Economics
Nash Equilibrium
Nash Equilibrium is a concept in game theory where each player's strategy is optimal given the strategies of the other players. In this state, no player has an incentive to unilaterally change their strategy. It is a key concept for understanding strategic decision-making and has applications in various fields, including economics, political science, and evolutionary biology.
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11 Key excerpts on "Nash Equilibrium"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 6 Nash Equilibrium In game theory, Nash Equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash Equilibrium. Stated simply, Amy and Bill are in Nash Equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision. Likewise, a group of players is in Nash Equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others. However, Nash Equilibrium does not necessarily mean the best cumulative payoff for all the players involved; in many cases all the players might improve their payoffs if they could somehow agree on strategies different from the Nash Equilibrium (e.g., competing businesses forming a cartel in order to increase their profits). Applications The Nash Equilibrium concept is used to analyze the outcome of the strategic interaction of several decision makers. In other words, it is a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation. Instead, we must ask what each player would do, taking into account the decision-making of the others.
eBook - PDF- Steven N. Durlauf, Lawrence E. Blume, Steven Durlauf(Authors)
- 2016(Publication Date)
- Palgrave Macmillan(Publisher)
Nash Equilibrium, refinements of Game theory studies decisions by several persons in situations with significant interactions. Two features distinguish it from other theories of multi-person decisions. One is explicit consideration of each person’s available strategies and the outcomes resulting from combinations of their choices; that is, a complete and detailed specification of the ‘game’. Here a person’s strategy is a complete plan specifying his action in each contingency that might arise. In non-cooperative contexts, the other is a focus on optimal choices by each person separately. John Nash (1950; 1951) proposed that a combination of mutually optimal strategies can be characterized mathematically as an equilibrium. According to Nash’s definition, a combination is an equilibrium if each person’s choice is an optimal response to others’ choices. His definition assumes that a choice is optimal if it maximizes the person’s expected utility of outcomes, conditional on knowing or correctly anticipating the choices of others. In some applications, knowledge of others’ choices might stem from prior agreement or communication, or accurate prediction of others’ choices might derive from ‘common knowledge’ of strategies and outcomes and of optimizing behaviour. Because many games have multiple equilibria, the predictions obtained are incomplete. However, equilibrium is a weak criterion in some respects, and therefore one can refine the criterion to obtain sharper predictions (Harsanyi and Selten, 1988; Hillas and Kohlberg, 2002; Kohlberg, 1990; Kreps, 1990). Here we describe the main refinements of Nash Equilibrium used in the social sciences. Refinements were developed incrementally, often relying on ad hoc criteria, which makes it difficult for a non-specialist to appreciate what has been accomplished. Many refinements have been proposed but we describe only the most prominent ones.
eBook - PDF- James D. Morrow(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
We define a Nash Equilibrium as a situation where each player's strategy is a best reply to the other player's strategy. When both players are playing best replies against each other's strategies, they have no incentive to change their strategies. The game in Figure 4.4 has two Nash equilibria in pure strategies, (Si;si) and (S2;s2). Si is Player l's best reply to sj, and s is Player 2's best reply to Si. Similarly, S2 is Player 1 's best reply to S2, and s? is Player 2's best reply to S2. Definition: A pair of strategies Sj and Sj forms a Nash equilib-rium iff the strategies are best replies to each other. Alternatively, a pair of strategies forms a Nash Equilibrium iff and M,(Si;Sj) > M!(S;SJ) for all S M2(SJ;SJ) > M 2 (Si;s) for all s CLASSICAL GAME THEORY 8 1 A Nash Equilibrium is stable because neither player has an incentive to de-viate unilaterally from its equilibrium strategy. Either player reduces its payoff if it defects from its equilibrium strategy. However, this observation does not imply that an equilibrium is the best outcome for either player. Nor are equilib-ria fair in any common meaning of fairness. Instead, Nash Equilibrium is a minimal condition for a solution to a game if the players can correctly antici-pate each other's strategies. Nash equilibria entail stable, mutual anticipations of other players' strategies. If such anticipations exist, neither player has an incentive to change its strategy unilaterally. To do so would reduce its payoff. Nash equilibria in pure strategies can be found easily from the strategic form. Player 1 can choose the row of the strategic form, and Player 2, the col-umn. Sometimes Players 1 and 2 are called Row and Column for this reason. Within a given column, that is, for a fixed strategy for Player 2, Player 1 's best reply is the strategy that produces the largest payoff for him in that col-umn. Within a given row, Player 2's best reply is the strategy that produces the largest payoff for her in that row.
- John Hoag(Author)
- 2007(Publication Date)
- WSPC(Publisher)
Perhaps the most complicated issue is the definition of equilibrium. One simple way to think about equilibrium is once equilibrium is achieved, there is no reason for either player to change strategy, b Bronowski (1973, p. 432). Used with permission. Game Theory 273 and the game is, in some sense, over. An equilibrium is, then, a stopping rule for the game. How do we define equilibrium? On the one hand, there is no obvious definition of equilibrium we would all find acceptable for all games. Still, in economics, there is perhaps one definition that dominates and forms a basis for the definition of equilibrium as the structure of the game becomes more complicated. In part, the domination comes from the fact that this definition of equilibrium accords with our previous work and does not require us to completely throw out what we have done so far. This definition is due to Nash who provided an elegant method for establishing its existence. It is called (guess what?) the Nash Equilibrium. Before we can offer a definition, we will need to introduce some notation. Suppose that we have n players of the game. Each player has a variety of actions that that player may take; for the i th player, this set is S i . From that set of possible actions, the player chooses one action, X i . In more sophisticated settings, we will not require that the player select one action, but the player will select a probability distribution (or density) over the possible actions. In this case, the player is faced with the set of all possible probability densities over the possible actions, and the player picks one probability density and follows that density when playing. In the simpler setting where probabilities do not play a role, the i th player’s set of possible actions, the strategy set, is S i . Each player has a payoff that depends on the actions of all players. The payoff could be dollars or utility or any other desired outcome.
eBook - PDFRational Choice Theory
Potential and Limits
- Lina Eriksson(Author)
- 2011(Publication Date)
- Red Globe Press(Publisher)
Perhaps there is one outcome that definitely is incompatible with the theory in question, and perhaps we can test whether that outcome obtains. If it does, that is good evidence against the theory. But if it does not, it is still hard to say just how much our confidence in the theory should be strengthened. Another problematic feature of games with multiple equilibria is that some of these equilibria seem unreasonable, even though they are as good Nash equilibria as the others. For example, Nash equi-libria in extensive form games might rely on non-credible threats that the threatening agent will have no incentive to carry out, should the bluff be called (Samuelson 1997). In response to these problems, game theorists have developed a host of refinements to the Nash Equilibrium, all constraining the set of Nash equilibria in different ways. These refinements narrow down the set of equilib-ria, and aim to do so in a way that selects all and only those equilib-ria that actually make sense in some particular way. For example, subgame perfect equilibrium has been developed to deal with, among other things, the problem of Nash equilibria that rely on non-credible threats. Non-equilibrium play and game theory As noted above, the use of the Nash Equilibrium concept presup-poses that agents know what their opponents’ pay-offs are or that there is some way to conceptualize their uncertainty in terms of judgements about risk. But in real life, these assumptions are at best reasonably good approximations, and often they are not even that. This raises issues for the validity or scope of RC explanations in general, but some of the issues are fundamental to game theory Equilibrium 211 itself. Suppose you play a game in an extensive form, and you have figured out what the equilibrium strategies of your opponent and yourself are. You thus make your first move accordingly, but then you watch, stunned, as your opponent makes a completely unex-pected move.
eBook - PDFModel Building in Economics
Its Purposes and Limitations
- Lawrence A. Boland(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
However, as I suggested earlier in this chapter, what game-theoretic models do provide is both the ability to avoid the unrealistic complications of using calculus-based characterizations of maximization and the ability to directly address one of the short-comings identified by critics of macro models which is that too often they do not recognize the diversity within any economy. Continuing this brief review, the first task of the model builder is to identify the eventual outcome of playing the posited game. And as is well known, to do so there are two considerations: dominance and what is called a Nash Equilibrium. Dominance is a primary means of eliminating from the list of possible joint options those that have no chance of being chosen – this would be simply because, given the game and the respective payoffs, there is always a better outcome for the players to change to without losing what would be obtained by the dominated joint option. In effect, a dominated pair of options in a two-person game is not Pareto optimal. 2 A Nash Equilibrium is an outcome that, if reached, leaves all players seeing no need to change and which defines the minimum necessary conditions for an equilibrium outcome that correspond to the idea of explanation dis- cussed in Chapter 2. If dominance does not eliminate all but one possible outcome, then identifying such a Nash Equilibrium is always the next essential step in applying the model to explain or predict. The only problem is that some game-theoretic models have more than one possible Nash Equilibrium, which means the model cannot be used to explain or predict, as I discussed in the previous chapter. And, of course, it is conceivable that 2 ‘Pareto optimality’ is an indirect and limited criterion of optimality. It simply says that if one person can gain without the other losing, then the state of trade is not optimal.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Game theoretic arguments of this type can be found as far back as Plato. Economics and business Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social network formation, and voting systems and to model across such broad classifications as behavioral economics and industrial organization. This research usually focuses on particular sets of strategies known as equilibria in games. These solution concepts are usually based on what is required by norms of rationality. In non-cooperative games, the most famous of these is the Nash Equilibrium. A set of strategies is a Nash Equilibrium if each represents a best response to the other strategies. So, if all the players are playing the strategies in a Nash Equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing. The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty. ____________________ WORLD TECHNOLOGIES ____________________ A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should this information be put. Economists and business professors suggest two primary uses: descriptive and prescriptive . Political science The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory.
- Julio González-Díaz, Ignacio García-Jurado, M. Gloria Fiestras-Janeiro(Authors)
- 2010(Publication Date)
- American Mathematical Society(Publisher)
, the Nash Equilibrium concept searches for rest points of the interactive situation described by the strategic game. Given a game G = ( A , u ) and a strategy profile a ∈ A , let ( a − i , ˆ a i ) denote the profile ( a 1 , . . . , a i − 1 , ˆ a i , a i − 1 , . . . , a n ) . Definition 2.2.1. Let G = ( A , u ) be a strategic game. A Nash Equilibrium of G is a strategy profile a ∗ ∈ A such that, for each i ∈ N and each ˆ a i ∈ A i , u i ( a ∗ ) ≥ u i ( a ∗ − i , ˆ a i ) . Next, we study the Nash equilibria of the strategic games we presented in the previous section. Example 2.2.1. The only Nash Equilibrium of the prisoner’s dilemma is a ∗ = ( D , D ) . Moreover, as we have already argued, this is the rational behavior in a noncooperative environment. Example 2.2.2. We now study the Nash equilibria in a Cournot model like the one in Example 2.1.2. We do it under the following assumptions: • We deal with a duopoly, i.e. , n = 2. • For each i ∈ { 1, 2 } , c i ( a i ) = ca i , where c > 0. • Let d be a fixed number, d > c . The price function is given by: π ( a 1 + a 2 ) = d − ( a 1 + a 2 ) a 1 + a 2 < d 0 otherwise. 2.2. Nash Equilibrium in Strategic Games 19 For each i ∈ { 1, 2 } and each a ∈ A , the payoff functions of the associated strategic game are u i ( a ) = a i ( d − a 1 − a 2 − c ) a 1 + a 2 < d − a i c otherwise. By definition, a Nash Equilibrium of this game, sometimes called a Cournot equilibrium, is a pair ( a ∗ 1 , a ∗ 2 ) ∈ A 1 × A 2 such that i) for each ˆ a 1 ∈ A 1 , u 1 ( a ∗ 1 , a ∗ 2 ) ≥ u 1 ( ˆ a 1 , a ∗ 2 ) and ii) for each ˆ a 2 ∈ A 2 , u 2 ( a ∗ 1 , a ∗ 2 ) ≥ u 2 ( a ∗ 1 , ˆ a 2 ) . Now, we compute a Nash Equilibrium of this game. For each i ∈ { 1, 2 } and each a ∈ A , let f i ( a ) : = a i ( d − a 1 − a 2 − c ) . Then, ∂ f 1 ∂ a 1 ( a ) = − 2 a 1 + d − a 2 − c and ∂ f 2 ∂ a 2 ( a ) = − 2 a 2 + d − a 1 − c . Hence, ∂ f 1 ∂ a 1 ( a ) = 0 ⇔ a 1 = d − a 2 − c 2 and ∂ f 2 ∂ a 2 ( a ) = 0 ⇔ a 2 = d − a 1 − c 2 .
eBook - PDF- Ngo Van Long(Author)
- 2010(Publication Date)
- World Scientific(Publisher)
Thus, I will exclude from consideration “repeated games” (such as the repeated prisoners’ dilemma etc.) since in such games there are no state variables and no transition equations that describe the changing environment in which the players operate. Economists have used dynamic games to analyze a variety of problems in various fields, such as dynamic oligopoly, dynamic contributions to a 1 2 Dynamic Games in Economics: A Survey public good, dynamic game of optimal tariffs and retaliation, redistributive taxation in the presence of forward-looking agents, exploitation of common property resources, non-cooperative environmental policies, and the arms race. In this survey, I intend to introduce the readers to the basic equilibrium concepts in dynamic games and present a number of interesting dynamic game models and results in various fields of economics. This chapter introduces the basic concepts and some ideas about solution techniques. In Sec. 1.2, I introduce two main equilibrium concepts, open-loop Nash Equilibrium (OLNE) and Markov-perfect Nash Equilibrium (MPNE), and illustrate their difference by means of simple examples. In Sec. 1.3, I introduce the concept of hierarchical dynamic games and two equilibrium concepts for such games, the open-loop Stackelberg equilibrium (OLSE) and the feedback Stackelberg equilibrium (FBSE). 1.2. Open-loop Nash Equilibirum and Markov-perfect Nash Equilibrium One of the most important distinctions in dynamic games is that between “open-loop strategies” (or pre-commitment strategies) and “Markov-perfect strategies” (or feedback strategies). 1 A player’s open-loop strategy is a planned time path of its actions. An OLNE is a profile of open-loop strategies (one for each player) such that each player’s open-loop strategy maximizes its payoff, given the open-loop strategies of other players. Some early articles analyze dynamic games using exclusively this equilibrium concept.
- Richard Quandt, Dusan Triska(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
PART SIX Strategies and General Equilibrium 20 Maximin vs. Nash Equilibrium: Theoretical Results and Empirical Evidence · Manfred J. Holler and Viggo H~st 1. Introduction Von Neumann and Morgenstern (1944) introduced the maximin solution into the theory of games where solution is plausibly a set of rules for each participant which tell him how to behave in every social situation which may conceivably arise (p. 31). It is understood that the rules ofrational behavior must provide definitely for the possibility of irrational behavior on the part of the others (p. 32). Accordingly, a solution contains the statement of how much the participant under consideration can get if he behaves 'rationally.' This 'can get' is, of course, presumed to be minimum; he may get more if the others make mistakes (p. 33). This contrasts with the widely-held view of contemporary game theory that the solution of a noncooperative game has to be a Nash Equilibrium (van Damme, 1987, p. 3). Needless to say, the two positions assume different qualities for the decision-makers: Nash Equilibrium presupposes that all agents act ratio-nally and are expected to act rationally while maximin allows for irrational behavior by fellow agents. As a consequence, the question of which of the • The authors gratefully acknowledge the conunents of Peter Skott on an earlier version of this paper 246 Holler and H~t two concepts is the right one cannot be decided on theoretical grounds alone. In this paper, we present the results of an empirical study which, however, builds strongly on the theoretical properties of the two concepts and their relations to one another. More specifically, the empirical study makes use of the fact that for all 2 x 2 matrix games the payoff values of the mixed-strategy Nash Equilibrium are equal to the maximin payoffs if the maximin solution implies mixed strategies.
eBook - PDFGame Theory
An Introduction
- E. N. Barron(Author)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter Five N-Person Nonzero Sum Games and Games with a Continuum of Strategies The race is not always to the swift nor the battle to the strong, but that’s the way to bet. —Damon Runyon, More than Somewhat 5.1 The Basics In previous chapters, the games all assumed that the players each had a finite or countable number of pure strategies they could use. A major generalization of this is to consider games in which the players have many more strategies. For example, in bidding games the amount of money a player could bid for an item tells us that the players can choose any strategy that is a positive real number. The dueling games we considered earlier are discrete versions of games of timing in which the time to act is the strategy. In this chapter, we consider some games of this type with N players and various strategy sets. If there are N players in a game, we assume that each player has her or his own payoff function depending on her or his choice of strategy and the choices of the other players. Suppose that the strategies must take values in sets Q i , i = 1, . . . , N and the payoffs are real-valued functions: u i : Q 1 × · · · × Q N → R, i = 1, 2, . . . , N. Here is a formal definition of a pure Nash Equilibrium point, keeping in mind that each player wants to maximize their own payoff. Definition 5.1.1 A collection of strategies q ∗ = (q ∗ 1 , . . . , q ∗ n ) ∈ Q 1 × · · · × Q N is a pure Nash Equilibrium for the game with payoff functions {u i (q 1 , . . . , q n )}, i = 1, . . . , N, if for each player i = 1, . . . , N, we have u i (q ∗ 1 , . . . , q ∗ i −1 , q ∗ i , q ∗ i +1 , . . . , q ∗ N ) ≥ u i (q ∗ 1 , . . . , q ∗ i −1 , q i , q ∗ i +1 , . . . , q ∗ N ), for all q i ∈ Q i . Game Theory: An Introduction, Second Edition. E.N. Barron. C 2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc. 213
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