Computer simulations indicate that the error quickly decays in few number of steps. When the number of bidders per item follows a Poisson distribution, it is shown that the seller can get a non-negligible revenue on several items, and hence making a partial revelation of the true value of the items. Finally, the attitude of the bidders towards the risk is considered. In contrast to risk-neutral agents who bids very small values, the cumulative distribution and the bidding support of risk-sensitive agents are more distributed.Comment: 47 pages, 13 figure

D-modules and complex foliations

Consider a complex analytic manifold $X$ and a coherent Lie subalgebra \shi of the Lie algebra of complex vector fields on $X$. By using a natural \shd_X-module \shm_\shi naturally associated to \shi and the ring (in the derived sense) \rhom[\shd_X](\shm_\shi,\shm_\shi), we associate integers which measure the irregularity of the foliation associated with \shi.Comment: 10 page

Empathy in Bimatrix Games

Although the definition of what empathetic preferences exactly are is still evolving, there is a general consensus in the psychology, science and engineering communities that the evolution toward players' behaviors in interactive decision-making problems will be accompanied by the exploitation of their empathy, sympathy, compassion, antipathy, spitefulness, selfishness, altruism, and self-abnegating states in the payoffs. In this article, we study one-shot bimatrix games from a psychological game theory viewpoint. A new empathetic payoff model is calculated to fit empirical observations and both pure and mixed equilibria are investigated. For a realized empathy structure, the bimatrix game is categorized among four generic class of games. Number of interesting results are derived. A notable level of involvement can be observed in the empathetic one-shot game compared the non-empathetic one and this holds even for games with dominated strategies. Partial altruism can help in breaking symmetry, in reducing payoff-inequality and in selecting social welfare and more efficient outcomes. By contrast, partial spite and self-abnegating may worsen payoff equity. Empathetic evolutionary game dynamics are introduced to capture the resulting empathetic evolutionarily stable strategies under wide range of revision protocols including Brown-von Neumann-Nash, Smith, imitation, replicator, and hybrid dynamics. Finally, mutual support and Berge solution are investigated and their connection with empathetic preferences are established. We show that pure altruism is logically inconsistent, only by balancing it with some partial selfishness does it create a consistent psychology.Comment: 24 pages, 9 figure

Quantile-based Mean-Field Games with Common Noise

In this paper we explore the impact of quantiles on optimal strategies under state dynamics driven by both individual noise, common noise and Poisson jumps. We first establish an optimality system satisfied the quantile process under jump terms. We then turn to investigate a new class of finite horizon mean-field games with common noise in which the payoff functional and the state dynamics are dependent not only on the state-action pair but also on conditional quantiles. Based on the best-response of the decision-makers, it is shown that the equilibrium conditional quantile process satisfies a stochastic partial differential equation in the non-degenerate case. A closed-form expression of the quantile process is provided in a basic Ornstein-Uhlenbeck process with common noise.Comment: 24 pages, 3 figure

Heterogeneous Learning in Zero-Sum Stochastic Games with Incomplete Information

Learning algorithms are essential for the applications of game theory in a networking environment. In dynamic and decentralized settings where the traffic, topology and channel states may vary over time and the communication between agents is impractical, it is important to formulate and study games of incomplete information and fully distributed learning algorithms which for each agent requires a minimal amount of information regarding the remaining agents. In this paper, we address this major challenge and introduce heterogeneous learning schemes in which each agent adopts a distinct learning pattern in the context of games with incomplete information. We use stochastic approximation techniques to show that the heterogeneous learning schemes can be studied in terms of their deterministic ordinary differential equation (ODE) counterparts. Depending on the learning rates of the players, these ODEs could be different from the standard replicator dynamics, (myopic) best response (BR) dynamics, logit dynamics, and fictitious play dynamics. We apply the results to a class of security games in which the attacker and the defender adopt different learning schemes due to differences in their rationality levels and the information they acquire

Mean-Field Learning: a Survey

In this paper we study iterative procedures for stationary equilibria in games with large number of players. Most of learning algorithms for games with continuous action spaces are limited to strict contraction best reply maps in which the Banach-Picard iteration converges with geometrical convergence rate. When the best reply map is not a contraction, Ishikawa-based learning is proposed. The algorithm is shown to behave well for Lipschitz continuous and pseudo-contractive maps. However, the convergence rate is still unsatisfactory. Several acceleration techniques are presented. We explain how cognitive users can improve the convergence rate based only on few number of measurements. The methodology provides nice properties in mean field games where the payoff function depends only on own-action and the mean of the mean-field (first moment mean-field games). A learning framework that exploits the structure of such games, called, mean-field learning, is proposed. The proposed mean-field learning framework is suitable not only for games but also for non-convex global optimization problems. Then, we introduce mean-field learning without feedback and examine the convergence to equilibria in beauty contest games, which have interesting applications in financial markets. Finally, we provide a fully distributed mean-field learning and its speedup versions for satisfactory solution in wireless networks. We illustrate the convergence rate improvement with numerical examples.Comment: 36 pages. 5 figures. survey styl

A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the value-function imposed in Lim and Zhou (2005) need not to be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.Comment: 20 pages, 2 figure

Optimal control and zero-sum games for Markov chains of mean-field type

We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the Total Variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.Comment: arXiv admin note: text overlap with arXiv:1603.0607