Answer: True. Weakly Max Dominated Strateges. For each player i, choose a subset Si of her set Ai of actions 3. Hence, a strategy is dominated if it is always better to play some other strategy, regardless of what opponents may do. Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy. In particular, each must yield the same expected payo . (a) [5 points] \A strictly dominated strategy can never be a best response." Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy. A strategy of player is weakly max dominated if for every strategy profile of the other players there is a strategy such that . A strategy is strictly dominated if choosing it always gives a worse outcome than choosing an alternative strategy, regardless of which moves other players make. The reason is that a strategy is dominated against all pure strategies of the opponents if and only if it is dominated against all their mixed strategies. Hence, a strategy is strictly dominant if it is always strictly better than any other strategy, for any profile of other players' actions. D vs. [εC + (1-ε)D] à (1-ε)1 + ε3 = (1-ε)+3ε C vs. [εC + (1-ε)D] à (1-ε)0 + ε2 = 2ε (1- Strategy B is strictly dominated if some other strategy exists that strictly dominates B. •A similar argument applies to player 2. L M R T 10 3 0 0 7 1 B 0 0 10 3 7 1 Note that neither the pure strategy Lnor Mis strictly dominated by R. The strategy which puts probability 1 So this procedure can be stronger than IESDS and IEWDS. The option elids performs an iterative elimination of all strictly dominated strategies. Eliminate strictly dominated actions from the game 2. Here no strategy is strictly or weakly dominated. Call it strategy2. Suppose there is some mixed strategy ms1 that assigns strategy1 non-zero probability p, and probability 1-p to some other mix of Rational players never play strictly dominated strategies, because such strategies can never Therefore, strategy D is strictly dominated by the mixed strategy (1/2, 1/2, 0). Strategy B is weakly dominated if some other strategy exists that weakly dominates B. Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game I For sequential games it is important to distinguish strategies from decisions or actions. I It turns out that in 2-player games, the two concepts coincide. It is also notable that even if is strictly dominated by a mixed strategy it is also max dominated. In the PD game, D strictly dominates C, and C is strictly dominated by D. Observe that we are using expected utilities even for the pure-strategy profiles because they may involve chance moves. Indeed, A is a unique best response to X and B is a unique best response to Y. Example 17 p. 241 1\2 10 10 4 4 24 3 0 510 2 → 1\2 10 10 4 0 510 2 Notice that neither player has any strictly dominated strategies, if we consider only pure strategies. (Dominated strategy) For a player a strategy s is dominated by strategy s 0if the payo for playing strategy s is strictly greater than the payo for playing s, no matter what the strategies of the opponents are. Eliminating a strictly dominated strategy should not affect the analysis of the game because this fact should be evident to all players in the game (common knowledge assumption). This notion can be generalized beyond the comparison of two strategies. If so, there must be another Nash equilibrium in mixed strategies. But a mixed that uses only undominated strategies can be dominated by a pure. ES strategies in the simple example • Is defection ES? Once strategy D is deleted because of being strictly dominated by the above mixed strategy, we have a (reduced) normal form game given by . B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory. I am looking at the proof of NE survives the iterated removal of strictly dominated strategies (MWG, ex 8.D.2) and in the solution manual, authors say something like if a mixed strategy is strictly dominated (by another mixed strategy say s), then we can find a pure strategy in the support of that strictly dominated mixed strategy that is strictly dominated (by s). Since it's dominated, there is some strategy that dominates it. Definition (Strict Domination by Mixed Strategies) An action si is strictly dominated if there exists a mixed strategy σ i ∈ Σi such that ui (σ i, s −i) > ui (si, s −i), for all s −i ∈ S −i. The problem is to define a Linear Program for strictly dominated strategy and never best response. Suppose that player i has a strictly dominated strategy s i: Explain why s i cannot be played in any pure strategy Nash equilibrium of the game G: Explain also why s i cannot be played with positive probability in any mixed strategy Nash equilibrium of G: b) Consider the game with payo⁄s as depicted in the table below. Also, if one strategy is strictly dominant, than all others are dominated. Call it strategy1. Strategy: A complete contingent plan for a player in the game. Show how you obtain this Nash equilibrium in mixed strategies. The main point I cannot get is how to iterate through infinite number of mixed strategies and how to convert the definition with sign equal for LP. A strictly dominant strategy is always played in equilibrium, and thus strictly dominated strategies never are. A strictly (weakly) dominated (dominant) strategy; Rational players do not adopt a strictly dominated strategy, hence iterative deletion of strictly dominated strategies (IDSDS). Definition Max-dominated strategies. Answer: True. B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. 2. to: A pure strategy, si, of player i is dominated if there is some strategy, pure or mixed, Ti, for player i, such that ri gives plunge i a strictly higher expected payoff than Si against all strategy … (Strategy A strictly dominates B). The strategy that strictly dominates it, by de nition, yields a strictly higher payo against all strategies and hence is a better response. In mixed strategies, L for player 2 is strictly dominated by a strategy that mixes between the pure strategies C and R (for instance: 0, 1/2, 1/2) -- L is never the best response. I This follows from the earlier comment that a strictly dominated strategy is never a best response. Mixed strategies; Continuous strategies: Cournot duopoly; Dominant strategies are considered as better than other strategies, no matter what other players might do. Main Lesson If a mixed strategy is a best response then each of the pure strategies involved in the mix must itself be a best response. However, M is strictly dominated by the mixed strategy (1 2,0, 1 2). Can a mixed strategy be strictly dominated, if it assigns positive probabilities only to pure strategies that are not even weakly dominated? If s i is a strictly dominant strategy, then si is a player’s strictly best response to any strategies the other players might pick, even to wildly irrational actions of the other players. Check if there is a mixed strategy profile α that (1) assigns positive probability only to actions in Si, and (2) satisfies the two conditions in the previous characterization. If si is a strictly dominated strategy for player i, then it cannot be a best response to any s i2S i. Clearly, the above game is solved by an iterated elimination of never best responses. Suppose there is some strategy that is dominated. A Mixed Strategy Dominated by a Pure Mixed strategies that use strictly dominated pure strategies are strictly dominated. The outcome does not change if we eliminate strictly dominated mixed strategies at every step. (b) [5 points] \In the candidate-voter model, if two people are standing, one to the left of This notion can be generalized beyond the comparison of two strategies. If a player has a strictly dominant strategy, than he or she will always play it in equilibrium. In this game, there are two strategy combinations that remain after elimination of strictly dominated strategies, and that they are Nash equilibrium combinations. A strategy ∈ of player is max-dominated if for every strategy profile of the other players − ∈ − there is a strategy ′ ∈ such that (′, −) > (, −).This definition means that is not a best response to any strategy profile −, since for every such strategy profile there is another strategy ′ which gives higher utility than for player . Before explaining why this must be true, let’s just try to rewrite this lesson formally, using our new notation: More Formal statement of the Same Lesson. On the other hand C is a never best response, that is, it is not a best response to any strategy of the opponent. Also, strategy s i is strictly dominated by s i. A strategy is weakly dominated if choosing it always gives an outcome that is as good as or worse than choosing an alternative strategy. IDSDS has “no bite.” Intermediate Microeconomic Theory 23 Player 2 Heads Tails Player 1 Heads 1,−1 −1,1 (Strategy A strictly dominates B). •No player has strictly dominated strategies. (To show it is strictly dominated, best to find the (possibly mixed) strategy strictly dominating) 1 No strategy of player 1 is dominated; clearly, it’s a best reply for some strategy of player 2. If a strictly dominated strategy is played with positive probability, the player can improve his payoff by reducing the probability of playing the dominated strategy and increasing the probability of playing the dominating strategy. I In this sense, rationalizability is (weakly) more restrictive than iterated deletion of strictly dominated strategies. In n-player games (n >2), they don’t have to. Thus, it is possible to describe the iterated elimination of strongly dominated strategies as the iterative elimination of strategies that are not a best response (see Myerson, 1991, 88{89). Method for finding all mixed strategy NE 1. •Player 1 does not find any strategy strictly dominated: •She prefers Heads when player 2 chooses Heads, but Tails when player 2 chooses Tails. 2 But R is dominated for player 2, so provided each player is rational, it is never played. es.Y [Example on p. 7 of S&T] When is a pure strategy weakly dominated? èA strictly dominated is never Evolutionarily Stable – The strictly dominant strategy will be a successful mutation 2,2 0,3 3,0 1,1 C D Cooperate Defect ε 1- ε Player 1 Player 2 1- ε For C being a majority For D being a majority 14. 4. T hen, represent the remaining (undeleted) strategies. Delete the strictly dominated strategies for player 1 that you found in the previous question. Exercise 16 Exercise 8.B.6 If a pure strategy is strictly dominated, then so is any mixed strategy that plays with positive probability. 3. 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. A weakly dominated strategy can be played with positive probability in a Nash equilibrium strategy. Example of a dominated strategy . If a player has a dominant strategy than all others are dominated, but the converse is not always true. 2.
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