*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 `$g$`

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.