The (strange) Mathematics of Game Theory | Are optimal decisions also the most logical?
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Hope you guys enjoy! So most of the 'games' mentioned in the video are in fact famous game theory examples but I changed the payouts to mainly involve money rather than something like 'years in jail' or hypothetical 'points'. If you want more info regarding the 'games' I pulled these payouts from though I've attached the wikipedia pages below.
Also if you're interested in the results of the 'iterated prisoner's dilemma' tournament I attached a link where a couple dozen programs each with a unique strategy played each other.
Battle of the Sexes: en.wikipedia.org/wiki/Battle_of_the_sexes_(game_theory)
Guess 2/3 of the Average: en.wikipedia.org/wiki/Guess_2/3_of_the_average
Prisoner's Dilemma: en.wikipedia.org/wiki/Prisoner%27s_dilemma
Iterated Prisoner's Dilemma Competition: www.lesswrong.com/posts/hamma4XgeNrsvAJv5/prisoner-s-dilemma-tournament-results
Not Mr Beast but someone has done this game with money
Once I find it I’ll edit my comment
Edit: jubilee have done it
it’s called human theory
well thanks for keeping your words. Hope you do the same with AI
I have been waiting for this video. Thank.
Nice video, a traditional matrix is easier to understand the strategies though. I found the reordering confusing
There is a TV show called Split or Steal, and the game is basically the same!
A note on the Prisoners' Dilemma tournaments: It's true that "Tit for Tat" was initially the most successful strategy. However, subsequent tournaments showed that stronger strategies exist.
For example, there was a tournament in which two A.I. players would be matched for 100 repetitions of the game. Matches were organized in a "Round Robin" fashion, where every algorithm played against every possible opponent for the same 100 rounds. Each bot would get a score based on the outcomes it achieved in each round. The winning strategy was really clever.
One programmer submitted several algorithms (the maximum number allowed) and gave them different roles so they would act as a team. The tournament rules made them play anonymously, so they had to get clever to communicate. They would use their first few moves to send a signal so they could identify each other.
One bot was the designated winner, and all the other team members would lose to it on purpose whenever they met, in order to give that one bot the best possible score. That same winner would play a more conventional strategy when paired against someone not on its own team.
Meanwhile, all the other bots were designed to lose on purpose, and to never cooperate with anyone else except for the designated winner. This way, they could sacrifice their own scores to harm everyone else.
This "teamwork" strategy proved successful against a large field of opponents.
Source on this? I would love to read about it.
@@RedYakshaEntertainment web.archive.org/web/20140421055745/www.southampton.ac.uk/mediacentre/news/2004/oct/04_151.shtml
You wrote:
"It's true that "Tit for Tat" was initially the most successful strategy. However, subsequent tournaments showed that stronger strategies exist.
For example, there was a tournament in which two A.I. players would be matched for 100 repetitions of the game. Matches were organized in a "Round Robin" fashion, where every algorithm played against every possible opponent for the same 100 rounds. Each bot would get a score based on the outcomes it achieved in each round. The winning strategy was really clever. [...]"
What you are describing doesn't showed that stronger strategies exist than "Tit for Tat". It just showed that in a different kind of game "Tit for Tat" isn't the strongest strategie. And that shouldn't be surprising, imho, because if you change the rules of a game significant, than the strongest strategie has to change also.
Say, the main goal of a chess changes from killing the king to one must kill all enemy figures. If the best player in the world wouldn't change they strategy, they would be easy to beat.
"Tit for Tat" seems to be the best strategy if two single player compete in a competition of several rounds in a row. The competition you are describing is a team gameplay in a "Round Robin" structure. That is a different kind of game. It would be very surprising if "Tit for Tat" would work here. It would show, that team work is a bad thing. And that sounds very unlikely for me.
_
Disclaimer: I neither studied mathematics nor game theory. It's not unlikely that I'm wrong. So if you see an error in my conclusion please tell me.
@@andarted The original finding that "Tit for Tat" was a strong strategy originated from exactly this kind of tournament. It grabbed headlines by defeating many other strategies in this same context, a few decades ago.
In some sense you are kind of right, though, because "Tit for Tat," like almost all other strategies, was focused on winning the match rather than winning the tournament. The team that succeeded in 2004 did so by taking a broader view of the rules of the game.
As far as your conclusion about chess, it's probably true with regard to the current best players. However, there probably does exist an optimal chess strategy which never loses even if the opponent knows it perfectly (his strategy would have to involve many contingency plans for different situations). Predictability is no disadvantage in a game with perfect information. And in theory, chess has perfect availability of information, although in practice no player (human or machine) actually has access to the whole of that information, nor has the kind of memory or intelligence necessary to understand it fully.
@@CricketStyleJ I wasn't aware, that the "Tit for Tat" originated from this kind of tournament.
The video didn't explained the whole ruleset. A short mention that the maker simplifies it for the sake of intelligibility, would have been nice.
Thank you for adding this information!
After 7 minutes I realized, that this is not about matpat
Da Aaron no, it’s better. It’s actual Mathematics.
ive been trying to find a game theory video without matpat.
I was waiting for something about matpat for the whole video
NO thats JUST a theory, A MATH THEORY
A MAT THEMATIC
You should make it clear the point of the card game is to maximize your payoff and it's not a contest to see who gets more, otherwise people might mistake it for a zero sum game.
This is a good point. I thought this was a competitive game, where the goal is to outscore the opponent. In this case, I would prefer 0-0, as we're still tied, over 1-2, which puts me 'behind' by one in ranked scoring.
For the second one, don't randomize, alternate. Once the other person realizes what you're doing, they can alternate with an opposite pattern, therefore maximizing the amount of money both of you make.
This was my first response to a strategy as well and kind of searched for this comment :D He even allowed making a strategy!
This should be super exploitable. If you alternate non-randomly, they can arrange their choice so they will get 100k or 10k meanwhile you get 10k or 0k.
@@joegreen4959 Doesnt really work over a large period of repetitions. Obviously, if you see that they aren't going to play nice, then you shift away and adopt another strategy.
@@joegreen4959 not that one, the one with 2 dollars and 1 dollar
"Let me win the 100,000 dollars and I'll give you 50,000, here's a signed contract."
@@Shikogo But cooperation is allowed. The only different thing is there is no signed contract. So, would you believe the other person if she/he told you "i win then i divide it"? she could totally just take the 100,000 dollars and leave you without a penny.
I feel like I saw a video of this with a guy and girl and the dude screwed the girl over
@@ethanschaefer8327 Yh I saw it. I feel like I got major trust issues from that video alone
@@Shikogo The actual game is the prisoner's dilemma so you can't exactly split the prison sentences.
I WILL GIVE YOU 10 TO 1 ODDS!!! U IN?!
have to say as somewhat colorblind the use of red and green in this video is not optimal on my eyes
Same, pretty much nope-d out of the video.
Sorry about that! Will keep this in mind for future videos, just wasn't thinking.
I thought you guys can make your screen change the color palette to fit your specific type of color blindness. Or is that not how it works?
as a person with no eyesight whatsoever, I find the use of video and motion pictures highly offensive
@You’re right because my dear god! just think of the atrocities committed against those persons without a brain! how could any youtuber with a modicum of self-respect and thus the acknowledgement of those less fortunate, dare to even discriminate against the diverse population of brain-less individuals! math should be accessible to every individual, including those without a well developed nervous system!
I shall be reporting this channel to Google HQ promptly for crimes committed against humanity!
There was a game show centered around a prisoner's dilemma like game, I think called "Split or Steal"
Golden Balls, iirc
@@cryme5 And the optimal outcome was to tell your opponent you were going to steal and then give them half after the show.
9:57 there actually is a gameshow called "golden balls" that does something very similar.
Ye but the golden balls got "solved" and got canceled soon after. If we would solve it like in the show I would say "I pick green no matter what you do. So you can choose red and I will share 50% of the money with you after the show or you can take green and we go home with a miserable 1000 $ both. "
In the show the host says than there is no binding contract and he does not have to share the money in any way after.
but any reasonable person would then still go for the chance of 50k instead of going with 1000 home.
The show because the numbers worked out had a switch back in the end def worth a watch:
czcams.com/video/S0qjK3TWZE8/video.html (I know Skyline prob watched it but that is for others)
@@Utoko yeah but the guy stated in this video you couldn't share the money afterwards, so that strategy won't work. I figured it was in direct reference to that gameshow.
Kintama
In theory, with the prisoner’s dilemma, if it can be repeated numerous times and the two can communicate, alternating red and green each would be a sustainable solution. Every other turn, each player would get the 100k, and if either player strayed from the pattern, it would likely devolve into both playing green every turn and thus only getting 1k every turn, essentially dividing the payout to both players by 50. There lies the incentive to hold the pattern.
You're right except there's generally an odd number of rounds then, for such a game. The incentive goes away on the last round!
@@vishwalohana6404 Actually, simply announcing the last round is the last will make the incentive go away, if you're trying to maximize own gain. Because whoever has to win nothing in the last case will betray the strategy to get something, knowing it is the last chance to do so.
I think people are missing the point of prisoners dilemma. The name of the game is "non cooperative games" which means that they can only choose the best strat one time, and they cant talk. Alternating because of communication would become cooperative game, which goes into a whole different area of game theory, and prisoners dilemma is exactly not that, hence ur theory make sense, but will never happen
It’s not Nash equilibrium though.
With your example of betrayal and winning money this was actually a British game show called Golden Balls. One really interesting example of how this ended is also on youtube where instead of agreeing to both split, one contestant just straight up told the other contestant he was going to pick steal no matter what but he would split the money after the show. They shook hands and when they both revealed their choices he actually picked the split option and they both split like 100 thousand.
I remember in a Brain Games episode where random* people played “Split” and “Steal”. If both split, then they get 50,000 each, if one says steal and the other says split than the stealer gets 100,000, if they both steal then nobody gets anything. It was interesting watching people take advantage of others optimism lol
I love that Terraforming Mars was used as one of the games shown at 13:10. I didn't love that the moving of the blue cubes done there don't have anything to do with how the game is actually played.
A good real-world example of a mixing strategy would be a pitcher in baseball. There are some situations where a certain pitch has a higher value than another pitch, but you can't just use that same pitch every time, otherwise batters will know what's coming. For example, a 1-2 count the pitcher might do 70% curveballs and 30% fastballs. I heard Greg Maddux used to check the clock in the stadium to randomly determine which pitch he would throw. Maybe other pitchers do that too.
+zachstar
One of my classes in University was Negotiations in business, and every class session we played a game similar to this. But a little more complex having 10 round each game with the option to "collaborate" or "compete", developed by Harvard. on the 5th and 10th round there was a BONUS if you chose to compete if everyone else collaborated (and vise versa).
100% of our class grade was based on how many points we acquired in the game throughout the semester. The element of self interest made the game extremely difficult, because when everyone agreed to "collaborate," the distributed points is much, much lower - essentially only guaranteeing a "C" for everyone in the class. For that reason, people would choose to compete and screw over everyone else and reap massive reward toward their grade on the bonus rounds. However, if everyone chose to compete, the distributed points would be 0, and the one person who collaborated would get all the points.
Statistically only 5% of the class could make an A, the majority would make B's and C's, and a few would make an F. What we learned from the professor who had a giddy excitement all through the semester as we became total savages, Is that there WAS in fact a way to play the game so that we ALL got an A. It was through strategy and negotiation. If we collaborated on HOW and WHEN we chose to "compete" on certain rounds each class period when we played the 15 minute game, then the distribution of points would be 100 for everyone come the end of the semester. the key was in the bonus rounds.
He was too cryptic and we never understood this concept until the end, and so he decided to give us all A's for trying our absolute best to understand the game. Make no mistake, we were SAVAGES to each other, and at one point we all HATED *each other* for betrayal, *ourselves* for being guilty of doing the same, and *the professor* for putting us in an "unfair" environment.
HAHA! It was an amazing life lesson. Everyone truly wins in capitalism if we work together!
Game theory: In a system of rules, cheaters reap the reward. On the contrary, In a system without rules, those who collaborate, create, and abide by agreed upon rules, reap the reward.
This is such an interesting anecdote, and that observation at the end might explain why corruption tends to arise in bureaucratic organizations that become overly bogged down with rules and red tape
BRUH, your class was literally an anime plot 💀💀
That game is known as the Win as Much as You Can Game. The payoff structures are designed to prevent collaboration. Other versions of the payoff structures will promote more collaboration.
Two years late, but I'd like to point out that everyone does not win in capitalism if we work together, as, if we all work together, it's no longer Capitalism, and it's, instead, Socialism.
Capitalism is a system of economics where businesses hold the "power of the product" (i.e. they are the ones who get to make most, if not all of the decisions on what happens).
Socialism is where the power of the product is given to the people THEMSELVES.
Lastly, Communism is where the power of the product is given to the state (aka the government).
@@deadersurvival4716 I hope your last line was intended as a joke, though it does not come off as one.
Zach, you're brilliant! I love your videos. I'm a physics major and I'm just learning to more comfortably thinking mathematically. Your videos make learning math soooo much fun!! Keep up the good work man.
Well done explanation. Loved it. Thanks for taking the time to share, put this together, and putting it out there.
Thank you for the video! I've been interested in game theory for a long time now but I haven't found a good video on it up until now.
The search is ruined by pretty much dumbest show on the internet.
4:42 when you wanted to make one door but accidentally made 12 doors in minecraft so you just drop the rest on the ground
10:31 The best strategy is you plays green then me plays red, then next round you plays red and me plays green. This way the average of 2 rounds for each person is 100k instead of 20k.
Well that isn't usually the case with this game cuz its a one off
HA! I said I would use a 3 sided die before you revealed the answer. I'm pretty proud of myself for figuring it out.
I remember a game show where some guy had the red-green card thing, but where red-red is 0-0, red-green is 100-0, green-red is 0-100, and green-green is 50-50. Some guys said he would play red no matter what and split the winnings after the show, so the other guy played green, but the first guy played green too, so they split the money at the show itself. big brain maneuvers.
Love the shot of Terraforming Mars at the end there :)
yeah, but what the hell os he doing there? hahahaha just randomly moving player markers from place to place
I noticed that too. TM is a great game.
Hey man, great content!! one note, however.The non-cooperative game with $100000 payout for green vs red should had a lower value for green(
Keep it up!! Love those videos
This isn’t the same as the prisoners dilemma because in that the most good comes from both cooperating, but in this it’s from one cooperating and one betraying
No betraying gives you a better payout than cooperating with another player who's cooperating. Otherwise there wouldn't be an incentive to betray if you were both cooperating and mutual cooperation would be an efficient nash equilibrium.
@@joshuafury5353 In the classic prisoners dilemma, the lowest total prison time comes from both sides cooperating.
@@cookiecakeeater6340 @cookiecakeeater6340 No that's not the prisoner's dilemma. While mutual cooperation is better than mutual defection and usually in how it's written mutual cooperation is less prison time between both players overall than 1 person betraying it is not the case that mutual cooperation is the best outcome to land on, betraying someone who is cooperating gives you less prison time or else there's no point in betraying.
In a game matrix player 1 picks a row and player 2 picks a column, player 1 receives the payoffs on the left. This is the prisoner's dilemma each player can cooperate or defect. Sorry for the shitty drawing but here: A>B>C>D
c d
c B,B | D,A
--------------
d A,D | C,C
Player 1 would like to play Defect as regardless of his opponents choice A>B and C>D
Player 2 would like to play Defect as regardless of her opponents choice A>B and C>D
Payoffs are good and prison time is bad so maybe it's more intuitive to think of the payoffs as the negative amount of years they spend in prison (so it's clear that more prison time is worse).
In the game you're describing we would likely see mutual cooperation. B>A>C>D
c d
c B,B | D,A
--------------
d A,D | C,C
There are actually 3 Nash equilibrium in this game but mutual cooperation is the only one that would make sense for you to play as it gives a better and actually the best possible payoff for both players than the other equilibria, a payoff of B every time.
A Nash equilibrium is a set of strategies for each player such that neither player has a profitable deviation given the other player's strategy. Mutual cooperation and mutual defection are both equilibria in this game.
if you are cooperating and the other player is cooperating you would not want switch to defect as B>A. This also applies to the other player.
if you are defecting and the other player is defecting you would not want to switch to cooperate as C>D. This also applies to the other player.
There is also a mixed strategy Nash equilibrium where each player is using the strategy that makes the other player indifferent between his choices. A Nash equilibrium by definition, as neither player can profitably deviate given the opponents strategy.
These other equilibria don't really matter though because it is in both players interest to play into the equilibrium where they both cooperate.
Hey Zach! Just one note about a statement you made about chess's Nash equilibrium - it turns out that every game can be split in 2 categories:1) a game where some player i has some set of moves such that they always win, or 2) a game where all players have a strategy such that no player wins. As it turns out, Chess is probably of game type (2) and therefore has no meaningful nash equilibrium (just all strategies that always result in a tie).
You sir, are my favorite CZcams of all time.
In my Number Theory class we spent 1 week (2 lectures) talking about how to divide assets. The way we did it was from one of the teacher’s colleagues (or maybe students?) and hadn’t been published yet, back in ‘13, I think. I really enjoyed it and have wanted to learn more about Game Theory. Do you have a text book you would recommend?
You make math cool, thanks for posting these videos
You should read up about the UK game show "Golden Balls" and then go and look out the most savage betrayal and, also, "the weirdest split or steal ever".
love the Terraforming Mars at the end.
I just wished they had used actual gameplay footage. The way the blue cubes are being moved makes no sense.
@@holgerchristiansen4003good catch! I did not see this before.
regarding the prisoner's dillema you set up at the end:
I'd get the other person to sign a contract saying they owe me 50k if I show red and they show green and make sure it's legally absolutely binding
11:15 To make this betrayal game spicier, both players should be married to each other or be best friends.
your videos are awesome keep up the good work
Honestly, I would just pick green everytime, because that way I get money for sure. I don't care what the other person gets.
I believe these feelings are going to change when you compare your progress with your oponent after each lap.
@@alexeysaphonov232 that is flawed thinking. Its not a zero sum game. You're just trying to win money.
The problem with this is that if you know your opponent will pick red, you get more money by playing red than by playing green.
@Pluto N. Uranus Of course I did. I did not talk against the video, all I said was what I would do.
And just picking green always gives me money, there's no need to gamble for me :)
@@yonatanbeer3475 if you know your opponent will pick red, then picking green gets you 90,000 more than if you pick red.
I would suggest that my opponent picks red and I pick green and then we split 50/50 after, then when he/she inevitably thinks I'm going to just take the money afterwards I'll suggest that we skip the roles making him/her more likely to split with me after
When you find out you've been using Game Theory to make decisions all your life:
There was a game show called Split or Steal that is very similar to the betraying for 100k game
We did a quiz grade in AP Microecon on this. We were randomly paired and had the choose the grade we wanted together. A or C. A/A gets F/F. C/C gets C/C. A/C gets A/F. Everyone said they were going to cooperate and then we waited to see the sharks.
Making a graded assignment rely on the outcome of a literal game sounds psychotic. Imagine narrowly losing a scholarship because your teacher thinks they're clever.
@@JhettJones Small assignment with minimal grade implication, if your scholarship depended on that then you didn’t deserve it in the first place 🤷🏼♂️😂
I don't think there's anything that I enjoy so much yet understand so little as game theory.
Got a link to the proof of your first example anywhere? I evaluated the minimax problem and got a slightly different answer: choose red 41.2% (not your 37.5%). But I did get that the opponent should do the opposite just like you did. Here is the calculation:
Your action is random variable a, can be Red or Green.
Your compeitor's action is random variable c, can be Red or Green.
Policy distributions are p(a) and p(c).
Your competitive reward function is,
r(a,c) =
R,R: 5-0 = 5
G,R: 2-3 = -2
G,G: 5-0 = 5
R,G: 0-5 = -5
Assume a is independent of c.
(Players don't model each other or read minds).
p(a,c) = p(a) * p(c)
Expected reward is,
E(r) = sum_{a,c} ( r(a,c) * p(a) * p(c) )
= 5 * p(a=R) * p(c=R) +
-2 * 1-p(a=R) * p(c=R) +
5 * 1-p(a=R) * 1-p(c=R) +
-5 * p(a=R) * 1-p(c=R)
Assume both players are trying to optimize their policies.
(I.e. competitor is doing the best they can).
Solve minmax problem for the above sum over parameters
x := p(a=R) and y := p(c=R). That is,
E(r|x,y) =
min_y(max_x(5*x*y - 2*(1-x)*y + 5*(1-x)*(1-y) - 5*x*(1-y)))
s.t. {x,y} in [0,1]^2
In this case it's the saddle point of a quadratic form.
Solution is x = 0.412, y = 0.558
I.e. choose action R about 41.2% of the time to optimize expected reward.
R: 41.2%
G: 58.8%
And interestingly y turns out to be the opposite, R at 58.8% and G at 41.2%.
This is because it is a zero-sum game (Nash equilibrium theory).
Alternative equivalent approach:
y = 1 y = 0
max_x(min(5*x - 2*(1-x), 5*(1-x) - 5*x))
=> R: 41.2% (same answer)
**So close to what you found Zack. What went wrong?**
So the thing that went wrong is that the solution Zach put up is optimal *for direct payouts* (where you don't care what your opponent makes), not for the competitive scenario.
I can't say anything about the mini-max tho, since with the adjusted payoffs the mixed nash equilibrium is exactly what you described (although I found it the traditional way by making the opponent indifferent).
You going to make video about math of higher dimensions, and np complete problems?
Great explanation. I appreciate that you got right to it.
This betrayal game was actually the final round of a game show called golden balls, check it out if you wanna see people playing!
I like how to mention mr. beast. xD I also like this math vid!
MajorPrep, I am currently in the 8 grade and I want to get a degree in electrical engineering and/or computer science. But the thing is, I don’t take advanced math classes and I have often struggled with mathematics in the past. But I have improved to the point where I am one of the best in my math class. What advice do you have for me if I still want to get a degree in electrical engineering and/or computer science?
this is a really great video
Great video, but of course we don't know a Nash equilibrium for chess, it's not impossible to find it but the computation power required is just absurd
As for the "screw over" game, my response would be: I would show red if it were someone I know, green if it was someone I didn't. My reasoning being: if I know the person it's likely I wouldn't want to screw them over, and vice versa, so by showing red we either get the best outcome for the both of us, or they're happy and might give me some money later when you can no longer enforce the no sharing thing. However if it's someone I don't know I have to assume that they are going to try and screw me over as there is no presumed trust, so I either get 1,000 and dollars which is pretty good for no work and we go our separate ways appreciating that we each have the same understanding that this is how it had to be, or I get 100,000 dollars, feel bad for them in the moment, then leave and never see or think about them again.
Ahh I love game theory back in Uni. I remember in one of our finance classes, the professor said that we can either put +5 or +3, if more than 95% of the students put +3, they don’t get a plus; if more than 6% puts +5 they get a minus 3; now there are like 200 students that are under the professor so basically only about 11 people or less can win on +5 and the test was really hard so most people put +3 because everyone was afraid or further lowering their scores but I knew this, also one of the reasons why I put +5 was also because this was the first time it was happening to us so everyone is scared to put +5, and lastly, that same professor keeps on repeating one of our first lessons in the principles of finance which is the risk-return tradeoff so I basically knew that even if there are more than 6% +5, the professor wouldn’t mind because he was subtly teaching us to take risks, idk if I was right tho but all of us got the plus that we put so I take it I was right. The next time he did it, I knew I had to put +3 because everyone else are now greedy because of the previous result and I was right so I basically got the best plus I could get.
Another time it happened again in our ethics class but this time if less than 5% puts +5 they get +5, if more than 95% puts +3, everyone gets a +3 and if more than 6% puts +5, everyone doesn’t get a plus. Everyone was understandably campaigning for a +3 because that’s the best result for everyone, but I knew one of us will put +5 and I’m also a greedy AH, well I kept it a secret since I don’t want to be hated but I know 2 other people that put +5 😂😂😂 never told anyone until of course after we finished taking that class.
If we’re allowed to strategize with each other beforehand, then there’s an alternative: “If you play red and I play green, then I’ll get 100,000 and I’ll give you 50,000, five times what you’d get if we both played red.” Or if there’s multiple rounds: “If we alternate who’s playing red and who’s playing green, we’ll both average 50,000 per round, which gives both of us maximum payout.”
Haven't you seen 'Golden Balls'?
This is a UK gameshow where they do exactly this!
Hate that show, it makes me lose faith in humanity.
Never heard of that show! Thanks for the heads up.
czcams.com/video/S0qjK3TWZE8/video.html check out this episode
@@skartdo that is an awesome episode but this one is even more messed up
czcams.com/video/p3Uos2fzIJ0/video.html
Those examples were very clearly presented and also very strong
My initial strategy without watching the rest of the video is to play green 70% of the time and red 30% of the time, chosen randomly.
Hey, not that far off!
The funny thing about the game at 5:14 is that both of us playing red every time is better for me than the 50:50 unstable equilibrium and stable fair equilibrium.
M8 I've got lots of question about college major, are you open to questions or are there sites which supports this?
Your channel is too interesting. It's taking time away from my full stack web development courses.
Omg the part where you mentioned Mr. Beast I nearly cried because of the thought of getting scammed out of 10,000 dollars as I would play red
Hey, i also wanna learn this kind of mathematics, is there some specific resources from where i can learn like you? Please help me out for learning.
I went through coursera's "game theory" course in preparation for this video and thought it was pretty good (some parts can get really technical but that's often if you just want more of the 'proofs' and rigorous mathematical explanation, the class is meant for beginners overall). I put a link in the description and it's a free course so you don't have to enter any payment information for this one. I'll also attach two youtube lecture series below.
Yale Game Theory Lectures: czcams.com/video/nM3rTU927io/video.html
William Spaniel Lectures: czcams.com/video/NSVmOC_5zrE/video.html
Thanx man
You severely moved the goalposts in the introduction. First, you said, "the goal is to end up with the most amount of money possible" but then you said "What would your strategy be to win this game over several rounds?" If the goal is to win as much money for myself as possible, then how much the other person wins is of no importance. However, if the goal is to beat the other person, then winning as much money as possible for myself is not the goal. The goal is simply to win more than you. I know it was just a slip of the tongue, but it was a big one!
However, wouldn’t earning the most amount possible inherently require beating the other person in this case?
This game is zero sum, though, so it's equivalent.
I remember running into the Prisoner's Dilemma for the first time in KOTOR 2 and it had me so confused for a good long time.
Was it in KOTOR 2? I don't remember that part.
@@raves8451 Sorry, KOTOR 1 not 2. It was on Kashyyyk, on the surface at the foot of all the trees. You had to answer a terminals questions to gain access to the star map and one of them was the Prisoner's Dilemma.
Great dude
Before watching this video: I'm choosing my card randomly without revealing it to you that I have chosen randomly, thus preventing you from using any game logic to deduce my choice. Worst case scenario for me, you know I'm going to do this and show green which gives us equal odds at winning $5 thereby making the bet have an average value of 0. Best case scenario is you don't know I'm choosing randomly and through some manner deduce that red is your best choice in which case the odds are in my favor. Sometimes the logical choice is chaos.
Chaosh isn't a pit. Chaosh is a laddah.
The Prisoner's Dilemma has been done in German television. One Guy ended up taking advantage over the other person...
Betrayal!
*[Betrayed]*
what's the link between game theory and the Laplace transform (a space combining sinusoidal functions and exponential growth/decay) ? I read that the latter was developed after Laplace's work on game theory.
They had a game show like that called Friend or Foe in the US. Also completely unrelated, as someone who plays Terraforming Mars, what is that player doing at 13:09? They paid 1 currency cube to lay "Industrial Microbes" which raises your power production and steel production each by 1. I probably wouldn't noticed except they then moved the token that tracks their titanium production every turn and put it on the Terraforming tack with is both your score and income (he has 2 tokens on there for some reason) and then puts it on the board also for... some reason. They has someone who knew how the board should be set up but couldn't bother to have that person play something that looked like the game?
That’s kinda useful to be honest.
Now imma play rock 41.76%, paper 30.197%, and scissors 28.043% next time
So I may be a little late to the Party. I've just been introduced to this channel today and I have been watching all videos in pretty rapid succession. But this reminds me of an Anime called Kakegurui. Its about gambling and games and the like. But on the second season episode 7 there's a game that I think is similar or related to the topic discussed here its called "The Greater Good Game" *Spoiler Warning Here* . I'll try to show the rules
There are 5 players. Each turn 1 player will be escorted to a room with 2 Boxes. A Personal Fund Box and a Tax Contribution Box. Each player is given 5 Silver Coins. Each player can choose the amount to put in each box. Once all turns have been taken the coins will then be counted and distributed as follows. The coins put into the personal fund box are theirs to keep. But the coins in the tax contribution box will be doubled and then distributed equally amongst the 5 people whether any player put any coins in the tax box or not. Before each round there is time for all players to be in a single room to discuss amongst themselves. But if at any time 3 or more playes agree, one player can be eliminated. The eliminated player will be excluded from the game and their coins confiscated. The goal is to acquire 40 silver coins in 5 rounds. If you reach 40 you win "Votes" (a type of desired currency in the show) depending how many coins they have collected. Obviously if they only put coins in the personal box all 5 rounds they won't have enough to win "Votes". And those who do not accumulate 40 silver coins by the end of the game lose all of their "Votes" that they had earned from previous games. There's a lot more to it if you watch the show but the limit of the winnings if you are the only one to get 40 silver coins is 133 Votes and if everyone gets 40 coins equally it turns into about 26 coins each. Im not sure what kind of mathematical equation or anything like that can be used to determine the Nash Equilibrium for this scenario and i might be missing some info but I thought it would be cool to mention :)
For the last one (the prisoners dilemna):
If you are able to discuss beforehand you should be able to win 50 000/round. Simply agree to alternate between me showing green while you show red and vice versa. You could deny me my 100 000 by playing a green when we agreed you would play red, but that would leave you worse off on future rounds because I would just play green for the rest of the game
His numbers are bad. The same logic applies to the Nash equilibrium if you replace 100k by any number larger than 1000. Thus we can pick 1500 which is still a decent incentive to betray, but makes alternating betrayal a suboptimal strategy.
Alternating works if you could play the game multiple times and cooperate, but for that example we are assuming you only play once and cannot split up the money afterwards.
@@zachstar Ah ok, I thought we were addressing a strategy for multiple games
Love Forsberg but if the [roze was only 1500 not 100k then wouldn't you just both choose red? there is no incentive to betray since the pay is now only 1500, instead of the 10k you would get for both picking red
You should agree to switch between showing red and green, always opposite to each other. Both of you get 100K, back and forth!
they played the betrayal game on German television. One betrayed the other and the audience hated him
Please make a video on William Lowell Putnam Mathematical Competition
10:00 This does exist- it’s a UK game show called Golden Balls
For the green-red game at the end, since I'm allowed to talk to the other person, I would agree with them that one of us would play green and the other would play red, and then we would split the $100,000 later.
Poker is all about dominating the Nash equilibrium and pushing it over the edge against players that don't understand it.
@13:10 that is not how terraforming mars works...
Certainly not HAHAHA
Don't know if this is interesting to anyone, but my uni does have a research lab for game theory experiments and I had the pleasure to take part in 6 of those experiments. The sample size is not too big but here's my experience: Eventhough cooperating usually gets a better combined outcome, betraying is common. In games where multiple rounds are played, the percentage of betraying increases from round to round.
Yes, people will jeopardise a great outcome for everybody to either benefit themselves more or worsen their position because everyone else gets tired of their shit eventually and act selfish too.
We humans really aren't the most clever species...
2:20 - what's the formula to get these percentages?
For the last game, you should add one more rule to actually have people play it with a strategy.
Make the rounds odd.
Whoever makes less money at the end off all rounds gets nothing.
This would be way more interesting.
My startegy would be secretly cursing matpat under my breath, and picking the green card because it seems ok.
There is a TV show called split, which is about dilemma.
9:50 this actually was based on a show “split or steal”.
Wow I thought 60-40 for the first one - very close :o
how did you calculate these percentages
If I was placed into that game, I'd tell the other player "I'm picking green regardless of what you do, since we'll both get 1,000 dollars if we show it, but I won't lose anything if you disagree, so the choice is yours." Simply because, 1,000 or 100,000, it doesn't matter, their actions won't dictate me coming out with nothing, therefore guarantees that'll they'll cooperate.
I paused the video at 7:21 and figured out what I would pick. I thought most people would assume that everyone's guesses would have an average of 50, so people would tend to pick ~33. Thus, my best bet would be 22. Sounds like I was pretty on base.
When you mentioned “fights” you did a weird thing with your eyes and now I’m worried.
This is a much for people who struggles with mathematics and general finance along with economics
Sir, Which Game theory book are you following for this kind of good stuff ?
He mentioned it in the notes under the video. It is the Coursera Stanford online course for Game Theory. You can take it for free. All his examples are from there. Check it. It is wonderful. I am taking it right now. That's why Youttube put this video on my list I guess. ;-)
@@joeah3479 Thanks Bro.
If MRBeast had people playing these games it'd be hilarious
For the modified prisoners dilemma I think there’s a clear best answer! If you’re allowed to cooperate with the person you’re playing against, tell them that you’ll pick green, and you’ll split the money with them if they pick red. That way you reach a Nash equilibrium: The other person will always make more money picking red than green, and you’ll always make more money picking green than red.
You can even split it unevenly (say, you take 80,000 and give them 20,000) and there’s no logical reason to refuse. Although that COULD end up with some mind games and stuff fighting over who’ll get the 80,000, so I’d just stick to 50-50.
Of course, this solution doesn’t work for the original prisoners dilemma bc you can’t “split” jail time once you’ve been sentenced, but there’s no reason given that you can’t exchange money outside the framework of the game!
Title:
"The mathematics of game theory"
3 minutes into the video:
"I'm not going to show the math"
If only we can cooperate more than being more greedy
Self-interest is a huge distraction for trust & it's hard to cooperate when trust is in uncertainty
Good video
this shit is actually pretty crazy. no wonder that alice in borderland numbers game was somehow both confusing & exciting at the same time lol
In the green red, 100,000 or 10,000 game you said they could communicate before hand & then betray eachother. Wouldn't they just agree to split the 100k 50/50 and then intentionally setup a betrayal? This actually seems a lot more likely than any of the other strategies seeing as I've actually seen it employed myself. At local card tournaments when you get two players in a finals match, the winner of which will win X (the 1st place sum), and the loser wins some lesser amount Y(2nd place sum), they might just agree to split x & y, and not even play the game, or just play a friendly game.
The idea works because each player might not have the confidence that they can win (in the case of the card game tournament it would be because they both know they are good players to have reached the final stage) and thus both assume cooperation is in their best interest (assuming after the intentional throwing they can't then betray eachother afterwards) e.g. I betray you because we agreed to split 100k 50/50 but then I just run off with the money.
In the example of the heads up card game they might assume that winning 100k is the best, and it's an obvious option that everyone would come up with, and thus expect both to play green, so cooperation on intentional betrayal is A LOT more likely to award a high payout then *hoping* for the other guy to blunder with a red card. Also most people, I think, Would upon realizing that 100k is a lot more tempting than any other option just play green regardless. I'd bet a small % of people actually do play red when trying to agree upon double red. Also if you're going to agree to cheat (play double red) why not just cheat & play green / red & split 100k.
Hell, they do this in real competition's too, which is probably why there is usually some rule against it which would see both teams forfeit the prize money.
Nice game of terraforming Mars going at around 13:11.
I love how they talk about one off games, and in the same breath equilibrium's. Is it just me or does an equilibrium necessarily imply multiplicity. And if some of the one off games were played multiple times then different strategies would be better!
Okay watched more of the clip and they started to talk about something similar. Had thought I'd watched all of the relevant part. Sigh...