# Definition:Two-Person Zero-Sum Game

## Definition

A **two-person zero-sum game** is a two-person game with zero sum.

That is, in which the total of the payoffs to all players is zero.

## Examples

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

Player $A$ has $m$ strategies: $A_1, A_2, \ldots, A_m$.

Player $B$ has $n$ strategies: $B_1, B_2, \ldots, B_n$.

The game is zero-sum in that a payoff of $p$ to $A$ corresponds to a payoff of $-p$ to $B$.

The game is completely defined by the table specifying the payoff to $A$ for each strategy of $A$ and $B$.

### Abstract Example 1

$B$'s strategy | ||

$A$'s strategy | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 \\ \hline A_1 & 5 & -2 & 1 \\ \hline A_2 & 6 & 4 & 2 \\ \hline A_3 & 0 & 7 & -1 \\ \hline \end{array}$ |

This has an equilibrium point at $\tuple {A_2, B_3}$ with a payoff to $A$ of $2$.

### Abstract Example 2

$B$'s strategy | ||

$A$'s strategy | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 \\ \hline A_1 & -2 & 1 & 1 \\ \hline A_2 & -3 & 0 & 2 \\ \hline A_3 & -4 & -6 & 4 \\ \hline \end{array}$ |

This has an equilibrium point at $\left({A_1, B_1}\right)$ with a payoff to $A$ of $-2$.

### Abstract Example 3

$B$'s strategy | ||

$A$'s strategy | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 & B_4 \\ \hline A_1 & -3 & 17 & -5 & 21 \\ \hline A_2 & 7 & 9 & 5 & 7 \\ \hline A_3 & 3 & -7 & 1 & 13 \\ \hline A_4 & 1 & -19 & 3 & 11 \\ \hline \end{array}$ |

This has an equilibrium point at $\tuple {A_2, B_3}$ with a payoff to $A$ of $5$.

### Abstract Example 4

$B$'s strategy | ||

$A$'s strategy | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 \\ \hline A_1 & 2 & -5 & -2 \\ \hline A_2 & 3 & -1 & -1 \\ \hline A_3 & -3 & 4 & -4 \\ \hline \end{array}$ |

This has an equilibrium point at $\tuple {A_2, B_3}$ with a payoff to $A$ of $-1$.

### Abstract Example 5

$B$'s strategy | ||

$A$'s strategy | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 \\ \hline A_1 & ? & ? & 3 \\ \hline A_2 & ? & ? & 4 \\ \hline A_3 & 7 & 6 & 5 \\ \hline \end{array}$ |

This has an equilibrium point at $\tuple {A_3, B_3}$ with a payoff to $A$ of $5$.

### Political Example 1

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 | $\begin{array}{r {{|}} c {{|}} c {{|}} c {{|}} } & \text X & \text Y & \text{No position} \\ \hline \text X & 45\% & 50\% & 40 \% \\ \hline \text Y & 60\% & 55\% & 50 \% \\ \hline \text{No position} & 45\% & 55\% & 40 \% \\ \hline \end{array}$ |

This is a two-person zero-sum game with an equilibrium point.

### Political Example 2

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.

### Political Example 3

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

This is a two-person zero-sum game with an equilibrium point.

## Also see

- Results about
**two-person zero-sum games**can be found**here**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**two-person zero-sum game**