it is possible you've played a game of matching pennies before but as I'm sure you're a child of modernity you probably had better games to play as a kid but the game goes like this you and a friend each have a pile of pennies you each pick one of your pennies up and secretly choose which side to show heads or tails then you reveal your choices at the same time if they match player a takes both pennies if they don't match player B wins both pennies we can put each of these outcomes into a payoff Matrix like this if player a picks tails and player B picks heads player B wins so a loses a penny but B gains one if player A and B both Jews Tails a wins one penny and B loses one if they both pick heads a wins one and B loses one and if a picks heads but B picks Tails a loses one penny and B wins one penny okay so what is the Nash equilibrium for this game well no matter which outcome we land on the loser always regrets their choice at Tails heads or heads Tails player a will wish they had chosen differently at heads heads or tails Tales player B will wish they had chosen differently so none of these outcomes work as a Nash equilibrium actually the reason Nash's work is so important is because he proved that there is always a Nash equilibrium in fact there are always an odd number of Nash equilibria it's just that sometimes they end up being what we call a mixed strategy equilibrium where the players want to choose their strategy randomly based on some probability in this case the best strategy is to be unpredictable to choose your strategy totally randomly I like just flipping the coin and choosing that way that's the Nash equilibrium here what about Pareto optimal outcomes though which ones are Pareto optimal at each outcome there's always a winner and a loser and the payouts always add to zero this is called a zero sum game which is a game where every outcome has payoffs for each player which add to zero since there's no way to make one player better off without making the other player worse off at every possible outcome all of the outcomes in this zero-sum game are Pareto optimal another well-studied game deals with advertising imagine we have two companies battling each other for customers if both companies choose not to advertise they split the market 50 50. and will say they each earn fifty dollars in profit but if one of them advertises they'll bring more customers to their side if B chooses to advertise but a does not we'll say they get 70 of the customers firm a only earns thirty dollars now and firm B earns seventy dollars but has to use some of that to pay for the advertisements we'll say advertising costs ten dollars so they end up with sixty dollars as their profit if player a advertises but player B does not then a gets the 60 and B gets the thirty dollars but if they both advertise they cancel each other out they split the market 50 50 again but now they're down the ten dollars they spent on Advertising leaving each of them with forty dollars what is the Nash equilibrium here one way to find it is to check for dominant strategies no matter what B chooses a will always wish they had chosen to advertise because sixty dollars is better than fifty dollars and forty dollars is better than thirty dollars the same is true for player B they will always wish they had chosen to advertise so the Nash equilibrium is at the intersection of these two dominant strategies is it Pareto optimal though no both can be made better off by agreeing not to advertise in fact the Nash equilibrium is the only outcome which is not Pareto optimal a quick way to spot a Pareto optimal outcome is to find each player's highest payout where they get sixty dollars is each player's best payout and so there's no other outcomes where they're not made worse off we would say that this game is a classic prisoner's dilemma because the Nash equilibrium and the Pareto optimal outcomes mirror the results of that game Game Theory can also help us understand why rebellions are so hard to coordinate when everyone is living under some oppressive regime if everyone Rebels they will be successful and will throw off their oppressors getting a big payout but if one Rebels but the other doesn't the Rebellion fails and the player who rebelled is punished severely while perhaps the rest are given reminders not to Rebel but if everyone just stays silent they can eke out a small reward living under the oppressive regime this game is interesting because there are two Nash equilibria if everyone Rebels no one regrets their decision and if no one Rebels no one wishes they had been the lone Rebel but only Rebellion is Pareto optimal because it makes both players better off compared to all other outcomes historians can confirm that this prediction of Game Theory matches reality people will go a long time in a repressive regime with no one rebelling but when they overcome the coordination problem and successfully Rebel they often can't stop rebelling they replace the old regime with a new one and immediately rebel against the new one in fact America is a rare example where we rebelled against our rulers and then we stopped rebelling for a period of time that's pretty rare in history lots of games have both players make choices at the same time but many more have players choose their strategy in turns sequential games are played in sequence where one player goes first and then the next player chooses based on that previous move we can't represent these games with a payoff Matrix so we use a different method we start with a decision node here's an example when an employer hires someone they have to decide if they want to offer that employee a high salary or a low one we will assume that the low salary is enough to get the employee to come on board and take the job and the high salary is more than that so the employee is going to take the job but now they have a choice they can either give the job their best effort or they can give the job low effort do they go above and beyond for their employer or do they do the minimum amount of work necessary to not get fired now we can match each outcome with the payoffs if the employer gives a high salary and the employee gives High effort they're essentially splitting the Surplus they get two each say but if the employer gives a high salary and the employee phones it in with low effort the employee gets all the reward while the employer barely gets what they paid for so the employers get no Surplus while the employee gets three if the employer chooses to give a low salary and the employee gives High effort anyway the employer gets all of that Surplus and if they give a low salary and the employee gives low effort they again split the Surplus but it's less than the high salary high effort Surplus finding the Nash equilibrium works just the same we want to find the No Regret outcome and that is the low salary low effort outcome no matter what the employer does the employee is better off choosing low effort but being able to see that clearly the employer's dominant strategy is to offer a low salary they'll always wish they had if you check them you can also see that every other outcome here is Pareto optimal so again we have another classic prisoners dilemma game theory has a lot of power when it comes to explaining the world around us biologists have applied it maybe more than economists because of how well it explains which traits will be favored by natural selection trees are another classic prisoner's dilemma if you are a tree it would take way fewer resources to be short rather than tall but if you grow a little taller than the trees around you you'll get more sunlight you'll get more resources so the dominant strategy is to be tall rather than short bad for the trees but I say it's good for those of us who love tall trees for nearly a century now Nations have employed a strategy called mutually assured destruction the U.S has stockpiled so many nuclear weapons that if we use them all the entire human race would die and the U.S threatens to do exactly that should any other Nation detonate a nuclear bomb on U.S soil mutually assured destruction is an attempt to set the payoff Matrix in a way where the dominant strategy is to not use nuclear weapons because the payoffs are so bad for even using just one less of an existential threat but more of an annoyance is how Game Theory applies to concerts everyone would be more comfortable sitting down but if you stand you can see better and if everyone else stands you need to stand to see it all so no matter what everyone else does your best strategy is to stand so everyone stands the prisoner's dilemma is all around us and Game Theory can help you navigate the complex world of competition