Algorithmic Game Theory Nicola Gatti and Marcello Restelli

Algorithmic Game Theory Nicola Gatti and Marcello Restelli {ngatti, restelli}@elet. polimi. it DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

Scientific Areas 2 The object of the study Situations in which selfish players interact, each aiming at maximizing its (expected) revenue • • • Game Theory • It studies strategic situations with rational and fully-informed players • Given a game protocol, the aim is to find players’ optimal strategies • Solution concepts: Nash equilibrium and refinements • No solving method (except for backward induction) Evolutionary Game Theory • It studies populations that evolve during time (no hypothesis of rationality and full information is made), playing repeatedly the game • Solution concepts: Evolutionary Stable Strategies (ESS) • Dynamical system analysis Algorithmic Game Theory • Algorithms for finding game theoretic solutions, usually based on operative research (e. g. simplex method, linear complementarity problem, heuristics) Multiagent Learning • It studies strategic situations with e–greedy and non-fully-informed players (players learn by exploring and exploiting) • Players repeatedly play the game • Algorithms for learning the optimal strategies Mechanism Design • It is the reverse of game theory: given players’ strategies searches for the protocol such that those strategies are optimal Nicola Gatti, Marcello Restelli

Intersections between Areas Game Theory 3 Algorithmic Game Theory Mechanism Design Evolutionary Game Theory Nicola Gatti, Marcello Restelli Multiagent Learning

Course Organization (1) • N. Gatti (10 hours) • Game theory groundings • • Modeling a game Game classes Non-equilibrium solutions Equilibrium concepts • • Solving a zero-sum strategic-form game Solving a general-sum two-player strategic-form game Computational issues in solving games Solving a general-sum two-player extensive-form game • Algorithms for basic solutions • Illustrations • Negotiations • Strategic patrolling • Selfish routing • Research directions Nicola Gatti, Marcello Restelli 4

Course Organization (2) • M. Restelli (10 hours) • Multi-agent learning • Reinforcement learning • Differences with single-agent learning • Differences with game-theoretical approaches • Equilibrium learning • Zero-sum games • Coordination games • General-sum games • Best response learning • Fictitious play • Independent learning • No regret learning • Learning to coordinate • Optimistic approaches • Collective intelligence • Evolutionary game theory • Evolutionary stable strategies • Replicator dynamics Nicola Gatti, Marcello Restelli 5

Examples of Strategic Settings: Bilateral Negotiations 6 • e. Commerce settings: electronic marketplace (e. g. , e. Bay) with • A software entity (agent) that sells items (e. g. , food) • A software agent that buys items • The items have some cost for the seller, say RPs • The buyer have a maximum budget, say RPb • The difference between (RPb – RPb) is a surplus produced by the transaction • What is the split of the surplus that is optimal for agents Nicola Gatti, Marcello Restelli

Examples of Strategic Settings: Web-Service Pricing 7 • e. Commerce settings: electronic marketplace (e. g. , e. Bay) with • A provider that sells services • Some customers that buy services • Services are characterized by price p and response time r • Customers are in competition with respect to the purchase of the services with the minimal response time • Agreements are pairs (p, r) • Extensions: • More providers are in competition with respect to the sale of services Nicola Gatti, Marcello Restelli

Examples of Strategic Settings: Strategic Patrolling Nicola Gatti, Marcello Restelli 8

Examples of Strategic Settings: Selfish Routing Sink Source Nicola Gatti, Marcello Restelli 9

10 And now, we can start! Nicola Gatti, Marcello Restelli