Betty and Billy are going on a date. Unfortunately, they've both forgotten what exactly they had decided to do. They know Betty wanted to go to the ballet and Billy wanted to go to the boxing match, but now they don't know what they chose. And they have no way of getting in touch with each other before they are supposed to meet at one of the venues.
This is the unlikely premise for the collective action problem known as Battle of the Sexes. (q.v. Prisoner's Dilemma, Chicken)
The name itself isn't really appropriate. The problem isn't really about one sex—one actor—bettering the other, even though their preferences are different. The problem is coordination: Both prefer going together over going alone, so if Billy thinks Betty is going to choose the ballet, he too will choose to go to the ballet, even though he would prefer going to see the fight. And the other way around.
If we put the problem in a game theory matrix, it looks like this:
| BILLY |
| Ballet | Boxing |
Ballet | 2,3p,n| 1,1 |
Boxing | 1,1 | 3,2p,n|
The rational strategy of both Betty and Billy is to comply to the other's strategy. There is no gain, no incentive, to unlilaterally defect from a situation where both actors choose the same venue. In other words, the strategy combination of compliance is a Nash equilibrium (n). Further, it is shown that whether they go to the Ballet or to see the boxing fight, the total utility gained is the same and as big as it gets: In both instances, it is Pareto efficient (p), even though the two situations favors the one or the other.
Now, there is no obvious way for Betty or Billy to know which venue to choose. It is a dilemma: A dilemma of common aversions. If they had been able to communicate they would probably end up at the same venue as nobody would gain anything by cheating. On the other hand, it would also create the possibility of one of them domineering the other to do what the one prefers.
In the end, the objective of the game is to get the other to comply to your strategy while remembering that if you act too selfish, your partner can always threaten to non-comply (defect) even though they themself loses by it. They might find solace in the fact that you stand to lose potentially relatively more than they.
Though abstract, The Battle of the Sexes isn't an uncommon problem. For instance, in International policy-making, Battle describes the problem of two countries trying to coordinate their tolls and taxes. The problems facing these international organisations when dealing with the risks of a Battle-like problem has spurred the creation of international organisations and regimes to help lift the veil of uncertainty.