Bayes-Nash Equilibria in Generalized Second Price Auctions with Allocative Externalities

1 Mathematical Background.- Integration of Banach-Valued Correspondences.- Set-Valued Functions of Two Variables in Economic Theory.- 2 Equilibria, Core, and Pareto Optimality.- A Theorem on the Existence of Competitive Equilibria in a Market with a Finite Number of Agents and Whose Commodity Space is L?.- The Core of an Economy Without Ordered Preferences.- Fundamental Theorems of Welfare Economics in Infinite Dimensional Commodity Spaces.- 3 Core Equivalence.- A Limit Theorem on the Core of an Economy with a Continuum of Commodities.- An Equivalence Theorem for the Core of an Economy with Commodity Space L? ? ?(L?, L1).- The Principle of Equivalence.- 4 The Existence of an Equilibrium in Economies with a Continuum of Agents.- A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities.- Equilibria in Markets with a Continuum of Agents and Commodities.- What is Perfect Competition?.- 5 Correlated Equilibria.- On the Existence of Correlated Equilibria.- Existence of Correlated Weak Equilibria in Discontinuous Games.- Communication Equilibria with Large State Spaces.- 6 Games with a Continuum of Players.- An Axiomatic Approach to the Efficiency of Non-Cooperative Equilibrium in Economies with a Continuum of Traders.- On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atomless Game.- Equilibria in Random and Bayesian Games with a Continuum of Players.- 7 Sequential Equilibria.- Recursive Utility Under Uncertainty.- Consistency and Continuity of Choice in a Sequence of Spot and Futures Markets.- 8 References.

Static Bayesian games with finite fuzzy types and the existence of equilibrium

In sponsored search auctions, advertisers compete for a number of available advertisement slots of different quality. The auctioneer decides the allocation of advertisers to slots using bids provided by them. Since the advertisers may act strategically and submit their bids in order to maximize their individual objectives, such an auction naturally defines a strategic game among the advertisers. In order to quantify the efficiency of outcomes in generalized second price auctions, we study the corresponding games and present new bounds on their price of anarchy, improving the recent results of Paes Leme and Tardos [16] and Lucier and Paes Leme [13]. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria. Given the existing lower bounds, this bound denotes that the number of advertisers has almost no impact on the price of anarchy. The proof exploits the equilibrium conditions developed in [16] and follows by a detailed reasoning about the structure of equilibria and a novel relation of the price of anarchy to the objective value of a compact mathematical program. For more general equilibrium classes (i.e., mixed Nash, correlated, and coarse correlated equilibria), we present an upper bound of 2.310 on the price of anarchy. We also consider the setting where advertisers have incomplete information about their competitors and prove a price of anarchy upper bound of 3.037 over Bayes-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments.

Traditional mobile crowdsensing has shown a great potential to address large-scale data sensing problems by allocating sensing tasks to pervasive mobile users with smart devices. As crowdsensing service provider involves data collection from mobile users, the issue of providing rewards to incentivize mobile users is fundamentally important to ensure the sustainability of crowdsensing system. Recently, socially-aware crowdsensing services have been introduced as the integration of social networks and crowdsensing platforms. For example, in health-related crowdsensing applications, a mobile user benefits from information regarding food, exercise, medicine and medical treatment collected and shared by her socially-connected friends and family members. In this thesis, incentive mechanism design is revisited in the context of socially-aware crowdsensing, where the game theory tool is used to study the interactions between crowdsensing service provider and mobile users. In the first part of the thesis, incentive mechanism is designed by considering the underlying social network effects amid mobile social networks, for motivating the participants. Namely, one MU will obtain additional benefits from information contributed or shared by local neighbors in social networks. The process of rewarding and participating is modeled as a two-stage game, and backward induction is used to analyze the mobile users' participation level and the crowdsensing service provider's optimal reward mechanism. The analytical expressions are derived for the discriminatory incentive as well as the uniform incentive mechanisms. In the second part of the thesis, information asymmetry challenges are investigated. A Bayesian Stackelberg game with incomplete information is formulated to analyze the interaction between the crowdsensing service provider and mobile users, where the social structure information, i.e., the social network effects, among mobile users is uncertain. The existence and uniqueness of the Bayesian Stackelberg equilibrium is analytically validated by identifying the best response strategies of the mobile users. To be more general and practical, different with the former two parts, incentive mechanism in presence of multiple crowdsensing service providers is studied in the third part of the thesis. Understanding the behaviors of mobile users and service providers in socially-aware crowdsensing is of paramount importance for incentive mechanisms. With this focus, a multi-leader and multi-follower Stackelberg game approach is proposed to model the strategic interactions among service providers and mobile users, where the social influence of mobile users and the strategic interconnections of service providers are jointly and formally integrated into the game modeling. Through backward induction methods, the existence and uniqueness of the Stackelberg equilibrium are theoretically proved. In summary, this thesis addresses a few urgent challenging problems with the tools of game theory in the real implementation of incentive mechanism design in the context of socially-aware crowdsensing. Extensive numerical simulations are also conducted to illustrate some important properties of the equilibrium. Finally, several promising research directions are outlined for the future work.

In part I of this study we established two existence theorems for mixed-strategy equilibrium in games with discontinuous payoff functions (see Theorems 4(4*) and 5(5*) in Dasgupta and Maskin (1986); hereafter D-M (1986)). In this second part we use the second of these two theorems to prove the existence of mixed-strategy equilibrium in some wellknown discontinuous economic games that fail to have pure-strategy equilibria. The models we analyse are the Bertrand-Edgeworth example of price setting duopolists with capacity constraints (see Section 2.2); price competition among firms producing differentiated products (Hotelling (1929) and d'Aspremont, Gabszewicz and Thisse (1979); see Section 2.3); spatial competition (Eaton and Lipsey (1975) and Shaked (1975); see Section 3); and models of market dependent information (Rothschild and Stiglitz (1976) and Wilson (1977); Section 4). The paper concludes (Section 5) with a classification of these models.

We study games with intransitive preferences that admit skew-symmetric representations. We introduce the notion of surrogate better-reply security for discontinuous skew-symmetric games and elucidate the relationship between surrogate better-reply security and other security concepts in the literature. We then prove existence of behavioral strategy equilibrium for discontinuous skew-symmetric games of incomplete information (and, in particular, existence of mixed-strategy equilibrium for discontinuous skew-symmetric games of complete information), generalizing extant results.

Uncertainty Aversion and Equilibrium Existence in Games with Incomplete Information

In this paper, differential games with incomplete information, or Bayesian games, are formulated for multiagent continuous-time dynamical systems in a communication graph. These new Bayesian graphical games model the situation where the agents are uncertain about their actual payoff functions and must employ the evidence observed from the behavior of other agents to improve their estimation of the real setting of their environment. Two different solutions are proposed for the game. First, the agents play their best response strategies with respect to the policies of their neighbors; then, the agents prepare themselves for the worst-case neighbor policies. A tight relationship between the beliefs of an agent and its distributed control policy is established. Two belief update methodologies are proposed for the agents to review their strategies using information obtained from the graph topology.

The generalized second price auction used in sponsored search has been analyzed for models where bidders value clicks on ads. However, advertisers do not derive value only from clicks, nor do they value clicks in all slots equally. There is a need to understand sponsored search auctions in a setting with more general bidder valuations, in order to encompass realistic advertising objectives such as branding and conversions. We investigate the practical scenario where bidders have a full spectrum of values for slots, which are not necessarily proportional to the expected number of clicks received, and report a single scalar bid to the generalized second price auction. We show that there always exists an equilibrium corresponding to the VCG outcome using these full vector values, under monotonicity conditions on the valuations of bidders and clickthrough rates. Further, we discuss the problem of bidding strategies leading to such efficient equilibria: contrary to the case when bidders have one-dimensional types, bidding strategies with reasonable restrictions on bid values do not exist.

In this paper, we consider the problem of traffic offloading in a two-tier downlink smallcell network with hybrid access. We formulate a Stackelberg game with pre-offloading decision, which includes two steps, to study the problem. The utility of the MBS is designed as a piecewise function, corresponding to the two possible decisions of the MBS. We design a bonus-based incentive mechanism to motivate smallcell base stations (SBSs) serve the MUEs. We prove the existence and uniqueness of the Stackelberg equilibrium. We also consider the incomplete information game, where the SBSs have private information. A Bayesian Stackelberg game with pre-offloading decision is formulated and the existence of Bayesian Stackelberg equilibrium is proved. Our proposed approach achieves much higher utilities than the existing approach does under some circumstances in both cases, which proves the necessity and superiority of our proposed game structure.

Intuitive and noncompetitive equilibria in weakly efficient auctions with entry costs

Since it is difficult to guarantee the completeness of information in the market competition, it is of great significance to study the operation and optimization of multi-energy hub(EH) systems under incomplete information for the stability and economic operation of the system. However, under the dynamic energy pricing mechanism, each EH restricts each other, and the optimal scheduling of each EH is affected by the seriously coupled energy flow and incomplete information. Therefore this paper analyzes the relationship between EHs under the dynamic energy pricing mechanism, and uses Bayesian game to solve the optimal scheduling problem of multi-EH systems under the condition of incomplete information. According to the incomplete information of CHP operation mode in EHs, each EH is modeled as different types, and then the multi-EHs optimal scheduling model based on Bayesian game is established, and the Bayesian Nash equilibrium is solved. The effectiveness of the model and solution method is verified by the case.

In sponsored search auctions advertisers typically pay a fixed amount per click that their advertisements receive. In particular, the advertiser and the publisher enter into a contract (e.g., the publisher displays the ad; the advertiser pays the publisher 10 cents per click), and each party’s subjective value for such a contract depends on their estimated click-through rates (CTR) for the ad. Starting from this motivating example, we define and analyze a class of contract auctions that generalize the classical second price auction. As an application, we introduce impression-plus-click pricing for sponsored search, in which advertisers pay a fixed amount per impression plus an additional amount if their ad is clicked. Of note, when the advertiser’s estimated CTR is higher than the publisher’s estimated CTR, both parties find negative click payments advantageous, where the advertiser pays the publisher a premium for the impression but the publisher then pays the advertiser per click.KeywordsPrice SchemeValuation FunctionPrice AuctionContract SpaceAuction DesignThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

In this paper, we study applying honeypots to protect social networks against DDoS attacks. Different from previous works that study honeypots for DDoS attacks, we consider attackers are rational and know to optimize attacking strategies based on the defender’s strategy. To deal with such strategic attackers, we propose a novel pseudo honeypot game model following the Bayesian game setting. In addition, we rigorously prove the existence of Bayesian Nash equilibriums (BNEs) and show how to find them in all different cases. Simulations show the BNEs achieved in the games not only reduce energy consumption but also improve efficiency of the defense.