Algorithmic Game Theory Nicola Gatti and Marcello Restelli {ngatti, DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

Nicola Gatti, Marcello Restelli 2 Scientific Areas 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  –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 3 Intersections between Areas Game Theory Algorithmic Game Theory Multiagent Learning Evolutionary Game Theory Mechanism Design

Nicola Gatti, Marcello Restelli 4 Course Organization (1) N. Gatti (10 hours) Game theory groundings Modeling a game Game classes Non-equilibrium solutions Equilibrium concepts Algorithms for basic solutions 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 Illustrations Negotiations Strategic patrolling Selfish routing Research directions

Nicola Gatti, Marcello Restelli 5 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 6 Examples of Strategic Settings: Bilateral Negotiations eCommerce settings: electronic marketplace (e.g., eBay) 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 RP s The buyer have a maximum budget, say RP b The difference between (RP b – RP b ) is a surplus produced by the transaction What is the split of the surplus that is optimal for agents

Nicola Gatti, Marcello Restelli 7 Examples of Strategic Settings: Web-Service Pricing eCommerce settings: electronic marketplace (e.g., eBay) 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 8 Examples of Strategic Settings: Strategic Patrolling

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

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