I don’t get anything from below except for (c). Please help me! Thanks.
Consider the following game matrix:
....................................................Player B............
............................................Left............Right.......
Player A.........Top..............(a, b)...........(c, d).......
.......................Bottom........(e, f)............(g, h).......
a) If (top, left) is a dominant strategy equilibrium, then we know that “a” is greater than ____, “b” is greater than _____, “c” is greater than _______ and “f” is greater than _______.
a > c , b > e , c > f , f > c
b) If (top, left) is a Nash equilibrium, then which of the inequalities from your answer in part (a) must be satisfied?
a > b and b > f
c) If (top, left) is a dominant strategy equilibrium, must it be a Nash equilibrium?
No
I'm having some trouble with your notation. Let me asssume that (x,y) means that x is the outcome going to player A, y is the outcome going to player B.
If top left is a dominant strategy equilibrium, it implies A perfers outcome Top and B prefers outcome Left. Ergo, I think a>e, b>d, c>g, f>h
Under a Nash equilibrium, a player cannot do better by switching. So if starting in Top-Left, The choice for player A is to move to Bottom-Left. If he does not move, it must be that a>e. Similarly, for B not to move, b>d.
We cant say if A would perfer c over g or vice-versa. Ditto, we cant tell if B would prefer f over h.
c) I think yes. A dominant strategy equilibrium must also be a Nash equilibrium.
To understand the answers to the questions, let's first have a brief explanation of dominant strategy equilibrium and Nash equilibrium.
In game theory, a dominant strategy equilibrium occurs when a player has a single strategy that yields a better outcome regardless of the strategy chosen by the other player(s). In this case, the player's dominant strategy is the one that they will always choose regardless of what the other player does.
On the other hand, a Nash equilibrium occurs when no player can improve their outcome by unilaterally changing their strategy, assuming all other players' strategies remain unchanged. In other words, it is a stable state of the game where no player has an incentive to deviate from their strategy.
Now let's look at each question and how to approach them:
a) If (top, left) is a dominant strategy equilibrium, then we know that "a" is greater than ____, "b" is greater than _____, "c" is greater than _______ and "f" is greater than _______.
To determine the inequalities, we need to compare the payoffs in the game matrix. Since (top, left) is a dominant strategy equilibrium, it means that regardless of Player B's strategy (left or right), Player A always chooses the strategy top. Therefore, we compare the payoffs in the first row (Player A's payoffs when choosing top).
From the game matrix, we can see that "a" is greater than "c" (a > c). Similarly, "b" is greater than "e" (b > e). Also, "c" is greater than "f" (c > f). However, we cannot compare "f" with any other payoff because it is in a different row.
Therefore, the correct answer is: a > c, b > e, c > f.
b) If (top, left) is a Nash equilibrium, then which of the inequalities from your answer in part (a) must be satisfied?
To determine the inequalities that must be satisfied for (top, left) to be a Nash equilibrium, we need to consider the payoffs for both players. In a Nash equilibrium, no player has an incentive to unilaterally change their strategy.
From part (a), we found that a > c, b > e, and c > f. To satisfy a Nash equilibrium, we need to ensure that Player A does not have an incentive to deviate from their strategy top. Since a > c, Player A will not want to change their strategy.
Moreover, we also need to ensure that Player B does not have an incentive to deviate from their strategy left. From the game matrix, there are no direct comparisons between the payoffs for Player B in the left column. However, we can infer that if a > c and b > e, then it must be the case that a > b as well. Otherwise, Player B would have an incentive to choose a different strategy.
Therefore, the correct answer is: a > b and b > f.
c) If (top, left) is a dominant strategy equilibrium, must it be a Nash equilibrium?
No, a dominant strategy equilibrium does not necessarily guarantee a Nash equilibrium. While a dominant strategy equilibrium is based on the best response of a player regardless of the opponent's strategy, a Nash equilibrium requires that both players simultaneously choose their best responses based on each other's strategies.
In other words, a dominant strategy equilibrium only considers the best response of one player, whereas a Nash equilibrium considers the best response of both players simultaneously. Therefore, a dominant strategy equilibrium can exist without being a Nash equilibrium.
I hope this explanation helps you understand the answers. If you have any further questions, feel free to ask!