In the payoff matrix below the rows correspond to player A\'s strategies and the

Question

In the payoff matrix below the rows correspond to player A's strategies and the columns correspond to player B's strategies. The first entry in each box is player A's payoff and the second entry is player B's payoff. (a) Find all pure strategy Nash equilibria of this game. (b) Notice from the payoff matrix above that Player A's payoff from the pair of strategies (U, L) is 3. Can you change player A's payoff from this pair of strategies to some non-negative number in such a way that the resulting game has no pure-strategy Nash equilibrium? Give a brief (1-3 sentence) explanation for your answer. (Note that in answering this question, you should only change Player A's payoff for this one pair of strategies (U, L). In particular, leave the rest of the structure of the game unchanged: the players, their strategies, the payoff from strategies other than (U, L), and B's payoff from (U, L).)

Explanation / Answer

For Player A:

If player B chooses L or R, player A must choose D (dominant strategy).

For Player B:

Player B does not have a dominant strategy. If player A chooses U, B should choose R and if A chooses D, B should choose L.

So, we find Nash equilibrium at the cell (2,1).

We won't have a Nash equilibrium, if A also doesn't have a dominant strategy, and he chooses U for B's L and D for B's R (then there would be no intersection of choices).

So, we can modify (3,3) to (5,3) to acquire that.

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