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I want an example of a completely mixed Bimatrix game. I have no clue how to approach. I guess it's a trial and error process. A completely mixed game is one where every optimal strategy (equilibrium strategy) of either player( considering 2 player game) is completely mixed. It can't have pure strategy, so the entries of the matrix have to be chosen accordingly.

It seems that Matching Pennies is one such celebrated example. Can someone come up with another such example just from the definition of completely mixed Bimatrix game? That is I want the intuition behind coming up with such an example. For example if the Payoff matrix is $3*3$ then keeping track of all mixed strategies become difficult.

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    $\begingroup$ Matching pennies is one such example. Perhaps you have in mind some additional constraints. $\endgroup$
    – mlc
    Mar 26, 2017 at 19:14
  • $\begingroup$ In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. $\endgroup$
    – mlc
    Mar 27, 2017 at 11:17
  • $\begingroup$ For $3 \times 3$ game, Rock-Paper-Scissors is a famous example. But see my former comment: it is not clear to me what is the goal of your question. $\endgroup$
    – mlc
    Mar 27, 2017 at 11:18

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Well, every game must have at least one equilibrium. So, to find a completely mixed game we just need to find a game with no pure strategy equilibria. All this means is that the set of squares on the matrix for player 1 that are their best responses are disjoint from the best responses for player two.

So consider a blank 3x3 matrix for player one. Then fill in player One's best responses. We will mark these with 1's. So for example we could have $$\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 &0 \end{bmatrix}$$ In that case player 1's best response is to play the middle unless player two plays the middle, whence player one should play the left. Now, just pick any other spots for player 2's best responses (choosing from the options top, middle, and bottom), the only key is that there can be no intersections. So, for example, the best responses for player 2 could be $$\begin{bmatrix}1 & 0 & 0 \\ 0& 1 & 0 \\ 0 & 0 &1 \end{bmatrix}$$

Or, anything else, as long as the two have no intersections. Then just fill in any numbers that satisfy the 1s the above matrices being the best responses. So for example:$$ \begin{bmatrix}(1,5 ) & (4,3) & (3,0) \\ (4,1)& (1,5) & (2,0) \\ (3,2) & (4,1) &(0,1) \end{bmatrix} $$

Is a 3x3 bimatrix game with no pure strategy equilibria. But this method could easily generate infinitely many.

(By the way the example game only has one equilibrium, where player 1 plays the middle $\frac15$ of the time, and right $\frac45$ of the time, and where player two plays the top $\frac34$ of the time while playing the middle $\frac14$ of the time.)

I hope that helps.

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