So i have this question
I have some learning disabilities and have no clue whatsoever how the best reply of each player is what it is or how all players will demand the values shown in the answer sheet .
Can somebody please explain in the simplest way possible how to get to these answers? Please please please make the explanation as simple as you can.
Thanks!
Ok. Let's say you are player $i$ demanding $x_i$ bandwidth.You and all other players need to send $P$ packets and of course you need to send it all in minimum time or minimum number of times. Let's say $P=1$. If you are given the whole channel($x_i =1$) you can send it in one shot. So $1$ is your answer. In general, since you have to split the packet(s) it will take $\frac{P}{x_iP} = \frac{1}{x_i}$ attempts. But, practically the full channel is not all yours. Assuming, only one other Player besides you, you can split the channel in half and have $x_i = \frac{1}{2}(1)$ and that would let you and the other player happy with his half as well. In general, your best reply is to have half. But the actual sending capability is $1 - \sum_{j=1}^n{x_j}$ instead of $1$. If you don't include your $x_i$ then $1 - \sum_{j \ne i}^n{x_j}$ would be the channel capacity so your best bet is that all players(symmetrically) go for a strategy profile which maximizes the throughput or minimize "sending" attempts which is $\frac{1}{2}(1 - \sum_{j \ne i}{x_j})$. if they deviate from this strategy then the packets $P$ cannot be sent in minimum number of tries and would violate the Nash Equilibrium. Somehow, you have to show that the Nash equilibrium is violated by proving that it takes more attempts, or not profitable for the player by selecting a different strategy for a different $x_i$. So, every player has the same strategy $x_i^*$. Now, with, this you can assume there is some $x_i = \frac{1}{k}$ and it is the same for all players. So, the best reply becomes $$x_i = \frac{1}{2}(1 - \sum_{j \ne i}{x_j})$$ $$=>\frac{1}{k} = \frac{1}{2}\left(1 - \sum_{j \ne i}{\frac{1}{k}}\right)$$ $$=>\frac{1}{k} = \frac{1}{2} - \frac{1}{2}{(n-1)}{\frac{1}{k}}$$ The last step there are $n-1$ excluding $x_i$. Simplifying $=> k = n+1$, giving $x_i^* = \frac{1}{n+1}$. Substituting for utility of each player $x_i$, $$u_i(x^*) = \frac{1}{n+1}(1 - \frac{n}{n+1}) = \frac{1}{(n+1)^2}$$...