Game Theory Feb. 4th, 2003 2. In general an elimination of strictly dominated strategies is not a one step process; it is an iterative procedure. 2. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. Procede with iterated elimination of strictly dominated strategies as usual, if possible. In Example 1.1, elimination of dominated strategies generates inessential players. Example 2 below shows that a game may have a … We call this process iterated elimination of strictly dominated strategies. Its use is justified by the assumption of common knowledge of rationality. Given a restriction G′ of G and a joint mixed Iterated Weaker-than-Weak Dominance Shih-Fen Cheng Michael P. Wellman Singapore Management University University of Michigan School of Information Systems Computer Science & Engineering 80 Stamford Rd 2260 Hayward St Singapore 178902 Ann Arbor, MI 48109-2121 USA sfcheng@smu.edu.sg wellman@umich.edu Abstract Elimination of a dominated strategy for one player may en- able new … Because players would never choose strictly dominated strategies, eliminating them from consideration should not affect the analysis of the game because this fact should be evi-dent to all players in the game. If a single set of strategies remains after eliminating all strictly dominated strategies, then we have a prediction for the game’s outcome. Iterated elimination of strictly dominated strategies cannot solve all games. EDIT: This is a Matrix that shows what I'm talking about: Examples show that this result is tight. Applying the Iterated Elimination of Strictly Dominated Strategies (IESDS) to a game resulted with the same solution of the Nash Equilibrium. This is called twice iterated elimination of strictly dominated strategies. In general, if a player is rational and knows that the other players are also rational (and the payoffs are as given), then he must play a strategy that survives twice iterated elimination of strictly dominated strategies. While iterated elimination of strictly dominated strategies seems to rest on rather firm foundation (except for the common knowledge requirement that might be a problem with more complicated situ-ations), eliminating weakly dominated strategies is more controversial because it is harder to argue that it should not affect analysis. Figure 4.2. Since these strategies Actually that specific "quadrant" of the matrix is the: Pareto optimal; Nash Equilibrium; Dominant strategies (through IESDS). Every Nash equilibrium survives iterated elimi-nation of strictly dominated strategies… Instead of the lengthy wording ‘the iterated elimination of strategies strictly dominated by a mixed strategy’ we write IESDMS. Iterated elimination of strictly dominated strategies (IESDS) The iterated elimination (or deletion) of dominated strategies (also denominated as IESDS or IDSDS) is one common technique for solving games that involves iteratively removing dominated strategies. The Order Independence of Iterated Dominance in Extensive Games Jing Chen CSAIL, MIT Cambridge, MA 02139, USA jingchen@csail.mit.edu Silvio Micali CSAIL, MIT Cambridge, MA 02139, If a mixture of two strategies strictly dominates a third strategy, you may eliminate the third strategy. Nash Equilibrium and Dominant Strategies Nash Equilibrium is a term used in game theory to describe an equilibrium where each player's strategy is optimal given the strategies of all other players. We offer a definition of iterated elimination of strictly dominated strategies (IESDS*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. 4.3 Exercises in Eliminating Dominated Strategies Apply the iterated elimination of dominated strategies to … Elimination of strictly dominated strategies will generally lack ... smaller than the set of strategies that survives iterated deletion of strictly dominated strategies. Backward induction eliminates incredible threats. In that case, one can rule out some Nash equilibria by eliminating weakly ... round of the iterated elimination of strictly dominated strategies. However the outcome of a successive eliminations may depend on the way in which weakly dominated actions are eliminated. Solution for In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? This problem does not arise for strictly dominated strategies: Proposition Iterated elimination of strictly dominated strategies produces the same nal residual game regardless of the order in which strategies are eliminated. elimination procedures used in (in)–nite games, for example, iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. Elimination of dominated strategies and inessential players 35 en game. each stage, weakly dominated strategies are eliminated.2 2A strategy is strongly dominated by a mixed strategy if and only if it is not a best response against any probability distribution on the opponents pro les. We used the proceeds of iterated elimination of dominated strategies, so we kept on cutting up the game to make it simpler and simpler to eventually find the optimal strategies for Piccola Osteria and Pizza Rosso, respectively. b) What is predicted by the iterated elimination of strictly dominated strategies? Now suppose the utility functions for Paul and John are given by 2 min {x, y}-x and 2 min {x, y}-y. nated, relative to opponents’ strategies which have not yet been eliminated. Proof If (a ;b ) is a strictly dominant strategy equilibrium, then in the IESDS process at stage 1 would eliminate all strategies except a and b , so (a ;b ) is the unique IESDS-equilibrium and hence the unique Nash-equilibrium. Abstract: We demonstrate that iterated elimination of strictly dominated strategies is an order dependent procedure. What are the pure-strategy Nash equilibria? And in order to do this we're going to look at an experiment that was done by Baldwin and Meese in the late 1970s and they were actually looking at social behavior in pigs. Leading fromStrength: Eliminating Dominated Strategies Strictly Dominated Strategy and Weakly Dominated Strategy Suppose si and s’i are two strategies for player i in a normal form game. Iterated elimination of dominated strategies: Eliminate all strictly (weakly) dominated strategies for all players in the original game. (Note this follows directly from the second point.) Iterated Deletion of Dominated Actions Iterated Deletion of Weakly Dominated Actions Iterated Elimination of Weakly Dominated Actions We can de ne iterated elimination of weakly dominated strategies. We also prove that order does not matter if strategy spaces are compact and payoff functions continuous. If so, delete these newly dominated strategies, and repeat the process until no strategy is dominated. Iterated deletion of strictly dominated strategies, or iterated strict dominance (ISD): after deleting dominated strategies, look at whether other strategies became dominated with respect to the remaining strategies. d) Find the best response functions. Under this condition, the best strategy for Player 2 is "Mid" giving his a payoff of "2" rather than "0" if he chooses "Left".Thus the unique solution for the game is (Up, Mid) giving a Payoff = (1,2) which is a Strictly Dominated IEDS Equilibrium.Note, it is Strictly Dominated Solution because all the strategies that were Eliminated were Strictly Dominated. Eliminate all strictly (weakly) dominated strategies for all players in the modified game where players cannot choose any strategy that was eliminated at Step 1. This example also illustrates that a Nash equilibrium can be in weakly dominated strategies. Then results that strategy R is dominated Iterated elimination of strictly dominated strategies (or iterated dominance: Example 1 Player 2 L R T 2,3 5,0 Player 1 M 3,2 1,1 Then player 1 knows that player 2 will never use R, so can evaluate her strategies only against strategy L. Player 2 L T 2,3 Player 1 M 3,2 example is the linear Cournot duopoly. An example of backward induction strategies ww and wc for LJ. In the above case, eliminations of the dominated strategies for girl 2 and of boy 3 as an inessential player constitute this process to obtain the 2-person battle of the sexes. L C R T 2,0 1,1 4,2 M 3,4 1,2… If iterated elimination of strictly dominated strate-gies yields a unique strategyn-tuple, then this strategyn-tuple is the unique Nash equilibrium (and it is strict). We have then the following counterpart of the IESDS Theorem 2, where we refer to Nash equilibria in mixed strategies. (Note that there are no other strictly dominated strategies in the game in the video.) We have called c;ww an incredible threat. e) State all Nash Equilibria of the game. Let's take a peek at a game now where we can begin to see whether iterative elimination of a strictly dominated strategies has any bite in, in application. Monotonicity* requires a Monotonicity property along any elimination path. c) What is predicted by the iterated elimination of weakly dominated strategies? Dominated Strategies & Iterative Elimination of Dominated Strategies 3. Iterated . Back to Game Theory 101 In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? A Nash Equilibrium exists when there is no unilateral profitable deviation from any of the players involved . Nash Equilibrium Dominant Strategies • Astrategyisadominant strategy for a player if it yields the best payoff (for that player) no matter what strategies the other players choose. What does this imply? Keywords: game theory, iterated strict dominance, order independence JEL code: C72 Iterated Delation of Strictly Dominated Strategies Iterated Delation of Strictly Dominated Strategies player 2 a b c player 1 A 5,5 0,10 3,4 B 3,0 2,2 4,5 We argued that a is strictly dominated (by b) for Player 2; hence rationality of Player 2 dictates she won’t play it. It was slightly more complicated than the first one we saw.
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