Two-Person Zero-Sum Game/Examples/Political Example 3

Example of Two-Person Zero-Sum Game

The two players are political candidates: $\text A$ and $\text B$.

Each may choose between the following three strategies:

supporting policy $\text X$

supporting policy $\text Y$

taking no position on the matter.

It is assumed that voting will be restricted to either $\text A$ or $\text B$. That is, no abstentions will take place.

The expected share of the vote for either candidate depends on which strategy they adopt.

The following table gives the expected share of the vote for $\text A$ based upon the combined strategies of $\text A$ and $\text B$:

		$B$'s Platform
$A$'s Platform		$\begin{array}{r {{\|}} c {{\|}} c {{\|}} c {{\|}} } & \text X & \text Y & \text{No position} \\ \hline \text X & 35 \% & 10 \% & 60 \% \\ \hline \text Y & 45 \% & 55 \% & 50 \% \\ \hline \text{No position} & 40 \% & 10 \% & 65 \% \\ \hline \end{array}$

The two players are $A$ and $B$.

The assumption is made at the start that the votes are divided solely between $A$ and $B$.

There is a fixed total share of the vote ($100\%$).

What is not gained by $A$ will be gained by $B$, whatever their strategies.

It can be seen by inspection that whatever $A$'s platform, $B$ should not take no position, because that is always inferior to supporting $X$.

Once it has been established that $B$ is either supporting $X$ or supporting $Y$, it can be seen that $A$ always does best when supporting $Y$.

It then follows that $B$ should support $X$, leaving $A$ with $45 \%$ of the vote.

If either party deviates from this strategy, their share of the vote will decrease.

Thus, by definition, this is an equilibrium point.

$\blacksquare$

1983: Morton D. Davis: Game Theory (revised ed.) ... (previous) ... (next): $\S 2$: The Two-Person, Zero-Sum Game with Equilibrium Points: A Political Example: Figure $2.9$