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I want to model the situation where there are two players, and they both have two types, they know their own type but not the other's, the only way I can think is to put two "nature" players, but it seems not to be right, so can anyone give me a suggestion about how to model this? I looked through textbooks but were unable to find the answers, please help.


Sorry for lack of details, I looked through the textbooks, the typical type of a game with incomplete information is: the nature choose a state(or the type of, say, player1), and the player 1 knows the type of himself(weak or strong to retaliate the other), player 2 doesn't know player 1's type, but I want to model that: two players know about their own type, but not the other's type, so they are uncertain about doing something bad because they fear the other player has strong power to retaliate.

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    $\begingroup$ You have not given us nearly enough information to help. Can you carefully describe a simplified version of your game that makes the problem clear? Do that by editing the question, not in comments. $\endgroup$ – Ethan Bolker Sep 21 '17 at 12:33
  • $\begingroup$ It's hard to say any definite with so little detail, but a mixed strategy might be useful. Both players pursue a strategy that is a mix of two strategies $x$ and $y$ (possibly weighing one strategy with odds zero and one with odds one). Both players know $x$ and $y$ but not the particular combination pursued by the opponent. $\endgroup$ – Stella Biderman Sep 21 '17 at 12:35
  • $\begingroup$ Is this similar to pokemon (for lack of a better example) where certain types are stronger or weaker against other types? $\endgroup$ – user472341 Sep 21 '17 at 12:38
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Assume individual type spaces $T_1 = \{t_1^1, t_2^1 \}$ and $T_2 = \{t_2^1, t_2^2 \}$. The state space is $T = T_1 \times T_2$.

You can visualize $T$ as a $2 \times 2$ matrix: \begin{array}{|c|c|} & t_2^1 & t_2^2 \\ \hline t_1^1 & & \\ \hline t_2^1 & & \\ \hline \end{array}

Now you can represent players' beliefs by means of an ex ante arbitrary probability distribution on the state space $T$. For example: \begin{array}{|c|c|} & t_2^1 & t_2^2 \\ \hline t_1^1 & a & b \\ \hline t_2^1 & c & d \\ \hline \end{array} with $a,b,c,d \ge 0$ and $a+b+c+d = 1$.

Given a type $t_k^i$, player $i$ has belief conditional on his type. For instance, type $t_1^1$ knows his type so his beliefs on the type of 2 (after conditioning on his knowledge) are given by \begin{array}{|c|c|} & t_2^1 & t_2^2 \\ \hline t_1^1 & \frac{a}{a+b} & \frac{b}{a+b} \\ \hline \end{array}

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  • $\begingroup$ This is brilliant, I understand how to represent the beliefs now, but how to represent this type of game in extensive-form ? because the player "nature" can only have two branches(correct me if i'm wrong), how can the player "nature" choose the combination of players of different type? for example, {t11,t12}, {t21, t12},{t11, t22} {t21, t22} $\endgroup$ – The R Sep 22 '17 at 8:17
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    $\begingroup$ In the initial node have start with Nature having four branches (one for each pair of types) with initial probabilities $a,b,c,d$. $\endgroup$ – mlc Sep 22 '17 at 8:40

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