In the Hollywood movie “A Beautiful Mind”, Russel Crowe plays John Nash who developed the Nash Equilibrium concept in his PhD thesis at Princeton University. In one of the early scenes of the movie, Nash finds himself in a bar with three of his fellow (male) mathematics PhD students when a group of five women enters the bar.1 The attention of the PhD students is focused on one of the five women, with each of the four PhD students expressing interest in asking her out. One of Nash’s fellow students reminds the others of Adam Smith’s insight that pursuit of self interest in competition with others results in the socially best outcome, but Nash — in what appears to be a flash if insight — claims “Adam Smith needs revision”.
A: In the movie, John Nash then explains that none of them will end up with the woman they are all attracted to if they all compete for her because they will block each other as they compete—and that furthermore they will not be able to go out with the other women in the group thereafter (because none of them will agree to a date once they know they are at best everyone’s second choice). Instead, he proposes, they should all ignore the woman they are initially attracted to and instead ask the others out—it’s the only way they will get a date. He quickly rushes off to write his thesis—with the movie implying that he had just discovered the concept of Nash Equilibrium.
(a) If each of the PhD students were to play the strategy John Nash suggests—i.e. each one selects a woman other than the one they are all attracted to, could this in fact be a pure strategy Nash Equilibrium?
(b) Is it possible that any pure strategy Nash equilibrium could result in no one pursuing the
woman they are all attracted to?
(c) Suppose we simplified the example to one in which it was only Nash and one other student encountering a group of two women. We then have two pure strategies to consider for each PhD student: Pursue woman A or pursue woman B. Suppose that each viewed a date with woman A as yielding a “pay off ” of 10 and a date with woman B as yielding a payoff of 5. Each will in fact get a date with the woman that is approached if they approach different women, but neither will get a date if they approach the same woman in which case they both get a payoff of 0. Write down the pay off matrix of this game.
(d) What are the pure strategy Nash Equilibria of this game?
(e) Is there a mixed strategy Nash Equilibriumin this game?
(f) Now suppose there is also a woman C in the group of women—and a date with C is viewed as equivalent to a date with B. Again, each PhD student gets a date if he is the only one approaching a woman, but if both approach the same woman, neither gets a date (and thus both get a payoff of zero). Now, however, the PhD students have 3 pure strategies: A, B and C. Write down the pay off matrix for this game.
(g) What are the pure strategy Nash Equilibria of this game? Does any of them involve woman A leaving without a date?
(h) In the movie, Nash then explains that “Adam Smith said the best result comes from everyone in the group doing what’s best for themselves." He goes on to say “...incomplete ... incomplete ... because the best result will come from everyone in the group doing what’s best for themselves and the group ... Adam Smith was wrong.” Does the situation described in the movie illustrate any of this?
(i) While these words have little to do with the concept of Nash Equilibrium, in what way does game theory—and in particular games like the Prisoners’ dilemma—challenge the inference one might draw from Adam Smith that self interest achieves the “best” outcome for the group?
B: Consider the 2-player game described in part A(c). (Note: Part (a) and (b) below can be done without having read Section B of the Chapter.)
(a) Suppose that the players move sequentially—with player 1 choosing A or B first—and player 2 making his choice after observing player 1’s choice. What is the sub game perfect Nash equilibrium?
(b) Is there a Nash equilibrium in which player 2 goes out with woman A? If so, is there a no credible threat that is needed to sustain this as an equilibrium?
(c) Next, consider again the simultaneous move game from A(c). Draw a game tree for this simultaneous move game — with player 1’s decision on the top. (Hint: Use the appropriate information set for player 2 to keep this game a simultaneous move game). Can you state different beliefs for player 2 (when player 2 gets to his information set) such that the equilibria you derived in A (d) and A(e) arise?
(d) Continue to assume that both players get payoff of 0 if they approach the same woman. As before, player 1 gets a payoff of 10 if he is the only one to approach woman A and a payoff of 5 if he is the only one to approach woman B. But player 2 might be one of two possible types: If he is type 1, he has the same tastes as player 1, but if he is of type 2, he gets a payoff of only 5 if he is the only one to approach woman A and a payoff of 10 if he is the only one to approach women B. Prior to the beginning of the game, Nature assigns type 1 to player 2with probability δ (and thus assigns type 2 to player 2 with probability (1−δ).) Graph the game tree for this game—using information sets to connect nodes where appropriate.
(e) What are the pure strategy equilibria in this game? Does it matter what value δ takes?