Download Algorithmic Applications of Game Theory

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:
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:
Rows/ Columns
C1
C2
C3
R1
1 /4
-1 / 6
2/7
R2
4 /3
3 / -2
3 /4
• Same idea for more than two players…
Notation
• Vector (profile) of strategies: s, or . That is
s = (s1,..., sn)
• Player i’s utility when s is played is denoted
Ui(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 as
Ui(s-i, )
Practicing notation on the example
•
•
•
•
Rows/ Columns
C1
C2
C3
R1
1 /4
-1 / 6
2/7
R2
4 /3
3 / -2
3 /4
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
Rows/ Columns
C1
C2
C3
R1
1 /4
-1 / 6
2/7
R2
4 /3
3 / -2
3 /4
• No matter what Rows does, Columns is better
off with C3
Analysis continued
Rows/ Columns
C1
C2
C3
R1
1 /4
-1 / 6
2/7
R2
4 /3
3 / -2
1 /4
• 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 have
Ui(s-i,i) ≥ Ui(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
Ui(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 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
Ui(s-i, i) ≥ Ui(s-i, i)
and
Ui(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
exam
?‫ובעברית – 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.