Mechanism Design without Money

Mechanism Design without Money Lecture 1 Avinatan Hassidim

Traditional computer science

Game Theory

Mechanism design • Engineering meets game theory • How do you design a game, such that: • Players will be happy • You can provably meet some goal?

Simple example • I have a pen, to give away to you. • Being your lecturer, I want to make us (me and you) as happy as possible • You can’t split the pen • Who do I give it to, to increase your happiness?

Assumptions about Happiness • Assumption: our happiness is the sum of happiness each one of you feels plus mine • Called Social Welfare • To maximize SW, we need to give the pen to the student who would maximally increase his or her happiness • Money transfers don’t change social welfare

Auction • Run an (ascending) auction for the pen. • The student who wins the auction, gets it, and pays the amount he should • Theorem: this maximizes social welfare

What if there is no money? • The winner can’t pay me • You can just go as high as you want in the auction • This will never end • Not clear who is the winner • Money was used to make us stand behind our words

Singing competition • We want to choose a singer • Each one gets how happy they are, with each singer chosen to be first and second • Each one gives a ranking on the singers. First name you say gets 5 points, second 4, etc.

Prediction • A set of agents (people) who are in a situation of conflict • Each agent has its own goals • Assumption – agents are rational + common knowledge of rationality • What will the agents do? • Nash equilibrium

Mechanism design examples • Auction theory • Ad auctions • Art auctions • Public projects • Dividing the rent between partners • Approximate solutions

Mechanism design without money • School choice • Labor markets • The match, הגרלת הסטאז' • Kidney exchange • Routing games

Administration • Lecture once a week, no recitation (TIRGUL) or homework • You need to be responsible and study (not just) before the test • Test in the end of the semester • Textbook: Algorithmic Game Theory by Nisan, Roughgarden, Tardos, and Vazirani • Also based on papers • Office hours on Thursday 9 am. Let me know if you are coming • I will have to skip a couple of classes, and will fill them another day \ Friday according to you

Let’s start from scratch

Games • Each player selects a strategy • Given the vector of strategies, each player gets a payoff • A game is summarized by the payoff matrix: • Same idea for more than two players…

Notation • Vector (profile) of strategies: s, or . That iss = (s1,..., sn) • Player i’s utility when s is played is denotedUi(s) • Suppose we want to state player’s i utility when all players play s, but instead of playing si he plays . This is denoted asUi(s-i, )

Practicing notation on the example • Denote s = (R1, C1) • URows(s) = 1 • URows(s-Rows,R2) = 4 • UColumns(R2,C2) = -2

But what will the players do? • I don’t know • We have a semester to talk about this • In some cases it’s obvious • No matter what Rows does, Columns is better off with C3

Analysis continued • Suppose player Columns plays C3. What will Rows do? • Play R1 • So the outcome will be 2 / 7

Dominant strategies • The last game was easy to analyze: no matter what Rows did, Columns played C3 • In this case we say that C3 is a Dominant strategy. • Formally: consider player i. If for any strategy profile s we haveUi(s-i,i) ≥ Ui(s)We say iis a dominant strategy for player i

Domination • A dominant strategy is the optimal action for a player i, no matter what the other players do. • Can we say that some strategy i is “better” than i even when i is not a dominant strategy? • We say that i dominates i if for every profile sUi(s-i, i) ≥ Ui(s-i, i)

Dominated strategies • We already know that if i is a dominant strategy we expect it will always be played. • Suppose i dominates i • Then we expect iwill never be played, since player i is always better off playing i • If for every other strategy i, we have that i dominates i then iis a dominant strategy

Relations between strategies • Suppose i dominates i. Can it be that i dominates i ? • Yes, but then player i is indifferent between them. Proof: • For every profile s we have Ui(s-i, i) ≥ Ui(s-i, i) andUi(s-i, i) ≥ Ui(s-i, i)gives Ui(s-i, i) = Ui(s-i, i) • Note that other players may get different utility if i plays i or I • In particular, player i can have multiple dominant strategies

Are dominant strategies an optimal predictor? • Well, only in theory • Think about chess • A strategy is what I will do in every board situation • Given white’s strategy and black’s strategy, the result is either white wins, black wins or tie • So in theory (and also in game theory), the game is “not interesting” and white will play a strategy which will let him always win or tie. • In practice (and taking a CS perspective) there is a computational question of finding the strategy…

Example – prisoner’s dilemma

Prisoner’s Dilemma is a theoretical concept with no real life interpretation • Show of hands: Please raise your hand if you did a preparation course for the psychometric examובעברית – מי עשה קורס הכנה לפסיכומטרי? • This is just a (multiplayer) prisoner’s dilemma

פסיכומטרי • Suppose there are n students A1…An ranked A1>A2>…An • If no one takes the course, the ranking is correct, and only the good students get to study CS. • No matter what the other students do, it’s dominant for Ai to take the course, and increase his chances of studying CS. • If all take the course, we get the same ranking again, but everyone wasted three months and a ton of money.

Mechanism Design without Money