leader, battle of the sexes, and the prisoner’s dilemma

Herewith a refresher on two-person mixed-motive games: The ordinal payoff structure of the game of Leader is shown in Matrix 6.3. This strategic structure arises in numerous everyday interactions, but the following familiar example will suffice to illustrate its essential properties. Two motorists are waiting to enter a heavy stream of traffic at an intersection, and both are in a hurry to get to their destinations. When a gap in the traffic occurs, each must decide whether to concede the right of way to the other (C) or to drive into the gap (D). If both concede, both will be delayed, which is the second-to-worst outcome for both, with payoffs of (2, 2). If both drive out together, a collision may occur (1, 1). But if one drives out while the other concedes the right of way, then the leader will be able to proceed immediately (the best payoff) and the follower will have to wait a few seconds longer (the second best payoff). Depending on who is leader, the payoffs are (4, 3) or (3, 4). The payoffs shown in Matrix 6.3 agree with these common-sense assumptions about the motorists’ orders of preference among the possible outcomes. There are no dominant strategies in the game of Leader …

The ordinal structure of the second of the four archetypal 2 x 2 games is depicted in Matrix 6.4 (which is actually an improved version of Luce and Raiffa’s model of the marital problem). The following well-known predicament, christened “Battle of the Sexes” by Luce and Raiffa (R.D. Luce & H. Raiffa, 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 …

The ordinal structure of the last and most interesting of the four archetypal 2 x 2 games, and the one that has generated by far the most empirical research, is shown in Matrix 6.7. Attention was first drawn to this peculiar game by Merrill Flood and Melvin Dresher at the RAND Corporation in 1950 or 1951, and shortly thereafter Albert W. Tucker named it “Prisoner’s Dilemma” for a seminar on game theory in the psychology department of Stanford University (H. Raiffa, Game theory at the University of Michigan, 1948-1952, in E.R. Weintraub (Ed.), 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