EconS Game Theory - Part 1

Transcription

1 EconS Game Theory - Part 1 Eric Dunaway Washington State University eric.dunaway@wsu.edu November 8, 2015 Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

2 Introduction Today, we are looking at Game Theory. This is how rms and individuals behave strategically. Normally, when decision makers make their choices, they only consider factors that they can control directly. Game Theory looks at how one rm s factors a ect another rms and how the two interract with one another. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

3 Game Theory What is a game? We re not talking about video games, or sports events, although both of them heavily use game theory. A game is any competition between two or more players where each player is able to choose some kind of strategy. Players can be individuals, rms, the government, etc. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

4 Game Theory We break down a players moves into actions and strategies. When it is a player s turn to move, they choose an action. When the player has multiple moves, all of their actions put together is the player s strategy. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

5 Game Theory After every player has moved the appropriate number of times, the game ends and each player receives a payo. The payo could be a utility level, a pro t level, money, anything, really. The key is that the payo is an ordinal number - The number itself doesn t really matter, just its ranking among the other payo s. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

6 Game Theory Games can have many di erent structures, and they determine the order in which players move, as well as if players actions can be observed. We break them into a few types. Move order: Simultaneous Move Games: This is when all players move at the same time. Sequential Move Games: This is when one player moves at a time. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

7 Game Theory Action Observability: Perfect Information: All players can observe everyone else s actions. Imperfect Information: At least one move for one player is not able to be observed by another player. Full knowledge: Complete Information: All players know everything about every other player. Incomplete Information: At least one player has some information that the other players do not know. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

8 Game Theory We are going to analyze games that focus on perfect and complete information. Moving away from those are for a more advanced class on Game Theory. That being said, I may relate some applications to these topics and use them for intuition questions. Let s start with simultaneous move games. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

9 The Prisoner s Dilemma Consider the case where two criminals, which we will call player 1 and player 2 rob a bank. They manage to get away from the scene of the crime clean, but get picked up the next day on unrelated crimes. The detectives are certain that players 1 and 2 were responsible for the bank robbery, but cannot prove it unless one player turns on the other. They proceed to put the players in seperate rooms and try to get one to betray the other. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

10 The Prisoner s Dilemma If both players remain silent, they both receive a 1 year prison sentence for their unrelated crime (a payo of -1). If one player betrays the other, the betraying player avoids prison all together (a payo of 0) while the betrayed players gets sentenced to 5 years in prison for the bank robbery (a payo of -5). If both players betray each other, they both receive a 3 year prison sentence for the bank robbery (a payo of -3). Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

11 The Prisoner s Dilemma Silence Player 2 Betray Player 1 Silence Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

12 The Prisoner s Dilemma We organize the game by putting all of the possible results in a table, then putting player 1 s strategies along the rows and player 2 s strategies along the columns. I have color coded everything in this example, but I won t always do this. For example, if player 1 chooses "Silence," we will only focus on the top row. If player 2 chooses "Betray," we will only focus on the right column. If they both make those choices, we end up in the top right cell of our table, and player 1 receives a payo of -5 (he gets betrayed and sent to prison) and player 2 receives a payo of 0 (he gets out of jail for free). Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

13 The Prisoner s Dilemma How do we nd the equilibrium? There are several ways, each with di erent levels of speci city. First, we want to look for strict dominance. This happens when the payo s for a player for a certain strategy are always higher than the payo s for another strategy regardless of what the opponent picks. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

14 The Prisoner s Dilemma Silence Player 2 Betray Player 1 Silence Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

15 The Prisoner s Dilemma As we can see, if Player 2 picks "Silence," Player 1 receives a payo of 0 for choosing "Betray" and -1 for choosing "Silence." Likewise, if player 2 pickes "Betray," Player 1 receives a payo of -3 for choosing "Betray" and -5 for choosing "Silence." Regardless of Player 2 s strategy, Player 1 always gets a better payo from choosing "Betray" over "Silence." We would say that "Betray" strictly dominates "Silence" for Player 1. This means that Player 1 will never choose to play "Silence." We can actually delete this strategy from our table since it will never be used. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

16 The Prisoner s Dilemma Player 1 Betray Player 2 Silence Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

17 The Prisoner s Dilemma Now, we can do the same thing for Player 2. Since Player 1 is choosing "Betray," Player 2 receives a payo of -3 from choosing "Betray" and -5 from choosing "Silence." Player 2 is obviously better o by picking "Betray," since it strictly dominates picking "Silence." We can now delete this strategy, as well. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

18 The Prisoner s Dilemma Player 1 Betray Player 2 Betray 3 3 Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

19 The Prisoner s Dilemma We are left with only one possible strategy for each player, and the makes it our equilibrium. This process is known as the Iterated Deletion of Strictly Dominated Strategies (IDSDS). Both Players will choose "Betray" and spend 3 years in prison for their crimes. A few notes: We started with player 1, but we would get the same result if we started with player 2. This is a nice property of IDSDS. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

20 The Prisoner s Dilemma This outcome is actually pretty bad. They both could have only spent 1 year in prison had they cooperated. Since they both had incentive to betray the other, they couldn t cooperate and ended up worse o. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

21 The Prisoner s Dilemma The Prisoner s Dilemma is the classic Game Theory example and has been studied to death in economics, psychology, and even biology. Interestingly, the theoretical results only hold in the experimental setting when the stakes are high enough. People are willing to take the cooperative outcome when only a few dollars are involved, but tend to act much more in line with the theory when thousands of dollars are involved. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

22 Golden Balls A British game show, Golden Balls, used a form of the Prisoner s Dilemma, known as the Weak Prisoner s Dilemma, during the climax of their show. The two players would work together to build up a cash pot (M) as a reward, then at the end, they both had to choose "Split" or "Steal" If they both chose "Split," they split the pot equally. If one chose "Split" and the other chose "Steal," the person who chose "Steal" gets the whole pot. If they both chose "Steal," they both get nothing. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

23 Golden Balls Split Player 2 Steal Player 1 Split Steal ½ M ½ M 0 M M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

24 Golden Balls In this game, we can t apply strict dominance. When one player is choosing "Steal," the other player is totally indi erent from choosing "Split" or "Steal." We can apply another solution concept, weak dominance, to nd a solution, but it gets a bit murky. The order in which we eliminate strategies matters in this case, and viable outcomes could accidentally get eliminated. Realistically, we need a better solution concept. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

25 John Nash This was the state of Game Theory until John Nash published his Dissertation in It was 27 pages long and had only 2 references (one of which was himself). The average dissertation today is 100 pages with 30+ references. It has been cited over 7000 times. Nash was awarded the Nobel prize in Economics in 1994 for his work. Nash focused on the idea of the Best Response Function. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

26 Best Response Function A best response function is where a player gures out their optimal move for every possible move that their opponent could make. For example, in the classic game "Rock, Paper, Scissors," The best response to an opponent playing "Rock" is "Paper," the best response to "Paper" is "Scissors" and the best response to "Scissors" is "Rock." We can gure out optimal responses for both of the players pretty easily regardless of the game. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

27 Best Response Function Nash took it one step further. In equilibrium, player 1 knows player 2 s best response. At the same time, player 2 knows that player 1 knows player 2 s best response. Also, player 1 knows that player 2 knows that player 1 knows player 2 s best response. That goes out to in nity. If both players can iterate their best response functions out to in nity and they end up having the best response for one player linked to the best response to the other player, they are in equilibrium. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

28 Nash Equilibrium To put it a bit more clearly, If for each player, they are choosing their optimal action, while at the same time every other player is choosing their optimal action, we are in equilibrium. This is known as Nash Equilibrium. Let s go back to the Prisoner s Dilemma. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

29 Nash Equilibrium Silence Player 2 Betray Player 1 Silence Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

30 Nash Equilibrium Starting with Player 1, The best response to player 2 playing "Silence" is to play "Betray" since its payo of 0 is greater than "Silence s" payo of -1 BR 1 (Silence) = Betray The best response to player 2 playing "Betray" is to play "Betray" since its payo of -3 is greater than "Silence s" payo of -5 BR 1 (Betray) = Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

31 Nash Equilibrium Likewise, for Player 2, BR 2 (Silence) = Betray BR 2 (Betray) = Betray Notice that the best response for both players when the other plays "Betray" is also "Betray." This implies that both players playing "Betray" is a Nash Equilibirum. We can see this graphically by underlining the best responses for both players. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

32 Nash Equilibrium Silence Player 2 Betray Player 1 Silence Betray Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

33 Nash Equilibrium When all of the payo s in a speci c cell are underlined, that cell is a Nash Equilibrium. Note that this was the same prediction that the iterated deletion of strictly dominated strategies (IDSDS) gave us. This brings up a neat property. If IDSDS gives a unique outcome (Only one cell left), that outcome is a Nash Equilibirum. Let s look at another example. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

34 Battle of the Sexes Consider a situation where a husband and a wife have to decide what to do for their evening entertainment. There s a catch, however. They cannot communicate and plan their evening together. They just have to individually choose whether to show up at the Boxing match or the Opera. Naturally, they would prefer to be at the event together, but they also have di erent tastes. The wife would prefer the boxing match to the opera, and the husband prefers the opera to the boxing match. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

35 Battle of the Sexes If both the husband and wife show up to the boxing match, the wife receives a payo of 3 and the husband receives a payo of 1. If both the husband and wife show up to the opera, the wife receives a payo of 1 and the husband receives a payo of 3. If the husband and wife don t show up at the same event, they both receive a payo of 0. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

36 Battle of the Sexes Fight Wife Opera Husband Fight Opera Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

37 Battle of the Sexes Let s look for strict dominance rst. If the wife chooses "Fight," The husband prefers "Fight" with a payo of 1 to "Opera" with a payo of 0. If the wife chooses "Opera," The husband prefers "Opera" with a payo of 3 to "Fight" with a payo of 0. Thus, no strategy is strictly dominated for the husband. If the husband chooses "Fight," The wife prefers "Fight" with a payo of 3 to "Opera" with a payo of 0. If the husband chooses "Opera," The wife prefers "Opera" with a payo of 1 to "Fight" with a payo of 0. Thus, no strategy is strictly dominated for the wife. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

38 Battle of the Sexes Well, that technique is out the window. Let s try best responses. If the wife chooses "Fight," The husband s best response is "Fight" with a payo of 1 over "Opera" with a payo of 0. BR H (Fight) = Fight If the wife chooses "Opera," The husband s best resonse is "Opera" with a payo of 3 over "Fight" with a payo of 0. BR H (Opera) = Opera Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

39 Battle of the Sexes Likewise, for the wife, If the husband chooses "Fight," The wife s best response is "Fight" with a payo of 3 over "Opera" with a payo of 0. BR W (Fight) = Fight If the husband chooses "Opera," The wife s best resonse is "Opera" with a payo of 1 over "Fight" with a payo of 0. BR W (Opera) = Opera We can underline these on our normal form game. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

40 Battle of the Sexes Fight Wife Opera Husband Fight Opera Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

41 Battle of the Sexes There are two Nash Equilibria! This should make sense. The best response for both players is to coordinate and show up where the other one is going. The challenge is that neither one knows which equilibria to pick. Can we resolve this? Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

42 Battle of the Sexes There is actually a third Nash Equilibria. Since there are two possible Nash Equilibria involving di erent strategies for each player, we can determine a third Nash Equilibrium by picking strategies randomly with some probability. This is known as a mixed strategy Nash Equilibrium. I am not going to go over them in this class, but I just want you to know that they exist. For those that are interested, the third Nash Equilibrium has the husband picking "Fight" with probability 0.25 and the wife picking "Fight" with probability If anyone would like to learn how this is derived, feel free to drop by my o ce hours. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

43 Simultaneous Move Games Solving simultaneous moves games almost always follows the same steps: 1. Check for strictly dominated strategies and eliminate them. 2. Determine the best response function. 3. Use the best response function to nd Nash Equilibria. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

44 Example L Player 2 C R U Player 1 M D Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

45 Example Starting o, we want to look for strictly dominated strategies. At rst, nothing is strictly dominated for player 2. But for player 1, playing "D" is strictly dominated by "M," and we can thus delete "D". Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

46 Example L Player 2 C R U Player 1 M D Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

47 Example L Player 2 C R U Player 1 M D Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

48 Example L Player 2 C R Player 1 U M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

49 Example Let s keep looking for strictly dominated strategies. Now, nothing is strictly dominated for player 1. Switching back to player 2, now that we have eliminated "D," "R" is now strictly dominated by "C," and we can thus delete "R" Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

50 Example L Player 2 C R Player 1 U M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

51 Example L Player 2 C R Player 1 U M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

52 Example L Player 2 C Player 1 U M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

53 Example Are there any more strictly dominated strategies? Nope! At this point, neither strategy for either player can be dominated. Now, we underline our best responses. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

54 Example L Player 2 C Player 1 U M Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

55 Example Thus, we have two Nash Equilibria. The rst is where player 1 picks "U" and player 2 picks "L." The second is where player 1 picks "M" and player 2 picks "C." Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

56 Summary Simultaneous move games show how players make strategic decisions when they move at the same time. Strategies that are strictly dominated will never be played by a player and should be deleted. A Nash Equilibrium occurs when all players are playing optimally at the same time. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

57 Preview for Friday Sequential Move Games What happens when the order of turns matters? If time, we ll see what happens when players interract multiple times. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

58 Assignment 6-4 & 6-5 In the two problems on the next slides, identify the following. Which (If any) strategies are strictly dominated (and by what are they dominated). The best response functions. All Nash Equilibria. Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

59 Assignment 6-4 & 6-5 (1 of 2) H Player 2 M L H Player 1 M L Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60

60 Assignment 6-4 & 6-5 (2 of 2) H Player 2 M L H Player 1 M L Eric Dunaway (WSU) EconS Lecture 28 November 8, / 60