Nash+equilibrium

A condition describing a set of strategies in which no player can improve their payoff by unilaterally changing their own strategy, given the other players' strategies. In this situation each player pursues the best possible strategy while possessing the knowledge of the strategies of the other players. Each player's strategy is an optimal response based on the anticipated rational strategy of the other players. The concept of nash equilibrium is very important because it represents a situation where every player is doing the best he or she can given what other players are doing.
 * Definition of Nash Equilibrium:**

Nash equilibrium for strategic cooperative games, was outlined in a 27-page dissertation by a 21-year old named John Nash. His existence proof was one of the first applications of Kakutani's fixed-point theorem. John Nash spawned much of the literature on non-cooperative game theory which has since grown at a prodigious rate-threatening, some claim, to overwhelm much of economics itself.
 * History of Nash Equilibrium:**

Based on the information in the table below, player B should reason as follows: "Player A will surely choose up, since up is a dominant strategy. Therefore, I should choose left." Assuming player A indeed chooses the dominant strategy (up), player B will earn 20 by choosing left, but only 8 by choosing the secure strategy (right).
 * Example of Nash Equilibrium:**


 * ||  |||| **Player B** ||
 * || **Strategy** || Left || Right ||
 * **Player A** || Up || 10,20 || 15,8 ||
 * Down || -10,7 || 10,10 ||

To understand the logic we must first define dominant strategy. A dominant strategy results in the highest payoff to the player regardless of the actions of the others players. In this situation player A will choose up because no matter what player B chooses, up results in the highest payoff for player A. If player A chooses up and player B chooses left, the payoff for player A is 10; however, If player A chooses down the payoff would be -10. Similarily, if player A chooses up and player B chooses right, the payoff for player A is 15; however, if player A chooses down the payoff is only 10. Clearly, player A will choose up regardless of the actions of player B.

Player B does not have a dominant strategy becasue if player A chooses up, player B will choose left and if player A chooses down, player B will choose right. If the player does not have a dominant strategy, look at the game from your rival's perspective. If your rival has a dominant strategy, anticipate that he or she will play it. In this situation, player B will anticipate that player A will choose up because it is their dominant strategy. Because player A will choose up, player B will choose left in order to receive the highest payoff.

When player A chooses up and player B chooses left this results in the highest payoff for each player given the other's strategy. This is nash equilibrium.

According to the ISID Ensyslopedia of Science and Philosophy, the concept of nash equilibrium is used when we predict things about people and are used in the following types of analysis:
 * Uses of Nash Equilibrium Strategy**


 * candidate's election strategies
 * causes of war
 * agenda manipulation in legislature
 * actions of interest groups

Additionally, nash equilibrium is used in business decisions. Nash equilibrium strategies can be used in the use of advertising decisions, quality decisions, and pricing and product mix decisions. Business managers also use nash equilibrium strategies to analyze interactions between workers and managers and to monitor the actions of employees.

Sources: Managerial Economics & Business Strategy, Bay, Michael R., 5th Ed, McGraw Hill 2006, Chapter 10 http://www.iscid.org/encyclopedia/Nash_Equilibrium http://en.wikipedia.org/wiki/Nash_equilibrium http://www.economyprofessor.com/economictherories/nash-equilibrium.php

a. Secure strategy b. Nash equilibrium c. Dominant strategy d. Mixed strategy The answer is b. The question is stating the definition for nash equilibrium.
 * Discussion Questions:**
 * 1. What is the condition describing a set of strategies in which no player can improve his or her payoff by unilaterally changing his or her own strategy, given the other players' strategies?**

2. **What is the nash equilibrium for the game above in the form of (Tazo, Cocoa)?**

a. Chai Tea, Iced Tea b. Iced Tea, Iced Tea c. Chai Tea, Chai Tea d. None of these options lead to a nash equilibrium The answer is d. Non of these options leads to a nash equilibrium because, no matter what the outcome, one of the producers will want to change its choice of tea if is believes its opponent will not change its choice.
 * ||  |||| **Cocoa** ||
 * ||  || Iced Tea || Chai Tea ||
 * **Tazo** || Iced Tea || 3,6 || 2.5,8 ||
 * Chai Tea || 2,8 || 3,5 ||

Please refer to questions 3, 4, and 5 for the chart presented below.


 * ||  |||| **Shocks** ||
 * ||  || Low || High ||
 * **Gel** || Low || 3,7 || 5,8 ||
 * High || -4,5 || 7,10 ||

Gel and Shocks are both companies that manufacture running shoes. Each must decide whether to set a high or low price for their running shoes. The matrix above provides each firm's profit in millions depending on which choice is made.

a. Gel sets low price, Shocks sets high price b. Gel sets low price, Shocks sets low price c. Gel sets high price, Shocks sets high price d. Gel sets high price, Shocks sets low price The answer is c. For Shock's, high price is a dominant strategy; Shock's profits will be higher choosing high price independent of Gel's choice. So Shocks will choose high. When Shocks chooses high, Gel's best choice is choosing high.
 * 3. What is the nash equilibrium?**

a. $3 million b. $5 million c. $4 million d. $7 million The answer is d. By referring to the matrix above, when both Gel and Shocks choose a high price, Gel's profits will be $7 million.
 * 4. What will Gel's profits be at nash equilibrium?**

a. $7 million b. $8 million c. $5 milion d. $10 million The answer is d. By referring to the matrix above, when both Gel and Shocks choose a high price, Shock's profits will be $10 million.
 * 5. What will Shock's profits be at nash equilibrium?**

Please answer questions 6 and 7 based on the following information: If you advertise and your rival advertises, you each will earn $0 million in profits. If neither of you advertise, you will each earn $1 million in profits. However, if one of you advertises and the other does not, the firm that advertises will earn $10 million and the non-advertising firm will lose $1 million.

a. A nash equilibrium is for both firms to advertise. b. A nash equilibrium is for you to advertise and your rival not to advertise. c. A nash equilibrium is for neither of you to advertise. d. A nash equilibrium does not exist. The answer is a. The dominant strategy is to advertise for both you and your rival. This would mean that the nash equilibrium would be for both firms to advertise.
 * 6. Which of the following is true?**

a. (0,0) b. (0,10) c. (10,0) d. The game does not have a nash equilibrium. The answer is a. Because we already know that the nash equilibrium is for both advertise, the text above informs us that both firms will make $0 million in profits when this occurs.
 * 7, What are the nash equilibrium payoffs for the firms in a one-shot game?**