Question

ECON 605 Economic Applications of Game Theory and Strategic Behavior Homework 5 Due date: at the BEGINING of the class on Friday, November 8. Maximum Total Score: 100*0.3=30 points Question 1. (30 Points in Total) In a football the offense can either run the ball or pass the ball, whereas the defense can either anticipate a run or a pass. Assume the payoff for the two team on any given down are as follows foto de Porto de este cam the ball or pas the ball, whereas the defense e Defense Anticipate Pass (1-0) 5.-5 Offense Anticipate Run 1,-1 9,-9 Run (p) Pass (1-P) -3.3 1.1. Show the game has no pure-strategy Nash Equilibrium (5 Points) 1.2 Suppose the two players use mixed-strategy, and their strategy, in terms of action probabilities, is Offense (p, 1-p). Defense (q, 1-9),find the unique mixed-strategy Nash Equilibrium where 0<p, q<1 by finishing the following steps Step 1. Write down the expected payoff from each of Offense's action as a function of Defense s q-mix, and the expected payoff from each of Defense' action as a function of Offense’s p-mix (10 Points) Step 2. According to the result in Step 1, derive each player's best-response function (5 Points) Step 3. Plot each player's best-response function in a diagram and find the mixed-strategy Nash equilibrium (5 Points)

Answer 1

4 This game has no pure strategy hash equilibrium as if player offenge chooses Run, Defense Defence will choose Anticipate dun as -17-5, but when defense chooses Anticipate Run, offense will choose pas as ati. If coffense chooses pass, Defence will choose Anticipate pass as 37-9. If Defense chooses Anticipate pass. then offense will choose Run as 57-3, so there is a loop and no pure shalegy nash equilibrium. 1.7 Expected pay off of offense if chooses Run EU (Run) = l.q+ 5(1-9) = 5-49 EU. ( Pass) = 9. q + (-3) (1-9) = 129-3 offene will choose fun iff Evo (home) Expected pay off of Defense if chooses Not eeporter AR) EUR (AR) = -lip + (-9) (1-P) = Dp-9-p. =80-9 EUD (AP) = -5p+ 3(1-P) = 3-89. offen Best response funderen offense will choose Rum iffa " EUo (Run) , & Uo (Pass) on 5-49 78 129 - 3 on -49-1297-3-5 an 1697-8 169 5 8 so, p=1 qa when ones Run)

Defense cuell choose anticipate eum iff. & UD (AR) T F Uo (AP) 80-9 73-8 p an 8p+8 9 7 12 o p 12/16 . P o P So, 9 - I detene 19) Anticipate oum) chooses ₂t (3/42 74 2(º) so, qb% then p=1, q> // then, Pzo руч Цам - ТРК «», то and q = 1½ then offense randamize between rum and pass P = 3/4 then, Defense randamize bekueen AR and AP