DECISION MAKING GAME THEORY

Transcription

1 DECISION MAKING GAME THEORY

2 THE PROBLEM Two suspected felons are caught by the police and interrogated in separate rooms. Three cases were presented to them.

3 THE PROBLEM CASE A: If only one of you confesses, one will go to jail for 1 year and the other will go to jail for 25 years. CASE B: If neither of you confess, you each get 3 years in jail. CASE C: If you both confess, you will each go to jail for 10 years.

4 THE PROBLEM If you are one of the suspects, what would be your best strategy?

5 WHAT IS GAME THEORY? GAME THEORY is the study of strategic interaction between participants (players) in a situation (games) that contains a set of rules and a set of outcomes.

6 WHAT IS GAME THEORY? It consists of a set of players and a set of actions for each of them. Each player receives a payoff that depends on the actions of all the players involved. The actions of the players are called moves. A sequence of moves is called a strategy.

7 BASIC IDEA IN GAME THEORY In analyzing or solving a game, what would be the player s best strategy to different moves of the other players to achieve the best or maximum payoff?

8 APPLICATIONS OF GAME THEORY commuters deciding how to go to work businesses competing in a market diplomats negotiating a treaty gamblers betting in a card game auctions voting evolutionary biology

9 APPLICATIONS OF GAME THEORY

10 CLASSIFICATION OF GAMES Simultaneous game Sequential game Perfect Information Game Complete Information Game Zero-Sum Game Cooperative and Non-cooperative games

11 Simultaneous Game Each player has only one move and all these moves are done simultaneously.

12 Sequential Game No two players move at the same time and players may have several moves.

13 A game of perfect information A sequential game has perfect information if each player knows all previous moves done by the other players and all random moves made so far.

14 A game of complete information In this classification, all players know the structure of the game the order in which player moves, all possible moves in each situation and each player s payoff for all possible outcomes.

15 Example: Complete Information MATCHING COINS GAME: One player is tagged as even and the other as odd. The two players each show a one-peso coin, showing up either head or tail. If both show the same side, then even keeps both coins. Otherwise, odd keeps both coins.

16 Perfect versus Complete In a game of complete information, players may not know the previous moves than by other players. In a game of perfect information, players do not know the payoffs of other players or some of the game structures.

17 Zero-Sum Game In every possible outcome of the game, the sum of the payoffs of each player equals zero.

18 Cooperative Game In cooperative games, players can cooperate (by forming coalitions) their strategies and share payoffs.

19 EXTENSIVE FORM A game in extensive form is a tree diagram in which each strategic decision is shown as a branch point.

20 NORMAL FORM A game in normal form is a table of numbers with the strategies listed along the margins of the table and the payoffs for the participants in the cells of the table.

21 Knowledge Check Is the Prisonner s Dilemma a simultaneous game or sequential game? Is it a game of perfect information or complete information? Is it a zero-sum game? Is it a cooperative or noncooperative?

22 Analyzing Simultaneous Games in Normal Form WORKING ON A PROJECT GAME: You and your classmate work together on a course requirement and each of you can choose to either work hard or take it easy. You both want to pass but both of you do not like working very hard.

23 Analyzing Simultaneous Games in Normal Form The normal form of the presented game is shown where the first number is your payoff and the second is your classmate s payoff.

24 Analyzing Simultaneous Games in Normal Form Assume that the two of you agreed to meet up tomorrow supposedly to combine your outputs. What would you do before that time? Work hard or take it easy? How would you explain your action rationally?

25 MAXIMIN SOLUTION This solution involves choosing the strategy that gives you the maximum payoff, even when your opponent is able to guess your choice. This was proposed by the mathematician John von Neumann.

26 PLAYER S MAXIMIN STRATEGY For a two-player game: Consider all your strategies. For each strategy, obtain your minimum payoff regardless of the action of the other player. Obtain the maximum among all the minimum payoffs from the previous step. The strategy corresponding to this maximum payoff is your maximin strategy.

27 Example: Maximin Strategy In the Working on a Project Game, what is your maximin strategy? What is your classmate s maximin strategy?

28 Example: Maximin Strategy Consider the following payoff table for a twoplayer game. Find the maximin strategy for both players.

29 DOMINANT STRATEGY Let M and U be strategies in a game. If all the payoffs of player A using strategy M are greater than or equal to all his payoffs using strategy U, regardless of the other player s action, then we say that M (strongly or weakly) dominates U (or U is dominated by M) for player A.

30 DOMINANT STRATEGY If one strategy dominates all the other strategies for a particular player, it is said to be a dominant strategy. If every player in the game has a dominant strategy, then we have a dominant strategy equilibrium.

31 Example: Dominant Strategy Consider again the following payoff table for a two-player game. For each player, which moves are dominated?

32 Example: Dominant Strategy Are there any dominant strategies in the Working on a Project Game?

33 NASH EQUILIBRIUM Suppose that a game consists of n players. Let M i be the action of player i in the game, where 1 i n. The strategies of the players are listed as (M 1, M 2,, M n ). If all the strategies in the given list are the best response of each player, then the list is said to be a Nash equilibrium.

34 NASH EQUILIBRIUM BASIC IDEA: A Nash equilibrium is a strategy for each player such that every players action is the best response to the other players actions. Each player is using his best response in the game. Then switching into another strategy would result into a lower payoff. No player can reach a better payoff by changing strategies.

35 NASH EQUILIBRIUM This solution concept was named after John Forbes Nash, Jr., an American mathematician. His theories are used in economics, evolutionary biology, military theory, artificial intelligence, computer science and many more.

36 Example: Nash Equilibrium Let us consider again the normal form of the Working on a Project Game. What would be your best response to your classmate s actions?

37 Maximin Solution vs Nash Equilibrium In the maximin solution, we think of the worst-case scenarios and then getting the best from all of these scenarios. In the Nash equilibrium, we try to be more optimistic. Assuming we know the actions of the other players, we get our best reply strategy.

38 Example BATTLE OF THE SEXES: Suppose, on a given night, you can either watch a sports program, or a telenovela. You prefer the sports program, but your partner prefers the telenovela. The worst possible thing to happen would be to have an argument and not watch together, or at all. What should you do?

39 Example The normal of this game follows: Determine your maximin solution. Is there a Nash equilibrium?

40 Knowledge Check Consider again the normal form of the Prisonner s Dilemma:

41 Knowledge Check Find the maximin solution of the game for each player. For each player, identify any dominant strategy. Is there a dominant strategy equilibrium? Identify any Nash equilibrium of the game.

42 Normal Form: IESDS A method of analyzing games in normal form is the Iterative Elimination of (Strongly) Dominated Strategies. This can be used to find any Nash equilibria.

43 Normal Form: IESDS Given the normal form of a game. Step 1: Identify all the dominated strategies for each player. Step 2: Delete the rows and columns of the table corresponding to these dominated strategies. Step 3: Repeat steps 1 and 2 until all the dominated strategies are eliminated. Step 4: The resulting table contains the Nash equilibria.

44 Example Consider the following payoff table for a twoplayer game: Use the IESDS to determine the Nash equilibrium.

45 Example Consider the following payoff table for a twoplayer game: Use the IESDS to determine the Nash equilibrium.

46 Remark In a simultaneous game, every dominant strategy equilibrium is an IESDS equilibrium. Every IESDS equilibrium is a Nash equilibrium.

47 Analyzing Sequential Games in Extensive Form Recall that games in extensive form are represented by tree diagrams. In analyzing games in extensive form, we breakdown the tree diagram to form subgames. A subgame includes a decision point and all the other parts emanating from it.

48 Example: Subgame Consider the game whose extensive form is as follows: List all the subgames of this game.

49 SUBGAME PERFECT EQUILIBRIUM In subgame perfect equilibrium, every player makes their decisions on the assumption that every subgame would be in equilibrium. It is a refinement of the Nash equilibrium. A game is in subgame perfect equilibrium if and only every subgame is a Nash equilibrium.

50 BACKWARD INDUCTION Step 1: Solve the subgames at the last stage in each branch to reduce the game into a smaller one. Step 2: We repeat until all subgames have been solved. Step 3: The resulting tree diagram gives a subgame perfect equilibrium.

51 Example: Backward Induction Find the backward induction solution of the following game:

52 BLOG YOUR THOUGHTS Some game theory situations are difficult to solve but the different solution concepts help me I find the Nash equilibrium solution concept more Game theory is significant because. I am interested to study Game theory because. I dislike Game theory because.