hi guys i would like to talk about uh one of my favorite examples it's a kind of a weird example but i think a perfect one to understand to check whether you understood the concept of nash equilibrium or not so here's how the game is uh there are two players uh player one and player two um so each player is simultaneously and independently choosing picking a number between 0 to infinity well infinity is not a number so basically they are picking a positive strictly positive real number so s1 and s2 the set of strategies are
equal to zero infinite open interval all right so any positive real number is is okay uh but again doesn't have to be integer it can be real number um by the way it could be natural numbers the set of naturals could be just fine but anyway just leave it as is as is um well what is the payoffs well the payoff for each player once they pick s1 s2 so these are just numbers is going to be s1 times s2 all right as simple as this so the bigger number they pick uh you know more
utilities they will or more payoffs they will get so the question is what is or are the nash equilibrium of this game well the answer is in pure strategies obviously i i i don't talk about mixed strategies there is no pure strategy nash equilibrium of this game well why so well some kind of simple there's a few reasons i mean there's there are few ways to explain this well first of all for any sj the you know the strategy of the player j the best response for player i given that his or her opponent has
picked sj is actually an empty set all right what does that mean well that means so to visualize things let's suppose uh we're talking about player one and player two selected hundred all right so question is what is the best response for player one if his opponent has selected hundred so don't forget the best response is basically the best strategy all right so here 100 is not the best response by the way the proper way is this because you know best response can be set but i'm going to ignore these brackets just for notational simplicity
so 100 is not a best response why not well because if player one selects hundred right so let's calculate player one's payoff if he selects hundred and his opponent also selected hundred well it's gonna be hundred times hundred so it's uh ten thousands all right so i mean imagine this is like you know whatever number you pick a bank is gonna pay you this much cash and let's suppose the bank has no limit for cash so you're gonna be paid ten thousand dollars well but the thing is if i for example select instead of 100
200 which i can select right my because my strategies are in between zero infinity well i would actually make twenty thousand dollars well you may say why two hundred you can select three hundred yeah of course you can select my point is whatever number you pick so given your opponent's strategy whatever number you pick there will always be a better number that you could pick and so for that reason there's going to be no best number you could pick and hence the best response is going to be an empty set all right um well because
of this remember the nash equilibrium says a strategy profile s1 s2 is a nash equilibrium if s1 is the best response to s2 but you know the best response for player 1 is an empty set so therefore no strategy is going to be best response and hence there can't be any nash equilibrium well for this you may have to you know think a bit more or sort of verify this uh you can say well why not really for example uh million i mean put as many zeros as you like and you know the other guy
is doing the same thing two four six zero uh well why not this in nash equilibrium well boy it's still the same thing there's always a bigger number that you could pick right so remember nash equilibrium is a stability concept is a is a outcome the nash equilibrium is picking out comes uh these profiles of outcomes where no player is going to regret so here or here all right whatever s2 your opponent picks you will always regret from your choice and that's simply because your payoffs are always increasing you will never say enough to the
you know additional money and so and because there is no upper limit obviously you are always going to find a better number and the better number and so you will never settle for the best okay so there's no pure nash equilibrium in this game well if i change this game slightly for example instead of saying you're allowed to choose any number but positive what if i tell you that you could choose zero infinity meaning you can choose zero or a you know a number higher than zero what would be different well this time the same
question is there in the nash equilibrium the answer is yes but this time the only the unique nash equilibrium is both players playing zero all right well why is this nash equilibrium well look my opponent let's say i'm player one my opponent i learned that my opponent selected zero what would be i mean would i regret from my choice no because because my opponent has selected zero i'm gonna get zero payoff whatever number i pick right so therefore i will never say should i should have selected say ten thousand or one million who cares the
numbers will be multiplied and i'm gonna get zero anyway so therefore i mean that means the best response for first player when his opponent is playing zero is actually um any number in this interval and similarly including zero and the so zero is in this set and the this the second player's best response is also zero infinity interval and again zero is in this set so therefore 0 0 is a nash equilibrium because these two hold all right well why is it unique maybe this is a bit more challenging but exactly the reason that i
sort of gave you why there's no nash equilibrium when zero is not included in this set remember any positive number s1 s2 where s1 and s2 they are different than um they're different than zero so s2 is different than zero so there's no such nash equilibrium no nash of this form i mean you can verify right just guess and verify for example why 000 is not nash well isn't that obvious the player 1 is going to regret from his choice he's going to say look i selected zero my opponents i just learned that my opponent
selected hundred so my payoff is zero times hundred zero dollars but if i instead selected hundred i would actually make ten thousand so again here my best response as player one given that my opponent is selecting hundred is an empty set all right but that doesn't mean that there's a better strategy better than zero you see what i mean so be careful this is not nash yes because br 100 is an empty set so zero is not a best response and hence it can't be nash equilibrium all right but that also means that 000 another
way of uh reading this you know 000 is not nash equilibrium is the following because player 1 has better strategy all right you know better than zero but she doesn't have the best strategy so this the idea of there's no best is making there's no nash equilibrium when somebody's selecting a positive number but when somebody's selecting zero well then there is a nash where both are actually playing zero okay well uh i give i mean i like this example uh partly because it shows that in a game like this all right uh i mean just
select a number uh higher than or equal to zero if two guys play this game uh they would probably select some large numbers but no one would select zero zero right i mean so it basically shows that nash equilibrium is a solution concept with its own flows i mean it's not an appropriate solution concept for all type of games so this is just one example where it really doesn't uh make sense so maybe we should be looking at the different solution concepts other than nash equilibrium but at least if we modify this game and say
you know what you can pick a number between 0 and 100 but no more than 100. what would be the nash equilibrium well then in this case yes we do have nash equilibrium one of them is still awkward zero zero is still nash but there's another sort of reasonable nash equilibrium outcome a hundred hundred is also nash well the question is why hundred hundred is a nash equilibrium all of a sudden well because the best response for player one given that his opponent has selected hundred is now hundreds all right well why is that so
there is no better number because better number hundred and one ten thousand they're not allowed strategies so you can't choose them so the best strategy the best strategy given that your opponent is choosing 100 or in fact any s2 other than zero right so any number other than zero uh so for example 80 or 10 my best response as player 1 is 100 okay so therefore same for the second player obviously best response for the second player given that the first guy selects hundred is hundred so you know what hundred hundred is a nash equilibrium
but again zero zero is still nash because given that my opponent is selecting zero uh playing zero is the best response because whether i choose positive number or not my my number will be multiplied by zero and hence i'm gonna get zero payoff and so i'm in different sort of between zero and any other positive number so zero is the best response so don't forget once again nash equilibrium is a stability concept are we going to regret from our choices once we learn the outcome of this game and here the zero zero outcome is regret
free code and code um 100 is obviously regret free but in a game like you can choose uh only positive real numbers while in this game you will always regret from the final choice because there's always a better number okay so be careful about all these
