Problem Set 2 (from the Yale Open Course in Game Theory Problem Sets)
Due Wednesday, January 19.
1. Recall the voting game we discussed in class. There are two candidates, each of whom
chooses a position from the set S
i
= {1, 2, 3,.
.., 10}. The voters are equally distributed across
these ten positions. So, 10% of the voters are at each position. Voters vote for the candidate
whose position is closest to theirs. If the two candidates are equidistant from a given position, the
voters at that position split their votes equally. The aim of the candidates is to maximize their
percentage of the total vote. Thus, for example, u
1
(8, 8) = 50 and u
1
(7, 8) = 70. (Where u
1
(7, 8)
means Player 1's payoff when Player 1 plays 7, Player 2 plays 8.)
[Hint: in answering this
question, you do not need to write out the full payoff matrices!]
(a) In class, we showed that strategy 2 strictly dominates strategy 1. In fact, other
strategies strictly dominate strategy 1. Find all the strategies that strictly dominate
strategy 1. Explain your answer. [Hint: try some guesses and see if they work.]
Solution: Strategies 2, 3, 4, 5, 6, and 7 strictly dominate 1. Strategies 8, 9, and 10 do not.
(e.g. u
1
(1, 7) = 35 >
u
1
(8, 7) = 30).
(b) Suppose now that there are three candidates. Thus, for example, u
1
(8, 8, 8) = 33.3 and
u
1
(7, 9, 9) = 73.3. (Where u
1
(7, 9, 9) means Player 1's payoff when Player 1 plays 7, Player 2
plays 9 and Player 2 plays 9.)
Is strategy 1 dominated, strictly or weakly, by strategy 2? How
about by strategy 3? Explain. Suppose we delete strategies 1 and 10. That is, we rule out the
possibility of any candidate choosing either 1 or 10, although there are still voters at those
positions. Is strategy 2 dominated, strictly or weakly, by any other (pure) strategy s
i
in the
reduced game? Explain.
Solution: Strategy 1 is now weakly dominated by strategy 2 and strategy 1 is weakly
dominated by strategy 3.
For example: u
1
(1, 2, 3) = 10
=
u
1
(2, 2, 3) = 10, u
1
(1, 2, 4) = 10
=
u
1
(3, 2, 4) = 10
If 1 and 10 are no longer options, then 2 is not dominated by any other pure strategy. For
example,
u
1
(2, 3, 4) = 20
>
u
1
(3, 3, 4) = 15 and the payoffs for playing 2 will be strictly larger for
any strategies 4 through 8 when it is "surrounded" by the other two players.
For example u