 In this module, we are going to look at the relationship between this dominant strategy implementability and dominant strategy incentive compatibility. So remember that we were discussing the indirect mechanisms and direct mechanisms. So if you can implement a social choice function using an indirect mechanism, we call that social choice function dominant strategy implementable. And if the direct mechanism is itself implementable, then we call it dominant strategy incentive compatible. Now, we have hinted at a result which actually shows some sort of a similarity or equivalence between these two things. And that result is known as the revelation principle. And here we are going to define the revelation principle for dominant strategy implementable social choice functions. And later we will see that we can actually look at a different version of this revelation principle for some other implementability as well. So what does this mean? It says that if there exists an indirect mechanism, remember those indirect mechanism where all those message spaces m1 to mn, g, so this message space and the corresponding decision rule that implements f in dominant strategies, then f is dsic. So you can as well find a direct mechanism to implement that social choice function. So the implication of this result is that we can without loss of generality focus only on dsic mechanisms. So the proof is fairly simple. You just have to do a very careful substitution. So it is just saying that if you have an indirect mechanism, so we start from that, we just write down the definition of dominant strategy implementability. So let us say this social choice function is implemented by this indirect mechanism. Then there must exist, according to the definition itself, there must exist some strategies Si which maps this theta is to Mi such that for all players in n and for all m minus i tilde's and Mi prime and theta i, this inequality should get satisfied. This is by the definition of implementability. This is by definition of dominant strategy implementability. Now, so this was the condition 1 and the second condition was that if everybody is picking their Si's accordingly because there exists such kind of strategies for all these players in that strategy profile, the outcome will be the same as the social choice outcome. So this is the second condition. Now we will be using this carefully. Now we know, so this equation 1 is going to hold for all Mi prime and M minus i tilde. So we are going to pick a very specific kind of messages. So the first message Mi prime is nothing but Si theta i prime. So you pick an arbitrary theta i prime for player i and apply the same strategy Si. The Si function is already given, you apply that, you get some message Mi prime. Similarly, you apply the same thing for all the other players for their corresponding theta i tilde's. So and theta minus i tilde is also arbitrary and for so this is essentially a couple, I mean a vector of all the strategies for all these players at their respective theta i tilde's, theta j tilde's and we are just using the shorthand to say that this is the message space chosen by the messages chosen by the other players. Of course, because this is going to get satisfied, this one is going to get satisfied for all Mi prime and M minus i tilde. So therefore, even if you pick this arbitrary theta i prime and theta minus i tilde and apply those strategies on them, this should also get satisfied for the same. So we rewrite this inequality 1 again, but now we are changing those things. So here we have made that substitution, this substitution and this substitution has been made in these two places and this inequality should get satisfied. Now we apply the second condition, we know that whenever you are looking at this Si's and S minus i's and the corresponding theta i's and theta minus i, they should implement the same. So the outcome should be the same as the social choice outcome as at those types. So we can write that this particular part here is nothing but f of theta i theta minus i tilde. Similarly, this one will be f of theta i prime theta minus i tilde. And now whatever you have is nothing but this and this is the definition of DSIC. So of course, the second condition is trivially satisfied in this case because you are essentially implementing the same social choice function. The direct mechanism in that case is the decision rule is f itself. All that we needed to satisfy is this condition here which is getting satisfied as we have shown here. So we can conclude that this function f is DSIC. So the whole phenomenon can be shown by by a schematic diagram here. We were initially looking at this message spaces. So remember this Si theta i where living in capital S1 theta 1 was living in M1. Similarly, Sn theta n was living in capital Mn and so on. So in this world we were so even though players were having their own types the mechanism the indirect mechanism was not asking them to reveal their theta ones rather they were asking them to give the messages in this message space. Something like purchase of those railway tickets of a certain class was their message that was being asked by the indirect mechanism not that type directly. So this indirect mechanism was taking those messages and applying this rule of G and that G of S theta 1, S1 theta 1 to Sn theta 1 was given as an output. Now what we can equivalently look at in this after we know this revelation principle is that if you are looking at only those indirect mechanisms which are DSIC then you can actually look at this dotted box here. So instead of looking at all these intricate details of this S1 and M1 same tools you can look at the direct mechanism theta 1 to theta n and this social choice function f directly and ask these agents to reveal their types. So they can report their types directly you apply this f on that and you can also find that this there exists a mechanism the direct mechanism which will also implement this in dominant strategy. So it is dominant strategy incentive compatible if you have a dominant strategy implementable social choice function. Similarly you can talk about the Bayesian extension so far we have actually spoken about only the types of others which can take any value. So we were saying that no matter what the message has been chosen by the other players or the types that has been chosen by reported by the other players the best response for this agent will be to report its type truthfully. So that is exactly what is meant by this inequality here that agent i is picking its type theta i and the other players are picking whatever they are trying to pick. So this is this theta minus i was arbitrary this inequality gets satisfied for all theta minus i tilde and of course for all theta i's and also for all i's. So here in the Bayesian extension we are weakening it a little bit rather than looking at the the inequality being satisfied for all theta minus i tilde we are taking the expectation with respect to theta i theta minus i tilde given theta i once. So this is something like the interim game where agent i has observed its type and it is trying to predict what is the type what could be the type of other agents and if it takes the expectation with respect to that then it is beneficial for that agent to report its type truthfully. So let us define it formally. So we are going to assume that these types are generated from a common prior and this common prior is a common knowledge you can see much similarity with this of this with the Bayesian game. So there also we had a common prior and that was also common knowledge and this type the type of a specific agent is revealed only to that respective agent. So we are looking at this interim stage where the types are realized but only you see the ith component if you are player i and you cannot see the other types even though you have a probabilistic belief about those types. So you can define this Bayesian game which has the same set of players the the message space is essentially taking the space place of the actions you have this original types capital theta i's now you have this common prior p and based on what type you are you have a different normal form game and this message mapping remains as before so all the setup and notation remains as before you are mapping these types into the message spaces. Now we can define the indirect mechanism in 1 to mn, g to be implementing a social choice function in Bayesian equilibrium in earlier case we were talking about dominant strategies now we have Bayesian equilibrium everything remains the same in particular this part also remains the same the change that happens is earlier we were having in this case we were having only m minus i tilde and we were so this both these cases were m minus i tilde and we are getting this satisfied for all m minus i tilde rather now what is going to happen is that we are taking this this quantity here that is player i is choosing this it's two two type is theta i and it is choosing this action or the strategy of s i other players are choosing their own strategies and this is becoming Bayesian equilibrium so you are taking the expectation with respect to theta minus i given theta i once you have observed theta i you have this belief you take the expectation with respect to that belief and once you take that reporting picking any other mechanism any other message in my prime is not beneficial for you why and you are still looking at the expected utility when you have observed your own type so this is quite naturally a little weaker than the previous definition of course if the previous definition was true that is some mechanism was dominant strategy implementable that is definitely Bayesian implementable you can do it as an exercise so in if this can two conditions get satisfied then we call this f to be Bayesian implementable via this indirect mechanism m 1 to m n comma g under this prior p so of course whenever we are talking about Bayesian incentive comp Bayesian implementability then we are also talking about this prior because that is very crucial we have just mentioned that this result that if your social choice function is dominant strategy implementable then it is also Bayesian implementable it it is very straightforward similarly to the direct mechanism in the in the previous case we can say that the direct mechanism is Bayesian incentive compatible similar to the dominant strategy incentive compatibility we have an equivalent definition for Bayesian incentive compatibility and we use this acronym BIC for it if the same thing happens that for every theta i and theta i prime notice that there is no theta minus i tilde anymore because this is expected over so you have this utility here when you are taking the expected utility after observing your own type that is going to be at least as much as whenever you are moving on reporting some other theta i prime this case so you are misreporting theta i prime that is not going to be any beneficial for you and you can also state a very similar result like the revelation principle for DSICs or DSi social choice functions here it is Bayesian implementable social choice functions that if the social choice function f is implementable in Bayesian equilibrium then it is also BIC so if you have an indirect mechanism which is implemented in Bayesian equilibrium then there must exist another the direct mechanism which is also Bayesian incentive compatible so that essentially demystifies the fact that we don't really need to look at if you are looking at only incentive compatibilities either Bayesian or dominant strategy you can rule out the indirect mechanisms altogether you can only without loss of generality look at the incentive compatible mechanisms or the direct mechanisms and the proof is very similar to DSi I recommend that you follow the same steps the only difference will happen is in the case where we are writing for all m minus i tilde here it will be expectations and that expectation has to be taken care of carefully but the steps are fairly straightforward so even though we have only defined all these results in terms of cardinal preferences you can also think of a very similar setup when these are all ordinal preferences and you can define all this dominant strategy implementation and dominant strategy incentive compatibility and similarly the Bayesian versions of it in the similar way and the corresponding revelation principle will also hold for those ordinal preferences and that I also leave as an exercise