Game Theory 101 (#31): Comparative Statics

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Does a striker aim to his weaker side more or less frequently as his accuracy improves on that weak side? Rather than just find equilibria, we should also analyze how equilibria change as a function of the game itself. We use comparative statics to study such changes. This lecture introduces the concept of comparative statics and goes through their method of calculation.
Unfortunately, this is the one lecture where calculus is necessary. If you have a quarter or semester's worth of calculus, you will be fine; we only need to calculate simple derivatives here. The lecture also includes a non-technical proof for the claim in case you have no calculus background.
Comparative statics are what make game theory interesting. If you need more practice running through the process, check lesson 3.4 of the textbook. I cover five examples in great detail. You should have a hang of it by the end.

[@PunmasterSTP]

I’ve really enjoyed all of your videos so far. You could say that watching them makes me…ec-static 😎

[@Gametheory101]

Thanks for the purchase! I don't think it is too weird. Keep in mind that when x = 0, the goalie can always dive right and guarantee that the kicker does not score. At that point, the striker is indifferent between his strategies and can mix in any way he pleases. This also implies that when x = 0, the equilibrium described is no longer unique. Indeed, in equilibrium, the striker could kick right as a pure strategy, which I think would satisfy you.

[@qweruilp]

I understand the math and all, but isn't it strange that if X=0 (i.e. he never scores kicking left) the probability of him kicking left becomes 100%? Pr(kick left) = 1/(1+X) if X=0 then Pr(kick left) = 1/(1+0) which is 1 (i.e. 100%) BTW, great clips. I bought your book (haven't used it yet, don't know if I will) just to support you and express my gratitude!

[@jdfoote2]

At the point that X = 0, it doesn't matter what he does; the goalie will always go right, and so his chance of scoring is 0% no matter what he does.

[@bobvance9519]

So when developing a model of a situation, it seems that the first thing to do is find an instance of a game type that represents the situation just to get an idea of what's going on and then exchange some of the payoffs with variables to do comparative statics.

[@sebastianhjarndal9110]

how would you analyze it if all the outcome is a variable? is that possible. And what if you add a timing variable, that changed the outcome depending on what shots has come before?

[@kaktotak8267]

I think there's an important practical distinction that is not told by simply stating the NE distribution. The NE here is not stable. It's not an equilibrium because you make the best choice given the opponent's choice, it's an equilibrium because the opponent's choice makes you indifferent between all of your options. Your NE strategy is only best for you when the opponent always perfectly counters you, which is never the case. If the opponent deviates even slightly from the NE, you are better off with a pure strategy. But then of course the opponent can beat you with one of his pure strategies. So in practice, in a repeated game, nobody would actually randomize their choices according to the NE. Instead, the players would constantly try to provoke, anticipate and outsmart each other. Though in the long run, the frequencies would probably still add up close to the NE probabilities, assuming the players are good enough at optimizing their choices.

[@adamhilmi761]

Awesome vid, was not expecting the final result hahaha

[@pulkiitsharma2665]

mind fking blown . Amazing video, your plot twists are better than crime dramas fr.

[@lamortexotique]

I am stuck with something here. Have listened to #30 again and again but did not quite understand it. (1) What is X exactly? Is it the striker's accuracy in any direction? Or to a specific direction such as right or left? I am a bit confused about this. (2) Why was the original matrix the way it is? Why is it 1,-1 at RL? Why is it not X,-X just like how it is for LR? Hope you see this, and thank you very much again. I ordered your book and I am super excited. It should arrive today :)

[@lamortexotique]

Oh wait. I am not sure if I understand this correctly now...does X mean the striker's accuracy when he kicks LEFT? That means his kicks to the LEFT are imperfect, but on the RIGHT they are perfect? Please correct me if I am wrong. But if I am correct, then #30 probably needs a small correction. But I am not 100% sure.

[@goclbert]

Yeah that’s it. X is his accuracy kicking left.

[@MinhNguyen-wx3zp]

I don't understand. If the probability that striker scores at left side is X, how can you assume that the payoff on left side of striker would also be X ?????

[@bobjoe258]

thats how we define x. like the more likely to score, the higher the payoff. think about the average outcome if he shot left 100 times.

[@SchamaliDhali]

Instead of going through the quotient rule, you could have just brought (1+X) up and made it (1+X)^-1 and derivated that to get -(X+1)^-2 or -1/(X+1)^2

[@MMcGui]

With the chain rule, indeed more straightforward. But this is good as well, though really not mentioned.

[@worldeconomicsbro]

This result makes no sense to me. Nash equilibrium needs improvement in the model.

[@alibehfarnia4782]

I think you made a mistake. -1/(1+X)^2 is an increasing function with sup = 0

[@Gametheory101]

The derivative is increasing, but the fact that it is negative tells you that the original function is decreasing. (The derivative increasing tells you that the function is convex.)

[@robinwang6399]

So if X approach infinity, probability go to 0, so that results in the kicker never using their ace in the hole. This doesn’t seem right.

[@toomanycharacter]

X doesn't approach infinity, as it is a value between 0 and 1. X signifies the average payoff of kicking right, which in this case equals the probability of scoring the shot. Probabilities are always between 0 and 1, so X must be between 0 and 1 also.