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105-399: If each player has chosen a strategy  – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B)

210-493: A k +1 , b k +1 , p k +1 and q k +1 satisfy conditions (1)–(6). We have a pair of sequences of intervals, and we would like to show them to converge to a limiting point with the Bolzano-Weierstrass theorem . To do so, we construe these two interval sequences as a single sequence of points, ( a n , p n , b n , q n ). This lies in the cartesian product [0,1]×[0,1]×[0,1]×[0,1], which

315-514: A Nobel Prize in Economics . In this case: In general equilibrium theory in economics, Kakutani's theorem has been used to prove the existence of set of prices which simultaneously equate supply with demand in all markets of an economy. The existence of such prices had been an open question in economics going back to at least Walras . The first proof of this result was constructed by Lionel McKenzie . In this case: Kakutani's fixed-point theorem

420-467: A behavior strategy assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the game tree , while a behavior strategy can be seen as a stochastic path. The relationship between mixed and behavior strategies

525-479: A fixed point , i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem . The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem

630-576: A game theory context stable equilibria now usually refer to Mertens stable equilibria. 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: Examples of game theory problems in which these conditions are not met: In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with

735-417: A n -simplex is the higher-dimensional version of a triangle. Proving Kakutani's theorem for set-valued function defined on a simplex is not essentially different from proving it for intervals. The additional complexity in the higher-dimensional case exists in the first step of chopping up the domain into finer subpieces: Once these changes have been made to the first step, the second and third steps of finding

840-455: A * and by condition (6), q * ≤ b *. But since ( b i − a i ) ≤ 2 by condition (2), So, b * equals a *. Let x = b * = a *. Then we have the situation that If p * = q * then p * = x = q *. Since p * ∈ φ( x ), x is a fixed point of φ. Otherwise, we can write the following. Recall that we can parameterize a line between two points a and b by (1-t)a + tb. Using our finding above that q<x<p, we can create such

945-399: A choice of 3 strategies and where each strategy is a route from A to D (one of ABD , ABCD , or ACD ). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via ABD experiences travel time of 1 + x 100 + 2 {\displaystyle 1+{\frac {x}{100}}+2} , where x {\displaystyle x}

1050-584: A compact Hausdorff range space Y , a set-valued function φ :  X →2 has a closed graph if and only if it is upper hemicontinuous and φ ( x ) is a closed set for all x . Since all Euclidean spaces are Hausdorff (being metric spaces ) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent. The Kakutani fixed point theorem can be used to prove

1155-424: A coordination game is the stag hunt . Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for

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1260-539: A fixed point: x = 0.5 is a fixed point, since x is contained in the interval [0,1]. The requirement that φ ( x ) be convex for all x is essential for the theorem to hold. Consider the following function defined on [0,1]: The function has no fixed point. Though it satisfies all other requirements of Kakutani's theorem, its value fails to be convex at x = 0.5. Consider the following function defined on [0,1]: The function has no fixed point. Though it satisfies all other requirements of Kakutani's theorem, its graph

1365-411: A function of the strategies. The strategy profile s ∗ {\displaystyle s^{*}} is a Nash equilibrium if A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak : a player might be indifferent among several strategies given the other players' choices. It is unique and called a strict Nash equilibrium if the inequality

1470-414: A game. A strategy profile must include one and only one strategy for every player. A player's strategy set defines what strategies are available for them to play. A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, a game of rock paper scissors comprises a single move by each player—and each player's move is made without knowledge of

1575-570: A large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents. Later, Aumann and Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in beliefs , rather than actions. For instance, in rock paper scissors an equilibrium in beliefs would have each player believing

1680-439: A limiting point and proving that it is a fixed point are almost unchanged from the one-dimensional case. Kakutani's theorem for n-simplices can be used to prove the theorem for an arbitrary compact, convex S . Once again we employ the same technique of creating increasingly finer subdivisions. But instead of triangles with straight edges as in the case of n-simplices, we now use triangles with curved edges. In formal terms, we find

1785-400: A line between p and q as a function of x (notice the fractions below are on the unit interval). By a convenient writing of x, and since φ( x ) is convex and it once again follows that x must belong to φ( x ) since p * and q * do and hence x is a fixed point of φ. In dimensions greater one, n -simplices are the simplest objects on which Kakutani's theorem can be proved. Informally,

1890-415: A payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of

1995-414: A player prefers "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 strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to the other players' strategies in that equilibrium. Formally, let S i {\displaystyle S_{i}} be

2100-430: A pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability 1 and every other strategy with probability 0 . A totally mixed strategy is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for equilibrium refinement such as trembling hand perfect equilibrium .) In

2205-446: A pure strategy for each player or might be a probability distribution over strategies for each player. Nash equilibria need not exist if the set of choices is infinite and non-compact. For example: However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players. Rosen extended Nash's existence theorem in several ways. He considers an n-player game, in which

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2310-446: A pure strategy for the game. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.) Since probabilities are continuous, there are infinitely many mixed strategies available to a player. Since probabilities are being assigned to strategies for a specific player when discussing the payoffs of certain scenarios the payoff must be referred to as "expected payoff". Of course, one can regard

2415-504: A sequence x n {\displaystyle x_{n}} such that f n ( x n ) = x n {\displaystyle f_{n}(x_{n})=x_{n}} , so ( x n , x n ) ∈ [ graph ⁡ ( φ ) ] 1 / n {\displaystyle (x_{n},x_{n})\in [\operatorname {graph} (\varphi )]_{1/n}} . Since S {\displaystyle S}

2520-402: A simplex which covers S and then move the problem from S to the simplex by using a deformation retract . Then we can apply the already established result for n-simplices. Kakutani's fixed-point theorem was extended to infinite-dimensional locally convex topological vector spaces by Irving Glicksberg and Ky Fan . To state the theorem in this case, we need a few more definitions: Then

2625-478: A small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold: If these cases are both met, then a player with the small change in their 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

2730-427: A soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game illustrates this situation, a simplified form of the game studied by Chiappori, Levitt, and Groseclose (2002). It assumes that if

2835-525: A strategy profile, a set consisting of one strategy for each player, where s − i ∗ {\displaystyle s_{-i}^{*}} denotes the N − 1 {\displaystyle N-1} strategies of all the players except i {\displaystyle i} . Let u i ( s i , s − i ∗ ) {\displaystyle u_{i}(s_{i},s_{-i}^{*})} be player i's payoff as

2940-457: A subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior , but 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 Nash in his 1951 article "Non-Cooperative Games"

3045-430: A unique prediction. They have proposed many solution concepts ('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not ' credible '. In 1965 Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats . Other extensions of

3150-416: A very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ x , accept any offer > x ; for x in ($ 0, $ 1, $ 2, ..., $ 20)}. A pure strategy provides a complete definition of how a player will play a game. Pure strategy can be thought about as a singular concrete plan subject to the observations they make during

3255-416: Is a compact set by Tychonoff's theorem . Since our sequence ( a n , p n , b n , q n ) lies in a compact set, it must have a convergent subsequence by Bolzano-Weierstrass . Let's fix attention on such a subsequence and let its limit be ( a *, p *, b *, q *). Since the graph of φ is closed it must be the case that p * ∈ φ( a *) and q * ∈ φ( b *). Moreover, by condition (5), p * ≥

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3360-496: Is a Cartesian product of convex sets S 1 ,..., S n , such that the strategy of player i must be in S i . This represents the case that the actions of each player i are constrained independently of other players' actions. If the following conditions hold: Then a Nash equilibrium exists. The proof uses the Kakutani fixed-point theorem . Rosen also proves that, under certain technical conditions which include strict concavity,

3465-472: Is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In a game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth. Nash showed that there

3570-411: Is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto efficient . As a result of these requirements, strong Nash

3675-552: Is a Nash equilibrium, possibly in mixed strategies , for every finite game. Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers . In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what

3780-432: Is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it. A game can have a pure-strategy or a mixed-strategy Nash equilibrium. In the latter, not every player always plays the same strategy. Instead, there is a probability distribution over different strategies. Suppose that in the Nash equilibrium, each player asks themselves: "Knowing

3885-469: Is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "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?" For instance if

3990-486: Is also true. A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997) with their Absent-Minded Driver game. Outcome equivalence combines the mixed and behavioral strategy of Player i in relation to the pure strategy of Player i’s opponent. Outcome equivalence is defined as the situation in which, for any mixed and behavioral strategy that Player i takes, in response to any pure strategy that Player I’s opponent plays,

4095-407: 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 payoff pair of the cell, is the maximum of

4200-436: Is compact, we can take a convergent subsequence x n → x {\displaystyle x_{n}\to x} . Then ( x , x ) ∈ graph ⁡ ( φ ) {\displaystyle (x,x)\in \operatorname {graph} (\varphi )} since it is a closed set. The proof of Kakutani's theorem is simplest for set-valued functions defined over closed intervals of

4305-519: Is demonstrated in the Absent-minded Driver game. With perfect recall and information, the driver has a single pure strategy, which is [continue, exit], as the driver is aware of what intersection (or decision node) they are at when they arrive to it. On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing at both intersections, maximized at p=2/3 (reference). This simple one player game demonstrates

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4410-414: Is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3). In equilibrium, the kicker kicks to their best side only 1/3 of

4515-445: Is helpful to think about a "strategy" as a list of directions, and a "move" as a single turn on the list of directions itself. This strategy is based on the payoff or outcome of each action. The goal of each agent is to consider their payoff based on a competitors action. For example, competitor A can assume competitor B enters the market. From there, Competitor A compares the payoffs they receive by entering and not entering. The next step

4620-453: Is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1. The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right

4725-436: Is not closed; for example, consider the sequences x n = 0.5 - 1/ n , y n = 3/4. Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem: This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article. We can show this by using the closed graph theorem for set-valued functions, which says that for

4830-400: Is often confused or conflated with that of a move or action, because of the correspondence between moves and pure strategies in most games : for any move X , "always play move X " is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strategies as being a different type of thing from actions, and therefore distinct. It

4935-617: Is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies . However, many games do have pure strategy Nash equilibria (e.g. the Coordination game , the Prisoner's dilemma , the Stag hunt ). Further, games can have both pure strategy and mixed strategy equilibria. An easy example

5040-443: Is strict so one strategy is the unique best response: The strategy set S i {\displaystyle S_{i}} can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g. S i = { Yes , No } . {\displaystyle S_{i}=\{{\text{Yes}},{\text{No}}\}.} Or,

5145-400: 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 N×N matrix may have between 0 and N×N pure-strategy Nash equilibria. The concept of stability , useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria. A Nash equilibrium for a mixed-strategy game is stable if

5250-408: Is the number of cars traveling on edge 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 their travel time. For

5355-535: Is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2. During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability. Game theorist Ariel Rubinstein describes alternative ways of understanding

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5460-405: Is the subject of Kuhn's theorem , a behavioral outlook on traditional game-theoretic hypotheses. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse

5565-451: Is to assume Competitor B does not enter and then consider which payoff is better based on if Competitor A chooses to enter or not enter. This technique can identify dominant strategies where a player can identify an action that they can take no matter what the competitor does to try to maximize the payoff. A strategy profile (sometimes called a strategy combination ) is a set of strategies for all players which fully specifies all actions in

5670-468: Is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium. A refined Nash equilibrium known as coalition-proof Nash equilibrium (CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by iterated strict dominance and on

5775-837: Is used in proving the existence of cake allocations that are both envy-free and Pareto efficient . This result is known as Weller's theorem . Brouwer's fixed-point theorem is a special case of Kakutani fixed-point theorem. Conversely, Kakutani fixed-point theorem is an immediate generalization via the approximate selection theorem : By the approximate selection theorem, there exists a sequence of continuous f n : S → S {\displaystyle f_{n}:S\to S} such that graph ⁡ ( f n ) ⊂ [ graph ⁡ ( φ ) ] 1 / n {\displaystyle \operatorname {graph} (f_{n})\subset [\operatorname {graph} (\varphi )]_{1/n}} . By Brouwer fixed-point theorem, there exists

5880-546: 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 . Nash proved that if mixed strategies (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be

5985-407: The expectation of the player who did the change, if the other player's mixed strategy is still (50%,50%)), 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 their actions in

6090-421: The minimax theorem in the theory of zero-sum games . This application was specifically discussed by Kakutani's original paper. Mathematician John Nash used the Kakutani fixed point theorem to prove a major result in game theory . Stated informally, the theorem implies the existence of a Nash equilibrium in every finite game with mixed strategies for any finite number of players. This work later earned him

6195-497: The subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. The coordination game is a classic two-player, two- strategy game, as shown in the example payoff matrix to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1. A famous example of

6300-604: The Kakutani–Glicksberg–Fan theorem can be stated as: The corresponding result for single-valued functions is the Tychonoff fixed-point theorem . There is another version that the statement of the theorem becomes the same as that in the Euclidean case: In his game theory textbook, Ken Binmore recalls that Kakutani once asked him at a conference why so many economists had attended his talk. When Binmore told him that it

6405-417: 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 complete information . However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others. A strategy profile

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6510-403: The behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy

6615-432: The benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation. Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with

6720-463: The case that p 0 ≥ 0 and hence condition (5) is fulfilled. Similarly condition (6) is fulfilled by q 0 . Now suppose we have chosen a k , b k , p k and q k satisfying (1)–(6). Let, Then m ∈ [0,1] because [0,1] is convex . If there is a r ∈ φ( m ) such that r ≥ m , then we take, Otherwise, since φ( m ) is non-empty, there must be a s ∈ φ( m ) such that s ≤ m . In this case let, It can be verified that

6825-581: The closed intervals [ a i , b i ] form a sequence of subintervals of [0,1]. Condition (2) tells us that these subintervals continue to become smaller while condition (3)–(6) tell us that the function φ shifts the left end of each subinterval to its right and shifts the right end of each subinterval to its left. Such a sequence can be constructed as follows. Let a 0 = 0 and b 0 = 1. Let p 0 be any point in φ(0) and q 0 be any point in φ(1). Then, conditions (1)–(4) are immediately fulfilled. Moreover, since p 0 ∈ φ(0) ⊂ [0,1], it must be

6930-404: 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 3×3 matrix: Using the rule, we can very quickly (much faster than with formal 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 the first column and 25 is the maximum of

7035-439: The concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally. The modern concept of Nash equilibrium is instead defined in terms of mixed strategies , where players choose a probability distribution over possible pure strategies (which might put 100% of the probability on one pure strategy; such pure strategies are

7140-400: The concept. The first, due to Harsanyi (1973), is called purification , and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors. A second interpretation imagines the game players standing for

7245-401: The course of the game of play. In particular, it determines the move a player will make for any situation they could face. A player's strategy set is the set of pure strategies available to that player. A mixed strategy is an assignment of a probability to each pure strategy. When enlisting mixed strategy, it is often because the game does not allow for a rational description in specifying

7350-557: The definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the solution. For instance, strictly speaking in the Ultimatum game a player can have strategies such as: Reject offers of ($ 1, $ 3, $ 5, ..., $ 19), accept offers of ($ 0, $ 2, $ 4, ..., $ 20) . Including all such strategies makes for

7455-442: The education process, regulatory legislation such as environmental regulations (see tragedy of the commons ), natural resource management, analysing strategies in marketing, penalty kicks in football (see matching pennies ), robot navigation in crowds, energy systems, transportation systems, evacuation problems and wireless communications. Nash equilibrium is named after American mathematician John Forbes Nash Jr . The same idea

7560-594: The equilibrium is unique. Nash's result refers to the special case in which each S i is a simplex (representing all possible mixtures of pure strategies), and the payoff functions of all players are bilinear functions of the strategies. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal . Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out. For such games

7665-515: The figure at the right, satisfies all Kakutani's conditions, and indeed it has many fixed points: any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of fixed points in this particular case. For example, x  = 0.72 (dashed line in blue) is a fixed point since 0.72 ∈ [1 − 0.72/2, 1 − 0.72/4]. The function: satisfies all Kakutani's conditions, and indeed it has

7770-537: 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

7875-426: The game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, 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). There

7980-443: 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. Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept . Mertens stable equilibria satisfy both forward induction and backward induction . In

8085-463: The game. Perfect recall is required for equivalence as, in finite games with imperfect recall, there will be existing mixed strategies of Player I in which there is no equivalent behavior strategy. This is fully described in the Absent-Minded Driver game formulated by Piccione and Rubinstein. In short, this game is based on the decision-making of a driver with imperfect recall, who needs to take

8190-470: The goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie). This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left)

8295-529: 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 ABD , 50 via ABCD , and 25 via ACD . Every driver now has a total travel time of 3.75 (to see this, a total of 75 cars take the AB edge, and likewise, 75 cars take the CD edge). Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via ABD and

8400-399: The importance of perfect recall for outcome equivalence, and its impact on normal and extended form games. Kakutani fixed-point theorem In mathematical analysis , the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions . It provides sufficient conditions for a set-valued function defined on a convex , compact subset of a Euclidean space to have

8505-413: The objective of showing how equilibrium points can be connected with observable phenomenon. Strategy (game theory) In game theory , a move , action , or play is any one of the options which a player can choose in a setting where the optimal outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting

8610-453: The other 50 through ACD , then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess's paradox . 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

8715-415: The other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy. While a mixed strategy assigns a probability distribution over pure strategies,

8820-399: The other's, not as a response—so each player has the finite strategy set {rock paper scissors}. A strategy set is infinite otherwise. For instance the cake cutting game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}. In a dynamic game , games that are played over a series of time, the strategy set consists of

8925-429: The outcome distribution of the mixed and behavioral strategy must be equal. This equivalence can be described by the following formula: (Q^(U(i), S(-i)))(z) = (Q^(β(i), S(-i)))(z), where U(i) describes Player i's mixed strategy, β(i) describes Player i's behavioral strategy, and S(-i) is the opponent's strategy. Perfect recall is defined as the ability of every player in game to remember and recall all past actions within

9030-479: The player expects the others to do. Nash equilibrium requires that one's choices be consistent: no players wish to undo their decision given what the others are deciding. The concept has been used to analyze hostile situations such as wars and arms races (see prisoner's dilemma ), and also how conflict may be mitigated by repeated interaction (see tit-for-tat ). It has also been used to study to what extent people with different preferences can cooperate (see battle of

9135-400: The player is indifferent between switching and not), then the equilibrium is classified as a weak or non-strict Nash equilibrium . The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium

9240-474: The player who changed. 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 their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage

9345-511: The possible rules a player could give to a robot or agent on how to play the game. For instance, in the ultimatum game , the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject. In a Bayesian game , or games in which players have incomplete information about one another, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information. In applied game theory,

9450-420: The probabilities for each player are (50%, 50%). An application 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 x {\displaystyle x} "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

9555-417: The real line. Moreover, the proof of this case is instructive since its general strategy can be carried over to the higher-dimensional case as well. Let φ: [0,1]→2 be a set-valued function on the closed interval [0,1] which satisfies the conditions of Kakutani's fixed-point theorem. Let ( a i , b i , p i , q i ) for i = 0, 1, … be a sequence with the following properties: Thus,

9660-430: The second exit off the highway to reach home but does not remember which intersection they are at when they reach it. Figure [2] describes this game. Without perfect information (i.e. imperfect information), players make a choice at each decision node without knowledge of the decisions that have preceded it. Therefore, a player’s mixed strategy can produce outcomes that their behavioral strategy cannot, and vice versa. This

9765-402: The second row. For (A,B), 25 is the maximum of the second column and 40 is the maximum of the first row; the same applies 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

9870-399: The set of all possible strategies for player i {\displaystyle i} , where i = 1 , … , N {\displaystyle i=1,\ldots ,N} . Let s ∗ = ( s i ∗ , s − i ∗ ) {\displaystyle s^{*}=(s_{i}^{*},s_{-i}^{*})} be

9975-422: The sexes ), and whether they will take risks to achieve a cooperative outcome (see stag hunt ). It has been used to study the adoption of technical standards , and also the occurrence of bank runs and currency crises (see coordination game ). Other applications include traffic flow (see Wardrop's principle ), how to organize auctions (see auction theory ), the outcome of efforts exerted by multiple parties in

10080-407: The smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have 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 strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if

10185-415: The strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?" If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium . If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e.

10290-465: The strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose. Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make

10395-457: The strategy of each player i is a vector s i in the Euclidean space R . Denote m := m 1 +...+ m n ; so a strategy-tuple is a vector in R. Part of the definition of a game is a subset S of R such that the strategy-tuple must be in S . This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when S

10500-572: The strategy set might be a finite set of conditional strategies responding to other players, e.g. S i = { Yes | p = Low , No | p = High } . {\displaystyle S_{i}=\{{\text{Yes}}|p={\text{Low}},{\text{No}}|p={\text{High}}\}.} Or, it might be an infinite set, a continuum or unbounded, e.g. S i = { Price } {\displaystyle S_{i}=\{{\text{Price}}\}} such that Price {\displaystyle {\text{Price}}}

10605-469: The time and goalies lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies. In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player

10710-487: The time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability. Chiappori, Levitt, and Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of

10815-453: Was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria . It has subsequently found widespread application in game theory and economics . Kakutani's theorem states: The function: φ ( x ) = [ 1 − x / 2 ,   1 − x / 4 ] {\displaystyle \varphi (x)=[1-x/2,~1-x/4]} , shown on

10920-419: 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 in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes [their] payoff if

11025-505: Was used in a particular application in 1838 by Antoine Augustin Cournot in his theory of oligopoly . In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the 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. Cournot also introduced

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