Okay, before going any further, maybe it's a good time to play a very simple game, to see what is the nature of strategic thought. Okay, so I'd like to ask you to prepare a deck of card and then. Okay, I am going, we are not going to use all of the cards. What we need is eight cards, okay? So first you prepare black king and black number one, ace, and black number two, and number three. Okay. Also you need red ones. Okay? Red king and red ace number one and red 2 number two, and a red 3. Okay. And you need to find a partner to play a simple card game with you. So let me explain the rules of the game. Okay. One player is the red player, okay? And red player holds four cards. King, ace, two, three. The other player is the black player who has black card king, one, two, and three. And each player carefully chooses one card and show it to the opponent, simultaneously with the opponent. Okay? And payoff is dependent on what cards those players chose. Okay, suppose red player and black player chose king. And in this case red player wins. Okay. And suppose those two players choose cards with different numbers. Okay, so red player chose one. And black player chose three cards with different numbers. And then again red player wins. Okay? In all other cases, the black player wins. This is the nature of this card game. And the go, I'm going to ask you and your partner to play this game for a certain number of iterations. And give me the result by means of first homework. So I'd like to ask you at your earliest convenience to do this game. And all the instructions are given in the homework page. And, take your note carefully about what kind of, cards, you chose, and report back to me, the results. Okay. Okay. So let me summarize the rules of the game. It's a very strange game but it's fun to play. So red player wins if. Both players choose K, King. Or players choose cards with different numbers such as one and three. Okay? In all other cases, the black player wins but let me explain when black player wins. Okay? A black player wins if. Only player chooses king. So one player chooses king, the other player chooses a card with a number. In that case, black player wins. Or if players choose cards with the same number, like one and one, then black player wins. Okay? So I'm going to ask you to you know, play this card game again and again with your partner. Okay, this is a perfect example of a strategic situation, because what is best for you crucially depends on what other player is going to do. Okay. And each player is trying to do their best, each player is trying to do his or her best against the other player. Okay, so by experiencing this card game you can personally see the nature of strategic thinking. So after playing this game, I'd like to ask you to think about the following three interesting questions. Okay? The rules of the game are not symmetric. Okay? So maybe the set of rules here favors red player. Well, maybe it favors black player. So the first question if the following, who has an advantage, red player or the black player. I would say that it, it, it's not so obvious. Okay? But after playing this game, think about this question first. Second question. What is the winning rate or winning probability of each player. Is it 50-50? Or, if red player has an advantage, is it 80-20 or 70-30 or some, anything like that? Again, this question, this question is more challenging because it's numerical. But let's try to think what would, what the answer would be. More challenging question is the following. Do you have any prediction about the distribution of cards you chose? Okay, maybe you choose card number more frequently. And maybe you can cho, you choose king less frequently. Okay? What is the nature of the distribution of the cards chosen by the players? Okay? This is a very, very tough question. And I'd like to say that I, if I don't know game theory, I have no clue to how to answer this last question. Very important thing about game theory is that game theory is going to give you very precise predictions about those three problems, okay? And I'm going to show you the answer possibly in the side work.