Game Theory Day 10

Example. Price Competition Duopoly: When two companies share exclusively the market for a certain product. How can a company set their prices in order to receive maximum profit? Compare NE (Nash equilibrium) to IDS (iterated dominance strategy)

Stage – Hare Hunting Problem Consider a group of 10 hunters who are trying to catch a stag. Each hunter has two options: she may remain attentive to the pursuit of the stag, or she may catch a hare instead. A hunter’s utility is simply the number of calories of food he gets from the hunt. If all 10 hunters pursue the stag, they catch it and share it equally, getting 100 calories each. However, if one or more hunters decide to catch a hare instead, the stag escapes and only each hare-hunter catches a hare worth 5 calories (which he eats for himself and does not share with those who pursued the stag). Describe the set of strategies for each player. Show that all players hunting the stag is a Nash equilibrium in this game. Find another Nash equilibrium. Is the game the same for any N number of hunters, given 1 < 𝑁 < ∞?

Employee\Employer Employee \ You Give in Hire attorney Settle 1, 2 0, 1 Be tough 3, 0 x, 1 In the cells of the matrix, your payoff is listed second; x is a number that both you and the employee know. Under what conditions can you rationalize selecting “give in”? Explain what you must believe for this to be the case. Situation: Suppose that you manage a firm and are engaged in a dispute with one of your employees. The process of dispute resolution is modeled by the following game, where your employee chooses either to “settle” or to “be tough in negotiation,” and you choose either to “hire an attorney” or to give in.

The Odd Couple Felix and Oscar share a flat. They have decidedly different views on cleanliness and hence, whether or not they would be willing to put in the hours of work necessary to clean the apartment. Suppose that it takes at least 12 hours of work per week to keep the flat clean, 9 hours to make it livable, and anything less than 9 hours the flat is filthy. Suppose that each person can devote either 3, 6, or 9 hours to cleaning. Felix and Oscar agree that a livable flat is worth 2 on the utility index. They disagree on the value of a clean apartment – Felix thinks it is worth 10 while Oscar it is only 5. They also disagree on the unpleasantness of a filthy apartment – Felix thinks it is (– 10) while Oscar thinks it is (-5). Each person’s payoff is the utility from the apartment minus the number of hours worked. Find N.E. Felix\Oscar 3 hours 6 hours 9 hours

The Odd Couple Find the Nash equilibrium strategy sets: Felix\Oscar 3 hours 6 hours 9 hours -13, -8 -1, -4 7, -4 -4, -1 4, -1 4, -4 1, 2 1, -1 1, -4 Find the Nash equilibrium strategy sets: (9 hours, 3 hours) (6 hours, 6 hours) (3 hours, 9 hours)

Steroids and Swimming Problem Evans\Smith s n 0, 0 1, -1 -1, 1 Game: Two athletes, Evans and Smith, are competitive swimmers who have the option to take performance-enhancing steroids (s) or not using it (n) before a meet. The two swimmers are equally good, and each has a 50 percent chance of winning if neither takes steroids or they both do. However, if only one swimmer uses steroids, then he will win. Evans and Smith do not know what the other has chosen to do. There is no drug- testing intervention. Let us denote the payoff to winning as 1 and the payoff to losing as -1. s is a dominant strategy This means swimmers are more likely to choose to take steroids. Thus, intervention must occur.

Intervention Testers (player 3) can test only one swimmer. Probability is fair (1/2 Evans, ½ Smith) If a swimmer tests positive, the Testers look good (utility 1) and the swimmer will receive a penalty greater than if he lost (-1-b where b>0) If the swimmer tests negative, the Testers are no worse or better (utility 0) Test Evans Evans\Smith S N -1-b, 1, 1 -1, 1, 0 0, 0, 0 Test Smith Evans\Smith S N 1, -1-b, 1 1, -1, 0 0, 0, 0

What should the testers do? Find N.E. Why does knowing who will be tested not work? How can we make n the dominant strategy for both players? Why not test both? Test Evans Evans\Smith S N -1-b, 1, 1 -1, 1, 0 0, 0, 0 Test Smith Evans\Smith S N 1, -1-b, 1 1, -1, 0 0, 0, 0