# Mechanism Design without Money Lecture 1 Avinatan Hassidim.

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Mechanism Design without Money Lecture 1 Avinatan Hassidim

Game Theory

Mechanism design Engineering meets game theory How do you design a game, such that: 1.Players will be happy 2.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… Rows/ ColumnsC1C2C3 R11 / 4-1 / 62 / 7 R24 / 33 / -23 / 4

Notation Vector (profile) of strategies: s, or . That is s = (s 1,..., s n ) Player i’s utility when s is played is denoted U i (s) Suppose we want to state player’s i utility when all players play s, but instead of playing s i he plays . This is denoted as U i (s -i,  )

Practicing notation on the example Denote s = (R1, C1) U Rows (s) = 1 U Rows (s -Rows,R2) = 4 U Columns (R2,C2) = -2 Rows/ ColumnsC1C2C3 R11 / 4-1 / 62 / 7 R24 / 33 / -23 / 4

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 Rows/ ColumnsC1C2C3 R11 / 4-1 / 62 / 7 R24 / 33 / -23 / 4

Analysis continued Suppose player Columns plays C3. What will Rows do? – Play R1 So the outcome will be 2 / 7 Rows/ ColumnsC1C2C3 R11 / 4-1 / 62 / 7 R24 / 33 / -21 / 4

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 have U i (s -i,  i ) ≥ U i (s) We say  i is 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 s U i (s -i,  i ) ≥ U i (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  i will 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  i is 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 U i (s -i,  i ) ≥ U i (s -i,  i ) and U i (s -i,  i ) ≥ U i (s -i,  i ) gives U i (s -i,  i ) = U i (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 exam ובעברית – מי עשה קורס הכנה לפסיכומטרי ? This is just a (multiplayer) prisoner’s dilemma

פסיכומטרי Suppose there are n students A 1 …A n ranked A 1 >A 2 >…A n 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 A i 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.