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”.
Let be a two-player game. Now construct a 3-player zero-sum game as following: Add another player , with one action, and let the utility of that player be .
Then the Nash equilibria of are the same as for : player can't deviate, and the utilities of the other players are not affected by the actions of . Therefore, the Nash equilibria in are the same as for , 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.