Periodic Reporting for period 2 - RAGT (Robust Algorithmic Game Theory) Reporting period: 2021-09-01 to 2023-02-28 Summary of the context and overall objectives of the project In what follows, we describe a few of the main focuses of the PI and team members of the RAGT ERC project, since its beginning. In the paper “On the Power and Limits of Dynamic Pricing in Combinatorial Markets” we consider the problem of resource allocation among heterogeneous agents, with the goal of maximizing social welfare. Specifically, we ask if maximizing welfare is possible by assigning resource prices in a dynamic and adaptive manner. We provide positive and negative results under different assumptions on the agents’ preferences. In the paper “Two-Price Equilibrium”, we study market equilibrium. The classical and one of the most appealing notions of market equilibrium is the Walrasian Equilibrium. In a Walrasian Equilibrium, all buyers are maximally happy with their set of items and the market clears. The downside of this notion is that it rarely exists. We introduce a new market equilibrium, Two-Price Equilibrium. It is a relaxation of Walrasian Equilibrium in which every item can have two prices. We show that in many markets in which a Walrasian equilibrium does not exist, a two-price equilibrium does exist. Moreover, we show an efficient way to find it. The paper “Almost Full EFX Exists for Four Agents” is concerned with the problem of discrete fair division, namely how to allocate a finite set of valuable items between different agents in a fair manner. Specifically, we consider the extensively studied fairness notion known as ``envy-freeness up to any good’’ (EFX), whereby an allocation is considered fair if no agent envies another following the removal of any item from the other agent’s bundle. We prove that EFX allocations exist for several different settings of interest.In the paper “A General Framework for Endowment Effects in Combinatorial Markets” we study the phenomenon of endowment effect in settings where there are multiple items. In particular, the paper tries to quantify the level of this effect that is needed in order to guarantee stable assignment of the items. In the two papers “Online Stochastic Max-Weight Matching: Prophet Inequality for Vertex and Edge Arrival Models”, and “General Graphs are Easier than Bipartite Graphs: Tight Bounds for Secretary Matching”, we study the problem of online matching in general graphs. An illustrating example is the problem of partitioning a class of students into pairs, where the students arrive online, and upon each arrival, one can measure the compatibility to former arriving students. We show how an online algorithm — one that makes assignments upon the arrival of a student, without knowing future arrivals — can approximate the best expected maximal matching. In the paper “Fair-Share Allocations for Agents with Arbitrary Entitlements”, we discuss the problem of fair allocations of a set of goods in cases where the agents have unequal entitlements to the goods. In particular, we suggest a new intuitive share that is inspired by the maximin share, and we present an algorithm that gives an allocation that approximates this share for every agent. In the two papers “Prophet Inequality with Competing Agents”, and “On a Competitive Secretary Problem with Deferred Selections”, we extend optimal stopping problems to the case of multiple agents. In particular, we consider settings where multiple agents want to purchase a house. Each time a house is presented to all of the agents with its price, and they need to decide whether to purchase it or not. We show how the presence of multiple agents influence the game, and how equilibria look in this multi-agent game. In the paper “Combinatorial Contracts”, we study the problem of how to incentivize people to work, in cases where we cannot directly observe their actions. An illustrating example is the case where a website owner wants to advertise his website, and hires an advertiser to do so. The website owner cannot see the actions the adviser takes, but can observe the traffic of the website. We present an algorithm that under some assumptions, calculates the best contract that maximizes the utility of the website owner. In the paper “On the Significance of Knowing the Arrival Order in Prophet Inequality”, we study the problem of measuring the importance of knowing the arrival order in online Bayesian stopping problems. We suggest a measurement that we call the order-competitive ratio, and we show how to use it to design better approximation algorithms to the more realistic benchmark of the best online algorithm (instead of the usual benchmark of the best offline algorithm). Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In “Two-Price Equilibrium” we introduce a new market equilibrium notion, called Two-Price Equilibrium. It is a relaxation of Walrasian equilibrium, where instead of a single price per item, every item has two prices. We define the discrepancy of a Two-Price Equilibrium which is a measure of distance from a Walrasian equilibrium and show a strong connection to the social welfare of the equilibrium. We use the discrepancy to prove efficiency guarantees for markets where no other equilibrium is guaranteed to exist. Additionally, we introduce novel techniques that provide new insights regarding valuation functions over identical items.In “Almost Full EFX Exists for Four Agents” we consider the discrete fair division problem that asks how to fairly distribute a finite set of items between agents with possibly different valuation functions. Specifically we consider the EFX notion of fairness, whereby an allocation is considered fair if for any two agents i,j, and any item g in agent j’s bundle, agent i does not prefer agent j’s bundle minus the item g over her own bundle. All of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations.In the paper “A General Framework for Endowment Effects in Combinatorial Markets” we generalize the framework of the paper “Combinatorial Auctions with Endowment Effect”, by Babaioff, Dobzinski and Oren, to arbitrary structures of endowments effects. This allows us to define a partial order over different endowment effects, and discuss the minimum requirements for an endowment effect to imply stability. This, in turn, allows us to show the existence of relatively small endowment effects that guarantee stability of agents with XOS valuations.In the two papers “Online Stochastic Max-Weight Matching: Prophet Inequality for Vertex and Edge Arrival Models”, and “General Graphs are Easier than Bipartite Graphs: Tight Bounds for Secretary Matching”, we generalize the online contention resolution scheme (OCRS) framework to batch arrival, and we show how to use it to construct prophet inequality algorithms for the constraint of matching. Along the way we show that surprisingly, for the secretary matching problem with general vertex arrival, one can achieve a better competitive ratio than 1/e.In the two papers “Prophet Inequality with Competing Agents”, and “On a Competitive Secretary Problem with Deferred Selections”, we combine ideas from two different fields, optimal stopping problems, and multi-agent games. The main theorem we show is that when the competition increases, then the online problem in which agents are allowed to select former arriving awards that were not selected by other agents, becomes more similar to the problem of a single agent that cannot select past awards.In the paper “Combinatorial Contracts”, we show how to use properties of valuations from the combinatorial auctions regime, to the combinatorial contracts regime. We show some key similarities, such as the immediate connection to demand queries, and use it to design optimal (respectively, almost optimal) contracts when the success probability of the underlying project is gross-substitute (respectively, monotone).In the paper “On the Significance of Knowing the Arrival Order in Prophet Inequality”, we propose an adaptive algorithm that can approximate the optimal online algorithm. The new algorithm is novel in that it uses one of two different thresholds. The first threshold works well for the case of an optimistic order of arrival, and the second threshold works well for the case of a pessimistic arrival order. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Tomer Ezra won the Deutsch Prize for excellent research in Tel Aviv University.Michal Feldman won the Kadar Award for Outstanding Research (Senior Category) (2022)Michal Feldman named Chair of Computation and Economics, Tel Aviv University (2021)Michal Feldman was selected to represent Israel in the World Laureates Forum (Young Scientists), Shanghai (2020)
