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!
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.
I already don't want to play the game
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
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:
ua-cam.com/video/S0qjK3TWZE8/v-deo.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
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
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.
+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.
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.
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.
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(
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.
I don't think there's anything that I enjoy so much yet understand so little as game theory.
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
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
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.
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.
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.
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.
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 🤷🏼♂️😂
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.
I would tell them I am going to play green. Also greens pay out is technically better for everyone since you could cooperate and split the $100,000, 50/50, the payouts should be 20k, 0, for betrayal, 15,15, for cooperation and 5,5 for both attempt to betray.
The rules stated that splitting was NOT allowed. Yes, that is an arbitrary rule, but that is true for the whole game, so what can you do?
@@holgerchristiansen4003 The you cannot divide the money afterwards rule is not just arbitrary but unneeded, there is a game like this but slightly different and yet it is a strategy and legitimate to split your winnings if you steal. There is no reason that a green steal should have a lot bigger payout since there are many reasons why letting someone steal by playing red in a roundabout way if you trust your opponent is more beneficial than splitting like asking for them to donate $10,001 to your charity of choose. In all honesty you can even tell them I will choose red have red already choose and if you trust the other player to spend that money in a way that is beneficial to you than loosing like this is the best outcome. In all fairness the order of magnitude more a solo win nets the community it is what these two players should conspire to achieve.
@@holgerchristiansen4003 also if my last comment seems confusing the whole point is in reality you don't trust the player outside the configs of the game so for all you know they would never split those winnings even if they said they would, my point was if you could convince the other player to do so you could convince them to benefit you in other ways that is straight or direct monetary compensation.
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
You sir, are my favorite UA-cam of all time.
11:15 To make this betrayal game spicier, both players should be married to each other or be best friends.
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.
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 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.
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.”
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".
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
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).
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.
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
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.
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.
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!
Well done explanation. Loved it. Thanks for taking the time to share, put this together, and putting it out there.
This betrayal game was actually the final round of a game show called golden balls, check it out if you wanna see people playing!
When you find out you've been using Game Theory to make decisions all your life:
The Prisoner's Dilemma has been done in German television. One Guy ended up taking advantage over the other person...
That’s kinda useful to be honest.
Now imma play rock 41.76%, paper 30.197%, and scissors 28.043% next time
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.
Game theory in a nutshell
When you mentioned “fights” you did a weird thing with your eyes and now I’m worried.
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.
with the 1,000 10,000 100,000 game you showed in the video with repeated interactions the optimal strategy is actually to alternate between red and green so it's
100,000 - 0
0 - 100,000
100,000 - 0
0 - 100,000
100,000 - 0
0 - 100,000
giving an average of 50,000 dollars as opposed to 10,000
You should agree to switch between showing red and green, always opposite to each other. Both of you get 100K, back and forth!
I like how to mention mr. beast. xD I also like this math vid!
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.
ua-cam.com/video/S0qjK3TWZE8/v-deo.html check out this episode
@@skartdo that is an awesome episode but this one is even more messed up
ua-cam.com/video/p3Uos2fzIJ0/v-deo.html
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.
There is a TV show called split, which is about dilemma.
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
this is a really great video
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?
This is a much for people who struggles with mathematics and general finance along with economics
they played the betrayal game on German television. One betrayed the other and the audience hated him
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!
If MRBeast had people playing these games it'd be hilarious
9:50 this actually was based on a show “split or steal”.
this shit is actually pretty crazy. no wonder that alice in borderland numbers game was somehow both confusing & exciting at the same time lol
My startegy would be secretly cursing matpat under my breath, and picking the green card because it seems ok.
Playing green have a chans of giving $100'000, all other combination is less then 10% and can be ignored, so green is the only option to get the jacpot.
There was a game show called Split or Steal that is very similar to the betraying for 100k game
Regarding the red and green cards, where it's 100k vs 10k vs 1k vs bust, what does game theory say to players who take an even bigger risk through cooperation to "game" the game? What's to stop them from going for one person winning the 100k and then splitting the reward and both walking away with 50k?
And then there's trolls who just want everybody else to lose, and they don't care about winning. How does game theory address that?
I HAVE SO MANY QUESTIONS!!
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
Your channel is too interesting. It's taking time away from my full stack web development courses.
But that's just a theory. A GAME THEORY, thanks for watching
10:00 This does exist- it’s a UK game show called Golden Balls
Poker is all about dominating the Nash equilibrium and pushing it over the edge against players that don't understand it.
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?
Betrayal!
*[Betrayed]*
I learned that you need to consider every option, people's psychology, and that algebra is beneficial to solving problems.
What about you?😁
Nice game of terraforming Mars going at around 13:11.
10:00 I would tell the other person, "No matter what, I'm going to pick green. If you pick red, I'll split the 100,000 with you 50/50. I'm trusting you to make sure we both walk away with as much money as possible, but we need to cooperate and now you need to trust me too."
Now if they think I'm lying, the difference in what they could get is only 1,000 (instead of 10,000 if we both agree to choose red), but if they think I'm trustworthy their payoff will be much bigger than the 1,000 maximum they can hope to achieve (since I've blatantly told them no matter what, I'm picking green). And yes, I would be true to my word if they chose red.
@@scythear That's the sort of assumption the gameshow bigwigs would like for us to believe to help their ratings. But once the money is mine and I walk away from the game, I should hope I'm free to do whatever I like with it, including but not limited to giving some of it to someone else.
No splitting was part of the rules of the game. If it's hard to get a situation that implies that restriction, imagine the prize being something else like food you have to eat right there or something you can't sell.
I would always show red if the other person says they are going to show red. I don't care if they are lying, trusting people always wins out in the long term and after all, relationships are always better than money.
What if you agree to split the 100,000 so that you both get 50,000?
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
If we are truly allowed to collaborate, the best strategy is actually to take turns winning $100,000. This yields an average of $50,000 > $10,000 > $1000.
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...
Keep it up!! Love those videos
Why is cooperating red and betraying green?
9:22 You and your counterpart discuss it in advance and agree to choose red, so that you each get 10K rather than 1K.
@@wlan246 what... Im talking about his choice of colors, why isnt cooperating green and betraying red
Because people are green when they are jealous. And cooperation may be related to a blood pledge or something.
It's because the situation is based on the Prisoner's Dilemma, choosing red might get you jail time while choosing green means freedom.
Treasure trove of information here. If it suits your taste, a kindred book might further satisfy. "Game Theory and the Pursuit of Algorithmic Fairness" by Jack Frostwell
For the second version of the game, I would hold up green-red-green-red to maximize our wins. I want the best for my homie. And if they don’t do the same I will simply start crying, guilting them into giving me my owed money or making them feel bad in general.
Title:
"The mathematics of game theory"
3 minutes into the video:
"I'm not going to show the math"
So I've just watched 1:03. I think my strategy is to pick green with probability 7/12, and red with probability 5/12.
2:20 - what's the formula to get these percentages?
I wouldve picked 22 too. Since people will think the average is 50% so the 2/3rds will be 33. This means most people will choose 33 and 2/3 of 33 is 22
As a poker player this question was very easy for me. If I play red 3/8 of the time I make you indifferent between playing red and green.
So in nutshell Nash equilibrium is all about optimal solution with minimal risk.
Great explanation. I appreciate that you got right to it.
As a true altruist I would totally choose red. Red gives the most money to the most people. IF we both get red than I benefit with 10,000 and so does the other player. If I am betrayed then I lose nothing but the other person does gain a lot. If I betray and also get betrayed we both get something but so little as to not make me care. I am disincentivized to choose green. Gaining little is worse than nothing.
Maybe that isn't altruism but it is how my mind works
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.
Mass Spellbreakers in every HU mirror.
Seriously thank you for not including prisoners' dilemma
10:05 Easy, say I'm going to play red, distract my opponent to look behind them, then swap the cards.