Pingzhong Tang

Computer-aided proofs of Arrow's and other impossibility theorems

Arrows Impossibility Theorem is one of the landmark results in social choice theory. Over the years since the theorem was proved in 1950, quite a few alternative proofs have been put forward. In this paper, we propose yet another alternative proof of the theorem. The basic idea is to use induction to reduce the theorem to the base case with 3 alternatives and 2 agents and then use computers to verify the base case. This turns out to be an effective approach for proving other impossibility theorems such as Sens and Muller-Satterthwaites theorems as well. Furthermore, we believe this new proof opens an exciting prospect of using computers to discover similar impossibility or even possibility results.

Optimal mechanisms with simple menus

We consider revenue-optimal mechanism design for the case with one buyer and two items. The buyers valuations towards the two items are independent and additive. In this setting, optimal mechanism is unknown for general valuation distributions. We obtain two categories of structural results that shed light on the optimal mechanisms. These results can be summarized into one conclusion: under certain conditions, the optimal mechanisms have simple menus. The first category of results state that, under a centain condition, the optimal mechanism has a monotone menu. In other words, in the menu that represents the optimal mechanism, as payment increases, the allocation probabilities for both items increase simultaneously. This theorem complements Hart and Renys recent result regarding the nonmonotonicity of menu and revenue in multi-item settings. Applying this theorem, we derive a version of revenue monotonicity theorem that states stochastically superior distributions yield more revenue. Moreover, our theorem subsumes a previous result regarding sufficient conditions under which bundling is optimal[Hart and Nisan 2012]. The second category of results state that, under certain conditions, the optimal mechanisms have few menu items. Our first result in this category says that, for certain distributions, the optimal menu contains at most 4 items. The condition admits power (including uniform) density functions. Our second result in this category works for a weaker (hence more general) condition, under which the optimal menu contains at most 6 items. This condition is general enough to include a wide variety of density functions, such as exponential functions and any function whose Taylor series coefficients are nonnegative. Our last result in this category works for unit-demand setting. It states that, for uniform distributions, the optimal menu contains at most 5 items. All these results are in sharp contrast to Hart and Nisans recent result that finite-sized menu cannot guarantee any positive fraction of optimal revenue for correlated valuation distributions.

Designing competitions between teams of individuals

We consider a setting with two teams, each with a number of players. There is an ordering of all players that determines outcome of matches between any two players from the opposing teams. Neither the teams nor the competition designer know this ordering, but each team knows the derived ordering of strengths among its own players. Each team announces an ordering of its players, and the competition designer schedules matches according to the announced orderings. This setting in general allows for two types of manipulations by a team: Misreporting the strength ordering (lack of truthfulness), and deliberately losing a match (moral hazard). We prove necessary and sufficient conditions for a set of competition rules to have the properties that truthful reporting are dominant strategies and maximum effort in matches are Nash equilibrium strategies, and certain fairness conditions are met. Extensions of the original setting are discussed.

Two equivalence results for two-person strict games

A game is strict if for both players, different profiles have different payoffs. Two games are best response equivalent if their best response functions are the same. We prove that a two-person strict game has at most one pure Nash equilibrium if and only if it is best response equivalent to a strictly competitive game, and that it is best response equivalent to an ordinal potential game if and only if it is best response equivalent to a quasi-supermodular game.

Reinforcement mechanism design

We put forward a modeling and algorithmic framework to design and optimize mechanisms in dynamic industrial environments where a designer can make use of the data generated in the process to automatically improve future design. Our solution, coined reinforcement mechanism design, is rooted in game theory but incorporates recent AI techniques to get rid of nonrealistic modeling assumptions and to make automated optimization feasible. We instantiate our framework on the key application scenarios of Baidu and Taobao, two of the largest mobile app companies in China. For the Taobao case, our framework automatically designs mechanisms that allocate buyer impressions for the e-commerce website; for the Baidu case, our framework automatically designs dynamic reserve pricing schemes of advertisement auctions of the search engine. Experiments show that our solutions outperform the state-of-the-art alternatives and those currently deployed, under both scenarios.

Mechanism Design for Personalized Recommender Systems

Strategic behaviour from sellers on e-commerce websites, such as faking transactions and manipulating the recommendation scores through artificial reviews, have been among the most notorious obstacles that prevent websites from maximizing the efficiency of their recommendations. Previous approaches have focused almost exclusively on machine learning-related techniques to detect and penalize such behaviour. In this paper, we tackle the problem from a different perspective, using the approach of the field of mechanism design. We put forward a game model tailored for the setting at hand and aim to construct truthful mechanisms, i.e. mechanisms that do not provide incentives for dishonest reputation-augmenting actions, that guarantee good recommendations in the worst-case. For the setting with two agents, we propose a truthful mechanism that is optimal in terms of social efficiency. For the general case of m agents, we prove both lower and upper bound results on the effciency of truthful mechanisms and propose truthful mechanisms that yield significantly better results, when compared to an existing mechanism from a leading e-commerce site on real data.

Reinforcement Mechanism Design for e-commerce

We study the problem of allocating impressions to sellers in e-commerce websites, such as Amazon, eBay or Taobao, aiming to maximize the total revenue generated by the platform. We employ a general framework of reinforcement mechanism design, which uses deep reinforcement learning to design efficient algorithms, taking the strategic behaviour of the sellers into account. Specifically, we model the impression allocation problem as a Markov decision process, where the states encode the history of impressions, prices, transactions and generated revenue and the actions are the possible impression allocations in each round. To tackle the problem of continuity and high-dimensionality of states and actions, we adopt the ideas of the DDPG algorithm to design an actor-critic policy gradient algorithm which takes advantage of the problem domain in order to achieve convergence and stability. We evaluate our proposed algorithm, coined IA(GRU), by comparing it against DDPG, as well as several natural heuristics, under different rationality models for the sellers - we assume that sellers follow well-known no-regret type strategies which may vary in their degree of sophistication. We find that IA(GRU) outperforms all algorithms in terms of the total revenue.

The multi-shop ski rental problem

We consider the multi-shop ski rental problem. This problem generalizes the classic ski rental problem to a multi-shop setting, in which each shop has different prices for renting and purchasing a pair of skis, and a consumer has to make decisions on when and where to buy. We are interested in the optimal online (competitive-ratio minimizing) mixed strategy from the consumers perspective. For our problem in its basic form, we obtain exciting closed-form solutions and a linear time algorithm for computing them. We further demonstrate the generality of our approach by investigating three extensions of our basic problem, namely ones that consider costs incurred by entering a shop or switching to another shop. Our solutions to these problems suggest that the consumer must assign positive probability in exactly one shop at any buying time. Our results apply to many real-world applications, ranging from cost management in IaaS cloud to scheduling in distributed computing.

The Price of Prior Dependence in Auctions

In the standard form of mechanism design, a key assumption is that the designer has reliable information and technology to determine a prior distribution over types of the agents. In the meanwhile, as pointed out by the Wilsons Principle, a mechanism should rely as little as possible on the prior type distribution. In this paper, we put forward a simple model to formalize this statement. In our model, each agent has a true type distribution, according to which his type is drawn. In addition, the agent is able to commit to a fake type distribution and bids rationally as if his type were from the fake distribution (e.g., plays a Bayes equilibrium under the fake distributions). We investigate the equilibria of the induced distribution-reporting games among bidders, under the context of single-item auctions. We obtain several interesting findings: (1) the game induced by Myerson auction under our model is strategically equivalent to the first-price auction under the standard model. Consequently, the two games are revenue-equivalent. (2) the second-price auction, a well known prior independent auction, yields (weakly) more revenue than several reserve-based and virtual-value-based truthful, prior-dependent auctions, under our model. Our results complement the current literature which aims to show the superiority of prior-independent mechanisms.

Non-clairvoyant Dynamic Mechanism Design

Despite their better revenue and welfare guarantees for repeated auctions, dynamic mechanisms have not been widely adopted in practice. This is partly due to the complexity of their implementation as well as their unrealistic use of forecasting for future periods. We address these shortcomings and present a new family of dynamic mechanisms that are simple and require no distribution knowledge of future periods. This paper introduces the concept of non-clairvoyance in dynamic mechanism design, which is a measure-theoretic restriction on the information that the seller can use. A dynamic mechanism is non-clairvoyant if the allocation and pricing rule at each period does not depend on the type distributions in future periods. We develop a framework (bank account mechanisms) for characterizing, designing, and proving lower bounds for dynamic mechanisms (clairvoyant or non-clairvoyant). This framework is used to characterize the revenue extraction power of non-clairvoyant mechanisms with respect to mechanisms that are allowed unrestricted use of distributional knowledge.