Dominance#
When solving a zero-sum matrix game, we begin by searching for dominated strategies. The rational play principle allows us to ignore dominated strategies as the logical, self-interested player will never play them.
Choosing Better Strategies#
When one strategy is inferior to another strategy that player can play, the rational player will discard the inferior one.
- Dominated Strategy
A strategy \(S\) dominates a strategy \(T\) if every entry for \(S\) is greater than or equal to the corresponding entry for \(T\).
If this is the case, we say that \(T\) is dominated by \(S\) and write:
Example 1
Given the following game, determine if any Colin strategy is dominated.
Remember, in this condensed zero-sum game matrix setup, Colin’s outcome values are the opposite of the ones diplayed. Colin is trying to minimize here.
Solution. In the original game, you should verify that neither Rose strategy is dominated by the other. Let’s scan instead for Colin dominance.
The best strategy for Colin is Colin A where For \(A\leq B\), we see that:
Similarly, for \(A\leq C\), we find:
Thus, we have shown that \(A\leq B\) and \(A\leq C\). Hence, Colin will never choose to play either Colin B or Colin C. His strategy \(A\) is at least as good or better for every outcome in \(B\) or \(C\) regardless of the strategy Rose employs. We therefore write the following:
Example 2a
Given the following game, determine if any Rose strategy is dominated.
Solution. In the original game, we notice that Rose C has values of \(11\) and \(8\). Yet, while these values are good for Rose, the last value is a \(-3\). This outcome prevents Rose C from dominating the other rows.
Let’s compare Rose A to Rose B instead. Since we have
We see that, for Rose, strategy \(B\) dominates \(A\), and so we write as follows:
Reduction by Dominance#
In Example 2a, we found that Rose A is dominated. Since Rose will never play it, we can discard it from our game analysis. The process is called reduction by dominance.
Example 2b
We showed that the game in Example 2 has strategy \(A\) for Rose included, but it is dominated. We can rewrite the game without it as Rose will never choose to play it.
Find any dominance in Colin’s strategies in the reduced game shown above.
Solution. The strategy Colin C appears at first glance to be very good as Colin is minimizing. When we check, we find that \(\text{Colin C}\leq\text{Colin A}\) due to the following comparisons:
We also find \(\text{Colin C}\leq\text{Colin B}\) because of these:
Pure Strategy Solutions#
Note how the game can once again be reduced by dominance into a single column:
Given the choices, Rose will maximize her outcome by always choosing strategy \(B\).
In vector form, we say that Rose’s optimal strategy set is given by
while Colin’s is given by
For a pure strategy solution (PSS), we often say the solution of the game is Rose B, Colin C. The value of the game is \(v=5\).