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.

  • $\begingroup$ The example in the third paragraph is clearly malformatted as it doesn't reflect what you said above. $\endgroup$ – G. Bach Apr 28 '14 at 17:25

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?


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 $2$.

  • $\begingroup$ No I dont understand. I am studying from J. Osborne's book. But it is not clear enough for me. Please teach me these clearly. Thanks $\endgroup$ – user315 Apr 28 '14 at 17:22
  • $\begingroup$ How to write this matrix as well? $\endgroup$ – user315 Apr 28 '14 at 17:23
  • $\begingroup$ The matrix is just a $3 \times 3$ matrix where $M_{(i,j)}$ equals the outcome when $A$ plays $i$ and $B$ plays $j$; e.g. $M_{(1,3)}$ would be $(1,-1)$ and $M_{(2,3)}$ would be $(-1,1)$. Since it's a zero sum game, you can just put the outcomes for one player. $\endgroup$ – G. Bach Apr 28 '14 at 17:30
  • $\begingroup$ Thank you so much:)) by the way, can you have any book suggestion on game theory for beginner level? Dear @Ploosu2 $\endgroup$ – user315 Apr 28 '14 at 17:42
  • 1
    $\begingroup$ This is very good video lecture in my opinion: oyc.yale.edu/economics/econ-159 $\endgroup$ – ploosu2 Apr 28 '14 at 17:44

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