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## Game Theory

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**Game Theory**Here we study a method for thinking about oligopoly situations. As we consider some terminology, we will see the simultaneous move, one shot game.**Oligopoly**An oligopoly is an industry structure. A key feature of this type of industry is that there are only a few firms in the industry. Let’s remember that in a competitive industry there were a whole lot of firms. Price was determined in the market through the interaction of supply and demand. Each firm was a price taker and went about its own business of maximizing profit. Each firm really did not have to worry about other firms because there is no way to have an impact. In a monopoly there is only one firm and so the firm did not have to worry about others. In perfect competition and monopoly the firms did not worry about what other firms are doing.**Have you every played the game rock, scissors, paper with**someone else? Basically you and the other player reveal at the same time each of your choices of the three mentioned alternatives. If both have the same it is a tie. Rock beats scissors, scissors beats paper and paper beats rock. Now, when you reveal your choice your outcome not only depends on what you do, but also what the other does. Sometimes the environment of an industry will be like this, particularly in oligopoly. The point is that firms in this type of environment have to worry about how their rivals might act. We say the firms are connected by strategic interaction. But perfect competition and monopoly are not like this. Examples of oligopoly industries might be the soft drink industry, the TV networks and automobile production. We will start off with a story called game theory as a way to model strategic interaction.**Simultaneous Move, One Shot Games**We will again assume there are only two decision makers - a duopoly. For now we assume both have to make their decisions at the same time. This is a simultaneous move game. Games could be sequential (like tic-tac-toe), where one player follows the other. Some games have only one round of decisions by each player involved (The game is over after each make their choice.) Other games involve repeated rounds of play.**Normal Form**In the normal form of a game, information about the options of each player is presented in a matrix. The “row” player’s options are described in each row and the payoff is the first number in each cell. The “column” player’s options are described in each column and the payoff is the second number in each cell. Since we have a simultaneous move game each player will choose its options without knowing what the other player will choose. But the choice of the other player may be anticipated.**On the next few slides I have a matrix set up. You will**notice I have called the two players the row player and the column player. The row player has to decide between going up or down and the column player has to decide between going left or right. Both will reveal their choice at the same time. In each cell of the matrix we see the outcome for each, with the row players outcome written first. For example, if they end up at up, left to row player gets 10 and the column player gets 20. Now, each player acts rationally. This typically means each wants the maximum profit they can possibly attain. Remember the profit of either depends on their own choice AND the choice of the other. The row player would like the profit of 15, but the only thing the row player can control is the choice of up or down. The column player picks left or right.**Generic Example**column player left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 The row player will think in the following way(look at each column). If column man picks left I can have 10 if I go up and –10 if I go down. So I will go up. If column man goes right I can have 15 if I go up and 10 if go down. So I will go up. In this example the row player sees it is best to go up no matter what the column player is doing. In this sense we say “up” is a dominant strategy for the row player. Players play their dominant strategy.**column player**left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 Now, here I have just repeated what I had on the last screen. I am illustrating the row player here. The row player has to decide between up or down. The final outcome for the row player will also depend on if the column player goes left of right. So, the row player will look at each option of the column player. Here I have highlighted the option left. The row player sees if the column player goes left the row player will get 10 or -10. In this case he would want to choose up.**column player**left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 Here I have highlighted the option right. The row player sees if the column player goes right the row player will get 15 or 10. In this case he would want to choose up. So, in both cases of what the column player can do the best for the row player is to go up. Up is a dominant strategy for the row player.**Example**column player left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 The column player will think in the following way(look at each row). If row man picks up I can have 20 if I go left and 8 if I go right. So I will go left. If row man goes down I can have 7 if I go left and 10 if go right. So I will go right. In this example the column player does not have a dominant strategy. What to do? If the other player has a dominant strategy, assume he will play it and then do the best you can. Column player should go left.**Nash Equilibrium**A set of choices by the players would be considered a Nash equilibrium if each player would NOT want to change their choice given the choice of the other player. The choices up and left is a Nash equilibrium because neither would choose to change given the choice of the other. The row player says – if the column player will be at left, then it is best for me to stay at up. The column player says - if the row player will be at up, then it is best for me to stay at left.**Let’s pick a different cell than up, left to see why**another cell might not be a Nash equilibrium. Let’s look at up, right. The row player says - if the column player is at right then I want to stay at up (Nash Equilibrium may still be in the running). The column player says – if the row player is up, then I want to change from right to left. Because the column player wants to change, the cell considered is not a Nash Equilibrium.