Each of them has exactly two strategies: International Club and Debate Club. Foe each their payoffs are shown in the boxes that correspond to each strategy pair. For example in the box for International Club, International Club the payoffs are 10 and The first payoff corresponds to Caitlin and the second corresponds to Malia. For Caitlin, her best response function is the following:. Since the best response changes for both players depending on the strategy choice of the other player, we know immediately that neither one has a dominant strategy.
What then? How do we think about the outcome of the game? The solution concept most commonly used in game theory is the Nash equilibrium concept. A Nash equilibrium is an outcome where, given the strategy choices of the other players, no individual player can obtain a higher payoff by altering their strategy choice.
The equilibrium is intuitive, if, when placed in a certain outcome, no player wishes to unilaterally deviate from it, then equilibrium is achieved — there are no forces within the game that would cause the outcome to change.
In the party game above, there are two outcomes where the best responses correspond: both players choosing IC and both players choosing DC. If Caitlin shows up at the International Club party and finds Malia there, she will not want to leave Malia there and go to the Debate Club party. Similarly if Malia shows up at the International Club party and finds Caitlin there, she will not want to leave Caitlin there and go to the Debate Club party.
This example illustrates the strengths and weaknesses of the Nash equilibrium concept. Intuitively, if they show up at the same party, both women would be content and neither one would want to leave to the other party if the other is staying. However, there is nothing in the Nash equilibrium concept that gives us guidance as to predicting at which party the women will end up.
In general, a normal form game can have zero, one or multiple Nash equilibrium. Now DC is a dominant strategy for Caitlin. Malia knows Caitlin does not like the International Club party and will correctly surmise that she will attend the Debate Club party no matter what.
In this case the Nash equilibrium concept is satisfying, it gives a clear prediction of the unique outcome of the game and it intuitively makes sense. Another possibility is a game that has no Nash equilibrium. Consider the following game called matching pennies. In this game two players, Ahmed and Naveen, each have a penny. They decide which side of the penny to have facing up and cover the penny until they are both revealed simultaneously. The players, strategies, and payoffs are given in the payoff matrix in Figure As you can clearly see, there is no correspondence of best response functions, no outcome of the game in which no player would want to unilaterally deviate and thus no Nash equilibrium.
The outcome of this game is unpredictable and no outcome creates contentment for both players. Finding the Nash equilibrium in normal form games is made relatively easy by following a simple technique that identifies the best response functions on the payoff matrix itself.
To do so, simply underline the maximum payoff for a given opponent strategy. Consider the example below. This is a representation of the best response to Pat playing left, Chris should play Up. So now that we have identified each players best response functions, all that remains is to look for outcomes where the best response functions correspond. Such outcomes are those where both payoffs are underlined.
Down is always the best strategy choice for Chris no matter what Pat chooses and Center is always the right strategy choice for Pat no matter what Chris chooses. Note also that the dominant strategy equilibrium Down, Center also fulfills the requirements of a Nash equilibrium. This will always be true of dominant strategy equilibria: all dominant strategy equilibria are Nash equilibria. From the example above, however, we know the reverse is not true: not all Nash equilibria are dominant strategy equilibria.
So dominant strategy equilibria are a subset of Nash equilibria, or to put it in another way, the Nash equilibrium is a more general concept than Dominant Strategy Equilibrium. So far we have concentrated on pure strategies where a player chooses a particular strategy with complete certainty.
We now turn our attention to mixed strategies where a player randomizes across strategies according to a set of probabilities he or she chooses. As was noted before and as we can see by using the underline strategy to illustrate best responses, there is no Nash equilibrium of this game in pure strategies. What about mixed strategies? Naveen can play either heads or tails with certainty or the same mixed strategy with equal results, he can expect to win half the time.
We know that the pure strategies have best responses as given in the matrix above so only the mixed strategy is a Nash equilibrium.
So this game, that did not have a Nash equilibrium in pure strategies, does have a Nash equilibrium in mixed strategies. Both players randomize over their two strategy choices with probabilities.
There are many situations where players of a game move sequentially. The game of checkers is such a game, one player moves, the other players observes the move and then responds with their own move. Sequential move games are games where players take turns making their strategy choices and observe their opponents choice prior to making their own strategy choices.
We describe these games by drawing a game tree , a diagram that describes the players, their turns, their choices at every turn, and the payoffs for every possible set of strategy choices. We have another name for this description of sequential games: extensive form games. Consider a game where two Indian food carts located next to each other are competing for many of the same customers for its Tikka Masala.
The extensive form, or game tree representation of this game is given in Figure The game tree describes all of the principle elements of the game: the players, the strategies and the payoffs. The players are described at each decision node of the game, each place where a player might potentially have to choose a strategy.
From each node there extend branches representing the strategy choices of a player. At the end of the final set of branches are the payoffs for every possible outcome of the game.
This is the complete description of the game. There are also subgames within the full game. Subgames are the all of the subsequent strategy decisions that follow from one particular node. How will the game resolve itself? To determine the outcome it is necessary to use backward induction: to start at the last play of the game and determine what the player with the last turn of the game will do in each situation and then, given this deduction, determine what the payer with the second to last turn will do at that turn, and continue this way until the first turn is reached.
Using backward induction leads to the Subgame Perfect Nash Equilibrium of the game. The Subgame Perfect Nash Equilibrium SPNE is the solution in which every player, at every turn of the game, is playing an individually optimal strategy.
For the curry pricing game illustrated in Figure Tridip is only concerned with his payoffs in red and will play Medium if Ashok plays High and get , will play Low if Ashok plays Medium and get , and will play Low if Ashok plays Low and get Because of common knowledge, Ashok knows this as well and so has only three possible outcomes.
Ashok knows that if he plays High, Tridip will play Medium and he will get ; if he plays Medium, Tridip will play Low and he will get , and if he plays Low, Tridip will play Low and he will get Since is best of the possible outcomes, Ashok will pick High. Since Ashok picks high, Tridip will pick Medium and the game ends. The game and the payoffs are given below in the normal form game in figure But would Vito really believe that Gino would fight if he entered?
Probably not. We call this a non-credible threat: a strategy choice to dissuade a rival that is against the best interest of the player and therefore not rational. Some of these situations can even be predicted by identifying the actions you should always take or the actions you should never take based on your behavior. It may seem strange to study this type of game. In particular, game theory can assist companies in making strategic choices within or outside of their organizations. In simple games, players are presented with scenarios that simulate real-world conditions and predict their behavior based on hypothetical scenarios.
In game theory, players are optimized for their outcomes by understanding the dynamics of the game. Rather than designing a game based on the strategies and aims of the players, Inverse Game Theory focuses on the strategies and goals of the players.
In game theory, two or more players are modeled to interact in a situation that has set rules and outcomes. The study of economics is one of the most notable uses of game theory, which is used in a number of disciplines. The study of business and economics is one of the most notable uses of game theory, which is used in a number of disciplines. John von Neumann , a Hungarian born mathematician and economist, was the first to apply game theory to economics. Essentially, game theory is the study of how individuals or organizations apply strategy to achieve an outcome that is to their advantage — namely, a reward.
In game theory, rational decision-makers are studied through mathematical models of strategic interactions. In addition to social science, it is applied to logic, systems science, and computer science as well. Robert Mockler views game theory as a mathematical technique that can be used to make decisions in situations of conflict, where the success of one part depends on the help of others, and where the individual decision maker is not fully involved in the decision-making process.
November 24, BY: Troy Helping business owners for over 15 years. Table of contents 1. A sequential game is one in which players move at different times or in turn. In business, sequential games are very important. It is possible to apply game theory to the science of managerial economics. A single game is also played in this game. One-shot games: games are played once. In the past, mathematical models of economics were unable to address crucial problems that game theory has solved.
The behavior of oligopoly firms is often understood by economists using game theory. In the case of price-fixing and collusion, it can help predict the likely outcomes. A business person can win a simultaneous-move, one-shot business game by following the same strategy. A repeated game, also known as a supergame, is a game that plays out over and over for a period of time, and therefore is usually represented using the extensive form of the game. In other words, the strategy space is greater in sequential or simultaneous games than in regular games.
The study of how economic agents interact with each other to produce outcomes with respect to their preferences or utilities is called game theory. In other words, the outcomes in question might not have been intended by any of the agents involved. In game theory, two or more players are modeled to interact in a situation that has set rules and outcomes. The study of economics is one of the most notable uses of game theory, which is used in a number of disciplines. Takeaways from the day.
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