 So, the plan today is to give you a general introduction to the issues that arise when you allow information exchange between agents. So, in this case this particular the first topic that I will talk to you about is what are called what is concerned itself with the problem of pre-play communication. So, players are communicating are allowed in this case to communicate with each other before the game, before actually taking a decision and the question that is asked in this paper is whether this actually leads them to play a particular Nash Equilibrium, ok. So, this is a classic paper by Robert Oman, the title of the paper is Nash Equilibrium Not Self-Enforcing. Now, what we will do today is we will do a reading of this paper because it is a very easy paper to read it is almost just like a little story that he has written and through that we will actually learn some lessons about pre-play communication. So, one of the things that one of the justifications that is used for the Nash Equilibrium is that it is said to be self-enforcing, self-enforcing means what that it means that if a pre-play agreement has been reached that play for in which players have agreed to play a particular Nash Equilibrium to play a Nash Equilibrium, then that Nash Equilibrium will be played during game play as well, ok. So, which means that the agreement to play a Nash Equilibrium actually enforces the Nash Equilibrium itself, ok. So, this is what the this concept is known this property is known as self-enforcing and it is one of the intuitive justifications for the Nash Equilibrium. As you as you see Nash Equilibria are not we cannot derive the concept of a Nash Equilibrium we can just define it and then save well justify it, yes. So, all those calculations are anyway part of the definition of the Nash Equilibrium. The Nash Equilibrium is defined by saying that well it is a point from which no player would want to deviate because no and no player would want to unilaterally deviate because only unilateral deviations are allowed. So, that is if anything is not a Nash Equilibrium then there is a problem. So, here the question is yeah that is a that is a different thing that is that is the problem of learning a Nash Equilibrium or arriving at a Nash Equilibrium you know through some kind of a dynamical process. There are various types of results I mean convexity based and I mean usually this leads to a dynamical systems analysis of the state or so the around the Nash Equilibrium has to be an attractor eventually your adjustment process should come get you back to the get you to the Nash Equilibrium. So, you need a kind of a stability study a dynamical system stability study for this. But here the question is not whether a Nash Equilibrium is meaningful the Nash Equilibrium is meaningful by itself, but one of the justifications for giving for why we you know why we say the Nash Equilibrium is meaningful is that we say that it enforces the self enforcing agreements. So, let us just read through what is written the intuitive basis for Nash's concept of strategic equilibrium in non-cooperative gains has recently received considerable attention a rationale that has been suggested is that Nash Equilibrium represents self enforcing agreements that a pre-play agreement to play a certain strategy tuple will be kept if and only if it is a Nash Equilibrium several years ago we came across an example that throws doubt on this contention. The example has been cited in various contexts, but has not but not here to for been discussed on its own merit. So, he is going to discuss this example. All right, so the example here is so the game of figure one has two pure strategy Nash Equilibrium C, C and DD. Let us just see what the Nash Equilibrium figure is. So, these are two player game completely symmetric player one on the plays rows player two plays columns C and D are the are the pure strategies and this is an econ paper. So, this is players are maximizing. So, both players are looking for the maximum pair. Now, in this case, if you see this matrix the there are two Nash Equilibria. The Nash Equilibria are C, C and DD. So, let us just study this carefully. So, if you see C, C is this 9 comma 9 that is obviously a Nash Equilibrium here you can compare 9 with 8 here and 9 with 8 here. All right, so that is a that is a Nash Equilibrium DD is 7 comma 7. So, this is also a Nash Equilibrium because playing 7 is better than 0 and 7 is better than 0 here. All right. Now, you can see that C, C is has some now out of these two Equilibrius both have some justification in addition to them being Nash Equilibrium or both have some additional property. So, here C, C is what is called Pareto Dominant. Pareto Dominant means it is better than for all players to play C, C than it is to play any other Nash Equilibrium. So, each player benefits by playing C, C. It is a Nash Equilibrium and it is uniformly better than all other Nash Equilibrium. So, this is one of the justifications that is used to eliminate an equilibrium like DD for example. So, say well C, C is better for everyone so since you are selecting one Nash Equilibrium may as well select C, C. But DD has another property DD is what is called risk dominant. So, I have not taught you this concept but roughly speaking the idea is each player tries to think of what would happen if the other player deviates from the Nash Equilibrium. If the other player deviates what is the what is the loss that I can incur. Now, in this case what is happening is you look at DD if the other player deviates then suddenly what happens is you end up getting any player. So, if suppose take the row player is playing D but the column player does not stick to D but deviates to actually this is I am being a little bit imprecise here. So, there is a notion by which you can by which you arrive at this by saying that essentially you ask how risky is it to play this means that suppose you have a certain projection about how the others play and if you go wrong in that projection how much of a risk are you taking. And it turns out that in this case DD is actually the one the equilibrium that has less risk as compared to the C, C equilibrium. The reason C, C has more risk than DD is the following. So, if the row player is playing C and the column player is playing C but suppose the column player does not play C and shifts to D then the row player suddenly goes from 9 to 0. So, the risk there is much higher as compared to DD where the risk is just 1. If the if they are playing DD and the other player shifts from D to C then you would then then you would lose not not 1 sorry the risk is 7 is this clear. So, for example here if the okay actually it is minus 1 so in fact this is even safer than right. So, the point is C, C is somehow is risky in the sense that it is a if you have gone wrong in some kind of projection you know there is a model behind all this okay this that helps you eliminate equilibria and C, C is more risky because if up it the idea is that if the other player does not stick to C and instead shifts to something else then the loss incurred is much higher is much higher than the 1 in D alright okay. So, DD is therefore what is called risk dominant it is safer in some sense okay. Now now indeed since players cannot communicate the row player Alice may as may well be uncertain that the column player Bob will play C alright. So, the since players cannot communicate there is no guarantee that they would in fact both coordinate on C they could they could as well coordinate on D right. She might therefore wish to play D which assures her 7 whereas with C she may get nothing. Moreover if she takes into account that Bob may reason the same way she is all the more likely to play D this makes it still more likely that Bob too will play D and so on alright this is effectively the reasoning that leads you to DD being the safer equilibria alright. We do not however assert that reasonable players must play D only that they may do so and that D is not unreasonable or foolish okay and for the time being we assert this only when there is no pre-play communication alright. Is this clear? So, in the absence of pre-play communication both equilibria there are 2 equilibria of this game and both equilibria have something attractive CC is Pareto dominant DD is safer alright. So, and the safer reasoning is basically just this that is that is given here alright alright. Now, let us now change the scenario by permitting pre-play communication and this is where things get interesting. On the face of it, it seems that the players can then agree to play CC so if you allow players to communicate in this game alright so suppose now you this is the game they are faced with and now players are allowed to discuss what they are going to play so what would they agree to do they would agree to play CC because after all it is better for everyone right. So, they would agree to play CC on the face of it it seems that players can then agree to play CC though the agreement is not enforceable it removes each players doubt about the other players playing other player playing C okay. Now, but the question is does it indeed remove this doubt suppose that Alice is a careful prudent person and in the absence of an agreement would play D okay. So, suppose the Alice is is in the absence of an agreement would play D suppose now that players agree on CC and each player retires to his corner in order to actually make a choice. So, they have verbally agreed to each other that they are going to play C and now comes the question of okay actually making the choice alright. So, you have an assurance that from the other one a verbal assurance that I am going to that the other player saying that he is going to play C the question is does he what information does that give you okay. So, Alice is about to choose C when she says to herself wait I have a few minutes let me think this over suppose that Bob does not trust me and so will play D in spite of our agreement then he would still want me to play C because that way he will get eight rather than seven and of course also if he does play C it is better for him that I play C thus he wants me to play C no matter what. So, he wants me he wants the agreement to play CC in any case it does not bind him and it might increase the chances of my playing C that does not imply that he will necessarily play D but he may since he wants the agreement no matter what he plays the agreement conveys no information about his play. In fact, he may as well have signed it without giving any thought as to how to actually play since he can reason about the same way about me neither of us gets any information from the agreement it is as if there were no agreement so I will now choose what I should have I would have chosen without an agreement namely D it is it is so that is that is basically the point the question is what so if you have agreed previously to do something and it is that agreement is on some on a point that is mutually beneficial okay beneficial uniformly beneficial as compared to the other point does it mean that that will actually get played. So, the question is when Bob says that let us let us do this let us both play C what does that I what that is a form if of that is a message that he is sending he is not actually binding himself but he is saying he is conveying the intent let us play C alright now form that form that communication what can Alice infer so this is not a question of psychology or any of that this is a question of what information content is present in the in the signal to say to play C and that is basically the point now why is there no new information so if that is the case if there is no new information then you know any sort of pre-play communication would typically carry no new information there is something special about this this matrix here yeah so so that is that is essentially the thing here so when Bob says that let us play C there are two things you do ways you can two pieces of information you could potentially get from this one is that Bob wants to play C the other is that Bob wants Alice to play C now you look at this cost function there is actually something very nice about this so Bob is the column player right and Alice is Alice is the role player is there anything dominating here any strategy dominates anything here nothing dominates anything right C 9 is greater than 8 but 0 is less than 7 likewise for the column so there is no dominant strategy here no there is no dominance but here there is a kind of a reverse type of dominance happening here in the in the following sense so if you look at what Bob is going to get as a function of what Alice plays ok now Alice playing C if Alice plays C Bob gets 9 and 8 and if Alice plays D Bob gets 0 and 7 so Alice playing C is better for Bob than Alice playing D so through this communication pre-play communication what Bob what Alice is saying is that what Bob has signaled is not that he wants to play C he may or may not want to play C it is that he wants Alice to play C not necessarily I mean see that is the point so this this is so the point so the point is from here form the signal that let us sign this agreement ok it is not conclusive that Bob wants to actually play what is said in that agreement ok when Bob says not let us sign this agreement what did I say when Bob says let us play C C it is from there you Alice cannot conclude that Bob means to say that I will play C it could also mean something else in this case it means that it means that Bob wants Alice to play C is this clear so this is basically the issue that so when you allow for communication like this right the communication from the communication what you are what you are inferring is some sort of hidden state that is available that is there at both players end so in this case the hidden state is say let us say which which of these two equilibrium strategies would the player want to want to play so we have reduced the player game down to these two equilibria now the question is between these two equilibria right so it is a question of estimating which one what is it that you would is your preference whereas now in this case what we have seen what we are seeing is that the intent to sign to play C C does not actually mean does not actually convey the intent to play C form on Bob's part it just it it just says that he is well let us play C C because that is exactly the you playing C is better for me and then of course there is nothing special about Alice Alice Bob could reason the same way and then effectively both both would go back and and do what they would do without an agreement now this is not saying that players should cannot will not play C okay this does not play say that players cannot play C C it just says that the agreement to play C C does not imply necessarily force them to play C that implication does not hold okay also the way it is written here it you may feel like well oh this is sort of trying to justify that one should always play D that is not what is being said either so for example it just says that what the players would play what they would play in the absence of an agreement so the agreement is non-informative is this clear okay so of course it so here you can read through this of course it may be that Alice is not careful and prudent but impulsive and optimistic and likes to think that Bob is also is so she may then choose C even without an agreement and and and so also with one we are not saying that rational players will never play C but only that agreeing to do so won't lead them to do it okay a player might play C or D whether or not he has agreed to C C the agreement has no effect one way or the other in such circumstances the agreement should not be called self-enforcing okay so the so this is basically his main point that you can the pre-play communication is may not actually lead you to a pre-play communication to play up and an agreement to play a particular equilibrium may not imply that you are in that you will in fact play that equilibrium okay so let's take another example so this is the this is the battle of sexes game this is a coordination game you can see here so the this is the above reasoning is not universal an agreement to play an equilibrium often is self-enforcing okay consider for example the familiar battle of sexes so this is figure two here so this this this game has has two players husband and a wife they want to both decide on whether they want to go to a ballet or to watch you know a fight or some see some boxing match or something but they would both want to do whatever it is they would want to do it together rather than do separate things so they get no payoff if they don't coordinate so if one guy goes to the ballet and the other goes to the fight they get both they both get zero if they both go to the ballet then the row player gets two and column player gets one if they both go to the fight then the row player gets one column player gets two okay this is the battle of sexes so essentially the players want to coordinate on one of these two either ff or bb so and both of these ff and bb are both Nash equilibria here okay so now the you can see here this is the sort of classic case where some some bit of communication would help replay communication would help because if you can signal your intent to say okay this is where this is where I would want to go then the other player would also want to coordinate on on that right but you have to be careful because not every time does not that intent is does not actually is not informative but in this in this particular case you see the numbers are such that in fact it is informative okay so consider for example the familiar battle of sexes without replay communications communication the players will be hard put to choose between ballet and fight but if they agree to bb then they are motivated to keep the agreement to explain to explain why consider again how Alice might reason it is not that she takes the agreement as a direct signal that Bob will keep it rather like in the previous section she realizes that by signing the agreement Bob is signaling that he wants her to keep it okay but unlike in the previous section here the fact that he wants her to keep it implies that he intends to keep it him himself okay so for her to it is worthwhile to keep it similarly for him this agreement is self enforcing okay so if you want to see why this is the case why why when the fact that he wants her to keep it implies that he intends to keep it himself if Bob plays B then he would prefer her to play B if he plays F he would prefer her to play F that is because of the coordination structure of this game so if he is playing B he would prefer that she she also plays B because because because in fact otherwise he is going to get zero it is not like he prefers ballet or prefers fight irrespective of what sorry he prefers her to play ballet irrespective of what he wants her to wants to do himself so if you see here look at Bob's payoffs one and zero one zero is not does not one zero is what he gets if Alice goes to the ballet if Alice goes to the fight one zero which does not dominate zero two right so he would want Alice to go to the ballet only if he himself also wants to go to the ballet all right okay so this is this is an example very very nice example of of what what happens of the you know once we start allowing communication within games okay so now let us we can this this whole problem of communication within games is actually interesting in its own in its own right so let us just read the last bit of discussion when then we can move because here there are there is a key there are a few key points that is that he makes here to say that that a game is non cooperative means that there is no external mechanism available for enforcing agreements thus when that when time comes to choose an action the players are assumed to act on the basis of existing incentives therefore an agreement is effective only if it changes the incentives that obtain in the presence of the agreement okay incentives can be changed in two ways either the payoffs are changed or the information okay either the payoffs are the change or the information of the players agreements being discussed here do not change the payoff the payoffs for any particular strategy tuple remain the same whether or not it violates the agreement okay so the agreement in that sense is not a binding agreement or there is no law or no other framework available to punish you in case you break the agreement so the payoff does not actually change even if you violate the agreement so therefore to be effective an agreement must change the players information okay specifically their information about how others will play an information about an event e is acquired by observing a parameter that depends on whether or not e obtains if the parameter does not really depend on e that is it has the same value whether or not e obtains then observing it yields no information about e okay in the games of figure of figure 1 and 3 this I skipped figure the other example 3 Alice is interested in knowing what Bob will play we may take e to be the event Bob will play C the parameter she observes is whether or not he agrees to C C okay but this parameter is the same no matter what Bob what Bob plays it is always to his advantage to agree to C C okay therefore the agreement yields no information about what he will really play since the agreement is important only for the information it yields and yields no information it is as if it had not been made this is clear okay so there are also some nice scenarios here maybe we should go through that also so the game of figure 1 is sometimes called the stag hunt game or the the dear rabbit game okay two men agree to hunt a stag to succeed they must go along separate paths giving that us undivided attention on the way each has the opportunity to abandon the stag hunt and and hunt rabbits instead okay if he does so the number of rabbits he bags increases if the other continues to hunt the stag both would prefer if it both hunted the stag since it is more valuable than a bag of rabbits but each fears that each mistrust the other and that mistrust breeds more mistrust and so on okay see in the international relations literature the game has been called a security dilemma two countries between which there is there are there is tension or each considering the deployment of a new expensive weapons system each of each is better best of if neither has the system but would be at a serious disadvantage if only the other had it can either side afford to not develop the system okay so this is some closely related game okay this is okay non-transferable utilities and all that we do not okay so this is this is again that sort of case so there are two equal two equilibria one is an equilibrium where where neither neither has you know this nuclear weapon the other is when both have the nuclear weapon neither is a better thing for both because it is it is very expensive to build have it but on the other hand the it is sort of safer risk wise the other equilibrium which in which both have the nuclear weapon is is risk wise safer right so again can can can either side afford to not develop the system and this is effectively what we see right okay a non-binding agreement can affect the outcome of a game only if it conveys information about what other what the players will do directly the information that such an agreement conveys is not that the players will keep it since it is not binding but that each player wants the other to keep it I mean we will have to formally write it out I mean there is a there is there is some he cited this Jarvis paper now we will have to look at exactly that so there is a particular pay security dilemma formulation so a non-binding agreement can have the effect of can affect the outcome of the game only if it conveys information of about what the other players will do directly the information that such an agreement conveys is not that the players will keep it okay because it is not a binding agreement but that each wants the other to keep it in the battle of sexes an agreement to play BB says BB say conveys the information that each player prefers the other to play B this implies that each play each will play B himself himself and so the agreement is self-enforcing but in the game of figures one and three each player always prefers the other to play C no matter what he himself plays therefore an agreement to play C C conveys no information about what the what the players will do and cannot be considered self-enforcing. So actually if you if you ever get a chance to be involved in some some of these decision situations in which let us say you want to collectively elect someone okay but if you want if that for that person to be collectively let us say a common situation which is arising you know in a lot of places is that there are like in in the city outside in Mumbai and in other cities also you will see this that there are old buildings that are being torn down and redeveloped into into newer buildings okay and now the but the for you for that proposal to go through a certain number of people from the existing building have to agree have to vote in favor of it all right now there could usually be competing proposals some that are better for some some that are better for the others etc etc and then there will always be situations where player people will want to convince the other to say you know to no no you you you sign the or part of this and so on you can see the risk there is that this is these these are high stakes gains because you are giving up your house effectively and there will be situation where player would want to convince the other player to sign a particular agreement and you would always ask yourself what is the meaning of that of you know why did he come and talk to me right what is he actually trying to say is it that he wants me to keep it or is it better or is it better for him that I keep it or is he telling me that he plans to keep it so these are all very closely related things and they are all closely intertwined you know sometimes it could there is there is another angle also here of in which a player may commit first okay but the commitment has to be enforceable as well if the commitment you there is a fine line between just me coming and telling you I am going to play C and me committing to play C when I commit I mean have I can really it means that I have in fact I have in fact you know that commitment can be enforced it is not just means you know sort of just just verbally telling you that I plan to play C is this clear so this is so these are all the kind of issues that start showing up once we allow for communication between between players okay so what is actually the information content in a particular communication and so on is eventually that influences the decision alright so what I want to do for the remaining part of the course is actually look at a few structured models that we know in which we know of you know the role of the in which we know a little bit about the role of communication okay.