Question

In a two-player, one-shot, simultaneous-move game, each player can choose strategy A or strategy B. If both players choose strategy A, each earns a payoff of $400. If both players choose strategy B, each earns a payoff of $200. If player 1 chooses strategy A and player 2 chooses strategy B, then player 1 earns $100 and player 2 earns $600. If player 1 chooses strategy B and player 2 chooses strategy A, then player 1 earns $600 and player 2 earns $100.

a.       Write this game in normal form.

 

b.       Find each player’s dominant strategy, if it exists.

 

c.       Find the Nash equilibrium (or equilibria) of this game.

 

d.       Rank strategy pairs by aggregate payoff (highest to lowest).

 

e.      Can the outcome with the highest aggregate payoff be sustained in equilibrium? Why or why not?

Answer 1

(a)

The normal-form game with the two-player, one-shot, simultaneous–move game with each player choosing strategy A or strategy B along with the possible outcome is given by:

(b)

It shall be noted that a strategy is a dominant strategy if it results in the highest payoff regardless of the action of the opponent.

In order to determine if Player 1 has a dominant strategy, consider the following argument:

If the Player 2 chooses a strategy A, the best choice by Player 1 is strategy B as choosing strategy B would fetch it a payoff of $600 which is higher than the payoff fetched by strategy A

If the Player 2 chooses a strategy B, the best choice by Player 1 is strategy B as choosing strategy B would fetch it a payoff of $200 which is higher than the payoff fetched by strategy A

This shows that no matter whatever strategy is chosen by Player 2, there is a specific strategy B for Player 1 which would provide it the highest payoff regardless of the strategy chosen by Player 2

Hence, the Player 1 has a dominant strategy, that is, strategy B

In order to determine if Player 2 has a dominant strategy, consider the following argument:

If the Player 1 chooses a strategy A, the best choice by Player 2 is strategy B as choosing strategy B would fetch it a payoff of $600 which is higher than the payoff fetched by strategy A

If the Player 1 chooses a strategy B, the best choice by Player 2 is strategy B as choosing strategy B would fetch it a payoff of $200 which is higher than the payoff fetched by strategy A

This shows that no matter whatever strategy is chosen by Player 1, there is a specific strategy B for Player 2 which would provide it the highest payoff regardless of the strategy chosen by Player 1

Hence, the Player 2 has a dominant strategy, that is, strategy B

(c)

Since, both the players 1 and 2 have strategy B as the dominant strategy, the Nash equilibrium of this game is {Player 1 choosing strategy B, Player 2 choosing strategy B}

(d)

The strategy pairs along with aggregate payoff (highest to lowest) are shown as below:

(e)

The outcome with the highest aggregate payoff is {Player 1 choosing strategy A, Player 2 choosing strategy A}

This outcome could not be sustained in equilibrium because; the case is that of one-shot simultaneous-move game where each player makes its decision in its self-interest without the knowledge of the other players’ decision, but in the environment of interdependence and absence of co-ordination. In such situation, the participating players know that one player can gain only at the expense of other player. Going further, each player’s dominant strategy is B and that in one-shot game each player would have an incentive to cheat on any collusive arrangement.