Game Theory 101 (#55): Discount Factors

gametheory101.com/courses/game-theory-101/
We need to explore infinite horizon games. However, if you add up an infinite string of a positive payoff, they all equal infinity regardless of the base value. Discount factors rescue us. Recognizing that consumption today is better than consumption tomorrow and the possibility that an interaction might not continue due to unforeseen circumstances, we use the discount factor to make tomorrow's payoffs smaller than today's.
Although we still end up with an infinite string of payoffs, they form something known as a geometric series. In the next lecture, we will see how geometric series have a clean closed-form solution that avoids the infinity problem.

[@rafaelgoncalves7443]

Just came from coursera. You made it way simpler to understand! Thanks :D

[@Nik-gs3gf]

You are amazing! Could not find any explanation of what a discount factor is in the German web... Thank you so much!

[@joselaloguzman8239]

bro you are such a crack. im from mexico and you are saving me during finals. please keep doing this.

[@PunmasterSTP]

How did your finals go?

[@ophrasbankaccount7716]

@@PunmasterSTP He graduated from his university with honors. Currently pursuing political science with hopes to run for Mayor with the end goal being Mexican president. Good stuff from this lad.

[@PunmasterSTP]

@@ophrasbankaccount7716 That's awesome to hear! Just to double-check, you were talking about Jose Guzman, right? kzbin.info/door/w1XkK4QgpNzVQLVOlXGlxQfeatured

[@PunmasterSTP]

Deltaaaawww yeah! I'm stoked for the rest of this series.

[@klairehoang667]

It's like every time a question pops up in my mind, you had already anticipated it and explained it so well in your video. I'm happier watching your video than some random Netflix show, wow the delicacy

[@KirklandBreiner]

Wish my professor taught this well!!! Haha wait you are my professor!!

[@PunmasterSTP]

I wasn't quite sure, but are you saying that William Spaniel is literally your professor at your university? Just curious...

[@KirklandBreiner]

@@PunmasterSTP uh, he sure was four years ago when I made that comment!

[@PunmasterSTP]

@@KirklandBreiner That's awesome! I've had some cool conversations and learned some neat stuff replying to old comments, and your response definitely did not disappoint me!

[@LDacic]

Wish my professor had a fraction of your teaching skills. Thank you.

[@PunmasterSTP]

I'm just curious; how'd the rest of college go?

[@DanZhukovin]

You have a handsome voice & your oral chakra is creating healing orgone-ki in my biceps, thank you

[@snazzybelt6535]

Please name your kid Cocker

[@orktv4673]

3:00 This is reminiscent of Pascal's wager. By the same reasoning, we can't be sure that we are visited by the mafia today and lose our money, in which case having the money tomorrow would be better. But I get the point that you're making. One question: how does the discount factor relate to the concept of positive affine transformations? A discount factor is very "numerical"; it doesn't seem in any way a product of our preference ordering.

[@HAAH999]

You explained the discount factor very rationally. Great work

[@huynhtho202]

very helpful, thanks a lot :')

[@NOVAsteamed]

Can someone explain to me why he puts an exponent to delta? Don't you simply subtract delta from Delta when you switch period? Let's say Delta is 0,02, then 0,04 then 0,,06 because you add Delta from period to period. This is why I don't understand the exponent over the delta.

[@Gametheory101]

Well, think of it this way. Suppose I get paid $1,000 every year. If we added it your way, then the value of the annuity for year 51 would be 0. It makes sense that the utility would be small---that's a LONG time away---but I would certainly rather have an extra $1,000 when I'm in my 83 years old than not. Using the geometric way, the annuity for the 51st iteration in today's terms is .98^50*1000, or roughly $364. Another way to see the problem with the suggested way is to put yourself in my shoes when I receive the 50th iteration. Should I think of the 50th iteration as infinitely more valuable than the 51st iteration? 82 year old me would be very confused why 83 year old me does not care at all about $1,000. The geometric way also solves this problem. 82 and 83 year old me internalize the same difference as 32 and 33 year old me.

[@NOVAsteamed]

@@Gametheory101 From what I understand, if delta dosent change the longer the time the less the money will be valuable. It basically does a horizontal asymptote to infiinity. So, the opportunity cost of doing another round of the game is less and less important as time goes. I'm only 18 and trying to learn by myself so what I said might be wrong.

[@Posei77]

Thanks William Spaniel this has saved me

[@dominiclebron6073]

this is genius- it makes me feel like a genius for cutting ties w/ my baby mama sooner than later. -> we keep getting the same payoffs for every PD we go through. i keep getting the same outcome. i always get the same low outcome, times delta. so, essetially im getting the 1 payoff, times delta.. and this continues infinitely. = I'm wasting alot of valuable time for these small payoffs. this man william spaniel is probably saving my life w/ these game theory lectures. thank u professor spaniel!!

[@sungodmoth]

So basically in MathJax it's: $\sum_{i=0}^\infty 3(\delta^i)$ And in python: k = 1 j = 1 delta = input('Enter a discount value. ') while k == 1: print 3*(delta**j) j += 1

[@ethanpetersen810]

Really should be called delay factors

[@Markd315]

So the irony I've noticed with these games is that if Yahweh or some other god descended and announced the date of the end of the world, he would break so many Prisoner's dilemma and Stag hunt agreements, like peace treaties and free trade agreements that he may be creating it by doing so.

[@beckerqueiroz]

Very helpful!

[@mesplin3]

Why not a limit of some average?

[@29yogev]

Thanks!

[@Gametheory101]

Thank you!

[@raikhanamir1488]

delta? are you kidding me? it is sigma. but thanks.. was helpful =)

[@harshitakitchen]

Raikhan Amir that is NOT sigma oof