Question

Two players find themselves in a legal battle over a patent. The patent is worth 20 for each player, so the winner would receive 20 and the loser 0. Given the norms of the country they are in, it is common to bribe the judge of a case. Each player can secretly oer a bribe of 0, 9 or 20, and the one whose bribe is the largest is 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 decides to bribe then the judge pockets it regardless of who gets the patent).

(a) Derive the game matrix.

(b) Is the game dominance solvable? If so, findnd the strategy prole surviving IDSDS.

(c) Now consider the case in which the allowed bribe amounts are instead 0, 9 and 15. Is the game dominance solvable? Find the best responses of each player to each of the pure strategies of the opponent.

Answer 1

In the game matrix above you can see the payoffs for the different possibilities. Equal chance of winning yields an expected payoff of 10.

(b) The game is dominance solvable. The bribe of 20 is the dominated strategy and so those rows and columns can be eliminated. This is because no matter what other player does, bribing 20 will yield lowest payoff Now, looking at the 2 x 2 matrix with bribing 0 or 9, bribing 0 is the dominated strategy since it yields the lowest payoff no matter what the other player does. Hence we are only left with the strategy for both players bribing 9.

(c) The game is no longer dominance solvable as shown in the game matrix below.

If player 1 chooses 0 worst strategy for player 2 is 15. If player 1 chooses 9 worst strategy is 0 and if player 1 chooses 15 worst strategy is 9. Hence there is no dominated strategy and the game is not dominance solvable.

The best responses of each player to each of the pure strategies of the opponent are as follows: optimum strategy for p2 if p1 chooses 0 is 9, if p1 chooses 9 it is 15, and if p1 chooses 15 it is 0.