 We have seen the effect of a iterative elimination of dominated strategy with the max mean strategy. And we have seen that for the player whose strategy has been eliminated, that person does not that agent does not have any effect on the on the max mean value. For other players, the max mean value might change. Now in this module, we are going to ask the very similar question, but for pure strategy Nash equilibrium, what happens to this PSNE after you do an iterative elimination of dominated strategy? First, let us look at one very simple, make one simple observation that even if the even if you forget about the domination, if you just remove one strategy for a specific player, then you can see that if that strategy if if you remove a specific strategy and you find out that the Nash equilibrium which was there in the original game continues to survive in the new game, the reduced game, then that particular strategy profile will also be a Nash equilibrium in that reduced game. Why? Because if you imagine that you have removed the strategy of player J. For all the players except player J, nothing has changed. Their strategy set remained the same. Everything else remained the same. So if there exists a Nash equilibrium in the original game, which survives after removal a specific removal of a specific strategy of player J, then for all the other players, the inequality for the pure strategy Nash equilibrium remains exactly the same. For that last player, the player J for which it has it has been removed a specific strategy has been removed, but not the strategy which is the SI star SJ star, so which is the Nash equilibrium strategy. Then by definition of Nash equilibrium, SJ star was the the column maxima for that particular player. By column I mean when the all other players are playing the strategy of S minus J star, which is the Nash equilibrium strategy profile of all the other players, then SJ star was the maxima and something that you have eliminated not SJ star, but that does not change the maxima property of SJ star. It was the maxima over a larger set. Now you have removed made the set smaller and you have kept the same maxima. It will be a maxima even over the smaller set. That is the intuition why if you remove a certain strategy for any player, not necessarily a dominated strategy and it happens that the strategy the Nash equilibrium strategy profile survives in that reduced game, then that is definitely going to be pure strategy Nash equilibrium in the reduced game. So this is the intuition, the proof I can live as an exercise. You can do it. It is very straightforward. But the more important question is that can you generate new equilibrium? So can it be possible that if you remove some of these strategies in particular the dominated strategies, then you can actually generate a new set of pure strategy Nash equilibrium. And in this module we are going to show the next result essentially shows that if you remove dominated, weakly dominated strategy, then nothing is going to change essentially. The strategy profile, the PSNE of that reduced game is also a pure strategy Nash equilibrium of the larger game of the original game. And what can happen when you are removing weakly dominated strategy is possibly you have removed some of the PSNEs. We have seen one example before where we have eliminated in a specific order, then you ended up in a specific Nash equilibrium, pure strategy Nash equilibrium. But if you eliminated in a different order, then you possibly end up in a different pure strategy Nash equilibrium. That is entirely possible. In one of those elimination, you have removed, you can verify this, you have removed some of the pure strategy Nash equilibrium in the original game. But you cannot introduce new pure strategy Nash equilibrium. And that is exactly what this theorem is saying. So let us try to prove this result. So let me just formally state that, state the result. Consider a normal form game G. Suppose SJ hat is a weakly dominated strategy of player J and if J hat, G hat is obtained from the original game G by eliminating SJ hat, which is a weakly dominated strategy of player J, every pure strategy Nash equilibrium of that reduced game is a pure strategy Nash equilibrium of the original game. So that is what the theorem statement says. And before I jump into the proof, let me just give you the broad intuition why it is true. So we are trying to prove that if you have a pure strategy Nash equilibrium in the reduced game, that is definitely going to be a pure strategy Nash equilibrium in the original game. Again, use the same intuition that for all the players except player J whose strategy has not been eliminated, there is no difference in the original game and the reduced game. Their strategy profile remains the same. Their strategy sets remain the same. So their inequalities for this PSNE will remain identical. We just need to show for that specific agent for SJ, for SJ star that that is whichever it is. If that is the pure strategy Nash equilibrium in the reduced game, then it should also be a pure strategy Nash equilibrium in the original game, including this strategy SJ hat. Now you know that SJ hat is a weakly dominated strategy. So therefore it means that there does exist another strategy which is weakly dominating it. And that strategy the SJ star which is the pure strategy Nash equilibrium that is maximizing the utility when other players are playing the Nash equilibrium strategy and that SJ star is even larger than that dominating strategy which is dominating SJ hat. So therefore it is definitely going to be larger. The utility is going to be larger than the utility at SJ hat. And when the other players are playing the strategy is minus SJ star. So it might sound a little mouthful at this moment as we go along the proof step by step this intuition essentially will be proved formally. So let us first look at the reduced game. So G hat in this game G hat what do we have? We have the strategy set for this player J. This is a distinguished player which does not have the strategy SJ hat. It has all the strategies except SJ hat. For all the other players there is no difference between the reduced game and the strategy sets in the reduced game and in the original game. So this is true for all the agent IS which is not equal to J. Now what do we need to show? We start with S star which is a PSNE in the reduced game and we will have to show that it is also PSNE in the original game. So for all the agents for all the agents I which is not equal to J we can write this inequality UI of S star by the definition of PSNE. This is going to be at least as much as the utility for any unilateral deviation by that player I and this should be true for all SI. So this SI which is living in this space SI hat and that is equal to SI because for this players nothing has changed their strategy sets did not change. For player J we can write the similar inequality but this is a little smaller set it has one strategy less than the original game. Now we will have to show that there is no profitable deviation for any player including the player J. Now for this players as we said for I not equal to J this is immediate there is nothing that is changing. For J this is true for all the strategies except SJ hat. Now since SJ hat is actually a dominated strategy what can you say about that? So because this is dominated there must be a strategy TJ which lives in the same set SJ hat in the reduced game set that means the strategy set without the strategy SJ hat which weakly dominates it which means that you are going to so we are just using one of the definition of course this there is another part of the weakly domination which is a strict inequality but we don't need that. So it must be true that the utility when player J is playing TJ and other players are playing whatever they want to play this is going to be at least as much as the utility of the same player when it is playing SJ hat. So this is by the definition of weak dominance and this should hold for all S-Js. So this is weak dominance therefore it should hold for all SJs and therefore in particular if you focus only on the Nash equilibrium strategy for all the other players this is definitely going to hold. Now since a star is a PSN in G hat the reduced game and because TJ is belonging to this reduced set so notice that it is belonging to this reduced set then what can we say this SJ star so because it's a pure strategy Nash equilibrium this inequality holds because of the definition of Nash equilibrium because TJ is one specific strategy there and we have just shown that this inequality is also holding for SJ hat because this is a weakly dominating dominated strategy SJ hat this strategy is being dominated by that but clearly because of the definition of pure strategy Nash equilibrium this is also going to be smaller than the SJ star. Note that we cannot say this for all SJs because this inequality is true only when the other players are playing S-J star this is by the definition of pure strategy Nash equilibrium but this inequality holds for every S-J and we are just using the fact that this also holds for S-J star but at least on the on this Nash equilibrium profile we can show that this inequality holds and therefore if we look at the the entire game when SJ hat is also present in the strategy space this is also going to be this inequality is also going to get satisfied. So S star SJ star S minus J star is also a pure strategy Nash equilibrium of the original and that completes the proof. So what did we learn from all these exercises so we have not explicitly shown it but elimination of strictly dominated strategies you can argue that very easily that if you if you just remove strictly dominated strategies then it will have no effect on the PSNE neither new PSNEs will be created of course that is not possible we have shown it neither you will delete any of the existing PSNEs but the the second thing is possible for weakly dominated strategies and as we said in the previous example we have seen some of the weakly dominated strategy removal has actually removed some of the PSNEs but you will never add new PSNEs and in the previous module we have also looked at the maximum value the player for which you are removing the weakly dominated strategy the maximum value does not change for that player.