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Consider the following voting game:

There are three candidates, each of whom chooses a position from the set $S_i = \lbrace 1, 2,...,10 \rbrace$. The voters are equally distributed across these ten positions. This means at each position, there are 10% of the available votes. Voters vote for the candidate whose position is closest to theirs. If the three candidates are equidistant from a given position, the voters at that position split their votes equally. Thus, for example, $u_1(8, 8,8) = 33.3$ ( this means the candidate $1$ choosing position $8$ with other two candidates also choosing position $8$ will get about 33.3% of the total available votes, and similarly $u_1(7,9,9) = 73.3$.

Now the problem is

Is strategy $1$ dominated, strictly or weakly, by strategy $2$? How about by strategy $3$? By "strategy $1$", I mean the strategy of choosing position $1$. Similarly for others.

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I'm not sure what's the question, because you got $u_1(7,9,9)$ right, so why not just compute the rest of the probabilities $u_n(i,j,k)$ where $i,j,k$ is ordered with repetitions and notice who wins/ties given $i,j,k$? That way I got this table

$$\begin{array}{|l|l|} \hline \text{} & 1 \\ \hline 1 & 2 \\ \hline \text{1 2} & 3 \\ \hline \text{1 2 3} & \text{4 10} \\ \hline \text{1 2 10} & 5 \\ \hline \text{1 9 10} & 6 \\ \hline \text{8 9 10} & \text{1 7} \\ \hline \text{9 10} & 8 \\ \hline 10 & 9 \\ \hline \text{} & 10 \\ \hline \end{array}$$ If in row $R$, column $C$ we have number $N$, it means that $N$ is dominated by $R$ (weakly if $C = 2$, strictly if $C = 1$)

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  • $\begingroup$ Can you please explain what your table means? I don't really understand. $\endgroup$ – PropositionX Aug 6 '17 at 7:45
  • $\begingroup$ @PropositionX The rows index $1,2,\ldots,10$ correspond to the strategy dominating other strategies. You see e.g. in row 4 that strategy 1 is dominated by 4. So is strategy 2, 3, and weakly 10. So if a candidate goes for strategy 1,2,3 he is doomed if someone else goes for strategy 4. $\endgroup$ – lasen H Aug 6 '17 at 7:49
  • $\begingroup$ I still don't follow. In row $4$, you have $1,2,3,10$. What does that even mean? $\endgroup$ – PropositionX Aug 6 '17 at 7:54
  • $\begingroup$ @PropositionX It means that all of strategy 1,2,3,10 are dominated by strategy 4 $\endgroup$ – lasen H Aug 6 '17 at 7:55

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