Algorithmic Game Theory (Lecture 9: Beyond Quasi-Linearity)

This lecture concludes the course's mechanism design section. It explores mechanism design with payment constraints, moving beyond the quasi-linear utility model. Constraints include budget limits and scenarios with no money. While harder, this area yields significant results, particularly relevant to keyword auctions. The lecture introduces the clinching auction, a mechanism for identical goods with public budgets, and the top trading cycle algorithm for house allocation without money. Both mechanisms are dominant-strategy incentive compatible but face challenges in achieving optimal surplus. This segment explains the concept of budget constraints in mechanism design, where each player has a maximum amount they can pay. It contrasts this with the typical quasi-linear utility model and discusses how this constraint affects the design of mechanisms. This segment introduces the concept of constraints on payments in mechanism design, moving beyond the quasi-linear utility model. It highlights two types of constraints: budget constraints and situations where money is not used at all, emphasizing the increased difficulty but also the significance of these problems in mechanism design theory. This segment illustrates the practical relevance of budget constraints in keyword auctions. It explains how advertisers set both a value per click and a daily budget, highlighting the real-world application of budget-constrained mechanism design. This segment explains the concept of market clearing price with a simple example of two bidders and two goods, illustrating how the price is determined when bidders bid truthfully. It sets the stage for understanding the complexities introduced by strategic bidding. This segment introduces budget constraints as a third type of constraint in mechanism design, alongside incentive compatibility and allocation constraints. It explains how this new constraint limits the designer's power and makes mechanism design more challenging, leading to impossibility results. This segment uses a single-item auction example to demonstrate the limitations of the Vickrey auction (and VCG mechanism) when budget constraints are present. It shows that maximizing surplus in the ex-post sense, as achieved by the Vickrey auction, is no longer possible with budget constraints. This segment describes a simple way to incorporate budget constraints into the quasi-linear model. It introduces the concept of utility becoming minus infinity if a player is asked to pay more than their budget, and briefly mentions smoother versions of this model. This segment uses a concrete example to demonstrate how demand reduction, a strategic bidding behavior, can manipulate the outcome of a poorly designed auction. It highlights the difference in utility for a bidder employing this strategy versus bidding truthfully. This segment compares the clinching auction with the market clearing price auction, emphasizing the differences in price determination and how the clinching auction handles bidders' budgets and demands. It clarifies the role of residual budgets in demand calculation.This segment walks through the same example used earlier, but this time with truthful bids, showing how the clinching auction determines prices and allocations. It sets the stage for proving the mechanism's incentive compatibility.This segment presents a proof that the clinching auction is dominant strategy incentive compatible (DSIC). It analyzes the impact of truthful versus non-truthful bidding on a bidder's utility, demonstrating that truthful bidding is always optimal.This segment discusses the optimality of the clinching auction, acknowledging the challenges in defining appropriate benchmarks for comparison. It explores Pareto optimality as one such benchmark and discusses ongoing research to better understand the auction's strengths and limitations. This segment details the Top Trading Cycle (TTC) algorithm, a mechanism for allocating houses among agents with preferences, iteratively assigning houses based on agent choices and deleting assigned agents. The algorithm's iterative nature, cycle identification, and reallocation process are clearly explained, providing a comprehensive understanding of its functionality.This segment focuses on evaluating the TTC algorithm's properties. It establishes that the algorithm is individually rational (agents are not worse off), providing a proof sketch. The discussion lays the groundwork for understanding the algorithm's incentive compatibility.This segment delves into the incentive properties of the TTC algorithm, demonstrating its dominant-strategy incentive compatibility. The explanation connects the algorithm's design to the strategic behavior of agents, highlighting its robustness against manipulation. This segment introduces the clinching auction mechanism as a solution to the problems highlighted in previous segments. It explains the algorithm's core logic, focusing on how it allocates goods piecemeal at different prices and manages bidders' budgets. This segment compares the TTC algorithm to other mechanisms, emphasizing the richness of the preference space in the housing allocation problem. It introduces the concept of the core, a stronger version of incentive compatibility, and explains why the TTC algorithm's allocation is in the core, making it uniquely efficient and resistant to coalitional deviations.