 Now that we know the normal form games, let us start with the first concept in normal form games. And the concept that we are going to start with is called domination. So, this is about domination of strategies. One strategy dominates other strategies. Let us start with an example to understand what I mean by domination. So, let us look at this game matrix player 1 and 2, player 1 has two strategies u and d and player 2 has three strategies l m and r. And the numbers represent what their payoffs are, utilities are. And now let us look at these numbers a little more carefully. And let us try to argue whether a player, a rational player who always tries to maximize its utility will ever play the strategy r. So, suppose you are player 2 and you are considering playing a strategy like r, would you ever like to do that? And I will try to argue, you will never like to play r. The reason being that if you play r, in both these cases, if player 1 chooses u or the other player chooses d, you get a payoff of 2 and 3 respectively. But at the same actions, if you choose this action m, then you get 3 which is larger than 2 and you get 5 which is larger than 3. So, there is no perceivable situation where you could have played r and you cannot improve that by playing m. So, therefore, there is no reason why player 2 will ever play r. And in the context, in this context, we are going to say that this strategy r is dominated by strategy m for player 2. So, let us make this definition a little more formal. So, first we are going to discuss about dominated strategy. So, we are going to say strategy Si prime of a player i is strictly dominated. So, notice the term strictly. If there exists another strategy Si such that for every other strategy profile, every strategy profile or the other player. So, in this case, we did not say that it is a dominated strategy. So, we did not say that L is dominated by m because even though when player 1 plays u, you might get a strict benefit. But while the other player is playing 6, you are not getting a strict benefit. So, therefore, you will not be calling that L is dominated by m. But m r is definitely dominated by m because for every other strategy profile of the other players, this inequality holds. That is, if you play, if player i plays Si, it is going to get strictly more utility than that when it plays Si prime. So, this is strict domination. There is a weakened notion called the weakly dominated strategy. So, we are going to say that a strategy Si prime weakly dominated if there are two conditions. So, for every strategy profile of the other players, you make this inequality a little weaker. So, it will, it is not just it is going to be strictly better, it will be greater than or equal to. And this happens for all strategy profiles s minus i for all the other players. But there is another condition, there exists some strategy profile of the other players such that this inequality is strict. What that means is that whenever you are looking at these two profiles, two strategies Si and Si prime of this player i, for every strategy profile, whatever the other players are picking, you will be at least better off than the in playing Si than playing Si prime. But there should be one such case, one such strategy profile of the other player where you are getting strictly better off. If you are not having that, then we will not even call it a weakly dominated strategy. If it happens to be same for all these players, then there is no reason to call it domination of any kind. So, let us come back to this example and try to see which kind of, which of these strategy pairs are strictly and weakly dominated. Clearly, we can see that for all the s minus i, so when we are considering i to be equal to two, the second player, then minus i is nothing but player one. And for all possible strategies, so for all s minus i, which is, which can take the values of u and d, we can see that m, the utility at m for player two is strictly better than r. So, m strictly dominates r. So, in some sense, r is strictly dominated. But if you look at player one and look at this strategy d, so you can see that here, utility of player one when it plays u is at least as much. So, let me just erase a little bit to make it a little clear. So, if you look at player one, you can see that this utility is strictly greater than this one. This utility is also strictly greater, but this utility is equal. So, you can clearly say that it is, I mean those inequality for all strategies of the other players, the utility is at least as much as the utility sd, utility at u is at least as much as the utility at d. But there are certain s minus i's, which can be either m or l, but this inequality is strict. So, certainly d is weakly dominated while r is strictly dominated. Now, once we have discussed what is a dominated strategy, now it is the right time to discuss about dominant strategy. So, just the direction essentially changes. So, dominated strategy is when this strategy is being dominated by some other strategy, dominant strategy is something which dominates all other strategies. And the definition essentially says the same thing. A strategy si is strictly or weakly dominant strategy for a player i, if that si strictly or weakly dominates all other si primes, not including that si of course, all other si prime. Again, let us look at examples and try to find out what are the dominant strategies. So, let us come back to this very old example of neighboring kingdoms dilemma. We remember that there was two strategies for each of these players, agriculture or warfare, which is equivalent to defense. And the numbers, the utilities were as shown in this figure, in this matrix. Now, let us try to understand whether there exists some strictly dominant strategy for any of these players. So, here we can see that if player 1 picks defense, you can see that this strategy, so the utility here is strictly more than the utility when he picks agriculture. And similarly, if the other player is also choosing defense, then also it is a better strategist. It is strictly better than the utility when it picks agriculture. So, you can say that this strategy defense for player 1 is essentially strictly dominating, strictly dominant strategy, because it dominates all other si primes. Just in this case, the si prime that said si minus si is just a singleton. So, there is only one other strategy, which is agriculture. And si dominates, so defense dominates, strictly dominates agriculture. And you can see that the same thing happens even for player 2. Defense strictly dominates agriculture. Okay. So, let us now look at another example. And presumably you can, you might have guessed by now that we are going to give an example of a weekly dominant strategy. So, the example is as follows. You have one indivisible item for sale. Let us say we are trying to sell a painting and there are two players who value this item with these numbers V1 and V2 respectively. So, you can imagine that if they get that item, how much satisfied or happy they become that is represented by this number. And the protocol is as follows. They can choose a number between 0 and M, in being a very large number, so much larger than V1 and V2. And within this range from 0 to M, each player can choose one number. Picking this number is essentially their strategy. Now, this player, the game is as follows. The rules of the game is as follows. The player who quotes the largest number essentially will win that object, this indivisible item and will break the tie in favor of player 1. And this is an arbitrary tie breaking. But it pays. So, quote unquote pays the losing players chosen number. So, if your valuation is let us say $100 and you quote a value of let us say $90 and suppose the other person has also built something like $80, then you win, you will be given that object, but you will be paying $80 and therefore your net pay off because you value that object as $100, but you are paying $80. So, this difference of $20 is essentially the utility of this player. And of course, the utility of the losing player who does not get the item, his valuation, his utility is going to be 0. He does not make any payment, neither the item is allocated to him. So, he gets a value of 0. Now, to represent this using the normal form game, we have two players here, 1 and 2. And the strategy space is a continuous interval between 0 and m. Now, the utility structure is a little involved. You have, so for player 1, if player 1 and 2 have picked this numbers, S1 and S2, which are the numbers that are the strategies of these two players, then if S1 happens to be larger than S2 or equal to, because we are breaking the tie in favor of player 1, then that player wins. So, player 1 actually wins in this in this case, and it pays the second highest number, which is S2 in this case. So, V1 minus S2 is going to be the utility of this particular player. And if it does not happen, that is if S1 is strictly less than S2, then player 1 does not win, and its utility is going to be 0. Now, the similarly opposite thing happens when S1 is less than S2, then player 2 wins, and then the utility is going to be V2 minus S1 and 0 otherwise. Now, I am going to argue, you can perhaps pause at this moment and think about which kind of strategy or which kind of dominant strategy it has. I am going to argue that the strategy where Si is actually equal to Vi, this is going to be a weakly dominant strategy. So, what does that weakly dominant strategy mean? So, let us go back to the definition. So, it means that for all the other Si primes, notice that it is a dominant strategy. Therefore, all the other strategies, all the other Si primes should be weakly dominated by this particular strategy Si, which is equal to Vi. For all those strategies, it is going to be at least as much as the utility that it gets when it replaces this Si with Vi. But for that pair of strategies Si, Si and Si prime, there must exist some S minus Si tilde such that this inequality is strict. I think this is clear. Let us see why this is true. So, imagine what can happen. So, without loss of generality, let us assume that we are just focusing on player 1. Let us say S1 is equal to V1, and we just claim that this is going to be weakly dominating any other strategy. So, let us say what could be a different strategy. It can be either larger than that, larger than V1, or it can be less than V1. And the argument that I give for one side when it is larger, so let us assume that S1 is strictly larger than V1. The argument that we are going to use, you can just use a very complementary argument to show that when S1 prime is less than V1, then also the same argument will hold. Now, we are done with these two pairs, so S1 and S1 prime. And we will have to show that for all S minus i tilde, so let us say for all minus S minus i, for all the choices of the other player, which in this case is just player 2, whatever strategy that the other player is picking, the utility of player 1 under this S1 given S minus 1, which is S2, would be at least as much as utility of the same player when it is picking S i prime, and the other players are picking whatever they are picking. Why is that true? So let us assume that the other players, the second player is, so on a line you can draw this particular thing, so S1 is here and which is equal to V1, and S1 prime is somewhere here, which is a little larger than that. So if the other player, this player 2, bids something above, above S1 prime, then that agent essentially wins the object. And the utility of both these cases, utility here or utility here is going to be zero, so in that case the equality holds. In the other extreme case where the bid, the S minus 1, which is the strategy chosen by the other player, S2, is smaller than S1, then also you win and you get the same utility, because whether you bid S1 prime your valuation is still going to be V1, and you are going to pay the same one, you are still becoming the winner. So your utility does not change, there also the inequality is satisfied with the equality. Now the interesting part happens when S minus 1 that is S2 sits somewhere in between. So let us say S2 is here, so let us say S1 is nothing but S1 plus epsilon and S2 is somewhere in between, which is S1 plus epsilon by 2 let us say. And what happens in this case, if you are S1, if you had reported S1, then the good thing is that you would have lost this item, you would not have been given this item and your utility would have been zero. But if you had reported S1 prime, which is larger than that value of S2, you still win the object and you pay this S2, which is actually larger than your valuation. So remember that S1 was nothing but V1, so therefore your actual valuation when you win at a higher reported Si, your valuation is going to be V1 minus the payment that you are making, which is V1 plus epsilon by 2. So in some sense you are getting a negative utility here. So in that case, you can see that this inequality is going to be satisfied with strict, in a strict manner. Here you will have minus epsilon by 2, but if you had reported your V1 in the same way, if your S1 was equal to V1, then you would have got zero. So you can clearly see that in this situation you have a case where you are having a strictly worse outcome for some choice of S minus i. For all the other S minus i's, you are going to be at least as good as the utility when you are reporting your Si to be equal to Vi. And you can use this argument even when you are reporting it if your Si prime was here, so which was smaller than S1, you can similarly use a very similar argument and show that that is also a weakly dominant strategy. So for no matter wherever your Si prime leaves, either it is larger than S1, smaller than S1, which is equal to V1, it is going to be weakly dominated by S1, which is equal to V1. And similar arguments holds for player 2 as well. So therefore, you can say that this strategy, strategy of reporting your Vi to be equal to your Si is a weakly dominant strategy. All right. So now that we have, we know what a dominance is, we also know what a dominant strategy is. And now we also come to the next definition, which is dominant strategy equilibrium. So this is the first time we are using this term equilibrium. So let's spend a little bit of time. So as in any other kind of systems like physical systems, equilibrium is a point from which you don't really want to deviate. So equilibrium is sort of a stable situation. And in this context also we are going to define it in a very similar way. We are going to say that a strategy profile, S1 star, S2 star up to Sn star is a strictly or weakly dominant strategy equilibrium. And we'll use the acronym SDAC or WDAC. If Si star is a strictly or weakly dominant strategy for that player i. And this holds for all players, for all n. So S1 star here is a weakly or strictly dominant strategy for player 1. Similarly, S2 star is the weakly or strictly dominant strategy for player 2 and so on. So let us look at an example and try to find out what is the strictly or weakly dominant strategy. It's fairly easy to figure that out. So here is the game matrix. Maybe you can just pause again and try to find it out yourself before I give the answer. So the answer is that if you look at this particular strategy of player 1, B, you can see that it weakly dominates A because you have this utility which is at least as much as A. And there is also another strategy for the other player where the inequality is strict. So therefore, B weakly dominates A. Similarly, you can see that for B also weakly dominates C because you have a strict inequality here. So notice that when Si and Si prime, this pair changes your S-i at S-i tilde where your strict inequality will hold, can also change. S-i tilde is essentially dependent on this pair Si and Si prime. It might not be the same for all Si Si primes. So here the strict inequality holds when the other player is choosing D and here the weak inequality holds when the other player is choosing E. So B weakly dominates A, B also weakly dominates C. So therefore, B is a weakly dominant strategy for player 1. Similarly, you can argue that E is a weakly dominant strategy. I leave that as an exercise for you. E weakly dominates strategy D and therefore, this strategy profile B, E is a weakly dominant strategy equilibrium. You can change this number slightly to make it strictly dominant and you can see that this is a strictly dominant strategy equilibrium. But this example is more to illustrate what it means to have a weakly dominant strategy equilibrium. So we can predict because game theory is all about predicting the outcome. We can predict as if this particular outcome B, E is the most probable outcome of this game.