Game Theory

1. 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.

2. Simultaneous Move, One Shot Games We will again assume there are only two decision makers. For now we assume both have to make their decisions at the same time. This is a simultaneous move game. The main point about a simultaneous move game is that although I may know all the information about what the other guy might do, I do not actually see what is done before I do my thing. Games could be sequential, where one player follows the other. Some games have only one decision by each player involved. Other games involve repeated rounds of play. Here we focus on the one shot games.

3. 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. Let’s check out an example on the next few screens.

4. Column player left right Row up a, A c, C Player down b, B d, D Let’s make sure we understand this table. Here we have two players: the Row Player and the Column Player (Miller and Bud, if you want). This is a generic game for now and all the Row Player can do is choose up or down (in rock, scissors, paper you can choose one of the three, but for the Row Player here the choice is up or down.) The Column Player can only choose left of right. Now, let’s say the row player picks up and the column player picks left. 4

5. Column player left right Row up a, A c, C Player down b, B d, D Now, let’s say the row player picks up and the column player picks left just as an example to see what each would get. If the game ended at up, left, the row player would get “a” and the column player would get “A.” Note all these letters usually are dollar amounts for us and can be negative amounts. But, sometimes the letters represent other concepts besides dollars. As an example the numbers could represent jail time. 5

6. Column player left right Row up a, A c, C Player down b, B d, D If the game ended at down, right, the row player would get “d” and the column player would get “D.” Caution: If as ROW Player you pick down thinking you get “b” remember that what you get is also influenced by Column Player. So, you could actually get “d” if the Column Player picks right. Next we want to think about how each player should decide what strategy to pick. 6

7. Row player should think this way: (As the row player) I do not know what the column player will do. But, I can look at each option of the column player, one strategy at a time. If the column player picks left my choices are up a down b As the row player I would pick the option that has the highest value: pick up if a > b, or pick down if a < b. If the column player picks right my choices are up c down d As the row player I would pick the option that has the highest value: pick up if c > d, or pick down if c < d. 7

8. For the row player the choice of strategy is up or down. A strategy is called a dominant strategy for a player if it has a higher payoff no matter what the other player chooses. So, up would be a dominant strategy for the row player if both a > b and c > d. Down would be a dominant strategy for the row player if both a < b and c < d. 8

9. 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. 9

10. Example column player left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 This is the same slide as before, but I show in each column what the ROW player will look at. If the highest value in each column is in the same row – the person has a dominant strategy. Again, up is a dominant strategy here for the row player. 10

11. Column player should think this way: (As the column player) I do not know what the row player will do. But, I can look at each option of the row player, one strategy at a time. If the row player picks up my choices are left right A C As the column player I would pick the option that has the highest value: pick left if A > C, or pick right if A < C. If the row player picks down my choices are left right B D As the column player I would pick the option that has the highest value: pick left if B > D, or pick right if B < D. 11

12. For the column player the choice of strategy is left or right. A strategy is called a dominant strategy for a player if it has a higher payoff no matter what the other player chooses. So, left would be a dominant strategy for the column player if both A > C and B > D. Right would be a dominant strategy for the column player if both A < C and B < D. 12

13. 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. 13

14. Example column player left right Row up 10, 20 15, 8 Player down -10, 7 10, 10 This is the same slide as before, but the column player looks at his possibilities in each row. 14

15. 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 represents 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. 15

16. 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. Note: You look for dominant strategies before the game is played and you see if you have a Nash equilibrium after the game is played. 16

17. Applications Here we look at several applications. We will see a classic example of a dilemma that can arise in such games. 17

18. Pricing Example column player low price high price Row low price 0, 0 50, -10 Player high price -10, 50 10, 10 The numbers in the matrix represent profit. The row player has a dominant strategy of a low price (0 is better than -10 and 50 is better than 10) and the column player has a dominant strategy of a low price as well (similar numbers). And, low low is a Nash equilibrium. A dilemma that arises here is that both could be better off if they both went with a high price. 18

19. Prisoner Dilemma Games The game we just saw has the characteristic that has come to be known as the prisoners’ dilemma. When the players act separately (competitively) the outcome is worse than if they cooperated. 19

20. Don’t be taken advantage of In the pricing example each might be tempted to get together and both charge the high price and both do better than the low price strategy. The problem of the high, high solution is that it is not a Nash equilibrium and thus each would have the incentive to change to the low price strategy. Your stockholders would not want you to be taken advantage of by a double crosser. Don’t collude here, for it is not likely to pay off. Let’s see why. Say you are the row player and you agree (illegally) to set prices high with the column player. After you finish the meeting you will note 1) if you as the row player see the column stay at a high price the it is better for you to have a low price (this is part of why high, high is not a Nash equilibrium), and 2) perhaps more importantly you will note the column player also has the incentive to switch to low (this is the other part of high, high not being a Nash equilibrium). So, if you stay at high you will look bad! 20

21. Advertising Example column player ad no ad Row ad 4, 4 20, 1 Player no ad 1, 20 10, 10 Profits are in the matrix. Each player has a dominant strategy of advertise. This outcome is a Nash equilibrium. A similar dilemma arises here as before. No ads would be better but each does not want to get taken advantage of in this situation. If you don’t have ads when the other does you won’t be on the minds of the consumers and thus you will not make enough. 21

22. Coordination example column player 120 volt 90 volt Row 120 volt 100, 100 0, 0 Player 90 volt 0, 0 100, 100 In this example say you have consumer appliance manufacturers who have the choice between making the appliances run on 120 or 90 volt plugs. If they both do not use the same plug they will earn less profit because consumers will have to spend on different plugs and thus not be willing to spend as much on the appliances. The game has two Nash equilibria – both doing the same plug. The next question is how do they get the same plug? 22

23. Coordination Example In this example, the firms need to figure out a way to get on the same page. Agreements to coordinate may be looked at negatively legally, so maybe companies lobby the government to set the standard for a product. 23