DOCSLIB.ORG

SEQUENTIAL GAMES with PERFECT INFORMATION Example

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Such a strategy profile is called subgame perfect. A solution found by backward induction is alway subgame perfect. There is another Nash equilibrium which is not subgame perfect. Assume P1 picks { R }. P2 should pick L00 and R0. We do not derive this strategy profile, but merely check that it satisfies the conditions necessary to be a Nash equilibrium. 0 00 0 What if P2 instead picks L and L . The choice L is “irrational” in terms of a choice at B. We claim 0 00 0 00 that { R } for P1 and { L ,L } is still a Nash equilibrium. For the strategy { L ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { L ,L }) = 4 π1(R, { L ,L }) = 4 ≥ 0 00 π1(L, { L ,L }) = 2. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { L ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { L ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0. Thus, neither player has an incentive to change their own strategy. This strategy is not subgame perfect 0 0 because starting at vertex B, P2 has an incentive to change from L to R . This did not affect the payoffs above because the path resulting from fixing P1’s strategy cannot reach vertex B. In summary, this example has two Nash equilibria. The first, found by backward induction, is subgame perfect. The second is not subgame perfect. 1 2 SEQUENTIAL GAMES WITH PERFECT INFORMATION Definition (page 96) A tree T is an n-player game tree provided that there are n players P1,... Pn and a. each nonterminal node of the tree is owned by either (i) exactly one player who needs to make a choice of an edge going out of the node, or (ii) “nature” with a probability on the edges going out of the node, b. each terminal node is not owned by any player but has a payoff vector, (p1(v), . , pn(v)), with a payoff for each player. In the above example, the two players know exactly which node they are at when they make a choice, and the choices at each node is made independently. Such a game is said to have perfect information.A sequential game with perfect information, or an extensive form game with perfect information, is a game tree where the players can make at each node independently other the choices at other nodes. We treat the case of a game with perfect information first, and then return to the case of “imperfect information”. Definition (page 99) Let V be the set of all the nodes in a game tree T , and let Nj be the set of all the nodes owned by Pj.A strategy for Pj is a choice of an outgoing edge at each v ∈ Nj, i.e., a function sj : Nj → V such that sj(v) is a child of v and (v, sj(v)) is an edge in the graph going out from v. Thus, a strategy can also be thought of as a function from Nj into the edges E such that (v, sj(v)) is an edge going out from node v. Let Sj be the set of tall the possible strategies for Pj.A strategy profile, (s1, . sn) is a strategy for each player, i.e., a choice of an edge at each node owned by each player. The set of all strategy profiles is the set S1 × · × Sn. Given the set of strategy profiles with the resulting payoffs expresses the game in strategic form. A strategy profile (s1, . sn) determines a path from the root to a terminal node. The payoff for Pj is the payoff at the terminal node determined by this path and is denoted by uj(s1, . sn). Definition (page 104) ∗ ∗ A strategy profile (s1, s2) for two-person sequential game with perfect information is called a Nash equilib- rium provided that ∗ ∗ ∗ u1(s1, s2) = max u1(s1, s2) and s1∈S1 ∗ ∗ ∗ u2(s1, s2) = max u2(s1, s2). s2∈S2 ∗ In this definition, the maximum of u1(s1, s2) corresponds to keeping all the choices of the second player fixed and varying the choices of edges going out of all the vertices owned by P1. Similarly, for the maximum ∗ of u2(s1, s2) by P2. Thus, a Nash equilibrium is not just a path in the game tree, but is a choice at each vertex owned by a player. The definition of a Nash equilibrium for an n-person sequential game is defined in a similar manner. Definition (page 116) A subgame of a game is formed by taking a branch of the game tree, using the same payoffs as the original tree and the nodes are owned by the same players. Definition (page 118) ∗ ∗ A strategy profile (s1, . , sn) of a strategic game is called a subgame perfect (Nash) equilibrium provided that it is a Nash equilibrium for each subgame and also for the whole game. Theorem 4.10 (Kuhn) (page 106) Every sequential game with perfect information has a subgame perfect Nash equilibrium. SEQUENTIAL GAMES WITH PERFECT INFORMATION 3 Idea of the proof: Backward induction constructs a subgame perfect Nash equilibrium. Take a node v1 that goes only to terminal nodes. This node v1 belongs to some player Pj. Take the maximum of uj over the terminal nodes ∗ which are children of v1. This defines sj (v1). Replace the tree with a game with v1 as a terminal node and ∗ the payoff at v1 the original payoff at sj (v1). Continue by induction. Example: Dueling In this duel, there are two players. The players alternate turns. If it is P1 turn, then the player can either fire (F) or wait (W) and take a step toward P2. At this stage the players are some distance d apart. If P1 fires, then there is a probability p1(d) that the player will hit the other player and a probability 1 − p1(d) that the player will miss. If the player hits the other player, then the payoff is 1 for P1 and 0 for P2. If P1 misses, then P2 can wait until they are close enough to hit, so the payoff is 1 for P2 and 0 for P1. Thus, the payoffs for the choice to fire by P1 is p1(d) · 1(1 − p1(d)) · 0 = p1(d) for P1 and p1(d) · 0(1 − p1(d)) · 1 = 1 − p1(d) for P2, or a payoff vector of (p1(d), 1 − p1(d)). In the same manner, if it is the turn for P2, this player can wait and take a step toward P1 or fire. If P2 fires, then there is probability of p2(d) that the player will hit P1, so the payoff P2 is p2(d) and 1 − p2(d) for P2, or a payoff vector of (1 − p2(d), p2(d)). (p1(d2n), 1 − p1(d2n)) (p1(d2n−2), 1 − p1(d2n−2)) (p1(d2), 1 − p1(d2)) F F F W (P2) W W (P2) (P1) (P1) (P1) F F (1 − p2n−1(d2), p2(d2n−1)) (0, 1) This game can be solved by backward induction. Assume that players start at a distance d2n = 2nδ apart. After 2n − 1 steps of size δ, the distance is d1 = δ. Thus, the game tree is as given in the following figure with 2n vertices that correspond to distances d2n = 2nδ,..., d1 = δ. At distance δ, assume that P2 is certain to hit and p2(δ) = 1, so P2 is certain to fire. Assume we know that P2 fires at distance d2j−1 = (2j − 1)δ. At this vertex the payoff vector is (1 − p2(d2j−1), p2(d2j−1))). At the previous vertex at distance d2j = (2j)δ, P1 can either fire or wait.
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]
  • 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]
  • 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]
  • The Extensive Form Representation of a Game
    The extensive form representation of a game Nodes, information sets Perfect and imperfect information Addition of random moves of nature (to model uncertainty not related with decisions of other players). Mas-Colell, pp. 221-227, full description in page 227. 1 The Prisoner's dilemma Player 2 confess don't confess Player 1 confess 2.2 10.0 don't confess 0.10 6.6 One extensive form representation of this normal form game is: 2 Player 1 confess don't confess Player 2 Player 2 confessdon't confess confess don't confess 2 10 0 6 payoffs 2 0 10 6 3 Both players decide whether to confess simultaneously (or more exactly, each player decides without any information about the decision of the other player). This is a game of imperfect information: The information set of player 2 contains more than one decision node. 4 The following extensive form represents the same normal form: Player 2 confess don't confess Player 1 Player 1 confessdon't confess confess don't confess 5 Consider now the following extensive form game: Player 1 confess don't confess Player 2 Player 2 confessdon't confess confess don't confess 2 10 6 0 6 payoffs 2 0 10 6 Now player 2 observes the behavior of player 1. This is a game of perfect information. Strategies of player 2: For instance "If player 1 confesses, I…; if he does not confess, then I…" Player 2 has 4 strategies: (c, c), (c, d), (d, c), (d, d) The normal form of this game is: Player 2 c, c c, d d, c d, d Player 1 confess 2.2 2.2 10.0 10.0 don't confess 0.10 6.6 0.10 6.6 There is only one NE, - strategy confess for player 1 and - strategy (c, c) for player 2.
    [Show full text]
  • What Is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?
    What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization? Chi Jin Praneeth Netrapalli University of California, Berkeley Microsoft Research, India [email protected] [email protected] Michael I. Jordan University of California, Berkeley [email protected] August 18, 2020 Abstract Minimax optimization has found extensive application in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises—“what is a proper definition of local optima?” Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting—local minimax—as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm—gradient descent ascent (GDA)—under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points. 1 Introduction arXiv:1902.00618v3 [cs.LG] 15 Aug 2020 Minimax optimization refers to problems of two agents—one agent tries to minimize the payoff function f : X × Y ! R while the other agent tries to maximize it.
    [Show full text]
  • Games with Hidden Information
    Games with Hidden Information R&N Chapter 6 R&N Section 17.6 • Assumptions so far: – Two-player game : Player A and B. – Perfect information : Both players see all the states and decisions. Each decision is made sequentially . – Zero-sum : Player’s A gain is exactly equal to player B’s loss. • We are going to eliminate these constraints. We will eliminate first the assumption of “perfect information” leading to far more realistic models. – Some more game-theoretic definitions Matrix games – Minimax results for perfect information games – Minimax results for hidden information games 1 Player A 1 L R Player B 2 3 L R L R Player A 4 +2 +5 +2 L Extensive form of game: Represent the -1 +4 game by a tree A pure strategy for a player 1 defines the move that the L R player would make for every possible state that the player 2 3 would see. L R L R 4 +2 +5 +2 L -1 +4 2 Pure strategies for A: 1 Strategy I: (1 L,4 L) Strategy II: (1 L,4 R) L R Strategy III: (1 R,4 L) Strategy IV: (1 R,4 R) 2 3 Pure strategies for B: L R L R Strategy I: (2 L,3 L) Strategy II: (2 L,3 R) 4 +2 +5 +2 Strategy III: (2 R,3 L) L R Strategy IV: (2 R,3 R) -1 +4 In general: If N states and B moves, how many pure strategies exist? Matrix form of games Pure strategies for A: Pure strategies for B: Strategy I: (1 L,4 L) Strategy I: (2 L,3 L) Strategy II: (1 L,4 R) Strategy II: (2 L,3 R) Strategy III: (1 R,4 L) Strategy III: (2 R,3 L) 1 Strategy IV: (1 R,4 R) Strategy IV: (2 R,3 R) R L I II III IV 2 3 L R I -1 -1 +2 +2 L R 4 II +4 +4 +2 +2 +2 +5 +1 L R III +5 +1 +5 +1 IV +5 +1 +5 +1 -1 +4 3 Pure strategies for Player B Player A’s payoff I II III IV if game is played I -1 -1 +2 +2 with strategy I by II +4 +4 +2 +2 Player A and strategy III by III +5 +1 +5 +1 Player B for Player A for Pure strategies IV +5 +1 +5 +1 • Matrix normal form of games: The table contains the payoffs for all the possible combinations of pure strategies for Player A and Player B • The table characterizes the game completely, there is no need for any additional information about rules, etc.
    [Show full text]
  • Extensive Form Games and Backward Induction
    Recap Perfect-Information Extensive-Form Games Extensive Form Games and Backward Induction ISCI 330 Lecture 11 February 13, 2007 Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 1 Recap Perfect-Information Extensive-Form Games Lecture Overview Recap Perfect-Information Extensive-Form Games Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 2 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 Recap Perfect-Information Extensive-Form Games Introduction I The normal form game representation does not incorporate any notion of sequence, or time, of the actions of the players Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 3 I Two variants: I perfect information extensive-form games I imperfect-information extensive-form games Recap Perfect-Information Extensive-Form Games 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. Extensive Form Games and Backward Induction ISCI 330 Lecture 11, Slide 3 Recap Perfect-Information Extensive-Form Games 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
    [Show full text]
  • Finding Strategic Game Equivalent of an Extensive Form Game
    Finding Strategic Game Equivalent of an Extensive Form Game • In an extensive form game, a strategy for a player should specify what action the player will choose at each information set. That is, a strategy is a complete plan for playing a game for a particular player. • Therefore to find the strategic game equivalent of an extensive form game we should follow these steps: 1. First we need to find all strategies for every player. To do that first we find all information sets for every player (including the singleton information sets). If there are many of them we may label them (such as Player 1’s 1st info. set, Player 1’s 2nd info. set, etc.) 2. A strategy should specify what action the player will take in every information set which belongs to him/her. We find all combinations of actions the player can take at these information sets. For example if a player has 3 information sets where • the first one has 2 actions, • the second one has 2 actions, and • the third one has 3 actions then there will be a total of 2 × 2 × 3 = 12 strategies for this player. 3. Once the strategies are obtained for every player the next step is finding the payoffs. For every strategy profile we find the payoff vector that would be obtained in case these strategies are played in the extensive form game. Example: Let’s find the strategic game equivalent of the following 3-level centipede game. (3,1)r (2,3)r (6,2)r (4,6)r (12,4)r (8,12)r S S S S S S r 1 2 1 2 1 2 (24,8) C C C C C C 1.
    [Show full text]
  • Revealed Preferences of Individual Players in Sequential Games
    Revealed Preferences of Individual Players in Sequential Games Hiroki Nishimura∗ January 14, 2021 Abstract This paper studies rational choice behavior of a player in sequential games of perfect and complete information without an assumption that the other players who join the same games are rational. The model of individually rational choice is defined through a decomposition of the behavioral norm assumed in the subgame perfect equilibria, and we propose a set of axioms on collective choice behavior that characterize the individual rationality obtained as such. As the choice of subgame perfect equilibrium paths is a special case where all players involved in the choice environment are each individually rational, the paper offers testable characterizations of both individual rationality and collective rationality in sequential games. JEL Classification: C72, D70. Keywords: Revealed preference, individual rationality, subgame perfect equilibrium. 1 Introduction A game is a description of strategic interactions among players. The players involved in a game simultaneously or dynamically choose their actions, and their payoffs are determined by a profile of chosen actions. A number of solution concepts for games, such as the Nash equilibrium or the subgame perfect equilibrium, have been developed in the literature in order to study collective choice of actions made by the players. In turn, these solution concepts are widely applied in economic analysis to provide credible predictions for the choice of economic agents who reside in situations approximated by games. ∗Department of Economics, University of California Riverside. Email: [email protected]. I am indebted to Efe A. Ok for his continuous support and encouragement. I would also like to thank Indrajit Ray, anonymous referees, and an associate editor for their thoughtful suggestions.
    [Show full text]