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I'm solving this question, where I'm supposed to find all the BNE for the following game. There are two players, 1 and 2, where 1 has type a and b, and 2 has just one type (so no private information). The probability that the type of player 1 will be a is 2/3. Then the payoff matrix is as follows (the left one is for type a, right one is for type b). enter image description here

My thoughts: I have first attempted to solve the pure strategies, which I think is (UU, R). I'm stuck on solving the mixed strategies part. I have found out that for both matrices, player 2 has a strictly dominating strategy (respectively L and R). But I cannot find a way to use this information since player 2 would not know the type of player 1. I've looked at other posts about solving mixed BNEs, but can't quite apply them here. Is there a way to find the mixed strategies?

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  • $\begingroup$ This is more or less a homework question. Please consider elaborating on what you have tried or read more on Bayes Nash games and work from first principles. See if you can follow along with the first example given in these lecture slides economics.mit.edu/sites/default/files/inline-files/… $\endgroup$
    – Kitsune Cavalry ♦
    Commented 2 days ago
  • $\begingroup$ I was confused because most slides- including the MIT one- did not consider mixed strategies, but I was able to get the answer using the below hint. $\endgroup$
    – Woonseo Paek
    Commented 2 days ago

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For this type of question, it may be easier to work with the ex ante normal form:

\begin{array}{|c|c|c|} \hline & L & R \\\hline UU & -1, 1 & 1, 1 \\\hline UD & -\frac23, \frac43 & \frac23, \frac53 \\\hline DU & -\frac13, \frac73 & \frac13, \frac73 \\\hline DD & 0, \frac83 & 0, 3 \\\hline \end{array}

where the payoff for, say $(UD,L)$, is calculated by \begin{equation} \frac23\underbrace{(-1,2)}_{u(U,L;a)} + \frac13\underbrace{(0,0)}_{u(D,L;b)} = \left(-\frac23, \frac43\right). \end{equation}

The BNE of the incomplete information game is the NE of this ex ante normal form game. I'll leave the rest to you.

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  • $\begingroup$ Thank you! I was able to solve the question by just finding the mixed strategy NE of the above matrix $\endgroup$
    – Woonseo Paek
    Commented 2 days ago
  • $\begingroup$ The question contains an error. It is the column player that has two types (as is obvious from looking at the payoff matrices). $\endgroup$
    – smcc
    Commented yesterday
  • $\begingroup$ @smcc: You are probably right in this case. However, I don't think, at least in principle, that it's necessary for the payoffs of the informed side to vary across states. I'll wait a bit for clarification from the OP to decide whether to remove/revise my answer. $\endgroup$
    – Herr K.
    Commented yesterday
  • $\begingroup$ I double-checked, player 1 (row player, player with {U,D}) has two types in this question. $\endgroup$
    – Woonseo Paek
    Commented 22 hours ago
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Your question says "1 has type a and b, and 2 has just one type", but I think it should be "where 2 has type a and b, and 1 has just one type". From the payoff matrices, the row player has the same payoffs in both matrices so must be the player with just one type. This means the strategy profile you describe $(UU, R)$ doesn't make sense as it is the column player for whom you have to specify two strategies.

As you noticed, each type of the column player has a strictly dominant strategy and so they must play it in equilibrium. Hence the equilibrium strategy profile of the column player is LR. For the row player, the payoff of $D$ is always $0$ and given the strategy of the column player, the expected payoff from $U$ is

$$ \frac{2}{3}(-1)+\frac{1}{3}(1)=-\frac{1}{3}$$

Therefore the only equilibrium is $(D,LR)$.

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