• An example: A challenger decides whether or not to enter (a market); if the challenger enters, the incumbent decides to ﬁght or acquiesce.. . We ﬁrst compute a Nash equilibrium of the subgame, then ﬁxing the equilibrium actions as they are (in this subgame), and example without any subgame-perfect equilibrium was presented in Harris et al. However, there are solutions in the class for which his statement does not hold (Example … 1 . Rubinstein bargaining game is extended to incorporate loss aversion, where the initial reference points are not zero. Subgame Perfect Nash Equilibrium Subgame Perfect Nash Equilibrium is a re nement of Nash Equilibrium It rules out equilibria that rely on incredible threats in a dynamic environment All SPNE are identi ed by backward induction 26/26 3 Telex vs. IBM, extensive form: subgame, perfect information Telex 0, 0 2, 2 1, 5 Enter Smash IBM Stay Out Accommodate Subgame … Let's construct subgame-perfect for our particular game model. Backward Induction and Subgame Perfection In extensive-form games, we can have a Nash equilibrium proﬁle of strategies where player 2’s strategy is a best response to player 1’s strategy, but where she will not want to carry out her plan at some nodes of the game tree. extensive-form game with perfect recall if it issequentially rationalandconsistent. 148 0 obj
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increasinglyfineapproximations,andasubgame—perfectequilibriumofeachofthe approximations,then itis natural to expectthat any limit point of thesequence of equilibriumpaths so obtained will be an equilibrium path of the original game. Example . 5 References: Watson, Ch. 1. We compute the subgame perfect equilibria as follows. 0000000016 00000 n
In the above example, ( E, A) is a SPE, while ( O, F) is not. It has three Nash equilibria but only one is consistent with backward induction. between a subgame perfect nash equilibrium and a nash equilibrium? 0000003708 00000 n
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This game has two subgames: one starts after player 1 plays E; the second one is the game itself. Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. We first play and then analyze wars of attrition; the games that afflict trench warfare, strikes, and businesses in some competitive settings.We find long and damaging fights can occur in class in these games even when the prizes are small in relation to the accumulated costs. Voting is simultaneous; the option that receives the majority will be implemented. 0000001411 00000 n
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A subgame perfect Nash equilibrium is a Nash equilibrium in which the strategy profiles specify Nash equilibria for every subgame of the game. xref
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Examples of perfect Bayesian equilibria Gift game 1. ECON 504 Sample Questions for Final Exam Levent Koçkesen Therefore,the set of subgame perfectequilibria is {(Rl,l),(Lr,r),(L3 4 l ⊕ 1 4 r, 1 4 l ⊕ 2 4 r)}. endstream
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The converse is not true. Notice that the den ition contains a subtlety. Some Examples Example 1. 0000002344 00000 n
We analyze three games using our new solution concept, subgame perfect equilibrium (SPE). • Example: Consider the following game, ... and #2 (subgame perfect Nash equilibrium) and will describe #3 (conditional dominance and forward induction) only briefly. Why subgame-perfect is better than the Nash equilibrium? So, the Nash equilibrium in the game Gamma is called subgame-perfect, if for any subgame of the initial game, the truncation of the Nash equilibrium, will be the Nash equilibrium in the subgame. Under the assumption that the highest rejected proposal of the opponent last periods is regarded as the associated reference point, we investigate the effect of loss aversion and initial reference points on subgame perfect equilibrium. In the previous chapter: 1. Consider the following extensive game: 1 2 ab c L RRL 1 … I A sequential equilibrium is a Nash equilibrium. In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. The players receive a reward upon termination of the game, which depends on the state where the game was terminated. In this chapter we’ll take a look at what happens when games are repeatedly infinitely. Example 1: (OUT&B, L) is a subgame perfect Nash equilibrium (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? Example 1 Consider three individuals, each strictly preferring option a to b. A subgame perfect equilibrium of a game G is a Nash Equilibrium of G that corresponds to a Nash Equilibrium in every subgame of G. Let's take a really simple example with two players, Russia and Ukraine. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. must have a unique starting point; • It . • It . 0000001630 00000 n
In market k, competitor C k chooses either \In" or \Out" given the histories in the previous k 1 … Subgame Perfect Equilibrium In practice you may use an algorithm similar to backward induction: 1 Find the Nash equilibria of the “smallest” subgame(s) 2 Fix one for each subgame and attach payoﬀs to its initial node 3 Repeat with the reduced game Levent Koc¸kesen … I offer an example extensive-form game to demonstrate that subgame perfection will not eliminate all undesirable equilibria of extensive-form games. *�����l��gW�l9s�d�I:�1���3 �ngo �p����>vy�� �n�E�;���4_w�F�������P it�h�Ў�%%e� -((f40� X��� r9�J��M�2@z�"� E�`��J. Let p* be the threshold such that when p≥p*, cooperation is sustainable as a subgame perfect equilibrium by the "grim trigger" strategy (under which each player cooperates as long as the other does and defects forever after if either player deviates), and when p

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