<p>Yes, you are correct. All types <span class="math-container">$t_{1}$</span> choose O (B) and all types <span class="math-container">$t_{2}$</span>
choose O (B) are both Bayesian equilibria.</p>
<p>Note that there are other Bayesian equilibrium in this game, if you are interested this is explained in detail <a href="https://users.auth.gr/kehagiat/Research/GameTheory/02Courses/Course3/06-incomplete-info.pdf" rel="nofollow noreferrer">here</a> (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 <span class="math-container">$Opera$</span> if <span class="math-container">$t_{1}$</span> exceeds some critical value <span class="math-container">$x_{1}$</span> and chooses <span class="math-container">$Fight$</span> otherwise, and player 2 chooses to <span class="math-container">$Fight$</span> if <span class="math-container">$t_{2}$</span> exceeds some critical value <span class="math-container">$x_{2}$</span> and goes to the <span class="math-container">$Opera$</span> otherwise. To find the values <span class="math-container">$x_{1}$</span>, <span class="math-container">$x_{2}$</span> 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.</p>
<p><em>Game Theory: Static and Dynamic Games of Incomplete Information
Branislav L. Slantchev Department of Political Science, University of California – San Diego</em></p>

CC BY-SA 4.0
2019-04-28T16:32:41.997Z
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.
Giskard
CC BY-SA 4.0
2019-04-28T16:33:37.573Z
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.
Student
CC BY-SA 4.0
2019-04-28T16:47:48.510Z
I have modified it to include the idea and reference @Giskard, thanks for the tip
Ali
CC BY-SA 4.0
2019-04-28T16:49:34.910Z
@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.
Ali
CC BY-SA 4.0
2019-04-28T22:54:25.403Z
Being new to Bayesian games, it seems like methods for solving such games (including auctions) rely on us assuming some form for the strategies and then proceeding (an example is the threshold equilibrium in the above game). Would there be any exhaustive methods for solving Bayesian games?
Student
CC BY-SA 4.0
2019-04-28T23:20:23.233Z
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 [...]
Ali
CC BY-SA 4.0
2019-04-28T23:28:42.757Z
so for some value of the types (affecting payoffs) the expected payoff of a certain action is higher than for the remaining possible strategies, giving you an equilibrium. Exhaustive methods for solving Bayesian games thus essentially involve considering all possibilities. At times this is not an easy task (at least speaking from my own exposure) but in my humble opinion it really boils down to considering every possible scenario.
Ali
CC BY-SA 4.0
2019-04-28T23:32:12.657Z
Another consideration of course is whether you are being asked to find A Bayesian equilibrium or all of them. Either way, remembering how you play in pure strategies with complete information (i.e. player 1 plays opera and player 2 plays opera) coupled with the calculation of the expected value given the a priori probabilities given incomplete information is a good rule of thumb to find such equilibria.
Ali
CC BY-SA 4.0
2019-04-28T23:53:55.450Z
Thanks, I essentially meant assuming things such as symmetry, monotonicity etc, which is common in the auction games I've encountered. Monotonicity is sometimes necessary for an equilibrium strategy, but symmetry is not. So such methods are not comprehensive, is what I meant.
Student
CC BY-SA 4.0
2019-04-28T16:18:16.670Z
2019-04-28T18:12:22.647Z
2020-06-17T09:40:54.643Z
Ali
<p>Consider the static Bayesian game as described below. <span class="math-container">$\ t_1$</span> and  <span class="math-container">$\ _2$</span><a href="https://i.sstatic.net/uvyEo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uvyEo.png" alt="enter image description here" /></a> are the types of the row and column player respectively, which are both uniformly distributed on the interval <span class="math-container">$[0,1]$</span>. 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.</p>

<game-theory><bayesian-game>
Bayesian Nash Equilibria in Battle of Sexes
Student
# Bayesian Nash Equilibria in Battle of Sexes

(Asked by: username_0 on 2019-04-28)

Consider the static Bayesian game as described below. $\ t\_1$ and $\ \_2$[![enter image description here](https://i.sstatic.net/uvyEo.png)](https://i.sstatic.net/uvyEo.png) 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.

**Tags:** game-theory, bayesian-game


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## 1 answers

### Answer by username_1 on 2019-04-28. Score: 4 | Accepted answer

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](https://users.auth.gr/kehagiat/Research/GameTheory/02Courses/Course3/06-incomplete-info.pdf) (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*

> **Comments:**
> **username_2**: 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.  
> **username_0**: 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.  
> **username_1**: I have modified it to include the idea and reference @username_2, thanks for the tip  
> **username_1**: @username_0 yes, there is a Bayesian equilibrium with threshold values. I included a quick hint here but it is very well explained in the link.  
> **username_0**: Being new to Bayesian games, it seems like methods for solving such games (including auctions) rely on us assuming some form for the strategies and then proceeding (an example is the threshold equilibrium in the above game). Would there be any exhaustive methods for solving Bayesian games?  
> **username_1**: 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 [...]  
> **username_1**: so for some value of the types (affecting payoffs) the expected payoff of a certain action is higher than for the remaining possible strategies, giving you an equilibrium. Exhaustive methods for solving Bayesian games thus essentially involve considering all possibilities. At times this is not an easy task (at least speaking from my own exposure) but in my humble opinion it really boils down to considering every possible scenario.  
> **username_1**: Another consideration of course is whether you are being asked to find A Bayesian equilibrium or all of them. Either way, remembering how you play in pure strategies with complete information (i.e. player 1 plays opera and player 2 plays opera) coupled with the calculation of the expected value given the a priori probabilities given incomplete information is a good rule of thumb to find such equilibria.  
> **username_0**: Thanks, I essentially meant assuming things such as symmetry, monotonicity etc, which is common in the auction games I've encountered. Monotonicity is sometimes necessary for an equilibrium strategy, but symmetry is not. So such methods are not comprehensive, is what I meant.  

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