Question

4) Each of two firms has a job opening. The firms offer different wages: firm i offers wage w; where 0.5 W 1 < W2 < 2.W 1. There are two workers that want to apply for a job. Each of whom can apply to only one firm. The workers simultaneously decide whether apply to firm 1 or to firm 2. If only one worker applies to a given firm, that worker gets the job. If both workers apply to one firm, the firm hires one worker at random and the other worker remains unemployed. a) Rapresent this game using the normal form b) Solve for the Nash equilibria (in pure and mixed strategies)

please answer with method working. thank you.

Answer 1

Soluation Given information 0.5w, <0,<2.W, Workers-2 0.5021 . 0.552 5o5w, oww w., walp workert W2, W, (1-P) q (1-9) Each worked has tut starteges F - Apply for form) f₂ - Apply son fim-2 - Ito both players chore & then each get onha, It both workers choose fr then each get oso Fim gien dat we can note following in Equalies = 0.5w, LW₂ <2a, & 05w, we can 0.5W, 40, Ib worker, choose & worker z's best strages is choose se and it player a chone Fr, workers is best stages is a So, F, te} is a pire storagey nach Equiliterieron. if worken choose a worked as best strategy is fi and (6)

= player 2, choos Fi, worker is. = best stratges is fr. - so, q Fq IF} A pune stratgy wash equilibrium. Mixed strategies = net worker I, play F with probability & Fe with probability (I-P). - Let worker 2 play in with probability of and te with probability (109) &, (G) = expected pupff of worker by playing F E, (Iz) - Expected puydt of worker, by playing.de Fe (5) - Expected say off it worked 2 bay playing F - Eg (Fe) – Expected safety of worked 2 by playing = F₂ - E,(5) = 0.5w,9 twi(1-2) = 0 - 0.5a, q + w, qu, E, ({2} , q + 0.5W2 (1-2) - w2q + 0.5w2 - 0590 = E(F) = E(6) 0.5w, qtw, W, 9 = 22 +05 W₂ - 0.5W₂q 9 (-0.5 w.) to = q (0.5us) + 0542 = = q (0.5w, 7 0.502) -6,-0.8W.

- 0.6 102 and 0.5w, toi5 w2 - W-0502 0.5w, to 502 o5w, +0.50,- w,+ 0.502 0.50 +0.5w2 W 2 015) 0.50 +0.502 E, (h) 2015w, P+w, (1-P) = = 0.50, P+w, w,P E₂(F₂) = wel +0.502 (1-P) - wep + 0.502 - 0.5 w₂P 2) E₂ (K) = 22 (6) = 0.5 wiptw, w, p=w2P to 502 -0.5421 - P(-0.5w, tw, = P(0.542) +0.50 - p(0.5w, to 5w.) – W, -0.542 po w. - 0.5W, 1054, 10.5ur (1-P) - w. - 0.5W, 0.5w, to 5w, 0.5w, toi5w - w, to5w, 0.5w, to swe (1-P) - welow) 0.5W, to.502 SO,

(P. 9): [ w,-0.5W2 w,-0.562 (050, +0.50 0.50 to 5w, The mixed stonteges Hash Equlibsium = pure stora tegy hash Equilibrium: { Fute} und {f2, 8} Mixed stoptegy NF [ w, -0.502 w, 0502 0.50 +0.502' 0.5497 0.502 mixed strategy Nach equilibriuos .. workeor , and worken-2 apply to finom-) with probability aud - Wt 0502 0.5w, to 5w, apply to fiom- 2 workers and worked 2 probability W₂ - 0.50) 0.509 +0.502