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Hi again folks. So, let's talk a little bit about a folk theorem now for this kind

Â of repeated game. So we're in the case where there is a discount factor and

Â people care more about today than the future, or than tomorrow and so forth. And

Â we want to think about the expansion of the logic that we just went through in

Â some examples but see whether that holds in a general setting of repeated games. So

Â what's the folk theorem, what's the extension of two repeated games? so take a

Â normal form game, and so that, there's actually very many versions of folk

Â theorems, and we'll do a very particular one which has, I think, the basic

Â intuition behind it, and a fairly simple proof. So the idea is we looking at, some

Â nash equilibrium of the stage game. So. Take a stage game, find a profile which is

Â a Nash equilibrium of the stage game, and then also look for some alternative

Â strategy and here we have a couple of typos that should be an a prime. So, look

Â for some alternative strategy, A prime, such that everybody gets a strictly higher

Â payoff from A prime than A, okay. where A is a Nash equilibrium. Then there exists

Â some discount factor below 1. Such that if everybody has a discount factor above that

Â level, then there exists a subgame perfect equilibrium of the infinite repetition of

Â the game that has a-prime played in every period on the equilibrium path. So what

Â this is telling us is, the logic that we went through in those two examples of

Â prisoner's dilemmas, where we found the discount factor of either 1/2, or at least

Â 7/9, etcetera. take any game find the Nash Equlibrium of that and find something

Â which is better than that which you'd like to sustain in an infinitely repeated game.

Â You can do the same logic that we did in those examples in the general case where

Â there'll be a high enough discount factor. That'll make that sustainable. Okay? And,

Â and basically, the proof the, of this theorem is, is very similar to what we

Â went through in those examples. So the idea is, is, you know, we'll play a prim

Â e, as long as everybody plays it. if anybody deviates from that, then we're

Â going to go to Grim Trigger. We're just going to threaten to play the Nash

Â Equilibrium a forever after, which is giving us a lower payoff than a prime. And

Â we just need to make sure that people care enough about the loss of the future to

Â offset the game the game from today. So in terms of the, the proof, checking that

Â this is a subgame equilibrium for high enough discount factors, what do have to

Â do? Well, playing a forever if everyone, anyone has ever deviated is part of a

Â subgame perfect continuation, if we If we ever have a deviation, because it's Nash

Â in every subgame. so we need to check, will anybody want to deviate from a prime

Â if nobody has in the past? And we can bound the gain. So an upper bound on the

Â gain is the maximum overall. Players and all possible deviations they could have of

Â the best, of the gain and payoff that they would get from that. So that gives us a

Â maximum possible gain. the minimum period, per-period loss, so this is the maximum

Â they can gain from today. We'll compare it to the minimum they could lose from

Â tomorrow. So, the minimum they could lose is, instead of getting a prime, they're

Â going to go to a. so that's that and, and take that, the, the minimum across

Â different players for this. And one question, sort of why this, the question

Â here is, you know, is, is, is really the minimum relative to the Nash equilibrium

Â or, or could they gain. So think about this a little bit. why wouldn't they want

Â to change from the Nash equilibrium in the future? Right, so the idea there is there,

Â there not going to be able to, to help themselves by trying to change away from

Â the punishment because that is a Nash Equilibrium so they're already getting the

Â best possible payoff, if the other people follow through with the punishment. So

Â we've got the maximum possible gain, the minimum possible loss. so if I deviate and

Â given what other players are doing the mox, maximum possible net gain overall is

Â I'll ga in the M today, but I could lose up to M tomorrow, in the future, tha I'll

Â lose at least m in the future. and this should be an i. Then we've got beta i over

Â 1 minus beta i. And, so, if you go ahead and, you know, set this has to be

Â non-negative, sorry, has to be negative in order for players not to want to deviate.

Â So, what, what do we need? We need, the m is, is, less than or equal to this. So m

Â over m is less than or equal to beta i over 1 minus beta i. and that gives us a

Â lower bound on beta i. It has to be at least m over m plus m. So that's not a

Â tight lower bound in the sense that we've went with fairly loose bounds here. But if

Â everybody has a high enough discount factor, then you can sustain cooperation.

Â So, this is just a straightforward generalization of the examples we looked

Â through before through before. And it's showing us that we can sustain cooperation

Â in an infinitely repeated setting ss, provided people have enough patience for

Â the future. now there's many bells and whistles on this. one thing to think about

Â you can, you can sustain fairly complicated play if you, if you'd like. So

Â let's take a look at the game we looked at before. So, the prisoner's dilemma, but

Â now we've got this very high payoff from deviating. one thing you can notice is the

Â total of the payoffs here, the players together get 10. if they cooperate they're

Â only getting 6 in total. So here, actually playing this makes one of the players

Â really well off. So if they played this in perpetuity, they'd get 3, 3. Suppose they

Â try and do the following. They say in odd periods will play c, d. Right? So in

Â periods 1, 3, 5, and so forth, will play co-operate by the first player, defect for

Â the other. So, the second player's going to get tens in those periods, but then

Â we'll play, we'll reverse it, in the even periods. Right? So now, hopefully on

Â average, players are getting 5 each instead of 3 each. Right? So what we'll do

Â is we'll, we'll alternate and as long as we're con, continuing to abide by these

Â rules where we nicely do this then in, in the future we, as long as everybody does

Â this we'll continue to do it. if anybody deviates from this, then we'll just go to

Â defect, defect. All right, and you can check and see what kinds of discount

Â factors you need, and you know, are there different discount factors you need for

Â the first player, the, for the player that's getting the CD in the first period,

Â versus the second player? And so forth. So you can go through that. And actually this

Â kind of thing is, is something that, that people worry about in, in regulatory

Â settings. So, for instance you know, imagine that you have a situation where.

Â You've got companies, bidding for government contracts and they're

Â repeating, you know, they're doing this repeatedly over time. and one way they

Â could do it is to say okay look, we could compete against each other and, and bid

Â very high every day or, or have to bid, you know, to give them, the government, a

Â low cost every day, if there's a procurement, contract. but what they could

Â do alternatively is say okay, look, I'll let you win the, the, contract today. You

Â let me win it tomorrow, and we'll just alternate. And as long as we keep

Â cooperating we won't compete with each other, we'll enjoy high payoffs. but if

Â that ever breaks down, then we're going to go back to competition. So they're

Â situations where regulators worry, and in fact, there's some various cases that have

Â some evidence that, companies will tend to do this, to try and game the system, and

Â increase payoffs. So you can see the kind of logic in what has to be true in order.

Â For that to happen. Okay. So, repeated games, we've had a fairly,

Â detailed look at these things. Players can condition their, future play on past

Â actions. That allows them to react to things in ways that they can't in a static

Â game. It produces new equilibria in the game. folk theorems, partly referring to

Â the fact that These were known for a long time in kind of folklore and game theory

Â before they were actually written down. there's many equilibria in these things

Â and they're based on, on key ingredients. having observation of what other players

Â do and being able to react to that and having sufficient value in the future,

Â either by limit of their means, which is an extreme value or high enough discount

Â factor so that players really care about the future. Now repeated games have

Â actually been a fairly active area of research recently. There's a lot of other

Â interesting questions on these. What happens if you don't always see what other

Â players do? You only see sometimes, there's some noise in this things. What

Â happens if there's uncertain payoffs over time? Our payoffs are varying. so there's

Â a whole series of, of issues there. There's also issues about things like

Â renegotiation. So you know, the logic here has been, okay, if, if we anybody ever

Â deviates, then we go to a bad. equilibrium forever after. so suppose that happens.

Â Somebody deviates, and then we, you know, after about a few, a few periods, we say,

Â well, this is kind of silly. Why are we, why are we hurting ourselves? let's go

Â back to the original agreement. Let's forget about things, bygones be bygones.

Â So we can do better by just starting all over, right? Okay, well that's wonderful,

Â the difficulty is that if we now believe that if we deviate eventually we're going

Â to be forgiven, then that changes the whole nature of the game and changes the,

Â the, The incentives at the beginning. And so, incorporating that kind of logic is,

Â is quite complicated and another area of research in these. So repeated games are

Â very interesting. They have lots of applications. There's some interesting

Â logic which comes out of them. sometimes you can sustain cooperation, or better

Â payoffs than you can in a static setting, sometimes you can't. and we've seen some

Â of the features that affect that.

Â