You can have very simple games up to very complicated games. Some games are so complicated that you cannot almost solve them at all even with the help of a computer. Today, we're going to talk about the principles of games, so we'll bother mostly with simple games. The first distinction between games is if those games are static, meaning that they happen all at once, one shot games, or they are dynamic, meaning that the players move in turns, first move one player, then the other player looks at the first player did, and they respond, and then maybe someone else responds, or the first player responds again. These games that we take turns are called dynamic. The games for which players play all together we call them static games. So, let's start with static games, with the one shot games, which are much simpler to analyze. So, what game is static game, and especially of completing formation that we will discuss mostly about today. So, the assumptions. First of all, players choose their actions simultaneously. This is something like the game, rock-scissors-paper. When you play this game, you choose your actions simultaneously, at the same moment. If one chooses first, that's not going to be very well for the game. So, players choose their actions. And then according to the combinations of actions that we observe, players receive payoffs. Like for example, if one guy plays rock and the other guy plays paper, the person who played the paper will get one point, the person who played the rock will get zero point. This is the structure of payoff and we have payoffs for every possible combination of actions selected. The payoff distribution with respect to the combinations of actions, is known to everybody. Or as we say in economic terminology, this is called common knowledge. It's common knowledge what the structure of payoffs is. Also, players care to maximize their payoffs. They care to get as possible as high payoffs in this game. Be careful here. You might have learned in microeconomics that players do not see the payoffs same every time. They might have a risk attitude in which some people are risk averse, some people are risk loving, some people are risks neutral. In games, we do not have that. All risk attitude of the agents of the players is incorporated in this system of payoffs. So, we just care to maximize the expected payoff in each round of the game and we care for no risk attitude in those games. How do we represent the game? We use something that is called the game bi-matrix. If you want to know everything about the static game, you need three things. The first thing is the players. Who plays? The second thing that you would need to know is the possible set of actions of each player. How many strategies is possible for this guy who plays to follow. The third is the payoffs for each possible combination of actions. If you know these three things, you will be okay. You know how to solve this game and you know most likely how to predict the outcome of this game. So, in order to put all of these three pieces of information together, we use something that is called the game bi-matrix. And this is a matrix that looks like that. We have now the luxury of using colors, usually imprinted in black and white, but now we have the luxury of using colors, so this will make your life much easier to understand who is doing what. So, I have two players. Let's say, I have Firm A, which is represented in the rows of the matrix, and then I have, you guessed it, Firm B, which is the second player, and is represented on the columns of the matrix. Now, be careful here. Two players, Firm A and Firm B. Firm A is on the rows of the matrix. Firm B is on the columns of the matrix. So, the strategies now. Firm A has two strategies. Assume for example, that they can do research and development to improve their product, or not spend the money to do research and development, R&D. So, these are the two possible strategies for Firm A. Firm B has also two possible strategies in this case. Let's assume that they do not have the same strategies. Let's get a different one. Let's say that they can advertise a product, or not spend money to advertise a product. So now, we have two strategies for Firm A and two strategies for Firm B. This can make four combinations of strategies that they can be played in this game and that's why we have four empty cells in this matrix. Let's try to put the payoffs in these four empty cells of this matrix. Here are the colors. If Firm A plays R&D, and Firm B plays advertise, the payoffs will be 10 for Firm A, and five for Firm B. If you are from a country that they use comma to denote fractions, be careful because we are not doing that here. We use a period to denote fractions. A comma separates the payoffs here. So, 10 is the red payoff for the red Firm A, and 5 is a green payoff for the green Firm B in order to get the notation with the coloring. If they go for R&D and not advertise, payoffs becomes 15 and 0. If they go for no R&D and advertise, payoffs becomes 6 and 8. And then payoffs become 10 and 2 if they play no R&D and no advertise. So, this is how the game is denoted. This is the three pieces of information that we need for the game. The players, Firm A and Firm B, the strategies R&D or no R&D for one firm, and advertise or not advertise for the other firm, and the payoffs, which is a set of numbers that I gave you in these four cells. Now, you have all possible pieces of information for this game. Let's now try to work on these matrices and let's try to see what is going on. How can we predict the result in these cases? First of all, let's start with domination. In some cases, you have different actions in your life, that some of the actions, you will never take because they are not very smart. You know from the beginning that no matter what everybody else does, these actions are not very good. These are called strictly dominated strategies. A strategy is strictly dominated when it yields a strictly lower payoff than another strategy, independently of the actions of the other players. No matter what the other players are doing, this action is not a smart action for you. This is the meaning of a strictly dominated strategy. Look at this matrix here. Take a moment. Observe the numbers very carefully because I guess, you are not very familiar yet with this notation. So, take a moment to look at this matrix very sharp. Observe the players, Player 1 and Player 2. Observe the strategies, up and down for Player 1, left and right for Player 2. And then there is a set of different payoffs depending on the combination. Don't forget that these guys will select actions simultaneously. When Player 1 selects between up and down, we'll not know what Player 2 has done, and vice versa. So, when we see at this matrix, if we look very carefully, we will observe that these payoffs that I underlined are better than the payoffs that are below them. So, Player 1 will never want to play down if he's a rational, smart player. This is because that if you play down, no matter what Player 2 is going to do, you're not going to do well. You will get two or three instead of nine or 10. And in this game, your objective is to maximize your payoffs. We don't need to say that in every game that we will meet. This is the objective as we said. So, rational players do not play strictly dominated strategy. And here, down is a strictly dominated strategy, so this strategy is not good for the game. I'm going to cross it over because no rational player will play it. Moreover, not only Player 1 will not play down, but also Player 2 who will understand that Player 1 will never play down. So, these payoffs for Player 2, the green payoffs on the bottom, will never be realized for Player 2 because Player 2 knows that this game will never end up there since Player 1 will not play the dominated strategy. Now, how can we use these concepts in order to derive a prediction for the game? We'll come to the next segment.