{ Eastern International University (EIU) - BINH DUONG. 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). A famous example of this type of game was called the stag hunt; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat (4 utility units) than the latter (1 utility unit). (1838) Researches on the Mathematical Principles of the Theory of Wealth. Δ and has a fixed point in {\displaystyle A_{i}}

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 the education process, [3] regulatory legislation such as environmental regulations (see tragedy of the commons), [4] natural resource management, [5] analysing strategies in marketing, [6] even penalty kicks in football (see matching pennies), [7] energy systems, transportation systems, evacuation problems [8] and wireless communications. , then For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. We claim that i It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). i A . ) If both firms agree on the chosen technology, high sales are expected for both firms.

2. not a perfect equilibrium. is a simplex and thus compact. 1 such that {\displaystyle {\text{Gain}}(i,\cdot )} A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. as needed.

But this is a clear contradiction, so all the gains must indeed be zero. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed indefinitely. If the firms do not agree on the standard technology, few sales result. ) In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. ", If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium.[14]. 4. a somewhat perfect equilibrium. B. Games of Strategy. Nash proved that if we allow mixed strategies (where a player chooses probabilities of using various pure strategies), 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 a pure strategy for each player or might be a probability distribution over strategies for each player). 94.). According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes his payoff if the strategies of the others are held fixed. , 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. 0 (In the latter a pure strategy is chosen stochastically with a fixed probability). (See Nasar, 1998, p. Since Mertens stable equilibria satisfy both forward induction and backward induction. One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that equilibrium. u The game hence exhibits two equilibria at (stag, stag) and (rabbit, rabbit) and hence the players' optimal strategy depend on their expectation on what the other player may do. Answer: B Difficulty: Hard 55. A We give a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible). The equilibria involving mixed strategies with 100% probabilities are stable. When analyzing the behavior of oligopolists, which of the following is crucial for the success of game theoretic analysis? {\displaystyle f_{i}} Another example of a coordination game is the setting where two technologies are available to two firms with comparable products, and they have to elect a strategy to become the market standard. For example, with payoffs 10 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. This solution concept is now called Mertens stability, or just stability. ( f {\displaystyle G=(N,A,u)} The stage game is usually one of the well-studied 2-person games. It has been used to study the adoption of technical standards,[citation needed] and also the occurrence of bank runs and currency crises (see coordination game). Not a perfect equilibrium C .

The (50%,50%) equilibrium is unstable. Condition 2. and 3. are satisfied by way of Berge's maximum theorem. i

σ is non-empty and upper hemicontinuous. Note then that. A Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). σ What has long made this an interesting case to study is the fact that this scenario is globally inferior to "both cooperating". {\displaystyle {\text{Gain}}_{i}(\sigma ^{*},a)=0} ∈

Nash equilibrium is named after American mathematician John Forbes Nash, Jr. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability. We add another where the probabilities for each player are (50%, 50%). denote the set of mixed strategies for the players.

, {\displaystyle \Delta } {\displaystyle f_{i}} {\displaystyle G}

Σ

A perfect equilibrium B . ) and so the left term is zero, giving us that the entire expression is 0{\displaystyle 0} as needed. The players have sufficient intelligence to deduce the solution. Δ {\displaystyle r_{i}(\sigma _{-i})} d. would be willing to undertake if in a position to do so. If a player A has a dominant strategy [17] However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. Convexity follows from players' ability to mix strategies. {\displaystyle \sigma ^{*}}

σ 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.

", If any player could answer "Yes", then that set of strategies is not a Nash equilibrium. 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? The players know the planned equilibrium strategy of all of the other players. A game can have a pure-strategy or a mixed-strategy Nash equilibrium. A is the number of players and {\displaystyle r\colon \Sigma \rightarrow 2^{\Sigma }} In game theory, a solution concept is a formal rule for predicting how a game will be played. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria. ( We have a game G=(N,A,u){\displaystyle G=(N,A,u)} where N{\displaystyle N} is the number of players and A=A1×⋯×AN{\displaystyle A=A_{1}\times \cdots \times A_{N}} is the action set for the players. In a game theory context stable equilibria now usually refer to Mertens stable equilibria. A famous example of this type of game was called the stag hunt; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat (4 utility units) than the latter (1 utility unit). ) Condition 1. is satisfied from the fact that Σ{\displaystyle \Sigma } is a simplex and thus compact. If both players chose strategy B though, there is still a Nash equilibrium. What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. i

{\displaystyle f} Which of the following is a valid critique of the use of game theory in economics? Suppose that in the Nash equilibrium, each player asks themselves: "Knowing 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? This idea was formalized by Aumann, R. and A. Brandenburger, 1995, Epistemic Conditions for Nash Equilibrium, Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly know, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. A Equilibrium will occur when the time on all paths is exactly the same. 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 ABD, ABCD, or ACD). Gain is the fixed point we have: Since × A A . Imagine two prisoners held in separate cells, interrogated simultaneously, and offered deals (lighter jail sentences) for betraying their fellow criminal. If either player changes their probabilities (which would neither benefit or damage 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%). ( Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. N σ

The players all will do their utmost to maximize their expected payoff as described by the 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 the education process, [3] regulatory legislation such as environmental regulations (see tragedy of the commons), [4] natural resource management, [5] analysing strategies in marketing, [6] even penalty kicks in football (see matching pennies), [7] energy systems, transportation systems, evacuation problems [8] and wireless communications. , then For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. We claim that i It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). i A . ) If both firms agree on the chosen technology, high sales are expected for both firms.

2. not a perfect equilibrium. is a simplex and thus compact. 1 such that {\displaystyle {\text{Gain}}(i,\cdot )} A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. as needed.

But this is a clear contradiction, so all the gains must indeed be zero. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed indefinitely. If the firms do not agree on the standard technology, few sales result. ) In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. ", If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium.[14]. 4. a somewhat perfect equilibrium. B. Games of Strategy. Nash proved that if we allow mixed strategies (where a player chooses probabilities of using various pure strategies), 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 a pure strategy for each player or might be a probability distribution over strategies for each player). 94.). According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes his payoff if the strategies of the others are held fixed. , 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. 0 (In the latter a pure strategy is chosen stochastically with a fixed probability). (See Nasar, 1998, p. Since Mertens stable equilibria satisfy both forward induction and backward induction. One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that equilibrium. u The game hence exhibits two equilibria at (stag, stag) and (rabbit, rabbit) and hence the players' optimal strategy depend on their expectation on what the other player may do. Answer: B Difficulty: Hard 55. A We give a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible). The equilibria involving mixed strategies with 100% probabilities are stable. When analyzing the behavior of oligopolists, which of the following is crucial for the success of game theoretic analysis? {\displaystyle f_{i}} Another example of a coordination game is the setting where two technologies are available to two firms with comparable products, and they have to elect a strategy to become the market standard. For example, with payoffs 10 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. This solution concept is now called Mertens stability, or just stability. ( f {\displaystyle G=(N,A,u)} The stage game is usually one of the well-studied 2-person games. It has been used to study the adoption of technical standards,[citation needed] and also the occurrence of bank runs and currency crises (see coordination game). Not a perfect equilibrium C .

The (50%,50%) equilibrium is unstable. Condition 2. and 3. are satisfied by way of Berge's maximum theorem. i

σ is non-empty and upper hemicontinuous. Note then that. A Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). σ What has long made this an interesting case to study is the fact that this scenario is globally inferior to "both cooperating". {\displaystyle {\text{Gain}}_{i}(\sigma ^{*},a)=0} ∈

Nash equilibrium is named after American mathematician John Forbes Nash, Jr. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability. We add another where the probabilities for each player are (50%, 50%). denote the set of mixed strategies for the players.

, {\displaystyle \Delta } {\displaystyle f_{i}} {\displaystyle G}

Σ

A perfect equilibrium B . ) and so the left term is zero, giving us that the entire expression is 0{\displaystyle 0} as needed. The players have sufficient intelligence to deduce the solution. Δ {\displaystyle r_{i}(\sigma _{-i})} d. would be willing to undertake if in a position to do so. If a player A has a dominant strategy [17] However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. Convexity follows from players' ability to mix strategies. {\displaystyle \sigma ^{*}}

σ 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.

", If any player could answer "Yes", then that set of strategies is not a Nash equilibrium. 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? The players know the planned equilibrium strategy of all of the other players. A game can have a pure-strategy or a mixed-strategy Nash equilibrium. A is the number of players and {\displaystyle r\colon \Sigma \rightarrow 2^{\Sigma }} In game theory, a solution concept is a formal rule for predicting how a game will be played. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria. ( We have a game G=(N,A,u){\displaystyle G=(N,A,u)} where N{\displaystyle N} is the number of players and A=A1×⋯×AN{\displaystyle A=A_{1}\times \cdots \times A_{N}} is the action set for the players. In a game theory context stable equilibria now usually refer to Mertens stable equilibria. A famous example of this type of game was called the stag hunt; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat (4 utility units) than the latter (1 utility unit). ) Condition 1. is satisfied from the fact that Σ{\displaystyle \Sigma } is a simplex and thus compact. If both players chose strategy B though, there is still a Nash equilibrium. What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. i

{\displaystyle f} Which of the following is a valid critique of the use of game theory in economics? Suppose that in the Nash equilibrium, each player asks themselves: "Knowing 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? This idea was formalized by Aumann, R. and A. Brandenburger, 1995, Epistemic Conditions for Nash Equilibrium, Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly know, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. A Equilibrium will occur when the time on all paths is exactly the same. 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 ABD, ABCD, or ACD). Gain is the fixed point we have: Since × A A . Imagine two prisoners held in separate cells, interrogated simultaneously, and offered deals (lighter jail sentences) for betraying their fellow criminal. If either player changes their probabilities (which would neither benefit or damage 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%). ( Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. N σ

The players all will do their utmost to maximize their expected payoff as described by the game.

.

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