# Game Theory - Hotelling Game with $n=4$ and 2 rounds of competition

Consider a modified Hotelling game, where there are players $a,a’,b,b’$, with $s_{i,\{a,a’\}} \in \{1..5\}$ and $s_{i,\{b,b’\}} \in \{5..9\}$.

The game has two rounds, where first $a$ and $a’$ compete with each other and $b$ and $b’$ compete with each other. This round is identical to the Hotelling game. Whichever player of each set has the highest payoff will advance to the next round. However, if there is any ties, then $a$ and $b$ will win in their own respective rounds. For example, if $s_a = 7$ and $s_{a’}=7$, then $a$ will advance to the second round.

The second round consists of players $a^*$ and $b^*$, which are the winners of the first round. This is the classic Hotelling game but with the players limited to their initial strategy sets. The players can not change their strategies in this round and will play the same strategies as their first round.

Question: Are there any strictly dominated strategies for $a’$?

Question: Are there any strictly dominated strategies for $a$?

Question 1: There are no strictly dominated strategies for $a'$. There is always a chance he will lose, so nothing is strictly dominated.
Question 2: Strategies $\{8,9\}$ are strictly dominated by $\{5,6,7\}$. Playing 8 or 9 for $a'$ will result in the same as 5 and 6 in Round 1. However, looking at the second round, playing 8 or 9 will equivocally result in a lower payoff than playing 5 or 6, for all things that $a$ could play. Therefore, it is strictly dominated.