Definition:Bach or Stravinsky?

Definition

Bach or Stravinsky? is an instance of a class of games whose mechanics are as follows:

There are two players: $\text A$lexis and $\text B$everley.

They wish to go out together to a musical concert, but $\text A$ prefers Bach and $\text B$ prefers Stravinsky.

The key points are:

$\text A$lexis and $\text B$everley wish to coordinate their behaviour

but:

they have conflicting interests.

Payoff Table

The payoff table of Bach or Stravinsky? is as follows:

		$\text B$
$\text A$		$\begin {array} {r {{|}} c {{|}} } & \text {Bach} & \text{Stravinsky} \\ \hline \text {Bach} & 2, 1 & 0, 0 \\ \hline \text {Stravinsky} & 0, 0 & 1, 2 \\ \hline \end {array}$

Analysis

Analysis of Bach or Stravinsky?

Solution

From the payoff table:

		$\text B$
$\text A$		$\begin {array} {r {{|}} c {{|}} } & \text {Bach} & \text{Stravinsky} \\ \hline \text {Bach} & 2, 1 & 0, 0 \\ \hline \text {Stravinsky} & 0, 0 & 1, 2 \\ \hline \end {array}$

There are two Nash equilibria:

$\tuple {\text {Bach}, \text {Bach} }$

$\tuple {\text {Stravinsky}, \text {Stravinsky} }$

Thus there are two steady states:

one in which both players always choose Bach

one in which both players always choose Stravinsky.

Either, but not both, experience their preferred music.

Also known as

This general class of game can be abbreviated BoS.

It is also referred to as the Battle of the Sexes, in which the two participants are one male, one female, and the options are a prizefight and a ballet.

Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $\text I$ Strategic Games: Chapter $2$ Nash Equilibrium: $2.3$: Examples: Example $15.3$