In a 2-player game the payoff function of player 1 is given by u subscript 1 left parenthesis s subscript 1 comma s subscript 2 right parenthesis equals left parenthesis 4 minus s subscript 1 right parenthesis s subscript 2 plus 8, where s subscript 1 denotes a strategy of player 1 and s subscript 2 a strategy of player 2. Suppose that S subscript 1 equals S subscript 2 equals left square bracket 7 comma 9 right square bracket (that is, all real values from 7 to 9, including 7 and 9) are the strategy sets of players 1 and 2, respectively. Write down the strategy of player 1 which is strongly dominant. You must format your answer as follows: Enter 8 for strategy s subscript 1 equals 8, enter 8.5 for strategy s subscript 1 equals 8.5, and so on. If none of player 1's strategies is strongly dominant, enter negative 1.
Question
In a 2-player game the payoff function of player 1 is given by u subscript 1 left parenthesis s subscript 1 comma s subscript 2 right parenthesis equals left parenthesis 4 minus s subscript 1 right parenthesis s subscript 2 plus 8, where s subscript 1 denotes a strategy of player 1 and s subscript 2 a strategy of player 2.
Suppose that S subscript 1 equals S subscript 2 equals left square bracket 7 comma 9 right square bracket (that is, all real values from 7 to 9, including 7 and 9) are the strategy sets of players 1 and 2, respectively.
Write down the strategy of player 1 which is strongly dominant. You must format your answer as follows: Enter 8 for strategy s subscript 1 equals 8, enter 8.5 for strategy s subscript 1 equals 8.5, and so on. If none of player 1's strategies is strongly dominant, enter negative 1.
Solution
A strongly dominant strategy is one that provides the highest payoff to a player, no matter what the other player does. In this case, the payoff function for player 1 is given by u1(s1, s2) = (4 - s1)s2 + 8.
To find the strongly dominant strategy for player 1, we need to find the value of s1 that maximizes this function, given that s2 can be any value between 7 and 9 (inclusive).
Taking the derivative of the payoff function with respect to s1, we get:
du1/ds1 = -s2
Setting this equal to zero, we find that the maximum occurs when s2 = 0. However, this is outside the strategy set for player 2, which is [7, 9]. Therefore, the payoff function is decreasing in s1 for all s1 in the strategy set of player 1.
This means that the value of s1 that maximizes player 1's payoff given the strategy set of player 2 is the smallest possible value of s1, which is 7.
Therefore, the strongly dominant strategy for player 1 is s1 = 7.
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