Two-Person Zero-Sum Game/Examples/Political Example 2
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 | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} }
& \text X & \text Y & \text{No position} \\ \hline \text X & 45\% & 10\% & 40 \% \\ \hline \text Y & 60\% & 55\% & 50 \% \\ \hline \text{No position} & 45\% & 10\% & 40 \% \\ \hline \end{array}$ |
This is a two-person zero-sum game with an equilibrium point.
Proof
Two-Person
The two players are $A$ and $B$.
Zero-Sum
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.
Equilibrium Point
It can be seen by inspection that whatever $B$'s platform, $A$'s share of the vote is a maximum when $A$ supports policy $Y$.
It is therefore clear to $B$ that this will be $A$'s decision.
So, while there may be a slim chance that $A$ will make the ridiculous decision to either support $X$ or take no position, this is sufficiently unlikely as to make $B$'s decision clear: to take no position.
If either party deviates from this strategy, their share of the vote will decrease.
Thus, by definition, this is an equilibrium point.
$\blacksquare$
Sources
- 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 $2.8$