*June 13, 1928 † May 23, 2015

Controversies in Game Theory III: GAME Theory and John F. NASH Heinrich H. Nax (hnax@ethz.ch) COSS, ETH Zurich May 30, 2016

This course is the third one in this series after last years’ courses on Social Preferences (2014) Mechanism Design (2015) Information about the course, and materials/slides of speakers, will be made available at http://www.coss.ethz.ch/education/controversies.html Also, please contact me (Heinrich) under hnax@ethz.ch if you have any questions about the course!

The course is organized by the GESS Professorship of Computational Social Science (COSS) which 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

Structure of the course

We have roughly 45-50 minutes talk plus 5-10 minutes discussion for each unit and would like you to actively ask and interrupt (unless the speaker says otherwise) with questions or comments at any moment in time!

“EXAM” Please prepare a self-explanatory presentation in 20-30 slides (which you will not be asked to hold) Topics: may include one or a combination of issues raised during the week, may prove a good understanding of topics covered during the week, combine several ideas, or propose fresh thoughts,… Deadline: June 15th! Send to: hnax@ethz.ch

AND Before we begin with the first talk… Let us clarify some basic ingredients of Game Theory…

WHAT IS “Game theory”? A mathematical language to express models “conflict and cooperation between intelligent rational decision-makers” (Myerson) In other words, “interactive decision theory” (Aumann) Dates back to von Neumann & Morgenstern (1944) Most important solution concept: the Nash (1950) equilibrium

UNDERLYING preference theory Perhaps we should have defined game theory as “interactive decision theory” involving “rational and SELFISH decision-makers” SELFISH = self-regarding in a narrow sense Social preference allows for other concerns such as altruism fairness considerations reciprocity etc.

Game theory No contracts can be written Players are individuals Noncooperative game theory Cooperative game theory No contracts can be written Players are individuals Main solution concepts: Nash equ Strong equ Binding contract can be written Players are individuals and coalitions of individuals Main solution concepts: Core Shapley value

cooperative game theory of 39

A cooperative game

The core

Shapley value of 39

Noncooperative game theory

A noncooperative game (normal-form) players: N={1,2,…,n} (finite) actions / strategies: (each player chooses s_i from his own finite strategy set; S_i for each i∈N) set of strategy combination: s= (s_1,…,s_n) >outcome of the game payoff: u_i=u_i(s) >payoff outcome of the game

equilibrium Equilibrium concept: An equilibrium solution is a rule that maps the structure of a game into an equilibrium set of strategies S*.

Nash equilibrium Definition: Best-response Player i's best-response (or, reply) to the strategies s_-i is the strategy s*_i ∈ S_i such that Definition: (Pure-strategy) Nash equilibrium All strategies are mutual best responses:

Strong equilibrium

application

Public goods game

K-strong equilibrium

THANKS EVERYBODY!