Introduction Social welfare maximization Revenue maximization Introduction to Mechanism Design Thodoris Lykouris National Technical University of Athens May 16, 2013 Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Contents Introduction Motivation Game Theory Mechanism Design Social welfare maximization Single-item auction VCG Mechanims Examples Revenue maximization Bayesian setting Optimal mechanism Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Algorithmic Game Theory 1. Theoretical Computer Science focuses on algorithms that are: ◮ fast ◮ (approximately) optimal Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Algorithmic Game Theory 1. Theoretical Computer Science focuses on algorithms that are: ◮ fast ◮ (approximately) optimal 2. Economics take into consideration people’s incentives Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Algorithmic Game Theory 1. Theoretical Computer Science focuses on algorithms that are: ◮ fast ◮ (approximately) optimal 2. Economics take into consideration people’s incentives Intersection of those two! ◮ Algorithms are usually run on people that have incentives and may not follow them if they can make things better for them by deviating (Algorithmic Game Theory) ◮ Thus we need to design the rules of the game so that we can deal with this (Mechanism Design) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments) ◮ Auctions Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Motivation Examples ◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments) ◮ Auctions (mechanism: in the next slides) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Prisoner’s dilemma Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Finite games ◮ n players ◮ Each player has a set of strategies S i ◮ Strategy profile is a vector of n strategies, one for each player: σ = ( σ 1 , σ 2 , . . . , σ n ) ◮ Pure strategy, when σ i is deterministically chosen from S i ◮ Mixed strategy, when σ i is a probability distribution on S i ◮ σ − i = ( σ 1 , σ 2 , . . . , σ i − 1 , σ i +1 , . . . , σ n ) ◮ For each player i , there is a payoff/utility function u i : S 1 × S 2 × · · · × S n → R Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Equilibria Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i , σ ′ i : u i ( σ ) ≥ u i ( σ − i , σ ′ i ) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Equilibria Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i , σ ′ i : u i ( σ ) ≥ u i ( σ − i , σ ′ i ) Dominant strategy equilibrium: A strategy profile in which every player’s strategy is at least as good as all other strategies, regardless of the actions of any other player ∀ i , σ ′ i , σ ′ − i : u i ( σ ′ − i , σ i ) ≥ u i ( σ ′ − i , σ ′ i ) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Equilibria Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i , σ ′ i : u i ( σ ) ≥ u i ( σ − i , σ ′ i ) Dominant strategy equilibrium: A strategy profile in which every player’s strategy is at least as good as all other strategies, regardless of the actions of any other player ∀ i , σ ′ i , σ ′ − i : u i ( σ ′ − i , σ i ) ≥ u i ( σ ′ − i , σ ′ i ) In Prisoner’s Dilemma, (Confess,Confess) is both a NE and a DSE Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Battle of the sexes ◮ Alice prefers going to ballet to going to soccer. ◮ Bob prefers going to soccer to going to ballet ◮ Both prefer doing anything together than doing anything alone. Alice \ Bob Ballet Soccer Ballet (10,3) (2,2) Soccer (0,0) (3,10) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Battle of the sexes ◮ Alice prefers going to ballet to going to soccer. ◮ Bob prefers going to soccer to going to ballet ◮ Both prefer doing anything together than doing anything alone. Alice \ Bob Ballet Soccer Ballet (10,3) (2,2) Soccer (0,0) (3,10) Here there are 2 NE [(Ballet, Ballet),(Soccer, Soccer)] but no DSE. Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Battle of the sexes (altered) Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10) Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Battle of the sexes (altered) Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10) Now there is just one NE (Soccer, Soccer). Introduction to Mechanism Design National Technical University of Athens
Introduction Social welfare maximization Revenue maximization Game Theory Battle of the sexes (altered) Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10) Now there is just one NE (Soccer, Soccer). ◮ Generally, in Mechanism Design, the designer (in this case, Bob) tries to change the game ◮ to maximize one objective function (in this case, Bob’s happiness). Introduction to Mechanism Design National Technical University of Athens
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