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

gametheory101.com/courses/game-theory-101/
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.

[@stahu17]

no cap

[@ralphmcralphie1276]

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.

[@julianfrank9673]

Same situation now for me. Now

[@bentongrayson]

Keeping this alive, mine is in 4 hours.

[@efggaming28]

Mine is in 15 minutes

[@ryaningold7737]

Mine is in 7 hours

[@sorooshalavi3156]

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 :))

[@Amicallef94]

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

[@PunmasterSTP]

How did your midterm go?

[@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....

[@PunmasterSTP]

How did the rest of your class go?

[@chloelivingston2552]

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

[@chiew1995]

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

[@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.

[@AM-em7gz]

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

[@yashikarao3168]

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 :')

[@tamimchamaa4145]

Why am I having other numbers in the (up, RL)-box?? Because I calculated them and had other values...

[@olihd2423]

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

[@realhumanbean21]

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!

[@kevinshujianyang4838]

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?

[@keikodes]

Thank you for teaching I love it!

[@vulcanraven9701]

Thank you William you're the best

[@OrcaChess]

Am I Right? That 4 x 2 Matrix is mi Normal Form representation of the Game.

[@asgv2895]

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.

[@jimcuddy7407]

Thanks a lot!

[@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.

[@Gametheory101]

It's limited to things you can actually put into a matrix form. Something like Hotelling's game wouldn't work with this because the strategies any given type can choose are infinitely many.

[@gladstonearantes]

@@Gametheory101 Thanks, William! If I understood well, this is still a powerful tool.

[@user-fe6vi3yg8i]

@@Gametheory101 Does having/not having dominated strategies matter? I haven't thought of an example yet, but if in the same matrix, without iterated elimination of any dominated strategies, if we were still left with 4 columns, we would still proceed to solve the mixed strategy BNE?

[@mjja61]

Hi Willliam. When will you upload Perfect Bayesian Equilibrium?

[@Gametheory101]

+mjja61 Probably months, and definitely not before this academic school year finishes---I have ~9 more lectures of BNE to do, and I also have a cross-country move to plan.

[@user-fe6vi3yg8i]

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)?

[@anweshadas9778]

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

[@kyleiong7311]

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

[@victorialiao3154]

Hi, thank you very much for your lecture. I am quite confused that why right-right-down is labelled as one of the best choices by player 1? Right Left Down is 1.8 and is larger than 1. Happy new year

[@Gametheory101]

Player 1 can only change his strategy. Down is the best he can do in response to RR. He can't control player 2's strategy to shift her to playing RL.

[@victorialiao3154]

LoL thank you very much for your prompt reply!

[@reyyan2462]

why did we choose up and rl exactly? I think I'm missing st

[@atan6105]

saver! much better than others

[@azzazidi6912]

Can you please do the signaling and bayesian learning in sequential games

[@Gametheory101]

+Azza Zidi I will be, but not before this academic school year finishes---I have ~9 more lectures of BNE to do, and I also have a cross-country move to plan.

[@keepmaking1309]

are you excited for the pittsburgh winter? hahaha

[@Gametheory101]

I lived in Rochester for five years, so I'm mostly indifferent at this point. More worried about how few sunny days Pittsburgh has every year.

[@keepmaking1309]

No, pitt gets sun alright. but this heat legit rises to kill. It's so hot here right now... :( So make sure you have an AC installed. hahahaa

[@Gametheory101]

I grew up in a desert, so I think I am covered there. :)

[@noegnoes]

have a small question, How did you get MSBNE : UP = 1/2 and RL = 5/8?

[@ken2cky]

Asking myself the same question...

[@loicjent8979]

For UP: By solving the following equation: U(RL) = p(UP)*3.2 + (1-p(UP))*0.4 = U(RR) = p(UP)*2.4 + (1-p(UP))*1.2 yields p(UP)=0.5 as a result. For RL: By solving the following equation: U(UP) = p(RL)*2.4 + (1-p(RL))*0 = U(DOWN) = p(RL)*1.8 + (1-p(RL))*1 yields p(RL)=0.625 as a result

[@valiantlydefault]

i don't think i get the idea behind these equations... we search for the probability of player 1 choosing up by using the payoffs of player 2??

[@ziyangfu2055]

The concept of making the other indifferent between his or her own strategies because you are averaging his utility using his utility values

[@lamortexotique]

Check previous videos on Mixed Strategies.

[@faith...5241]

Thank you so much!

[@theentertainer1097]

Sorry, i didn't see the previous vids; I don't just get which game is the "SH".

[@lamortexotique]

Stag Hunt

[@asmit2543]

IS there any example on Perfect Bayesian?? or a video ?

[@Gametheory101]

Haven't gotten to that yet.

[@asmit2543]

thank you anyway.... your videos are really helpful!

[@jamesfranco2844]

Does Player one know its own typ?

[@avikasliwal1541]

yes

[@danielklein5560]

if both players have two types, could i just make an 4x4 bimatrix? like LL LR RL RR and XX XY YY YX

[@Gametheory101]

Correct. And best of luck trying to solve such a monstrosity!

[@elietteadolfsen8618]

@@Gametheory101 what about a 3x3 where one player has two types?

[@marishaghosh6611]

Not sure but U,RL cell should be (2.4, 2.4) since 4*0.2 + 2*0.8 = 0.8+1.6 = 2.4 Am I wrong?

[@user-fe6vi3yg8i]

U, RL for player one will be 0.2*0+0.8*3=2.4. But for player two, it will be 0.2*4+0.8*3 (and not 0.8*2)=3.2. So, (2.4, 3.2) is accurate for (U, RL). The payoff for player two, when playing L for SH is 3 and not 2. Could be a visual trick, where 2 is the payoff for playing R in the SH game for player two, which is in the same row when player one plays U. I had made a similar error in one of the cells I worked through too :D. The matrix fights when you dig into it...

[@satibldrcn1277]

Teşekkürler, çok güzel anlatım.

[@fernandaroig2964]

What would happen if

[@PunmasterSTP]

if what?

[@ben2591]

„Stag Hunt“?? What the hell does that mean

[@TheOtherOne122]

Apyrenum watch previous video, I think number 5. Game in which both players have highest payoff by cooperating

[@stahu17]

LETS LEARN SOME GAME THEORY!!!

[@truth1901]

We all know and agree through observation that physical is slowed down energy. And we all agree that energy can become conscious. But some have a belief with no proof that such happened by chance and without intelligent manipulation.

[@HimanshuSharma-zu2ku]

Exactly the reason to believe that it happened by chance.

[@PunmasterSTP]

Man there were a lot of calculations and concepts in this video, and I have to admit that it's all a bit of a...baze. But in all seriousness, I might have to watch this a few more times...

[@user-ef5ib6kw8q]

you are fkin genius

[@sandeepsikarwar9779]

kyu bacho ke future ke sath khel rha hai ?

[@user-fe6vi3yg8i]

Kya aap bataa sakte ho ki kidhar ye tarika thik nahi hai? Genuinely poochh rahe hain hum, taaki seekhne mein kami naa reh jaye.