I'm studying Game theory, I saw the question:
Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference between the number chosen by player A and player B is less than 2, than player A pays 1 dollar to player B; if not then player B pays 1 dollar to player A.
For example, if player A chooses 1 and player B chooses 3 then difference is 2, so player B pays to player B. If instead of 3, player A chooses 1 or 2 than player A would pay player B.
Firstly, how to model this situation as a strategic game and find the Nash equilibria of this game. And how should I explains why it is reasonable for player B to play according to the Nash equilibria strategy for his part.
Thank you for helping.
Start by writing out the pay-off matrix (the table where you have the strategies for both players). In this case the stategies are just $1, 2$ and $3$ for both players, so it's a $3\times 3$-matrix. Do you know how to find out the equlibria from there?
EDIT:
The pay-off matrix tells you what is the pay-off for both player given the strategies the played (the number chosen in this case). The matrix looks like this:
$ \left( \begin{array}{ccc} -1/1 & -1/1 & 1/-1 \\ -1/1 & -1/1 & -1/1 \\ 1/-1 & -1/1 & -1/1 \end{array} \right) $
Here the row index is the player $A$'s choise and column index player $B$'s choise. The pay-offs (dollars gained or lost) are given as $A/B$.
To find out the Nash equilibrium go through the choises and check if the player's pay-off can be made greater by the act of the other player changing his/her strategy. If not, then you know that that is equilibrium. (Check some source for better explanation of Nash equilibrium... :D)
Quite clearly, you can see that in this case player $B$ should pick $2$, since he wins always by picking $2$.