And that's why you should never trust women!
What the woman did, however, makes complete sense. If the man had decided to steal, it would make no difference to the amount of money she would take back (which would be zero) whether she splits or steals. If on the other hand the man had decided to split, it would obviously be favorable for her to steal. Thus whatever the man might have chosen to do, split or steal, the outcome is more favorable to the woman if she chooses to steal.
But what if the man had also done this simple calculation before taking his decision? The outcome would be the worst possible for either of the participants! Thus if both players are rational and try to maximize their own payoffs they lose everything. Thus sometimes altruism can be more beneficial than selfishness. This is a version of the prisoner's dilemma (PD) (although there are subtle differences). The wikipedia article mentions that in experiments 40 % of people choose to trust each other and cooperate. Thus people do not always make the selfish, rational choice.
I read about a similar paradox called the traveler's dilemma (TD) in an article written in Scientific American by Kaushik Basu who used this paradox to argue that
The game and our intuitive prediction of its outcome also contradict economists' ideas. Early economics was firmly tethered to the libertarian presumption that individuals should be left to their own devices because their selfish choices will result in the economy running efficiently. The rise of game-theoretic methods has already done much to cut economics free from this assumption. Yet those methods have long been based on the axiom that people will make selfish rational choices that game theory can predict. TD undermines both the libertarian idea that unrestrained selfishness is good for the economy and the game-theoretic tenet that people will be selfish and rational.
You might wonder (as I did when I first read the above article) when in real life do people have to make such decisions (other than contrived situations in game shows)? Let me give you two examples,
1)A recession like the current one is worsened due to a lack of confidence. Consumers because of lower income and out of fear of unemployment choose to spend less which leads to companies doing badly which leads to further unemployment and lowering of incomes. Thus although it may be rational for each individual to save money in a recession, this leads to a situation which is bad for the population as a whole. This is called the paradox of thrift.
2) In an arms race between two countries both the countries have the choice to invest more in weapons or reduce arms. For either country military expansion is the 'rational' choice irrespective of what the other country does. This leads to a situation which makes both countries less safe than if both had chosen to reduce arms.
As the second example suggests PD is not limited to just economics. In fact the iterated PD (a version where two players play PD many times) has been used to explain among other things morality and the evolution of altruism (subscription required)!