In game theory, the Nash equilibrium (namedafter John Forbes Nash) is a solution concept of a game involving two or more players, inwhich each player is assumed to know the equilibriumstrategies of the other players, and no player has anything togain by changing only his own strategy unilaterally.^[1]:14If each player has chosen a strategy and no player can benefit bychanging his or her strategy while the other players keep theirsunchanged, then the current set of strategy choices and thecorresponding payoffs constitute a Nash equilibrium.

Stated simply, Amy and Phil are in Nash equilibrium if Amy ismaking the best decision she can, taking into account Phil'sdecision, and Phil is making the best decision he can, taking intoaccount Amy's decision. Likewise, a group of players are in Nashequilibrium if each one is making the best decision that he or shecan, taking into account the decisions of the others.

Applications:

Game theorists use the Nash equilibrium concept to analyze theoutcome of the strategic interaction of several decision makers. In other words, it providesa way of predicting what will happen if several people or severalinstitutions are making decisions at the same time, and if theoutcome depends on the decisions of the others. The simple insightunderlying John Nash's idea is that we cannot predict the result ofthe choices of multiple decision makers if we analyze thosedecisions in isolation. Instead, we must ask what each player woulddo, taking into account the decision-making of theothers.

Nash equilibrium has been used to analyze hostile situationslike warand arms races^[2](see Prisoner's dilemma), and also howconflict may be mitigated by repeated interaction (see Tit-for-tat). It has also been used to study towhat extent people with different preferences can cooperate (seeBattle of the sexes),and whether they will take risks to achieve a cooperative outcome(see Stag hunt). It has been used to study the adoptionof technical standards^{[citationneeded]}, and also the occurrence of bankruns and currency crises (see Coordination game). Other applicationsinclude traffic flow (see Wardrop's principle), how to organizeauctions (see auction theory), the outcome of effortsexerted by multiple parties in the education process,^[3]regulatory legislation such as environmental regulations (seeTragedy of theCommons),^[4]and even penalty kicks in soccer(see Matching pennies).^[5]

A version of the Nash equilibrium concept was first used byAntoine Augustin Cournot inhis theory of oligopoly (1838). In Cournot's theory, firms choosehow much output to produce to maximize their own profit. However,the best output for one firm depends on the outputs of others. ACournot equilibrium occurs when eachfirm's output maximizes its profits given the output of the otherfirms, which is a pure strategy Nash Equilibrium.

The modern game-theoretic concept of Nash Equilibrium is insteaddefined in terms of mixed strategies, where players choose aprobability distribution over possible actions. The concept of themixed strategy Nash Equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book TheTheory of Games and Economic Behavior. However, their analysiswas restricted to the special case of zero-sum games. They showed that a mixed-strategyNash Equilibrium will exist for any zero-sum game with a finite setof actions. The contribution of John Forbes Nash in his 1951 articleNon-Cooperative Games was to define a mixed strategy NashEquilibrium for any game with a finite set of actions and provethat at least one (mixed strategy) Nash Equilibrium must exist insuch a game.

Since the development of the Nash equilibrium concept, gametheorists have discovered that it makes misleading predictions (orfails 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 theNash concept. One particularly important issue is that some Nashequilibria may be based on threats that are not 'credible'. Therefore, in 1965 Reinhard Selten proposed subgame perfect equilibrium as arefinement that eliminates equilibria which depend on non-credible threats. Other extensionsof the Nash equilibrium concept have addressed what happens if agame is repeated, or what happens if a game is playedin the absence of perfect information. However,subsequent refinements and extensions of the Nash equilibriumconcept share the main insight on which Nash's concept rests: allequilibrium concepts analyze what choices will be made when eachplayer takes into account the decision-making of others.

Informal definition

Informally, a set of strategies is a Nash equilibrium if noplayer can do better by unilaterally changing his or her strategy.To see what this means, imagine that each player is told thestrategies of the others. Suppose then that each player askshimself 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 isnot a Nash equilibrium. But if every player prefers not to switch(or is indifferent between switching and not) then the set ofstrategies is a Nash equilibrium. Thus, each strategy in a Nashequilibrium is a best response to all other strategies in thatequilibrium.^[6]

The Nash equilibrium may sometimes appear non-rational in athird-person perspective. This is because it may happen that a Nashequilibrium is not Pareto optimal.

The Nash equilibrium may also have non-rational consequences insequential games because players may "threaten" each other withnon-rational moves. For such games the subgame perfect Nashequilibrium may be more meaningful as a tool of analysis.

Formal definition

Let (S, f) be a game with n players, whereS_i is the strategy set for player i,S=S₁ × S₂ ... × S_n is theset of strategy profiles andf=(f₁(x), ..., f_n(x)) is the payofffunction for x S. Let x_i be a strategy profile of playeri and x_-i be a strategy profile of allplayers except for player i. When each player i {1, ..., n} chooses strategy x_i resulting instrategy profile x = (x₁, ..., x_n)then player i obtains payoff f_i(x). Notethat the payoff depends on the strategy profile chosen, i.e., onthe strategy chosen by player i as well as the strategieschosen by all the other players. A strategy profilex^* S is a Nash equilibrium (NE) if no unilateral deviation instrategy by any single player is profitable for that player, thatis

A game can have either a pure-strategy or a mixed Nash Equilibrium, (in the latter a purestrategy is chosen stochastically with a fixed frequency). Nash proved that if we allow mixed strategies, then everygame with a finite number of players in which each player canchoose from finitely many pure strategies has at least one Nashequilibrium.

When the inequality above holds strictly (with instead of ) for all players and all feasible alternative strategies, thenthe equilibrium is classified as a strict Nash equilibrium.If instead, for some player, there is exact equality between and some other strategy in the set , then the equilibrium is classified as a weak Nashequilibrium.

Prisoner's dilemma

The Prisoner's Dilemma has a similar matrix as depicted for theCoordination Game, but the maximum reward for each player (in thiscase, 5) is only obtained when the players' decisions aredifferent. Each player improves his own situation by switching from"Cooperating" to "Defecting," given knowledge that the otherplayer's best decision is to "Defect." The Prisoner's Dilemma thushas a single Nash Equilibrium: both players choosing to defect.

What has long made this an interesting case to study is the factthat this scenario is globally inferior to "Both Cooperating." Thatis, both players would be better off if they both chose to"Cooperate" instead of both choosing to defect. However, eachplayer could improve his own situation by breaking the mutualcooperation, no matter how the other player possibly (or certainly)changes his decision.

Coordination game

The coordination game is a classic (symmetric) two player, two strategy game, with an examplepayoff matrix shown to the right. The playersshould thus coordinate, both adopting strategy A, to receive thehighest payoff; i.e., 4. If both players chose strategy B though,there is still a Nash equilibrium. Although each player is awardedless than optimal payoff, neither player has incentive to changestrategy due to a reduction in the immediate payoff (from 2 to1).

A famous example of this type of game was called the Staghunt; in the game two players may choose to hunt a stag or arabbit, the former providing more meat (4 utility units) than thelatter (1 utility unit). The caveat is that the stag must becooperatively hunted, so if one player attempts to hunt the stag,while the other hunts the rabbit, he will fail in hunting (0utility units), whereas if they both hunt it they will split thepayload (2, 2). The game hence exhibits two equilibria at (stag,stag) and (rabbit, rabbit) and hence the players' optimal strategydepend on their expectation on what the other player may do. If onehunter trusts that the other will hunt the stag, he should hunt thestag; however if he suspects that the other will hunt the rabbit,he should hunt the rabbit. This game was used as an analogy forsocial cooperation, since much of the benefit that people gain insociety depends upon people cooperating and implicitly trusting oneanother to act in a manner corresponding with cooperation.

Another example of a coordination game is the setting where twotechnologies are available to two firms with compatible products,and they have to elect a strategy to become the market standard. Ifboth firms agree on the chosen technology, high sales are expectedfor both firms. If the firms do not agree on the standardtechnology, few sales result. Both strategies are Nash equilibriaof the game.

Driving on a road, and having to choose either to drive on theleft or to drive on the right of the road, is also a coordinationgame. For example, with payoffs 100 meaning no crash and 0 meaninga crash, the coordination game can be defined with the followingpayoff matrix:

In this case there are two pure strategy Nash equilibria, whenboth choose to either drive on the left or on the right. If weadmit mixed strategies (where a pure strategy ischosen at random, subject to some fixed probability), then thereare three Nash equilibria for the same case: two we have seen fromthe pure-strategy form, where the probabilities are (0%,100%) forplayer one, (0%, 100%) for player two; and (100%, 0%) for playerone, (100%, 0%) for player two respectively. We add another wherethe probabilities for each player is (50%, 50%).

以及基于网络的Nash均衡:

Network traffic

An application of Nash equilibria is in determining the expectedflow of traffic in a network. Consider the graph on the right. Ifwe assume that there are "cars" traveling from A to D, what is the expected distribution oftraffic in the network?

This situation can be modeled as a "game" where every travelerhas a choice of 3 strategies, where each strategy is a route from Ato D (either , , or ). The "payoff" of each strategy is the travel time of each route.In the graph on the right, a car travelling via experiences travel time of , where is the number of cars traveling on edge . Thus, payoffs for any given strategy depend on the choices ofthe other players, as is usual. However, the goal in this case isto minimize travel time, not maximize it. Equilibrium will occurwhen the time on all paths is exactly the same. When that happens,no single driver has any incentive to switch routes, since it canonly add to his/her travel time. For the graph on the right, if,for example, 100 cars are travelling from A to D, then equilibriumwill occur when 25 drivers travel via , 50 via , and 25 via . Every driver now has a total travel time of 3.75.

Notice that this distribution is not, actually, sociallyoptimal. If the 100 cars agreed that 50 travel via and the other 50 through , then travel time for any single car would actually be 3.5 whichis less than 3.75. This is also the Nash equilibrium if the pathbetween B and C is removed, which means that adding an additionalpossible route can decrease the efficiency of the system, aphenomenon known as Braess's Paradox.

Competition game

This can be illustrated by a two-player game in which bothplayers simultaneously choose an integer from 0 to 3 and they bothwin the smaller of the two numbers in points. In addition, if oneplayer chooses a larger number than the other, then he/she has togive up two points to the other.

This game has a unique pure-strategy Nash equilibrium: bothplayers choosing 0 (highlighted in light red). Any other choice ofstrategies can be improved if one of the players lowers his numberto one less than the other player's number. In the table to theright, for example, when starting at the green square it is inplayer 1's interest to move to the purple square by choosing asmaller number, and it is in player 2's interest to move to theblue square by choosing a smaller number. If the game is modifiedso that the two players win the named amount if they both choosethe same number, and otherwise win nothing, then there are 4 Nashequilibria (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 apayoff matrix. It is especially helpful in two-person games whereplayers have more than two strategies. In this case formal analysismay become too long. This rule does not apply to the case wheremixed (stochastic) strategies are of interest. The rule goes asfollows: if the first payoff number, in the duplet of the cell, isthe maximum of the column of the cell and if the second number isthe maximum of the row of the cell - then the cell represents aNash equilibrium.

We can apply this rule to a 3×3 matrix:

Using the rule, we can very quickly (much faster than withformal analysis) see that the Nash Equilibria cells are (B,A),(A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of thefirst column and 25 is the maximum of the second row. For (A,B) 25is the maximum of the second column and 40 is the maximum of thefirst row. Same for cell (C,C). For other cells, either one or bothof the duplet members are not the maximum of the corresponding rowsand columns.

This said, the actual mechanics of finding equilibrium cells isobvious: find the maximum of a column and check if the secondmember of the pair is the maximum of the row. If these conditionsare met, the cell represents a Nash Equilibrium. Check all columnsthis way to find all NE cells. An N×N matrix may have between 0 andN×N pure strategy Nash equilibria.

Stability

The concept of stability, useful in the analysis of manykinds of equilibria, can also be applied to Nash equilibria.

A Nash equilibrium for a mixed strategy game is stable if asmall change (specifically, an infinitesimal change) inprobabilities for one player leads to a situation where twoconditions hold:

1. the player who did not change has no better strategy in the newcircumstance
2. the player who did change is now playing with a strictly worsestrategy.

If these cases are both met, then a player with the small changein his mixed-strategy will return immediately to the Nashequilibrium. The equilibrium is said to be stable. If condition onedoes not hold then the equilibrium is unstable. If only conditionone holds then there are likely to be an infinite number of optimalstrategies for the player who changed. John Nash showed that the latter situationcould not arise in a range of well-defined games.

In the "driving game" example above there are both stable andunstable equilibria. The equilibria involving mixed-strategies with100% probabilities are stable. If either player changes hisprobabilities slightly, they will be both at a disadvantage, andhis opponent will have no reason to change his strategy in turn.The (50%,50%) equilibrium is unstable. If either player changes hisprobabilities, then the other player immediately has a betterstrategy at either (0%, 100%) or (100%, 0%).

Stability is crucial in practical applications of Nashequilibria, since the mixed-strategy of each player is notperfectly known, but has to be inferred from statisticaldistribution of his actions in the game. In this case unstableequilibria are very unlikely to arise in practice, since any minutechange in the proportions of each strategy seen will lead to achange in strategy and the breakdown of the equilibrium.

The Nash equilibrium defines stability only in terms ofunilateral deviations. In cooperative games such a concept is notconvincing enough. Strong Nash equilibrium allows fordeviations by every conceivable coalition.^[7]Formally, a Strong Nash equilibrium is a Nashequilibrium in which no coalition, taking the actions of itscomplements as given, can cooperatively deviate in a way thatbenefits all of its members.^[8]However, the Strong Nash concept is sometimes perceived as too"strong" in that the environment allows for unlimited privatecommunication. In fact, Strong Nash equilibrium has to be Pareto efficient. As a result of theserequirements, Strong Nash is too rare to be useful in many branchesof game theory. However, in games such as elections with many moreplayers than possible outcomes, it can be more common than a stableequilibrium.

A refined Nash equilibrium known as coalition-proof Nashequilibrium (CPNE)^[7]occurs when players cannot do better even if they are allowed tocommunicate and make "self-enforcing" agreement to deviate. Everycorrelated strategy supported by iterated strict dominance and on thePareto frontier is a CPNE.^[9]Further, it is possible for a game to have a Nash equilibrium thatis resilient against coalitions less than a specified size, k. CPNEis related to the theory of the core.

Finally in the eighties, building with great depth on such ideasMertens-stable equilibriawere introduced as a solution concept. Mertens stable equilibria satisfy bothforward induction and backward induction. In a Gametheory context stable equilibria now usually refer toMertens stable equilibria.

Occurrence

If a game has a unique Nash equilibrium and is played amongplayers under certain conditions, then the NE strategy set will beadopted. Sufficient conditions to guarantee that the Nashequilibrium is played are:

1. The players all will do their utmost to maximize their expectedpayoff as described by the game.
2. The players are flawless in execution.
3. The players have sufficient intelligence to deduce thesolution.
4. The players know the planned equilibrium strategy of all of theother players.
5. The players believe that a deviation in their own strategy willnot cause deviations by any other players.
6. There is common knowledge that all playersmeet these conditions, including this one. So, not only must eachplayer know the other players meet the conditions, but also theymust know that they all know that they meet them, and know thatthey know that they know that they meet them, and so on.

Where the conditions are not met

Examples of game theory problems in which theseconditions are not met:

1. The first condition is not met if the game does not correctlydescribe the quantities a player wishes to maximize. In this casethere is no particular reason for that player to adopt anequilibrium strategy. For instance, the prisoner’s dilemma is not adilemma if either player is happy to be jailed indefinitely.
2. Intentional or accidental imperfection in execution. Forexample, a computer capable of flawless logical play facing asecond flawless computer will result in equilibrium. Introductionof imperfection will lead to its disruption either through loss tothe player who makes the mistake, or through negation of thecommon knowledge criterionleading to possible victory for the player. (An example would be aplayer suddenly putting the car into reverse in the game of chicken, ensuring a no-loss no-winscenario).
3. In many cases, the third condition is not met because, eventhough the equilibrium must exist, it is unknown due to thecomplexity of the game, for instance in Chinese chess.^[10]Or, if known, it may not be known to all players, as when playingtic-tac-toe with a small child who desperatelywants to win (meeting the other criteria).
4. The criterion of common knowledge may not be met even if allplayers do, in fact, meet all the other criteria. Players wronglydistrusting each other's rationality may adopt counter-strategiesto expected irrational play on their opponents’ behalf. This is amajor consideration in “Chicken” or an armsrace, for example.

Where the conditions are met

Due to the limited conditions in which NE can actually beobserved, they are rarely treated as a guide to day-to-daybehaviour, or observed in practice in human negotiations. However,as a theoretical concept in economics and evolutionary biology, the NE hasexplanatory power. The payoff in economics is utility (or sometimesmoney), and in evolutionary biology gene transmission, both are thefundamental bottom line of survival. Researchers who apply gamestheory in these fields claim that strategies failing to maximizethese for whatever reason will be competed out of the market orenvironment, which are ascribed the ability to test all strategies.This conclusion is drawn from the "stability" theory above. In these situationsthe assumption that the strategy observed is actually a NE hasoften been borne out by research^{[citationneeded]}.^{[verificationneeded]}

NE and non-credible threats

The Nash equilibrium is a superset of the subgame perfect Nashequilibrium. The subgame perfect equilibrium in addition to theNash Equilibrium requires that the strategy also is a Nashequilibrium in every subgame of that game. This eliminates allnon-credible threats, that is,strategies that contain non-rational moves in order to make thecounter-player change his strategy.

The image to the right shows a simple sequential game thatillustrates the issue with subgame imperfect Nash equilibria. Inthis game player one chooses left(L) or right(R), which is followedby player two being called upon to be kind (K) or unkind (U) toplayer one, However, player two only stands to gain from beingunkind if player one goes left. If player one goes right therational player two would de facto be kind to him in that subgame.However, The non-credible threat of being unkind at 2(2) is stillpart of the blue (L, (U,U)) Nash equilibrium. Therefore, ifrational behavior can be expected by both parties the subgameperfect Nash equilibrium may be a more meaningful solution conceptwhen such dynamic inconsistencies arise.

Proof of existence

Proof using the Kakutani fixed point theorem

Nash's original proof (in his thesis) used Brouwer's fixed pointtheorem (e.g., see below for a variant). We give a simpler proofvia the Kakutani fixed point theorem,following Nash's 1950 paper (he credits DavidGale with the observation that such a simplification ispossible).

To prove the existence of a Nash Equilibrium, let be the best response of player i to the strategies of all otherplayers.

Here, , where , is a mixed strategy profile in the set of all mixed strategiesand is the payoff function for player i. Define a set-valued function such that . The existence of a Nash Equilibrium is equivalent to having a fixed point.

Kakutani's fixed point theorem guarantees the existence of afixed point if the following four conditions are satisfied.

1. is compact, convex, and nonempty.
2. is nonempty.
3. is convex.
4. is upper hemicontinuous

Condition 1. is satisfied from the fact that is a simplex and thus compact. Convexity follows from players'ability to mix strategies. is nonempty as long as players have strategies.

Condition 2. is satisfied because players maximize expectedpayoffs which is continuous function over a compact set. TheWeierstrass Extreme Value Theorem guarantees thatthere is always a maximum value.

Condition 3. is satisfied as a result of mixed strategies.Suppose , then . i.e. if two strategies maximize payoffs, then a mix between thetwo strategies will yield the same payoff.

Condition 4. is satisfied by way of Berge's maximum theorem. Because is continuous and compact, is upper hemicontinuous.

Therefore, there exists a fixed point in and a Nash Equilibrium.^[11]

When Nash made this point to John von Neumann in 1949, von Neumannfamously 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-pointtheorem

We have a game where is the number of players and is the action set for the players. All of the action sets are finite. Let denote the set of mixed strategies for the players. The finitenessof the s ensures the compactness of .

We can now define the gain functions. For a mixed strategy , we let the gain for player on action be

The gain function represents the benefit a player gets byunilaterally changing his strategy. We now define where

for . We see that

We now use to define as follows. Let

for . It is easy to see that each is a valid mixed strategy in . It is also easy to check that each is a continuous function of , and hence is a continuous function. Now is the cross product of a finite number of compact convex sets,and so we get that is also compact and convex. Therefore we may apply the Brouwerfixed point theorem to . So has a fixed point in , call it .

I claim that is a Nash Equilibrium in . For this purpose, it suffices to show that

This simply states the each player gains no benefit byunilaterally changing his strategy which is exactly the necessarycondition for being a Nash Equilibrium.

Now assume that the gains are not all zero. Therefore, , , and such that . Note then that

So let . Also we shall denote as the gain vector indexed by actions in . Since we clearly have that . Therefore we see that

Since we have that is some positive scaling of the vector . Now I claim that

.To see this, we first note that if then this is true by definition of the gain function. Now assumethat . By our previous statements we have that

and so the left term is zero, giving us that the entireexpression is as needed.

So we finally have that

where the last inequality follows since is a non-zero vector. But this is a clear contradiction, so allthe gains must indeed be zero. Therefore is a Nash Equilibrium for as needed.

Computing Nash equilibria

If a player A has a dominant strategy then there exists a Nash equilibrium in which A plays . In the case of two players A and B, there exists a Nashequilibrium in which A plays and B plays a best response to . If is a strictly dominant strategy, A plays in all Nash equilibria. If both A and B have strictly dominantstrategies, there exists a unique Nash equilibrium in which eachplays his strictly dominant strategy.

In games with mixed strategy Nash equilibria, the probability ofa player choosing any particular strategy can be computed byassigning a variable to each strategy that represents a fixedprobability for choosing that strategy. In order for a player to bewilling to randomize, his expected payoff for each strategy shouldbe the same. In addition, the sum of the probabilities for eachstrategy of a particular player should be 1. This creates a systemof equations from which the probabilities of choosing each strategycan be derived.^[6]

Examples

In the matching pennies game, player A loses a point to B if Aand B play the same strategy and wins a point from B if they playdifferent strategies. To compute the mixed strategy Nashequilibrium, assign A the probability p of playing H and (1−p) ofplaying T, and assign B the probability q of playing H and (1−q) ofplaying 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 eachplayer to randomly choose H or T with equal probability.

关于Nash的生平:From Wikipedia, the freeencyclopedia

JohnForbes Nash, Jr.

John Forbes Nash, Jr.
Born	June13, 1928 (age84) Bluefield, West Virginia,U.S.
Residence	United States
Nationality	American
Fields	Mathematics, Economics
Institutions	MassachusettsInstitute of Technology Princeton University
Alma mater	Princeton University, Carnegie Institute ofTechnology (now part of Carnegie Mellon University)
Doctoral advisor	Albert W. Tucker
Knownfor	Nash equilibrium Nash embedding theorem Algebraic geometry Partialdifferential equations
Notable awards	Nobel MemorialPrize in Economic Sciences (1994)
Spouse	Alicia Lopez-Harrison de Lardé (m. 1957–1963;2001–present)

John Forbes Nash, Jr. (born June 13, 1928) is an Americanmathematician whose works in gametheory, differential geometry, and partialdifferential equations have provided insight into the forcesthat govern chance and events inside complex systems in daily life.His theories are used in market economics,computing, evolutionary biology, artificial intelligence, accounting,politics and military theory. Serving as a Senior ResearchMathematician at Princeton University during the latterpart of his life, he shared the 1994 Nobel MemorialPrize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

Nash is the subject of the Hollywood movie A Beautiful Mind. The film,loosely based on the biography of the same name, focuseson Nash's mathematical genius and struggle with paranoidschizophrenia.^[1][2]

In his own words, he states,

I later spent times of the order of five to eight months inhospitals in New Jersey, always on an involuntary basis and alwaysattempting a legal argument for release. And it did happen thatwhen I had been long enough hospitalized that I would finallyrenounce delusional hypotheses and revert to thinking of myself asa human of more conventional circumstances and return tomathematical research. In these interludes of, as it were, enforcedrationality, I did succeed in doing some respectable mathematicalresearch. Thus there came about the research for "Le problème de Cauchy pour les équationsdifférentielles d'un fluide général"; the idea that Prof. Hironakacalled "the Nash blowing-up transformation"; and those of "ArcStructure of Singularities" and "Analyticity of Solutions ofImplicit Function Problems with Analytic Data".

But after my return to the dream-like delusional hypotheses inthe later 60's I became a person of delusionally influencedthinking but of relatively moderate behavior and thus tended toavoid hospitalization and the direct attention ofpsychiatrists.

Thus further time passed. Then gradually I began tointellectually reject some of the delusionally influenced lines ofthinking which had been characteristic of my orientation. Thisbegan, most recognizably, with the rejection ofpolitically-oriented thinking as essentially a hopeless waste ofintellectual effort.^[3]

Early lifeand career

Nash was born on June 13, 1928, in Bluefield, West Virginia. Hisfather, after whom he is named, was an electrical engineer for theAppalachian Electric PowerCompany. His mother, born Margaret Virginia Martin and known asVirginia, had been a schoolteacher before she married. Both parentspursued opportunities to supplement their son's education,providing him with encyclopedias and even allowing him to takeadvanced mathematics courses at a local college while still in highschool. After attending Carnegie Institute ofTechnology (now Carnegie Mellon University) andgraduating in 1948 with bachelor's and master's degrees inmathematics, he accepted a scholarship to Princeton Universitywhere he pursued his graduate studies in Mathematics.^[3]

Post-graduate career

Nash's advisor and former Carnegie Tech professor R.J. Duffinwrote a letter of recommendation consisting of a single sentence:"This man is a genius."^[4]Nash was accepted by Harvard University, but the chairman ofthe mathematics department of Princeton, Solomon Lefschetz, offered him the John S. Kennedyfellowship, which was enough to convince Nash that Harvard valuedhim less.^[5]Thus he went to Princeton where he worked on his equilibrium theory. He earned a doctorate in 1950 with a 28-pagedissertation onnon-cooperative games.^[6]The thesis, which was written under the supervision of Albert W. Tucker, contained the definitionand properties of what would later be called the "Nash equilibrium". These studies led to fourarticles:

Nash did ground-breaking work in the area of real algebraic geometry:

His work in mathematics includes the Nash embedding theorem, which showsthat any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space. He also made significantcontributions to the theory of nonlinear parabolic partialdifferential equations and to singularity theory.

In the book A Beautiful Mind, author Sylvia Nasarexplains that Nash was working on proving a theorem involving ellipticpartial differential equations when, in 1956, he suffered a severedisappointment when he learned of an Italian mathematician,Ennio de Giorgi, who had published a proof acouple of months^[vague]before Nash achieved his proof. Each took different routes to getto their solutions. The two mathematicians met each other at theCourant Institute of Mathematical Sciences of New York University during the summer of1956. It has been speculated that if only one of them had solvedthe problem, he would have been given the Fields Medal for the proof.^[3]

In 2011, the National Security Agencydeclassified letters written by Nash in 1950s, in which he hadproposed a new encryption-decryption machine.^[8]The letters show that Nash had anticipated many concepts of moderncryptography, which are based on computationalhardness.^[9]

Personallife

In 1951, Nash went to the MassachusettsInstitute of Technology as a C. L. E. Moore Instructor in themathematics faculty. There, he met Alicia Lopez-Harrison de Lardé(born January 1, 1933), a physicsstudent from El Salvador, whom he married in February1957 at a Catholic ceremony, although Nash was an atheist.^[10]She admitted Nash to a mental hospital in 1959for schizophrenia; their son, John Charles MartinNash, was born soon afterward, but remained nameless for a yearbecause his mother felt that her husband should have a say in thename.

Nash and de Lardé divorced in 1963, though after his finalhospital discharge in 1970 Nash lived in de Lardé's house. Theywere remarried in 2001.

Nash has been a longtime resident of West Windsor Township,New Jersey.^[11]

Mentalillness

Schizophrenia

Nash began to show signs of extreme paranoia and his wife later described his behavioras erratic, as he began speaking of characters like Charles Hermanand William Parcher who were putting him in danger. Nash seemed tobelieve that all men who wore red ties were part of a communistconspiracy against him. Nash mailed letters to embassies inWashington, D.C., declaring that they wereestablishing a government.^[12][13]

He was admitted to the McLean Hospital, April–May 1959, where he wasdiagnosed with paranoidschizophrenia. The clinical picture is dominated by relativelystable, often paranoid, fixed beliefs that are either false,over-imaginative or unrealistic, usually accompanied by experiencesof seemingly real perception of something not actuallypresent— particularly auditory and perceptionaldisturbances, a lack of motivation for life, and mild clinicaldepression.^[14]Upon his release, Nash resigned from MIT, withdrew his n,and went to Europe, unsuccessfully seeking political asylum inFrance and East Germany. He tried to renounce his U.S.citizenship. After a problematic stay in Parisand Geneva, he was arrested by the French police anddeported back to the United States at the request of the U.S.government.

In 1961, Nash was committed to the New Jersey State Hospital at Trenton. Over thenext nine years, he spent periods in psychiatric hospitals, where, aside fromreceiving antipsychotic medications, he was administered insulin shock therapy.^[14][15][16]

Although he sometimes took prescribed medication, Nash laterwrote that he only ever did so under pressure. After 1970, he wasnever committed to the hospital again and he refused anymedication. According to Nash, the film A Beautiful Mind inaccuratelyimplied that he was taking the new atypical antipsychotics during thisperiod. He attributed the depiction to the screenwriter (whosemother, he notes, was a psychiatrist), who was worried aboutencouraging people with the disorder to stop taking theirmedication.^[17]Others, however, have questioned whether the fabrication obscured akey question as to whether recovery from problems like Nash's canactually be hindered by such drugs,^[18]and Nash has said they are overrated and that the adverse effectsare not given enough consideration once someone is deemed mentally ill.^[19][20][21]According to Sylvia Nasar, author of the book A Beautiful Mind, on which themovie was based, Nash recovered gradually with the passage of time.Encouraged by his then former wife, de Lardé, Nash worked in acommunitarian setting wherehis eccentricities were accepted. De Lardé said of Nash, "it's justa question of living a quiet life".^[13]

Nash dates the start of what he terms "mental disturbances" tothe early months of 1959 when his wife was pregnant. He hasdescribed a process of change "from scientific rationality ofthinking into the delusional thinkingcharacteristic of persons who are psychiatrically diagnosed as'schizophrenic' or 'paranoid schizophrenic'"^[22]including seeing himself as a messenger or having a specialfunction in some way, and with supporters and opponents and hiddenschemers, and a feeling of being persecuted, and looking for signsrepresenting divine revelation.^[23]Nash has suggested his delusional thinking was related to hisunhappiness and his striving to feel important and be recognized,and to his characteristic way of thinking such that "I wouldn'thave had good scientific ideas if I had thought more normally." Hehas said, "If I felt completely pressureless I don't think I wouldhave gone in this pattern".^[24]He does not see a categorical distinction between terms such asschizophrenia and bipolar disorder.^[25]Nash reports that he did not hear voices until around 1964, laterengaging in a process of rejecting them.^[26]He reports that he was always taken to hospitals against his will,and only temporarily renounced his "dream-like delusionalhypotheses" after being in a hospital long enough to decide tosuperficially conform - to behave normally or to experience"enforced rationality". Only gradually on his own did he"intellectually reject" some of the "delusionally influenced" and"politically-oriented" thinking as a waste of effort. However, by1995, although he was "thinking rationally again in the style thatis characteristic of scientists," he says he also felt morelimited.^[22][27]

Recognition and latercareer

At Princeton, campus legend Nash became "The Phantom of FineHall" (Princeton's mathematics center), a shadowy figure who wouldscribble arcane equations on blackboards in the middle of thenight. The legend appears in a work of fiction based on Princetonlife, The Mind-Body Problem, by Rebecca Goldstein.

In 1978, Nash was awarded the John von Neumann Theory Prizefor his discovery of non-cooperative equilibria, now calledNash equilibria. He won the Leroy P. Steele Prize in 1999.

In 1994, he received the Nobel MemorialPrize in Economic Sciences (along with John Harsanyi and Reinhard Selten) as a result of his gametheory work as a Princeton graduate student. In the late 1980s,Nash had begun to use email to gradually link with workingmathematicians who realized that he was the John Nash andthat his new work had value. They formed part of the nucleus of agroup that contacted the Bank of Sweden's Nobelaward committee and were able to vouch for Nash's mental healthability to receive the award in recognition of his earlywork.^{[citationneeded]}

As of 2011 Nash's recent work involves ventures in advanced gametheory, including partial agency, which show that, as in his earlycareer, he prefers to select his own path and problems. Between1945 and 1996, he published 23 scientific studies.

Nash has suggested hypotheses on mental illness. He hascompared not thinking in an acceptable manner, or being "insane"and not fitting into a usual social function, to being "on strike" from an economic point of view. He hasadvanced evolutionary psychology views aboutthe value of human diversity and the potential benefits ofapparently nonstandard behaviors or roles.^[28]

Nash has developed work on the role of money in society. Withinthe framing theorem that people can be so controlled and motivatedby money that they may not be able to reason rationally about it,he has criticized interest groups that promote quasi-doctrinesbased on Keynesian economics that permitmanipulative short-term inflation and debt tacticsthat ultimately undermine currencies. He has suggested a global"industrial consumption price index" system that would support thedevelopment of more "ideal money" that people could trust ratherthan more unstable "bad money". He notes that some of his thinkingparallels economist and political philosopher Friedrich Hayek's thinking regarding moneyand a nontypical viewpoint of the function of theauthorities.^[29][30]

Nash received an honorary degree, Doctor of Science andTechnology, from Carnegie Mellon University in1999, an honorary degree in economics from the University of NaplesFederico II on March 19, 2003,^[31]an honorary doctorate in economics from the University ofAntwerp in April 2007, and was keynote speaker at a conferenceon Game Theory. He has also been a prolific guest speaker at anumber of world-class events, such as the Warwick Economics Summit in 2005held at the University of Warwick.

Nash has an Erdős number of 3.^[32][33][34]

博弈论大师：约翰·福布斯·纳什（John Forbes Nash Jr.）

(from:http://forum.e2002.com/forum.php?mod=viewthread&tid=325578)

约翰·纳什1928年出生在美国西弗吉尼亚州工业城布鲁菲尔德的一个富裕家庭。他的父亲是受过良好教育的电子工程师，母亲则是拉丁语教师。纳什从小就很孤僻，他宁愿钻在书堆里，也不愿出去和同龄的孩子玩耍。但是那个时候，纳什的数学成绩并不好，小学老师常常向他的家长抱怨纳什的数学有问题，因为他常常使用一些奇特的解题方法。而到了中学，这种情况就更加频繁了，老师在黑板上演算了整个黑板的习题，纳什只用简单的几步就能解出答案。
　　

中学毕业后，约翰·纳什进入了匹兹堡的卡耐基技术学院化学工程系。1948年，大学三年级的纳什同时被哈佛大学、普林斯顿大学、芝加哥大学和密执安大学录取，而普林斯顿大学则表现得更加热情，当普林斯顿大学的数学系主任莱夫谢茨感到纳什的犹豫时，就立即写信敦促他选择普林斯顿，这促使纳什接受了一份1150美元的奖学金。

　　当时的普林斯顿已经成了全世界的数学中心，爱因斯坦等世界级大师均云集于此。在普林斯顿自由的学术空气里，纳什如鱼得水，他21岁博士毕业，不到30岁已经闻名遐迩。1958年，纳什因其在数学领域的优异工作被美国《财富》杂志评为新一代天才数学家中最杰出的人物。

　　约翰·纳最重要的理论就是现在广泛出现在经济学教科书上的“纳什均衡”。而“纳什均衡”最著名的一个例子就是“囚徒困境”，大意是：一个案子的两个嫌疑犯被分开审讯，警官分别告诉两个囚犯，如果两人均不招供，将各被判刑一年；如果你招供，而对方不招供，则你将被判刑三个月，而对方将被判刑十年；如果两人均招供，将均被判刑五年。于是，两人同时陷入招供还是不招供的两难处境。两个囚犯符合自己利益的选择是坦白招供，原本对双方都有利的策略不招供从而均被判刑１年就不会出现。这样两人都选择坦白的策略以及因此被判５年的结局被称为“纳什均衡”，也叫非合作均衡。“纳什均衡”是他21岁博士毕业的论文，也奠定了数十年后他获得诺贝尔经济学奖的基础。

　　那时的纳什“就像天神一样英俊”，1.85米的个子，体重接近77公斤，手指修长、优雅，双手柔软、漂亮，还有一张英国贵族的容貌。他的才华和个人魅力吸引了一个漂亮的女生——艾里西亚，她是当时麻省理工学院物理系仅有的两名女生之一。1957年，他们结婚了。之后漫长的岁月证明，这也许正是纳什一生中比获得诺贝尔奖更重要的事。

　　就在事业爱情双双得意的时候，纳什也因为喜欢独来独往，喜欢解决折磨人的数学问题而被人们称为“孤独的天才”。他不是一个善于为人处世并受大多数人欢迎的人，他有着天才们常有的骄傲、自我中心的毛病。他的同辈人基本认为他不可理喻，他们说他“孤僻，傲慢，无情，幽灵一般，古怪，沉醉于自己的隐秘世界，根本不能理解别人操心的世俗事务。”
普林斯顿的幽灵
　　1958年的秋天，正当艾里西亚半惊半喜地发现自己怀孕时，纳什却为自己的未来满怀心事，越来越不安。系主任马丁已答应在那年冬天给他永久教职，但是纳什却出现了各种稀奇古怪的行为：他担心被征兵入伍而毁了自己的数学创造力，他梦想成立一个世界政府，他认为《纽约时报》上每一个字母都隐含着神秘的意义，而只有他才能读懂其中的寓意。他认为世界上的一切都可以用一个数学公式表达。他给联合国写信，跑到华盛顿给每个国家的大使馆投递信件，要求各国使馆支持他成立世界政府的想法。他迷上了法语，甚至要用法语写数学论文，他认为语言与数学有神秘的关联……

　　终于，在孩子出生以前，纳什被送进了精神病医院。

　　几年后，因为艾里西亚无法忍受在纳什的阴影下生活，他们离婚了，但是她并没有放弃纳什。离婚以后，艾里西亚再也没有结婚，她依靠自己作为电脑程序员的微薄收入和亲友的接济，继续照料前夫和他们惟一的儿子。她坚持纳什应该留在普林斯顿，因为如果一个人行为古怪，在别的地方会被当作疯子，而在普林斯顿这个广纳天才的地方，人们会充满爱心地想，他可能是一个天才。

　　于是，在上世纪70和80年代，普林斯顿大学的学生和学者们总能在校园里看见一个非常奇特、消瘦而沉默的男人在徘徊，他穿着紫色的拖鞋，偶尔在黑板上写下数字命理学的论题。他们称他为“幽灵”，他们知道这个“幽灵”是一个数学天才，只是突然发疯了。如果有人敢抱怨纳什在附近徘徊使人不自在的话，他会立即受到警告：“你这辈子都不可能成为像他那样杰出的数学家！”

　　正当纳什本人处于梦境一般的精神状态时，他的名字开始出现在70年代和80年代的经济学课本、进化生物学论文、政治学专著和数学期刊的各领域中。他的名字已经成为经济学或数学的一个名词，如“纳什均衡”、“纳什谈判解”、“纳什程序”、“德乔治－纳什结果”、“纳什嵌入”和“纳什破裂”等。

　　纳什的博弈理论越来越有影响力，但他本人却默默无闻。大部分曾经运用过他的理论的年轻数学家和经济学家都根据他的论文发表日期，想当然地以为他已经去世。即使一些人知道纳什还活着，但由于他特殊的病症和状态，他们也把纳什当成了一个行将就木的废人。
传奇仍在继续
　　有人说，站在金字塔尖上的科学家都有一个异常孤独的大脑，纳什发疯是因为他太孤独了。但是，纳什在发疯之后却并不孤独，他的妻子、朋友和同事们没有抛弃他，而是不遗余力地帮助他，挽救他，试图把他拉出疾病的深渊。

　　尽管纳什决心辞去麻省理工学院教授的职位，但他的同事和上司们还是设法为他保全了保险。他的同事听说他被关进了精神病医院后，给当时美国著名的精神病学专家打电话说：“为了国家利益，必须竭尽所能将纳什教授复原为那个富有创造精神的人。”越来越多的人聚集到纳什的身边，他们设立了一个资助纳什治疗的基金，并在美国数学会发起一个募捐活动。基金的设立人写到：“如果在帮助纳什返回数学领域方面有什么事情可以做，哪怕是在一个很小的范围，不仅对他，而且对数学都很有好处。”对于普林斯顿大学为他做的一切，纳什在清醒后表示，“我在这里得到庇护，因此没有变得无家可归。”

　　守得云开见月明，妻子和朋友的关爱终于得到了回报。80年代末的一个清晨，当普里斯顿高等研究院的戴森教授像平常一样向纳什道早安时，纳什回答说：“我看见你的女儿今天又上了电视。”从来没有听到过纳什说话的戴森仍然记得当时的震惊之情，他说：“我觉得最奇妙的还是这个缓慢的苏醒，渐渐地他就越来越清醒，还没有任何人曾经像他这样清醒过来。”

　　纳什渐渐康复，从疯癫中苏醒，而他的苏醒似乎是为了迎接他生命中的一件大事：荣获诺贝尔经济学奖。当1994年瑞典国王宣布年度诺贝尔经济学奖的获得者是约翰·纳什时，数学圈里的许多人惊叹的是：原来纳什还活着。

　　纳什没有因为获得了诺贝尔奖就放弃他的研究，在诺贝尔奖得主自传中，他写道：从统计学看来，没有任何一个已经66岁的数学家或科学家能通过持续的研究工作，在他或她以前的成就基础上更进一步。但是，我仍然继续努力尝试。由于出现了长达25年部分不真实的思维，相当于提供了某种假期，我的情况可能并不符合常规。因此，我希望通过目前的研究成果或以后出现的任何新鲜想法，取得一些有价值的成果。”

　　而在2001年，经过几十年风风雨雨的艾里西亚与约翰·纳什复婚了。事实上，在漫长的岁月里，艾里西亚在心灵上从来没有离开过纳什。这个伟大的女性用一生与命运进行博弈，她终于取得了胜利。而纳什，也在得与失的博弈中取得了均衡。

　　2005年6月1日晚，诺贝尔北京论坛在故宫东侧菖蒲河公园内的东苑戏楼闭幕。热闹的晚宴结束后，纳什没有搭乘主办方安排的专车，而是一个人夹着文件夹走出了东苑戏楼。他像一个普通老人一样步行穿过菖蒲河公园，然后绕到南河沿大街路西的人行横道上等待红绿灯。绿灯亮起，老人隅隅独行的背影在暮色中渐行渐远，终于消失不见。

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