Dominant Strategy

4 Key excerpts on "Dominant Strategy"

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  eBook - ePub

  Game Theory in the Social Sciences

  A Reader-friendly Guide

  • Luca Lambertini(Author)
  • 2011(Publication Date)
  • Routledge

    (Publisher)

  This refinement is based on an idea that we have already encountered at the outset of this chapter, that is, the possibility that a player's best reply be invariant to the rivals’ strategies. If this sort of automatic pilot does work, the resulting best reply qualifies as that player's Dominant Strategy. The rigorous definition is as follows.

  DEFINITION 3.2 Strategy i ∈ Si is (at least weakly) dominant for player i if it maximizes his/her payoff πi ( i , s− i ) irrespective of the opponents’ behaviour s− i , that is, if and only if

  If the above inequality holds strictly, then i is strictly dominant; otherwise, if it holds as an equality for at least some admissible s− i , then it is weakly dominant.

  In agreement with Definition 3.2, if all players do have a Dominant Strategy (or at least one that is weakly so) and adopt it, the outcome identified by the combination of (at least weakly) dominant strategies will qualify as an equilibrium in dominant strategies, as follows.

  DEFINITION 3.3 Given a game G ≡ N, Si ,πi (s) , the outcome ≡ ( 1 , 2 ,..., N ), with i ∈ Si for all i ∈ N, is an equilibrium in (at least weakly) dominant strategies if and only if

  This concept deserves a few comments. In particular, it is worth dwelling upon the difference between the condition that must be satisfied for an outcome to be a Nash equilibrium and the one contained in Definitions 3.2 and 3.3 in order for a Dominant Strategy equilibrium to arise. While the former requires a player to identify the best reply to every possible strategy chosen by any rival – whereby no ex post regrets may exist – the latter poses a much stronger requirement, which consists in finding a strategy yielding a payoff systematically at least as high as that for any other strategies the same player could adopt, no matter what the opponents do. Relatedly, as we shall see below, the existence of an at least weakly Dominant Strategy for a player does not necessarily entail that this player's best reply will be unique. To put it differently, this indeed entails that we commonly observe games with several Nash equilibria in pure strategies, generated by the crossing of best replies, and, if we're lucky, one of these equilibria will emerge as an equilibrium in at least weakly dominant strategies. Equivalently, we may say that while every Dominant Strategy equilibrium is also a Nash equilibrium, the opposite is not true. In the light of these considerations, it should be clear that the dominance criterion is much more demanding than the no ex post

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  Behavioral Game Theory

  Experiments in Strategic Interaction

  • Colin Camerer(Author)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  5

  Dominance-Solvable Games

  DOMINANCE IS THE MOST BASIC PRINCIPLE Strategy A strictly dominates B if the payoff from choosing A is higher than the payoff from B, for any strategy choice by other players' strategy. A weakly dominates B if A's payoffs are higher for some choices by others, and never lower.1 Dominance is extremely appealing because, if A dominates B, A will turn out as least as good as B no matter what you think other players will do. This also means that you should choose a Dominant Strategy over a dominated one even if you don't know what other players' payoffs are, or how rational they are.

  Assuming that other players obey dominance gives a player a way to make a simple, conservative guess about what others will do. Assuming others obey dominance can then enable a player to infer that certain payoffs from her own undominated strategies will never be realized—because they come about only if other players violate dominance. This inference can make a strategy that is initially undominated in effect dominated. Dominance can therefore be applied iteratively: First eliminate dominated strategies for all players; then check whether that first round of elimination makes some (initially undominated) strategies dominated; eliminate those (iteratedly) dominated strategies, and repeat. Games in which this process of iteratively deleting dominated strategies leads to a unique equilibrium are called “dominance solvable.”

  Two examples will illustrate. Suppose driving the wrong way down a one-way street is akin to a violation of dominance. Then a pedestrian who thinks drivers obey dominance will expect cars to come from only one direction—the correct one—and need look only one way for oncoming cars. Looking both ways before crossing one-way streets therefore implies that the pedestrian thinks the driver may violate dominance.

  The importance of a second level of mutual reasoning is illustrated by the words painted on the back of some large eighteen-wheeler trucks. The words are not painted very large, so drivers with normal vision (which excludes Superman and Mr. Magoo) can see them only as they get close to the truck. The words are: “If you can see this, I can't see you.” That means drivers who are close enough to read the words are in the truck driver's “blind spot” (an area behind the truck not visible in the truck driver's side rear-view mirrors). The words install knowledge in the car driver's head about what the truck driver knows (or actually, what the truck driver doesn't

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  Game Theory

  A Nontechnical Introduction to the Analysis of Strategy

  • Roger A McCain(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  Table 3.12 shows the best responses for Dr. Boingboing, depending on Prof. Heffalump's strategy choice. What we see is that Dr. Boingboing will want to choose a different strategy when Dr. Heffalump chooses a 400 page text than he will want to choose if Prof. Heffalump chooses a 600 or 800 page text. Dr. Boingboing's idea is to write a text just one step longer than Dr. Heffalump's text, if he can. It follows that there is no one strategy that is Dr. Boingboing's best response to each of the different strategies Prof. Heffalump might choose. In other words, there is no Dominant Strategy for Dr. Boingboing. And since the game is symmetrical, we can reason in just the same way and find that there is no Dominant Strategy for Prof. Heffalump either.

  Table 3.12. Best Responses for Dr. Boingboing.

  So the textbook writing game gives us an example of a game in which there is no Dominant Strategy. If we are to find a “solution” for a game like this one, it will have to be a different kind of solution. We will go on to investigate that in the next chapter. So we set this example aside for now, and will return to it in Chapter 4 .

  9. SUMMARY

  One objective of game theoretic analysis is to discover stable and predictable patterns of interactions among the players. Following the example of economics, we call these patterns “equilibria.”

  Since we assume that players are rational, their choices of strategies will only be stable if they are best response strategies — the player's best response to the other players’ strategies. If there is one strategy that is the best response to every strategy the other player or players might choose, we call that a Dominant Strategy. If every player in the game has a Dominant Strategy, then we have a Dominant Strategy equilibrium.

  A Dominant Strategy equilibrium is a noncooperative equilibrium, which means that each player acts independently, not coordinating the choice of strategies. If the players in the game are able to commit themselves to a coordinated choice of strategy, the strategies they choose are called a cooperative equilibrium. It is possible that the cooperative equilibrium may be the same as a Dominant Strategy equilibrium, but then again it may not be.

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  The Economics of Tourism

  • Mike J. Stabler, Andreas Papatheodorou, M. Thea Sinclair(Authors)
  • 2009(Publication Date)
  • Routledge

    (Publisher)

  an optimal position can be achieved by an enterprise irrespective of what its rivals choose to do. This basic notion in game theory is illustrated by the case of a tour operator deciding whether to advertise. The example presupposes that advertising is both competitive by each tour operator to capture a larger market share as well as informative to extend the market and increase the pay-off for both firms. Figure 4.9 shows a typical 2×2 matrix in a two person (firm) game representing the duopoly form of oligopoly, in which the pay-offs to tour operator X with respect to advertising or not advertising (rows) are given in bold while those for tour operator Y (columns) are in italics in parentheses. Tour operator X will advertise as this is the best strategy (best net pay-off) which can be adopted irrespective of what Y does. If Y does not advertise the pay-off for X is 25, whereas if Y does advertise X will achieve a pay-off of 20. Likewise for Y, the pay-off is respectively 10 if both advertise and 15 if X does not advertise. It can be seen, therefore, that both will advertise where the total market pay-off is 30 (20X, 10Y) in the top left cell. This position is a stable one. Figure 4.9 Advertising: Dominant Strategy case If X does not have a Dominant Strategy then the optimal decision depends on what Y does. For example, in the advertising case given in Figure 4.9, if the pay-off to X is 40 if neither advertises (bottom right cell) then X’s strategy is determined by what Y does. If Y advertises then so must X; nonetheless, if Y does not advertise, neither should X because the pay-off is much greater in adopting this strategy. Therefore, X must guess what Y will do. As Y has the same Dominant Strategy as before it will obviously advertise because whatever X does this is its best action. If X correctly guesses Y’s action, which it should given the assumptions of the model, then a stable equilibrium can still be attained

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