subgame perfect equilibrium backward induction

stream At the node h where x can be adopted: Let y be the alternative that will be chosen if x is not chosen. � Q��n�֘ ��. In games with perfect information, the Nash equilibrium obtained through backwards induction is subgame perfect. Extensive Games Subgame Perfect Equilibrium Backward Induction Illustrations Extensions and Controversies Subgame perfect Nash equilibrium (SPNE) •A subgame perfect Nash equilibrium (子博弈完美均衡) is a strategy proﬁle s with the property that in no subgame can any player i do better by choosing a strategy diﬀerent from s >> /Subtype /Form 39 0 obj As an example, assume Player A goes first and has to decide if he should “take” or “pass” the stash, which currently amounts to $2. If both players always choose to pass, they each receive a payoff of $100 at the end of the game. 36 0 obj In its standard formulation, backward induction applies only to ﬁnite games of perfect information. 3- All Backward induction solutions are sequentially rational. Clearly every SPE is a NE but not conversely. /Matrix [1 0 0 1 0 0] The equilibrium concepts that we now think of as various forms of backwards induction, namely, subgame perfect equilibrium (Selten, 1965), perfect equilibrium (Selten, 1975), sequential equilibrium (Kreps and Wilson, 1982), and quasi-perfect equilibrium (van Damme, 1984), while formally well defined in a wider class of games, are explicitly restricted to games with perfect recall. << >> 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. << endobj IFind all nonterminal histories of L … As with solving for other Nash Equilibria, rationality of players and complete knowledge is assumed. Experimental studies have shown that “rational” behavior (as predicted by game theory) is seldom exhibited in real life. 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. In a perfect information game without payoff ties, the unique SPNE coincides with the strategy profile indentified by backward induction. 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. endobj 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. stream BackwardInductionandSubgamePerfection CarlosHurtado DepartmentofEconomics UniversityofIllinoisatUrbana-Champaign hrtdmrt2@illinois.edu June13th,2016 $\endgroup$ – step Dec 16 '17 at 13:46 Use backward induction to –nd the subgame perfect equilibrium. 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. Effectively, one is determining the Nash equilibrium of each subgame of the original game. This logic can be generalized to general nite horizon extensive games with perfect information. 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) Then s∗ is a backward induction equilibrium of Γ. << /Length 15 Why is your answer different than in (a)? 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. FØlix Muæoz-García (WSU) EconS 424 - Recitation 5 March 24, 2014 10 / 48. to subgame perfection. 30 0 obj endobj A subgame perfect equilibrium is a strategy prole that induces a Nash equilibrium in each subgame. Thus, player 1 selects N. The full backwards induction reasoning is shown in ﬁgure 7. Subgame Perfect Nash Equilibrium. Backward induction and subgame-perfect Nash equilibria Backward induction is a useful tool while solving for the subgame-perfect Nash equilibrium (SPNE) of a sequential game. Determine the Nash equilibria of each subgame. Mark Voorneveld Game theory SF2972, Extensive form games 6/25 Subgame perfect equilibria via backward induction /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0 1] /Coords [4.00005 4.00005 0.0 4.00005 4.00005 4.00005] /Function << /FunctionType 2 /Domain [0 1] /C0 [0.5 0.5 0.5] /C1 [1 1 1] /N 1 >> /Extend [true false] >> >> 2- Not all Nash equilibria are sequentially rational. >> Subgame Perfect Nash Equilibrium A strategy speci es what a player will do at every decision point I Complete contingent plan Strategy in a SPNE must be a best-response at each node, given the strategies of other players Backward Induction 10/26. endobj /FormType 1 %�� << 4- SPNE solutions are Nash equilibria . /Resources 35 0 R Will Company 2 release a similar competing product? Subgame Perfect Equilibria Questions Use backward induction to determine the subgame perfect equilibrium of the following games: Question 1 Question 2 Question 3. 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. After this reduction, Player 1 can maximize its payoffs now that Player 2's choices are made known. Below is a simple sequential game between two players. It has three Nash equilibria but only one is consistent with backward induction. Using Backward Induction - Entry and Predation GameEntrant In Out Accommodate Entry Fight Entry /FormType 1 In that case you should write down all possible strategies: There are 2^3 strategies for A, 2^2 strategies for B. /BBox [0 0 8 8] In that sense we say that SPE is a reﬁnement of NE. If B takes, she gets $3 (i.e., the previous stash of $2 + $1) and A gets $0. Below is the solution to the game with the equilibrium path bolded. << In the game on the previous slide, only (A;R) is subgame perfect. 29 0 obj Please help, it is for a very important assignment, thank you so very much! 22 0 obj (Extensive Form Refinements of Nash Equilibrium) Solutions Question 1 { S ; t } with payoffs of (1,0). We can find such equilibria by starting using backward induction , which instructs us to start at the last action and work our way progressively backward from there. the equilibrium computed using backward induction remains an equilibrium (computed again via backward induction) of the subgame. 21 0 obj 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. /Filter /FlateDecode endstream The ad- 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. Solving Sequential Games Using Backward Induction. endobj 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.’ /FormType 1 For instance, for dynamic games with perfect information, (a_1^*,R_j (a_1^*)) is the solution for player j with backwards induction, where (a_1^*,R_j (a_1)), is the solution for the subgame perfect Nash-equilibrium? 17 0 obj Notice that every SPNE must also be a NE, because the full game is also a subgame. A subgame perfect equilibrium is an equilibrium in which all actions are Nash equilibria for all subgames. endobj 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). x��P(�� << /S /GoTo /D [31 0 R /Fit] >> Subgame Perfect Equilibrium Proposition Let Γ be an extensive form game with perfect information and s∗ be a subgame perfect equilibrium of Γ. The Nash Equilibrium is a concept within game theory where the optimal outcome of a game is where there is no incentive to deviate from their initial strategy. Make a matrix using these as row and column labels. /ProcSet [ /PDF ] stream But if B passes, A now gets to decide whether to take or pass, and so on. 26 0 obj /Resources 39 0 R endstream /Length 15 38 0 obj /Resources 37 0 R For ﬁnite games of perfect information, any backward induction solution is a SPNE and vice-versa. << /S /GoTo /D (Outline0.5) >> Any finite extensive-form game has a subgame perfect Nash equilibrium. (Backward Induction) The traveler's dilemma demonstrates the paradox of rationality—that making decisions illogically often produces a better payoff in game theory. x��P(�� x��P(�� Model the game with a strategic grid. Find all Nash Equilibrium to the normal-form game. "oﬀ-the-equilibrium-path"behaviorcanbeimportant, be-cause it aﬀects the incentives of players to follow the equilibrium. >> Use backward induction to find the subgame perfect nash equilibrium to the game. Answer to 7 Using backward induction, find the subgame perfect equilibrium (equilibria) of the following game. /Length 15 playing C – giving player 1 a payoﬀ of 4. 43 0 obj Algorithm Consider the normal forms of all subgames. endobj But First! 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. Some comments: Hopefully it is clear that subgame perfect Nash equilibrium is a refinement of Nash equilibrium. The centipede game in game theory involves two players alternately getting a chance to take the larger share of an increasing money stash. The labels with Player 1 and Player 2 within them are the information sets for players one or two, respectively. 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. subgame perfect equilibrium outcome of any binary agenda Proof: By backwards induction, we can determine alternative that will result at any node. endobj endobj << /S /GoTo /D (Outline0.2) >> stream Backward induction solutions are special cases of the more powerful concept of subgame … Thus the only subgame perfect equilibria of the entire game is ${AD,X}$. 10 0 obj In this way, we will mark the lines in blue that maximize the player's payoff at the given information set. /ProcSet [ /PDF ] 34 0 obj /BBox [0 0 5669.291 8] We show the other two Nash equilibria are not subgame perfect: each fails to induce Nash in a subgame. << /Type /XObject Game theory is a framework for modeling scenarios in which conflicts of interest exist among the players. Irrational players may actually end up obtaining higher payoffs than predicted by backward induction, as illustrated in the centipede game. x��XKs�6��W�H�0ޏ['N�:�Z��F[SS�i��. /Matrix [1 0 0 1 0 0] Behavioral Economics is the study of psychology as it relates to the economic decision-making processes of individuals and institutions. >> endobj endobj This describes player 1’s, player 2’s and 6. 18 0 obj However, the results inferred from backward induction often fail to predict actual human play. �B;��کh ��.��- ��k N��^ ��kn�Nj@W�k Q� ILet L <1be the maximum length of all histories. The numbers in the parentheses at the bottom of the tree are the payoffs at each respective point. endstream 25 0 obj /Filter /FlateDecode /Length 1184 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 (Formalizing the Game) Backward induction is ‘the process of analyzing a game from the end to the beginning. (Subgame Perfect Nash Equilibrium) Consider the two-player “ centipede ” game in Figure 2, in which each player sequentially chooses either to … If each player has a strict preference over his possible terminal node payoﬀs (no ties), then backward induction gives a unique sequentially rational strategy proﬁle. /Matrix [1 0 0 1 0 0] >> (Note that s1, 2 could be a sequence, e.g. Let’s introduce a way of incorporating the timing of actions /BBox [0 0 16 16] 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). �uH}J��s�ϧ�Vq��hF}j�1R�[�-K�k�Ԙ�;��;lt��ͪG!5�D3��2�0�T�s�{��h�"i�֡|�=>��ʸ�_+�R��!��̀��S�8��r�(�vk�B��*L\��;÷�WdA�8��z��M�r��$$��h�@�%��&3�{�.��4�&�� :�ZͶ��r��lU�}�~�͋y|��[��. However, in reality, relatively few players do so. endobj b. 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Not contribute, while players 2 and 3 do without payoff ties, the results inferred from induction... Reasoning is shown in ﬁgure 7 2014 10 / 48 ﬁgure 7 proper sub.! It aﬀects the incentives of players and complete knowledge is assumed row and labels! Case you should write down all possible strategies: There are 2^3 strategies for a very important assignment thank. A product, what might Company 2 do in response the tree are the information for! That There is something ﬁshy about the equilibrium path bolded of ( 1,0 ) the full strategy (. Can determine alternative that will result at any node with payoffs of ( )... All histories if game has a subgame perfect equilibrium ( SPE ) may. Game between two players the previous slide, only ( a ) speciﬁes the full is! By backward induction solutions are Nash equilibria are not subgame perfect equilibrium ( SPE.... Not conversely selects N. the full strategy: ( N, NC, NCCN.! Equilibrium outcome is that Player 2 within them are the information sets for players one or,...