Skip to main content
Loading Explore guides…
Nash Equilibrium · Acedra
Home
Explore
Economics
Nash Equilibrium
Sign in to save
Chat
If YouTube asks you to sign in, open the video on YouTube.
Watch on YouTube
Reload player
Transcript
61 segments
Copy
Hide video
0:01
You've heard the term Nash equilibrium thrown around. Maybe in economics class, maybe in a beautiful mind, maybe from that friend who's trying to sound smart. But what is it really? And for that matter, what even is game theory? Here's what most people get wrong. They think game theory is about, well, games. You know, chess, poker, that kind of thing. It's not about games. It's about strategic decisions. Anytime your best move depends
0:36
on what someone else does, that's game theory. Think about it. Should you ask for that race? Should you cooperate with a competitor? Should you trust your partner on a project? These aren't games. They're real life. and Nash equilibrium. It's the answer to what happens when everyone makes their best move. But to truly understand it, you need to build up to it. Now, check out our learning path on the screen. We're going to start with dominant strategies, then move
1:11
to iterated deletion, and finally, we'll arrive at Nash equilibrium. It's that sweet spot where no one wants to change their strategy. By the end of this video, you won't just know what it means, you'll see it everywhere. And more importantly, you'll know how to use it. All right, before we can even find the winning move, we need to learn how to see the game itself. And to do that, we're
1:38
going on a game show. [Music] Imagine it. You're under the bright lights. The audience is roaring. You and one other person, Sarah, have just won $100,000. But there's a catch. You're in separate rooms. You can't communicate. And you each get two choices: split or steal. The rules are simple. If you both choose to split, you each get a cool $50,000. It's a happy ending. But if you choose to split and Sarah steals, she gets the whole $100,000. You get nothing.
2:18
And if you both choose to steal, you both walk away with zero. So the host is counting down. What do you do? This is a strategic game. And to solve it, we need to map it out. We need to build the game board. Every game has three core components. First, the players. That's you and Sarah. Second, the strategies, the choices each player can make, which are split steel. And third, the outcomes. What happens for every single combination of those choices? We can map this all out
2:49
in a simple table. Check this out. This is our game board. You can see your choices on the side and Sarah's on the top. The boxes show who gets what money. So, what's the right move? Looking at this, you can't really decide because this board is missing the most important piece of information. What do you actually want? This is where we introduce the idea of payoffs. Payoffs are just numbers that represent what each outcome is worth to you. A higher number simply means
3:23
a better outcome. Let's assume for a moment that you're a rational maximizer. Your only goal is to get the most money possible. So, your preferences are pretty clear. The best thing is getting 100 grand. Let's give that a payoff score of two. Getting 50 grand is good, so that gets a one. And getting zero is the worst, so that's a zero. Now, let's assume Sarah is the same kind of player. Look at the game board. Now, we've replaced the dollar amounts with these payoff
3:48
numbers. The first number in each box is your payoff. The second is Sarah's. With this board, your choice becomes much clearer. Let's walk through it. First, let's say Sarah decides to split. If you also split, you get a payoff of one. But if you steal, you get a payoff of two. Well, two is better than one, so you should steal. Okay. Now, what if Sarah decides to steal? If you split, you get betrayed, which is a payoff of zero. If you also steal, you both get nothing,
4:23
which is also a payoff of zero. In this case, zero is no worse than zero. So, stealing is still a perfectly logical move. No matter what Sarah does, stealing looks like the logical choice. But here's the twist. What if you're not a rational maximizer? What if you're a trusting idealist? The situation hasn't changed. The money is the same. The rules are the same. But you have changed. Your preferences as a trusting idealist look totally different. The absolute best thing for you is
4:59
that you both split. That feeling of mutual trust is everything. So, we'll give that a huge payoff of three. Getting the money but feeling bad about betraying someone. That's next. A payoff of two. Getting betrayed yourself but knowing you kept your integrity. That gets a payoff of one. And the absolute worst outcome, you both steal. A total failure of trust. That's a zero. Now, let's look
5:31
at the new game. You're the idealist. And we'll assume Sarah is still that maximizer. Let's think it through again. If Sarah splits, what should you do? Splitting gets you your best case payoff of three. Stealing only gets you a two. Three is better than two, so you should split. Okay, but what if Sarah steals? If you split, you get betrayed, but walk away with your integrity, which has a payoff of one. If you also steal, you both fail, which has a payoff of zero. Well, one
5:59
is better than zero, so you should still split. Your rational choice has completely flipped. It went from steal to split. This is the huge aha moment. The physical situation doesn't define the game. The player's motivations do. You can't solve a strategic problem until you know what everyone is truly playing for. By defining the players, strategies, and now the payoffs, you've officially built the game board. Let's go back to our splitter seal game. You've got your payoff
6:38
matrix. You know what everyone wants, but you're still stuck on the actual decision. [Music] Here's what expert game theorists do. First, they look for a dominant strategy.
6:53
This is a move so powerful, it's always the right choice, no matter what your opponent does. Think about that. If you find one, the game is instantly solved. Now, check out the formal definition on this slide. A strategy is strictly dominant if for any opponent strategy, your payoff from that move is strictly greater than from any other move. I know, I know that looks like academic nonsense. Let's use a simple rule instead. A dominant strategy is
7:22
a move that's just always better. That's all you need to remember. Let's make this real. Look at this first example. We're finding Sarah's dominant strategy. Remember principled Sarah who valued trust? She's up against rational Steven. Here's their payoff matrix. To find her best move, we have to check all of Steven's options. We don't guess, we check. So, step one, let's imagine Steven chooses to split. We're looking at that first column. If Sarah chooses to split,
7:49
her payoff is three. If she chooses to steal, her payoff is two. Well, three is bigger than two, so in this case, her best move is split. Easy enough. Okay, now for step two. What if Steven chooses to steal? We're looking at the second column now. If Sarah chooses split, her payoff is one. If she chooses steal, her payoff is zero. One is bigger than zero, so her best move is still split. This is the aha moment. It doesn't matter what Steven does. If he splits,
8:18
she should split. If he steals, she should still split. Split is always her best move. That means split is a strictly dominant strategy for principled Sarah. The game is solved for her. Her decision is made. She doesn't need to worry or guess. Okay, but there's a slight variation on this. Look at this next formal definition for weak dominance. Again, don't get hung up on the symbols. The key idea here is that your payoff is never worse and
8:44
sometimes better than any other move. It's not always strictly better, but it's never a bad idea. Let's see this in action. Example two. What about rational Sarah, the one who only cared about the money? Here's her payoff matrix against Steven. Let's do the same analysis. Step one. If Steven chooses to split, what should Sarah do? If she splits, she gets a payoff of one. If she steals, she gets a payoff of two. Two is greater than one, so her best move is to steal. Okay, step
9:14
two. What if Steven chooses to steal? If Sarah splits, she gets a payoff of zero. If she steals, she also gets a payoff of zero. It's a tie, so steal isn't strictly better here, but it's just as good. So, let's put it all together. Her steel strategy is never worse than split, and sometimes it's better. This is a weakly dominant strategy. For any rational player, the choice is still obvious. You play the move that can't hurt you and might help you. For
9:42
rational Sarah, that move is steel. This is your new superpower. Before you overthink any decision, before you start tying yourself in knots, just ask yourself, do I have a dominant strategy? If you do, the game is already solved. You have found the master move. But life is rarely that simple. Most of the time, there is no dominant strategy. Your best move genuinely depends on what your opponent does. So now what? Are you stuck analyzing every single crazy move that would ruin
10:20
everything? No. Because rational players don't make crazy moves. And once you realize that, you can literally delete those threats. We're going to use a technique called iterated deletion of dominated strategies. I know it's a mouthful. So, let's call it the rationality cascade. It's based on one simple powerful idea. First, you are rational. Second, you assume your opponent is also rational. Third, and this is the key, they will never choose a move that's obviously terrible
11:00
for them. And fourth, knowing this, you can delete that threat and see the real game hiding underneath. Let's see this in action with the tech company game. You're Innovate Core and you're up against Build It, Inc. Check out this table. It shows the profits for both of you based on the strategies you each choose. You're looking at your options and you see that third column for Build It, the Moonshot Project. And it's terrifying. If they choose that, your profits get absolutely
11:30
crushed. That single threat makes it impossible to choose. You're stuck. But wait, you're a game theorist now. Before you panic, you're going to analyze the board from their perspective. Okay, step one. Let's analyze Build It Inc.'s choices. Look at their payoffs and compare their new feature strategy to their moonshot strategy. Let's go rowby row. If you play strategy A, they get a payoff of seven from new feature, but a miserable -10 from moonshot. They'd obviously
12:04
choose new feature. If you play strategy B, they get five from new feature versus -8 from moonshot. Again, new feature is way better for them. And if you play strategy C, they get three versus -9. It's no contest in every single case. new feature is better for them. This is the first aha moment. The moonshot project, the move that was giving you nightmares is an unmitigated disaster for them. It is strictly dominated by new feature. The most terrifying threat just
12:39
vanished. Because it was never a rational move for them to make in the first place. Okay, step two. The cascade continues. The crazy threat is gone. The game is simpler. Look at the board now. Now, let's re-evaluate your options. Let's compare your strategy A against your strategy C. If Build It chooses safe bet, strategy A gets you a five, but C only gets you a four, so A is better. What if Build It chooses new feature? Strategy A gets you a four while C only gets you a two. A
13:13
is better again. Another insightful chime. Another aha moment. Your own strategy A strictly dominates strategy C. As a rational player, you would never choose C, so you can delete it from your own mind. Step three, the final reveal. Build it in knows you're rational, so they know you just eliminated strategy C. The game in their mind simplifies again. Look at this tiny 2x2 box we're left with. One last time, analyze, build its choice.
13:49
If you play strategy A, they'd prefer new feature. Seven is better than six. If you play strategy B, they'd still prefer new feature. Five is better than four. So, new feature now strictly dominates safe bet for builded inc. You have just predicted their move with certainty. They must choose new feature. So, what's your best move? Well, if they're definitely choosing new feature, you just look at that column. Your strategy B gets you a payoff of five, while strategy
14:19
A only gets you four. The choice is obvious. The single logical outcome of this complex, scary game is strategy B, new feature. Iterated deletion is your tool for cutting through the noise. The move that's terrible for your opponent isn't a threat, it's a distraction. By peeling away the impossible layers, you reveal the true heart of the game and find your winning move. Sometimes a game is so perfectly balanced that our first two tools just don't work.
14:58
There are no dominant strategies, nothing to delete. You're stuck in an endless loop.
15:07
Your best move depends on theirs and their best move depends on yours. This is where we need a completely different approach. Instead of looking for the single best move, we're going to look for a stable outcome. This is the legendary Nash equilibrium, and it's all about finding the no regrets point. Okay, here's the formal math. Check out this formula on the screen. A set of strategies is
15:39
a Nash equilibrium if no player can improve their payoff by unilaterally changing their strategy. I know that looks intense, but the idea behind it is incredibly simple. It's called the no regrets test. An outcome is a Nash equilibrium if no player can look back and say, "Darn it, knowing what they did, I wish I had chosen something else." Everyone is happy with their choice given what everyone else chose. No regrets. Let's prove it with a classic game our old tools
16:10
can't solve. The battle of the sexes. Imagine a couple, Alex and Bailey. Alex prefers they both go to a football game. Bailey prefers they both go to the opera, but their biggest priority is to be together. Going alone is the worst possible outcome for both of them. Here's the game matrix. will give a payoff of two for their favorite event, one for the other event, and a big fat zero for being alone. As you can see, if Alex goes to football and Bailey goes to the opera,
16:41
they both get zero. Bad outcome. Now, let's try our old tools. Look at the board. Does Alex have a dominant strategy? No. His best move completely depends on what Bailey does. Can we delete anything? Nope. No move is always worse than another. Our first two tools are useless here. So, how do we find the no regrets point? I'm going to show you a simple trick, a visual hack that instantly reveals the Nash equilibrium. It's called best response analysis, but we'll just
17:16
call it the underlining trick. Okay, let's walk through this together. Step one, find Alex's best responses. We go column by column. For each of Bailey's choices, we find Alex's best move. First, let's look at the first column. If Bailey chooses football, what's Alex's best move? He can get a two by also choosing football or a zero by choosing opera. Two is better than zero. So, let's underline Alex's payoff of two. All right. Now, the second column. If Bailey chooses opera,
17:52
what's Alex's best move? He can get zero from football or a one from opera. One is better than zero. So we underline Alex's payoff of one. Okay. Step two, find Bailey's best responses. Now we do the same thing for Bailey, but we go row by row. Why don't you pause and try it yourself? All right, you back. Let's check your work. First row, if Alex chooses football, what's Bailey's best move? She can get a payoff of one from football or zero from opera. One is
18:27
better than zero, so we underline her payoff of one. Second row, if Alex chooses opera, what's Bailey's best move? She can get a zero from football or a two from opera. Two is way better than zero. So, we underline her payoff of two. And now for step three, the final reveal. A Nash equilibrium is any cell where both payoffs are underlined. Why? It's simple. Because if a payoff is underlined, that player is making their best choice. If both are underlined, both
19:01
players are making their best choice at the same time. No one has any regrets. So in this game, we have two Nash equilibria. Football Football and Opera Opera. Let's test the first one. Football Football for regrets. Look at the top left cell. Alex asks, "Given Bailey went to football, was football my best move?" Yes. A payoff of two is better than zero. No regrets. Then Bailey asks,
19:30
"Given Alex went to football, was football my best move?" Yes. A payoff of one is better than zero. No regrets. It's a stable point. This is your universal tool. When dominant strategies fail, the underlining trick finds you the stable points in any strategic game. It narrows down the world of possibilities to only the rational outcomes. But what happens when there are no stable points at all? Let's look at a penalty kick game that breaks our tools. Okay, check this out. You're a
20:03
right-footed kicker. These numbers you see in the table are the probabilities of you scoring. If you kick left with power and the goalie dives left, you have a.3 chance. If the goalie dives right, it's a guaranteed goal, a probability of one. Let's use our underlining trick to see if there's a Nash equilibrium. Step one, find the kicker's best responses. We'll go column by column to find
20:25
where the kicker score is highest. If the goalie dives left, what should the kicker do? Kick right for a.9 probability, which is way better than.3. Let's underline that 0.9. Now, if the goalie dives right, what's the kicker's best move? Kicking left gives a probability of one, while right gives a zero. Nob brainer. Underline the one. [Music] Okay, step two. Find the goalie's best responses.
20:52
We go rowby row. But remember, the goalie wants to maximize their own score, which is the save rate. So, they want to choose the option where the kicker's score is lowest. If the kicker goes left, what should the goalie do? Diving left holds the kicker to a.3 score while diving right gives up a one. The goalie definitely prefers the.3. So we underline the goalie's payoff there which is 7. And if the kicker goes right, the goalie can dive left and give up a 0.9 goal probability or
21:20
dive right and give up a zero. The goalie wants the zero. So we underline their payoff of one. Step three, the final reveal. A Nash equilibrium is where both payoffs are underlined. Look at the table. There's no cell where both players are making their best choice. This confirms it. There's no pure strategy Nash equilibrium. If the goalie knows you're kicking left, they'll dive left. But then you'd want to kick right. But if they know you're kicking right,
21:51
they'll dive right, but then you'd want to kick left. It's an endless cycle. Your best move is to be unpredictable. We just saw a game where being predictable means instant defeat. This is where game theory gets really interesting. Pure strategies are for predictable games. The real world is messy. To win, you must become unpredictable. Let's go back to our penalty kick example to see how this works. Check out the table again. Remember,
22:18
we looked for a pure strategy Nash equilibrium and there wasn't one. Is an endless cycle of outguing each other. But just how unpredictable should we be? We need the math to find the perfect answer. Okay, here's the mathematical foundation. We're talking about something called a mixed strategy. Take a look at this formula. All it's saying is that you assign a probability to each of your
22:44
possible moves. These probabilities have to add up to 100%. Because you have to do something, and of course, none of them can be negative. And so for example, a mixed strategy for the kicker might be to kick left with a 60% probability and kick right with a 40% probability. You can see 6 +4 equals 1. It's a valid strategy. But is this the right mix? We'll need a bit more math to find out. Now,
23:13
when players are using these mixed strategies, how do we figure out our expected payoff? Check out this formula for expected utility. Okay, it looks a little scary, but here's the plain English version. We just sum up all the possible outcomes. For each outcome, we multiply your payoff by the total probability of that specific outcome happening. And that probability is just the product of each player's chance of choosing their part of that move. Let's make that concrete
23:38
with an example. What's the kicker's expected score if they play that 60% left, 40% right mix, and the goalie just guesses playing 50% left, 50% right? Look at the table. There are four possible outcomes. First, they both go left. The probability of that is 6 *.5, which is.3. The kicker score in that case is also.3. Second, kicker goes left, goalie goes right. Again, that's a 6 *.5 probability, so.3. The kicker's payoff here is a full 1.0. Third, kicker goes right,
24:10
goalie goes left. Probability is4 *.5 or2. The payoff is.9. And finally, they both go right. That's a .04 *.5 probability again 2. And the payoff is zero. So we multiply each payoff by its probability and add them all up.3 *.3 +.3 * 1 and so on. And we get an expected score of 0.57. But again, is this the best we can do? Is this the right mix? Here's the secret, the core intuition. To find your perfect mix, you calculate the probability that makes your opponent indifferent.
24:52
Think about it. If your opponent gets the exact same expected payoff no matter what they do, they have no best response. They cannot exploit you. They are stuck guessing. That point of perfect balance is the mixed strategy Nash equilibrium. But how unpredictable? A 50/50 split, 7030. Just guessing isn't good enough. Your opponent will exploit any pattern. This is the most advanced technique we'll cover. We are going to use math to calculate the perfect bluff. A mathematically
25:27
perfect mix of strategies that makes you completely unexploitable. And here's the secret. The core intuition that makes it all click. It's completely counterintuitive. To find your perfect mix, you calculate the probability that makes your opponent indifferent. If your opponent gets the exact same expected payoff no matter what they do, they have no best response. They can't exploit you. They are stuck. That point of perfect balance is the mixed strategy Nash equilibrium. So
26:02
what's the formal math behind this? It's called the indifference condition. Look at this formula. In plain English, all it's saying is that you must be equally happy with each of your moves that you're randomizing between. Because think about it, if you preferred one move over another, you just always choose the better one, right? You wouldn't be randomizing at all. So, let's test our
26:24
60/40 example. Is the goalie indifferent? First, let's calculate the goalie's expected payoff from diving left. There's a 60% chance the kicker goes left where the goalie's save payoff is 7. And there's a 40% chance the kicker goes right where the goalie's payoff is.1. So that's 6 * 7 plus4 *.1 which equals 0.46. Okay. Now what's their payoff for diving right? There's a 60% chance the kicker goes left. Payoff is zero. And a 40% chance the kicker goes right. Payoff is one. That
27:03
comes out to 0.4. So 0.46 is not equal to 0.4. The goalie is not indifferent. The goalie actually prefers diving left here. And we'll exploit our 60/40 mix every single time. So now let's use those formulas for real. We're going to use that indifference condition. And remember the secret. To find your probabilities, we make your opponent indifferent. Let's prove it. Grab a pen and paper.
27:32
We're going to solve this together. First, let's find the kicker's perfect mix. Okay, look at the table. We're going to assign a variable P to be the probability that the kicker kicks left. That means the probability they kick right has to be 1 minus P. Now, we set up the indifference equation for the goalie. We want the goalie's expected utility from diving left to be exactly equal to
27:53
their utility from diving right. So, the goaliey's payoff from diving left is P * 0.3 + 1 - P * 0.9. And the goalie's payoff from diving right is P * 1.0 + 1 - P * 0. We set those equal to each other. Now, we just do the algebra. We get 0.3 P + 0.9US 0.9 P= P. A little rearranging. 0.9als 1.6 P. and we solve for P. P equals 0.5625. That's it. That's the perfect bluff. The kicker's optimal strategy
28:33
is to kick left 56.25% of the time and kick right 43.75% of the time. Okay. Now, let's find the goalie's perfect mix. We use the exact same logic. Let's use Q for the probability the goalie dives left. We need to make the kicker indifferent. So, the kicker's expected score from kicking left has to equal their expected score from kicking right. Let's set up the equation. 0.3 Q + 1.0 * 1 - Q = 0.9 Q + 0. We solve that out. 1 = 1.6 Q. So, Q = 0.625. The goalie's perfect strategy
29:10
is to dive left 62.5% of the time. And now the grand reveal, the calculated solution. The mixed strategy Nash equilibrium is the kicker kicks left with 56.25% probability and the goalie dives left with 62.5% probability. This isn't a guess. It's not a gut feeling. It is the definitive calculated solution. It's the only rational outcome when two players are trying to outsmart each other.
29:38
We have proven that even in a world of bluffs and randomness, there is a perfect logical answer. You've just calculated the most advanced solution in basic game theory. With this final tool, your strategic toolkit is now complete. You came here wondering what Nash equilibrium really was. Now you don't just know the definition. You can find it, calculate it, and use it. Think about it. The world is filled with games. A price war between two companies. a negotiation with your boss,
30:09
even just deciding where to go for dinner. Before you were just a player, now you're an analyst. So, to recap your new toolkit, here's your strategic flowchart for any game you face. Okay, look at the flowchart with me. Number one, always look for dominant strategies. If you can find your master move, the game is solved. Simple as that. If that doesn't work, move to number two. Try iterated deletion. This is where you simplify
30:41
the game by eliminating the irrational threats, peeling back the layers until the answer becomes clear. And finally, number three, find the Nash equilibrium. This is your universal tool. It's how you find the stable points of a game or even calculate the perfect bluff. You've now mastered static games where everyone moves at the same time. But what about games like chess or business negotiation when you can see your opponent's move and react to it? That's called
31:08
a sequential game and it's a whole new level of strategic thinking and it's exactly what we're breaking down in part two. So if you want to master that, make sure you're subscribed and have notifications on. Now go out there and start seeing the strategy in everything.
Here to help you learn
Sign in to keep a private conversation with your course tutor.
Ask me anything about the material…
Sign in to chat