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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”.

# 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.