Question

2. [20 points] An old lady is looking for help crossing the street. Only one person is needed to help her; if more people help her, this is no better. A and B are the two people in the vicinity who can help; we have to choose simultaneously whether to do so. Each of us will get pleasure worth 3 units from her success (no matter who helps her). But each one who goes to help will bear a cost of 1 unit, this being the value of one's time taken up in helping. If neither player helps, she cannot cross the street, and the payoff for each player is 0. If only one of the players chooses to help her, then the one who chooses to help incurs an additional cost of 1 unit, say because she gets angry at the other player. a. [8 points] Set up this game as a simultaneous-move game. Write the payoff matrix and find the pure strategy Nash equilibrium. Use the net benefit (pleasure-cost) to a player in an outcome as their corresponding payoff number. You can calculate the net benefit using the information provided above. b. [8 points) Conduct the best-response analysis and find the Nash equilibrium/equilibria. c. [4 points) Which class of games does this game fit into? Explain. (Consider the following types of games: Prisoners' Dilemma, Games of Coordination & Conflict, Games of Pure Coordination, Games of Chicken, Games of Assurance, Zero-sum Games. Do not analyze each category. Just mention the type of game this game fits into and briefly explain why.)

Answer 1

Part a) There are two pure strategy Nash Equilibria.

Part b) There are two pure strategy Nash equilibria( and one more mixed strategy Nash equilibrium as a sidenote.)

Part c) Game of Chicken.
wohet both A & B deced ve both A & B Then the payoff Tet us write the decide to not help the lady for each one of them is o. payant in the form of (x,y); where Ta = Pasaff of A DO in the case mentioned above, the payoff is. Tience 110,0) Now let us suppose that A decides to help B decides to not help and Hence B's payoff is 3 from the lady's sucess. LA's payoff, however = 3 (from the lady's success) - 1 ( time in helping) - As payoff 1 lady angry at other player) 1. The payoff is (1,3). Similarly of B decides to help and A decide to not help the payoff is (3,1). Suppose both decide to help As payoff = 3 (hom the lady's success) -16 time taken in helping) - As payoff = 2 Smilarly, f's payaff = 2. i. The payoff = (2, 2)

We can construct the payets nativ here as follow - Doesn't A help Doesn't Help 10,0) (1₂3) Help CH) 63,1) (2, 2) HelpH we can see that if 3 decides to help. A would choose to not help whereas if B doen't decide to help, then A will help Similarly if A decides to help, B would choose to not help and it A chooses to not help, then B will decide to help Hence in this case, we find that both the payoffs et of (3,1) and (1,3) are the pure strategy Nash Equilibria b) we can write the best response tables as- & B's B: Best Response A! Best Response #

where D: Doesn't help Hi Help of a Even via the best response analysis, we can see that there exist st multiple Nash equilibria. of CD,H) and CH, o) for $3,1) and cl, 3) Jespectively These are pure strategy Nash equilibrio

DATE We can also see that other than the two pure strategy Nash equilibria, there also exists a mixed strategy Lequilibrium ; as both the players are indifferent between the tuo strategies you - Hence any one player can choose to randompse so as to keep the other player guessing. Each player can assige probabilities to what the other player will play and then decide his more accordingly.

a This particular game fits the class of games called as the game of chicken DU The game of chicken is a special case of the Volunteer's dilemma, where the number of dolor volunteers is 2 . We have also proved in part b that there exists I a mixed strategy equilibrium. Also, we can say that both the pure strategy pure equilibria are Pareto optimal Note . Pareto optimal means that there is no way we can chay deviate from one pure strategy Nash equilibsia without hurting the payoff of at least one player