In the text, we discussed the “Matching Pennies” game and illustrated that such a game only has a mixed strategy equilibrium.
A: Consider each of the following and explain (unless you are asked to do something different) how you might expect there to be no pure strategy equilibrium—and how a mixed strategy equilibrium might make sense.
(a) A popular children’s game, often played on long road trips, is “Rock, Paper, Scissors”. The game is simple: Two players simultaneously signal through a hand gesture one of three possible actions: Rock, Paper or Scissor s. If the two players signal the same, the game is a tie. Otherwise, Rock beats Scissor s, Scissor s beats Paper and Paper beats Rock.
(b) One of my students objects: “I understand that Scissor s can beat Paper, and I get how Rock can beat Scissor s, but there is no way Paper should beat Rock. What ... Paper is supposed to magically wrap around Rock leaving it immobile? Why can’t Paper do this to Scissor s? For that matter, why can’t Paper do this to people? I’ll tell you why: Because Paper can’t beat anybody!”3 If Rock really could beat Paper, is there still a mixed strategy Nash Equilibrium?
(c) In soccer, penalty kicks often resolve ties. The kicker has to choose which side of the goal to aim for, and, because the ball moves so fast, the goalie has to decide simultaneously which side of the goal to defend.
(d) How is the soccer example similar to a situation encountered by a professional tennis player whose turn it is to serve?
(e) For reasons I cannot explain, teenagers in the 1950’s sometimes played a game called “chicken”. Two teenagers in separate cars drove at high speed in opposite directions on a collision course —and whoever swerved to avoid a crash lost the game. Sometimes, the cars crashed and both teenagers were severely injured (or worse). If we think behavior in these games arose within an equilibrium, could that equilibrium be in pure strategies?
B: If you have done part B of exercise 24.4, appeal to incomplete information games with almost complete information to explain intuitively how the mixed strategy equilibrium in the chicken game of A (e) can be interpreted.