# Strategic game with complete informaation

Consider the following strategic game with complete information played by three players. Each player $$i ∈ {1, 2, 3}$$ chooses her action from $$A = \{1, 2, . . . , 10\}$$. Utility functions, mapping each action profile $$(a_1, a_2, a_3) ∈ A^3$$ into utils, of the three players are as follows: $$u_1(a_1, a_2, a_3)=-|a_3-a_1|+|a_2-a_1|$$ $$u_2(a_1, a_2, a_3)=-|a_1-a_2|+|a_3-a_2|$$ $$u_3(a_1, a_2, a_3)=-|a_2-a_3|+|a_1-a_3|$$

The given solution is as follows:

Suppose $$a_1 < a_3$$. Straightforward argument shows that the set of actions that constitute pure best response for player 2 is $$\{1, . . . , a_1\}$$. When $$a_1 > a_3$$, the set is $$\{a_1, . . . , 10\}$$ and when $$a_1 = a_3$$, then the set is $$\{1, . . . , 10\}$$.

This solution is too short for me to understand how to start solving it. I understand the conditions when one is >,< or = but I do not seem to follow how the BR is calculated

• So you don't understand "Suppose $a_1 < a_3$. Straightforward argument shows that the set of actions that constitute pure best response for player 2 is $\{1, . . . , a_1\}$."? Have you tried substituting numbers to see if that makes it easier for you to understand? E.g., do you see why if $a_1 = 5, a_3 = 8$, and Player $2$'s payoff function is $$u_2(5, a_2, 8)=-|5-a_2|+|8-a_2|$$ they would never play numbers larger than 5? Jan 24 at 15:52
• "When $a_1 > a_3$, the set is $\{a_1, . . . , 10\}$", so yes. Jan 24 at 16:27
• I’m voting to close this question because it is a specific self-study problem and was resolved in the comments. Jan 24 at 18:26
• If the OP has solved the question, they could answer it themselves so we have a completed Q and A instead of closing off the question? Jan 25 at 16:26
• @Giskard I am slightly lost for words. Jan 26 at 14:49

Suppose $$a_1 < a_3$$. Straightforward argument shows that the set of actions that constitute pure best response for player 2 is $$\{1, . . . , a_1\}$$. When $$a_1 > a_3$$, the set is $$\{a_1, . . . , 10\}$$ and when $$a_1 = a_3$$, then the set is $$\{1, . . . , 10\}$$.
Suppose we set $$a_1=5$$ and $$a_3=3$$ as in the case when $$a_1>a_3$$, as described in the comment as well $$u_2(5, a_2, 3)=-|5-a_2|+|3-a_2|$$ then player 2 would be better playing from $$\{a_1,....10\}$$ and vice versa.