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Consider the static Bayesian game as described below. $\ t_1$ and $\ 𝑡_2$enter image description here are the types of the row and column player respectively, which are both uniformly distributed on the interval [0,1]. The first part of the question is asking us to find a Bayesian Nash Equilibrium. Trivially, don't the the top left and bottom right corners correspond to equilibrium outcomes? Unless I've misunderstood the definition of a BNE.

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Yes, you are correct. All types $t_{1}$ choose O (B) and all types $t_{2}$ choose O (B) are both Bayesian equilibria.

Note that there are other Bayesian equilibrium in this game, if you are interested this is explained in detail here (p. 10, see reference below) for this particular battle of the sexes with two-sided incomplete information. The basic idea is to note that in this game, each player has a continuum of types, and so the set of types is infinite. You can look for a Bayesian equilibrium in which player 1 goes to the $Opera$ if $t_{1}$ exceeds some critical value $x_{1}$ and chooses $Fight$ otherwise, and player 2 chooses to $Fight$ if $t_{2}$ exceeds some critical value $x_{2}$ and goes to the $Opera$ otherwise. To find the values $x_{1}$, $x_{2}$ that make these strategies a Bayesian equilibrium you can calculate each player's expected payoffs given the other player's strategy and find the optimal values based on this.

Game Theory: Static and Dynamic Games of Incomplete Information Branislav L. Slantchev Department of Political Science, University of California – San Diego

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    $\begingroup$ Ideally you would post a short description of the linked content, because links break over time. You can give a name that people can google, quote, etc. $\endgroup$ – Giskard Apr 28 at 16:32
  • $\begingroup$ Fantastic. I must point out that the question had a hint about threshold values which is what confused me. Perhaps that is covered in your link. $\endgroup$ – Student Apr 28 at 16:33
  • $\begingroup$ I have modified it to include the idea and reference @Giskard, thanks for the tip $\endgroup$ – user20105 Apr 28 at 16:47
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    $\begingroup$ @Student yes, there is a Bayesian equilibrium with threshold values. I included a quick hint here but it is very well explained in the link. $\endgroup$ – user20105 Apr 28 at 16:49
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    $\begingroup$ Well I am not sure what you mean by assuming some form of the strategies, as the set is fixed by the very game setup. The essential idea that you want to ask yourself, as with any Nash equilibrium, is in what situation will neither of the players (in this case 2) have a profitable deviation? So obviously the most straight forward answer is, as you pointed out, the situation in which {Opera, Opera} and {Fight, Fight}. However, rational choice implies that you take into account the strategy of the other player given a priori knowledge of the probability of his type [...] $\endgroup$ – user20105 Apr 28 at 23:20

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