Dear all, the following five games are the problems that I couldn't finish due to my lack of knowledge of "Nash equilibria" as well as "Mixed Strategy Nash equilibria". Therefore, if you can solve these games by using Nash equilibria could you please explain me how to solve these games by using Nash equilibria?

1.2 Simultaneous or Sequential In simultaneous games, participants act at the same time and only know the decision of their counterpart when / after they have made their own strategic choice. In sequential games, there is a first-mover and a second-mover. The rst-mover makes his choice rst; but, he can still anticipate the response of the second-mover if he knows the payoffs that the second-mover can earn under each scenario. 1.3 Normal Form or Extensive Form Normal form games are presented as a matrix, and extensive form games are presented as decision trees. Each node of the decision tree represents an individual's decision point. Each branch leads from a decision point either (1) to another participant's decision point, or (2) to ultimate payoffs. 1.4 Symmetric or Asymmetric Games are symmetric if the payoffs to each player are mirrors of one another. Games are asymmetric if the payoffs to the players are not equal under identical outcomes. As can be seen in Panel (1), the payoffs are symmetric when the players face the same set of outcomes; in Panel (2), the players face different payoffs for the same set of outcomes. --| EA Iii-- Panel (1) ['3 Iii-54 Panel (2) 2 Nash Equilibrium A Nash Equilibrium is an outcome or set of conditions under which no participant can im- prove her circumstances by choosing a different strategy, holdin all other pla ers' strate ies fixed. This can come in the form of a single outcome, as will be discussed in section 2.1, or it can come in the form of a probabilistic strategy, as will be discussed in section 2.2. 2.1 Pure-Strategy Nash Equilibrium A set of players' strategies is a Pure Strategy Nash Equilibrium (PSNE) iff no participant could do strictly better by changing strategies, holding all other players' strategies xed. A simple game can have one PSNE, as is seen in Panel (1) above. If P1 chooses strategy A, P2 is best off choosing strategy A, as well. When P1 chooses strategy B, P2 is best off choosing strategy A once again. (No matter what P1 does, P2 is best off choosing strategy A.) When P2 chooses strategy A, P1 is best off choosing strategy A; and, when P2 chooses strategy B, P1 is best off choosing strategy A. (No matter what P2 does, P1 is best off choosing strategy A.) Clearly, we settle into a condition in which P1 and P2 both choose A, and can do no better by deviating. Somewhat more complicated games can have more than one PSNE, as can be seen in Panel (2) If P1 chooses A P2 would be best off choosing strategy A, but if P1 chooses B P2 would be best off choosing strategy B. If P2 chooses A P1 would be best off choosing strategy A but if P2 chooses strategy B P2 would choose strategy B. If both settled on strategy A, neither would choose to deviate, cetem's paribus; if both settled on strategy B, neither would choose to deviate, ceteris paribus. As can be seen in many of the games below including the French Connection game in section 3.7 sometimes there are no PSNE, and instead we must look for an optimal mixed strategy. 2.2 Mixed-Strategy Nash Equilibrium If no PSNE exists, an equilibrium probability of each player choosing a given strategy can be identied: a Mixed-Strategy Nash Equilibrium (MSNE). To arrive at a MSNE, a player chooses to play a specic strategy with a specic probability so that his counter- part / opponent / partner is indifferent between the strategies he can choose from. Instead of just asking whether any player has an incentive to change strategies, you could ask whether any player has an incentive to change his probability of playing various strategies. Numerical examples of MSNEs will be discussed in detail below. 3 Simple Games There are a handful of very famous archetypal games. Studying these stylized games will help us to identify circumstances in the real world which have similar characteristics to these games. Though not everyone is part of a criminal duo, or a hunting team, or an international conspiracy avoiding the police, the simplied conditions explained below will shed light on the decision-making process of utility-maximizing non-cooperative people. 3. 1 Prisoner's Dilemma Consider two criminals * Bonnie and Clyde 7 who travel the country robbing banks together. One day they are caught by the police and are put into separate rooms where they will each speak with detectives. The detectives are trying to get each of the criminals to confess to robbing banks. But, Bonnie and Clyde both know that if neither of them confess to robbing the banks, and they choose to cooperate with one another, the detectives will not be able to convict them of any crimes, and they will both go free at a relatively low cost. If one cooperates with the police, the other goes to prison and incurs a huge cost while the confessor walks away with the lowest possible cost. If they both confess, they both spend some time in prison and incur a high cost. ##Lmel - Do Not Confess C, ,1, Do Not Confess y As can be seen, whether Bonnie confesses or not, Clyde chooses to confess; and, the same goes for Bonnie. For each bank robber, the worst outcome is if they do not confess, while their partner-incrime does confess. So, they know they cannot trust their partner-in-crime, and both ultimately confess. While they would both be better off if they cooperated with one another and neither confessed, if either one chose not to confess, the other would confess, making Confess/ Confess the PSNE. So, Bonnie and Clyde should each expect a payout of 6 (which we can think of as 6 years in prison). 3. 2 Industry Entry Let's look at the computer industry: There was a point in time in which IBM dominated the personal and corporate/research computer industry, and Apple was just a wily startup company operating out of a garage. At some point, Apple's founders had to consider whether they would truly enter the computer industry and try to compete with IBM. If Apple stayed out of the industry, it would earn no income or incur any costs, and IBM would continue to dominate the market. If, however, Apple decided to enter the industry, IBM would then need to choose whether to accommodate Apple and allow it into the industry as a competitor, or try to ght Apple by cutting prices, and trying to drive Apple out of the industry. IBM (Incumbent) Accommodate Fight Apple (Mam, Do Not Enter This is a classic game often described as a sequential game drawn in extensive form. Apple can be thought of as the rst mover, and IBM as the second mover. Yet we can look at the payoffs, anticipate what IBM would do under each set of conditions, and, through backward induction, identify whether Apple would enter the market (and what IBM will choose to do after Apple makes its decision). If Apple entered the industry, IBM would be best off by accommodating; so, Apply should be expected to enter the industry (earning 5) and IBM should be expected to accommodate Apple and earn 5 (rather than a monopoly payoff of 10). 3.3 Stag Hunt There are two hunters who go into the woods with the agreement that they will try to nd and catch a stagl. To catch the stag, the two hunters separate from one another with the agreement that they will each stalk the stag and eventually trap it as a team. It is impossible for a single hunter to trap the stag by himself. When the two hunters separate from one 1A large male deer. 1 Introduction Game theory is a eld of study Within economics that focuses on human action and inter action in noncooperative situations. Most of economics considers conditions under which equilibrium prevails, outcomes are efcient and people cannot act strategically to improve their outcomes. Game theory advances the eld of study by considering the circumstances under which people do act strategically because they can improve their outcomes. We face these circumstances all of the time: Children play \"rock-paperscissors\Prompt: Find all of the Nash equilibria of each of the following five normal form games. Explain why the games have no other Nash equilibria beyond those you identified . If the optimal strategy is a Mixed Strategy Nash Equilibrium, make sure to show your work. Given the Nash equilibria that you identify, determine what the expected payout is for each player. If any of the following games is similar to any of the simple stylized games from the handout, identify which stylized form the game below takes.Game 1 Game 2 Game 3 Player 1 Strategy A Strategy B Strategy A 4, 4 8, 2 Player 2 Strategy B 2, 8 7, 7 Player 1 Strategy A Strategy B Strategy A O, 8 4, 0 Player 2 Strategy B 2, 0 0, 1 Player 1 Strategy A Strategy B Strategy A 7, 10 3, 5 Player 2 Strategy B 2, 1 4, 6 Game 4 Game 5 Player 1 Strategy A Strategy B Strategy A 3, 5 5, 4 Player 2 Strategy B 1, 7 7, 6 Player 1 Strategy A Strategy B Strategy C Strategy A 6, 6 4, 7 8, 5 Player 2 Strategy B 3, 4 1, 5 5, 3 Strategy C 7', 2 3, 4 7, 1