Question

1.Use the following payoff matrix for a one-shot game to answer the accompanying question.

a. Determine the Nash equilibrium outcomes that arise if the players make decisions independently, simultaneously, and without any communication. Which of these outcomes would you consider most likely? Explain.

b. Suppose player 1 is permitted to “communicate” by uttering one syllable before the players simultaneously and independently make their decisions. What should player 1 utter, and what outcome do you think would occur as a result?

c. Suppose player 2 can choose its strategy before player 1, that player 1 observes player 2’s choice before making her decision, and that this move structure is known by both players. What outcome would you expect? Explain.

Short answer

1. Strategy A, X (25,25) and strategy B, Y (15,15)
2. (25,25).
3. The maximum possible payoff is (25.25)

Step-by-step solution

To Finding the Nash equilibria 

On the payoff matrix for a one-shot game with two players who establish their decision independently, simultaneously and without any information about the opponent, we can establish two Nash equilibria.

• If Player 1 chooses option A, Player 2 will choose option X, otherwise he would lose. Therefore, in this strategy A, X (25,25) will be the first Nash equilibrium since it is the most optimal decision for both players.

• If Player 1 chooses B, Player 2's best option will be to choose Y. Similarly in this strategy both players have the highest possible payoff.Therefore, the strategy B, Y (15,15) will be the second Nash equilibrium.

Given that the players do not have any type of contact or know the other's move, the safest and most reliable decision is the second Nash equilibrium with strategy B, Y(15,15). Here, both players obtain benefits, since on equilibrium A, X one of the players can cheat and generate losses to the rival player.

Finding the result outcome

1. If player 1 is allowed to communicate with player 2 using a single syllable, he must say 'A' if he seeks to obtain the highest payoff possible. The player 2 already knows that player 1 will choose A, so he can choose option X and obtain the highest payoff. Thus, we obtain a Nash equilibrium where both players obtain (25,25).

Finding the expected outcome

If Player 2 can make his decision before Player 1 but Player 1 can see Player 2's strategy, we will be in the presence of a backward induction Nash equilibrium where the last decision will be observed and based on this the first decision will be made.

Therefore, Player 2 will choose option X in which he has a maximum payoff, and Player I, observing this, will choose option A which also generates the maximum possible payoff (25.25). Thus, this condition is generating a perfect subgame in the Nash equilibrium.