16. value of information Two competitors are fighting it out as the only merchants on a remote island. Each has two strategies, simultaneous play is the order of the day, and Nature will provide one of two states (with equal probability). If state one obtains, the players’ payoffs are given by the following bimatrix game.

left right up 10,10 0,12 down 40,-40 2,2 Conversely, if state two obtains, the players’ face the following bimatrix game.

left right up 4,4 10,0 down -40,12 10,10

(a) Suppose neither player can gather any additional information.

Verify that 7 for Row and 7 for Column are equilibrium payoffs.

(Here they play the game defined by the expected value of the two matrices.)

(b) Suppose Row obtains perfect information before acting. Column knows this and Row knows that Column knows, and so on. Verify that an equilibrium has Row play down no matter what state occurs, and Column play right, with expected payoffs of 6 for each. (Here Row has four strategies: up no matter what; down no matter what; up in state one and down in state two; and vice versa.) How do you explain this equilibrium? Is Row better off with the information?

(c) Suppose both Row and Column acquire perfect information. Determine and interpret the resulting equilibrium behavior.