www Assignment Point com Game Theory studies strategically
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Game Theory • studies strategically inter-dependent situations. • provides us tools for analyzing most of problems in social science. • employs Nash equilibrium as a solution concept. • creates a revolution in Economics. www. Assignment. Point. com
Games in Two Forms • Static games Þ The normal/strategic-form representation • Dynamic games Þ The extensive-form representation • In principle, static (/ dynamic) games can also be analyzed in an extensive-form (/a normal-form) representation. www. Assignment. Point. com
Static Games Þ In a normal-form representation, each player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. The players do not necessarily act simultaneously: it suffices that each chooses her own action without knowing others’ choices. We will also study dynamic games in an extensive-form representation later. www. Assignment. Point. com
Normal-form Games The normal-form (strategic-form) representation of a game specifies: 1. 2. 3. The players in the game. The strategies available to each player. The payoff received by each player (for each combination of strategies that could be chosen by the players). www. Assignment. Point. com
STRATEGIC GAME is a model of interacting decision-makers. In recognition of the interaction, we refer to the decision-makers as players. The model captures interaction between the players by allowing each player to be affected by the actions of all players, not only her own action. Specifically, each player has preferences about the action profile—the list of all the players’ actions. www. Assignment. Point. com
the players may be firms, the actions prices, and the preferences a reflection of the firms’ profits. Or the players may be candidates for political office, the actions campaign expenditures, and the preferences a reflection of the candidates’ probabilities of winning. Or the players may be animals fighting over www. Assignment. Point. com some prey, the actions concession times,
it is frequently convenient to specify the players’ preferences by giving payoff functions that represent them. Bear in mind that these payoffs have only ordinal significance. If a player’s payoffs to the action profiles a, b, and c are 1, 2, and 10, for example, the only conclusion we can draw is that the player prefers c to b and b to a; The numbers do not imply that the player’s preference between c and b is stronger than her preference between a and b. www. Assignment. Point. com
The idea is that each player chooses her action once and for all, and the players choose their actions “simultaneously” in the sense that no player is informed, when she chooses her action, of the action chosen by any other player. www. Assignment. Point. com
the Prisoner’s Dilemma Two suspects in a major crime are held in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they both stay quiet, each will be convicted of the minor offense and spend one year in prison. If one and only one of them finks, she will be freed and used as a witness against the other, who will spend four years in prison. If they both fink, each www. Assignment. Point. com will spend three years in prison.
This situation may be modeled as a strategic game: Players The two suspects. Actions Each player’s set of actions is {Quiet, Fink}. Payoff values for player one u 1(Fink, Quiet) = 3, u 1(Quiet, Quiet) = 2, u 1(Fink, Fink) = 1, and u 1(Quiet, Fink) = 0. Payoff values for player two u 2(Quiet, Fink) = 3, u 2(Quiet, Quiet) = 2, u 2(Fink, Fink) = 1, and u 2(Fink, Quiet) = 0. www. Assignment. Point. com
How to Use Bi-Matrices Þ Any two players game (with finite number of strategies) can be expressed as a bi-matrix. The payoffs to the two players when a particular pair of strategies is chosen are given in the appropriate cell. The payoff to the row player (player 1) is given first, followed by the payoff to the column player (player 2). How can we solve this game? www. Assignment. Point. com
Payoff Bi-Matrix The Prisoner’s Dilemma models a situation in which there are gains from cooperation (each player prefers that both players choose Quiet than they both choose Fink) but each player has an incentive to “free ride” (choose Fink) whatever the other player does. www. Assignment. Point. com
Terminology: Dominant Strategy When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of: B dominates A: choosing B always gives at least as good an outcome as choosing A. There are 2 possibilities: www. Assignment. Point. com
B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do. B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give B at least the same payoff as A www. Assignment. Point. com
General Tricks A Strategy is strictly dominant if this strategy strictly dominates every other possible strategy. A Strategy is weakly dominant if this strategy dominates all other strategies, but some are only weakly dominated. A Strategy is strictly dominated if some other strategy exists that strictly dominates this. A Strategy is weakly dominated if some other strategy exists that weakly dominates this www. Assignment. Point. com
Terminology Dominant strategy: A strategy s is called a dominant strategy if playing s is optimal for any combination of other players’ strategies. Pareto efficiency: An outcome of games is Pareto efficient if it is not possible to make one person better off (through moving to another outcome) without making someone else worse off. www. Assignment. Point. com
Applications of PD Examples Players Arms races Countries Silent Disarm Confess Arm International trade policy Countries Lower trade barriers No change Marital cooperation Couple Obedient Demanding Provision of public goods Citizen Contribute Free-ride Deforestation Woodmen Restrain cutting Cut down maximum www. Assignment. Point. com
Working on a joint project You are working with a friend on a joint project. Each of you can either work hard or goof off. If your friend works hard then you prefer to goof off (the outcome of the project would be better if you worked hard too, but the increment in its value to you is not worth the extra effort). www. Assignment. Point. com
You prefer the outcome of your both working hard to the outcome of your both goofing off (in which case nothing gets accomplished), and the worst outcome for you is that you work hard and your friend goofs off (you hate to be “exploited”). www. Assignment. Point. com
Duopoly www. Assignment. Point. com
Bach or Stravinsky Two people wish to go out together. Two concerts are available: one of music by Bach, and one of music by Stravinsky. One person prefers Bach and the other prefers Stravinsky. If they go to different concerts, each of them is equally unhappy listening to the music of either composer. www. Assignment. Point. com
Payoff table of Bach vs Strs. www. Assignment. Point. com
Matching Pennies Two people choose, simultaneously, whether to show the Head or the Tail of a coin. If they show the same side, person 2 pays person 1 a dollar; if they show different sides, person 1 pays person 2 a dollar. Each person cares only about the amount of money she receives, and (naturally!) prefers to receive more than less. www. Assignment. Point. com
The Payoff Table www. Assignment. Point. com
Nash Equilibrium In game theory, Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. www. Assignment. Point. com
If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision. www. Assignment. Point. com
Computing Nash equilibrium If a player A has a dominant strategy s. A then there exists a Nash equilibrium in which A plays s. A. In the case of two players A and B, there exists a Nash equilibrium in which A plays s. A and B plays a best response to s. A. If s. A is a strictly dominant strategy, A plays s. A in all Nash equilibrium. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays his strictly dominant strategy. www. Assignment. Point. com
Dominance and Nash equilibrium If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibrium. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium. Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibrium. www. Assignment. Point. com
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