Game Theory 101 (#65): Solving for Bayesian Nash Equilibrium

This lecture shows how to use Nash equilibrium to find Bayesian Nash equilibrium. The trick is converting the multiple potential matrices into a single one, creating new strategies by using all combinations of strategies among the types. Once you have converted everything into a proper matrix, you can solve the game by using Nash equilibrium as you normally would.

[kyrilcouda]

You are contributing to the society way more than an average individual. Thank you.

[sorooshalavi]

man your explanation is very smooth for all the game theory courses! I always watch them when I have not understood something from the books. I wish most of the professors was like you :))

[ralphmcralphie]

William Spaniel my game theory exam is in an hour and I think I finally worked out BNE. I'm going to name my second child 'SpanielBNE2022' after you.

[realhumanbean]

The amount of students that this guy has taught is enormous. Hopefully I pass my micro class exam because the explanation in your videos is simpler than the one my professor gave us. Thank you!

[Amicallef]

Wonderful timing on this upload, since I have a midterm on it tomorrow and you are the best on youtube.

[olihd]

Student here doing a microeconomics course, I pay 10000£ a year to attend uni, and your 20minute videos have taught me more than ive learnt in the past 6 months at university. Safe g

[kevinshujianyang]

Hi William, thank you so much for uplaoding your lecture! They are amazing! I have a quick question though: Let's say I have transferred the original game to the 2x4 matrix game with columns labeled LL, LR, RL, RR. However, this time, the 1st column is not strictly dominated by the 3rd column, neither is the 2nd column strictly dominated by the 4th column. But we have that the 2nd column is strictly dominated by the 3rd column, and that this is the only strict domination in this 2x4 matrix game. In that case, should we eliminate the 2nd column and find all Nash Equilibria for the remaining 2x3 game? If so, what would be the intuition then?

[whatitmeans]

I have two questions about this example:
i) What is the interpretation of the value RL = 5/8 of the point of view of player 2??? I don't understand the relation with their strategy. Given player 1 have the same payoffs on both games, when calculating the mixed strategy of player 2, for both PD or SH kinds yoy find q_pd = q_sh = 1/2, so I was expecting that Player 2 will be playing half the time L and the other half R indepently from its own kind. Because of this, I don't understand what really means the RL = 5/8 value. Hope you can explain what I am doing wrong.
ii) It is possible to have the same game but with different payoffs for player 1 in each matrix?? or this will automatically leads to a 4x4 set of game matrices???
As an exercise to apply what you presented in the video, I was trying to solve a similar exercise with other probabilities and payoffs:
In a game of two players, Player 1 can choose Up or Down (U or D), and Player 2 can choose Left or Right (L or R). But Player 1 would confront a Player 2 of "kind I" with probability p(I)=0.2 in a game with payoffs [[(2, 1);(0, 0)][(0, 0);(1, 2)]], or confront a Player 2 of "kind U" with probability p(U)=0.8 in a game with payoffs [[(0, -1);(-1, 1)][(-1, 1);(1, -1)]].
Using the method shown in the video I reach a matrix for Player 1 where there are no dominant strategies' columns for Player 2, so they can't be deleted of the 2x4 matrix. And I don't know how to calculate the mixed strategy for Player 1 in a 2x4 matrix (there are 4 different linearly independent equations to find only one unknown variable). Hope you can explain this. Beforehand, thanks you very much.

[nawwafmj]

Thank you so much. I was struggling in my Game Theory Class, and you have helped me to understand every thing. I appreciate. Keep going....

[yashikarao]

Thank you soo much for explaining it in a simple way. i have my game theory exam tomorrow and your game theory playlist literally saved me from failing :')

[chiew]

Hi Sir, when will you do a video for signaling game?

[kyleiong]

constructing a new 2x4 payoff matrix by calculating the expected payoff 6:20
eliminate the ones that have been strictly dominated 11:10
left with a 2x2 matrix and calculate the ne/bne 12:50

[AM-emgz]

why don't we solve for the BNE at the previous video, ( nr.64) in this way as well?

[PhDHopeful]

This is a great approach in finding the MSBNE. As so many have noted, thanks so much for your time and efforts in making such great videos. Had a quick query. At around 1:20, one of the points you mentioned was that player one doesn't know which type player two is. Then, at around 5:23, there is the example clarifying when the combined matrix can apply, where the boss mentions to the solver, that we don't know which type we are. Would be kind of you if you can clarify which perspective was the example from. As in, the example, when the boss says that they don't know if we are of type PD or SH, are the boss and solver both player two? In that case, along with player one, is it that in this example, even player two doesn't know his/her type (ex-ante)?

[brianward]

This video is great, but I think one thing is confusing. Although player 1 has “prisoner dilemma preferences” in the PD game, they game is itself not a prisoner’s dilemma.

It has the same result as a PD tho: (down, right) is the unique NE, and the players would prefer to “cooperate” to (up, left). But the reason isn’t the same: only player 2’s “right” strategy is strictly dominant and player 1’s “down” strategy is the choice given that but it’s not strictly dominant.

Correct me if I’m wrong, but I found that part confusing and thought this would help others!

[chloelivingston]

This video helped me understand so much better! Thank you!

[vulcanraven]

Thank you William you're the best

[gladstonearantes]

Hi, William! Your classes are simply amazing! Years ago you published it and there are still people learning from them! Thank you.
I have a doubt: you mentioned that this is a method for solving a subset of all possible Bayesian games. How do you characterize this subset? Did you mention that? Thanks again.

[asgv]

What if for example player 1's move up strictly dominated down? How would this look in the combined matrix? Can we use this technique in that case? How can we approach this if we can't use this technique? I am doing a question where that occurs and I cannot figure out how to make a 2x2 from the combined matrix.

[anweshadas]

what happens if player 2 does not have strictly dominant strategy in any one of the type? how to proceed