Mixed Strategy

12 Key excerpts on "Mixed Strategy"

• 

  eBook - PDF

  Microeconomic Theory

  Basic Principles and Extensions

  • Walter Nicholson, Christopher Snyder(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  For example, a history professor might decide to ask an exam question about World War I because, unbeknownst to the students, she recently read an interesting journal article about it. See John Harsanyi, “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed-Strategy Equilibrium Points,” International Journal of Game Theory 2 (1973): 1–23. Harsanyi was a co-winner (along with Nash) of the 1994 Nobel Prize in economics. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 Chapter 8: Game Theory 257 In our terminology, a Mixed Strategy is a general category that includes the special case of a pure strategy. A pure strategy is the special case in which only one action is played with positive probability. Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies . Returning to the examples from the previous paragraph of mixed strategies in the Battle of the Sexes, all four strategies (1/3, 2/3), (1/2, 1/2), (1, 0), and (0, 1) are mixed strategies. The first two are strictly mixed, and the second two are pure strategies. With this notation for actions and mixed strategies behind us, we do not need new defi-nitions for best response, Nash equilibrium, and dominant strategy. The definitions intro-duced when s i was taken to be a pure strategy also apply to the case in which s i is taken to be a Mixed Strategy. The only change is that the payoff function U i 1 s i , s 2 i 2 , rather than being a certain payoff, must be reinterpreted as the expected value of a random payoff, with probabilities given by the strategies s i and s 2 i . Example 8.2 provides some practice in computing expected payoffs in the Battle of the Sexes. EXAMPLE 8.2 Expected Payoffs in the Battle of the Sexes Let’s compute players’ expected payoffs if the wife chooses the Mixed Strategy (1/9, 8/9) and the husband (4/5, 1/5) in the Battle of the Sexes.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Mixed strategies Chapter summary Given a game in strategic form we extend the strategy set of a player to the set of all probability distributions over his strategies. The elements of the new set are called mixed strategies, while the elements of the original strategy set are called pure strategies. Thus, a Mixed Strategy is a probability distribution over pure strategies. For a strategic-form game with finitely many pure strategies for each player we define the mixed extension of the game, which is a game in strategic form in which the set of strategies of each player is his set of mixed strategies, and his payoff function is the multilinear extension of his payoff function in the original game. The main result of the chapter is the Nash Theorem, which is one of the milestones of game theory. It states that the mixed extension always has a Nash equilibrium; that is, a Nash equilibrium in mixed strategies exists in every strategic-form game in which all players have finitely many pure strategies. We prove the theorem and provide ways to compute equilibria in special classes of games, although the problem of computing Nash equilibrium in general games is computationally hard. We generalize the Nash Theorem to mixed extensions in which the set of strategies of each player is not the whole set of mixed strategies, but rather a polytope subset of this set. We investigate the relation between utility theory discussed in Chapter 2 and mixed strategies, and define the maxmin value and the minmax value of a player (in mixed strategies), which measure respectively the amount that the player can guarantee to himself, and the lowest possible payoff that the other players can force on the player. The concept of evolutionary stable strategy, which is the Nash equilibrium adapted to Darwin’s Theory of Evolution, is presented in Section 5.9.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Intermediate Microeconomics

  An Intuitive Approach with Calculus

  • Thomas Nechyba(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In simultane-ous move games, however, pure strategies have involved picking an action, but this is not true for mixed strategies . Consider the following game. Two people, James and Abbie, are asked to put a one-euro coin on a table. If their coins match in the sense that they both have the same side of the coin showing, James ends Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 738 CHAPTER 24 STRATEGIC THINKING AND GAME THEORY up getting Abbie’s coin. If, on the other hand, the coins do not match, i.e. one shows Heads and the other Tails , Abbie gets James’ euro. This simple game, known as matching coins , is illustrated in Table 24.9. In this game there is no pure strategy Nash equilibrium; James’ best response to any move of Abbie’s is to match it while Abbie’s best response to any move of James is to contradict it. In such a game, there is no way to predict for sure what will happen because the very structure of the game prohibits such predict-ability. A common way to think of this formally is through the use of mixed strategies. A Mixed Strategy for a player is a probability distribution over the pure strategies . Even though we will only explore mixed strategies for simultaneous move games, the same definition holds for sequential move games. James has two pure strategies in the matching coins game: Heads and Tails . A Mixed Strategy is a set of two probabilities ( r ,1 2 r ) such that 0 # r # 1. If James decide to play the Mixed Strategy (0.5,0.5), it means that he will play Heads with probability 0.5 and Tails with probability 0.5.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  The Mathematics of Collective Action

  • James Coleman(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  A pure strategy here would be a strategy with the probability of acting as leader equal to 0 or 1, which means that conditional probabilities of action are 0 or 1. In the single-play game, a pure strategy meant that the probability of A i was 1.0 for some i, 0 for all others. Here, the pure strategy means that the probability of A i given A i on the last move, or given a particular B j, is 1.0. A Mixed Strategy in the present sense means the conditional probabilities are less than 1: the actor acts both as leader and as follower. The optimal strategy employed in this game at the present level of analysis is one in which each actor selects a probability mixture of 0.5 of being follower and 0.5 of being leader. A strategy which carried the analysis to a deeper level would be to allow this probability itself to change over time as a function of the other’s tendency to take the leader or follower role, or to bias the probability of taking the leader’s role to begin with, to inform the other that one was taking the leader or follower role. The first of these deeper-level strategies could be described as a sensitivity to the other, or following the other, at a deeper level—a sensitivity not to his preceding action, but to a conditional probability inferred from it. The second could be described as leading the other in a deeper sense than before: informing the other that one is not likely to change one’s action or is quite ready to change one’s action. These deeper-level strategies begin to sound somewhat more like the actual considerations that persons use in social situations—not only where there is complete coincidence of interest, but also where there is conflict of interest (where the use of one deeper-level strategy or the other may have differentially beneficial consequences to the two parties)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Principles of Automated Negotiation

  • Shaheen Fatima, Sarit Kraus, Michael Wooldridge(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  Most conflict situations are in fact non-zero-sum games. The famous Pris- oner’s Dilemma is a non-zero-sum game. Figure 2.2 illustrates the Prisoner’s Dilemma game in normal form. The two players simultaneously choose one of the two possible strategies confess or deny, and the utilities they get from each of the four possible choice combinations are as shown in the figure. (We discuss this fascinating game in more detail in Section 2.5.1.) In a normal-form game, Player i can choose to play any strategy from S i . Al- ternatively, i can randomise over the strategies in S i and play a Mixed Strategy. In the next section we will provide details about these two alternatives. 2.2 Pure and mixed strategies Recall that, for a game G, S i denotes the set of player i’s strategies. Techni- cally, we say that each element of S i is a pure strategy. In contrast, a Mixed Strategy is obtained by randomly choosing between a set of, say m, pure strate- gies (Osborne and Rubinstein, 1994, pp. 31–32). Thus, mixed strategies are an extension of pure strategies. Mixed strategies are important because, for some 2.3 Rational behaviour in strategic settings 21 games, although a solution cannot be determined in terms of pure strategies, it can be if we allow mixed strategies. Formally, a Mixed Strategy is defined as follows. Let there be m strategies in S i . Then, a randomised or Mixed Strategy for i is a probability distribution over the set S i . We will let MS i denote the set of all possible mixed strategies for Player i, where MS i = {( p i,1 ,..., p i,m ) ∈ (0, 1) m | m ∑ k=1 p i,k = 1, p i,k ≥ 0, ∀k ∈ P}. Thus, Player i will play strategy s i,k with probability p i,k . A strategy ( p i,1 ,..., p i,m ) for Player i is called pure if there is a k with p i,k = 1. In other words, in a pure strategy one of the choices is selected with certainty.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus and Techniques of Optimization with Microeconomic Applications

  • John Hoag(Author)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  We extend the idea of a Nash equilibrium to games without a pure strategy equilibrium. 282 Calculus and Techniques of Optimization with Microeconomic Applications MIXED STRATEGIES XI.5. Definition An n -person game in normal form involves n persons. Each person has a set of pure strategies she could play, S i . Each player has a payoff function, U i ( x 1 , x 2 , . . . , x n ) for the i th player, where the payoff functions depend on the choices of other players, and, for each player, a set of pos-sible probability densities over the strategies that player could play. XI.6. Definition A Mixed Strategy Nash equilibrium is a set of probability den-sities, one for each player, over the alternative possible strate-gies the player could play so that the expected utility for each player is maximum given that the other players play their Nash probability density. Remark First, this idea of equilibrium is distinguished from the case where we do not need to rely upon probability densities to decide what strategy to play. If we can find an equilibrium without using probabilities, we say we have an equilibrium in pure strategies or a pure strategy equilibrium . The definition of a Mixed Strategy Nash equilibrium is quite general; in the case where there is a pure strategy equilibrium as in the first game we considered, the probabilities would be 1 on the pure strategy giving rise to equilibrium and 0 on all other strategies. Can we be sure that there is always a Nash equilibrium of this variety? The answer is yes; that is in fact what Nash proved. This kind of equilibrium is called a Mixed Strategy equilibrium . XI.7. Theorem (Nash’s Theorem) In any finite game, there is at least one Nash equilibrium which may be a Mixed Strategy equilibrium. Remark We will not prove this theorem as it employs a tool just beyond the content of these notes, a fixed point theorem. The proof is relatively straightforward, given that theorem, and is outlined in Gibbons (1992, p. 45ff).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Decision Sciences

  Theory and Practice

  • Raghu Nandan Sengupta, Aparna Gupta, Joydeep Dutta, Raghu Nandan Sengupta, Aparna Gupta, Joydeep Dutta(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  2.13 Formal Definition of a Mixed-Strategy Equilibrium Let the Mixed Strategy employed by player i be denoted by σ i , which randomly chooses action a i ∈ A i with probability σ i ( a i ) . For instance, in the BoS game, σ b ( G ) = p and σ b ( M ) = 1 − p . Let each σ i ∈ i , where i is the space of probability distributions over the action set A i of the i th user. Let σ = ( σ 1 , σ 2 , . . . , σ n ) denote the mixed-strategy profile employed by the n players, where σ ∈ defined as = 1 × 2 × · · · × n , the Cartesian product of the mixed strategies of all the users. This is similar to the outcome space A in the context of pure strategies. The average payoff to the i th user for the mixed-strategy profile σ is given as u i ( σ ) = a ∈ A ⎛ ⎝ n j = 1 σ j ( a j ) ⎞ ⎠ u i ( a ) . (2.8) where a = ( a 1 , a 2 , . . . , a n ) denotes an action profile such that a i ∈ A i . Thus, the above computation essentially weights the u i ( a ) to each user i for the action profile a , with the probability that a is chosen randomly corresponding to the mixed-strategy profile σ . Further, similar to the case of pure Introduction to Game Theory 101 strategies, the quantity u i ( σ ) can be more naturally represented in the context of game theory as u i ( σ ) = u i (( σ 1 , σ 2 , . . . , σ n )) = u i ⎛ ⎜ ⎝ σ i , ( σ 1 , σ 2 , . . . , σ i − 1 , σ i + 1 , . . . , σ n ) σ − i ⎞ ⎟ ⎠ = u i ( σ i , σ − i ) , where σ − i represents the strategy profile of all the users other than the i th user. A strategy profile σ = ( σ 1 , σ 2 , . . . , σ n ) ∈ is a mixed-strategy Nash equilibrium if u i ( σ i , σ − i ) ≥ u i ( σ i , σ − i ) , for all mixed strategies σ i ∈ i of the i th user and for all users 1 ≤ i ≤ n . This basically means that no user can benefit by unilaterally deviating from his Nash strategy σ i to any strategy σ i ∈ i .

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Behavioral Game Theory

  Experiments in Strategic Interaction

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

    (Publisher)

  3

  Mixed-Strategy Equilibrium

  IN GAMES WITH MIXED -STRATEGY EQUILIBRIA (MSE) players are predicted to choose probabilistic “mixtures” in which no single strategy is played all the time. Common examples are zero-sum games in which your win is my loss—as in sports, perhaps war and diplomacy, and some other domains. In these games, if I always choose a particular strategy, and you anticipate that strategy, then you will win; so I shouldn't behave so predictably. The only equilibrium will involve unpredictable mixing. Randomizing is also sensible when a little genuine unpredictability will deter another player from doing something you dislike. (Think of random searches of passengers boarding U.S. airlines after the September 11, 2001, attacks, or the liquor store in the South with a large sign outside—“This establishment is guarded by a vicious dog three nights a week…. See if you can guess which nights.”)

  Games with mixed equilibria do not have some of the complications of other games described in this book. The games reported in this chapter do not have multiple equilibria, so there is no question about which equilibria are selected (as in Chapter 7 ). And in constant-sum games it is not possible for one player to help another without hurting herself, so social preferences and reciprocation (recall Chapter 2 ) play no role.

  Furthermore, in constant-sum MSE games the maximin solution—in which players just choose strategies that maximize the minimum they can get—is also a Nash equilibrium.1 Maximin leads straight to Nash equilibrium in zero-sum games because players' interests are strictly opposed: If others will do whatever they can to get the most, their actions will give me the least, so I should try to maximize the least I can get. Since maximin is a particularly simple decision rule and coincides with Nash, in these games there is a good chance that the Nash equilibrium will describe what players do in these games.

  That's the good news. The bad news is that, from a behavioral point of view, MSE games raise difficult challenges for Nash equilibrium and for learning.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  An Introductory Course on Mathematical Game Theory

  • Julio González-Díaz, Ignacio García-Jurado, M. Gloria Fiestras-Janeiro(Authors)
  • 2010(Publication Date)
  • American Mathematical Society

    (Publisher)

  Although we have referred above to the mixed extension of a game as a “theoretical trick”, mixed strategies are natural in many practical situa-tions. We briefly discuss this point in the following example (and also later in this book), in which we informally introduce the mixed extension of a strategic game. After the example, we provide the formal definition. For a much more detailed discussion on the importance, interpretations, and appropriateness of mixed strategies, Osborne and Rubinstein (1994, Sec-tion 3.2) can be consulted. Example 2.4.1. Consider the matching pennies game (see Example 2.2.6). Suppose that the players, besides choosing E or O , can choose a lottery L that selects E with probability 1/2 and O with probability 1/2 (think, for instance, of a coin toss). The players have von Neumann and Morgenstern utility functions, i.e. , their payoff functions can be extended to the set of Mixed Strategy profiles computing the mathematical expectation (Defini-tion 1.3.4). Figure 2.4.1 represents the new game we are considering. Note E O L E 1, − 1 − 1, 1 0, 0 O − 1, 1 1, − 1 0, 0 L 0, 0 0, 0 0, 0 Figure 2.4.1. Matching pennies allowing for a coin toss. that the payoff functions have been extended, taking into account that play-ers choose their lotteries independently (we are in a strategic game). For instance: u 1 ( L , L ) = 1 4 u 1 ( E , E ) + 1 4 u 1 ( E , O ) + 1 4 u 1 ( O , E ) + 1 4 u 1 ( O , O ) = 0. Observe that this game has a Nash equilibrium: ( L , L ) . The mixed exten-sion of the matching pennies is a new strategic game in which players can choose not only L , but also any other lottery over { E , O } . It is easy to check that the only Nash equilibrium of the mixed extension of the matching pen-nies is ( L , L ) . One interpretation of this can be the following. In a matching pennies situation it is very important for both players that each one does not have any information of what will be his final choice ( E or O ). In order

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  This is in marked contrast to the maxmin strategy of a player, which is determined solely by the player’s own payoffs. This is yet another expression of the significant difference between the solution concepts of Nash equilibrium and maxmin strategy, in games that are not two-player zero-sum games.  162 Mixed strategies 5.2.4 Dominance and equilibrium The concept of strict dominance (Definition 4.6 on page 86) is a useful tool for computing equilibrium points. As we saw in Corollary 4.36 (page 109), in strategic-form games a strictly dominated strategy is chosen with probability 0 in each equilibrium. The next result, which is a generalization of that corollary, is useful for finding equilibria in mixed strategies. Theorem 5.20 Let G = (N, (S i ) i ∈N , (u i ) i ∈N ) be a game in strategic form in which the sets (S i ) i ∈N are all finite sets. If a pure strategy s i ∈ S i of player i is strictly dominated by a Mixed Strategy σ i ∈  i , then in every equilibrium of the game, the pure strategy s i is chosen by player i with probability 0. Proof: Let s i be a pure strategy of player i that is strictly dominated by a Mixed Strategy σ i , and let  σ = (  σ i ) i ∈N be a strategy vector in which player i chooses strategy s i with positive probability:  σ i (s i ) > 0. We will show that  σ is not an equilibrium by showing that  σ i is not a best reply of player i to  σ −i . Define a Mixed Strategy σ  i ∈  i as follows: σ  i (t i ) =   σ i (s i ) · σ i (s i ) t i = s i ,  σ i (t i ) +  σ i (s i ) · σ i (t  i ) t i = s i . (5.52) In words, player i , using strategy σ  i , chooses his pure strategy in two stages: first he chooses a pure strategy using the probability distribution  σ i . If this choice leads to a pure strategy that differs from s i , he plays that strategy. But if s i is chosen, player i chooses another pure strategy using the distribution σ i , and plays whichever pure strategy that leads to.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Graphs, Dynamic Programming and Finite Games

  • Kaufmann(Author)
  • 1967(Publication Date)
  • Academic Press

    (Publisher)

  The assumptions will be such that only pure strategies can have a concrete meaning, and we shall compare the intuitive methods of two actual teams of players to a system based on the theoretical concepts which have been discussed in the preceding pages. The second example, which is of a completely different type, will give substance to the concept of a Mixed Strategy in the case of a duel: one between the police and thieves. A. EXAMPLE 1: A SIMPLIFIED BUSINESS GAME^ Let us remember that “a business game” is the artificial reproduction of the activities of several firms who are competitors in the same market. In it, the relation of cause and effect, both between the firms, and also between them and the market, is reproduced by means of a mathematical model. In a business game the teams are allowed to receive certain information and have to take decisions for a series of stages, each of which may be of 3-6 months’ duration. In our example the duration is of one year. The reader should not be surprised at the choice of a very simplified example, since we are more concerned with explaining how a certain form of reasoning can be used than with applying it to cases which are more general and closer to reality. Where business games are detailed and complex enough to give a simulation of real situations,2 the use of This example is taken from the thesis by Simon-Pierre Jacot, Strate’gie et con- currence, Facultk de Droit et des Sciences Cconomiques de Lyon, April, 1961. In this work, one of the best on the subject, the reader will find a very detailed study of the employment of the theory of games in the analysis of competition. See, for example, the model OMNILOC in A. Kaufmann, R. Faure, and A. Le GarfT, Les jeux d’entreprises, Series Que sais-je ?, Presses Universitaires de France, 1950. 170 111. THE THEORY OF GAMES OF STRATEGY the theory of games of strategy for the purpose of obtaining optimal strategies has often proved difficult, and even impossible.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Game Theory in Business and Economics

  • Thomas J. Webster(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  Game theory is a powerful tool for analyzing strategic situations precisely because a unique Nash equilibrium does, in fact, exist for all games, although not always in pure strategies. Recall that a pure strategy is a complete and nonrandom game plan. By contrast, a Mixed Strategy involves randomly mixing pure strategies, which expands the usefulness of game theory to a much broader range of strategic applications. We will begin our discussion of mixed strategies by examining noncooperative, static games with zero-sum payoffs. The reader may recall that such games do not have pure-strategy Nash equilibria. ZERO-SUM GAMES Nearly a quarter of a century before John Nash demonstrated the existence of a “fixed-point” equilibrium in noncooperative games, John von Neumann was developing his own theories of strategic behavior. As a junior faculty member at Princeton University, von Neumann became intrigued with the strategic behavior of poker players. Poker is an example of a zero-sum game in which one player’s gain is another player’s loss. MIXING PURE STRATEGIES 133 An important aspect of poker games is the bluff, which occurs when a player bets big on a bad hand. The objective of bluffing is to get opponents to “fold” (quit) by persuading them that they are sitting on losing hands, when, in fact, the opposite may be true. Not surprisingly, von Neumann found that for bluffing to be successful, big bets must be made randomly on both good and bad hands. The underlying logic is obvious. Rivals will eventually see through a player who repeatedly bets big on inferior hands. When this happens, not only will bluffing become ineffective, but rivals will exploit the attempt. This observation led von Neumann (1928) to publish the proof of his famous minimax theorem, which will be discussed at greater length below. Unfortunately, von Neumann’s discovery had few real-world applications because it was limited to zero-sum games.

  Sign up to read

  Learn more about book