(Equilibrium for pairwise interactions in a single population) Find all the Nash equilibria of the game in Figure 51.3. Which of the equilibria, if any, correspond to a steady state if the game models pairwise interactions between the members of a single population?
The notion of a strategic game originated in the work of Borel (1921) and von Neumann (1928). The notion of Nash equilibrium (and its interpretation) is due to Nash (1950a). (The idea that underlies it goes back at least to Cournot (1838, Ch. 7).) The Prisoner’s Dilemma appears to have first been considered by Melvin Dresher and Merrill Flood, who used it in an experiment at the RAND Corporation in January 1950 (Flood 1958/59, 11–17); it is an example in Nash’s PhD thesis, submitted in May 1950. The story associated with it is due to Tucker (1950) (see Straf- fin 1980). O’Neill (1994, 1010–1013) argues that there is no evidence that game theory (and in particular the Prisoner’s Dilemma) influenced US nuclear strategists in the 1950s. The idea that a common property will be overused is very old (in Western thought, it goes back at least to Aristotle (Ostrom 1990, 2)); a precise modern analysis was initiated by Gordon (1954). Hardin (1968) coined the phrase “tragedy of the commons”. BoS, like the Prisoner’s Dilemma, is an example in Nash’s PhD thesis; Luce and Raiffa (1957, 90–91) name it and associate a story with it. Matching Pennies was first considered by von Neumann (1928). Rousseau’s sentence about hunting stags is interpreted as a description of a game by Ullmann-Margalit (1977, 121) and Jervis (1977/78), following discussion by Waltz (1959, 167–169) and Lewis (1969, 7, 47). The information about John Nash in the box on p. 20 comes from Leonard (1994), Kuhn et al. (1995), Kuhn (1996), Myerson (1996), Nasar (1998), and Nash (1995). Hawk–Dove is known also as “Chicken” (two drivers approach each other on a narrow road; the one who pulls over first is “chicken”). It was first suggested (in a more complicated form) as a model of animal conflict by Maynard Smith and Price (1973). The discussion of focal points in the box on p. 30 draws on Schelling (1960, 54–58). Games modeling voluntary contributions to a public good were first considered by Olson (1965, Section I.D). The game in Exercise 31.1 is studied in detail by Palfrey and Rosenthal (1984). The result in Section 2.8.4 is due to Warr (1983) and Bergstrom, Blume, and Varian (1986).