21 0 obj In that case you should write down all possible strategies: There are 2^3 strategies for A, 2^2 strategies for B. Using Backward Induction - Entry and Predation GameEntrant In Out Accommodate Entry Fight Entry Backward induction solutions are special cases of the more powerful concept of subgame … game of the game, the equilibrium computed using backwards induction remains to be an equilibrium (computed again via backwards induction) of the subgame. The payoffs are arranged so that if the pot is passed to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. ILet L <1be the maximum length of all histories. /Filter /FlateDecode Backward Induction Backward induction refers to starting from the last subgames of a finite game, then finding the best response strategy profiles or the Nash equilibria in the subgames, then assigning these strategies profiles and the associated payoffs to be subgames, and moving successively towards the beginning of the game. Subgame Perfect Equilibrium Proposition Let Γ be an extensive form game with perfect information and s∗ be a subgame perfect equilibrium of Γ. Below is an example of how one might model such a game. << The equilibrium outcome is that player 1 does not contribute, while players 2 and 3 do. endobj >> Clearly every SPE is a NE but not conversely. endstream << /S /GoTo /D (Outline0.3) >> endobj If he takes, then A and B get $1 each, but if A passes, the decision to take or pass now has to be made by Player B. Find all Nash Equilibrium to the normal-form game. >> /Resources 37 0 R The game is also sequential, so Player 1 makes the first decision (left or right) and Player 2 makes its decision after Player 1 (up or down). 5- SPNE solutions are sequentially rational if game has at least one proper sub game. As a result, they get a higher payoff than the payoff predicted by the equilibria analysis. Backward induction in game theory is an iterative process of reasoning backward in time, from the end of a problem or situation, to solve finite extensive form and sequential games, and infer a sequence of optimal actions. 17 0 obj Experimental studies have shown that “rational” behavior (as predicted by game theory) is seldom exhibited in real life. /FormType 1 The game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion. � Q���n�֘ ���. /Length 15 The first game involves players’ trusting that others will not make mistakes. to subgame perfection. For finite games of perfect information, any backward induction solution is a SPNE and vice-versa. Determine the Nash equilibria of each subgame. If Company 1 wanted to release a product, what might Company 2 do in response? << Because no player has an incentive to deviate at any infor- Effectively, one is determining the Nash equilibrium of each subgame of the original game. endobj subgame perfect equilibrium outcome of any binary agenda Proof: By backwards induction, we can determine alternative that will result at any node. A subgame perfect equilibrium is an equilibrium in which all actions are Nash equilibria for all subgames. ;��EO"T� 4- SPNE solutions are Nash equilibria . The result is an equilibrium found by backward induction of Player 1 choosing "right" and Player 2 choosing "up." Solving Sequential Games Using Backward Induction. But $(ahj,de)$ can be an equilibrium because the history given by this strategy profile never reaches these nodes. Mark Voorneveld Game theory SF2972, Extensive form games 6/25 Subgame perfect equilibria via backward induction Use backward induction to –nd the subgame perfect equilibrium. But if they distrust the other player and expect them to “take” at the first opportunity, Nash equilibrium predicts the players will take the lowest possible claim ($1 in this case). A majority prefers x to y; so x will be adopted at h. In that sense we say that SPE is a refinement of NE. Describe the backward induction outcome of this game for any –nite integer k. FØlix Muæoz-García (WSU) EconS 424 - Recitation 5 March 24, 2014 12 / 48. (Extensive Form Refinements of Nash Equilibrium) /ProcSet [ /PDF ] /FormType 1 FØlix Muæoz-García (WSU) EconS 424 - Recitation 5 March 24, 2014 10 / 48. endobj Will Company 2 release a similar competing product? %PDF-1.5 /Subtype /Form Let™s do a few examples together.! playing C – giving player 1 a payoff of 4. x��XKs�6��W�H�0ޏ['N�:�Z����F[SS�i����. There, every backward induction equilibrium (BIE), i.e., a strategy profile that survives backward pruning, is also a subgame perfect equilibrium (SPE), and all SPEs result from backward pruning. However, the results inferred from backward induction often fail to predict actual human play. Any finite extensive-form game has a subgame perfect Nash equilibrium. 29 0 obj Behavioral Economics is the study of psychology as it relates to the economic decision-making processes of individuals and institutions. /Matrix [1 0 0 1 0 0] (Note that s1, 2 could be a sequence, e.g. x���P(�� �� Backward induction finds the optimal actions of the players in the “ last ” subgame first, and then, given these actions, works backward to the beginning to find the SPE of the game. 43 0 obj ) is a Subgame Perfect Nash Equilibrium (SPNE) of the game since it speci–es a NE for each proper subgames of the game. << 16) Player 2 Left Right Player 1 Up 0, 1 1, 0 Down 1, c Consider the two-player “ centipede ” game in Figure 2, in which each player sequentially chooses either to … Algorithm Consider the normal forms of all subgames. 26 0 obj Model the game with a strategic grid. << /S /GoTo /D (Outline0.4) >> /Type /XObject The offers that appear in this table are from partnerships from which Investopedia receives compensation. Subgame perfection generalizes this notion to general dynamic games: Definition 11.1 A Nash equilibrium is said to be subgame perfect if an only if it is a Nash equilibrium in every subgame of the game. Backward induction has been used to solve games since John von Neumann and Oskar Morgenstern established game theory as an academic subject when they published their book, Theory of Games and Economic Behavior in 1944. 39 0 obj A subgame perfect equilibrium is a strategy prole that induces a Nash equilibrium in each subgame. /Subtype /Form stream << /S /GoTo /D (Outline0.2) >> >> /Filter /FlateDecode in extensive form representation, process of backward induction to find path relies on both firms having perfect info about decisions that will be made in each subgame (a Nash equilibrium for each subgame in the larger representation) In the centipede game, two players alternately get a chance to take a larger share of an increasing pot of money, or to pass the pot to the other player. /FormType 1 =�acI(�)���@���0'A(��6�S4��)D��{� �� >> Effectively, one is determining the Nash equilibrium of each subgame of the original game. It has three Nash equilibria but only one is consistent with backward induction. %���� 10 0 obj The labels with Player 1 and Player 2 within them are the information sets for players one or two, respectively. Thus, player 1 selects N. The full backwards induction reasoning is shown in figure 7. (Formalizing the Game) << >> Do you want to find all the equilibria that are not subgame perfect? endobj endobj This process continues backward until the best action for every point in time has been determined. ���y?g?b�I};~�Q��ҭ����D�E��F�T2>uv+…�Z�(��S���z�7��+ה����eD2�#´�����;ͨS���c��&���Hm��T���z�/b�4����!o�j +�C�I��4< P�� `�� In this way, we will mark the lines in blue that maximize the player's payoff at the given information set. /BBox [0 0 8 8] stream endobj The point of the game is if A and B both cooperate and continue to pass until the end of the game, they get the maximum payout of $100 each. << /S /GoTo /D (Outline0.1) >> The concept of backwards induction corresponds to this assumption that it is common knowledge that each player will act rationally with each decision node when she chooses an option — even if her rationality would imply that such a node will not be reached.’ 14 0 obj Let’s introduce a way of incorporating the timing of actions 35 0 obj As with solving for other Nash Equilibria, rationality of players and complete knowledge is assumed. However, in reality, relatively few players do so. /ProcSet [ /PDF ] stream (Exercises) Some comments: Hopefully it is clear that subgame perfect Nash equilibrium is a refinement of Nash equilibrium. However, the results inferred from backward induction often fail to predict actual human play. 25 0 obj << /S /GoTo /D [31 0 R /Fit] >> At either information set we have two choices, four in all. /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 8.00009] /Coords [0 0.0 0 8.00009] /Function << /FunctionType 3 /Domain [0.0 8.00009] /Functions [ << /FunctionType 2 /Domain [0.0 8.00009] /C0 [1 1 1] /C1 [0.5 0.5 0.5] /N 1 >> << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [0.5 0.5 0.5] /N 1 >> ] /Bounds [ 4.00005] /Encode [0 1 0 1] >> /Extend [false false] >> >> /Length 15 Entrant In Out AF If B takes, she gets $3 (i.e., the previous stash of $2 + $1) and A gets $0. Irrational players may actually end up obtaining higher payoffs than predicted by backward induction, as illustrated in the centipede game. /Filter /FlateDecode 2- Not all Nash equilibria are sequentially rational. The traveler's dilemma demonstrates the paradox of rationality—that making decisions illogically often produces a better payoff in game theory. /Matrix [1 0 0 1 0 0] /Type /XObject This logic can be generalized to general nite horizon extensive games with perfect information. /Matrix [1 0 0 1 0 0] endobj endobj In the game on the previous slide, only (A;R) is subgame perfect. This describes player 1’s, player 2’s and Make a matrix using these as row and column labels. /BBox [0 0 16 16] Thus the only subgame perfect equilibria of the entire game is \({AD,X}\). /Length 1184 >> stream A set of strategies is a subgame perfect Nash equilibrium (SPNE), if these strategies, when confined to any subgame of the original game, have the players playing a Nash equilibrium within that subgame (s1, s2) is a SPNE if for every subgame, s1 and s2 constitute a Nash equilibrium within the subgame. Backward induction • Backward induction refers to elimination procedures that go as follows: 1 Identify the “terminal subgames” (ie those without proper subgames) 2 Pick a Nash equilibrium for each terminal subgame 3 Replace each terminal subgame with a terminal node where players get the payoffs from the corresponding Nash equilibrium /BBox [0 0 5669.291 8] By forecasting sales of this new product in different scenarios, we can set up a game to predict how events might unfold. IFind all nonterminal histories of L … << If both players always choose to pass, they each receive a payoff of $100 at the end of the game. In a perfect information game without payoff ties, the unique SPNE coincides with the strategy profile indentified by backward induction. 37 0 obj The Nash equilibrium of this game, where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice, suggests the first player would take the pot on the very first round of the game. /Resources 35 0 R A situation in which one person’s gain is equivalent to another’s loss, so that the net change in wealth or benefit is zero. endstream 38 0 obj Answer to 7 Using backward induction, find the subgame perfect equilibrium (equilibria) of the following game. Every finite game of perfect information can be solved using backward induction. Below is a simple sequential game between two players. This game could include product release scenarios. 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