2. The Extensive Form of a Game

Transcription

1 2. The Extensive Form of a Game In the extensive form, games are sequential, interactive processes which moves from one position to another in response to the wills of the players or the whims of chance. It will justify and give meaning to more abstract concepts such as strategy. Three new concepts make their appearance in the extensive form of a game: the game tree, chance moves, and information sets.

2 Basically, we will represent a game by a directed graph, a graph such that the edges are assigned directions. (Mathematically, a graph is consisted of vertices and edges.) Example: Cuban Missile Crisis

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4 The Game Tree. The extensive form of a game is modeled using a directed Graph. Roughly, a graph consists of vertices and edges joining the vertices. In a directed graph, each edge is assigned a direction. In our case, the vertices represent the positions of a game, and the edges represent the moves from one vertex along the assigned. direction to the other vertex. We do not want CYCLES.

5 A directed graph is a pair (T,F) where T is a nonempty set of vertices and F is a function that gives for each x T a subset, F(x) of T called the followers of x. The vertices represent positions of the game. The followers, F(x), of a position, x, are those positions that can be reached from x in one move. A path from a vertex t 0 to a vertex t 1 is a sequence, x 0, x 1,..., x n, of vertices such that x 0 = t 0, x 0 = t 1 and x j is a follower of x j-1 for all j = 1,..., n. The path from a vertex to one of its followers is called an edge and we may think of edges to represent moves from a given position.

6 Definition. A tree is a directed graph, (T,F) in which there is a special vertex, t 0, called the root or the initial vertex, such that for every other vertex t T, there is a unique path beginning at t 0 and ending at t. A path represents a course of play from the START to the present position. The existence and uniqueness of the path implies that a tree is connected, has a unique initial vertex, and has no circuits or loops.

7 The Kuhn tree of a game: We will use a tree with appropriate labeling of the vertices and the edges to describe a game. It is called a game tree or a Kuhn tree. When a tree is used to define a game, the vertices are interpreted as positions and the edges as moves. Each non-terminal position is assigned either to the player responsible to choosing the next move or to chance.

8 How to model poker?

9 Chance Moves. Many games involve chance moves. Examples include the rolling of dice in board games like monopoly or backgammon or gambling games such as craps, the dealing of cards as in bridge or poker, the spinning of the wheel of fortune, or the drawing of balls out of a cage in lotto. It is assumed that the players are aware of the probabilities of the various outcomes resulting from a chance move. In the Kuhn tree, the probabilities assigned to the edges leading out of each chance vertex must be displayed.

10 Example: A biased coin with probability 0.8 showing Head is tossed by a referee. The result is then shown to Player I.

11 The root of the game starts the game and the terminal vertices represent various possible ending of a game. Play starts at the initial vertex and continues along one of the paths eventually ending in one of the terminal vertices. Each directed path from the root to terminal vertex describes a possible course of play. Each terminal vertex should carry some indication of the outcome of the game; it typically takes the form of a payoff vector, indicating how much each player wins or loses.

12 Example: Draw the game tree for Paper-Scissor- Rock.

13 Information. Another important aspect we must consider in studying the extensive form of games is the amount of information available to the players about past moves of the game. When a player choosing a move is uninformed about some of the previous moves (concealed moves, simultaneous moves etc.), we draw little balloons around set of vertices which the player making the choice cannot discriminate. These are called information sets, and the player in question must make the same choice for all vertices in the set. Edges issuing from vertices in the information set must be labeled the same.

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15 In summary, the following are the basic elements of a Kuhn tree. 1. A finite set N, representing the players. 2. A finite rooted tree representing positions and moves. The root represents the start of the game and the terminal vertices the possible conclusions. 3. Terminal vertices represent the possible conclusions of the game. The payoff to each player is labeled at this vertex as a vector.

16 4.Vertices are labeled to name of the player and Chance. A move by them at these positions means choosing a particular edge leading out from this vertex. 5. Edges coming out from Chance should be labeled with the probability of this particular occurrence. 6. Information sets: The vertices of each player are partitioned into information sets. The player cannot distinguished the vertices within this information set. Hence, there are same number of edges with the same label leading out from each vertex in the information set.

17 Games of Perfect Information. A game of perfect information is a game in extensive form in which each information set of every player contains a single vertex. Games in which players remember all past information they once knew and all past moves they made are called games of perfect recall.

18 Games in which both players know the rules of the game, that is, in which both players know the Kuhn tree, are called games of complete information. Games in which one or both of the players do not know some of the payoffs, or some of the probabilities of chance moves, or some of the information sets, or even whole branches of the tree, are called games with incomplete information.

19 Imperfect recall: example from movie Memento Suggested by Larry Blume One of the most common assumptions made by game theorists is that players have perfect recall - they remember everything that happened in the game up to now. Larry Blume writes "Rubenstein and Piccione convinced us that imperfect recall might be pretty weird. The movie Memento shows how strategies in such a game might have to be implemented. It's a stretch, but my first year graduate students thought you should know." In the movie, the main character suffers from short-term memory loss, and adopts several unique strategies to remind him of what has so far transpired. Since, without perfect recall, segments of a game are somewhat independent, the movie tells his story backwards.

20 1. Pick up bricks: A pile of 5 bricks has been stacked on the ground. Two players take turn to pick up either one or two bricks from the pile. The player who picks up the last brick loses $1 to the other player.

21 2. The Ultimatum Game: There are 5 gold coins for Colin and Rose to divide. Colin will offer Rose from 1 to 5 coins. If Rose rejects the offer, each will get nothing.

22 3. Two cards marked H (for high) and L (for low) are placed in a hat. Rose draws a card and inspects it. She may then fold in which case she pays Colin $a, or she may bet in which case Colin may either fold, paying $a to Rose, or call. If Colin calls he receives from or pays to Rose $b according to Rose s card is marked H or L respectively.

23 4. From a deck of three cards numbered 1, 2, 3, Player I picks a card at will. Player II tries to guess the card. After each guess, Player I signals either High or Low or Correct depending on the guess of the opponent. The game is over as soon as the card is correctly guessed by Player II. Player II pays Player I an amount equal to the number of trials he made. 5. Each player places $a in the pot. In each of two hats two cards marked H and L are placed (so that there is a total of 4 cards). Each player draws a card at random from his designated hat. Player I has two alternatives now: she can see (i.e. challenges Player II) or she can raise by adding $b to the pot. If she sees, the higher hand wins and equal hands split the pot. If she raises, Player II has two options: he can fold or he can add $b to the pot and call. If he folds, Player I wins the pot. If he calls, then again the higher hand wins and equal hand splits the pot.

24 6. Truth Game: A biased coin with probability 0.8 showing H (Head) is tossed by a referee. The result is shown to Player I. Then Player I will announce to Player II about the result being H or T (Tail). Player II, having heard the announcement from Player I but not seen the result, must guess what the result was (either H or T). This ends the game. Payoffs are made as follows: Player II gets $1 if his guess matches and he gets $0 otherwise. Player I gets $2 if Player II guesses H and $0 if Player II guesses T. In addition to this, Player I gets $1 more if what he says to Player II matches the result of the coin toss and $0 if his message to Player II is different than the coin toss. 7. Votes by veto: A winner from a set of four candidates {a, b, c, d} is being selected by a committee of {I, II, III}. The committee will select a winner by taking turn, starting with I, he veto one of the candidates. Then the others take turn to veto the remaining candidates.

25 Example 8: Basic Endgame in Poker. Both players put 1 dollar, called the ante, in the center of the table. The money in the center of the table, so far two dollars, is called the pot. Then Player I is dealt a card from a deck. It is a winning card with probability ¼ and a losing card with probability 3/4. Player I sees this card but keeps it hidden from Player II. (Player II does not get a card.) Player I then checks or bets: If he checks, then his card is inspected; if he has a winning card he wins the pot and hence wins the 1 dollar ante from II, and otherwise he loses the 1 dollar ante to II. If I bets, he puts 2 dollars more into the pot. Then Player II not knowing what card Player I has must fold or call. If she folds, she loses the 1 dollar ante to I no matter what card I has. If II calls, she adds 2 dollars to the pot. Then Player I s card is exposed and I wins 3 dollars (the ante plus the bet) from II if he has a winning card, and loses 3 dollars to II otherwise.

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27 Can you draw the game tree for Chess?

28 Assignment 3: 6. Votes by veto: A winner from a set of four candidates {a, b, c, d} is being selected by a committee of {I, II, III}. The committee will select a winner by taking turn, starting with I, he veto one of the candidates. Then the others take turn to veto the remaining candidates. 7. Draw the game tree for the following Hidden Pearl Game. The Hidden Pearl: There are two dark boxes. Player I hides a pearl in one of them. Then Player II, not knowing which box contains the pearl, peeks into one of them. If the pearl is in Box 1 and she looks there, she sees it with probability 1/2. If it is in box 2 and she looks there, she sees it with probability 1/3. But if she looks into the wrong box she sees nothing (and is not even told that the box is empty). The payoff is $5 to II and -$5 to I if II sees the hidden pearl; otherwise there is no payment.

29 How will the game play out?

30 Reduction of a Game in Extensive Form to Strategic Form. Pure strategy. A pure strategy is a player s complete plan for playing the game. It should cover every contingency. A pure strategy for a Player is a rule that tells him exactly what move to make in each of his information sets. It should specify a particular edge leading out from each information set.

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32 Remark: Even for a simple game, there may be a large number of pure strategies. Example: (Two move chess) There are 20 possible moves for Player I and 20 possible moves for Player II. How many pure strategies are there for Player II?

33 Reduced pure strategy: Specification of choices at all information sets except those that are eliminated by the previous moves. Remark: It is not too easy to find the set of reduced pure strategies. Also we need to use full pure strategies for another important concept.

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35 Now that we have set of pure strategies for each player, we need to find the payoffs to put the game in strategic form. Random payoffs. The actual outcome of the game for given pure strategies of the players depends on the chance moves selected, and is therefore a random quantity. We represent random payoffs by their average values.

36 Example: Suppose Player I uses BCF and Player II uses ac. Then, the payoff will be (-5, 1) with probability 0.7 and (4,5) with probability 0.3. Then, the expected payoff is 0.7(-5,1) + 0.3(4,5) = (-2.3, 2.2)

37 Remark: In representing the random payoffs by their averages, we are making a rather subtle assumption. We are saying that receiving $5 outright is equivalent to receiving $10 with probability 0.5. The proper setting for this concept is Utility Theory developed by von Neumann and Morgenstern.

38 =related

39 Game in strategic form (2 players): Set of players: {1, 2} A set of pure strategies, X i, for player i=1, 2. Payoff function for the i th player. u i : X 1 x X 2 R, i=1, 2.

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43 How will the game play out? One natural concept is that each player will use the BEST strategy to counter that used by his/her opponent. This strategy is called the Best Response (BR). Example: If the Row Chooser uses R 1 D 2, the BR for the Column Chooser is either d 1 r 2 or d 1 d 2. If the Column Chooser uses r 1 d 2, the BR for the Row Chooser is R 1 D 2.

44 Definition. A vector of pure strategy choices (x 1, x 2 ) with x 1 X 1, x 2 X 2 is said to be a pure strategic equilibrium, or PSE for short, if u 1 (x 1, x 2 ) u 1 (x, x 2 ), for any x X 1, u 2 (x 1, x 2 ) u 2 (x 1, x), for any x X 2, The first inequality says that x 1 is the BR to x 2, and the second inequality says that x 2 is a BR to x 1. Therefore, (x 1, x 2 ) is BR to each other. It is then reasonable to expect that (x 1, x 2 ) is a possible outcome of the game. We are then interested to find PSE s.

45 Remark: It is reasonable to assume that a player should use a Best Response strategy and if a player is using a Best Response strategy, then he/she will not change to other strategy. Therefore, a reasonable concept for the outcome of a game should for each player to adopt a strategy that is BR to the strategies that other players are using. This is the basic concept of Nash Equilibrium of John Nash.

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47 Finding All PSE s. (2 Person) Put an asterisk after each of Player I s payoffs that is a maximum of its column. Put an asterisk after each of Player II s payoffs that is a maximum of its row. Then any entry of the matrix at which both I s and II s payoffs have asterisks is a PSE, and conversely.

48 Example: Prisoner s Dilemma Confess Confess Silent * * * ( 9, 9 ) (0, 10) Silent * ( 10,0 ) ( 1, 1)

49 Remark: Many games have no PSE s. Example: * * ( 2,2 ) (3, 3) * * (3, 3) ( 4,4 )

50 Games of perfect information always have at least one PSE that may be found by the method of backward induction. Method of Backward Induction: Starting from any terminal vertex and trace back to the vertex leading to it. The player at this vertex will discard those edges with lower payoff. Then, treat this vertex as a terminal vertex and repeat the process. Then, we get a path from the root to a terminal vertex Theorem: The path obtained by the method of backward induction defines a PSE.

51 SØREN KIERKEGAARD ( ) Life can only be understood backwards; but it must be lived forwards.

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53 Zermelo s Theorem Theorem (Zermelo (1912)): In chess either white can force a win, or balck can force a win, or both can force at least a draw.

54 In fact, the PSE obtained by the method of backward induction satisfies stronger properties so that it is called a perfect pure strategy equilibrium. Definition: A subgame of a game presented in extensive form is obtained by taking a vertex in the Kuhn tree and all the edges and paths originated from this vertex.

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56 Definition: A PSE of a game in extensive form is called a Perfect Pure Strategy Equilibrium (PPSE) if it is a PSE for all subgames. Theorem: The path obtained by the method of backward induction defines a PPSE.

57 Remark: Subgame perfect implies that when making choices, a player look forward and assumes that the choice that will subsequently be made by himself and by others will be rational. Threats which would be irrational to carry through are ruled out. It is precisely this kind of forwardlooking rationality that is most suited to economic applications. Example: Incredible Threats and Incredible Promises

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60 Incredible Threats PSE allow players to make noncredible threats provided they never have to carry them out. Decisions on the equilibrium path are driven in part by what the players expect will happen off the equilibrium path.

61 The following examples show that PPSE is not so Perfect.

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63 Assignment 4: 8. Reduce the following Kuhn tree to strategic form. Find all PSE s and PPSE.

64 9. Reduce the following Kuhn tree to strategic form and find all PSE s.