 Hi again, folks. So let's talk a little bit about a folk theorem now for discounted repeated games. So we're in a case where there is a discount factor and people care more about today than the future or than tomorrow and so forth. And we want to think about the expansion of the logic that we just went through in some examples, but see whether that holds in a general setting of repeated games. So what's the folk theorem? What's the extension to repeated games? So take a normal form game and so there's actually very many versions of folk theorems and we'll do a very particular one which has I think the basic intuition behind it and fairly simple proof. So the idea is we're looking at some Nash equilibrium of the stage game. So take a stage game, find a profile which is a Nash equilibrium of the stage game and then also look for some alternative strategy and here we have a couple of typos that should be an A prime. So look for some alternative strategy A prime such that everybody gets a strictly higher payoff from A prime than A. Okay, where A is a Nash equilibrium. Then there exists some discount factor below one such that if everybody has a discount factor above that level then there exists a sub game perfect equilibrium of the infinite repetition of the game that has A prime played in every period on the equilibrium path. So what this is telling us is the logic that we went through in those two examples of prisoner's dilemmas where we found the discount factor either a half or at least seven nines, etc. Take any game, find a Nash equilibrium of that and find something which is better than that which you'd like to sustain in an infinitely repeated game and you can do the same logic that we did in those examples in the general case where there'll be a high enough discount factor that'll make that sustainable. And basically the proof of this theorem is very similar to what we went through in those examples. So the idea is we'll play A prime as long as everybody plays it. If anybody deviates from that then we're going to go to Grim Trigger. We're just going to threaten to play the Nash equilibrium A forever after which is giving us a lower payoff than A prime. And we just need to make sure that people care enough about the loss of the future to offset the game from today. So in terms of the proof, checking that this is a sub-game equilibrium for high enough discount factors, what do we have to do? Well, playing A forever if anyone has ever deviated is part of a sub-game perfect continuation if we ever have a deviation because it's Nash in every sub-game. So we need to check will anybody want to deviate from A prime if nobody has in the past? And we can bound the gain. So an upper bound on the gain is the maximum over all players and all possible deviations they could have of the best of the gain and payoff that they would get from that. So that gives us a maximum possible gain. The minimum per period loss, so this is the maximum they can gain from today well compared to the minimum they could lose from tomorrow. So the minimum they could lose is instead of getting A prime they're going to go to A. So that's that and take the minimum across different players for this. And one question is sort of why this? The question here is really the minimum relative to the Nash equilibrium or could they gain, so think about this a little bit, why wouldn't they want to change from the Nash equilibrium in the future? So the idea there is they're not going to be able to help themselves by trying to change away from the punishment because that is a Nash equilibrium so they're already getting the best possible payoff if the other people follow through with the punishment. So we've got the maximum possible gain, the minimum possible loss. So if I deviate then given what other players are doing the maximum possible net gain overall is I'll gain the M today but I could lose up to M tomorrow in the future I'll lose at least M in the future and this should be an I. Then we've got beta I over 1 minus beta I and so if you go ahead and set this has to be non-negative sorry, has to be negative in order for players not to want to deviate so what do we need? We need that M is less than or equal to this so M over M is less than or equal to beta I over 1 minus beta I and that gives us a lower bound on beta I it has to be at least M over M plus M so that's not a tight lower bound in the sense that we went with fairly loose bounds here but if everybody has a high enough discount factor then you can sustain cooperation so this is just a straightforward generalization of the examples we looked through before and it's showing us that we can sustain cooperation in an infinitely repeated setting provided people have enough patience for the future now there's many bells and whistles on this one thing to think about you can sustain fairly complicated play if you'd like let's take a look at the game we looked at before so the prisoner's dilemma but now we've got this very high payoff from deviating one thing you can notice is the total of the payoffs here the players together get 10 if they cooperate they're only getting 6 in total so here actually playing this makes one of the players really well off so if they play this in perpetuity they get 3-3 suppose they try and do the following they say in odd periods we'll play CD so in periods 1, 3, 5 and so forth we'll play cooperate by the first player, defect for the other so the second player is going to get 10s in those periods but then we'll reverse it in the even periods so now roughly on average players are getting 5 each instead of 3 each so what we'll do is we'll alternate and as long as we're continuing to abide by these rules where we nicely do this then in the future as long as everybody does this we'll continue to do it if anybody deviates from this then we'll just go to defect-defect and you can check and see what kind of discount factors you need are there different discount factors you need for the first player the player is getting a CD in the first period versus the second player and so forth so you can go through that and actually this kind of thing is something that people worry about in regulatory settings so for instance imagine that you have a situation where you've got companies bidding for government contracts and they're doing this repeatedly over time and one way they could do it is to say look we can compete against each other and bid very high every day or have to bid to give them the government a low cost every day if there's a procurement contract but what they could do alternatively is say okay look I'll let you win the contract today you let me win it tomorrow and we'll just alternate and as long as we keep cooperating we won't compete with each other we'll enjoy high payoffs but if that ever breaks down then we're going to go back to competition so there's situations where regulators worry and in fact there's some various cases that have some evidence that companies will tend to do this to try and game the system and increase payoffs so you can see the kind of logic and what has to be true in order for that to happen okay so repeated games we've had a fairly detailed look at these things players can condition their future play on past actions that allows them to react to things in ways that they can't in a static game it produces new equilibria in the game folk theorems partly referring to the fact that these were known for a long time in kind of folklore and game theory before they were actually written down there's many equilibria in these things and they're based on key ingredients having observation of what other players do and being able to react to that and having sufficient value on the future either by limit of the means which is an extreme value or high enough discount factors so that players really care about the future now repeated games have actually been a fairly active area of research recently there's a lot of other interesting questions on these what happens if you don't always see what other players do you only see sometimes there's some noisiness in things what happens if there's uncertain payoffs over time our payoffs are varying there's a whole series of issues there there's also issues about things like renegotiation so the logic here has been if anybody ever deviates then we go to a bad equilibrium forever after so suppose that happens somebody deviates and then after about a few periods we say well this is kind of silly why are we hurting ourselves let's go back to the original agreement let's forget about things by bonds be bygones so we can do better by just starting all over that's wonderful the difficulty is that if we now believe that if we deviate eventually we're going to be forgiven then that changes the whole nature of the game and changes the incentives at the beginning and so incorporating that kind of logic is quite complicated and another area of research in these so repeated games are very interesting they have lots of applications there's some interesting logic which comes out of them sometimes you can sustain cooperation or better payoffs than you can in a static setting sometimes you can't and we've seen some of the features that affect that