DOCSLIB.ORG

Extensive Form Games and Backward Induction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Extensive Form Games and Backward Induction Recap Perfect-Information Extensive-Form Games Subgame Perfection Backward Induction Introduction I The normal form game representation does not incorporate any notion of sequence, or time, of the actions of the players I The extensive form is an alternative representation that makes the temporal structure explicit. I Two variants: I perfect information extensive-form games I imperfect-information extensive-form games Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 3 Recap Perfect-Information Extensive-Form Games Subgame Perfection Backward Induction Perfect-Information Extensive Form Representation I Represents a finite sequential game as a rooted tree I Each internal node represents a particular player’s ’turn’ I Each branch from this internal node represents a different choice for that player I Each terminal node represents a possible final outcome of the game I Usually written with the basal root on top I Parts of the tree are labeled as follows: I Internal nodes (including the root) are labeled with player identifiers I Branches are labeled with player choices I Terminal nodes are labeled with utility outcomes Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 4 Recap Perfect-Information Extensive-Form Games Subgame Perfection Backward Induction Example: the sharing game q 1 ¨¨HH ¨ H ¨¨ HH ¨ H 0–21–12–0 ¨¨ HH ¨ H ¨ Hqqq 222 ¡A ¡A ¡A ¡ A ¡ A ¡ A ¡ A ¡ A ¡ A yesnoyesnoyesno ¡ A ¡ A ¡ A ¡ A ¡ A ¡ Aqqqqqq (0,2)(0,0)(1,1)(0,0)(2,0)(0,0) Get with a partner and decide on a simple sequential game (e.g. tic-tac-toe) and represent it in extended form Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 5 Recap Perfect-Information Extensive-Form Games Subgame Perfection Backward Induction Pure Strategies I In the sharing game (splitting 2 coins) how many pure strategies does each player have? I player 1: 3; player 2: 8 I Overall, a pure strategy for a player in a perfect-information game is a complete specification of which deterministic action to take at every node belonging to that player. I Can think of a strategy as a complete set of instructions for a proxy who will play for the player in their abscence Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 8 5.1 Perfect-information extensive-form games 109 ¨H1 ¨¨ HH ¨ q H 0–2 ¨ 1–1 H 2–0 ¨¨ HH ¨¨ HH 2 ¨ 2 H 2 ¡A ¡A ¡A ¡ q A ¡ q A ¡ q A no ¡ A yes no ¡ A yes no ¡ A yes ¡ A ¡ A ¡ A ¡ A ¡ A ¡ A (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) q q q q q q Figure 5.1 The Sharing game. Notice that the definition contains a subtlety. An agent’s strategy requires a decision Recapat each choicePerfect-Information node, regardless Extensive-Form of whether Games or not itSubgame is possib Perfectionle to reach thatBackward node given Induction the other choice nodes. In the Sharing game above the situation is straightforward— Pureplayer Strategies 1 has three pure Example strategies, and player 2 has eight (why?). But now consider the game shown in Figure 5.2. 1 A B 2 2 C D E F 1 (3,8) (8,3) (5,5) G H (2,10) (1,0) Figure 5.2 A perfect-information game in extensive form. What are the pure strategies for player 2? InI orderS2 to= define{(C, a E complete); (C, F strategy); (D, for E); this (D, game, F )} each of the players must choose anWhat action are at each the of pure his two strategies choice nodes. forplayer Thus we 1? can enumerate the pure strategies of the players as follows. I S1 = {(B, G); (B, H), (A, G), (A, H)} S1 = (A, G), (A, H), (B, G), (B, H) I This{ is true even though, conditional} on taking A, the choice S = (C, E), (C,F ), (D, E), (D,F ) 2 between{ G and H will never} have to be made ExtensiveIt is Form important Games and to Backward note that Induction we have to include the strategies (A, G) andISCI(A, 330 H Lecture), even 11, Slide 9 though once A is chosen the G-versus-H choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfect- information game can be converted to an equivalent normal form game. For example, the perfect-information game of Figure 5.2 can be converted into the normal form im- age of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are Multi Agent Systems, draft of September 19, 2006 Recap Perfect-Information Extensive-Form Games Subgame Perfection Backward Induction Nash Equilibria Given our new definition of pure strategy, we are able to reuse our old definitions of: I mixed strategies I best response I Nash equilibrium Theorem Every perfect information game in extensive form has a PSNE This is easy to see, since the players move sequentially. Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 10 5.1 Perfect-information extensive-form games 109 ¨H1 ¨¨ HH ¨ q H 0–2 ¨ 1–1 H 2–0 ¨¨ HH ¨¨ HH 2 ¨ 2 H 2 ¡A ¡A ¡A ¡ q A ¡ q A ¡ q A no ¡ A yes no ¡ A yes no ¡ A yes ¡ A ¡ A ¡ A ¡ A ¡ A ¡ A (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) q q q q q q Figure 5.1 The Sharing game. Notice that the definition contains a subtlety. An agent’s strategy requires a decision Recapat each choicePerfect-Information node, regardless Extensive-Form of whether Games or not itSubgame is possib Perfectionle to reach thatBackward node given Induction the other choice nodes. In the Sharing game above the situation is straightforward— Inducedplayer 1 has Normal three pure Form strategies, and player 2 has eight (why?). But now consider the game shown in Figure 5.2. 1 I In fact, the connectionA to the normalB form is even tighter I we can “convert”2 an extensive-form game2 into normal form C D E F 1 CE CF DE DF (3,8) (8,3) (5,5) AG G3, 8 3H, 8 8, 3 8, 3 AH 3, 8 3, 8 8, 3 8, 3 BG (2,10)5, 5 (1,0)2, 10 5, 5 2, 10 BH 5, 5 1, 0 5, 5 1, 0 Figure 5.2 A perfect-information game in extensive form. In order to define a complete strategy for this game, each of the players must choose an actionI this at each illustrates of his two the choice lack nodes. of compactness Thus we can enumerat of thee normal the pure form strategies of the playersI games as follows. aren’t always this small even here we write down 16 payoff pairs instead of 5 S = I(A, G), (A, H), (B, G), (B, H) 1 { } S = (C, E), (C,F ), (D, E), (D,F ) 2 { } ExtensiveIt is Form important Games and to Backward note that Induction we have to include the strategies (A, G) andISCI( 330A, LectureH), even 11, Slide 11 though once A is chosen the G-versus-H choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfect- information game can be converted to an equivalent normal form game. For example, the perfect-information game of Figure 5.2 can be converted into the normal form im- age of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are Multi Agent Systems, draft of September 19, 2006 5.1 Perfect-information extensive-form games 109 ¨H1 ¨¨ HH ¨ q H 0–2 ¨ 1–1 H 2–0 ¨¨ HH ¨¨ HH 2 ¨ 2 H 2 ¡A ¡A ¡A ¡ q A ¡ q A ¡ q A no ¡ A yes no ¡ A yes no ¡ A yes ¡ A ¡ A ¡ A ¡ A ¡ A ¡ A (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) q q q q q q Figure 5.1 The Sharing game. Notice that the definition contains a subtlety. An agent’s strategy requires a decision Recapat each choicePerfect-Information node, regardless Extensive-Form of whether Games or not itSubgame is possib Perfectionle to reach thatBackward node given Induction the other choice nodes. In the Sharing game above the situation is straightforward— Inducedplayer 1 has Normal three pure Form strategies, and player 2 has eight (why?). But now consider the game shown in Figure 5.2. 1 In fact, the connection to the normal form is even tighter I A B I we can “convert” an extensive-form game into normal form 2 2 C D E F CE 1 CF DE DF (3,8) (8,3) (5,5)AG 3, 8 3, 8 8, 3 8, 3 AH G3, 8 3H, 8 8, 3 8, 3 BG 5, 5 2, 10 5, 5 2, 10 BH (2,10)5, 5 (1,0)1, 0 5, 5 1, 0 Figure 5.2 A perfect-information game in extensive form. InI orderwhile to define we can a complete write anystrategy extensive-form for this game, each game of th ase players a NF, must we choose can’t an actiondo at the each reverse. of his two choice nodes. Thus we can enumerate the pure strategies of the players as follows. I e.g., matching pennies cannot be written as a S1 = (A,perfect-information G), (A, H), (B, G), (B, extensive H) form game { } S = (C, E), (C,F ), (D, E), (D,F ) 2 { } ExtensiveIt is Form important Games and to Backward note that Induction we have to include the strategies (A, G) andISCI( 330A, LectureH), even 11, Slide 11 though once A is chosen the G-versus-H choice is moot.
Load more

Recommended publications
  • Lecture 4 Rationalizability & Nash Equilibrium Road
    Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i).
    [Show full text]
  • 1 Sequential Games
    1 Sequential Games We call games where players take turns moving “sequential games”. Sequential games consist of the same elements as normal form games –there are players, rules, outcomes, and payo¤s. However, sequential games have the added element that history of play is now important as players can make decisions conditional on what other players have done. Thus, if two people are playing a game of Chess the second mover is able to observe the …rst mover’s initial move prior to making his initial move. While it is possible to represent sequential games using the strategic (or matrix) form representation of the game it is more instructive at …rst to represent sequential games using a game tree. In addition to the players, actions, outcomes, and payo¤s, the game tree will provide a history of play or a path of play. A very basic example of a sequential game is the Entrant-Incumbent game. The game is described as follows: Consider a game where there is an entrant and an incumbent. The entrant moves …rst and the incumbent observes the entrant’sdecision. The entrant can choose to either enter the market or remain out of the market. If the entrant remains out of the market then the game ends and the entrant receives a payo¤ of 0 while the incumbent receives a payo¤ of 2. If the entrant chooses to enter the market then the incumbent gets to make a choice. The incumbent chooses between …ghting entry or accommodating entry. If the incumbent …ghts the entrant receives a payo¤ of 3 while the incumbent receives a payo¤ of 1.
    [Show full text]
  • Notes on Sequential and Repeated Games
    Notes on sequential and repeated games 1 Sequential Move Games Thus far we have examined games in which players make moves simultaneously (or without observing what the other player has done). Using the normal (strategic) form representation of a game we can identify sets of strategies that are best responses to each other (Nash Equilibria). We now focus on sequential games of complete information. We can still use the normal form representation to identify NE but sequential games are richer than that because some players observe other players’decisions before they take action. The fact that some actions are observable may cause some NE of the normal form representation to be inconsistent with what one might think a player would do. Here’sa simple game between an Entrant and an Incumbent. The Entrant moves …rst and the Incumbent observes the Entrant’s action and then gets to make a choice. The Entrant has to decide whether or not he will enter a market or not. Thus, the Entrant’s two strategies are “Enter” or “Stay Out”. If the Entrant chooses “Stay Out” then the game ends. The payo¤s for the Entrant and Incumbent will be 0 and 2 respectively. If the Entrant chooses “Enter” then the Incumbent gets to choose whether or not he will “Fight”or “Accommodate”entry. If the Incumbent chooses “Fight”then the Entrant receives 3 and the Incumbent receives 1. If the Incumbent chooses “Accommodate”then the Entrant receives 2 and the Incumbent receives 1. This game in normal form is Incumbent Fight if Enter Accommodate if Enter .
    [Show full text]
  • Lecture Notes
    GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies ..................................
    [Show full text]
  • SEQUENTIAL GAMES with PERFECT INFORMATION Example
    SEQUENTIAL GAMES WITH PERFECT INFORMATION Example 4.9 (page 105) Consider the sequential game given in Figure 4.9. We want to apply backward induction to the tree. 0 Vertex B is owned by player two, P2. The payoffs for P2 are 1 and 3, with 3 > 1, so the player picks R . Thus, the payoffs at B become (0, 3). 00 Next, vertex C is also owned by P2 with payoffs 1 and 0. Since 1 > 0, P2 picks L , and the payoffs are (4, 1). Player one, P1, owns A; the choice of L gives a payoff of 0 and R gives a payoff of 4; 4 > 0, so P1 chooses R. The final payoffs are (4, 1). 0 00 We claim that this strategy profile, { R } for P1 and { R ,L } is a Nash equilibrium. Notice that the 0 00 strategy profile gives a choice at each vertex. For the strategy { R ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { R ,L }) = 4 π1(R, { R ,L }) = 4 ≥ 0 00 π1(L, { R ,L }) = 0. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { R ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { R ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0, where ∗ means choose either L0 or R0. Since no change of choice by a player can increase that players own payoff, the strategy profile is called a Nash equilibrium. Notice that the above strategy profile is also a Nash equilibrium on each branch of the game tree, mainly starting at either B or starting at C.
    [Show full text]
  • 1 Bertrand Model
    ECON 312: Oligopolisitic Competition 1 Industrial Organization Oligopolistic Competition Both the monopoly and the perfectly competitive market structure has in common is that neither has to concern itself with the strategic choices of its competition. In the former, this is trivially true since there isn't any competition. While the latter is so insignificant that the single firm has no effect. In an oligopoly where there is more than one firm, and yet because the number of firms are small, they each have to consider what the other does. Consider the product launch decision, and pricing decision of Apple in relation to the IPOD models. If the features of the models it has in the line up is similar to Creative Technology's, it would have to be concerned with the pricing decision, and the timing of its announcement in relation to that of the other firm. We will now begin the exposition of Oligopolistic Competition. 1 Bertrand Model Firms can compete on several variables, and levels, for example, they can compete based on their choices of prices, quantity, and quality. The most basic and funda- mental competition pertains to pricing choices. The Bertrand Model is examines the interdependence between rivals' decisions in terms of pricing decisions. The assumptions of the model are: 1. 2 firms in the market, i 2 f1; 2g. 2. Goods produced are homogenous, ) products are perfect substitutes. 3. Firms set prices simultaneously. 4. Each firm has the same constant marginal cost of c. What is the equilibrium, or best strategy of each firm? The answer is that both firms will set the same prices, p1 = p2 = p, and that it will be equal to the marginal ECON 312: Oligopolisitic Competition 2 cost, in other words, the perfectly competitive outcome.
    [Show full text]
  • Economics 201B Economic Theory (Spring 2021) Strategic Games
    Economics 201B Economic Theory (Spring 2021) Strategic Games Topics: terminology and notations (OR 1.7), games and solutions (OR 1.1-1.3), rationality and bounded rationality (OR 1.4-1.6), formalities (OR 2.1), best-response (OR 2.2), Nash equilibrium (OR 2.2), 2 2 examples × (OR 2.3), existence of Nash equilibrium (OR 2.4), mixed strategy Nash equilibrium (OR 3.1, 3.2), strictly competitive games (OR 2.5), evolution- ary stability (OR 3.4), rationalizability (OR 4.1), dominance (OR 4.2, 4.3), trembling hand perfection (OR 12.5). Terminology and notations (OR 1.7) Sets For R, ∈ ≥ ⇐⇒ ≥ for all . and ⇐⇒ ≥ for all and some . ⇐⇒ for all . Preferences is a binary relation on some set of alternatives R. % ⊆ From % we derive two other relations on : — strict performance relation and not  ⇐⇒ % % — indifference relation and ∼ ⇐⇒ % % Utility representation % is said to be — complete if , or . ∀ ∈ % % — transitive if , and then . ∀ ∈ % % % % can be presented by a utility function only if it is complete and transitive (rational). A function : R is a utility function representing if → % ∀ ∈ () () % ⇐⇒ ≥ % is said to be — continuous (preferences cannot jump...) if for any sequence of pairs () with ,and and , . { }∞=1 % → → % — (strictly) quasi-concave if for any the upper counter set ∈ { ∈ : is (strictly) convex. % } These guarantee the existence of continuous well-behaved utility function representation. Profiles Let be a the set of players. — () or simply () is a profile - a collection of values of some variable,∈ one for each player. — () or simply is the list of elements of the profile = ∈ { } − () for all players except . ∈ — ( ) is a list and an element ,whichistheprofile () .
    [Show full text]
  • Chapter 16 Oligopoly and Game Theory Oligopoly Oligopoly
    Chapter 16 “Game theory is the study of how people Oligopoly behave in strategic situations. By ‘strategic’ we mean a situation in which each person, when deciding what actions to take, must and consider how others might respond to that action.” Game Theory Oligopoly Oligopoly • “Oligopoly is a market structure in which only a few • “Figuring out the environment” when there are sellers offer similar or identical products.” rival firms in your market, means guessing (or • As we saw last time, oligopoly differs from the two ‘ideal’ inferring) what the rivals are doing and then cases, perfect competition and monopoly. choosing a “best response” • In the ‘ideal’ cases, the firm just has to figure out the environment (prices for the perfectly competitive firm, • This means that firms in oligopoly markets are demand curve for the monopolist) and select output to playing a ‘game’ against each other. maximize profits • To understand how they might act, we need to • An oligopolist, on the other hand, also has to figure out the understand how players play games. environment before computing the best output. • This is the role of Game Theory. Some Concepts We Will Use Strategies • Strategies • Strategies are the choices that a player is allowed • Payoffs to make. • Sequential Games •Examples: • Simultaneous Games – In game trees (sequential games), the players choose paths or branches from roots or nodes. • Best Responses – In matrix games players choose rows or columns • Equilibrium – In market games, players choose prices, or quantities, • Dominated strategies or R and D levels. • Dominant Strategies. – In Blackjack, players choose whether to stay or draw.
    [Show full text]
  • Dynamic Games Under Bounded Rationality
    Munich Personal RePEc Archive Dynamic Games under Bounded Rationality Zhao, Guo Southwest University for Nationalities 8 March 2015 Online at https://mpra.ub.uni-muenchen.de/62688/ MPRA Paper No. 62688, posted 09 Mar 2015 08:52 UTC Dynamic Games under Bounded Rationality By ZHAO GUO I propose a dynamic game model that is consistent with the paradigm of bounded rationality. Its main advantages over the traditional approach based on perfect rationality are that: (1) the strategy space is a chain-complete partially ordered set; (2) the response function is certain order-preserving map on strategy space; (3) the evolution of economic system can be described by the Dynamical System defined by the response function under iteration; (4) the existence of pure-strategy Nash equilibria can be guaranteed by fixed point theorems for ordered structures, rather than topological structures. This preference-response framework liberates economics from the utility concept, and constitutes a marriage of normal-form and extensive-form games. Among the common assumptions of classical existence theorems for competitive equilibrium, one is central. That is, individuals are assumed to have perfect rationality, so as to maximize their utilities (payoffs in game theoretic usage). With perfect rationality and perfect competition, the competitive equilibrium is completely determined, and the equilibrium depends only on their goals and their environments. With perfect rationality and perfect competition, the classical economic theory turns out to be deductive theory that requires almost no contact with empirical data once its assumptions are accepted as axioms (see Simon 1959). Zhao: Southwest University for Nationalities, Chengdu 610041, China (e-mail: [email protected]).
    [Show full text]
  • Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction
    Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction Renato Paes Leme Eva´ Tardos Department of Computer Science Department of Computer Science Cornell University, Ithaca, NY Cornell University, Ithaca, NY [email protected] [email protected] Abstract—The Generalized Second Price Auction has for advertisements and slots higher on the page are been the main mechanism used by search companies more valuable (clicked on by more users). The bids to auction positions for advertisements on search pages. are used to determine both the assignment of bidders In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full to slots, and the fees charged. In the simplest model, information setting, socially optimal Nash equilibria are the bidders are assigned to slots in order of bids, and known to exist (i.e., the Price of Stability is 1). This paper the fee for each click is the bid occupying the next is the first to prove bounds on the price of anarchy, and slot. This auction is called the Generalized Second Price to give any bounds in the Bayesian setting. Auction (GSP). More generally, positions and payments Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated in the Generalized Second Price Auction depend also on strategies. In the full information setting we prove a bound the click-through rates associated with the bidders, the of 1.618 for the price of anarchy for pure Nash equilibria, probability that the advertisement will get clicked on by and a bound of 4 for mixed Nash equilibria.
    [Show full text]
  • Extensive Form Games with Perfect Information
    Notes Strategy and Politics: Extensive Form Games with Perfect Information Matt Golder Pennsylvania State University Sequential Decision-Making Notes The model of a strategic (normal form) game suppresses the sequential structure of decision-making. When applying the model to situations in which players move sequentially, we assume that each player chooses her plan of action once and for all. She is committed to this plan, which she cannot modify as events unfold. The model of an extensive form game, by contrast, describes the sequential structure of decision-making explicitly, allowing us to study situations in which each player is free to change her mind as events unfold. Perfect Information Notes Perfect information describes a situation in which players are always fully informed about all of the previous actions taken by all players. This assumption is used in all of the following lecture notes that use \perfect information" in the title. Later we will also study more general cases where players may only be imperfectly informed about previous actions when choosing an action. Extensive Form Games Notes To describe an extensive form game with perfect information we need to specify the set of players and their preferences, just as for a strategic game. In addition, we also need to specify the order of the players' moves and the actions each player may take at each point (or decision node). We do so by specifying the set of all sequences of actions that can possibly occur, together with the player who moves at each point in each sequence. We refer to each possible sequence of actions (a1; a2; : : : ak) as a terminal history and to the function that denotes the player who moves at each point in each terminal history as the player function.
    [Show full text]
  • UC Merced Proceedings of the Annual Meeting of the Cognitive Science Society
    UC Merced Proceedings of the Annual Meeting of the Cognitive Science Society Title Language as a Tool for Thought: The Vocabulary of Games Facilitates Strategic Decision Making Permalink https://escholarship.org/uc/item/2n8028tv Journal Proceedings of the Annual Meeting of the Cognitive Science Society, 28(28) ISSN 1069-7977 Authors Keller, Josh Loewenstein, Jeffrey Publication Date 2006 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Language as a Tool for Thought: The Vocabulary of Games Facilitates Strategic Decision Making Jeffrey Loewenstein ([email protected]) McCombs School of Business, 1 University Station B6300 Austin, TX 78712 USA Josh Keller ([email protected]) McCombs School of Business, 1 University Station B6300 Austin, TX 78712 USA Abstract thereby influence key decision makers to favor their desired policies. More than 60 years ago, the sociologist Mills People in competitive decision-making situations often make (1939) argued that problems are perceived relative to a poor choices because they inadequately understand the vocabulary. For example, what counts as murder seems decision they are to make, particularly because they fail to straightforward, but should differ substantially across consider contingencies such as how their opponent will react vegans, anti-abortion activists, lawyers, soldiers and the to their choice (Tor & Bazerman, 2004). Accordingly it would be useful to have a generally applicable vocabulary to guide (hopefully extinct) ritual practitioner of human sacrifice people towards effective interpretations of decision situations. (Clark, 1998). The vocabulary of games provides one such toolkit. We One role of language is to invoke particular framings or presented 508 participants with words from the vocabulary of interpretations.
    [Show full text]