Lecture 11 Strategic Form Games

Transcription

1 Lecture 11 Strategic Form Games Jitesh H. Panchal ME 597: Decision Making for Engineering Systems Design Design Engineering Purdue (DELP) School of Mechanical Engineering Purdue University, West Lafayette, IN October 23, 2014 c Jitesh H. Panchal Lecture 11 1 / 25

2 Lecture Outline 1 Overview of Game Theory 2 3 Source: Dutta, P. K. (1999). Strategies and Games: Theory and Practice. Cambridge, M, The MIT Press. c Jitesh H. Panchal Lecture 11 2 / 25

3 What is Game Theory? Game Theory Game theory is the study of strategic situations situations in which an entire group of people is affected by the choices made by every individual within that group. It is about interdependence between decisions. Can you think of some examples? c Jitesh H. Panchal Lecture 11 3 / 25

4 Let us play a game Prisoner s Dilemma Consider two prisoners - and B B stays silent (cooperates) B betrays (defects) stays silent (cooperates) Each serves 1 year : 3 years, B: goes free betrays (defects) : goes free, B: 3 years Each serves 2 years c Jitesh H. Panchal Lecture 11 4 / 25

5 Interdependence between Decisions 1 What will each individual guess about the others choices? 2 What action will each person take? 3 What is the outcome of these actions? Is this outcome good for the group as a whole? 4 Does it make any difference if the group interacts more than once? 5 How do the answers change if each individual is unsure about the characteristics of others in the group? c Jitesh H. Panchal Lecture 11 5 / 25

6 Formal Definition of a Game 1 Players: In any game, there is more than one decision maker; each decision maker is referred to as a player. 2 Interaction: What any one individual player does directly affects at least one other player in the group. 3 Strategic: n individual player accounts for this interdependence in deciding what action to take. 4 Rational: While accounting for this interdependence, each player chooses her best action. c Jitesh H. Panchal Lecture 11 6 / 25

7 Rules of the Game To define a game, we need to specify four things: 1 Who is playing the group of players that strategically interacts 2 What they are playing with the alternative actions or choices, the strategies, that each player has available 3 When each player gets to play (in what order) 4 How much they stand to gain (or lose) from the choices made in the game. c Jitesh H. Panchal Lecture 11 7 / 25

8 Common Knowledge bout the Rules If you asked any player about who, what, when, and how much, they would give the same answer. This does not mean that all players have the same information when they make choices, or are equally influential, or that all have the same choices. It simply means that everyone knows the rules. Common knowledge goes a step further: everyone knows the rules... everyone knows that everyone knows the rules... everyone knows that everyone knows that everyone knows the rules ad infinitum c Jitesh H. Panchal Lecture 11 8 / 25

9 Forms of a game 1 Extensive form - generally used for sequential games 2 Normal (strategic) form - generally used for simultaneous games c Jitesh H. Panchal Lecture 11 9 / 25

10 1. The Extensive Form of a Game The extensive form is a pictorial representation of the rules of a game. lso called a game tree. Nodes are decision nodes. Choices are branches. Strategies: pair of strategies (one for each player determines the way in which the game is actually played.) c Jitesh H. Panchal Lecture / 25

11 Example of Extensive Form of a Game Theater game: b = bus, c = car, s = subway, T = Ticket, N = No ticket. Figure : 2.5 on page 24 (Dutta) c Jitesh H. Panchal Lecture / 25

12 Example of Extensive Form of a Game Game of Nim: Suppose there are two matches in one pile and a single match in the other pile (2,1). The player who removes the last match wins the game. Player 2 l Player 1 (1,0) (1,-1) u (1,1) r (0,1) (1,-1) Player 1 (2, 1) m Player 2 (0,1) c (-1, 1) d Player 2 L (1,0) (1,-1) (2,0) R (0,0) (-1,1) Figure : 2.6 on page 25 (Dutta) c Jitesh H. Panchal Lecture / 25

13 Example of Extensive Form of a Game Strategic Committee Voting: Voter thinks through what the other voters are likely to do rather than voting simply according to the preferences. Scenario: Suppose there are two competing bills (, B) and three voters 1, 2, 3. Possible outcomes are either bill passes or no bill passes (N). Process: First, bill is pitted against bill B. The winner is then pitted against the status quo (N). The legislators have the following preferences: 1 N B 2 B N 3 N B c Jitesh H. Panchal Lecture / 25

14 Example of Extensive Form of a Game Voter 1 Voter 2 B B B Voter 3 B B Voter 1 B B Voter 2 N Voter 3 N N N N N N 1,0,0 0, -1, 1 Figure : 2.8 on page 26 (Dutta) c Jitesh H. Panchal Lecture / 25

15 Information Sets and Strategies Representing simultaneous moves within the extensive form. Example: Player 2 is unable to distinguish between the two nodes (i.e., whether Player 1 chose c or n). Player 2 c c n Player 1 n c n Figure : 2.4 on page 21 (Dutta) c Jitesh H. Panchal Lecture / 25

16 2. The Normal Form of a Game Example: Prisoner s Dilemma 1 / 2 Confess Not Confess Confess 0, 0 7, 2 Not Confess 2, 7 5, 5 c Jitesh H. Panchal Lecture / 25

17 Notation Overview of Game Theory Consider a set of players in a game labeled 1, 2,..., N i th player: representative player s i : player i s strategies si : player i s specific strategy s i : a strategy choice of all players other than player i s1, s2,..., sn: a strategy vector (one strategy for each player) π i (s 1, s 2,..., s N): Player i s payoff for strategy vector s 1, s 2,..., s N c Jitesh H. Panchal Lecture / 25

18 Example: Prisoner s dilemma Prisoner s Dilemma 1 / 2 Confess Not Confess Confess 0, 0 7, 2 Not Confess 2, 7 5, 5 c Jitesh H. Panchal Lecture / 25

19 Dominant Strategy Dominant Strategy Strategy s i strongly dominates all other strategies of player i if the payoff to s i is strictly greater than the payoff to any other strategy, regardless of which strategy is chosen by the other player(s). In other words, for all s i and all s i π i (s i, s i ) > π i (s i, s i ) c Jitesh H. Panchal Lecture / 25

20 Weakly Dominant Strategy Weakly Dominant Strategy Strategy s i (weakly) dominates another strategy, say s # i, if it does atleast as well as s # i against every strategy of the other players, and against some it does strictly better, i.e., π i (s i, s i ) π i (s # i, s i ), for all s i π i (s i, ŝ i ) > π i (s # i, ŝ i ), for some ŝ i In this case, s # i is a dominated strategy. If s i weakly dominates every other candidate strategy s i, then s i is said to be a weakly dominant strategy. c Jitesh H. Panchal Lecture / 25

21 Iterated Elimination of Dominated Strategies Player 1 / Player 2 Left Right Up 1, 1 0, 1 Middle 0, 2 1, 0 Down 0, -1 0, 0 c Jitesh H. Panchal Lecture / 25

22 nother example Bertrand Competition Firm 1 / Firm 2 High Medium Low High 6, 6 0, 10 0, 8 Middle 10, 0 5, 5 0, 8 Low 8, 0 8, 0 4, 4 c Jitesh H. Panchal Lecture / 25

23 Iterated Elimination of Dominated Strategies: dvantages and Disadvantages dvantage: simplicity Disadvantages: 1 Layers of rationality 2 Order of elimination matters 3 Non-unique outcomes 4 Nonexistence 1 / 2 Left Right Top 0, 0 0, 1 Bottom 1, 0 0, 0 1 / 2 Left Middle Bad Top 1, -1-1, 1 0, -2 Middle -1, 1 1, -1 0, -2 Bad -2, 0-2, 0-2, -2 c Jitesh H. Panchal Lecture / 25

24 Summary Overview of Game Theory 1 Overview of Game Theory 2 3 c Jitesh H. Panchal Lecture / 25

25 References 1 Dutta, P. K. (1999). Strategies and Games: Theory and Practice. Cambridge, M, The MIT Press. c Jitesh H. Panchal Lecture / 25

26 THNK YOU! c Jitesh H. Panchal Lecture 11 1 / 1