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- 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 decision of each one 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. Nash equilibrium has been used to analyze hostile situations like war and arms races, and also how conflict may be mitigated by repeated interaction. It has also been used to study to what extent people with different preferences can cooperate, and whether they will take risks to achieve a cooperative outcome. It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises. Other applications include traffic flow, how to organize auctions, and even penalty kicks in soccer. History A version of the Nash equilibrium concept was first used by Antoine Augustin Cournot in his theory of oligopoly (1838). In Cournot's theory, firms choose how much output to produce to maximize their own profit. However, the best output for one firm depends on the outputs of others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure strategy Nash Equilibrium. The modern game-theoretic concept of Nash Equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy Nash Equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior. However, their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash Equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of John Forbes Nash in his 1951 article Non-Cooperative Games was to define a mixed strategy Nash Equilibrium for any game with a finite set of actions and prove that at least one (mixed strategy) Nash Equilibrium must exist. Since the development of the Nash equilibrium concept, game theorists have discovered that it makes misleading predictions (or fails to make a unique prediction) in certain circumstances. Therefore they have proposed many related solution concepts (also called 'refinements' of Nash equilibrium) designed to overcome perceived flaws in the Nash concept. One particularly important issue is that some Nash equilibria may be based on threats that are not 'credible'. Therefore, in 1965 Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeated, or what happens if a game is played in the absence of perfect information. However, subsequent refinements and extensions of the Nash equilibrium concept share the main insight on which Nash's concept rests: all equilibrium concepts analyze what choices will be made when each player takes into account the decision-making of others. Definitions Informal definition Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks himself or herself: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" If any player would answer "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because it may happen that a Nash equilibrium is not Pareto optimal. The Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with non-rational moves. For such games the Subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. Formal definition Let (S, f) be a game with n players, where Si is the strategy set for player i, SS1 X S2 ... X Sn is the set of strategy profiles and f(f1, ... , fn) is the payoff function. Let <math>x_{-i} be a strategy profile of all players except for player i. When each player i <math>\in {1, ... , n} chooses strategy xi resulting in strategy profile x (x1, ... , xn) then player i obtains payoff fi(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile x <math>\in S is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is \forall i,x_i\in S_i, x_i \neq x^*_{i} : f_i(x^*_{i}, x^*_{-i}) \geq f_i(x_{i},x^*_{-i}). A game can have a pure strategy NE or an NE in its mixed extension (that of choosing a pure strategy stochastically with a fixed frequency). Nash proved that if we allow mixed strategies, then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium. When the inequality above holds strictly (with <math>> instead of <math>\geq) for all players and all feasible alternative strategies, then the equilibrium is classified as a strict Nash equilibrium. If instead, for some player, there is exact equality between <math>x^*_i and some other strategy in the set <math>S, then the equilibrium is classified as a weak Nash equilibrium. Examples Coordination game The coordination game is a classic two player, two strategy game, with an example payoff matrix shown to the right. The players should thus coordinate, both adopting strategy A, to receive the highest payoff, i.e. , 4. If both players chose strategy B though, there is still a Nash equilibrium. Although each player is awarded less than optimal payoff, neither player has incentive to change strategy due to a reduction in the immediate payoff (from 3 to 1). An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game. Driving on a road, and having to choose either to drive on the left or to drive on the right of the road, is also a coordination game. For example, with payoffs 100 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix: In this case there are two pure strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player is (50%, 50%). Prisoner's dilemma Main article: Prisoner's dilemma (note differences in the orientation of the payoff matrix) The Prisoner's Dilemma has the same payoff matrix as depicted for the Coordination Game, but now C > A > D > B. Because C > A and D > B, each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. The Prisoner's Dilemma thus has a single Nash Equilibrium: both players choosing strategy #2 ("betraying"). What has long made this an interesting case to study is the fact that D < A (ie. , "both betray" is globally inferior to "both remain loyal"). The globally optimal strategy is unstable; it is not an equilibrium. Network traffic An extension of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are <math>n "cars" traveling from A to D, what is the expected distribution of traffic in the network? This situation can be modeled as a "game" where every traveler has a choice of 3 strategies, where each strategy is a route from A to D (either <math>ABD, <math>ABCD, or <math>ACD). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via <math>ABD experiences travel time of <math>(1+x/100)+2, where <math>x is the number of cars traveling on edge <math>AB. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal in this case is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to his/her travel time. For the graph on the right, if, for example, 100 cars are travelling from A to D, then equilibrium will occur when 25 drivers travel via <math>ABD, 50 via <math>ABCD, and 25 via <math>ACD. Every driver now has a total travel time of 3.75. Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via <math>ABD and the other 50 through <math>ACD, then travel time for any single car would actually be 3.5, which is less than 3.75. Competition game This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then he/she has to give up two points to the other. This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number. In the table to the left, for example, when starting at the green square it is in player 1's interest to move to the purple square by choosing a smaller number, and it is in player 2's interest to move to the blue square by choosing a smaller number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria (0,0...1,1...2,2... and 3,3). Nash equilibria in a payoff matrix There is an easy numerical way to identify Nash Equilibria on a Payoff Matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the duplet of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium. We can apply this rule to a 3x3 matrix: Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash Equlibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and NxN pure strategy Nash equilibria. Stability The concept of stability, useful in the analysis of many kinds of equilibrium, can also be applied to Nash equilibria. A Nash equilibrium for a mixed strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold: the player who did not change has no better strategy in the new circumstance the player who did change is now playing with a strictly worse strategy If these cases are both met, then a player with the small change in his mixed-strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. John Nash showed that the latter situation could not arise in a range of well-defined games. In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed-strategies with 100% probabilities are stable. If either player changes his probabilities slightly, they will be both at a disadvantage, and his opponent will have no reason to change his strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes his probabilities, then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%). Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy of each player is not perfectly known, but has to be inferred from statistical distribution of his actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium. A Coalition-Proof Nash Equilibrium (CPNE) (similar to a Strong Nash Equilibrium) occurs when players cannot do better even if they are allowed to communicate and collaborate before the game. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core. Occurrence If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are: The players all will do their utmost to maximize their expected payoff as described by the game. The players are flawless in execution. The players have sufficient intelligence to deduce the solution. The players know the planned equilibrium strategy of all of the other players. The players believe that a deviation in their own strategy will not cause deviations by any other players. There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on. Where the conditions are not met Examples of game theory problems in which these conditions are not met: The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner’s dilemma is not a dilemma if either player is happy to be jailed indefinitely. Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the common knowledge criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the game of chicken, ensuring a no-loss no-win scenario). In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria). The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in “Chicken” or an arms race, for example. Where the conditions are met Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics and evolutionary biology, the NE has explanatory power. The payoff in economics is money, and in evolutionary biology gene transmission, both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that agents failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research. NE and non-credible threats The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash Equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change his strategy. The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L) Nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise. Proof of existence As above, let <math>\sigma_{-i} be a mixed strategy profile of all players except for player <math>i. We can define a best response correspondence for player <math>i, <math>b_i. <math>b_i is a relation from the set of all probability distributions over opponent player profiles to a set of player <math>i's strategies, such that each element of b_i(\sigma_{-i}) is a best response to <math>\sigma_{-i}. Define b(\sigma) b_1(\sigma_{-1}) \times b_2(\sigma_{-2}) \times \cdots \times b_n(\sigma_{-n}). One can use the Kakutani fixed point theorem to prove that <math>b has a fixed point. That is, there is a <math>\sigma^* such that <math>\sigma^* \in b(\sigma^*). Since <math>b(\sigma^*) represents the best response for all players to <math>\sigma^*, the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium. When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed point theorem. " (See Nasar, 1998, p. 94. ) Alternate proof using the Brouwer fixed point theorem We have a game <math>G(N,A,u) where <math>N is the number of players and <math>A A_1 \times \ldots \times A_N is the action set for the players. All of the action sets <math>A_i are finite. Let <math>\Delta \Delta_1 \times \ldots \times \Delta_N denote the set of mixed strategies for the players. The finiteness of the <math>A_is insures the compactness of <math>\Delta. We can now define the gain functions. For a mixed strategy <math>\sigma \in \Delta, we let the gain for player <math>i on action <math>a \in A_i be Gain_i(\sigma,a) \max \{0, u_i(a_i, \sigma_{-i}) - u_i(\sigma_{i}, \sigma_{-i})\} The gain function represents the benefit a player gets by unilaterally changing his strategy. We now define <math>g (g_1,\ldots,g_N) where g_i(\sigma)(a) \sigma_i(a) + Gain_i(\sigma,a) for <math>\sigma \in \Delta, a \in A_i. We see that \sum_{a \in A_i} g_i(\sigma)(a) \sum_{a \in A_i} \sigma_i(a) + Gain_i(\sigma,a) 1 + \sum_{a \in A_i} Gain_i(\sigma,a) > 0 We now use <math>g to define <math>f: \Delta \rightarrow \Delta as follows. Let f_i(\sigma)(a) \frac{g_i(\sigma)(a)}{\sum_{b \in A_i} g_i(\sigma)(b)} for <math>a \in A_i. It is easy to see that each <math>f_i is a valid mixed strategy in <math>\Delta_i. It is also easy to check that each <math>f_i is a continuous function of <math>\sigma, and hence <math>f is a continuous function. Now <math>\Delta is the cross product of a finite number of compact convex sets, and so we get that <math>\Delta is also compact and convex. Therefore we may apply the Brouwer fixed point theorem to <math>f. So <math>f has a fixed point in <math>\Delta, call it <math>\sigma^*. I claim that <math>\sigma^* is a Nash Equilibrium in <math>G. For this purpose, it suffices to show that \forall 1 \leq i \leq N, ~ \forall a \in A_i, ~ Gain_i(\sigma^*,a) 0 \text{. } This simply states the each player gains no benefit by unilaterally changing his strategy which is exactly the necessary condition for being a Nash Equilibrium. Now assume that the gains are not all zero. Therefore, <math>\exists i, <math>1 \leq i \leq N, and <math>a \in A_i such that <math>Gain_i(\sigma^*, a) > 0. Note then that \sum_{a \in A_i} g_i(\sigma^*, a) 1 + \sum_{a \in A_i} Gain_i(\sigma^*,a) > 1 So let <math>C \sum_{a \in A_i} g_i(\sigma^*, a). Also we shall denote <math>Gain(i,\cdot) as the gain vector indexed by actions in <math>A_i. Since <math>f(\sigma^*) \sigma^* we clearly have that <math>f_i(\sigma^*) \sigma^*_i. Therefore we see that \sigma^*_i \frac{g_i(\sigma^*)}{\sum_{a \in A_i} g_i(\sigma^*)(a)} \Rightarrow \sigma^*_i \frac{\sigma^*_i + Gain_i(\sigma^*,\cdot)}{C} \Rightarrow C\sigma^*_i \sigma^*_i + Gain_i(\sigma^*,\cdot) \left(C-1\right)\sigma^*_i Gain_i(\sigma^*,\cdot) \Rightarrow \sigma^*_i \left(\frac{1}{C-1}\right)Gain_i(\sigma^*,\cdot) Since <math>C > 1 we have that <math>\sigma^*_i is some positive scaling of the vector <math>Gain_i(\sigma^*,\cdot). Now I claim that \sigma^*_i(a)(u_i - u_i) \sigma^*_i(a)Gain_i(\sigma^*, a) <math>\forall a \in A_i. To see this, we first note that if <math>Gain_i(\sigma^*, a) > 0 then this is true by definition of the gain function. Now assume that <math>Gain_i(\sigma^*, a) 0. By our previous statements we have that \sigma^*_i(a) \left(\frac{1}{C-1}\right)Gain_i(\sigma^*, a) 0 and so the left term is zero, giving us that the entire expression is <math>0 as needed. So we finally have that 0 u_i(\sigma^*_i, \sigma^*_{-i}) - u_i(\sigma^*_i, \sigma^*_{-i}) \left(\sum_{a \in A_i} \sigma^*_iu_i\right) - u_i(\sigma^*_i, \sigma^*_{-i}) \sum_{a \in A_i} \sigma^*_i(a) (u_i - u_i) \sum_{a \in A_i} \sigma^*_i(a) Gain_i(\sigma^*, a) \quad \text{ by the previous statements } \sum_{a \in A_i} \left(C -1 \right) \sigma^*_i(a)^2 > 0 where the last inequality follows since <math>\sigma^*_i is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore <math>\sigma^* is a Nash Equilibrium for <math>G as needed. Computing Nash equilibria If a player A has a dominant strategy <math>s_A then there exists a Nash equilibrium in which A plays <math>s_A. In the case of two players A and B, there exists a Nash equilibrium in which A plays <math>s_A and B plays a best response to <math>s_A. If <math>s_A is a strictly dominant strategy, A plays <math>s_A in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays his strictly dominant strategy. In games with mixed strategy Nash equilibria, the probability of a player choosing any particular strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, his expected payoff for each strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived. Examples In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed strategy Nash equilibrium, assign A the probability p of playing H and (1−p) of playing T, and assign B the probability q of playing H and (1−q) of playing T. E[payoff for A playing H] (−1)q + (+1)(1−q) 1−2q E[payoff for A playing T] (+1)q + (−1)(1−q) 2q−1 E[payoff for A playing H] E[payoff for A playing T] ⇒ 1−2q 2q−1 ⇒ q 1/2 E[payoff for B playing H] (+1)p + (−1)(1−p) 2p−1 E[payoff for B playing T] (−1)p + (+1)(1−p) 1−2p E[payoff for B playing H] E[payoff for B playing T] ⇒ 2p−1 1−2p ⇒ p 1/2 Thus a mixed strategy Nash equilibrium in this game is for each player to randomly choose H or T with equal probability. See also Adjusted Winner procedure Best response Braess's paradox Complementarity theory Conflict resolution research Cooperation Evolutionarily stable strategy Game theory Glossary of game theory Hotelling's law Mexican Standoff Minimax theorem Optimum contract and par contract Prisoner's dilemma Relations between equilibrium concepts Self-confirming equilibrium Solution concept Equilibrium selection Stackelberg competition Subgame perfect Nash equilibrium Wardrop's principle References Game Theory textbooks . Suitable for undergraduate and business students. Fudenberg, Drew and Jean Tirole (1991) Game Theory MIT Press. . An 88-page mathematical introduction; see Chapter 2. Free online at many universities. Morgenstern, Oskar and John von Neumann (1947) The Theory of Games and Economic Behavior Princeton University Press . A modern introduction at the graduate level. . A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online. . Lucid and detailed introduction to game theory in an explicitly economic context. Original Papers Nash, John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences 36(1):48-49. Nash, John (1951) "Non-Cooperative Games" The Annals of Mathematics 54(2):286-295. Other References Mehlmann, A. The Game's Afoot! Game Theory in Myth and Paradox, American Mathematical Society (2000). Nasar, Sylvia (1998), "A Beautiful Mind", Simon and Schuster, Inc. Notes External links Complete Proof of Existence of Nash Equilibria Yale Economics lecture on Nash Equilibria
- Das Nash-Gleichgewicht, teils auch (wie im Englischen) Nash-Equilibrium genannt, ist ein zentraler Begriff der mathematischen Spieltheorie. Es beschreibt in nicht-kooperativen Spielen einen Zustand eines strategischen Gleichgewichts, von dem ausgehend kein einzelner Spieler für sich einen Vorteil erzielen kann, indem er einseitig von seiner Strategie abweicht. Es ist ein grundlegendes Lösungskonzept der Spieltheorie. Definition und Existenzbeweis des Nash-Gleichgewichts gehen auf die 1950 veröffentlichte Dissertation des Mathematikers John Forbes Nash Jr. zurück.
- En teoria de jocs, un equilibri de Nash és un tipus de concepte de solució d'un joc en què participen dos o més jugadors i on cap d'ells té res a guanyar pel fet de canviar unilateralment la seva estratègia. En altres paraules, si cada jugador ha escollit una estratègia i cap jugador es pot beneficiar d'un canvi en la seva pròpia estratègia mentre els altres no la canviïn, llavors el conjunt actual d'estratègies escollides pels jugadors i llurs beneficis corresponents són un equilibri de Nash. Com que l'equilibri de Nash es centra en les preferències de cada individu, es poden produir resultats antiintuïtius. Pot haver-hi un equilibri de Nash quan, si els jugadors poguessin coordinar-se, tots canviarien d'estratègia. El joc de la «caça del cérvol» és un exemple d'aquest fet. El concepte d'equilibri de Nash no és original de John Forbes Nash, ja que Antoine Augustin Cournot demostrà com trobar allò que avui en dia anomenem equilibri de Nash per a la competició de Cournot. Per tant, alguns autors es refereixen a aquest concepte com a equilibri de Cournot-Nash o equilibri de Nash-Cournot. Tot i així, fou Nash qui demostrà per primera vegada, a la seva tesi Non-cooperative games (1950) que han d'existir equilibris de Nash per a qualsevol joc finit amb qualsevol nombre de jugadors. Fins llavors, el resultat només s'havia demostrat per a jocs de suma nul·la per a dos jugadors, gràcies a John von Neumann i Oskar Morgenstern (1947).
- Nashova rovnováha je v teorii her stav, kdy žádný z hráčů nemůže jednostranným krokem zlepšit svoji situaci. Příkladem takové rovnováhy je situace ve vězňově dilematu, kdy se oba zadržení navzájem udají. Když jeden z nich vinu druhého zapře, dostane vyšší trest, zatímco druhý bude osvobozen. Proto je pro oba dva výhodnější kolegu udat, přestože v případě, že budou oba spolupracovat a zapírat, bude pro ně v součtu výsledek lepší. Pojem je pojmenován po matematiku Johnu Nashovi, který roku 1950 ve své dizertaci dokázal, že rovnováha existuje ve všech konečných hrách.
- En teoría de juegos, se define el equilibrio de Nash como un modo de obtener una estrategia óptima para juegos que involucren a dos o más jugadores Si hay un conjunto de estrategias tal que ningún jugador se beneficia cambiando su estrategia mientras los otros no cambien la suya, entonces ese conjunto de estrategias y las ganancias correspondientes constituyen un equilibrio de Nash El concepto de equilibrio de Nash apareció por primera vez en su disertación Non-cooperative games (1950) John Forbes Nash demostró que las distintas soluciones que habían sido propuestas anteriormente para juegos tienen la propiedad de producir un equilibrio de Nash Un juego puede no tener equilibrio de Nash, o tener más de uno Nash fue capaz de demostrar que si permitimos estrategias mixtas (en las que los jugadores pueden escoger estrategias al azar con una probabilidad predefinida), entonces todos los juegos de n jugadores en los que cada jugador puede escoger entre un número finito de estrategias tienen al menos un equilibrio de Nash con estrategias mixtas Si un juego tiene un único equilibrio de Nash y los jugadores son completamente racionales, los jugadores escogerán las estrategias que forman el equilibrio
- Nashin tasapaino on peliteorian ongelmien ratkaisukonsepti, jolla kuvataan tilannetta, jossa peliteoreettisen pelin, jossa on vähintään kaksi pelaajaa lopputulos on sellainen, ettei yksikään pelaajista voi saavuttaa parempaa lopputulosta itselleen muuttamalla yksipuolisesti omaa strategiaansa. Jos kukin pelaaja on valinnut strategiansa eikä kukaan pelaajista voi saavuttaa parempaa lopputulosta muuttamalla strategiaansa, muodostavat pelaajien valitsemat strategiat Nashin tasapainotilan. Esimerkki Nashin tasapainosta on Vangin dilemman ratkaisu. Nashin tasapaino on nimetty yhdysvaltalaisen matemaatikon ja talousnobelisti John Forbes Nashin mukaan.
- John Nash a défini une situation d'interaction comme stable si aucun agent n'a intérêt à changer sa stratégie. La formalisation de ce constat simple a été essentielle pour la théorie des jeux.
- A játékelméletben Nash-egyensúlynak nevezzük a résztvevő játékosok egyéni stratégiáinak olyan stratégiaegyüttesét, amelyre igaz, hogy minden egyes játékos aktuális stratégiája egy parciálisan legjobb választ ad a többi játékos aktuális stratégiájára. Pontosabban: amennyiben a többi játékos egyike sem változtat az aktuális stratégiáján, akkor az adott játékosnak sem érdemes változtatnia, mert csak rontana a változtatással.
- In teoria dei giochi si definisce equilibrio di Nash un profilo di strategie (una per ciascun giocatore) rispetto al quale nessun giocatore ha interesse ad essere l'unico a cambiare.
- ナッシュ均衡（ナッシュきんこう、Nash equilibrium）は、ゲーム理論における非協力ゲームの解の一種であり、いくつかの解の概念の中で最も基本的な概念である。数学者のジョン・フォーブス・ナッシュにちなんで名付けられた。 ナッシュ均衡は、他のプレーヤーの戦略を所与とした場合、どのプレーヤーも自分の戦略を変更することによってより高い利得を得ることができない戦略の組み合わせである。ナッシュ均衡の下では、どのプレーヤーも戦略を変更する誘因を持たない。 ナッシュ均衡は、必ずしもパレート効率的ではない。その良い例が、囚人のジレンマである。
- In de speltheorie is een Nash-evenwicht een centrale uitkomst in een spel voor twee of meer spelers. In een Nash-evenwicht kan geen van de spelers, door individueel te veranderen, zijn winst vergroten. Elke speler speelt dus de strategie die optimaal is, er van uit gaande dat de ander zijn strategie niet verandert. Doordat alle spelers dit doen, heeft geen van de spelers een reden om af te wijken, dus zijn alle strategieën in het evenwicht optimaal (niemand kan zijn situatie verbeteren). De combinatie van optimale strategieën, vormt een Nash-evenwicht. Een evenwicht kan zowel uit pure strategieën als uit gemengde strategieën (waarbij de verschillende pure strategieën met een bepaalde kans gespeeld worden) bestaan. Het concept werd in 1950 door John Forbes Nash geïntroduceerd in zijn dissertatie. Hij bewees toen dat er voor elk spel (met een paar voorwaarden) minstens één evenwicht bestaat als gemengde strategieën toegestaan worden. Een Nash-evenwicht betekent lang niet altijd dat de totale opbrengst van alle spelers gemaximaliseerd wordt. Doordat iedere speler voor zichzelf, gegeven de strategie van de anderen, zijn opbrengst maximaliseert, kan het zo zijn dat de totale uitkomst niet gemaximaliseerd wordt. Het bekendste voorbeeld daarvan is het Prisoner's dilemma.
- John Nash innførte begrepsforskjellen mellom kooperativ spillteori, som behandler spill der det kan treffes bindende avtaler, og ikke-kooperativ spillteori. For den ikke-kooperative spillteorien utviklet Nash et likevektsbegrep som senere er blitt kalt "Nash-likevekt" og som er blitt en standard verktøy innenfor alle områder av økonomisk teori. Nash-likevekten er den rekken av trekk i et spill som ender med at når du ser hva motstanderen din har gjort, så angrer du ikke på dine egne trekk.
- Równowaga Nasha jest to profil strategii teorii gier, w którym strategia każdego z graczy jest optymalna, przyjmując wybór jego oponentów za ustalony. W równowadze żaden z graczy nie ma powodów jednostronnie odstępować od strategii równowagi. W tym sensie równowaga jest stabilna. Każda skończona gra ma przynajmniej jedną równowagę Nasha, niekoniecznie w strategiach czystych. Równowaga Nasha nie musi być efektywna w sensie Pareto. Klasycznym przykładem tej nieefektywności jest paradoks znany jako dylemat więźnia. Rozważmy grę dwuosobową. Równowagą Nasha jest następujący wybór. Wybór gracza A jest optymalny dla wyboru gracza B i wybór gracza B jest optymalny przy danym wyborze A. Inaczej: Wybieram to, co jest dla mnie najlepsze, gdy ty robisz to, co robisz. Ty robisz to, co jest dla ciebie najlepsze, gdy ja robię to, co robię.
- O Equilíbrio de Nash representa uma situação em que, em um jogo envolvendo dois ou mais jogadores, nenhum jogador tem a ganhar mudando sua estratégia unilateralmente. Para melhor compreender esta definição, suponha que há um jogo com n participantes. No decorrer deste jogo, cada um dos n participantes seleciona sua estratégia ótima, ou seja, aquela que lhe traz o maior benefício. Então, se cada jogador chegar à conclusão que ele não tem como melhorar sua estratégia dadas as estratégias escolhidas pelos seus n-1 adversários (estratégias dos adversários não podem ser alteradas), então as estratégias escolhidas pelos participantes deste jogo definem um "equilíbrio de Nash".
- Echilibrul Nash este un termen central al teoriei matematice a jocului. Prin jocuri este descrisă o stare a echilibrului strategic, plecând de la care un jucător nu are nici un avantaj, schimbând de unul singur strategia. Definiţia si demonstrarea existenţei echilibrului Nash au fost făcute in anul 1950 în disertaţia publicată de matematicianul John Forbes Nash Jr.
- В теории игр равновесием Нэша (названным в честь Джона Форбса Нэша, который предложил его) называется тип решений игры двух и более игроков, в котором ни один участник не может увеличить выигрыш, изменив своё решение в одностороннем порядке, когда другие участники не меняют решения. Такая совокупность стратегий выбранных участниками и их выигрыши называются равновесием Нэша. Концепция равновесия Нэша (РН) впервые использована не Нэшем; Антуан Огюст Курно показал, как найти то, что мы называем равновесием Нэша, в игре Курно. Соответственно, некоторые авторы называют его равновесием Нэша-Курно. Однако Нэш первым показал в своей диссертации Некооперативные игры (1950), что равновесия Нэша должны существовать для всех конечных игр с любым числом игроков. До Нэша это было доказано только для игр с 2 участниками с нулевой суммой Джоном фон Нейманом и Оскаром Моргенштерном (1947).
- Oyun Teorisi'nin en önemli araçlarından biri olan Nash dengesi, oyuncuların belli özellikler taşıyan strateji seçimlerine verilen isimdir. Her oyuncu, oyun içinde elinde olan eylemlerden birini seçmiş olsun, ve tüm oyuncuların böyle bir seçim yaptığını düşünelim. Bir oyuncu için seçilmiş eylem, diğer oyuncuların seçtikleri eylem gözetildiğinde oynanabilecek (getiri anlamında) en iyi eylem ise, ve bu özellik tüm oyuncular için sağlanıyorsa, bu eylemler bir Nash Dengesi oluşturur. Oyuncular, yalın eylemler seçebilecekleri gibi, birden çok eylemi, belli bir olasılıkla oynamayı da tercih edebilirler. Nash Dengesi, Oyun Teorisi kavramına önemli katkıları olan Amerikalı matematikçi John Nash'in adıyla anılmaktadır. Analitik anlamda benzer bir gözlemde bulunan ilk kişi Antoine Augustin Cournot isimli Fransız bir matematikçidir. Cournot bu olguyu iki firmanın, eş zamanlı olarak üretim miktarını belirledikleri kuramsal bir pazar için gözlemlemiş ve detaya dökmüştür. John Nash, 1950 yılında yazdığı doktora bitirme tezinde, bu dengenin, oyuncuların fayda fonksiyonlarının belli özellikleri sağladığı tüm oyunlarda var oduğunu ispatlayarak 1994 Ekonomi Nobel Ödülü'nü almıştır.
- В теорії ігор рівновагою Неша (названою на честь Джона Форбса Неша, який запропонував цей термін) називається у грі з двоми чи більше гравцями така сукупність статегій та виграшів, при якій жоден із учасників не може збільшити виграш, змінивши вибір стратегії в односторонньому порядку, коли інші учасники не змінюють свого вибору.
- 納什平衡，又稱為非合作賽局平衡，是博弈论的一個重要概念，以约翰·納什命名。 如果某情況下無一參與者可以獨自行動而增加收益，則此策略組合被稱為納什均衡點。
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