COMPUTATIONALSOCIAL SCIENCE

SEMINAR

HEINRICH H. NAX (HNAX@ETHZ.CH)COSS, ETH ZURICH

SEE WWW.NAX.SCIENCE

TWO PARTSadmin some game

theory intro

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ADMIN

• Information about the course, • Updated materials/slides of speakers, • Program, links, etc

• will be made available at • https://coss.ethz.ch/education/computational.html

• ETH students: please contact me (Heinrich) under hnax@ethz.ch or Nino under nino.antulov@gess.ethz.ch with questions about the course!

• Information about the course, • materials/slides of speakers, • Links, etc

• will be made available at • https://coss.ethz.ch/education/computational.html

• ETH students: please contact me (Heinrich) under hnax@ethz.ch or Nino under nino.antulov@gess.ethz.ch with questions about the course!

The course is co-organized by

Nino Antulov-Fantulin (Data Science, Machine Learning)https://coss.ethz.ch/people/postdocs/nantulovfantulin.html

We are part of the chair ofDirk Helbing (COSS, ETH) … who is on sabbatical

• Computational Social Science aims at• bringing modeling and computer simulation of social processes and

phenomena together with related empirical, experimental, and data-driven work

• combining perspectives of different scientific disciplines (e.g. socio-physics, social, computer and complexity science)

• bridging between fundamental and applied work using game theory to model individual-level decision-making

+ me

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PRELIMINARY SCHEDULE…

THANKS TO SPEAKERS

“COURSEWORK”Please prepare, in “diverse” groups of 4 – 8 students (less is not encouraged, neither is more),

1. Highly polished presentation in 15-20 slides (which you mightbe asked to hold)

2. Accompanying notes/essay outlining your arguments ofthe presentation including at least one “data”component (max. 60 pages).

Topics: You choose! You may include one or a combination of issues raised during the course, may prove a good understanding of recurring topics covered during the week, combine several ideas, or propose fresh thoughts related to issues raised in the course,… in English or German.

Deadline: December 12!December 1: Propose a project with 1-2 slides, and finalize teams (including one team leader responsible for project delivery).

• A mix of brief introductory and research talks (hopefully exciting)• introducing concepts and methods• some review and research• new ideas

• Each talk will come with some discussion time and we encourage active participation and questioning (unless the speaker says otherwise) – i.e. questions or comments at any moment in time!

• Also use the breaks to form “teams”

YOUR MARKS

WWW.DVSN.APP

SHARING GAINS FROM A JOINT VENTURE

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THE BASIC PROBLEM• n people bake a cake together• the cake is worth 1 dollar

• how should it be split amongst the people?

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1/N?

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1/N?• What if some people did more than others?

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RELATIVE EFFORTS?• Who know this?• Who can verify it?

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CONSIDER THE SCENARIO• n people bake a cake together• the cake is worth 1 dollar• a third party holds it but has no idea of who did what• people submit proposals about how it should be split• the third party aggregates these proposals and pays

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HOW?

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HOW?

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• Please form teams of 4-5 people and think about this for 5 min

• Then every team will have 1 min to present their proposal

“THE SOLUTION”

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“THE SOLUTION”

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3 properties

1. Strategy-proof: requires that your own claim about yourself does not matter for your own share

2. Objective: requires that a partner be unable to affect the share of any other partner by reporting a different belief about his own contribution

3. Consensual: if there is a set of shares that is consistent with all of the inputs that the partners provide, the rule needs to assign those shares.

ILLUSTRATION OF THE FORMULA

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ILLUSTRATION OF THE FORMULA

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ILLUSTRATION OF THE FORMULA

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ILLUSTRATION OF THE FORMULA

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ILLUSTRATION OF THE FORMULA

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ILLUSTRATION OF THE FORMULA

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residual

ILLUSTRATION OF THE FORMULA

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residual

ILLUSTRATION OF THE FORMULA

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residuali

j

k

l

THE FORMULA

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Average relative contribution jk

Average RC jk without i’s opinion

Auxiliary function assigning share to i when j excluded

Final payment

share in the other slicesi’s residual in his slice

THINK ABOUT IT

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AND READ R THEN L THEN R

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EXAMPLE (MADE SIMPLE)

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L 33, 33, 33

ML 50, 25, 25

MR 50, 25, 25

R 50, 25, 25

OUTPUT – NOTE CONSENSUALITY MUST BITE HERE

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say\get L ML MR R

L 33 33 33

ML 50 25 25

MR 50 25 25

R 50 25 25

Gets 40 20 20 20

JUST AVERAGING WOULD GIVE

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say\get L ML MR R

L 33 33 33

ML 50 25 25

MR 50 25 25

R 50 25 25

Gets 50/133 28/133 28/133 28/133=37 =21 =21 =21

DVSN.APP

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THE SITUATION• E.g. 5 students do a course project together• the project gets –for example- a 5.5

• What should the individual marks be?

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5.5 TO ALL?

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5.5?• What if some people did more than others?

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RELATIVE EFFORTS?• Who know this? The examiner doesn’t.• Who can verify it? The examiner cannot.

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YOU DO!• You each specify what the contributions of everyone were

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I USE

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FINAL GRADE

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2 parts

1. The total is group size times group mark

2. Individual marks are based on the mutual evaluation exercise based on this mechanism by de Clippel et al. via www.DVSN.app

GOOD LUCK.

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SOME GAME THEORY

Lecture 1: Introduction

A game

Rules:

1 Players: All of you:

IKUuwQ2 Actions: Choose a number between 0 and 1003 Outcome: The player with the number closest to half the average of all

submitted numbers wins.

4 Payoffs: He/she will receive CHF, which I will pay out

right after the game.

5 In case of several winners, divide payment by number of winners andpay all winners.

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Lecture 1: Introduction

Others before you did

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Lecture 1: Introduction

Soziologisches Institut

“…It is not a case of choosing those [faces] that, to the best of one’s judgment, are really the prettiest, nor even those that average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the fourth, fifth and higher degrees.” (John Maynard Keynes, General Theory of Employment, Interest, and Money, 1936, p.156).

Analogy between stock markets and newspaper contest in which people guess what faces others will guess are most beautiful.

Lecture 1: Introduction

Lecture 1: Introduction

Soziologisches Institut

Diekmann, Andreas. "Rational choice, evolution and the “Beauty Contest”." Raymond Boudon. A Life in Sociology. Oxford: Bardwell (2009), p,.8 ff.

K=0K=1K=2K=3

Beliefs and learning

Lecture 1: Introduction

Level 0 (“no reasoning”)random guess or simple rules

Level 1 reacts to base strategy at level 0Guesses of 50 = 33Level 2 reacts to level 1Guesses of of 50 = 22Level k reacts to level k-1

Guesses ( ) k 0…

: Cognitive Hierarchy Theory

Soziologisches Institut

Diekmann, Andreas. "Rational choice, evolution and the “Beauty Contest”." Raymond Boudon. A Life in Sociology. Oxford: Bardwell (2009), p,.8 ff.

K=0K=1K=2K=3

Beliefs and learning

Soziologisches Institut

Diekmann, Andreas. "Rational choice, evolution and the “Beauty Contest”." Raymond Boudon. A Life in Sociology. Oxford: Bardwell (2009), p,.8 ff.

K=0K=1K=2K=3

Beliefs and learning

Soziologisches Institut

Diekmann, Andreas. "Rational choice, evolution and the “Beauty Contest”." Raymond Boudon. A Life in Sociology. Oxford: Bardwell (2009), p,.8 ff.

K=0K=1K=2K=3

Beliefs and learning

Soziologisches Institut

Diekmann, Andreas. "Rational choice, evolution and the “Beauty Contest”." Raymond Boudon. A Life in Sociology. Oxford: Bardwell (2009), p,.8 ff.

K=0K=1K=2K=3

Beliefs and learning

Soziologisches Institut

Bosch-Domènech et al. (2002, AEA)

Soziologisches Institut

Bosch-Domènech et al. (2002, AEA)

Lecture 1: Introduction

Lecture 1: Introduction

Acknowledgments

Bary Pradelski (ETHZ)

Peyton Young (Oxford, LSE)

Bernhard von Stengel (LSE)

Francoise Forges (Paris Dauphine)

Paul Duetting (LSE)

Jeff Shamma (Georgia Tech, KAUST)

Joergen Weibull (Stockholm, TSE)

Andreas Diekmann (ETHZ)

Dirk Helbing (ETHZ)

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Lecture 1: Introduction

Game theory

A tour of its people, applications and concepts

1 von Neumann

2 Nash

3 Aumann, Schelling, Selten, Shapley

4 Today

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Lecture 1: Introduction

John von Neumann (1903-1957)7 / 47

Lecture 1: Introduction

What is game theory?

A mathematical language to express models of, as Myerson says:

“conflict and cooperation between intelligent rational decision-makers”

In other words, interactive decision theory (Aumann)

Dates back to von Neumann & Morgenstern (1944)

Most important solution concept: the Nash (1950) equilibrium

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Lecture 1: Introduction

Games and Non-Games

What is a game? And what is not a game?

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Lecture 1: Introduction

Uses of game theory

Prescriptive agenda versus descriptive agenda

“Reverse game theory”/mechanism design:

“in a design problem, the goal function is the main given, while the

mechanism is the unknown.” (Hurwicz)

The mechanism designer is a game designer. He studies

What agents would do in various games

And what game leads to the outcomes that are most desirable

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Lecture 1: Introduction

Game theory revolutionized several disciplines

Biology (evolution, conflict, etc.)

Social sciences (economics, sociology, political science, etc.)

Computer science (algorithms, control, etc.)

game theory is now applied widely (e.g. regulation, online auctions,

distributed control, medical research, etc.)

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Lecture 1: Introduction

Its impact in economics (evaluated by Nobel prizes)

1972: Ken Arrow − general equilibrium

1994: John Nash, Reinhard Selten, John Harsanyi − solution concepts

2005: Tom Schelling and Robert Aumann − evolutionary game theory

and common knowledge

2007: Leonid Hurwicz, Eric Maskin, Roger Myerson − mechanism

design

2009: Lin Ostrom − economic governance, the commons

2012: Al Roth and Lloyd Shapley − market design

2014: Jean Tirole − markets and regulation

2016: Oliver Hart and Bengt Holmström − contract theory

2017: Richard Thaler − limited rationality, social preferences

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Lecture 1: Introduction

Part 1: game theory

“Introduction” / Tour of game theory

Non-cooperative game theory

No binding contracts can be

written

Players are individuals

Main solution concepts:

Nash equilibrium

Strong equilibrium

Cooperative game theory

Binding contract can be written

Players are individuals and

coalitions of individuals

Main solution concepts:

Core

Shapley value

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Lecture 1: Introduction

Noncooperative game theory

John Nash (1928-2015)

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Lecture 1: Introduction

A noncooperative game (normal-form)

players: N = {1, 2, . . . , n} (finite)

actions/strategies: (each player chooses si from his own finite strategyset; Si for each i ∈ N)

resulting strategy combination: s = (s1, . . . , sn) ∈ (Si)i∈N

payoffs: ui = ui(s)payoffs resulting from the outcome of the game determined by s

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Lecture 1: Introduction

Some 2-player examples

Prisoner’s dilemma − social dilemma, tragedy of the commons,free-riding

Conflict between individual and collective incentives

Harmony − aligned incentives

No conflict between individual and collective incentives

Battle of the Sexes − coordination

Conflict and alignment of individual and collective incentives

Hawk dove/Snowdrift − anti-coordination

Conflict and alignment of individual and collective incentives

Matching pennies − zero-sum, rock-paper-scissor

Conflict of individual incentives

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Lecture 1: Introduction

Player 2Heads Tails

Player 1Heads 1,-1 -1,1Tails -1,1 1,-1

Matching pennies

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Lecture 1: Introduction

Confess Stay quietA A

Confess-6 -10

B -6 0

Stay quiet0 -2

B -10 -2

Prisoner’s dilemma

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Lecture 1: Introduction

WOMANBoxing Shopping

MANBoxing 2,1 0,0

Shopping 0,0 1,2

Battle of the sexes

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Lecture 1: Introduction

Player 2Hawk Dove

Player 1Hawk -2,-2 4,0Dove 0,4 2,2

Hawk-Dove game

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Lecture 1: Introduction

Company BCooperate Not Cooperate

Company ACooperate 9,9 4,7

Not Cooperate 7,4 3,3

Harmony game

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Lecture 1: Introduction

Equilibrium

Equilibrium/solution concept:

An equilibrium/solution is a rule that maps the structure of a game into

an equilibrium set of strategies s∗.

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Lecture 1: Introduction

Nash Equilibrium

Definition: Best-response

Player i’s best-response (or, reply) to the strategies s−i played by all

others is the strategy s∗i ∈ Si such that

ui(s∗i , s−i) � ui(s′i, s−i) ∀s′i ∈ Si and s′i �= s∗i

Definition: (Pure-strategy) Nash equilibrium

All strategies are mutual best responses:

ui(s∗i , s−i) � ui(s′i, s−i) ∀s′i ∈ Si and s′i �= s∗i

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Lecture 1: Introduction

Confess Stay quietA A

Confess-6 -10

B -6 0

Stay quiet0 -2

B -10 -2

Prisoner’s dilemma: both players confess (defect)

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Lecture 1: Introduction

WOMANBoxing Shopping

MANBoxing 2,1 0,0

Shopping 0,0 1,2

Battle of the sexes: coordinate on either option

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Lecture 1: Introduction

Player 2Heads Tails

Player 1Heads 1,-1 -1,1Tails -1,1 1,-1

Matching pennies: none (in pure strategies)

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Lecture 1: Introduction

Player 2Hawk Dove

Player 1Hawk -2,-2 4,0Dove 0,4 2,2

Hawk-dove: either of the two hawk-dove outcomes

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Lecture 1: Introduction

Company BCooperate Not Cooperate

Company ACooperate 9,9 4,7

Not Cooperate 7,4 3,3

Harmony: both cooperate

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Lecture 1: Introduction

Pure-strategy N.E. for our 2-player examples

Prisoner’s dilemma − social dilemma

Unique NE − socially undesirable outcome

Harmony − aligned incentives

Unique NE − socially desirable outcome

Battle of the Sexes − coordination

Two NE − both Pareto-optimal

Hawk dove/Snowdrift − anti-coordination

Two NE − Pareto-optimal, but perhaps Dove-Dove “better”

Matching pennies − zero-sum, rock-paper-scissor

No (pure-strategy) NE

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Lecture 1: Introduction

How about our initial game

Remember the rules were:

1 Choose a number between 0 and 100

2 The player with the number closest to half the average of all submitted

numbers wins

What is the Nash Equilibrium?

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Lecture 1: Introduction

0

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Lecture 1: Introduction

Finally, let’s play again!

You remember the game:

1 Choose a number between 0 and 100

2 The player with the number that is closest to half the average

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This is what others did the 2nd time:

Lecture 1: Introduction

THANKS EVERYBODY

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