Ex-post Nash Equilibrium Research Articles

Delay-Optimal Multi-User Wireless-Powered Relay Networks With Protection Against the Risk of Distributed Algorithm Manipulation

We consider simultaneous-wireless-information-and-power-transfer (SWIPT) in a relay network, where multiple users with stochastic traffic flows are serviced by a common wireless-powered relay. A two-phase forwarding protocol is deployed along with power splitting. In the first phase, the relay splits its received signal from sources into two parts: one for information decoding and the other for energy harvesting. In the second phase, the relay allocates the total harvested energy from the first phase to forward each user’s data. To optimize the system’s long-run delay performance, we formulate the relay’s power control problem as a Markov decision process (MDP) with per-stage (energy) budget constraints. We realize that the Bellman’s equations associated with the formulated MDP have a weakly-coupled structure, and assuming that users act as price-takers, the global optimum can be computed distributively via Lagrangian decomposition. We then abandon this assumption to face the more realistic case where the users are price-anticipating and may manipulate the rules of the distributed algorithm to favor themselves. To protect against the risk of unfaithful implementation, we design a dynamic economic mechanism that builds on the notions of transfer payments and periodic ex-post Nash equilibrium to disarm the users’ self-interest and ensure adherence to the algorithm.

Multi-Resource Allocation of Shared Energy Storage: A Distributed Combinatorial Auction Approach

This paper proposes a combinatorial auction approach for multi-resource allocation of an energy storage (ES) shared by multiple electricity end users in a residential community. Through the auction, a user buys a group of ES resources, including capacity, energy, charging power, and discharging power, from the ES operator. With the ES resources, users store grid energy during low-electricity-price hours, so that they can consume the cheap stored energy during high-electricity-price hours to reduce their electricity bills. In the auction, users submit their resource demands and corresponding bid prices, based on which the ES operator determines the winners and the final payments that the winners must pay. To solve the NP-hard winner determination problem, a fully polynomial time approximation scheme (FPTAS) is developed, which can optimize social welfare but may violate resource supply constraints. To deal with the constraint violation, the ES operator may buy extra energy outside the system to meet the winners’ actual demands. Further, a distributed implementation of the auction is designed to offload the auction computation onto the users while preventing the users from manipulating the auction outcomes in the course of computation. The proposed distributed auction can ensure that all users faithfully complete the assigned computation tasks in an ex-post Nash equilibrium. A real time-of-use (TOU) electricity tariff and actual home load data are used in the simulation, in which the proposed auction approach is evaluated in terms of social welfare and computational efficiency.

A Simple, Fast, and Safe Mediator for Congestion Management

Congestion is a severe problem in cities. A large population with little information about each other's preferences hardly reaches equilibrium and causes unexpected congestion. Controlling such congestion requires us to collect information dispersed in the market and to coordinate actions among agents. We aim to design a mediator that a) induces a game with high social welfare in equilibrium, b) computes an equilibrium efficiently, c) works without common prior, and d) performs well even when only some of the agents in the market use the mediator. We propose a mediator based on a version of best response dynamics (BRD). We prove that, in a simple setting with two resources, “good behavior” (reporting truthfully and following the recommendation) forms an (approximate) ex-post Nash equilibrium in the mediated game; in the equilibrium, the welfare is close to the first-best when preferences diverge enough. Furthermore, under a certain behavioral assumption, those who are not using the mediator can always enjoy non-negative payoff gain by joining it even without the full participation of others. Additionally, our experimental results suggest that such results remain valid for more general settings.

An Optimal ICO Mechanism

Initial Coin Offerings (ICOs) are raising billions in funding using multiple strategies, none justified from the point of view of mechanism design, resulting in severe underpricing and high volatility. In the present paper, an optimal ICO mechanism is proposed for the first time: a truthful multi-unit Vickrey-Dutch auction of callable tokens (i.e., a new hybrid security of tokens packaged with callable warrants). Truthful bidding is an ex-post Nash equilibrium strategy and the auction terminates with an ex-post efficient allocation; additionally, the callability of the warrants eliminates the winner’s curse of the auction and its underpricing. An implementation demonstrates its practical viability.

New Privacy-Preserving Ascending Auction for Assignment Problems

We introduce a new ascending auction that allocates heterogeneous objects among bidders with purely private unit demands. Our auction design differs from existing dynamic auctions in a number of ways: it economizes on information solicited from bidders by requiring marginal bidders to reveal a single new bid at a time; it uses a transparent price adjustment process; and it allows the seller to set starting prices above his reservation valuations. Despite these new features, (i) the auction stops in a finite time, (ii) sincere bidding is an ex-post Nash equilibrium, (iii) the auction ending prices and revenue depend only on bidders valuations and starting prices, and (iv) the auction is efficient if it starts with the seller’s valuations.

Maximizing Revenue with Limited Correlation: The Cost of Ex-Post Incentive Compatibility

In a landmark paper in the mechanism design literature, Cremer and McLean (1985) (CM for short) show that when a bidder’s valuation is correlated with an external signal, a monopolistic seller is able to extract the full social surplus as revenue. In the original paper and subsequent literature, the focus has been on ex-post incentive compatible (or IC) mechanisms, where truth telling is an ex-post Nash equilibrium. In this paper, we explore the implications of Bayesian versus ex-post IC in a correlated valuation setting. We generalize the full extraction result to settings that do not satisfy the assumptions of CM. In particular, we give necessary and sufficient conditions for full extraction that strictly relax the original conditions given in CM. These more general conditions characterize the situations under which requiring ex-post IC leads to a decrease in expected revenue relative to Bayesian IC. We also demonstrate that the expected revenue from the optimal ex-post IC mechanism guarantees at most a (|Θ| + 1)/4 approximation to that of a Bayesian IC mechanism, where |Θ| is the number of bidder types. Finally, using techniques from automated mechanism design, we show that, for randomly generated distributions, the average expected revenue achieved by Bayesian IC mechanisms is significantly larger than that for ex-post IC mechanisms.

A Simple Ascending Auction for Assignment Problems

We introduce a simple ascending auction that allocates heterogeneous objects among bidders with purely private unit demands. Our auction design differs from existing dynamic auctions in a number of ways: it solicits a single new bid from selected bidders at a time, thus minimizing bidder information revelation; it uses a simple and intuitive price adjustment procedure; the seller can set starting prices above his valuations. Despite these new features, (i) the auction stops in a finite time, (ii) sincere bidding at every stage of the auction is an ex-post Nash equilibrium, and (iii) for given valuations, the auction ending prices and revenue depend only on starting prices. We establish sincere bidding and path-independent ending prices using simple combinatorial arguments.

A near Pareto optimal auction with budget constraints

In a setup where a divisible good is to be allocated to a set of bidders with budget constraints, we introduce a mechanism in the spirit of the Vickrey auction. In the mechanism we propose, understating budgets or values is weakly dominated. Since the revenue is increasing in budgets and values, all kinds of equilibrium deviations from true valuations turn out to be beneficial to the auctioneer. We also show that ex-post Nash equilibrium of our mechanism is near Pareto optimal in the sense that all full winnersʼ values are above all full losersʼ values.

Approximately Optimal Mechanism Design Via Differential Privacy

We study the implementation challenge in an abstract interdependent values model and an arbitrary objective function. We design a mechanism that allows for approximate optimal implementation of insensitive objective functions in ex-post Nash equilibrium. If, furthermore, values are private then the same mechanism is strategy proof. We cast our results onto two specific models: pricing and facility location. The mechanism we design is optimal up to an additive factor of the order of magnitude of one over the square root of the number of agents and involves no utility transfers. Underlying our mechanism is a lottery between two auxiliary mechanisms — with high probability we actuate a mechanism that reduces players influence on the choice of the social alternative, while choosing the optimal outcome with high probability. This is where differential privacy is employed. With the complementary probability we actuate a mechanism that may be typically far from optimal but is incentive compatible. The joint mechanism inherits the desired properties from both.

Multi-Unit Auctions with Budget Constraints

Motivated by sponsored search auctions, we study multi-unit auctions with budget constraints. In the mechanism we propose, Sort-Cut, understating budgets or values is weakly dominated. Since Sort-Cut's revenue is increasing in budgets and values, all kinds of equilibrium deviations from true valuations turn out to be beneficial to the auctioneer. We show that the revenue of Sort-Cut can be an order of magnitude greater than that of the natural Market Clearing Price mechanism, and we discuss the efficiency properties of its ex-post Nash equilibrium.

On the robustness of non-linear personalized price combinatorial auctions

Though the VCG auction assumes a central place in the mechanism design literature, there are a number of reasons for favoring Iterative Combinatorial Auctions (ICAs). Several promising ICA formats were developed based on primal–dual and subgradient algorithms. Prices are interpreted as a feasible dual solution and the provisional allocation is interpreted as a feasible primal solution. iBundle(3), dVSV and Ascending Proxy Auction result in VCG payoffs when the coalitional value function satisfies buyer submodularity and bidders bid straightforward, which is an ex-post Nash equilibrium in this case. iBEA and CreditDebit auctions do not even require the buyer submodularity and achieve the same properties for general valuations. Often, however, one cannot assume straightforward bidding and it is not clear from the theory how these non-linear personalized price auctions (NLPPAs) perform in this case. Robustness of auctions with respect to different bidding behavior is a critical issue for any application. We conducted a large number of computational experiments to analyze the performance of NLPPAs with respect to different bidding strategies and valuation models. We compare NLPPAs with the VCG auction and with ICAs with linear prices, such as ALPS and the Combinatorial Clock Auction. While NLPPAs performed very well in case of straightforward bidding, we observe problems with revenue, efficiency, and speed of convergence when bidders deviate.

On purification of equilibrium in Bayesian games and expost Nash equilibrium

Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also expost stable. Through examples we demonstrate the ‘bounds obtained on the approximation’ are tight.

Incentive compatible resource allocation in concurrent design

This article presents an approach that a manager can use to allocate resources needed to design a system among the members of a concurrent design team. The system being designed is assumed to be composed of a number of subsystems, each designed by a different engineer. These engineers possess private information about the performance of their subsystem as a function of the design resources that they are allocated. This article shows how the manager of such a concurrent design project can induce rational self-interested engineers to reveal truthfully their private subsystem performance functions. This is accomplished through an incentive contract that ties each engineer’s pay to the contribution of their subsystem to the performance of the overall system. The approach builds from a Vickrey–Clarke–Groves mechanism to achieve, as an ex-post Nash equilibrium, truthful reporting of private subsystem performance functions by risk-neutral agents.