The introduction of game theory（lecture notes 3）

1、Simultaneous Move Games Simultaneous Move Games Arise when players have to make their strategy choices simultaneously, without knowing the strategies that have been chosen by the other player(s). Student studies for a test; the teacher writes questions. Two firms independently decide whether or not

2、to develop and market a new product. While there is no information about what other players will actually choose, we assume that the strategic choices available to each player are known by all players. Players must think not only about their own best strategic choice but also the best strategic choi

3、ce of the other player(s).Normal or Strategic Form A simultaneous move game is depicted in “Normal or “Strategic form using a game table that relates the strategic choices of the players to their payoffs. The convention is that the row players payoff is listed first and the column players payoff is

4、listed second. For example, if Row player chooses R2 and Column player chooses C1, the Row players payoff is e and the Column players payoff is f. a,b c,de,f g,hColumn PlayerRow PlayerStrategy C1Strategy C2Strategy R1Strategy R2Strategy Types: Pure versus Mixed A player pursues a pure strategy if sh

5、e always chooses the same strategic action out of all the strategic action choices available to her in every round e.g. Always refuse to clean the apartment you share with your roommate. A player pursues a mixed strategy if she randomizes in some manner among the strategic action choices available t

6、o her in every round. e.g. Sometimes choose Head, sometimes choose Tail (“mix it up, “keep them guessing). We focus for now on pure strategies only.Example: Battle of the Networks Suppose there are just two television networks. Both are battling for shares of viewers (0-100%). Higher shares are pref

7、erred (= higher advertising revenues). Network 1 has an advantage in sitcoms (situation comedy). If it runs a sitcom, it always gets a higher share than if it runs a game show. Network 2 has an advantage in game shows. If it runs a game show it always gets a higher share than if it runs a sitcom.55%

8、,45% 52%,48%50%,50% 45%,55%Network 2Network 1SitcomGame Show SitcomGame ShowNash Equilibrium We cannot use rollback in a simultaneous move game, so how do we find a solution? We determine the “best response of each player to a particular choice of strategy by the other player. We do this for both pl

9、ayers. If each players strategy choice is a best response to the strategy choice of the other player, then we have found a solution or equilibrium to the game. This solution concept is know as a Nash equilibrium, after John Nash who first proposed it. A game may have 0, 1 or more pure strategies Nas

10、h equilibria.Cell-by-Cell Inspection Method The cell-by-cell inspection method is the most reliable for finding Nash equilibria. First Find Network 1s best response. If Network 2 runs a sitcom, Network 1s best response is to run a sitcom. If Network 2 runs a game show, Network 1s best response is to

11、 run a sitcom. 55%,45% 52%,48%50%,50% 45%,55%Network 2Network 1SitcomGame Show SitcomGame ShowCell-by-Cell Inspection Method, Continued Next, we find Network 2s best response. If Network 1 runs a sitcom, Network 2s best response is to run a game show. If Network 1 runs a game show, Network 2s best r

12、esponse is to run a game show. The unique Nash equilibrium is for Network 1 to run a sitcom and Network 2 to run a game show. 55%,45% 50%,50% 45%,55%Network 2Network 1SitcomGame Show SitcomGame Show52%,48%Dominant Strategies A player has a dominant strategy if it outperforms (has higher payoff than)

13、 all other strategies regardless of the strategies chosen by the opposing player(s). For example, in the battle of the networks game, Network 1 has a dominant strategy of always choosing to run a sitcom. Network 2 has a dominant strategy of always choosing to run a game show. Why? Elimination of non

14、-dominant or “dominated strategies can help us to find a Nash equilibrium. Successive Elimination of Dominated Strategies Another way to find Nash equilibrium. Draw lines through (successively eliminate) each players dominated strategies. If successive elimination of dominated strategies results in

15、a unique outcome, that outcome is the Nash equilibrium of the game. We call such games dominance solveable55%,45% 50%,50% 45%,55%Network 2Network 1SitcomGame Show SitcomGame Show52%,48%Some Classical Games We have already discussed the prisoners dilemma game. There are many other classical games ana

16、lyzed by game theory experts. Zero-Sum Games (Matching Pennies) Coordination game Battle of the Sexes Chicken or Hawk versus Dove Stag Hunt Boxed PigsZero-Sum Games The first games analyzed by game theorists were just the opposite -zero sum games, where the sum of agents utilities in each outcome su

17、ms up to zero (or a constant).Zero-sum games are true games of conflict. Any gain on my side comes at the expense of my opponents. Think of dividing up a pie. The size of the pie doesnt change - its all about redistribution of the pieces between the players (tax policy is a good example). The simple

18、st zero sum game is matching pennies. This is a two player game where player 1 get a Dollar from player 2 if both choose the same action, and otherwise loses a Dollar: This game has no pure strategy Nash equilibrium and one mixed Nash equilibrium. 1,-1-1,1-1,11,-1 H THTCoordination Game Consider two

19、 students who want to meet in the campus, each student i can choose actions from a strategy/action set Si =Library; Cafeteria. The normal-form representation game is as follows. This game has two pure strategy Nash equilibrium and one mixed Nash equilibrium. 1,10,00,01,1 L CLCBattle of the Sexes Thi

20、s game is interesting because it is a coordination game with some elements of conflict. The idea is that a couple want to spend the evening together. The wife wants to go to the Opera, while the husband wants to go to a football game. Each get at least some utility from going together to at least on

21、e of the venues, but each wants to go their favorite one (the husband is player 1- the column player). This game has two pure strategy Nash equilibrium and one mixed Nash equilibrium. 2,10,00,01,2 F OFOChicken or Hawk versus Dove This game is an anti-coordination game. The story is that two teenager

22、s drive home on a narrow road with their bikes, and in opposite directions. None of them wants to go out of the way - whoever chickens out loses his pride, while the tough guy wins. But if both stay tough, then they break their bones. If both go out of the way, none of them is too happy or unhappy.T

23、his game has two pure strategy Nash equilibrium and one mixed Nash equilibrium. -1,-110,00,105,5 T CTCStag Hunt A sentence in Discourse on the origin and foundation of inequality among men (1755) by the philosopher Jean-Jacques Rousseau discusses a group of hunters who wish to catch a stag. They will succeed if they all remain sufficiently attentive, but if anyone is tempted to desert his post and catch a hare, the stag escapes, and the hare belongs to the defecting hunter alone. Each hunter prefers a share of the stag to a hare. This game has two pure strategy Nash equi