 Now, that we have defined the Bayesian equilibrium in Bayesian games, let us look at some examples and try to find out what the Bayesian equilibrium in those games are. So, this is the same example that we have looked at in the previous module. This is the sealed bit auction, there are two players, both are willing to buy one single indivisible object and their values and their bids lie in this interval 0 to 1. So, the allocation function is fairly simple, it is just whoever bids the highest and breaking the tie in favor of player 1. So, this function is nothing but the indicator function which says whether player 1 wins or not and because it breaks the tie in favor of 1, it is the indicator that b1 is at least as much as b2. If b2 is strictly greater than b1, then player 2 wins. And the common prior in this case is uniform and we are assuming that these types are actually independent. So, therefore, if you look at the posterior, so from the point of view of player 1, when it knows its own tie, the distribution of over theta 2 is actually uniform again. Alright, so that is the setup. Now, we are looking at two different kinds of auctions. The first one is what is known as the first price auction and this is the one that we have discussed in the previous example as well. So, what is the difference in this case? So, this first price auction is essentially saying defining the player's payments. So, if player 1 wins, that is b1 is less than greater than equal to b2, then it should pay her own bid. If the other thing is true, that is b1 is less than b2, then player 2 will win and it will pay its own bid. So, that is the rule of the first price auction. So, we can write down the utility for this player when player 1 is choosing this action b1, player 2 is choosing b2 and their types of theta 1 and theta 2. Then we know that player 1 becomes the winner only when this condition is true. And in that case, it gets an utility of theta 1 minus b1. Similarly, for player 2, it wins under this situation and its utility is this. Therefore, its utilities can be written as a product of these two terms. Now, let us look at a very simple bidding mechanism. So, bid is nothing but a strategy chosen by each of these players, which depends on their own type. So, this is the x-interim decision that they are making. So, we are assuming that these strategies are simply a multiplicative factor of their own types. So, if their type is theta i, they are just choosing their strategy or their bid to be alpha i times theta i, where alpha is positive and of course, it should be less than or equal to 1. Now, the question is that in order to find this patient equilibrium, we need to find that si star or in this case, this alpha i star that maximizes the x-interim utility for this player i. So, the strategy is sigma i and this strategy should maximize this x-interim utility for player i when other players are actually choosing their appropriate s minus i or sigma minus i stars. So, we can write this problem in the form of an optimization from the point of view of player 1. So, what is this? So, this is essentially the utility. So, we have just expanded it. b1 is greater than equal to b2, where we have replaced b2 with alpha 2 theta 2. And this, because this theta 2 is a random variable, we will have to take the expectation. So, this is the same thing. So, this term is nothing but this, that with respect to the common prior, you are taking the expectation with respect to theta minus i when your own type is theta i. So, your expected utility, expected utility of player 1 when player 2 is choosing theta 1 is given by this and this will be, so player 1 will try to maximize this expected utility for all the choices of b1, which should lie between 0 to alpha 2. Note that you don't really need to go anything beyond that because theta 2 is always lying between 0 and 1. So, b1 never needs to be larger than alpha 2. If it is equal to alpha 2, that means it has already chosen the maximum value that it can achieve. Alpha 2 will just be a multiplying factor, which is less than 1. So, therefore, if b1 is larger than b2, then it is sufficient to look for only those numbers between 0 to alpha 2 for b1. All right, so now we can actually take a look at this. So, this, we know that this stuff, so the inner argument can be also written as theta 2 less than or equal to b1 by alpha 2 and as long as this condition is true, this is going to be 1, this indicator function. So, we can just restrict this integration from 0 to b1 by alpha 2 and because this is equal to 1 and this term is independent of theta 2, we can just take that out. All that you are left with is this b1 by alpha 2. So, that is the value of this integration. Now, if you want to solve this, you can see that the functional form of this is nothing but a concave function which has a maxima. And based on whether that maxima, so this maxima occurs at theta 1 by 2, it depends on whether the maximum value, this b1 star will be equal to this theta 1 by 2 or not depending on where alpha 2 lives. If alpha 2 is larger than that, that value, then of course, theta 1 by 2 is the maximum value, otherwise alpha 2 itself is going to be the maximum value. So, we can write this down explicitly and therefore, in a very similar way, we can also find the strategy, the optimizing strategy for player 2, the optimal bid for player 2. Now, these two things, if you choose this numbers alpha 1 and alpha 2 to be exactly equal to half, then this minimum value essentially becomes equal to theta 1 by 2 and theta 2 by 2 respectively, which becomes the Bayesian equilibrium in this case. So, therefore, this particular thing, this strategy, this bidding profile theta 1 by 2 and theta 2 by 2 is a Bayesian equilibrium. So, in the Bayesian game induced by the uniform prior on first price auction, bidding half the true value is a Bayesian equilibrium. That is what we have realized from the whole exercise. Now, let us move on to a different type of auction. The difference here is that the payment here, the highest bidder still wins, but it pays the second highest bid. So, the only difference is in the payment part. So, how does this change the utility? So, the utility for player 1 in that case will be under the same condition, player 1 wins, but the utility is theta 1 minus B2, because it is now going to pay the other player's bid. Similarly, if player 2 is the winner, then this difference is going to be the utility. So, now we can do a very similar exercise as before. The player's bidding problem, player 1's bidding problem is to maximize this quantity with respect to B1. Now, how does this B1 look like? Now, so this B1 now will be a little different function than the previous one. This now a function of theta 2 itself. So, unlike earlier where it was just a function of B1, it will be a function of the other player's bid. And therefore, so this is this S2 theta 2 is nothing but alpha 2 theta 2. So, if you'll write that down, you can actually get this expression here. And this term can be replaced with alpha 2 theta 2. Now, you can do this integration by restricting this term between 0 and B1 by 2. So, this upper bound will be restricted to this, because for everything else it is going to be equal to 0. Do this integration and you will find this expression here. And now this expression is going to get maximized only when B1 is equal to theta 1. And notice that this is independent of alpha 2. So, similarly, you can solve the problem, the same bidding problem for player 2 and find that B2 will be equal to theta 2. So, what it says is that under this assumption, the uniform common prior assumption and independence, bidding its true value, true type is essentially the Bayesian equilibrium for both these players. You can actually extend this idea if the distributions of theta 1 and theta 2 were arbitrary. Instead of this uniform thing that we have assumed, if you assume that they are arbitrary, but independent, then the maximization problem will slightly change. Earlier we were just using one in this case, but in this new setup where the distribution is arbitrary, we cannot use that to be equal to 1. You can put that arbitrary f. But when you do the integration, what you get is the corresponding CDF here and the other stuff remains as before. Now, if you do the differentiation, so the first order condition with respect to B1, you get this expression and again we get the same stuff. So, B1 minus theta 1 multiplied by some terms is going to be equal to 0. That means that whenever B1 is actually equal to theta 1, you get the maxima of this point. Of course, I have only shown the first order condition, but you can do the second order condition and convince yourself that this is certainly true. The second order derivative is actually negative. So, it is indeed a maxima point. But it also assumes the fact that the probability distribution at that point is positive. So, this is one of the assumptions that we have made that this probability mass is going to be positive everywhere. And the similar exercise can be done for player 2 and you can also find out that B2 will be equal to theta 2 under this condition as well. So, we can actually generalize this result and say that for any independent positive prior, so this positive prior is important, bidding the true type is a Bayesian equilibrium of the induced Bayesian game in a second prize auction. So, second prize auction is essentially good in this sense that it is revealing the true types as their bids. And we will see that this is going to be very interesting and we will revisit the second prize option and the first prize auction again when we discuss mechanism design.