Battle of the Sexes resembles Leader in many ways. Neither player has a dominant strategy, the maximin strategies intersect in the non-equilibrium (C, C) outcome …
The Prisoner’s Dilemma game presents a genuine paradox. The D strategies are dominant for both players, because each receives a larger payoff by choosing D than by choosing C against either counter-strategy of the other player. The maximin strategies intersect in the (D, D) outcome, which is the only Nash equilibrium in the game: neither player has any incentive to deviate from a D choice if the other also chooses D. In other words, it is in the interest of each player to disclose the incriminating evidence (or to leave an empty bag) whatever the other player does. But — and this is the paradox — if both players adopt this individualistic approach, the payoffs (2, 2) are worse for both of them than if they both chose their inadmissible (dominated) C strategies, in which case the payoffs are (3, 3). In game theory terminology, the dominant strategies intersect in a Pareto deficient equilibrium point. In the logic of this game there is a curious clash between individual and collective rationality. According to purely individualistic criteria, it is clearly rational for both players to choose their D strategies, but if both opt to be martyrs by choosing C, then the outcome is preferable for both. What is clearly required in order to ensure a better outcome for both is some principle of choice based on collective interests. The best-known principle of this type is the Golden Rule …
— Andrew M. Colman, Game Theory and Its Applications: In the Social and Biological Sciences, Routledge; 2nd Rev edition (September, 1995, pp. 108-109, 110, 115-116