Bribes: Two players find themselves in a legal battle over a patent. The patent is worth 20 to each player, so the winner would receive 20 and the loser 0. Given the norms of the country, it is common to bribe the judge hearing a case. Each player can offer a bribe secretly, and the one whose bribe is the highest will be awarded the patent. If both choose not to bribe, or if the bribes are the same amount, then each has an equal chance of being awarded the patent. If a player does bribe, then the bribe can be valued at either 9 or 20. Any other number is considered very unlucky, and the judge would surely rule against a party who offered a different number.
a. Find the unique pure-strategy Nash equilibrium for this game.
b. If the norm were different, so that a bribe of 15 was also acceptable, is there a pure-strategy Nash equilibrium?
c. Find the symmetric mixed-strategy Nash equilibrium for the game with possible bribes of 9, 15, and 20.