Consider a gunfight between Bat Masterson and William "Curly Bill" Brocius. Both men have private... Consider a gunfight between Bat Masterson and William "Curly Bill" Brocius. Both men have private information regarding their skill with a six-shooter. Nature moves first by determining each gunfighter's skill. He can have either a fast draw or a slow draw. There is 65% chance that Bat is fast and 60% chance that Curly Bill is fast. After each gunfighter learns his type - thought remaining uncertain about the other gunfighter's type - he chooses between draw and wait. If both wait, then the payoff is 50. If both draw and (1) they are the same type (either both fast or both slow), then each had a payoff of 20; and (2) they are different types, then the faster gunfighter has a payoff of 30 and the slow one of -40. If one draws and the other waits and (1) they are of the same type, then the one who drew has a payoff of 30 and the other a payoff of -40; (2) the one who draws is fast and the other is slow, then the one who drew has a payoff of 30 and the other a payoff of -40; and (3) the one who drew is slow and the other is fast, then each has a payoff of 20. If at least one chooses to draw, then there is a gunfight.

Is it consistent with BNE for there to be no gunfight for sure? (That is, both gunfighters wait regardless of their type)

Using it means a payoff of 50—as a gun fight is prevented — and all the other outcomes produce a lower payoff, so every drawing's estimated payoff must be less.

More specifically, the conditions for balancing Bat are,

Fast Type (Wait): .6 x 50 + .4 x 50 >= .6 x 30 + .4 x 30 = 50 >= 30

Slow Type (Wait): .6 x 50 + .4 x 50 >= .6 x 20 + .4 x 30 = 50 >= 24.

The equilibrium conditions for Curly Bill are as follows;

Fast Type (Draw): .65 x 50 + .35 x 50 >= .65 x 30 + .35 x 30 = 50 >= 30

Slow Type (Draw): .65 x 50 + .35 x 50 >= .65 x 20 + .35 x 30 = 50 >= 23.5.