**Herewith a refresher on two-person mixed-motive games:**

*Games and Decisions: Introduction and Critical Survey*, New York: Wiley, 1957, p. 91), provides an interpretation of Matrix 6.4. A married couple has to choose between two options for an evening’s entertainment. The man prefers one kind of entertainment and the woman the other (a prize fight and a ballet, in Luce and Raiffá s stereotyped version), but both would rather go out together than alone. If both opt for their first choices (C, C), each ends up going out alone, with payoffs (2, 2), and a worse outcome with payoffs of (1, 1) results if both make the heroic sacrifice of going to the entertainments they dislike (D, D). If one chooses his or her preferred option and the other plays the role of hero, however, the payoffs (4, 3) or (3, 4) are better for both, but not quite as good for the hero as for the hero’s partner.

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 …

*Toward a History of Game Theory: Annual Supplement to Volume 24 History of Political Economy*(pp. 165-175), Durham, NC & London, England: Duke University Press, 1992; P. Straffin, The Prisoner’s Dilemma,

*UMAP Journal*, 1, 1980, pp. 101-103). The name derives from the following imaginary strategic interaction. Two people are arrested and charged with involvement in a serious crime. They are held in separate cells and prevented from communicating with each other. The police have insufficient evidence to obtain a conviction unless at least one of the them discloses certain incriminating information. Each prisoner is faced with a choice between concealing information from the police (C) and disclosing it (D). If both conceal, both will be acquitted (3, 3). If both disclose, both will be convicted (2, 2). If only one prisoner discloses the information, that prisoner will not only be acquitted but will receive a reward for giving Queens evidence, while the “martyr” who conceals the information will receive an especially heavy sentence from the court: (4, 1) or (1, 4) depending on who discloses and who conceals. These payoffs are assumed to take into account the players’ moral attitudes towards obstructing the course of justice, betraying a comrade, and so on; for some people in the situation described in the story, the payoffs would not correspond to the Prisoner’s Dilemma game. It is customary to interpret the C strategies as cooperative and the D strategies as defecting choices. […]

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