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author: niplav, created: 2022-04-05, modified: 2022-04-05, language: english, status: in progress, importance: 2, confidence: likely

Solutions to the textbook “Algorithmic Game Theory”.

Contents

Solutions to “Algorithmic Game Theory”

Chapter 1

1.3

Let g be a two-player game. Now construct a 3-player zero-sum game g' as following: Add another player 3 , with one action, and let the utility of that player be u_3'(a_3, a_{-3})=0-(u_1(a_{-3})+u_2(a_{-3}) .

Then the Nash equilibria of G' are the same as for g : player 3 can't deviate, and the utilities of the other players are not affected by the actions of 3 . Therefore, the Nash equilibria in g' are the same as for g , and equally hard to find—which means that Nash equilibria for three-player zero-sum games are at least as hard to find as for two-player games.